Samson Lasaulce - SupélecChannel Estimation and Multiuser Detection for TD-CDMA Systems Soutenue le...

Thèseprésentée pour obtenir le grade de docteur

de l’Ecole Nationale Supérieuredes Télécommunications

Spécialité : Signal et Images

Samson Lasaulce

Channel Estimation and MultiuserDetection for TD-CDMA Systems

Soutenue le 19/11/2001 devant le jury composé de

Jean-Claude BELFIORE Président

Dirk SLOCK RapporteursPierre DUHAMEL

Soodesh BULJORE ExaminateursArnauld TAFFIN

Philippe LOUBATON Directeurs de thèseÉric MOULINES

Ecole Nationale Supérieure des Télécommunications, Paris

Contents

Acronyms and Abbreviations iii

Notations v

Introduction 1

1 Problem statement and motivations 3

1.1 Overview of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Multipath and ISI canceller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 CDMA and MAI canceller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2.1 Multiple access methods . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2.2 CDMA concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 The need for channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 The UMTS-TDD Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 The UMTS air interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1.1 Why CDMA? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1.2 Comparison of UMTS-TDD and UMTS-FDD physical layer key pa-rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1.3 Why UMTS-TDD? . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Physical layer description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2.1 The UMTS-TDD options . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2.2 Frame and time slot structures . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2.3 Modulation, spreading, scrambling and shaping filtering . . . . . . . . 9

1.2.2.4 Channel coding and interleaving . . . . . . . . . . . . . . . . . . . . 10

1.2.2.5 Transmit diversity (downlink) . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2.6 ETSI test propagation environments . . . . . . . . . . . . . . . . . . 10

1.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Signal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1.1 From the physical problem to the MIMO system modelling . . . . . . 11

1.3.1.2 Expression of the received signal . . . . . . . . . . . . . . . . . . . . 14

ii Contents

1.3.2 Starting point: the initial UMTS-TDD receiver . . . . . . . . . . . . . . . . . . 14

1.3.2.1 First receiver stage: conventional channel estimator . . . . . . . . . . 14

1.3.2.2 Second receiver stage: conventional symbol detector . . . . . . . . . . 16

1.3.3 Towards a new receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.3.1 Prior information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.3.2 Road map and contributions . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.4 The problems to solve in two figures . . . . . . . . . . . . . . . . . . . . . . . . 20

2 The optimum semi-blind receiver 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 EM algorithm principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 General statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Convergence of the EM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 The EM algorithm in the UMTS-TDD mode context . . . . . . . . . . . . . . . . . . . 26

2.3.1 Channel estimation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1.1 General expression of the auxiliary function . . . . . . . . . . . . . . 26

2.3.1.2 The structured auxiliary function . . . . . . . . . . . . . . . . . . . . 27

2.3.1.3 Parameters estimation procedure . . . . . . . . . . . . . . . . . . . . 28

2.3.1.4 Adaptation of the forward-backward algorithm to the UMTS-TDD Mode 28

2.3.2 Symbol detection procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.3.1 Simulation assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.3.2 Simulation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Semi-blind channel estimation techniques based on second-order blind methods 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Semi-blind techniques based on second-order blind methods . . . . . . . . . . . . . . . 40

3.2.1 Why second-order statistics based semi-blind techniques ? . . . . . . . . . . . . 40

3.2.2 Second-order blind methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2.1 Subspace method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2.2 Linear prediction method . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2.3 Weighted linear prediction method . . . . . . . . . . . . . . . . . . . 44

3.2.3 Semi-blind estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Asymptotic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 The regularized approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1.1 The subspace case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.1.2 The linear prediction case . . . . . . . . . . . . . . . . . . . . . . . . 47

Contents iii

3.3.2 The projected approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Road map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


3.4.3.1 Unicity of the solution of blind criteria . . . . . . . . . . . . . . . . . 51

3.4.3.2 Performances of the blind estimators . . . . . . . . . . . . . . . . . . 51

3.4.3.3 Tuning of the regularizing constants . . . . . . . . . . . . . . . . . . 52

3.4.3.4 Performance of the regularized approach . . . . . . . . . . . . . . . . 54

3.4.3.5 Robustness of semi-blind estimators to channel overmodelling . . . . 54

3.4.3.6 Influence of the midamble length . . . . . . . . . . . . . . . . . . . . 56

3.4.3.7 Influence of the number of blind observations . . . . . . . . . . . . . 58

3.4.3.8 The projected approach . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Low complexity channel estimation schemes based on continuous time slot transmissions 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 The strongest-paths method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.2 Major drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 A revisited Wiener approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.2 The different scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.3 The optimum weighting procedure . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.4 The different schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.4.1 The low rank approximation (S1) . . . . . . . . . . . . . . . . . . . . 70

4.3.4.2 How to exploit the ideal correlation properties . . . . . . . . . . . . . 70

4.3.4.3 One-dimensional hybrid weighting (S6) . . . . . . . . . . . . . . . . 71

4.3.4.4 Low rank approximation and ideal correlation properties (S7) . . . . . 72

4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4.1 Determination of the scenario corresponding to the UMTS-TDD context . . . . . 73

4.4.2 Comparison between the strongest-paths method and the one-dimensional weight-ing method based on the Wiener approach . . . . . . . . . . . . . . . . . . . . . 77

4.4.2.1 Influence of the assumed number of significant paths . . . . . . . . . . 77

4.4.2.2 Comparison between the strongest-paths method and the one-dimensionalweighting based on the Wiener approach in typical situations . . . . . 79

4.4.2.3 Influence of the number of active users . . . . . . . . . . . . . . . . . 79

4.4.2.4 Influence of the number of available time slots . . . . . . . . . . . . . 79

iv Contents

4.4.2.5 Influence of noise variance estimation accuracy . . . . . . . . . . . . 80

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Implementable receiver structures 85

5.1 Uplink and downlink cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Downlink solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.1 The two a priori candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.2 Transmit adaptive antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.2.1 The TxAA idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.2.2 Weights calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.1 Simulation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88


5.3.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Conclusion 95

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

A Multiple access methods: an analogy (Motorola SPS) 99

B Review of polyphase components 101

C Properties of the EM algorithm 103

D Convergence in probability 105

E Kronecker product 107

F Proofs of chapter 3 109

F.1 Regularized semi-blind case (proposition 1) . . . . . . . . . . . . . . . . . . . . . . . . 109

F.2 Regularized semi-blind subspace (proposition 2) . . . . . . . . . . . . . . . . . . . . . . 110

F.3 Regularized semi-blind linear prediction (proposition 3) . . . . . . . . . . . . . . . . . . 112

F.4 Projected semi-blind case (proposition 4) . . . . . . . . . . . . . . . . . . . . . . . . . 115

Acronyms and Abbreviations

3G Third Generation of mobile4G Fourth Generation of mobile3GPP Third Generation Partnership Project3GPP TSG RAN WG1 3GPP Technical Specifications Group Radio Acess Network Working Group 13GPP TSG RAN WG4 3GPP TSG RAN Working Group 4AMPS Analog Mobile Phone SystemAPP A Posteriori ProbabilityARIB Association of Radio Industries and BusinessesAWGN Additive White Gaussian NoiseBER Bit Error RateBS Base StationCDMA Code Division Multiple AccessCDMA2000 3G American StandardCWTS China Wireless Telecommunication StandardDFE Decision Feedback EqualizerDL Down-LinkEDGE Enhanced Data-rates for GSM Evolutione.g. Exempli gratia (for instance)EM Expectation Maximization AlgorithmETSI European Telecommunication Standard InstituteFDD Frequency Division DuplexFDMA Frequency Division Multiple AccessFIR Finite Impulse ResponseFSC Finite Samples ConvergenceGSM Global System for Mobile communicationsIA Indoor A propagation environment (ETSI)IB Indoor B propagation environment (ETSI)iid Independent and identically distributedi.e. Id est (that is)IMT2000 International Mobile Telecommunications-2000IR Impulse ResponseIS-95 Interim Standard-95ICI Inter-Chip InterferenceISI Inter-Symbol InterferenceJD Joint DetectionJP Joint Pre-distortionkbps Kilobit per secondL1 Physical Layer (Layer 1)LOS Line Of Sight

vi Acronyms and Abbreviations

LP Linear Prediction methodLS Least-Squares estimationMAI Multiple Access InterferenceMAP Maximum A PosterioriMF Matched FilterMIMO Multiple Inputs Multiple OutputsML Maximum LikelihoodMSE Mean Square ErrorMMSE Minimum Mean Square ErrorMMSE-BLE MMSE Block Linear EqualizerMMSE-JD MMSE Joint DetectionMS Mobile StationMUD Multi-User DetectionNB-TDD Narrow Band TDD (Chinese option)Node B Base Station in 3G SystemsNLOS Non Line Of SightOSI Open System Interconnect modelPB Pedestrian B propagation environment (ETSI)PDF Probability Density FunctionPIC Parallel Interference CancellationPN Pseudo NoiseQPSK Quaternary Phase Shift KeyingRRC Root-Raised CosineSIC Serial Interference CancellationSIMO Single Input Multiple OutputsSNR Signal-to-Noise RatioSOS Second-Order StatisticsSS SubSpace methodSTD Selection Transmit DiversitySTTD Space Time Transmit DiversitySVD Singular Value DecompositionT1 American standard committee for TelecommunicationsTD-CDMA Time Division CDMATDD Time Division DuplexTDMA Time Division Multiple AccessTS Training Sequencetr Trained (training-based)TTA Telecommunication Technology AssociationTTC Telecommunication Technology CommitteeTxAA Transmit Adaptive AntennaUE User Equipment (mobile station)UL Up-LinkUMTS Universal Mobile Telecommunication SystemVA Vehicular A propagation environment (ETSI)WB-TDD Wide Band TDD (European option)W-CDMA Wide band CDMAWLP Weighted Linear Prediction methodw.r.t With respect toZF Zero ForcingZF-BLE ZF Block Linear Equalizer

Notations

Mathematical conventions

(.)∗ Complex conjugate(.)T Transpose(.)H Hermitian transpose(.)# Moore-Penrose pseudo-inverse|.| Absolute value< ., . > Scalar (or inner, or dot) product‖.‖ Euclidean normvec(.) Column vectorization of matrices (Appendix E)⊗ Kronecker product of matrices (Appendix E), By definition equalsδk,l Kronecker deltas Scalarv Column vectorM or M Matricess, v,M Estimates of s,v and M respectivelyspan(M) Column space of matrix Mnull(M) Orthogonal complement of span(M)Det(M) Determinant of matrix MRank(M) Rank of the matrix MTrace(M) Trace of the square matrix MM d= m×n Matrix M has m rows and n columns

diag(M) Let M d= n×n be a square matrix, then diag(M) , (M1,1 . . .Mn,n)T

Diag(v) Diag(v) is the diagonal matrix which diagonal is the vector vIn n×n identity matrixIm,n m×n identity matrix0n Zeros n×n matrixE[.] Expectation operatorxn = o(yn) Convergence of deterministic series, limn→+∞

xnyn

= 0xn = O(yn) The serie xn

ynis bound

Xn = oP(.) Xn converges in probability (Appendix D)Xn = OP(.) Xn is bounded in probability (Appendix D)Cov(x) limT→∞T ×E[xxH ]x(t) Value of the tth element of the discrete sequence xx(z) z-transform of xZT[.] z-transform operatordeg(X(z)) Degree of polynomial matrix X(z) = ∑n Xnz−n is

viii Notations

the largest index n for which Xn is non zero[ f (z)]x(n) The discrete-time sequence x(n) is filtered by the filter f (z)

Main notations

Scalar constants and variables

d d = dim(null(Q(α)))K Number of active users per time slotk User indexL Degree of the transfer function H(z), which is also the ISI duration` Maximum length of the chip-rate-defined physical channels`k Channel length of the chip-rate-defined physical channel of user kM Smoothing factor or regression order in blind methodsm Number of observations used by the midamble-based estimation procedureN Spreading factorNs Number of possible states that can be taken by xL(t)n Chip indexPy Useful received power in the TxAA algorithmp Number of significant channel parameters in chapter 4q Number of antennas at the base station in chapter 5r Number of iterations of the EM algorithmT Number of unknown symbols in a time slotTc Chip durationTs Symbol durationt Symbol indexW Number of available time slots in chapter 4 (continuous transmissions)w Midamble length

ck(.) Code of user kd(.) Output sequence of the spreading operatorEk(i) Energy of physical channel coefficient #if (z) Shaping filter transfer functiongk(z) Physical channel of user k (emission, propagation and reception filters)hk(z) Overall channel of user k (spreading and physical channel)h(n)

k nth polyphase component of the physical channel of user kQ(θ;θ(p)) Objective function in the EM algoritm at iteration pth

sk(.) Chips of the training sequence of user kv(.) Additive White Gaussian Noisexk(.) QPSK symbol sequence of user kxk Oversampled version of the symbol sequence of user ky(.) Received signal (observations)

αk Regularizing constant for user k in semi-blind estimation proceduresδk(i) Weigth for channel coefficient #i of user kη Threshold value in the hard thresholding methodθ Unkown parameter (scalar) in the EM algorithmλ(i) Eigenvalue #i of the decomposition on matrix Rg in chapter 4ρ ρ = T/m

Notations ix

σ Noise standard deviationσ2

est(i) Estimation noise variance for channel coefficients #iφk(z) Propagation channel of user k

Multidimensional constants and variables (1/2)

1l Column vector of K onesck Vector code of user kg g = [gT

1 . . .gTK]T

gtr

Trained estimate of the channelg

kVector impulse response of the physical channel of user k

hk Vector impulse response of the overall channel of user kRSY RSY = m−1SY trui ui = (0 . . .0 1 0 . . .0)T , the one is on position (i+1)X Burst vector defined by X =

[xT (0) . . .xT

(T2 −1

)]T

X This vector is the concatenation of XA and XB which arethe symbol vectors associated with bursts A and B of each time slot

V tr Noise vector associated with the training sequence: V tr = (v(`) . . .v(w−1))T

V Noise vector associated with bursts A and BY tr Observations generated by the training sequence: Y tr = (y(`) . . .y(w−1))T

Y This vector comprises the observations generated by X

w Weight vector in the TxAA algoritm

α Vector of regularizing constants α = (α1 . . .αK)T

γ Vector of principal components of the channel in chapter 4

DP(eiω) For a given P ∈ N, DP(eiω) = (1 e−iω . . .e−iPω)T

hk(z) hk(z) = (h(0)k (z) . . .h(N−1)

k (z))T

v(t) ∀t ∈ Z,v(t) = v(t) = (v(tN)...v(tN +N−1))T

vM(t) Noise vector associated with the vector vM(t)x(t) ∀t ∈ Z,x(t) = (x1(t) . . .xK(t))T

xL(t) Symbol vector comprising the ISI and defined by xL(t) = [xT (t) . . .xT (t−L)]T

xM+L(t) Symbol vector defined by xM+L(t) = [xT (t) . . .xT (t− (M +L))]T

y(t) ∀t ∈ Z,y(t) = y(t) = (y(tN)...y(tN +N−1))T

yM

(t) Regression vector defined by[yT (t) . . .yT (t−M)

]T

Ck Code matrix associated with user k defined in (2.13)D D = H(0)HH(0)D(π) Type 1 Sylvester matrix defined in (3.7)Gq Physical channels matrix when a q-sensor antenna is implemented at

the base stationH Overall channels matrix H = [h1 . . .hK ]H(0) First polyphase component of H(z)P Orthogonal projector onto null(Q(α))Qk,sub Qk,sub = ∆H

k (π)∆k(π)Qk,lin Qk,lin = ∆H

k (A,D)∆k(A,D)Qk,wlin Qk,wlin = ∆H

k (A,D)Wk(D,σ2)∆k(A,D)Q(α) Q(α) = Diag(α1Q1, . . . ,αKQK) (see equation (3.27))RA See equation (3.34)

x Notations

Rg Physical channel covariance matrix in chapter 4RM Covariance matrix of observation vector y

MRSS RSS = m−1SHSR(∞)

SS R(∞)SS = limm→+∞ m−1RSS

Rγ Covariance matrix of vector γRπ See equation (3.33)

Multidimensional constants and variables (2/2)

S Training sequences matrix defined in (1.9)S(A) Type 1 Sylvester matrix defined in (3.15)S(H) Type 1 Sylvester matrix defined in (1.18)T(H) Type 2 Sylvester matrix defined in (3.3)W Weighting matrix for the training-based estimated vector channelWk(D,σ2) Wk(D,σ2) = IM+L+1⊗ (πD⊥ +σ2D#)W(D,σ2) W(D,σ2) = Diag(W1(D,σ2), . . . ,WK(D,σ2))

∆k(π) ∆k(π) = D(π)Ck (subspace method)∆k(A, D) ∆k(A, D) = Diag(πD⊥ ,IN , . . . ,IN)S(A)Ck (linear prediction)Θ Unkown parameter (matrix) in the EM algorithmΠ Matrix for the low rank approximation in chapter 5π Noise subspace of matrix RM (3.4)πD⊥ Orthogonal projection matrix on the orthogonal complement of Range(D)Σ Σ = Cov(δ∆ g)Υ Unitary matrix defined in (3.18)

A(z) Prediction filter defined by (3.13)H(z) H(z) = [h1(z) . . .hK(z)]

Important note!

The introduced notations are the same for all the chapters of this thesis.

List of Figures

1.1 Multipath propagation channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Different multiple access schemes: power versus time and frequency for FDMA (left),TDMA (middle) and CDMA (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Multipath effect on spreading codes: orthogonality is lost . . . . . . . . . . . . . . . . . 6

1.4 UMTS-TDD frames are divided into 15 time slots; each frame has a 10 ms duration andeach time slot comprises 2560 chips . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Type 1 UMTS-TDD time slot comprises two data parts, one 512-chip midamble and one96-chip guard period (GP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Type 2 UMTS-TDD time slot comprises two data parts, one 256-chip midamble and one96-chip guard period (GP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Equivalence between the conventional (left) and Tsatsanis (right) representations of CDMAoperation for N-periodic CDMA codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.8 The MIMO system under consideration in this study; there are K inputs and N outputs(the latter being the N samples associated with a given symbol) . . . . . . . . . . . . . . 13

1.9 The receiver is considered as a 3-stage structure . . . . . . . . . . . . . . . . . . . . . . 15

1.10 The initial receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.11 How we want to design a new receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1 Convergence of the EM algorithm: the objective function (Q) saturates after 6 iterationsif Eb/No = 0 dB and 4 iterations if Eb/No = 3 dB . . . . . . . . . . . . . . . . . . . . 33

2.2 Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noise ratio;receiver structures 1 (MMSE block linear joint detection + training-based estimation) and5 (receiver based on the EM algorithm) are compared; simulation conditions: VehicularA, 2 active users, mobile station speed = 120 km/h, 1 single antenna, no channel coding;for the voice service BER target (1%) the conventional receiver loses 3 dB w.r.t the semi-blind optimum receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xii List of Figures

2.3 Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noise ratio;the symbol detection strategy is fixed (MMSE block linear detection); the receiver per-formance is evaluated for four different channel estimation schemes: the training-basedestimation, the training-based estimation when the channel length is known, the iterativechannel estimation based on the EM algoritm (initialized with the trained estimate) andthe case where the channel is known from the detector; simulation conditions: VehicularA, 2 active users, mobile station speed = 120 km/h, 1 single antenna, no channel cod-ing; the flawed channel knowledge costs 2.5 dB to the trained estimation and the flawedchannel length knowledge causes a 2 dB loss to the trained estimation . . . . . . . . . . 34

2.4 Bit error rates (averaged over the 8 users and 2000 time slots) versus signal-to-noiseratio; the symbol detection strategy is fixed (MMSE block linear detection); the detectorperformance is evaluated when the trained estimate is used and when the channel isknown; simulation conditions: Vehicular A, 8 active users, mobile station speed = 120km/h, 1 single antenna, no channel coding; the channel knowledge becomes a moreand more critical point as the multiple access interference level increases; 4 dB are lostbecause of the flawed channel knowledge . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noiseratio; the receiver stuctures 4 (maximum a posteriori joint detection+ trained estimation +known channel length), 5 (maximum a posteriori joint detection + EM based on iterativechannel estimation) and 6 (maximum a posteriori joint detection + known channel) arecompared; simulation conditions: Vehicular A, 2 active users, mobile station speed = 120km/h, 1 single antenna, no channel coding; even if the channel length is assumed to beknown, the flawed knowledge of the coefficients costs a 0.8 dB loss to the MAP detection 35

2.6 Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noiseratio; the channel is assumed to be known; the receiver performance is evalutated withthe matched filter, the MMSE block linear joint detection and the MAP joint detection;simulation conditions: Vehicular A, 2 active users, mobile station speed = 120 km/h, 1single antenna, no channel coding; with one antenna, even in a low MAI scenario, thematched filter does not perform well; the MMSE and the MAP performances are veryclose; if K/N is small, MAP detection capability is not fully exploited . . . . . . . . . . 36

3.1 Bit error rates versus assumed ISI duration (L); three blind estimators are compared:the subspace method (SS), the weighted linear prediction (WLP) and the conventionallinear prediction (LP); for every user, the strongest channel coefficient is assumed tobe known; simulation conditions: Vehicular A, the physical channel is truncated at 16chips (L = 1), 4 active users, Eb/No = 17.5 dB, T = 122 blind observations; the linearprediction approaches are more robust to channel overmodelling than the subspace method 52

3.2 Bit error rates versus signal-to-noise ratio; comparison of the linear prediction approachesbetween themselves in a blind context; for every user, the strongest channel coefficient isassumed to be known; simulation conditions: Vehicular A, the physical channel is trun-cated at 16 chips (L = 1), 4 active users, L = 1; in the case where channel lengths areknown, if the number of blind observations (250 and 500 in the figure) and the signal-to-noise ratio (> 10 dB in the figure) are high enough, the weighted linear predictionperforms better than its non-weighted counterpart . . . . . . . . . . . . . . . . . . . . . 53

List of Figures xiii

3.3 For a given time slot: the asymptotic error covariance matrix is estimated (top curve) andthe distance between the true and the estimated channel (bottom curve) is calculated; thisis done for several values of the regularizing constant (RC); the figure represents theseerrors as a function of the regularizing constant; a hyperbolic scale has been chosen inorder to facilitate the research of the optimum values; simulation conditions: the semi-blind subspace case is considered, Vehicular A, true physical channel length = 19 chips,LN = 32 chips, 4 users are assumed, presented results correspond to user 1, 200 valuesof α are tested ranging from 0 to 3, Eb/No = 10 dB; the corresponding functions areconvex and the two underlying optimum values of α are very close when the subspacemethod is used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Gain in terms of signal-to-noise ratio versus bit error rate target; this figure shows the in-fluence of regularizing constant accuracy on the MMSE joint detector performance; sim-ulation conditions: see figure ??; the top curves correspond to the semi-blind subspacecase and the two bottom curves correspond to the semi-blind weighted linear predictioncase; for each case, we compare the proposed tuning strategy with the case where α istuned from the Euclidean distance between the true and estimated vector channels; themain result is that the regularizing constant is very well tuned even though the optimumvalue is derived from an asymptotic approach; this holds for the subspace case; however,for the weighted linear prediction, 0.2 dB are lost . . . . . . . . . . . . . . . . . . . . . 55

3.5 Bit error rates versus signal-to-noise ratio; the MMSE joint detector performance is eval-uated with four different channel estimates: the trained estimate, the semi-blind estimatebased on the subspace method, the semi-blind estimate based on the weighted linearprediction approach and the true channel; simulation conditions: Vehicular A, physicalchannel length = 19 chips, LN = 32, 4 users, 1 antenna, no channel coding; for a 1%BER target the semi-blind linear prediction provides a 0.5 dB gain while the semi-blindsubspace reaches a 1 dB gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Gain on signal-to-ratio versus assumed ISI duration (L); simulation conditions: see figure??, BER target = 1%; the top curve corresponds to the gain that would be achieved if thechannel was known; the middle one is the performance of the semi-blind subspace chan-nel estimate; the bottom curve represents the performance of the semi-blind weightedlinear prediction based channel estimate; as a result, the proposed semi-blind estimatorsare robust to channel overmodelling but the semi-blind solution based on the subspacemethod is more robust than the semi-blind weighted linear prediction solution . . . . . . 56

3.7 Bit error rates versus midamble length; for a given signal-to-noise ratio (Eb/No = 12.5dB) the MMSE joint detection performance is evaluated with the trained and the semi-blind channel estimates; simulation conditions: Vehicular A, physical channel length =19 chips, LN = 32, 4 users; 1 single antenna, no channel coding; for short midambles(m = 152) the semi-blind schemes provide very significant improvements over the training-based approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.8 Bit error rates versus signal-to-noise ratio; simulation conditions: see figure ??, m = 152;when using the trained approach, more than 5 dB are lost w.r.t the semi-blind strategies(at BER = 6%) because of the reduced midamble length . . . . . . . . . . . . . . . . . 57

3.9 Gain (over the trained approach) in terms of signal-to-noise ratio versus number of blindobservations; simulation conditions: BER target = 1%, midamble length = 512 chips,Vehicular A, physical channel length = 19 chips, LN = 32 chips, 4 users, 1 single antenna,no channel coding; if the number of blind samples is multiplied by 4 (from T = 122 to4T = 488), we can benefit of a 0.2 dB gain for the semi-blind subspace based estimationand 0.5 dB for the semi-blind weighted linear prediction . . . . . . . . . . . . . . . . . 58

xiv List of Figures

3.10 Gain (over the trained approach) in terms of signal-to-noise ratio versus number of blindobservations; simulation conditions are the same as for figure ?? except for midamblelength, which equals 184 chips (m = 152); if the number of blind samples is multipliedby 4 (from T = 122 to 4T = 488), we can benefit of a 0.5 dB gain for the semi-blindsubspace based estimation and 1 dB for the semi-blind weighted linear prediction . . . . 59

3.11 For a given time slot: Euclidean distance between the true and estimated channels versusdim(null(Qk)); simulation conditions: Vehicular A, physical channels are truncated at16 chips, channel lengths are assumed to be known, the projected semi-blind subspaceapproach is considered, 4 users, user 1 is considered, Eb/No = 10 dB; for a given user,if the dimension of the null space of the matrix Qk (denoted by d) is assumed to be 1, theestimation error (Euclidean distance) is worse than in the trained case (d = 16) . . . . . 59

3.12 Bit error rates versus signal-to-noise ratio; the MMSE joint detector performance is eval-uated in the conventional UMTS-TDD context except for the channel length, which isassumed to be known from the receiver; simulation conditions: Vehicular A, physicalchannel is truncated at 16 chips, 4 users, 1 antenna, no channel coding; the influenceof the assumed dimension of the null space of the matrices Qk is studied; if d = 1,the semi-blind projected approaches perform dramatically worse than the training-basedapproach; however, if d could be optimally tuned (by choosing the d minimizing theEuclidean distance for each time slot), the semi-blind schemes performance gains wouldreach 40% of the available margin gain (1 dB at BER = 1%) . . . . . . . . . . . . . . . 60

3.13 Bit error rates versus signal-to-noise ratio; this figure is similar to figure ?? but 4T = 488blind observations are used; the choice of d is not as crucial as in the figure above; butif the best dimension could be known, the performance would be significantly improved(0.8 dB for the semi-blind projected subspace approach while the maximum gain is 1 dB(at BER=1%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.14 Mean square estimation error versus signal-to-noise ratio; simulation conditions: seefigure ??; for d = 1 and T = 122, the semi-blind projected subspace approach beats thetraining-based approach if SNR> 21 dB . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Power-delay profile of Indoor B channel; physical channel length = 12 chips; this fig-ure represents the channel coefficient energies in the canonical basis (top curve) and theKarhunen-Loeve basis (bottom curve); the KL basis is obtained by performing an eigen-value decomposition on the covariance matrix of the channel; in both bases, the channelenergy is concentrated on three coefficients; as a result, considering a new basis is useless 73

4.2 Power-delay profile of Vehicular A channel; physical channel length = 19 chips; thisfigure represents the channel coefficient energies in the canonical basis (top curve) andthe Karhunen-Loeve basis (bottom curve); the KL basis is obtained by performing aneigenvalue decomposition on the covariance matrix of the channel; in both bases, thechannel energy is concentrated on six coefficients; as a result, considering a new basis isuseless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 Power-delay profile of Pedestrian B channel; physical channel length = 24 chips; thisfigure represents the channel coefficient energies in the canonical basis (top curve) andthe Karhunen-Loeve basis (bottom curve); the KL basis is obtained by performing aneigenvalue decomposition on the covariance matrix of the channel; in both bases, thechannel energy is concentrated on nine coefficients; as a result, considering a new basisis useless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

List of Figures xv

4.4 This figure depicts a three-dimensional view of the covariance matrix of the channel(Vehicular A) for a given user; we see that it is "almost" diagonal . . . . . . . . . . . . 75

4.5 This figure depicts a three-dimensional view of a diagonal block of the matrix of trainingsequences RSS; it represents the autocorrelation properties of the training sequence of agiven user . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6 This figure depicts a three-dimensional view of a non-diagonal block of the matrix oftraining sequences RSS; this figure has to be compared with the figure above; it repre-sents the intercorrelation properties between two given users; we see that the maximummagnitude of the block entries under consideration is less than 25 while the diagonal co-efficients of the matrix depicted in figure ?? equal 456, which means that RSS is "almost"diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.7 For a given BER target (3%): gain in terms of signal-to-noise ratio as a function of thenumber of selected channel coefficients (p) for the strongest-paths method; simulationconditions: Indoor B channel, MMSE block linear joint detection, 4 active users, M = 11time slots are available, mobile station speed = 3 km/h, no channel coding; the optimumnumber of paths to select is 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.8 For a given BER target (3%): gain in terms of signal-to-noise ratio as a function of thenumber of selected channel coefficients (p) for the strongest-paths method; simulationconditions: Vehicular A channel, MMSE block linear joint detection, 4 active users,M = 11 time slots are available, mobile station speed = 3 km/h, no channel coding;the optimum number of paths to select is about 7 or 8; if p is not properly chosen,say p = 3 the training-based estimation works better without the strongest-paths methodbased selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.9 For a given BER target (3%): gain in terms of signal-to-noise ratio as a function of thenumber of selected channel coefficients (p) for the strongest-paths method; simulationconditions: Pedestrian B channel, MMSE block linear joint detection, 4 active users,M = 11 time slots are available, mobile station speed = 3 km/h, no channel coding; theoptimum number of paths to select is about 9; if p is not properly chosen, say p = 3 thetraining-based estimation works better without the strongest-paths method based selection 78

4.10 Bit error rates versus signal-to-noise ratio; the performance of the MMSE joint detector isevaluated in Indoor B with 5 different channel estimates: the conventional training-basedestimate, the training-based estimate followed by the strongest-paths selection with p =3, the training-based estimate followed by the strongest-paths selection with p = 6, thetraining-based estimate followed by the proposed one-dimensional hybrid weighting andthe true channel; simulation conditions: 4 active users, M = 61 time slots are available,known noise variance, mobile station speed = 3 km/h; in any cases, the post-processingoperation is beneficial to performance: the hard and the hybrid weighting operationsprovide a 2 dB gain for any Eb/No . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xvi List of Figures

4.11 Bit error rates versus signal-to-noise ratio; the performance of the MMSE joint detec-tor is evaluated in Vehicular A with 5 different channel estimates: the conventionaltraining-based estimate, the training-based estimate followed by the strongest-paths se-lection with p = 3, the training-based estimate followed by the strongest-paths selectionwith p = 6, the training-based estimate followed by the proposed one-dimensional hy-brid weighting and the true channel; simulation conditions: 4 active users, M = 61 timeslots are available, known noise variance, mobile station speed = 3 km/h; the Vehic-ular A propagation environment allows us to highlight one of the major defaults of thestrongest-paths method, which is its consistency; behavior of the one-dimensional hybridweighting method is very regular for any tested SNR . . . . . . . . . . . . . . . . . . . 81

4.12 Bit error rates versus signal-to-noise ratio; the performance of the MMSE joint detec-tor is evaluated in Pedestrian B with 5 different channel estimates: the conventionaltraining-based estimate, the training-based estimate followed by the strongest-paths se-lection with p = 3, the training-based estimate followed by the strongest-paths selectionwith p = 6, the training-based estimate followed by the proposed one-dimensional hy-brid weighting and the true channel; simulation conditions: 4 active users, M = 61 timeslots are available, known noise variance, mobile station speed = 3 km/h; the observationdone for Vehicular A is more apparent; strongest-paths method behavior is related bothto the SNR and the number of selected paths whereas the hybrid weighting is "always"efficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.13 For a given BER target (3%): gain in terms of signal-to-noise ratio versus number ofactive users; simulation conditions: see figure ??; the top curve is the maximum gainthat would be achieved if the channel was known; the bottom curve represents the gainprovided by the one-dimensional hybrid weighting; the relative gain is quite stable withregard to the number of users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.14 Bit error rates versus signal-to-noise ratio; simulation conditions are the same as in figure?? except for the number of time slots; the top curve is the performance of the trainedestimation while the bottom curve corresponds to the case where the channel is assumedto be known from the receiver; the two other curves are the performance of the hybridweighting with M = 11 and M = 121; as a conclusion, we do not need a large number oftime slots to take profit of the proposed channel estimate post-processing . . . . . . . . . 83

4.15 Bit error rates versus signal-to-noise ratio; the influence of estimation noise varianceaccuracy on performance is studied; simulation conditions: Vehicular A channel, 1 activeuser, M = 11 time slots are available, noise estimation is based on the 6 weakest paths,mobile station speed = 3 km/h, no channel coding; the difference between the curvescorresponding to the case where the noise variance is assumed to be known and the casewhere it is estimated by using MQ = 66 samples is negligible . . . . . . . . . . . . . . . 83

5.1 TxAA transmission; there are q antennas at the base station; the data sequences of theK active users are transmitted with the q antennas; each antenna is weighted by wi withi = 1, . . . ,q; the data transmitted from the antenna #i experience the equivalent channelhi(z); the antenna weights are calculated in order to maximize the useful power of thereceived signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Symmetrical allocation: the traffic between uplink and downlink is alternated . . . . . . 89

List of Figures xvii

5.3 Bit error rates versus signal-to-noise ratio; performances of the matched filter and theMMSE joint detection are compared in Indoor A; simulation conditions: the channelestimation strategy is fixed (training-based estimation), 8 active users, 2 transmit anten-nas, mobile speed = 3 km/h, uplink channel estimates are performed at Eb/No = 6 dB,symmetrical allocation, no channel coding; the joint detection capability is not fully ex-ploited over Indoor A-like channels because they are nearly flat fading channels; as aconsequence, with 2 antennas, the joint detection provides a 0.8 dB gain at BER = 2%and 1.5 dB at BER = 1% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Bit error rates versus signal-to-noise ratio; performances of the matched filter and theMMSE joint detection are compared in Pedestrian B; simulation conditions are the sameas for figure ??; in high-selective channels such as Pedestrian B, the use of joint detectionis mandatory; the matched filter detection performance reaches an error floor; for a 3%BER target, more than 4 dB are lost because of the use of the matched filter . . . . . . . 90

5.5 Bit error rates versus signal-to-noise ratio; performances of the matched filter and theMMSE joint detection are compared in Indoor A; simulation conditions: the channel es-timation strategy is fixed (training-based estimation), 8 active users, 4 transmit antennas,mobile speed = 3 km/h, uplink channel estimates are performed at Eb/No = 6 dB, sym-metrical allocation, no channel coding; observations from figure ?? are confirmed with4 transmit antennas; at BER = 1% the joint detection outperforms the matched filter byonly 0.3 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.6 Bit error rates versus signal-to-noise ratio; performances of the matched filter and theMMSE joint detection are compared in Pedestrian B; simulation conditions are the sameas for figure ??; even with 4 transmit antennas, the joint detection is justified in highfrequency selective propagation environment . . . . . . . . . . . . . . . . . . . . . . . 91

5.7 Bit error rates versus signal-to-noise ratio; performances of the matched filter and theMMSE joint detection are compared in Indoor A; simulation conditions: the channel es-timation strategy is fixed (training-based estimation), 8 active users, 8 transmit antennas,mobile speed = 3 km/h, uplink channel estimates are performed at Eb/No = 6 dB, sym-metrical allocation, no channel coding; the curves corresponding to the matched filterand the MMSE joint detector are very close; in indoor environment, when 8 transmitantennas are used, joint detection is not worth being implemented; notice also that the1% BER target is achieved for Eb/No = 3 dB, which is due to the diversity gain . . . . . 91

5.8 Bit error rates versus signal-to-noise ratio; performances of the matched filter and theMMSE joint detection are compared in Pedestrian B; simulation conditions: the channelestimation strategy is fixed (training-based estimation), 8 active users, 8 transmit anten-nas, mobile speed = 3 km/h, uplink channel estimates are performed at Eb/No = 6 dB,symmetrical allocation, no channel coding; when 8 antennas are used at the base station,the use of joint detection schemes in the mobile station is questionable; this means thatthe detection stage complexity can be significantly reduced (by using a matched filterbased detection) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.9 For a given BER target (2%) joint detection is compared with the matched filter detectionin a nearly flat fading channel and a high frequency selective channel; 2, 4 and 8 antennasare used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.10 For a given time slot, the impulse responses of the physical channels of 8 antennas arerepresented. The last waveform (bottom right) shows the equivalent physical channelprovided by the TxAA algorithm. The latter channel tends to become flat as the numberof antennas increses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xviii List of Figures

List of Tables

1.1 Comparison between UMTS-TDD and UMTS-FDD physical layers . . . . . . . . . . . 7

1.2 Filter coefficients of the root raised-cosine filter with roll-off factor 0.22 (at Tc/2) . . . . 10

1.3 Power-delay profile of Indoor A and Indoor B propagation channels . . . . . . . . . . . 11

1.4 Power-delay profile of Vehicular A and Pedestrian B propagation channels . . . . . . . . 11

1.5 Symbol detection issues are addressed in chapters 2 and 5; channel estimation issues areaddressed in chapters 2, 3 and 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 Number of states of the trellis corresponding to the forward-backward algorithm, whichis denoted by Ns; K is the number of active users per time slot and L is the assumed ISIduration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Number of iterations (r) of the EM algorithm versus signal-to-noise ratio (SNR) . . . . . 31

2.3 Receiver structures under consideration in this chapter; three channel estimation schemesare considered: the case where the channel is known from the receiver, the training-based channel estimation (least-squares estimation) and the iterative channel estimationof the EM procedure (initialized by the trained estimate); three symbol detection schemesare considered: the matched filtering, the MMSE block linear joint detection and themaximum a posteriori symbol-by-symbol joint detection . . . . . . . . . . . . . . . . . 32

3.1 Tuning accuracy of the regularizing constant for semi-blind subspace and weighted lin-ear prediction; the regularizing constant values derived from the proposed semi-blindschemes are compared to those obtained by minimizing the Euclidean distance betweenthe true and estimated channels on each time slot; accuracy level of the regularizing con-stants is higher in the semi-blind subspace case than in the weighted linear predictionone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 Possible scenarios when assumptions (a), (b) and (c) are considered . . . . . . . . . . . 68

4.2 Noise variance estimation accuracy as a function of the number of samples . . . . . . . . 81

5.1 The proposed receiver structures based on the results from this study . . . . . . . . . . . 97

xx List of Tables

Introduction

By the early 1980s, less than one million mobile phone subscribers were registered worldwide. Twodecades later, the number of subscribers is set to reach one billion. As mobile users become more andmore numerous and demanding, mobile phone makers and operators and related companies have to con-tinuously upgrade their cellular systems. This is the main reason why the third generation (3G) cellularsystems, known collectively as International Mobile Telecommunications-2000 (IMT-2000), came intoexistence and are currently being deployed and implemented.

The most important IMT-2000 proposals are UMTS as the successor to GSM (European standard) andCDMA2000 as the successor to IS-95 (Northern American standard). The main features of 3G cellularsystems will be compatible standards, high data rate services, high flexibility and intelligent networks.Transmission quality and system capacity, which were among the 2G cellular systems developementneeds, still remain important and challenging issues to focus on.

As we will see, the ultimate goals of the study that this report is related to, are to improve transmissionquality and data rates of 3G systems and especially the UMTS-TDD mode based systems. Indeed, theThird-Generation Partnership Project (3GPP), which comprises ETSI (Europe), ARIB/TTC (Japan), T-1 (USA), TTA (South Korea) and CWTS (China), has decided that UMTS terrestrial radio access willsupport both Frequency Division Duplex (FDD) and Time Division Duplex (TDD).

With regard to transmission quality and data rate improvements, the main efforts have to be concentratedon radio access and core networks. As far as this study is concerned, it will focus on radio accessaspects since its main aim is to design advanced UMTS-TDD receivers by devising, implementing andtesting signal processing and digital communications algorithms. The goal is of course to design themost efficient and simple receiver.

From a baseband processing point of view, receiver performance is mainly linked to the three followingreceiver stages:

• the propagation channel estimator,

• the symbol detector and

• the channel decoder.

Indeed, in radio systems, there is a need to estimate the effect caused by the radio communication channelon the data being transmitted. Propagation channel estimation is required so that the received data canbe equalized in order to reduce, restore or minimize signal degradation caused by transmission channelimpairments. From a given knowledge of the propagation channel, the symbol detector (or equalizer)purpose is then to recover the transmitted data. Finally, the channel decoder purpose is to eliminate the

2 Introduction

redundancy, which has been introduced at the transmitter side in order to insure better protection againstchannel degradations, from the detected data.

In the framework of this PhD project, our main objective was twofold:

• to enhance channel estimation accuracy and symbol detection reliability and

• to reduce receiver complexity

with respect to conventional TDD receivers (as those recommended by IMT-2000).

These three problems (how to enhance channel estimation accuracy, how to improve symbol detectionreliability and how to reduce receiver complexity) are analyzed over four chapters. This PhD thesis com-prises five chapters. The first one aims at reviewing basical concepts of radio communications systems(multiple access interference and multiple access methods in particular) and describing the physical layerof the second operating mode of the Universal Mobile Telecommunications System (UMTS) namely thetime division duplex mode (TDD). At last, this chapter presents the signal model that is used for thewhole study, outlines the thesis content and provides the motivations of this research.

Chapter 2 focuses on receiver performance by considering the optimum semi-blind channel estimatorand the maximum a posteriori symbol-by-symbol symbol detection. This receiver is implemented viathe well-known Expectation-Maximization algorithm.

In chapter 3, only the channel estimation problem is addressed. The goal is also to exploit all the ob-servations generated by the data time slot (semi-blind principle) but less complex solutions are studied.More specifically, this chapter deals with suboptimum semi-blind channel estimation schemes, which arebased on second-order blind channel estimation techniques.

In chapter 4, an additional assumption is made. Time slots transmissions are assumed to be continu-ous. We show how to exploit the well-known Wiener approach in order to propose channel estimationsolutions both simple and very efficient. The basic idea is to directly remove estimation noise from thetraining-based channel estimates by using several data time slots and all the available information onthe system. At this point, we have to precise that, in chapters 2, 3 and 4, performance evaluations andtheoretical derivations are done for the most general uplink communications case (transmissions fromthe mobiles to the base station) but presented algorithms are also applicable to downlink transmissions(from the base station to the mobile).

Chapter 5 specifically addresses the downlink case. The aim is to know whether a single-user symboldetector can be implemented in the user equipment while providing reasonable performance. To this end,use is made of an adaptative multiple antenna in the base station, which is implemented via the TxAAalgorithm (Transmit Adaptative Antennas). Based on a comparison between performances of matchedfilter based and linear multiuser detection schemes, we answer to the raised question.

Finally, concluding remarks are made in order to summarize important results from this study and opennew research perspectives.

Chapter 1

Problem statement and motivations

As mentioned in the introduction, for the whole study, we focus on the physical layer (Layer 1 in theOpen Systems Interconnect model) and more particularly on the propagation channel estimation andthe symbol detection problems. Before describing more precisely in what sense channel estimation andsymbol detection can be improved in the UMTS-TDD mode, we give a general view of the context ofthe study by briefly reviewing the multipath problem, the code division multiple access principle (whichis the main multiple access technology that has been chosen for IMT-2000), the concepts of intersymbolinterference (ISI) and multiple access interference (MAI). Next, we describe the UMTS-TDD modephysical layer. At last, we outline the thesis by providing a brief overview of the chapter contents andthe motivations of this research.

1.1 Overview of the problem

1.1.1 Multipath and ISI canceller

Radio propagation in the land mobile channel is characterized by multiple reflections, diffractions and at-tenuation of the transmitted signal energy. These are caused by natural obstacles such as buildings, hills,and so on, resulting in so-called multipath propagation. More specifically, multipath is the compositionof a primary signal plus duplicate or echoed images caused by reflections of signals off objects betweenthe transmitter (e.g. the base station) and receiver (e.g. the mobile station). In figure 1.1, the receiver"hears" the primary signal sent directly from the transmission facility, but it also sees secondary signalsthat are bounced off nearby objects. These bounced signals will arrive at the receiver later than the inci-dent signal. These signals are overlapped and combined into a single one. Because of this misalignment,the "out-of-phase" signals will cause intersymbol interference (ISI) or distortion of the received signal.Correction algorithms (equalization step or equivalent step) must be put in place to compensate for theeffect of multipath on transmitted symbols (ISI).

The importance of the ISI cancellation step depends on propagation environment. In line-of-sight en-vironments, there is a direct path between the base station (BS) and the mobile station (MS) so thatmultipath is usually minor and can be overcome easily. The amplitudes of the echoed signals are muchsmaller than the primary one and can be effectively filtered out using standard equalization techniques.However, in non-line-of-sight environments such as those considered in this study, the echoed signals

4 Problem statement and motivations

may have higher power levels, because the primary signal may be partially or totally obstructed, andgenerally more multipath is present. This makes the equalization design more difficult.

Figure 1.1: Multipath propagation channel

1.1.2 CDMA and MAI canceller

1.1.2.1 Multiple access methods

In the description of multipath reception made above, there was only one mobile phone user communi-cating with the base station. In real situations, several users are exchanging data with the base station.As the communication resources are finite, they have to be shared amongst the active users. This leadsto the most important concept to any cellular telephone system, which is the "multiple access" concept,meaning that multiple, simultaneous users can be supported. In other words, a large number of usersshare a common pool of radio channels and any user can gain access to any channel (each user is not al-ways assigned to the same channel). A channel can be thought of as merely a portion of the limited radioresource which is temporary allocated for a specific purpose, such as someone’s phone call. A multipleaccess method is a definition of how the radio spectrum is divided into channels and how channels areallocated to the many users of the system.

A number of multiple access techniques exist (figure 1.2), whereby a finite communication resource isdivided into any number of physical parameters, such as:

• Frequency division multiple access (FDMA) whereby the total number of frequencies used in thecommunication system are shared.Example: AMPS system uses 30 kHz "slices" of spectrum for each channel. With FDMA, onlyone subscriber at a time is assigned to a channel. No other conversations can access this channeluntil the subscriber’s call is finished, or until that original call is handed off to a different channelby the system.

• Time division multiple access (TDMA) whereby each communication resource, say a frequencyused in the communication system, is shared amongst users by dividing the resource into a numberof distinct time periods (time slots, frames, etc.).Example: GSM system uses 8 time slots for each frequency.

1.1 Overview of the problem 5

• Code division multiple access (CDMA) whereby communication is performed by using all of therespective frequencies, in all of the time periods, and the resource is shared by allocating eachcommunication a particular code on any frequency at any time, to differenciate desired signalsfrom undesired signals.Example: IS-95 system divides the radio spectrum into carriers which are 1.25 Mhz wide andmakes use of 64 spreading codes for each carrier.

A more "illustrative" explanation of multiple access principles is provided in Appendix A for non-experts.

Figure 1.2: Different multiple access schemes: power versus time and frequency for FDMA (left),TDMA (middle) and CDMA (right)

1.1.2.2 CDMA concept

The CDMA concept is the following: each user is assigned a unique code sequence (spreading code)it uses to encode its information-bearing signal. The receiver, knowing the code sequences of the user,decodes the received signal and recovers the original data. This is possible since the crosscorrelations (orinner products) between the codes of the desired user and the codes of the other users are small or zero.

In fact, in an environment without multipath (so-called flat fading channels), performing a simple corre-lation at the receiver side suffices to separate the desired user from the other users. If the various usersare (re-)synchronized at the receiver side and the codes are orthogonal, the user of interest can be totally"isolated" from undesired ones. If the codes are not perfectly orthogonal, there remains a residual inter-ference, which is named multiple access interference (MAI). Note that MAI is a concept proper to theso-called multiuser systems such as CDMA system.

On the other hand, if there is multipath, even if the transmitted codes are perfectly orthogonal, the desireduser cannot be totally separated from the undesired users. This is due to the fact that codes are filtered bythe multipath channel. Therefore, orthogonality can no longer be maintained (see figure 1.3). In order tocompensate for the effect of multipath on codes (high MAI), MAI cancellation step (or equivalent step)must be put in place.


Figure 1.3: Multipath effect on spreading codes: orthogonality is lost

1.1.3 The need for channel estimation

To remove intersymbol and multiple access interferences from CDMA signals that have been transmittedover multipath channels, equalization and MAI cancellation steps are required. Usually, these steps areperformed in one step, which is the multiuser detection step. Performing the detection step requiresthe estimation of the radio propagation conditions. This is why a procedure estimating the channelpropagation has to be put in place.

1.2 The UMTS-TDD Mode

1.2.1 The UMTS air interface

1.2.1.1 Why CDMA?

As mentioned before, CDMA is the main multiple access technology selected by IMT-2000 for 3Gsystems (UMTS and CDMA2000). The two UMTS air interfaces are commonly referred as W-CDMA(wideband) and TD-CDMA (time division). Among the reasons that have been claimed for selectingCDMA as the multiple access scheme for 3G are: capacity increase, simplified system planning throughthe use of the same frequency in every sector of every cell, improved coverage characteristics allowingfor the possibility of fewer cell sites and bandwidth on demand. We will not comment on IMT-2000 airinterface selections since they did not depend on technical factors only but also political and businessones [?].

In order to better understand the features of the TD-CDMA mode, we make a simplified and technicalcomparison between the FDD and the TDD physical layers. Note that the W-CDMA mode is also namedFDD mode since duplex is made in the frequency-domain while the TD-CDMA mode is called TDD

1.2 The UMTS-TDD Mode 7

mode because duplex is made in the time-domain.

1.2.1.2 Comparison of UMTS-TDD and UMTS-FDD physical layer key parameters

Table 1.1 shows the main differences between the UMTS-TDD and UMTS-FDD physical layers. Asfar as this study is concerned (focus on the UMTS-TDD mode), at least three points are worth beingemphasized:

• users are synchronized in uplink (UL) and downlink (DL),

• spreading codes are short and

• multiuser detection is recommended (joint detection).

TDD mode FDD mode

Frequency bands 1900 MHz - 1920 MHz UL: 1920 MHz - 1980 MHz2010 MHz - 2025 MHz DL: 2110 MHz - 2170 MHz

Duplex method Time division Frequency divisionCells synchronization Supposed to be acquired Not necessaryUsers synchronization Required for both UL and DL Not requiredCodes length 1 to 16 4 to 512BS reception Joint detection recommended Support for advanced receiversMS reception Joint detection recommended Low-complexity receivers possiblePower control UL = 100 Hz / 200 Hz 1500 Hz

DL ≤ 800 HzIntra-freq. handover Hard handover Soft handover

Table 1.1: Comparison between UMTS-TDD and UMTS-FDD physical layers

1.2.1.3 Why UMTS-TDD?

The aim of this section is to know more about the reasons why UMTS-TDD will be useful for 3Gsystems. At least four issues are worth being highlighted.

1. Asymmetric services: services such as the Internet often set different capacity requirements for theuplink and downlink. The utilization of a TDD frequency band is not fixed between the uplinkand downlink (unlike with FDD) and this flexibility in resource allocation can be used if the airinterface design is flexible enough.

2. Reciprocal channel: based on the received signal, the TDD transmitter is able to know the fastfading of the multipath channel (fast fading depends on frequency). This assumes that the TDDframe length is shorter than the coherence time of the channel. This assumption holds if TDDmobiles are slowly moving terminals. For instance, reciprocal channel can be utilized for closedloop power control and adaptive antenna techniques.


3. Spectrum allocation: providing high bit rate services, even with high spectral efficiency, requiresa large bandwidth. Efficient and flexible utilization of all the available bandwidth, including theUMTS-TDD band, is essential for viable 3G systems.

4. Standards cohabitation: as the UMTS-TDD mode only needs a single band for both uplink anddownlink, it can operate in countries where FDD bands are not available. For example, the uplinkFDD band is not available in the US since it is occupied by Personal Communications Services(1850 MHz - 1990 MHz).

1.2.2 Physical layer description

1.2.2.1 The UMTS-TDD options

The UMTS-TDD mode comprises two options, the 3.84 Mcps TDD (wide band TDD) and the 1.28 McpsTDD (narrow band TDD) [?]. The first one is the historical European UMTS-TDD option, which waschosen at the same time as UMTS-FDD. The Narrow Band option has been proposed by China WirelessTelecommunications Standard (CWTS) within 3GPP TSG RAN WG1 and accepted as second optionduring the year 2000. The main differences between these two options are mainly located in the physicallayer (chip rate three times lower for narrow band TDD, antenna arrays with 8 elements recommendedfor narrow band TDD, etc). Thus, from now on, the option under consideration is the 3.84 Mcps TDDoption. However, almost all the algorithms presented in this report can be used for the 1.28 Mcps option.Of course, performance gains have to be reevaluated.

1.2.2.2 Frame and time slot structures

All physical channels take on a three-layer structure with respect to time slots, radio frame and systemframe numbering. Depending on resource allocation, the configuration of radio frames and time slotsbecomes different. Time slots are used in the sense of a TDMA component to separate signals from dif-ferent users in the time and code domains. Each TDMA frame has a duration of 10 ms and is subdividedinto 15 time slots, as we can see in figure 1.4.

Radio Frame (10 ms)

Time slot 1 Time slot 2 Time slot 3 Time slot 13 Time slot 14 Time slot 15

Frame n+1Frame n

Time Slot (2560 Tc)

Figure 1.4: UMTS-TDD frames are divided into 15 time slots; each frame has a 10 ms duration and eachtime slot comprises 2560 chips

A physical channel in UMTS-TDD is a particular code within a given time slot allocated within radioframes. The allocation can be continuous, i.e. the time slot in every frame is allocated to the physicalchannel, or discontinuous, i.e. the time slot in a subset of all frames is allocated only.

1.2 The UMTS-TDD Mode 9

Every time slot can carry a maximum of 16 users differentiated in the code-domain by their allocatedspreading code (channelization code). One user can use several codes in one time slot in order to achievea higher data rate, as defined in [?] for the UE (User Equipment) of for the Node B (Base station).

A time slot is made up of four parts (figures 1.5 and 1.6), one midamble (training sequence for channelestimation purpose) located between the two data parts and one guard period at the end of the time slot inorder to absorb the channel memory. The duration of one time slot is 10ms/15 = 666.66 µs. Each timeslot can be allocated either to the uplink or to the downlink. In any configuration, at least one time slotmust be allocated for the uplink, and this applies to the downlink as well. There are two types of timeslot (for dedicated data traffic) that differ from each other by the length of the midamble, that of courseimplies different number of chips in both data parts. The time slot type 1 is defined in figure 1.5 and thetime slot type 2 in figure 1.6. The midamble length equals 512 chips for the type 1 or 256 for the type 2.

Data Symbols976 chips

Midamble512 chips

Data symbols976 chips 96 chips

GP

2560 chips

Figure 1.5: Type 1 UMTS-TDD time slot comprises two data parts, one 512-chip midamble and one96-chip guard period (GP)

Data Symbols Data symbols96 chips GP

2560 chips

1104 chips 1104 chipsMidamble256 chips

Figure 1.6: Type 2 UMTS-TDD time slot comprises two data parts, one 256-chip midamble and one96-chip guard period (GP)

The data rate of the physical channel depends on the length of the midamble used since a longer midamblelength implies a shorter data field. Short midambles are therefore used in service providing higher datarate. In 3GPP RAN WG4 technical specifications [?], [?] short midambles are used for data rates of 144kbps and 384 kbps and long midambles for data rates of 12.2 kbps and 64 kbps.

1.2.2.3 Modulation, spreading, scrambling and shaping filtering

The data modulation scheme in UMTS-TDD is QPSK. The modulated data symbols are spread with aspecific channelization code of length ranging from 1 to 16 (1 or 16 for the downlink and 1, 2, 4, 8 or 16for the uplink).The chip duration is:

Tc =666.66 µs

2560= 0.26042 µs.

The maximum symbol duration is:

Ts = 16×Tc = 4.1667 µs.


The 16 orthogonal binary CDMA codes are generated from Walsh-Hadamard codes. Data spreading isthen followed by scrambling with a cell-specific scrambling sequence, which is used to mitigate inter-cell interference. The scrambling process is a chip-by-chip multiplication (the scrambling codes lengthis equal to the spreading factor). The scrambling sequences are Gold sequences (pseudo noise sequence).The spreading codes are finally obtained by a multiplication of these elements (a(n)) with a 4-periodicsequence:

∀ n ∈ [1,N], c(k)(n) = (i)n.a(k)(n)

where N is the spreading factor and k the user index. Finally, pulse shape filtering is applied to eachchip at the transmitter: each chip is filtered with a root raised cosine filter with roll-off factor 0.22. Inorder to be implemented, the discrete impulse response f (i) of this filter has to be truncated. The ETSIspecifies to keep 17 coefficients of the Tc/2-rate impulse response as described in table 1.2:

i 0 +/−1 +/−2 +/−3 +/−4 +/−5 +/−6 +/−7 +/−8f(i) 0.7550 0.4401 −0.0449 −0.1215 0.0366 0.0480 −0.0239 −0.0166 0.0083

Table 1.2: Filter coefficients of the root raised-cosine filter with roll-off factor 0.22 (at Tc/2)

1.2.2.4 Channel coding and interleaving

For the channel coding in UMTS-TDD, three options are supported:

• convolutional coding;

• turbo coding;

• no channel coding.

Channel coding selection is indicated by higher layers. In the technical specifications of 3GPP RANWG1 [?], it is shown the usage of channel coding schemes and coding rates in transport channels. Even-tually, in order to randomize transmission errors, bit interleaving is performed further.

1.2.2.5 Transmit diversity (downlink)

Two transmit diversity techniques with 2 elements have been standardized in downlink:

• closed loop for dedicated channels: adaptative transmit diversity (TxAA, [?], [?],) and selectivetransmit diversity (STD);

• open loop for common channels: space-time transmit diversity (block STTD, [?], [?], [?], [?]).

1.2.2.6 ETSI test propagation environments

In this subsection, we present four propagation environments that are specified by the European Telecom-munications Standardization Institute. The four propagation channels under consideration comprise 6

1.3 Problem statement 11

paths (taps). Each tap is characterized by its relative delay and its average power, the first tap beingconsidered as the reference path both in terms of delay and power. That is why delay and power of thelatter are zero and unitary (or 0 in dB) respectively.

Tap Indoor A Indoor B

Relative Average Relative AverageDelay[ns] Power[dB] Delay[ns] Power[dB]

1 0 0 0 02 50 −3.0 100 −3.63 110 −10.0 200 −7.24 170 −18.0 300 −10.85 290 −26.0 500 −18.06 310 −32.0 700 −25.2

Table 1.3: Power-delay profile of Indoor A and Indoor B propagation channels

Tap Pedestrian B Vehicular A

Relative Average Relative AverageDelay[ns] Power[dB] Delay[ns] Power[dB]

1 0 0 0 02 200 −0.9 310 −1.03 800 −4.9 710 −9.04 1200 −8.0 1090 −10.05 2300 −7.8 1730 −15.06 3700 −23.9 2510 −20.0

Table 1.4: Power-delay profile of Vehicular A and Pedestrian B propagation channels

1.3 Problem statement

1.3.1 Signal model

1.3.1.1 From the physical problem to the MIMO system modelling

The aim of this section is to introduce the Multiple Inputs Multiple Outputs (MIMO) system model thatwill be used for the whole study. This model will be the same for uplink and downlink except for the factthat propagation channel is common to every user in downlink, which is not the case in uplink. We wantour system model to take into consideration:

1. the multipath propagation;


2. the emission and reception filterings;

3. the code division multiple access;

4. the presence of noise;

5. the mobility of the terminal(s).

Multipath effect of propagation channels on transmitted data is generally represented by a linear filter [?][?]. This means so far that transmitted data undergo three filterings: the shaping filtering, the propagationchannel filtering and the reception filtering (for more information on shaping and receiving filtering seereferences [?] [?]). In our study, the transfer function of the resulting filter(s) will be denoted by gk(z),k being the user index. In downlink g1(z) = g2(z) = . . .gK(z), where K is the number of active users pertime slot.

Basically, for a given user, CDMA operation consists in modulating each transmitted symbol by the sig-nature sequence or spreading code that is assigned to the considered user. Additionnally, if the spreadingsequence period equals the symbol duration, which is the case in the UMTS-TDD mode, one can showthat the spreading process is equivalent to an oversampling followed by a linear filtering [?].

c(z)

c0, . . . ,cN−1

⇔x(t)t∈Z c0x(t), . . . ,cN−1x(t)t∈Z x(t)

Nx(n) d(n)

Figure 1.7: Equivalence between the conventional (left) and Tsatsanis (right) representations of CDMAoperation for N-periodic CDMA codes

In this figure, which corresponds to a given user, x(t) is the sequence of QPSK symbols to be transmittedand N is the spreading factor. c0, . . . ,cN−1 is the spreading sequence of the considered user from whichwe define c(z) , ∑i=N−1

i=0 c(i)z−i. One can easily check that sequence d(n) (on the right) coincidewith the output of the spreading operator (on the left) from the definition of oversampling: ∀(t, j) ∈Z× [0,N−1], x(tN + j) = δ jx(t). Therefore, ∀(t,m) ∈ Z× [0,N−1],

d(tN +m) =i=N−1

∑i=0

c(i)x(tN +m− i) (1.1)

=i=N−1

∑i=0

c(i)δm−ix(t) (1.2)

= c(m)x(t) (1.3)

which is exactly the output of the spreading operator.

As for noise, it is modelled by an additive white and Gaussian random process (AWGN). Recall thatnoise both comprises the internal noise, which is essentially due to components of the receiver, and theexternal noise coming from other systems [?].

Finally, mobility is taken into account by updating propagation channels at the beginning of each timeslot. Channel update is made according to the Jakes model [?]. Over a given time slot however, propa-gation channels are fixed and modeled according to [?].


For conventional CDMA systems, received signal is sampled at the chip rate. When the received signal issampled at the chip rate, it is generally assumed a baseband equivalent discrete-time model. In summary,the synchronized CDMA system under consideration (the UMTS-TDD system) can be represented bythe classical multirate discrete-time equivalent model [?] given in figure 1.8.

AWGN

N

N

x1(n)x1(t)

xK(n)xK(t)

y(n)

g1(z)

gK(z)

c1(z)

cK(z)

h1(z)

hK(z)

Figure 1.8: The MIMO system under consideration in this study; there are K inputs and N outputs (thelatter being the N samples associated with a given symbol)

where:

• N is the spreading factor, which value is constant (N = 16);

• K is the number of active users per time slot;

• for each k ∈ 1, · · · ,K, xk(t)t∈Z is a sequence of independent and identically distributed QPSKsymbols and xk(n) is the corresponding up-sampled sequence. For 1 ≤ k ≤ K, 1 ≤ k′ ≤ K andk 6= k′, the sequences xk(t)t∈Z and xk′(t)t∈Z are independent;

• for each k∈1, · · · ,K, (ck(n))n=N−1n=0 denotes the spreading code of user k, and ck(z)= ∑n=N−1

n=0 ck(n)z−n

is the corresponding degree (N−1) polynomial vector;

• the polynomial gk(z)k=1,...,K are the z-transform associated with the discrete-time equivalent ofunknown channels sampled at the chip-rate. For the user #k, let us define `k as `k = deg(gk(z)). Inthe sequel, we will use also ` = maxk(`k). These channels are assumed to be causal. Various usersbeing synchronized, lim|z|→∞ gk(z) 6= 0 for k ∈ 1, . . . ,K;

• hk(z) denotes the transfer function defined by hk(z) = ck(z)gk(z);

• v(n) is an additive white complex circular Gaussian noise;

• y(n) is the received signal.

For the whole study, a single antenna is assumed at the mobile station. Beside chapter 5, the sameassumption will be done for the base station. Note that each of the filters gk(z) include the emission filter(a root raised cosine filter), the propagation channel and the reception filter (a root raised cosine as well).These filters are different for the uplink but identical for the downlink because the propagation channel 1

is common to every user (∀k ∈ [1,K]Z,gk(z) = g(z)).1Propagation environment that will be considered are those specified by the ETSI. As for propagation channel modelling, it

is based on the Jakes fading model revisited by Dent et al [?].


1.3.1.2 Expression of the received signal

The received signal y(n) (sampled at the chip rate) may be expressed as

y(n) =k=K

∑k=1

[hk(z)]xk(n)+ v(n). (1.4)

It is often more convenient to represent the received signal by the stationary N–dimensional signal y(t) =(y(tN) . . .y(tN +N−1))T with t ∈ Z. It is easily seen [?] that

y(t) = [H(z)] x(t)+ v(t) (1.5)

=τ=L

∑τ=0

H(τ) x(t− τ)+ v(t) (1.6)

where:

• x(t) = (x1(t) . . .xK(t))T ;

• v(n) = (v(tN) . . .v(tN +N−1))T ;

• H(z) = [h1(z) . . .hK(z)] is a N×K polynomial matrix of degree L;

• ∀τ∈ [0,L],H(τ) is such that [HT (0) . . .HT (L)]T = [h1 . . .hK ] with ∀k∈ [1,K],hk =(hk(0) . . .hk(l))T ;

• hk(z) = (h(0)k (z) . . .h(N−1)

k (z))T ;

• h(0)k (z) . . .h(N−1)

k (z) are the polyphase components of hk(z) = ∑i=`k+N−1i=0 hk(i)z−i (Appendix B).

1.3.2 Starting point: the initial UMTS-TDD receiver

We already mentioned in the introduction that the receiver structure mainly comprises three stages: thechannel estimation stage, the detection stage and the decoding stage (figure 1.9). This work is focusedon improvement of the first and the second stages. Even though the receiver structure of the mobilestation is the same as the base station one, complexity is a much more crucial issue at the user equipmentside. That is why, in general, implemented algorithms in the mobile station differ from those imple-mented in the base station.

In this study, we will focus more on the uplink case (chapters 2, 3 and 4) because this choice was madeaccordingly to a strategic decision of Motorola and it makes more sense to implement advanced receiver(inducing generally an additional complexity cost) in the base station. By default, all algorithms underconsideration will be studied in the uplink case, even though many of them are applicable to the downlinkcase.

1.3.2.1 First receiver stage: conventional channel estimator

In order to compensate for signal degradations, which are caused by the propagation channel, there isthe need for estimating the propagation channel impulse response. Transmitted time slots durations are

ChannelEstimator Detector

ChannelSymbolDecoder

ReceivedSignal

EstimatedData

Stages under consideration in this study

Figure 1.9: The receiver is considered as a 3-stage structure

generally small enough to assume the channel impulse response to be constant over a time slot. A part ofeach time slot is dedicated to channel estimation purpose. This part is named Training Sequence (TS)and comprises only symbols known from the receiver. In the UMTS-TDD mode, the training sequenceis located in the middle of time slot (see 1.5, 1.6) and is therefore named midamble.

The most commonly used channel estimation procedure is based on a Least-Squares approach, whichconsists in minimizing the mean square error between the received signal during the emission of thetraining sequence and the filtered version of the known chip sequence.

Unlike UMTS-FDD, UMTS-TDD training sequences are not obtained by spreading known symbols.Indeed, midambles are defined at the chip rate. In uplink, midamble length is fixed at w = 512 chips(while in downlink use of 256- or 512-chip midamble is possible [?]). Denote by ` the maximum channellength (expressed in number of chips) that is ` = max(`1, . . . , `K). Define also m by m , w− `. At last,denoting by sk the chip-rate-defined training sequence for user #k, the received signal is given by:

ytr(n) =k=K

∑k=1

i=`

∑i=0

gk(i)sk(n− i)+ v(n). (1.7)

As the unknown parameters, which are the channel coefficients gk(i), can be stacked into one vector g =[gT

1 . . .gTK]T (g

k= (gk(0) . . .gk(`))T ), it is convenient to make use of the multi-dimensional counterpart

of (1.7):

Y tr = Sg+V tr (1.8)

where:

• Y tr is the m−dimensional vector obtained by stacking the observations generated by the mi-dambles. By definition, Y tr = (y(`) . . .y(w− 1))T . Note that the use of y(i) for i < ` is not pos-sible because it would require to know the chips that have been transmitted before the trainingsequences;


•

S =

s1(`) . . . s1(0) . . . sK(`) . . . sK(0)...

s1(w−1) . . . s1(w−1− `) . . . sK(w−1) . . . sK(w−1− `)

d= m×K(`+1);

(1.9)

• V tr = (v(`)...v(w−1))T ;

recall that notation S d= m×K(`+1) () means that the matrix S has m columns and K(`+1) rows.

Noise being white and Gaussian, maximum likelihood estimates of g based on the observation Y tr isgiven by:

gtr

= argminf‖ Y tr−S f ‖2 . (1.10)

From equation (1.10), it is also possible to explicit the trained channel estimate as:

gtr

= S#Y tr (1.11)

which is the form used for implementation. Superscript "#" denotes the Moore Penrose pseudo inverse:

gtr

= (SHS)−1SHY tr. (1.12)

In the sequel we will also make use of RSS and RSY defined by:

RSS , 1m

SHS and RSY , 1m

SHY tr. (1.13)

1.3.2.2 Second receiver stage: conventional symbol detector

In CDMA systems, the most conventional way of decoding transmitted data is to implement the Rakereceiver [?] as it is done in IS-95 systems and in many UMTS-FDD systems. However, it is indicatedin table 1.1 that joint detection schemes [?] are recommended for UMTS-TDD systems. In fact, one ofthe major difference between UMTS-TDD and UMTS-FDD/IS-95 is the operating spreading factor. ForUMTS-TDD, spreading factor ranges from 1 to 16 while ranging from 4 to 512 for UMTS-FDD. Asthe Rake receiver performance is strongly related to processing gain, which is quite low in UMTS-TDD(maximum value: 10× log10(16) = 12 dB ), it cannot be expected from the Rake receiver to performwell in such a context, at least if no pre-processing is made at the base station (e.g. transmit diversity).We will discuss this point more deeply in chapter 5.

Moreover, one of the major concern with CDMA systems is the near far problem [?]. Because of thecoarse fading variations due to shadowing by obstacles in the line of sight between receiver and trans-mitter, the differences in power received at the base station from different users may be very high (e.g.90 dB in typical urban cells). The Rake receiver is very sensitive to this effect since it is based on a"single-user" structure and requires accurate implementation of fast power control to combat fast fadingand the resulting near far effect.

At least for these two reasons (low spreading factor and near far problem), alternatives to the Rakereceiver are desirable in UMTS-TDD systems. Joint detection, which is the multiuser scheme recom-mended by IMT-2000 specifications, shows better performance than the Rake receiver. The Zero Forcing


(ZF) and the Minimum Mean Square Error (MMSE) equalizers are the two reference linear Joint De-tectors (JD). The ZF joint detector is known for "killing" both ISI and MAI2 while the MMSE detectorcombats ISI, MAI and noise [?].

The type of joint detector that will be considered in this thesis is those developed in [?], [?]. Theseare in particular zero-forcing block linear equalizer and minimum mean square error block linearequalizer. These detection schemes have been designed to adapt conventional ZF and MMSE equalizersto block transmission CDMA systems. Let us describe what these equalizers consist in.

By looking at the UMTS-TDD time slot structure (figures 1.5 and 1.6) it can be seen that there are twodata blocks in every time slot. For sake of simplicity, we will consider these blocks as isolated, whichmeans there is no interchip interference (ICI) corrupting the first chips of each block. In summary,it is assumed that they can be decoded individually. Note that this assumption is only needed for thesecond block. Indeed, even in the case where time slots are continuously transmitted, the first block isnot corrupted by ICI because of the guard period at the end of each time slot (no energy over 96 chipdurations).

Zero forcing block linear equalizer (ZF-BLE)

Denote by T the number of data symbols in one time slot. For type 1 time slots, T = 122 and T = 138for type 2 time slots. For a given data part (comprising T

2 QPSK symbols) stack the transmitted symbolsvector x(t) (equation (1.5)) comprising data from all active users into one combined data vector:

X ,[

xT (0) . . .xT(

T2−1

)]Td=

T2

K×1. (1.14)

In the same way, let us define:

V =[

vT (0) . . .vT(

T2−1+L

)]Td= (

T2

+L)N×1 (1.15)

Y ,[

yT (0) . . .yT(

T2−1+L

)]Td= (

T2

+L)N×1. (1.16)

It is then easily seen that X , V and Y are related by:

Y = S(H)X +V (1.17)

2If the channel was perfectly known, ISI and MAI would be totally removed by the ZF operation


where S(H) is the Sylvester matrix given by:

S(H) =

H(0) 0...

. . .

H(L). . .

. . . H(0). . .

...0 H(L)

d=(

T2

+L)

N× T2

K. (1.18)

The zero forcing solution is obtained by eliminating the effects of the multipath channel (ISI and MAI),which corresponds to pseudo-invert the filtering matrix S(H):

XZF = S#(H)×Y (1.19)

(1.20)

XZF = (SH(H)S(H))−1SH(H)Y . (1.21)

Minimum least mean square error block linear equalizer (MMSE-BLE)

The MMSE-BLE minimizes E[(XMMSE −X)H(XMMSE −X)] and leads to:

XMMSE = (SH(H)S(H)+σ2IKT/2)−1SH(H)Y . (1.22)

Note that performing the ZF or the MMSE block linear joint detections requires estimation of H.

1.3.3 Towards a new receiver

When designing a receiver, the main concern is to improve data rates, quality of service and powerconsumption. It is a hard task to jointly optimize these three parameters . Almost all the time, thereis a trade-off to find between quality of service (or data throughput) and power consumption. In orderto find the "best" trade-off between performance and complexity it is of a great importance to takeinto account the whole information that we have on the studied system. In this section is provided theprior information (subsection 1.3.3.1) on which our study is based for enhancing the initial UMTS-TDDreceiver either in terms of performance or in terms of complexity. Then we precisely define our objectivesand provide the thesis outline (subsection 1.3.3.2).

1.3.3.1 Prior information

−P1− Observations generated by the unknown symbols also contain information on the channel.

−P2− UMTS-TDD midambles have "good" correlation properties (RSS is almost diagonal).


−P3− Under some conditions the matrix S is right circulant. Conditions: the assumed channel lengthhas to be 57 chips or 64 chips for type 1 and type 2 time slots respectively.

−P4− Mobile radio channels are sparse. Channels can be decribed by a reduced number of significantcoefficients.

−P5− Assumed propagation channel lengths are overestimated almost all the time. For instance, fortype 1 time slots, the assumed physical channel length is 57, which correspond to a 15 µs duration.This has to be compared to delays from tables 1.3 and 1.4.

−P6− Reciprocal channel. At a given time, uplink and downlink channels can be considered as iden-tical. This assumption is justified in the UMTS-TDD mode because the duplex is done in thetime-domain.

1.3.3.2 Road map and contributions

As for performance, our objective is twofold: we want to improve both accuracy of channel estimatesand detection reliability. On the other hand, we have to reduce as much as possible receiver complexity.As we will see, this is possible for the detection stage only.

When considering the channel estimation problem, it is useful to define two types of transmissions:

• the sporadic transmission, which is the case where only one time slot can be used for the channelestimation procedure. In the sporadic case, the time interval between two successive time slotsundergoes large variation and is too high to allow the estimation procedure to make use of morethan one time slot;

• the continuous transmission, which corresponds to the case where several time slots can be ex-ploited in the estimation process. This time, transmission is roughly regular and time intervalbetween two successive time slots is small enough regarding channel stationarity.

Correspondingly, chapters 2 and 3 are devoted to the first type of transmission while chapter 4 makes thecontinuous transmission assumption.

In chapter 2, we study a receiver based on the Expectation Maximization (EM) algorithm. This algorithmexploits the observations generated by the whole time slot in an opimum way. Essentially, what we areasking for is:

• In terms of performance, how well does the initial UMTS-TDD receiver perform when comparedto the "best" reception scheme (maximum likelihood receiver) and the "worse one" (matched filteror Rake receiver)? (Q1)

• Is there a stage (estimation stage or detection stage) which deserves more attention than the other?Which stage has the stronger impact on overall receiver performance? (Q2)

Chapter 3 is devoted to semi-blind channel estimation based on second-order blind methods (subspaceand linear prediction methods). The chief idea is also to take into account the observations generated bythe unknown symbols in order to improve channel estimates accuracy. In this case however, the goal is


to provide semi-blind solutions less complex than the EM algorithm-based solution. That is why in thischapter we study suboptimum semi-blind approaches. In this chapter, the main issues are:

• Implementing the semi-blind approach that is considered in this thesis requires the minimizationof a criterion resulting of the combination between the training sequence based on criterion andthe blind criterion. The problem is that this (linear) combination gives rise to a weighting problem.The question is then: is that possible to properly tune the regularizing constant introduced by thesemi-blind criterion? (Q3)

Reference [?] shows how to deal with this tuning problem for the semi-blind subspace case for SIMOsystems. In our study, we show how to extend these results to CDMA systems and semi-blind estimationbased on second-order blind techniques. In particular we answer the two following questions:

• In an overmodelling context, which is a situation where propagation channel lengths are overesti-mated, what is the best second-order blind method to make use of to design a semi-blind channelestimator? How robust are semi-blind channel estimators to channel overmodelling? (Q4)

• How well are the proposed semi-blind estimators performing in UMTS-TDD-like realistic con-texts? (Q5)

In chapter 4, assuming continuous time slot transmissions, we propose improving performances of theconventional training-based channel estimation by adding a post-processing stage to the training-basedchannel estimation stage. Unlike chapters 2 and 3, the channel is no more considered as a deterministicprocess but as a random one. This chapter aims at taking into account statistics of the channel andprior information (−P2− to −P5−) (Q6). Our approach is based on the well-known Wiener filtering.Depending on the admitted system complexity and the channel nature, we show how to exploit theWiener approach for UMTS-TDD systems and implement it in a simple and efficient way.

Finally, in chapter 5 we propose implementable receiver structures. Chapter 5 addresses more particular-ily downlink issues. For uplink, a short summary of results obtained in chapters 2, 3 and 4 is done. Asfor downlink, we present performance evaluation of the matched filter and MMSE joint detection withTransmit Adaptative Antennas (TxAA) in order to know how necessary it is to implement a multiuserdetection scheme in the user equipment (Q7). This allows us to know if the mobile station detectorcomplexity can be reduced or not. Additionally, we justify why the TxAA operation can be viewed, to acertain extent, as a pre-equalization step.

The main issues that are dealt with in this study are summarized in table 1.5:

1.3.4 The problems to solve in two figures

In order to be more illustrative, we summarized our strategy to tackle the two considered problems(channel estimation and symbol detection) in figures 1.10 and 1.11. Figure 1.10 corresponds to theinitial UMTS-TDD receiver structure according to 3GPP recommendations for uplink and downlink.Figure 1.11 illustrates the philosophy of our approach. Note in particular how the a priori information(−P1−, . . . ,−P6−) is taken into account and how performance and complexity are modified.


Question Chapter Channel estimation Symbol detection

1 2 • •2 2 • •3 3 •4 3 •5 3 •6 4 •7 5 •

Table 1.5: Symbol detection issues are addressed in chapters 2 and 5; channel estimation issues areaddressed in chapters 2, 3 and 4

Figure 1.10: The initial receiver


Figure 1.11: How we want to design a new receiver

Chapter 2

The optimum semi-blind receiver

2.1 Introduction

From a performance point of view, the goal of this study is of course to design the most efficient receiverwhile keeping reasonable complexity. One of the first concerns that arises is to know how well the bestchannel estimation and symbol detection schemes perform regardless of complexity. The underlyingquestion is to know how far the conventional channel estimation and symbol detection performances arefrom performances of the optimum estimation and detection strategies respectively. The problem is thento know what the optimum strategies are.

As for symbol detection, we will consider the Maximum A Posteriori (MAP) symbol-by-symbol jointdetection as the optimum joint detection scheme.

With regard to channel estimation the answer is more context dependent. In a context where no additionalinformation is available, the conventional Least-Squares channel estimate is also the maximum likelihoodestimate based on the observations generated by the midamble. But in our context, a priori information isavailable (−P1−, . . . ,−P6−). Consequently, it is likely that better performance can be achieved throughnew schemes taking into account this information. In this chapter and chapter 3, we propose takinginto account −P1−. This is particularly adapted to sporadic transmissions (chapters 2 and 3) for whichonly one time slot is used for the channel estimation process. Indeed, traditional channel estimationtechniques are based on the training sequence only. However, the observations corresponding to theunknown part of the time slot also contain some information on the channel since unknown data havebeen filtered by the channel as well. The idea is to derive channel estimates based on the total duration ofthe slot. The interest of this is twofold. In certain adverse scenarios (low signal to noise ratio, high-levelof interference), the training sequence alone does not suffice to obtain reliable estimates of the channel.In this case, there is a need to design more robust channel estimators. On the other hand, for typicalsituations, one may ask if it would be possible to reduce the length of the training part by taking intoaccount additional information provided by the unknown symbols part of the slot. This would allow thetransmitter to increase data throughputs. The corresponding strategies, which consist in exploiting boththe "trained" and the "blind" observations, are often referred to as semi-blind schemes [?]. Therefore, thequestion is to know what the performances of the optimum semi-blind channel estimation scheme are(ML estimation based on the total duration of the time slot), which is equivalent to evaluate the potentialperformance improvement that can be provided by semi-blind approaches.

24 The optimum semi-blind receiver

This is why this chapter aims, in particular, at both evaluating MAP symbol-by-symbol joint detectionand ML channel estimation based on the whole time slot. The objective of this chapter is twofold:

1. to evaluate the performance of the statistically optimal semi-blind channel estimate;

2. to separate the influence of multiuser detection reliability from that of channel estimation qualityon the overall performance.

In the literature, the performance of efficient semi-blind channel estimators, such as the stochastic ML[?], has been studied, based on the evaluation of Cramer-Rao bounds [?]. Here, we propose evaluating theperformance improvement in terms of bit error rates, that could be provided by semi-blind approaches,by studying a specific receiver based on ML estimations. Note that some numerical evaluations havebeen presented for GSM [?]. In our case, the receiver is developed in a TD-CDMA context. Reference[?] deals also with a CDMA receiver, based on the SAGE algorithm and a wavelet decomposition of thesignal, but for a single-user system.

In the work reported in this chapter [?], we have implemented a UMTS-TDD receiver based on theclassical Expectation Maximization (EM) algorithm. This chapter briefly reviews the EM algorithm(section 2.2), shows the adaptation to the special context of the UMTS-TDD mode (section 2.3) andanswers the two questions asked above.

2.2 EM algorithm principle

2.2.1 General statement

The EM algorithm [?] provides an iterative approach to maximum likelihood based estimations. Sinceeach iteration of the algorithm consists of an Expectation step followed by a Maximization step, it iscalled the EM algorithm. It is philosophically similar to the ML detection in the presence of unknownparameters (propagation channel and noise variance). The likelihood function is averaged with respectto the unknown quantity (the transmitted data also called the hidden data) before detection, which is amaximization step. To do that, we only use the observations (received signal samples): we focus on thecomplete data (observations and hidden data) taking into account the fact that the available informationon the hidden data comes necessarily from the observations.

We note :

• x the hidden data (transmitted symbols)

• θ the unknown parameter (propagation channel and noise variance )

• y the observations (received signal samples)

• z = x,y the complete data.

The basic idea behind the EM algorithm is that we would like to find θ maximizing1 logP(z/θ), but wedo not have the data z to compute the log-likelihood [?]. So, instead, we maximize the expectation of

1It is worth highlighting that even though we cannot evulate the function P(z/θ), the EM algorithm still allows us tomaximize it

2.2 EM algorithm principle 25

logP(z/θ) given the observations y and our current estimate of θ. This can be expressed in two steps.We assume that we have initialized the algorithm with a value of the parameter θ = θ(0). Let θ(p) be ourestimate of the parameter at the pth iteration.

1. For the E-step compute :

Q(θ;θ(p)) = E[logP(z/θ)/ y,θ(p).] (2.1)

It is important to distinguish between the first and second arguments of the Q function. Thefirst argument θ conditions the likelihood of the complete data. The second argument θ(p) is aconditioning argument to the expectation and is regarded as fixed and known at every E-step. Theexpectation is made over the hidden data.

2. For the M-step let θ(p+1) be the value of θ which maximizes Q(θ;θ(p+1)):

θ(p+1) = argmaxθ

Q(θ;θ(p)). (2.2)

We note that the maximization is done with respect to the first argument of the Q function, theconditioner of the complete data likelihood.

The EM algorithm consists in choosing an initial θ(0), then performing the E-step and the M-step suc-cessively until convergence. Convergence may be determined by examining when the parameter is nolonger changing, i.e. , stop when |θ(p)−θ(p−1)| < ε for some ε and some appropriate distance measure|.|.

2.2.2 Convergence of the EM algorithm

At every iteration of the algorithm, a value of the parameter is computed so that the likelihood functiondoes not decrease (Appendix C). The estimated parameter provides an increase in the likelihood functionuntil a local maximum is reached. So, there is no guarantee that the convergence will be to a globalmaximum [?]. For functions with multiple maxima, convergence will be to a local maximum whichdepends on the initial starting point θ(0).

The convergence rate of the EM is also of interest. Based on mathematical and empirical examinations,it has been determined that the convergence rate is usually slower than a quadratic convergence typicallyavailable with a Newton’s type method. However, as observed by Dempster [?], the convergence nearthe maximum (at least for the exponential families) depends upon the eigenvalues of the Hessian ofthe update function M (M is the function defined by θ(p+1) = M(θ(p))), so that the rapid convergencemay be possible. In any case, even with potentially slow convergence there are some advantages to EMalgorithms over Newton’s algorithms. In the first place, no Hessian needs to be computed. Also, there isno chance of overshooting the target or diverging away from the maximum. It is guaranteed to be stableand to converge to an ML estimate. Regarding stable points of the auxiliary function (such as Q reachesa maximum), it is of high importance to precise that these also are stationary points of the log-likelihood(such as its derivative vanishes). The corresponding proof is provided in Appendix C.


2.3 The EM algorithm in the UMTS-TDD mode context

2.3.1 Channel estimation procedure

For every time slot, denote by XA and XB the QPSK symbols vectors of the first and the data partsrespectively (section 1.2.2.2). According to equation (1.15) each data part of QPSK symbols X generatesan observation vector Y . Stacking XA and XB into one vector X = [XT

A XTB ]T and doing the same for Y A

and Y B, the observation equation associated with the unknown part of the time slot can be expressed by:

Y = I2⊗S(H)X+V (2.3)

where S(H) is the generalized Sylvester matrix defined in equation (1.18) and I2 is the 2-dimensionalidentity matrix. From this equation, it is possible to derive the expression of the Q function. At last, therest of the time slot, which is the midamble, is used for initializing the estimation procedure.

2.3.1.1 General expression of the auxiliary function

We define the matrix Θ by: Θ , H(0), . . . ,H(L),σ2. To obtain the estimation at the (p+1)th iterationwe have to explicit the function Q(Θ,Θ(p)) that is:

Q(Θ,Θ(p)) = E[logP(X,Y/Θ)/Y,Θ(p).] (2.4)

The Bayes’s formula allows us to extract the log-likelihood and therefore makes the computation possi-ble:

P(X,Y/Θ) = P(Y/X,Θ)P(X/Θ) (2.5)

= P(Y/X,Θ)P(X) (2.6)

where we used the fact that the transmitted data are X are independent on the channel Θ.

Since maximization of the auxiliary function is made with respect to Θ, it is equivalent to maximizeE[logP(Y/X,Θ) / Y,Θ(p)].

Since noise is assumed additive white and Gaussian of variance σ2 we have:

logP(Y/X,Θ) =−T Nlog(2πσ2)− 12σ2

t=T−1

∑t=0

n=N−1

∑n=0

|y(tN +n)−uTn [H(0) . . .H(L)]xT

L (t)|2 (2.7)

with un = (0 . . .0 1 0 . . .0)T d= N×1 and

xL(t) , [xT (t) . . .xT (t−L)]T d= K(L+1)×1. (2.8)

We also recall that T is the number of QPSK symbols in a time slot and N is the spreading factor.

Finally, the problem is then to minimize:

E[t=T−1

∑t=0

n=N−1

∑n=0

|y(tN +n)−uTn [H(0) . . .H(L)]xT

L (t)|2 / Y,Θ(p)] (2.9)

2.3 The EM algorithm in the UMTS-TDD mode context 27

with respect to Θ.

This minimization still allows for estimation of the overall impulse responses h1, . . . ,hK while we arelooking for the propagation channels g1, . . . ,gK

. This is why we have to take into account the structureof the overall channels, which are structured by the spreading codes (subsection 1.3.1.1).

2.3.1.2 The structured auxiliary function

It is easily shown that:

uTn [H(0) . . .H(L)] = gT C(n) (2.10)

= [gT1...gT

K]C(n) (2.11)

where :

•

C(n) =

c1,n 0 c1,LN+n 0. . . . . .

. . .0 cK,n 0 cK,LN+n

(2.12)

• ck,m is the column vector associated with the mth row (m ∈ [1, l +1]N) of the matrix Ck:

Ck =

ck(0) 0

ck(N−1). . .

ck(0). . .

0 ck(N−1)

(2.13)

since physical channels and overall channels are related by :

∀k ∈ [1,K]N, hk = Ckgk. (2.14)

After some elementary mathematical manipulations, we obtain the codes-structured auxiliary function:

Q(g,g(p)) =n=N−1

∑n=0

t=T−1

∑t=0

|y(tN +n)|2 +gT C(n)E[xL(t)xHL (t) / Y,g(p)]CH(n)g∗ (2.15)

−gHy(tN +n)C∗(n)E[x∗L(t) / Y,g(p)] (2.16)

−gT y∗(tN +n)C(n)E[xL(t) / Y,g(p)]


2.3.1.3 Parameters estimation procedure

For the (p + 1)th iteration, the minimizations of Q with respect to g and σ lead to channel and noisevariance estimates:

g(p+1) = (R(p)s )−1 µ(p)

s(2.17)

(σ(p))2 =1

2T N

n=N−1

∑n=0

t=T−1

∑t=0

|y(tN +n)− (g(p))T C(n)xL(t)|2 (2.18)

where:

R(p)s =

[n=N−1

∑n=0

t=T−1

∑t=0

C(n)E[xL(t)xHL (t) / Y, g(p)]CH(n)

]∗(2.19)

µ(p)s

=

[n=N−1

∑n=0

t=T−1

∑t=0

C(n)y∗(tN +n)E[xL(t) / Y, g(p)]

]∗. (2.20)

Since the emission symbol alphabet is finite, xL(t) belongs to a finite set (I) too. Hence,

E[xL(t)xHL (t) / Y,g(p)] = ∑

i∈I

iiH P(xL(t) = i / Y,g(p)) (2.21)

E[xL(t) / Y,g(p)] = ∑i∈I

i P(xL(t) = i / Y,g(p)). (2.22)

To compute the A Posteriori Probability (APP) P(xL(t) = i / Y,g(p)), we use the forward-backwardalgorithm [?], [?], [?].

2.3.1.4 Adaptation of the forward-backward algorithm to the UMTS-TDD Mode

Before expressing the APP, a few definitions are in order. We denote by Yt1:t2 , [yT (t1)...yT (t2]T . Re-calling that the symbol modulation is QPSK we define:

• the initial state probabilities :πi , P(xL(0) = i) = 4−K

• the transition probabilities :ai, j , P(xL(t +1) = j / xL(t) = i) ∈ 0,4−KL,1

• the probability density function :

fi(y(t)) , P(y(t) / xL(t +1) = i,g(p))

= (2πσ2)−N

[exp[− 1

2σ2

n=N−1

∑n=0

|y(tN +n)− [H(0) . . .H(L)]T i|2]


• the forward probabilities :αt(i) , P(Y1:t ,s(t) = i)

• the backward probabilities :βt(i) , P(Yt+1:T ,xL(t) = i).

It has been shown [?] that :

P(xL(t) = i / Y,g(p)) =αt(i)βt(i)

∑i αt(i)βt(i)(2.23)

where ∀t ∈ [0,T −2],

α0(i) = fi(y(1))πi (2.24)

αt+1( j) = f j(y(t +1))∑i

ai, jαt(i) (2.25)

βT−1(1) = 1 (2.26)

βt(i) = ∑j

ai, jβt+1( j f j(y(t +1))). (2.27)

For a given ε, when channel estimation procedure has converged, we do the detection, which is addressesnext.

2.3.2 Symbol detection procedure

Once the a posteriori probabilities have been stored from the last iteration, they can be used to estimatethe symbols. There are several detection strategies. Since we know the APP, the most straightforwardstrategies are:

• to perform directly a symbol-by-symbol MAP joint detection

• to exploit them for the computation of the trellis metrics to perform the ML Sequence Estimation(Viterbi Algorithm), which requires additional computations.

To perform the optimal joint detection scheme [?], we have implemented the MAP joint detection. Toestimate x(t), we just need to convert joint propabilities P(xL(t) / Y,g(p)) into marginal probabilitiesP(x(t) / Y,g(p)) by summing over the ISI.

Symbol by symbol MAP detection:

∀t ∈ [0,T −1]N, x(t) = argmaxx

P(x(t) / Y,g(p)) (2.28)

where

P(x(t) / Y,g(p)) = ∑x(t−1)...x(t−L)

P(xL(t) / Y,g(p)). (2.29)


2.3.3 Simulation results

2.3.3.1 Simulation assumptions

Number of users, channel lengthThe computational cost of the Expectation step, which is the most demanding step, is imputable tothe number of states that xL(t) can take, because it is also the size of the trellis induced by the MAPprocedure. Since the data modulation is QPSK, this number is given by:

Ns = 4K(L+1) (2.30)

where L is the assumed degree of H(z). In real situations, true channel degree is not known from thereceiver and therefore has to be assumed. That is why L is replaced by L in the expression 2.30. In table2.1, the number of possible states has been computed for different number of active users and channeldegrees. The cases for which K = 1 or L = 0 have not been considered since we want to study a multiusersystem in a multipath context. As for the other cases, the only case we were able to implement in Matlabis the case (K, L) = (2,1). This is still a case of interest. From section 1.2.2.6, it can be deduced thatthree propagation environments meet the condition L = 1 provided that not more than 14 (13 in fact sincethis number is odd) significant coefficients are kept for the truncated version of the shaping filter (whichis defined at Tc/2). In order to justify this, let us define:

• h Tc2(z) be the transfer function of the half chip rate impulse response of the overall channel;

deg(h Tc2(z)) = 2(L+1)N−1

• g Tc2(z) be the transfer function of the half chip rate impulse response of the physical channel;

deg(g Tc2(z)) = 2LN

• p Tc2(z) be the transfer function of the half chip rate impulse response of the shaping filter; deg(p Tc

2(z)),

lp

• φ Tc2(z) be the transfer function of the half chip rate impulse response of the propagation channel;

deg(φ Tc2(z)) = 2LN− lp.

Since N = 16, we haveL≤ 1

⇒ deg(h Tc2(z))≤ 63

⇒ deg(g Tc2(z))≤ 32

⇒ deg(φ Tc2(z))≤ 32− lp.

Finally, we see that if lp = 13 then deg(φ(z)) has to be less than 19, which corresponds to 20 coefficientsat Tc/2 or equivalenty 10 chips.

ChannelThe test environment is assumed to be the Vehicular A (section 1.2.2.6). It includes 6 taps, which delaysare in the time interval [0,2.51µs]. The chip rate counterpart of the propagation channel impulse responsecomprises 10 chips. Mobile station speed is fixed at 120 km/h. Carrier frequency is 2000 MHz.

Ns L = 1 L = 2 L = 3K = 2 256 4096 65536K = 3 4096 262144 . . .

K = 4 65536...

. . .

Table 2.1: Number of states of the trellis corresponding to the forward-backward algorithm, which isdenoted by Ns; K is the number of active users per time slot and L is the assumed ISI duration

SNR [dB] 0 3 6 9 12r 6 4 3 2 2

Table 2.2: Number of iterations (r) of the EM algorithm versus signal-to-noise ratio (SNR)

Noise varianceNoise variance is assumed to be known for two reasons: for the MMSE the noise variance has to beassumed known from the receiver unless a special estimator of the noise variance is put in place. Inorder to make a fair comparison between the EM receiver and the conventional receiver, we have to dothis assumption. Additionally, the EM algorithm uses NT = 16×120' 2000 samples for estimating thenoise variance. Therefore, when the algorithm has converged, noise variance accuracy is very high.

Assumption on the training sequenceFor channel estimation purpose, we use UMTS-TDD midambles. In the UMTS-TDD mode, midamblesare defined at chip rate but they are not constructed by spreading symbol rate-defined training sequences.

For the MAP procedure, we still assume that training sequences are spread, which means that the wholetime slot is spread. This allows us to perform the forward-backward procedure over the whole time slotduration.

Time slotsIn simulations, only type 1 time slot is used (subsection 1.2.2.2). This means that for each time slot, two61-symbol data parts are transmitted. In order to evaluate bit error rates, we made use of 2000 time slotsfor each signal-to-noise ratio value ranging from 0 dB to 12 dB.

Number of iterationsFor each signal-to-noise ratio value, ranging from 0 dB to 12 dB, the iteration process is stopped at therth iteration where r verifies |Q(r+1)−Q(r)|< 10−4. As expected, the value of r is dependent on the SNRand for the proposed experimental setup, corresponding values have been recorded in table 2.2:

2.3.3.2 Simulation analysis

OverviewAs mentioned in the beginning of this chapter, what we are asking for, is essentially:

• to know how far the initial receiver performance (training-based channel estimation + MMSE-JD)is from the optimum semi-blind receiver performance (optimum semi-blind estimator + MAP-JD)

• to study separately the influence of channel estimation accuracy and symbol detection reliability


on overall performance.

For this purpose, we compare receivers, for which three options are possible for channel estimation(Least-Squares based on the training sequence, EM-based iterative procedure and the case where channelis known from the receiver) and three options are possible for symbol detection (Matched Filter, MMSEJoint Detection and MAP Joint Detection). The different receiver structures of interest, corresponding tothe possible combination of these options, are summarized in table 2.3:

Receiver structure Channel estimation scheme Joint Detection scheme

Structure 0 Known channel MFStructure 1 Least-Squares MMSE-JDStructure 2 Iterative (EM) MMSE-JDStructure 3 Known channel MMSE-JDStructure 4 Least-Squares MAP-JD (EM)Structure 5 Iterative (EM) MAP-JD (EM)Structure 6 Known channel MAP-JD (EM)

Table 2.3: Receiver structures under consideration in this chapter; three channel estimation schemes areconsidered: the case where the channel is known from the receiver, the training-based channel estimation(least-squares estimation) and the iterative channel estimation of the EM procedure (initialized by thetrained estimate); three symbol detection schemes are considered: the matched filtering, the MMSEblock linear joint detection and the maximum a posteriori symbol-by-symbol joint detection

Comparison between initial and EM-based receiversThe initial receiver corresponds to structure 1 while the "EM receiver" corresponds to structure 5. Recallthat in the initial receiver, channel estimation is based on a 512-chip midamble, which enables an esti-mation of a 57-chip physical channel per user. In the optimum receiver, the EM algorithm is initializedwith a 16-chip trained estimate.

In these conditions (figure 2.2), the EM receiver brings a 3 dB improvement on the Eb/No for a 1%(uncoded voice service) BER target. This shows that even in a low MAI scenario (2 users), withoutchannel coding and multiple antenna, performance of the conventional receiver may be significantlyimproved by using suitable estimation and detection strategies.

Comparison of channel estimation techniquesIn this section, the detection strategy is fixed and four channel estimates are used to perform joint detec-tion:

• the trained estimate with an a priori channel length of 57 chips

• the trained estimate when channel length is known (16 chips),

• the estimate provided by the EM algorithm (iterative procedure with known channel order)

• the true channel (channel is known from the receiver).

First let us choose the MMSE-BL-JD as symbol detector (structures 1,2 and 3). The results depicted infigure 2.3 show that the channel order knowledge has a noteworthy influence on the channel estimation


process. We see in figure 2.3 that the EM estimation procedure outperforms the trained 57-chip one, withan observed gain of about 2.5 dB. However, when channel order is known from both the EM receiver andthe initial receiver, the corresponding gain is only of 0.5 dB. At last, if the channel was perfectly known,we could benefit of a total gain of 2.6 dB, compared to the worst 57-chip trained estimation. In presenceof higher multiple access interference, the gain obtained with a more accurate channel estimation is evenmore significant. Indeed, for the 8-user case, channel knowledge provides at least a 4 dB gain in termsof SNR (figure 2.4).

When the symbol-by-symbol MAP detection is used (structures 4, 5 and 6), the same kind of observationsis obtained. Channel accuracy is very influential on detector behaviour (figure 2.5). Channel orderknowledge is also very important especially for short channels, which is the studied case (Vehicular A).

Comparison of detection techniquesHere, for a given channel estimation strategy, the aim is to evaluate the influence of the detection schemeson receiver performance. We have made comparisons between structures 1 and 4, 2 and 5, 3 and 6 andwe have also evaluated performance of the Matched Filter detection.

Conclusions are the followings (figure 2.6). Whatever the channel estimation strategy is (conventionalLeast-Squares or known channel), the general statement is that symbol detection has less importancethan channel estimation. Indeed, when comparing MMSE-JD with MAP-JD, the difference in termsof signal-to-noise ratio reaches only 0.7 dB while flawed channel knowlege costs a 2 dB loss on theSNR. Note that this statement holds for multiuser detectors. Indeed, in the same figure we also see howwell the matched filter (receiver stucture 0) performs in such a context. As a result, the matched filterdetection does not perform well even in the low MAI scenario. We still note that this statement holds fora single-antenna.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 63000

3100

3200

3300

3400

3500

3600

3700

Number of iterations

|Q|

Eb/No = 0 dB

Eb/No = 3 dB

Figure 2.1: Convergence of the EM algorithm: the objective function (Q) saturates after 6 iterations ifEb/No = 0 dB and 4 iterations if Eb/No = 3 dB

0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

Comparison between initial and EM−based receivers

Eb/N

0 [dB]



Vehicular A2 active users1 antennano channel codingMS speed = 120 km/h2000 time slots

Structure 1: MMSE−JD + LS channel estimation Structure 5: MAP−JD + iterative channel estimation

Figure 2.2: Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noise ratio;receiver structures 1 (MMSE block linear joint detection + training-based estimation) and 5 (receiverbased on the EM algorithm) are compared; simulation conditions: Vehicular A, 2 active users, mobilestation speed = 120 km/h, 1 single antenna, no channel coding; for the voice service BER target (1%) theconventional receiver loses 3 dB w.r.t the semi-blind optimum receiver

0 2 4 6 8 10 12

10−3

10−2

10−1

100

Comparison of channel estimation techniques (MMSE−JD)

Eb/N

0 [dB]



Vehicular A2 active users1 antennaNo channel codingMS speed = 120 km/h2000 time slots

Structure 1: Training−based estimation (assumed channel length = 57 chips)Training−based estimation (known channel length) Structure 2: Iterative estimation (EM) Structure 3: Known channel

Figure 2.3: Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noise ratio;the symbol detection strategy is fixed (MMSE block linear detection); the receiver performance is eval-uated for four different channel estimation schemes: the training-based estimation, the training-basedestimation when the channel length is known, the iterative channel estimation based on the EM algo-ritm (initialized with the trained estimate) and the case where the channel is known from the detector;simulation conditions: Vehicular A, 2 active users, mobile station speed = 120 km/h, 1 single antenna,no channel coding; the flawed channel knowledge costs 2.5 dB to the trained estimation and the flawedchannel length knowledge causes a 2 dB loss to the trained estimation

3 4 5 6 7 8 9 10 11 1210

−3

10−2

10−1

100

Eb/N

0 [dB]



Influence of channel accuracy on MMSE−JD performance (K=8)


MMSE + TSMMSE + Known channel

Figure 2.4: Bit error rates (averaged over the 8 users and 2000 time slots) versus signal-to-noise ratio; thesymbol detection strategy is fixed (MMSE block linear detection); the detector performance is evaluatedwhen the trained estimate is used and when the channel is known; simulation conditions: VehicularA, 8 active users, mobile station speed = 120 km/h, 1 single antenna, no channel coding; the channelknowledge becomes a more and more critical point as the multiple access interference level increases; 4dB are lost because of the flawed channel knowledge

0 1 2 3 4 5 6 7 8 9

10−2

10−1

100

Eb/N

0



Comparison of channel estimation techniques (MAP−JD)

Vehicular A2 active users1 antennaNo channel codingMS speed = 120km/h2000 time slots

TS−based estimation (+ known channel length)Iterative estimation (EM) Known channel

Figure 2.5: Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noise ratio; thereceiver stuctures 4 (maximum a posteriori joint detection+ trained estimation + known channel length),5 (maximum a posteriori joint detection + EM based on iterative channel estimation) and 6 (maximum aposteriori joint detection + known channel) are compared; simulation conditions: Vehicular A, 2 activeusers, mobile station speed = 120 km/h, 1 single antenna, no channel coding; even if the channel length isassumed to be known, the flawed knowledge of the coefficients costs a 0.8 dB loss to the MAP detection

0 1 2 3 4 5 6 7 8 910

−3

10−2

10−1

100

Eb/N

0 [dB]



Comparison of detection techniques (Known channel)


Structure 0: Matched Filter + Known channelStructure 3: MMSE−Joint Detection + Known channelStructure 6: MAP symbol−by−symbol + Known channel

Figure 2.6: Bit error rates (averaged over the 2 users and 2000 time slots) versus signal-to-noise ratio;the channel is assumed to be known; the receiver performance is evalutated with the matched filter,the MMSE block linear joint detection and the MAP joint detection; simulation conditions: VehicularA, 2 active users, mobile station speed = 120 km/h, 1 single antenna, no channel coding; with oneantenna, even in a low MAI scenario, the matched filter does not perform well; the MMSE and the MAPperformances are very close; if K/N is small, MAP detection capability is not fully exploited

2.4 Conclusions

In this chapter, our aim was to show that applying semi-blind receiver techniques to the UMTS-TDDcontext can bring significant improvement over classical approaches. Here, our purpose was not topropose new semi-blind algorithms, but to quantify their potential benefits by studying the Expectation-Maximization algorithm.

Indeed, the EM algorithm provides a maximum-likelihood estimation of the unknown channel, relyingon the observation sequence. It performs an iterative, blind estimation of the channel in the sense that itdoes not rely on any explicit knowledge of some transmitted symbols. Moreover, in our study, it is usedin a semi-blind context where the channel value is initialized thanks to the training sequence.

Simulations were conducted in the case of 2 users, experiencing the Vehicular A channel and a singlereception antenna. The classical receiver consists of a training-based least-squares channel estimationand an MMSE block linear joint detection scheme. The EM receiver functions with an iterative channelestimation procedure initialized with the trained estimate and a symbol-by-symbol MAP detection. Theobtained results showed that there is an improvement of about 3 dB thanks to the optimum semi-blindEM approach.

It has to be emphasized that this comparison involves two different channel estimation methods as well astwo different detection techniques. Thus, in a second step, we tried to separate the contribution of channelestimation accuracy from the one of the detection performance, on the overall 3 dB gain. Simulationsclearly showed that most improvement was obtained by enhancing the channel estimation procedure.When the channel is perfectly known from the receiver, the MMSE-JD performance is improved by 2.6

2.4 Conclusions 37

dB for a 1% uncoded BER. On the other hand, this gain is only of 0.4 dB when comparing the case wherethe MMSE is fed with the EM channel estimate with the case where the channel is known, which meansthat the EM estimation is very good! For a given channel estimation strategy, the influence of detectionscheme on receiver performance confirms that point.

The results obtained with the EM approach showed the significant impact of the channel estimation accu-racy in the final receiver performance. This conclusion can be extended to a more realistic configuration:the 8-user case. Unfortunately, the EM could not be assessed in this realistic context, due to implemen-tation complexity, and we only present results for the MMSE JD receiver. The impact of the channelestimation accuracy was found to be even more significant in this scenario, with 4 dB gain brought by a"perfect" channel estimation.

It is also of importance to notice that there is a "great opportunity" to improve channel estimate accu-racy in the UMTS-TDD mode because channel lengths are overestimated; 57-chip channel lengths areassumed for type 1 time slots whereas typical vehicular propgation environments generate 15 to 20-chipphysical channels. This point has been confirmed by considering the case where the channel lengthsare known from the receiver. In Vehicular A, with 2 active users, the perfect channel order knowledgeprovides a 2 dB gain in terms of SNR for BER = 1%.

From now on, the detection strategy is fixed at the MMSE joint detection solution and we investigate thechannel estimation in first place. As the complexity of the EM algorithm grows exponentially with thenumber of users and the ISI length, we have to envisage suboptimum semi-blind schemes as we do inchapter 3, which deals with second-oder semi-blind channel estimation techniques.

Chapter 3

Semi-blind channel estimation techniquesbased on second-order blind methods

3.1 Introduction

As we mentioned in the previous chapter, the idea of semi-blind approaches is to exploit not only theobservations generated by the training sequence but also those generated by the unknown symbols of thetime slot. In order to take advantage of these additional observations, we want to make use of second-order blind estimation procedures such as the subspace [?], [?], [?] and linear prediction algorithms[?], [?]. Semi-blind methods generally improve the estimates based on the training sequence alone,while avoiding the pitfalls of blind methods (and in particular, the lack of consistency of certain blindestimators).

The most efficient semi-blind approach is based on the maximum likelihood estimation, which in practicecan be implemented by the expectation maximization algorithm (chapter 2). In chapter 2, we have seenthat, for synchronized CDMA communications, the computational complexity of this algorithm growsexponentially with the number of users and the assumed channel length (in symbol duration), and it isthen impossible to implement (except when the number of users and the channel length are very limited).Another kind of semi-blind approach, based on deterministic or Gaussian Maximum Likelihood methods,have been proposed by D. Slock et al. [?], [?]. These methods lead to the minimization of a compositecriterion defined as the sum of the classical training based least-squares criterion and the cost functionassociated with the blind deterministic or Gaussian maximum likelihood method. Minimizing such acomposite criterion was also proposed in [?] and [?]. In the latter cases, the blind criterion was derivedfrom the subspace method originally introduced in [?]. Minimizing a composite criterion can be usednot only to estimate the channel, but also to identify an equalizer. In particular, [?], [?], and [?] proposemixing a classical least-squares criterion with the Constant Modulus Algorithm (CMA, [?] ). Finally, [?]proposes adapting the blind algorithm of [?] to a semi-blind context.

Here, we focus more on the approach introduced in [?] and studied more recently in details in [?] for sin-gle user systems. The corresponding channel estimation scheme consists in minimizing a cost functiondefined as a weighted sum of the training sequence based criterion and the quadratic criterion associatedwith the blind subspace estimator. This estimator can be interpreted as a regularized least-squares solu-tion where the blind subspace criterion plays the role of the regularization constraint. We will see howto use results provided by [?] for tuning the balance between the least-squares criterion and the blind

40 Semi-blind channel estimation techniques based on second-order blind methods

subspace criterion, without making the deterministic or Gaussian assumption as it is done in [?]. In fact,our results are derived under asymptotic assumptions: the time slot and the midamble sizes tend towardsthe infinity. In this chapter, we extend the approach of [?] to the context of synchronized CDMA systemsfor uplink channel estimation [?], [?]. We discuss the best choice of which second-order blind methodto implement and we also study the robustness of the proposed semi-blind channel estimators to channelovermodelling.

3.2 Semi-blind techniques based on second-order blind methods

3.2.1 Why second-order statistics based semi-blind techniques ?

In pure blind contexts, many studies have been devoted to Second Order Statistics (SOS) based estimationtechniques [?], [?], [?], [?]. There are several reasons for this. First, the preference to second-ordermethods is mainly related to historical reasons: there is a close relationship with classical Fourier analysisand strong similarity of SOS system identification to the narrow-band array processing settings, whichattracted the attention of many experts in array processing [?], [?]. From a technical point of view, themain advantage of SOS-based identification is global identification issues and the possibility of learningabout (partial) system identification from the output signals in the absence of information or even clearunderstanding of the statistical nature of the inputs [?].

In semi-blind contexts, the main advantage of second-order techniques stems from their computationalsimplicity. As most of SOS methods (subspace and linear prediction methods in particular) boil down toquadratic optimization, they can also be easily coupled with training sequence based estimation, withoutcompromising the attractive simplicity of the method [?]. Corresponding semi-blind criteria are alsomore natural from a statistical perspective. Indeed, the linear combination of the blind and training basedcriteria has not only the advantage of computational simplicity but also an appealing Bayesian flavor:the SOS based semi-blind criterion corresponds to the log-likelihood for Gaussian noise and the blindcriterion may be seen as a zero-mean Gaussian log-prior.

When considering more particularly the subspace and linear prediction approaches, we see that both haveattractive properties for the aimed context, namely the UMTS-TDD mode. It is well known that subspacemethod has the convenient Finite Samples Convergence (FSC) property in the noiseless case [?], [?]whereas the linear prediction approach is interesting for its robustness to flawed channel order knowledge[?] [?]. These are good arguments for implementing these algorithms in UMTS-TDD-like contextswhere time-slots comprise no more than 150 samples and propagation channel lengths are intrinsicallyunknown. Of course, the presence of the training part in the estimation procedure modifies somehowthese theoretical results. One of the goals of this chapter is to know to what extent this is done.

3.2.2 Second-order blind methods

3.2.2.1 Subspace method

In this subsection, we recall how the subspace method of [?] can be adapted to the context of CDMAsystems (see e.g. [?], [?], [?], [?]). Let M be an integer (classically called the smoothing factor) and

3.2 Semi-blind techniques based on second-order blind methods 41

yM

(t) denote the N(M +1)–dimensional regression vector defined by

yM

(t) , [yT (t) . . .yT (t−M)]T (3.1)

= TM(H)xM+L(t)+ vM(t) (3.2)

where TM(H) is the Sylvester matrix, associated with the matrix H(z), defined by

TM(H) =

H(0) . . . H(L) 0. . . . . .

0 H(0) . . . H(L)

d= N(M +1)×K(L+M +1) (3.3)

and where xM+L(t) , [xT (t) . . .xT (t − L−M)]T and vM+L(t) , [vT (t) . . .vT (t −M)]T . Let RM be thecovariance matrix of the vector y

M(t). Transmitted data and noise samples being white sequences, RM is

then given by

RM = TM(H)THM(H)+σ2IN(M+1). (3.4)

If M is chosen large enough, the Sylvester matrix TM(H) is tall, and σ2 is the smallest eigenvalue ofRM . Denote by π the orthogonal projection matrix on the eigensubspace associated with the smallesteigenvalue σ2, which is usually called the noise subspace of the matrix RM . The noise subspace methodis based on the simple observation that the column space of the Sylvester matrix is orthogonal to π, i.e.that

π TM(H) = 0. (3.5)

Let us split the projection matrix π into N + 1 matrices such as π = [π0 π1 . . . πN ] where each matrixπk (k = 1, . . . ,K) is N(M + 1)× (M + 1). Then, the subspace equation (3.5) can be more convenientlyrewritten as:

π TM(H) = 0⇔D(π)

H(0)...

H(L)

= 0 (3.6)

where

D(π) =

π0 0. . .

... π0

πM...

. . .0 πM

d= N(L+M +1)(M +1)×N(L+1). (3.7)

Equation (2.14) implies that the matrix [H(0)T . . .H(L)T ]T can be written as

[H(0)T . . .H(L)T ]T = [C1g1 . . .CKgK]. (3.8)

Therefore, (3.5) is equivalent to the linear equations

∆k(π)gk= 0 (3.9)


for each k = 1, . . . ,K, with∆k(π) = D(π)Ck. (3.10)

Under (admittedly rather non explicit) appropriate conditions (see e.g. [?] and [?]) which are assumedto hold from now on1, for each k = 1, . . . ,K, the vector g

kis the unique solution (up to a scaling factor)

of the structured subspace equation (3.9). Therefore, each vector channel impulse response gk

can beestimated consistently by the vector:

gk,sub

= argminf k

f Hk∆H

k (π)∆k(π)︸︷︷︸Qk,sub

fk

(3.11)

where π represents a consistent estimate of π. The notation "sub" stands for subspace method. In thefollowing, we denote by Qk,sub and Qk,sub the matrices defined by

Qk,sub , ∆H

k (π)∆k(π)Qk,sub , ∆H

k (π)∆k(π)(3.12)

Of course, the minimization (3.11) has to be performed under a non-trivial constraint, e.g. ‖ fk‖= 1 for

each k = 1, . . . ,K, at least in the pure blind context.

3.2.2.2 Linear prediction method

We first recall that the linear prediction approach is based on the observation that if H(z) is irreducible(i.e. Rank(H(z)) = K for all z 6= 0), then it exists a (non unique) K ×K polynomial matrix A(z) =I +∑i=M

i=1 Aiz−i (where M is a large enough integer) satisfying

A(z)H(z) = H(0) (3.13)

or equivalently in matrix formS(A) H = [H(0)T 0 . . . 0]T (3.14)

where S(A) denotes the N(M +L+1)×N(L+1) block Toeplitz matrix

S(A) =

A(0) 0...

. . .A(M) A(0)

. . ....

0 A(M)

d= N(M +L+1)×N(L+1). (3.15)

H being the N(L +1)×K matrix given by H = [H(0)T . . . H(L)T ]T (see e.g. [?]) and A(0) = I. Let usdenote by (R(i))i∈Z the autocovariance function of y(t) defined by R(i) = E[y(t + i)yH(t)] and, for eachinteger i, by Ri the (i + 1)N× (i + 1)N covariance matrix of the vector Y i(t) = [yT (t) . . . yT (t − i)]T .

1Advanced identifiability results for CDMA downlink are provided in [?]. One of the major results is that if H(z) isirreducible and if the codes of the various users satisfy a certain non restrictive assumption, then the code-structured subspacemethod provides a consistent estimate of the channel if the assumed channel order ˆ is not greater than `+ N (recall that thechannel length is `+ 1 chips and N is the spreading factor). However, for uplink identifiability remains an open issue. That iswhy we will merely justify the underlying assumptions through experimental results (3.4.3.1).

3.2 Semi-blind techniques based on second-order blind methods 43

Then, matrices A , [A(0)A(1) . . . A(M)] and D , H(0)HH(0) can be identified by the well-knownYule-Walker equations:

[A(1) . . . A(M)] = − [R(1) . . . R(M)] (RM−1−σ2I)#

D = R(0)+∑i=Mi=1 A(i)RH(i)

(3.16)

Here, # stands for the pseudo-inverse (for M large enough, (RM−1−σ2I) is singular). From A(z) andD, it is possible to identify H(z) up to a constant unitary matrix, i.e. it is possible to extract from D adefined N×K matrix H(0) satisfying D = H(0)HH(0). The matrix H(0) is given by H(0) = H(0)Υ forsome unitary K×K constant matrix Υ. Finally, the polynomial matrix H(z) is uniquely defined fromA(z) and H(0) by

A(z)H(z) = H(0) (3.17)

where H(z) = H(z)Υ, a polynomial relation which is equivalent to the matrix relation

H = HΥ. (3.18)

Under appropriate conditions (similar to the conditions stated in [?] and [?] for the subspace method), themissing factor Υ can be identified by exploiting the specific algebraic structure of H(z) (see equations(2.14), (3.8)).

Of course, in practice, A, D and H(0) have to be estimated. Equation (3.16) gives the estimates A and D(this is obtained by truncating to its largest K eigenvalues the matrix R(0)+ ∑i=M

i=1 A(i)RH(i)) of A andD). An estimate H(0) of a N×K square root of D is then immediately obtained from D.

This procedure allows for the estimation of H(z) up to a constant unitary matrix. In order to identify themissing factor, a simple approach consists in minimizing jointly (with respect to f = [ f T

1. . . f T

K] and Φ)

the cost function ψ defined by

ψ( f ,Φ) = ‖S(A)[C1 f

1. . .CK f

K

]− [

H(0)T 0 . . .0]T

Γ‖2 (3.19)

where f1, . . . , f

Kare N(L+1)–dimensional vectors and where Φ is a K×K matrix (see [?] in the down-

link case). Minimizing first with respect to Φ, we get immediately that the vectors g1, . . . , gKminimizing

in f1, . . . , f

Kthe cost function (3.19) minimize the quadratic forms

gk,lin

= argminf k

f Hk∆H

k (A, D)∆k(A, D)︸︷︷︸Qk,lin

fk. (3.20)

Here, ∆k(A, D) represents the N(M +L+1)×NL matrix defined by

∆k(A, D) = Diag(πD⊥ ,IN , . . . ,IN)S(A)Ck (3.21)

where πD⊥ denotes the orthogonal projection matrix on the orthogonal complement of null(D). In thesequel, Qk,lin and Qk,lin denote the matrices

Qk,lin , ∆H

k (A,D)∆k(A,D)Qk,lin , ∆H

k (A, D)∆k(A, D).(3.22)

The notation "lin" stands for linear prediction method.


3.2.2.3 Weighted linear prediction method

For single-user systems, the linear prediction approach is known to have poor statistical performance.In contrast to the subspace method, the asymptotic covariance matrix of the linear prediction estimateis non-zero in absence of noise (σ2 = 0). Motivated by this observation, [?] proposed using a weightedlinear prediction estimate and addressed the blind identification of unstructured MIMO FIR transferfunction. The idea of [?] was recently adapted to the context of the blind channel identification ofdownlink CDMA systems ([?]), where it was shown that the use of a simple admissible2 weighting matrixcan produce significant performance improvements. More specifically, the corresponding asymptoticestimation error decreases towards 0 when σ2 → 0 if the assumed channel length does not exceed thetrue one by more than one symbol duration. In this case, the blind subspace method and the blindweighted linear prediction approach have nearly the same performance. However, if the assumed degreeof H(z) (say L) is greater than the true degree of H(z) by more than one (L≥ L+1), the subspace methodis no longer consistent whereas the weighted linear prediction scheme still provides satisfying results.In the uplink context considered here, the results of [?] can be adapted by estimating each vector as thesolution of the following minimization problem:

gk,wlin

= argminf k

[f H

k∆k(A, D)HWk(D,σ2)∆k(A, D) f

k

]. (3.23)

where the weighting matrix Wk(D,σ2) is defined by

∀k = 1, . . . ,K, Wk(D,σ2) = IM+L+1⊗ (πD⊥ +σ2D#) (3.24)

and is replaced in practice by the matrix Wk(D,σ2); "⊗" stands for the Kronecker product (Appendix E).Replacing Wk(D,σ2) by Wk(D,σ2) does not modify the asymptotic performance. More insights on theconstruction of the weighting matrix are given in subsection 3.3.1 (see also [?] for corresponding resultin the downlink case). In the following, we denote by Qk,wlin and Qk,wlin the matrices

Qk,wlin , ∆H

k (A,D)Wk(D,σ2)∆k(A,D)Qk,wlin , ∆H

k (A, D)Wk(D,σ2)∆k(A, D)(3.25)

The notation "wlin" stands for weighted linear prediction method.

3.2.3 Semi-blind estimation

In order to introduce our results, we first remark that the previous blind estimates are obtained by min-imizing for each k = 1, . . . ,K a quadratic form f H

kQk f

kunder the constraint ‖ f

k‖ = 1. The matrix Qk

respectively equals Qk,sub, Qk,lin or Qk,wlin in the context of the subspace method, the linear predictionapproach or the weighted linear prediction approach. Note that the main motivation for adapting the lin-ear prediction approaches was the attractive properties of the weighted estimate derived in [?]. In orderto simplify the exposition of the results, we have first to introduce some notations. We denote by Q theblock diagonal matrix

Q = Diag(Q1, . . . ,QK) (3.26)

and for each K–dimensional vector α = (α1, . . . ,αK), by Q(α) the matrix

Q(α) = Diag(α1Q1, . . . ,αKQK). (3.27)2W is admissible for M if and only if null(MW) = null(W).

3.3 Asymptotic analysis 45

Moreover, W(D,σ2) is defined by

W(D,σ2) = Diag(W1(D,σ2), . . . ,WK(D,σ2)) = IK(M+L+1)⊗ (πD⊥ +σ2D#). (3.28)

Finally, we define in the same way block diagonal matrices ∆,∆(α) = Diag(α1∆1, . . . ,αK∆K), ∆ and∆(α) from matrices (∆k)k=1,...,K (each ∆k equals ∆k(π) in the subspace method or ∆k(A,D) in thelinear prediction approaches) and (∆k)k=1,...,K .

For every matrix Qk, we study the semi-blind estimates of g1, . . . ,gKobtained by minimizing the com-

posite cost function

‖Y tr−S f‖2 +T

(k=K

∑k=1

αk f Hk

Qk fk

)= ‖Y tr−S f‖2 +T f HQ(α) f (3.29)

where f = ( f T1. . . f T

K)T and where the components of the vector α = (α1, . . . ,αK)T are positive constants

weighting the contribution of each cost function in the global cost function (3.29). The correspondingestimate is given by

g =[RSS +ρQ(α)

]−1RSY (3.30)

where ρ is defined by ρ = Tm (T is the number of symbols per time slot and m is the midamble length);

recall that RSS = m−1SHS and RSY = m−1SHY . One of the crucial point here is to derive the "best" valueof α. In the single-user context of [?] (there is a single parameter α to adjust), it has been in particularnoticed that the choice of α influenced dramatically the performance of the estimate. It is therefore ofpractical importance to adjust the vector α in a relevant way.

In this chapter, we follow the approach proposed in [?] that consists in choosing the parameter α inorder to minimize the asymptotic estimation error of the semi-blind estimate (trace of the correspondingcovariance matrix). By asymptotic, we mean that both m and T converge towards the infinity.

In the next section, we calculate in closed-form this asymptotic autocovariance matrix for each semi-blind method (subspace method, weighted and conventional linear prediction approaches) under twodifferent assumptions. First, we assume that ρ is a constant. We call this case the regularized semi-blindcase. The corresponding asymptotic autocovariance matrix allows for tuning α. Second, we assumethat m = o(T N), which is called the projected semi-blind case. In this case, we show that the studiedsemi-blind procedure is asymptotically equivalent to a projection of the trained channel estimate onto acertain subspace.

3.3 Asymptotic analysis

In this section, our main goal is to evaluate theoretically the asymptotic covariance of estimation errorfor the studied semi-blind channel estimates in a CDMA uplink case. In this purpose, we generalize themethod introduced by [?] for the single-user semi-blind subspace case.


3.3.1 The regularized approach

The main interest of the present approach lies in the fact that an optimum value of α can be provided.Indeed, the regularizing vector α is determined by minimizing the asymptotic covariance matrix of thechannel estimation error. When m ∼ NT , the convergence rate of the blind and the training based esti-mates are of comparable order. Then, an asymptotic analysis, for which the number of observations Tand the midamble length m grow to infinity with some prescribed rate, allows us to evaluate this matrix.

Proposition 1 provides the general expression of the asymptotic covariance of the estimation error for thesemi-blind schemes studied in this chapter.

Proposition 1: Let ρ = T/m be a real constant. The asymptotic covariance matrix of estimation error(δg = g− g) is given by:

Cov(δg) = limT→∞,T/m=ρ

T E[δg δgH ]

= M−1

[ρσ2R(∞)

SS +ρ2∆H(α)WΣW∆(α)]

M−1 (3.31)

where Σ , Cov(δ∆ g), δ∆ , ∆−∆ and M , R(∞)SS +ρQ(α). Q = Qsub,Qlin or Qwlin in the subspace,

the linear prediction and the weighted linear prediction cases respectively. The weighting matrix equalsI in the subspace and the non-weighted linear prediction schemes, and W(D,σ2) in the context of theweighted linear prediction approach. Moreover, the matrix R(∞)

SS represents limm→+∞ m−1RSS

Proof of Proposition 1 is provided in Appendix F . In the sequel, we denote Γ(α) the asymptotic co-variance matrix of δg defined in (3.31). Our approach consists in selecting the value of α minimiz-ing Trace(Γ(α)). However, the function α → Trace(Γ(α)) depends on unknown terms which haveto be estimated in practice. The Matrix R(∞)

SS is of course estimated by the matrix RSS itself. Ma-trices Q(α), ∆(α) and W(D,σ2) are simple functions of the projection matrix π and (or) matricesD and A. Therefore, they can be estimated consistently by replacing π, D and A by their consis-tent estimates defined above. The estimation of the matrix Σ is however more difficult. In order tosolve this problem, we have to consider separately the subspace case and the linear prediction case.Before providing the corresponding results, we first define the following notations: δk,l is the Kro-

necker symbol, uk = (0 . . .0 1 0 . . .0)T d= K × 1 ; the one is on the position kth, Jk = (IM+L ⊗ uk),T#

M−1(H) , [T(0) . . . T(M− 1)], T(eiω) = ∑p=P−1p=0 T(p)e−ipω. For each (k, l) ∈ [1,2, . . . ,K]2Z, we also

denote by Σ(k, l) the N(M +L+1)×N(M +L+1) block (k, l) of Σ. Let also define the N(M +1)×N–valued FIR transfer function π(z) as

π(z) =i=M

∑i=0

πiz−i (3.32)

and denote by Rπ the (M +L)N(M +1)× (M +L)N(M +1) block-Toeplitz matrix defined by

Rπ =∫ π

−π

[DM+L(e

iω)DHM+L(e

iω)]∗⊗π(eiω)πH(eiω) dω (3.33)

with DP(eiω) , (1 e−iω . . .e−iPω)T for a given P ∈ N. At last, we define in the same way RA as:

RA =∫ π

−π

[DM+L(e

iω)DHM+L(e

iω)]∗⊗A(eiω)AH(eiω) dω (3.34)


3.3.1.1 The subspace case

We first give the closed-form expression of Σ. The corresponding result can be shown by generalizingthe calculations of [?] (Appendix F).

Proposition 2: Assuming that the filtering matrix TM(H) has full column rank, Σ can be written as

Σ = σ2Σ(1) +σ4Σ(2) (3.35)

where ∀(k, l) ∈ [1,K]2Z,Σ(1)(k, l) = δk,lRπ (3.36)

and

Σ(2)(k, l) =∫ π

−π[JT

k T(eiω)TH(eiω)Jl]∗⊗π(eiω)π(eiω)dω (3.37)

We can see that Σ(2) depends on the unknown channels via the filtering matrix TM(H) and via thepseudo-inverse of TM(H)TH

M(H). This means that the estimation of Σ(2) requires an initial estimate ofthe channels. Moreover, the estimation of the pseudo-inverse of TM(H)TH

M(H) is not an easy task in thecase where the Sylvester matrix is ill-conditioned. However, proposition 2 shows that if the term in σ4 isneglected (for high enough signal-to-noise ratios in particular), then Σ can be approximated by σ2Σ(1),which is a simple function of π, and therefore can be estimated quite easily. Proposition 2 implies inparticular that Γ(α) can be written as

Γ(α) = σ2[R(∞)

SS +ρQ(α)]−1 [

ρR(∞)SS +ρ2∆(α)HΣ(1)∆(α)

][R(∞)

SS +ρQ(α)]−1

+O(σ4). (3.38)

In practice, this matrix is replaced by its empirical estimate Γ(α) obtained by replacing R(∞)SS by the

matrix RSS and the projection matrix π by its estimate π; π is constructed from the Singular ValueDecomposition of RY = (1/T )∑t y

M(t)Y H

M(t). It is worth noting that if RSS is block diagonal (or closeto be block diagonal), then Γ(α) is block diagonal because Σ(1) is itself block diagonal. In this case,

Trace(Γ(α)) = σ2k=K

∑k=1

TraceN−1k

[ρRSS,k +ρ2α2

kCHk DH(π)RπD(π)Ck

]N−1

k +O(σ4) (3.39)

where we recall that Qk,sub = CHk DH(π)D(π)Ck and Nk , (RSS,k + αkQk,sub). The minimization of

Trace(Γ(α)) is thus equivalent to the minimization of the K one-variable functions at the righthandsideof (3.39), which is much easier than minimizing the corresponding multivariable function, which isprobably not convex.

3.3.1.2 The linear prediction case

We first evaluate the closed-form expression of Σ = Cov[δ∆(A,D) g

](defined in Proposition 1). For

this purpose, we partition each block Σ(k, l) of Σ as follows:

Σ(k, l) =[

Σ11(k, l) Σ12(k, l)Σ21(k, l) Σ22(k, l)

]d= N(M +L+1)×N(M +L+1) (3.40)


where blocks Σ11(k, l),Σ12(k, l) and Σ22(k, l) are N ×N,N ×N(M + L) and N(M + L)×N(M + L)respectively. Using straightforward but tedious perturbation theory calculations, it is possible to showthe following result (Appendix F).

Proposition 3: Assuming that the filtering matrix TM(H) has full column rank, Σ can be written as

Σ = Σ(0) +σ2Σ(1) +σ4Σ(2) (3.41)

where matrices Σ(0), Σ(1) and Σ(2) are defined as follows: for each (k, l) ∈ [1,2, . . . ,K]2Z,

Σ(k, l) , Σ(0)(k, l)+σ2Σ(1)(k, l)+σ4Σ(2)(k, l) (3.42)

where matrices Σ(0)(k, l), Σ(1)(k, l) and Σ(2)(k, l) are given by

Σ(0)11 (k, l) = 0,Σ(0)

12 (k, l) = 0,Σ(0)21 (k, l) = 0 (3.43)

Σ(0)22 (k, l) = δk,l (IM+L⊗D)

Σ(1)11 (k, l) = δk,l

∫ π−π πD⊥A(eiω)AH(eiω)πD⊥dω

Σ(1)12 (k, l) = δk,l

∫ π−π e−iωDT

M+L−1(ω)⊗πD⊥A(eiω)AH(eiω)dω

Σ(1)21 (k, l) = Σ(1),H

12 (k, l)

Σ(1)22 (k, l) =

∫ π−π[JT

k T(eiω)TH(eiω)Jl]∗⊗Ddω

+δk,l∫ π−π[DM+L−1(ω)DH

M+L−1(ω)]∗⊗A(eiω)AH(eiω) dω

Σ(2)11 =

∫ π−π uT

k [H#(0)A(eiω)AH(eiω)H#,H(0)]∗ul π⊥D A(eiω)AH(eiω)π⊥D dω

Σ(2)22 =

∫ π−π[JkT(eiω)TH(eiω)Jl]∗⊗A(eiω)AH(eiω)dω

Σ(2)12 =

∫ π−π e−iω[uT

k H#(0)A(eiω)TH(eiω)Jl]∗⊗π⊥D A(eiω)AH(eiω)dω

Σ(2)21 = Σ(2),H

12 .

(3.44)

As in the subspace case, it is easily seen that

Γ(α) =[R(∞)

SS +ρQ(α)]−1 [

ρR(∞)SS +ρ2∆(α)HW(Σ(0) +σ2Σ(1))W∆(α)

][R(∞)

SS +ρQ(α)]−1

+O(σ4).(3.45)

From now on, we assume that the SNR is high enough to neglect the terms in σ4. Therefore, the matrixΓ(α) does not depend on the matrix Σ(2).

We now discuss the advantages of the weighted semi-blind estimate. For this, we first study the behaviourof Γ(α) in the non-weighted case. We first remark that Γ(α) is difficult to estimate because the matrix


Σ(1) depends on the true channel via the matrix T (see Σ(2)22 in (3.44)). The estimation of Γ(α) thus

requires the use of a good accuracy initial estimate of the channel. Moreover, we remark that, in contrastwith the subspace case, limσ2→0 Γ(α) 6= 0. Also, the matrix Σ(1) is not block diagonal (see (3.44)).Hence, even if R(∞)

SS is block diagonal, the matrix Γ(α) is not the sum of K cost functions depending onthe single parameters α1, . . . ,αK . The consistent estimate of Γ(α) is therefore more difficult to minimizewith respect to the vector α.

We now consider the weighted case where the matrix W coincides with IK(M+L+1)⊗ (πD⊥ + σ2D#).Firstly, we notice that the contribution of the first term of the righthandside of (3.44) to σ2WΣ(1)W isa O(σ4) term : this is because (πD⊥ + σ2D#)D = O(σ2). As O(σ4) terms are neglected, the matrixΓ(α) depends only on matrices D and A, and not on the true channel. Estimating consistently Γ(α)(up to O(σ4) terms) is thus quite easy. Next, we remark that limσ2→0 Γ(α) = 0 because WΣ(0)W = 0.Therefore, as in the subspace case, the variance of the weighted estimate converges to 0 when σ2 → 0.Finally, WΣ(1)W is a block diagonal matrix because the right and left multiplications by W cancelthe non block diagonal entries of Σ(1). The minimization of the consistent estimate of Γ(α) is thusequivalent to K minimizations of K one-variable functions associated with each individual user. It turnsout that the proposed weighted estimate has more attractive statistical properties than the non-weightedone.

3.3.2 The projected approach

When m = o(NT ), the convergence rate of the semi-blind estimator is higher than that of the trainingbased estimator and the blind criterion overrides the least-squares fit in the dominant subspace Q. Thebasic idea is then to say that, as we can “trust” the blind procedure, the semi-blind estimate of the vectorchannel impulse response should be confined in the null-space of Q(α). This means that the semi-blindestimate is obtained by projecting the trained estimate onto the null-space of Q(α). The latter coincideswith the null-space of Q(1l) because Q(α) is block diagonal (see (3.27)). In this subsection, Q(1l) issimply denoted by Q.

This is proved by the following proposition (the proof is provided in Appendix F).

Proposition 4: Assume that the filtering matrix TM(H) has full column rank and m = o(NT ). Let g be thesemi-blind channel estimate provided by whether the semi-blind subspace method or linear predictionapproaches. Then, m1/2(g−g) is asymptotically normal with covariance matrix given by

Γ =σ2

mPR−1

SS P+O(σ4) (3.46)

and g is asymptotically equivalent to Pgtr

such that

g = Pgtr

+OP(m−1 +T−1/2) (3.47)

where P is the orthogonal projector onto the null-space of the matrix Q; Q = Qsub,Qlin or Qwlin.

If TM(H) has full column rank and if codes respect certain non restrictive conditions [?], each channel gk

is identifiable, which means that dim(null(Q)) = K. But in practice, it is not efficient to construct P fromthe K eigenvectors associated with the K smallest eigenvalues. Indeed, the number of available obser-vations is generally not high enough to ensure very good conditioning of the matrix Q. This means thatthe K smallest estimated eigenvalues are not necessarily those associated with the K different channels.


As a consequence, it is generally more efficient to peform an individual singular value decomposition oneach matrix Qk.

3.4 Simulations

3.4.1 Road map

In the theoretical part of this chapter we have distinguished two kinds of semi-blind approaches: theregularized approach, which corresponds to the case where the training sequence length is of the sameorder as the time slot length and the projected approach which corresponds to transmissions based onvery short training sequences. For the blind part of the estimator, three blind techniques have beenconsidered: the subspace method, the conventional linear prediction and the weighted linear prediction.In summary, it makes six possibilities to study. In this section however, we focus more particularly ontwo of them. Indeed, our observations showed us that the projected approach is not suited at all forUMTS-TDD systems and the weighted linear approach is in general more efficient than its non-weightedcounterpart.

Firstly, we study the pure blind estimators. Then we study in details the regularized approach and itsrobustness and sensitivity to important parameters such as the channel length, the midamble and the timeslot size. At last, some performance evaluation of the projected approach are presented.

3.4.2 Experimental setup

The parameters of our simulation chain are the followings:

– uplink case (a single antenna is assumed);

– fixed spreading factor: N = 16;

– fixed number of active users per time slot: K = 4;

– no channel encoding;

– propagation channel: Vehicular A;

– physical channel: it is the convolution of the propagation channel by the shaping filter. Time durationsof physical channels are about one symbol duration (16 chips);

– the mobile speed is fixed at 120 km/h;

– the noise standard deviation is σ =√

N2 10−

Eb/No|dB20 and is assumed known from the receiver;

– only the terms in σ2 are taken into consideration in order to estimate the covariance matrices ofestimation error;

– Bit Error Rates (BER) are averaged over 2000 time slots and over the 4 users.

3.4 Simulations 51


3.4.3.1 Unicity of the solution of blind criteria

As it is explained in subsection 3.2.2.1, we have indicated under what conditions the structured subspace(3.11) or linear prediction equations (3.20), (3.23) have a unique solution. This is why we proposestudying experimentally this issue. To this end, we denote by gk(z) the physical channel of user k andv(0)

k (z) the z-transform of the eigenvector associated with the smallest eigenvalue of Qk.

In the noiseless case of the subspace method or the weighted linear prediction approach, for short chan-nels like Indoor A (10 chips3 for the physical channel) or Indoor B (12 chips for the physical channel),we observed that

• if L = 1 then v(0)k (z) = α0gk(z);

• if L = 2 then v(0)k (z) = (α0 +α1z−N)gk(z);

• if L = 3 then v(0)k (z) = (α0 +α1z−N +α2z−2N)gk(z).

In the noiseless case of the subspace method or the weighted linear prediction approach, for mediumchannels like Vehicular A (19 chips for the physical channel) or Pedestrian B (24 chips for the physicalchannel), we observed that

• if L = 2 then v(0)k (z) = α0gk(z);

• if L = 3 then v(0)k (z) is no longer proportional to the vector channel of user k. It is impossible to

recover the channel.

3.4.3.2 Performances of the blind estimators

Here we compare the three considered blind estimators between themselves (figure 3.1). To get rid of thewell-known ambiguity problem, we assumed known from the receiver the strongest channel coefficient.In figure 3.1, we provide the BERs of the MMSE-JD for three different assumed ISI durations L = 1, L =2 and L = 3. The true physical channel length is 16 chips (truncated Vehicular A), which corresponds toL = 1. When L = 1, the best estimator is the subspace method-based estimator followed by the weightedlinear prediction approach-based estimator and the linear prediction approach. But the situations ofpractical interest are cases for which L = 2 and L = 3, which is the case where channels are overmodelled.In these cases, the linear prediction performs better than the subspace approach. This confirms the resultsshowed in [?]. This is one of the reasons why it seemed relevant to us to exploit both the semi-blindsubspace and linear prediction approaches. The problem is then to know to what extent this observationcan be applied to the semi-blind case.

As for the comparison of the linear prediction approaches between themselves, we observed that theweighted linear prediction was performing better in almost all the simulated contexts. The benefit of

3In this section the chip duration equals 0.244 µs, which corresponds to ETSI specifications v1.2.0 (release 1999)

weighting the linear prediction was particularly obvious in the semi-blind context. In the pure blindcontext, the weighted approach can be beneficial even it the channel lengths are assumed to be known.Gains are really significant if certain conditions are met as we see in figure 3.2. Typically, the weightingoperation is worth being implemented for Eb/No > 12 dB and for T > 250. As a result, only theperformance of the weighted approach is presented in the sequel.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Assumed channel length (in symbol duration)



Comparison of the blind estimators

Vehicular ATrue physical channel length = 16 chips (L = 1)4 active usersE

b/N

0 = 17.5 dB

gmax

is assumed known for every user

Subspace method Weighted linear predictionLinear prediction

Figure 3.1: Bit error rates versus assumed ISI duration (L); three blind estimators are compared: thesubspace method (SS), the weighted linear prediction (WLP) and the conventional linear prediction (LP);for every user, the strongest channel coefficient is assumed to be known; simulation conditions: VehicularA, the physical channel is truncated at 16 chips (L = 1), 4 active users, Eb/No = 17.5 dB, T = 122blind observations; the linear prediction approaches are more robust to channel overmodelling than thesubspace method

3.4.3.3 Tuning of the regularizing constants

In all the simulation section, we assume block diagonal the matrix RSS, which allows us to split the K-variable minimization into K 1-variable minimizations as seen in (3.39). This assumption is well verifiedin practice as confirmed in chapter 4. The propagation environment is always the Vehicular A but thecorresponding physical channel is not truncated, which means that the physical channel length is 19chips (L = 2). For each time slot, the values of the regularizing constants are chosen in order to minimizethe trace of the estimated asymptotic covariance matrix of channel estimation error. In the followingexperiments, we first show the convexity of the derived semi-blind criteria for the subspace for example(figure 3.3) and then we compare our semi-blind estimates when the values of parameters (αk)k=1,...,K aregiven by our procedure, and when the values of (αk)k=1,...,K are given by an oracle minimizing the truechannel estimation error on each time slot (∀k = 1, . . . ,K, αk = argminαk ‖g

k− g

k(αk)‖2). The various

parameters of the simulation correspond all to the specifications of the UMTS-TDD mode. In particular,there are T = 122 unknown symbols in each time slot and the midamble length is 512. Moreover, theassumed channel length is two symbol durations (L = 2, LN = 32 chip durations).

The table 3.1 provides some statistics on the regularizing constant associated with a given user. Notation” < . > ” stands for averaging over all the generated slots. Notations "SS" and "LP" stand for subspaceand (weighted) linear prediction respectively. As mentioned above, the oracle under consideration is

3.4 Simulations 53

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15

10−2

10−1

Eb/No [dB]



conventional LP with 250 observationsconventional LP with 500 observationsweighted LP with 250 observations weighted LP with 500 observations

Figure 3.2: Bit error rates versus signal-to-noise ratio; comparison of the linear prediction approachesbetween themselves in a blind context; for every user, the strongest channel coefficient is assumed tobe known; simulation conditions: Vehicular A, the physical channel is truncated at 16 chips (L = 1), 4active users, L = 1; in the case where channel lengths are known, if the number of blind observations(250 and 500 in the figure) and the signal-to-noise ratio (> 10 dB in the figure) are high enough, theweighted linear prediction performs better than its non-weighted counterpart

EbNo

5 dB 7.5 dB 10 dB 12.5 dB< α > (SS/SS,oracle) 0.793 / 0.629 0.783 / 0.599 0.772 / 0.610 0.763 / 0.659< α > (LP/LP,oracle) 0.623 / 0.241 0.638 / 0.219 0.630 / 0.222 0.660 / 0.2594

Table 3.1: Tuning accuracy of the regularizing constant for semi-blind subspace and weighted linearprediction; the regularizing constant values derived from the proposed semi-blind schemes are comparedto those obtained by minimizing the Euclidean distance between the true and estimated channels on eachtime slot; accuracy level of the regularizing constants is higher in the semi-blind subspace case than inthe weighted linear prediction one

based on the knowledge of the true channel estimation error for every time slot. It turns out that theregularizing constant is quite well tuned in the regularized semi-blind subspace case. This does not seemto be the case in the regularized semi-blind weighted linear prediction case.In figure 3.4 we represent the gain on Eb/No provided by semi-blind schemes under consideration asa function of the targeted BER. We compare the regularized semi-blind subspace method with the reg-ularized semi-blind subspace method equipped with the oracle under consideration; the correspondingcurves are very close. In the weighted linear prediction case, for a targeted BER ranging from 1% to10%, the loss due to the non-optimality of the tuning is about 0.2 dB. It should also be noted that for thespecified parameters of the UMTS-TDD mode and for a 1% BER target, our semi-blind methods providegains of 0.4 dB and 1 dB in the weighted linear prediction and the subspace cases respectively.


0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Regularizing constant (hyperbolic scale)

Cha

nnel

est

imat

ion

erro

r

User # 1/4Regularized Semi−Blind Subspace200 values of the RC are testedRange of RC [0,3]a

lin = 4 * atanh(a

hyp)

Eb/N

0 = 10 dB

case 1: RC tuned from the error covariance matrixcase 1: optimum value case 2: RC tuned from the instantaneous error case2: optimum value

Figure 3.3: For a given time slot: the asymptotic error covariance matrix is estimated (top curve) and thedistance between the true and the estimated channel (bottom curve) is calculated; this is done for severalvalues of the regularizing constant (RC); the figure represents these errors as a function of the regularizingconstant; a hyperbolic scale has been chosen in order to facilitate the research of the optimum values;simulation conditions: the semi-blind subspace case is considered, Vehicular A, true physical channellength = 19 chips, LN = 32 chips, 4 users are assumed, presented results correspond to user 1, 200 valuesof α are tested ranging from 0 to 3, Eb/No = 10 dB; the corresponding functions are convex and the twounderlying optimum values of α are very close when the subspace method is used

3.4.3.4 Performance of the regularized approach

In this subsection, the physical channel is always 19 chips (Vehicular A propagation environment). Theassumed channel length is 32 chips. Four active users are assumed. In these conditions, the regularizedsemi-blind subspace provides a gain of 1 dB for BERs of interest while the semi-blind weighted linearprediction allows for a benefit of 0.5 dB (figure 3.5). Obtained gains are significant but in fact UMTS-TDD time slots are not very well adapted to semi-blind estimation. Indeed, semi-blind capability is notfully exploited in a context where the "two sources of information" (trained and blind parts of the timeslot) are of the "same reliability". This is why we give in subsection 3.4.3.6 simulation examples wherethe potential of semi-blind techniques is more apparent.

3.4.3.5 Robustness of semi-blind estimators to channel overmodelling

Robustness of channel estimator to channel overmodelling is an important issue since the propagationenvironment is intrinsically unknown. Usually, in the literature of blind channel estimation, channel isassumed to be known because of the identifiability problem. However, as we mentioned in the theoreticalpart of this chapter, for CDMA systems the conditions to be met are less restrictive. Additionally it canbe expected from semi-blind estimators to be less sensitive to this problem.

Once again, parameters are those of the UMTS-TDD mode. The impact of channel overmodelling onreceiver performance is evaluated in figure 3.6. For a targeted BER of 1%, it represents the gains providedby semi-blind approaches on Eb/No versus the assumed channel length in symbol duration. In dashdot

3.4 Simulations 55

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Quality of the proposed regularizing constant tuning

Targeted BER

Gai

n on

Eb/

No

[dB

]

true channel length: 1 symbol duration (19 chip durations)assumed channel length: 2 symbol durations (32 chip durations)

Subspace based Semi−Blind Estimation equipped with an oracle Subspace based Semi−Blind Estimation Linear Prediction based Semi−Blind Estimation equipped with an oracleLinear Prediction based Semi−Blind Estimation

Figure 3.4: Gain in terms of signal-to-noise ratio versus bit error rate target; this figure shows the influ-ence of regularizing constant accuracy on the MMSE joint detector performance; simulation conditions:see figure 3.3; the top curves correspond to the semi-blind subspace case and the two bottom curvescorrespond to the semi-blind weighted linear prediction case; for each case, we compare the proposedtuning strategy with the case where α is tuned from the Euclidean distance between the true and esti-mated vector channels; the main result is that the regularizing constant is very well tuned even thoughthe optimum value is derived from an asymptotic approach; this holds for the subspace case; however,for the weighted linear prediction, 0.2 dB are lost

5 6 7 8 9 10 11 12 1310

−3

10−2

10−1

100

Eb/N

0 [dB]



Performance of the regularized semi−blind approaches

4 active usersVehicular A1 antennaNo channel codingMS speed = 120km/h1000 time slots

Training−based estimation Semi−blind weighted linear predictionSemi−blind subspace Known channel

Figure 3.5: Bit error rates versus signal-to-noise ratio; the MMSE joint detector performance is evaluatedwith four different channel estimates: the trained estimate, the semi-blind estimate based on the subspacemethod, the semi-blind estimate based on the weighted linear prediction approach and the true channel;simulation conditions: Vehicular A, physical channel length = 19 chips, LN = 32, 4 users, 1 antenna, nochannel coding; for a 1% BER target the semi-blind linear prediction provides a 0.5 dB gain while thesemi-blind subspace reaches a 1 dB gain


line, we have represented the gain that could be achieved if channels were known from the symboldetector. We see that semi-blind approaches provide gains on Eb/No between 0.8 dB (weighted linearprediction approach) and 1.2 dB (subspace method) for an assumed channel length of 48 chip durations,which roughly corresponds to the maximum size of channels in the context of UMTS-TDD mode (see[?]). Although the weighted linear prediction is more robust to channel overmodelling than the subspacemethod (see figure 3.1), this does not seem to be an advantage we can take of for semi-blind estimationin UMTS-TDD-like systems.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.5

1

1.5

2

2.5

3

Assumed channel length (symbol duration)

Gai

n on

Eb/

No

[dB

]

Influence of the assumed channel length

Targeted BER: 1%4 userstrue channel length: 1 symbol duration

Oracle: channel is perfectly knownSubspace based Semi−Blind Linear Prediction based Semi−Blind

Figure 3.6: Gain on signal-to-ratio versus assumed ISI duration (L); simulation conditions: see figure3.5, BER target = 1%; the top curve corresponds to the gain that would be achieved if the channel wasknown; the middle one is the performance of the semi-blind subspace channel estimate; the bottomcurve represents the performance of the semi-blind weighted linear prediction based channel estimate;as a result, the proposed semi-blind estimators are robust to channel overmodelling but the semi-blindsolution based on the subspace method is more robust than the semi-blind weighted linear predictionsolution

3.4.3.6 Influence of the midamble length

In this experiment, parameters are always the same as those used in the UMTS-TDD mode, except for thelength of the training sequence. We fix Eb/No at 12.5 dB (the results are similar for lower signal-to-noiseratios). The assumed channel length equals 32 chip durations.Figure 3.7 represents the BER provided by semi-blind approaches versus m (recall that the size of themidamble is equal to m+LN). For m = 480 chips, the BER associated with the trained channel estimatorequals 1%. The BER is about 0.85% for the weighted linear prediction method and 0.65% for thesubspace method. But for shortest training sequences, BERs are no longer so close. Indeed, for m = 152,the BER equals 10% in the trained case and 1% in the regularized semi-blind subspace case. We alsonote that in order to achieve a 1% BER, the trained estimate needs m = 480 chips while the semi-blindsubspace estimate only requires m = 152. The use of this semi-blind estimate thus allows us to reducethe midamble size by about 3. This corresponds roughly to a gain of 15% for the data rate.

In order to confirm the results presented in figure 3.7, we compare in figure 3.8 the performance of

3.4 Simulations 57

150 200 250 300 350 400 450 500

10−2

10−1

Influence of the midamble size on MMSE−BL−JD performance

m



true channel length: 1 symbol duration (16 chip durations)assumed channel length: 2 symbol durations (32 chip durations)Eb/No = 12.5 dB

Training Sequence based Estimation Linear Prediction based Semi−Blind EstimationSubspace based Semi−Blind Estimation

Figure 3.7: Bit error rates versus midamble length; for a given signal-to-noise ratio (Eb/No = 12.5dB) the MMSE joint detection performance is evaluated with the trained and the semi-blind channelestimates; simulation conditions: Vehicular A, physical channel length = 19 chips, LN = 32, 4 users; 1single antenna, no channel coding; for short midambles (m = 152) the semi-blind schemes provide verysignificant improvements over the training-based approach

the proposed semi-blind estimates with the trained one versus Eb/N0 in the case where m = 152. Theduration of the channel is always 32 chip durations and T = 122. It is clear that the semi-blind approachesoutperform quite significantly the classical trained estimate.

8 9 10 11 12 13 14 1510

−3

10−2

10−1

Eb/No [dB]



Semi−Blind schemes −−vs−− TS based scheme for short midambles

true channel length: 1 symbol duration (16 chip durations)assumed channel length: 2 symbol durations (32 chip durations)m = 152

Training Sequence based Estimation Linear Prediction based Semi−Blind EstimationSubspace based Semi−Blind Estimation

Figure 3.8: Bit error rates versus signal-to-noise ratio; simulation conditions: see figure 3.7, m = 152;when using the trained approach, more than 5 dB are lost w.r.t the semi-blind strategies (at BER = 6%)because of the reduced midamble length


3.4.3.7 Influence of the number of blind observations

On the other hand, if we fix the length of the training sequence and consider time slots comprising moreunknown symbols (122, 244 and 488 QPSK symbols) we can observe (figure 3.9) that gains are in-creased. Increasing the number of blind samples is especially beneficial for the linear prediction method.In figure 3.9 the gain on signal-to-noise ratio is represented for a 1% BER target as a function of the timeslot size for a 512-chip midamble. In the same way, figure 3.10 represents the gain on SNR versus thetime slot size for a 184-chip midamble.

100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

1.2

1.4Influence of time slot size (midamble length is fixed at 512)

Number of blind observations

Gai

n on

Eb/N

0 [dB

]

Regularized semi−blind subspace Regularized semi−blind linear prediction

Target BER = 1%Midamble length = 512Vehicular A Assumed channel length = 32 chips4 active users

Figure 3.9: Gain (over the trained approach) in terms of signal-to-noise ratio versus number of blindobservations; simulation conditions: BER target = 1%, midamble length = 512 chips, Vehicular A,physical channel length = 19 chips, LN = 32 chips, 4 users, 1 single antenna, no channel coding; if thenumber of blind samples is multiplied by 4 (from T = 122 to 4T = 488), we can benefit of a 0.2 dB gainfor the semi-blind subspace based estimation and 0.5 dB for the semi-blind weighted linear prediction

3.4.3.8 The projected approach

At last we have evaluated the performance of the projected approach. Even though there is no moreregularizing constant to tune, it still remains a discrete hyperparameter to choose. Theoretically, thedimension of the null-space of the matrix Qk (for user k) equals 1. In practice, it is seen that it is bet-ter to make use of more eigenvectors in the construction of the matrix Pk as it is shown in figure 3.11.This curve represents the instantaneous estimation error (for a given time slot) versus the assumed di-mension of null(Qk), which is denoted by d. If d equals 1, the performance of the projected approachboth for the subspace method and the linear prediction approach is worse than the training-based esti-mator performance (see figure 3.12). However, this figure also shows the performance of the projectedapproach-based estimators equipped with an oracle. The oracle provides the estimator the best d, whichminimizes the estimation error on each time slot. In this case, the performance of the projected approachis dramatically improved. The same kind of simulation has been made for 4T = 488 blind observa-tions (see figure 3.13). In summary, projected semi-blind approaches are able to perform better thanpure trained approach but this holds only when the channel length and the best d are known from thereceiver. As a conclusion, this approach is not suited to UMTS-TDD-like systems. In this respect, figure

3.4 Simulations 59

100 150 200 250 300 350 400 450 5005

5.5

6

6.5

7

7.5Inflence of time slot size (midamble length is fixed at 184)

Number of blind observations

Gai

n on

Eb/N

0 [dB

]

Regularized semi−blind subspace Regularized semi−blind linear prediction

Target BER = 5%Midamble length = 184Vehicular AAssumed channel length (l+1) = 32 chips 4 active users

Figure 3.10: Gain (over the trained approach) in terms of signal-to-noise ratio versus number of blindobservations; simulation conditions are the same as for figure 3.9 except for midamble length, whichequals 184 chips (m = 152); if the number of blind samples is multiplied by 4 (from T = 122 to 4T =488), we can benefit of a 0.5 dB gain for the semi-blind subspace based estimation and 1 dB for thesemi-blind weighted linear prediction

3.14 shows that in the UMTS-TDD mode the projected semi-blind approach provides a better estimationerror variance than that of the training-based estimation for Eb/No greater than 21 dB!

0 2 4 6 8 10 12 14 160.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Assumed dimension of the projection matrix P

|| g k −

gk,

est||2

Influence of the choice of dim(P) on the estimation performance

User #1/4Projected Semi−Blind SubspaceE

b/N

0 = 10 dB

Known channel length

Figure 3.11: For a given time slot: Euclidean distance between the true and estimated channels versusdim(null(Qk)); simulation conditions: Vehicular A, physical channels are truncated at 16 chips, channellengths are assumed to be known, the projected semi-blind subspace approach is considered, 4 users, user1 is considered, Eb/No = 10 dB; for a given user, if the dimension of the null space of the matrix Qk(denoted by d) is assumed to be 1, the estimation error (Euclidean distance) is worse than in the trainedcase (d = 16)

2 3 4 5 6 7 8 9 10 11 12 13

10−2

10−1

100

Projected approaches with T=122 samples

Eb/N

0 [dB]

 K

MMSE−JD4 active usersVehicular AL = 1L

assumed = 1

1 antenna2000 time slots

WLP with dim(P) = 1 SS with dim(P) = 1 Training−based estimationWLP with dim(P) = d* SS with dim(P) = d* Known channel

Figure 3.12: Bit error rates versus signal-to-noise ratio; the MMSE joint detector performance is eval-uated in the conventional UMTS-TDD context except for the channel length, which is assumed to beknown from the receiver; simulation conditions: Vehicular A, physical channel is truncated at 16 chips,4 users, 1 antenna, no channel coding; the influence of the assumed dimension of the null space of thematrices Qk is studied; if d = 1, the semi-blind projected approaches perform dramatically worse thanthe training-based approach; however, if d could be optimally tuned (by choosing the d minimizing theEuclidean distance for each time slot), the semi-blind schemes performance gains would reach 40% ofthe available margin gain (1 dB at BER = 1%)

2 4 6 8 10 12 1410

−3

10−2

10−1

100

Eb/N

0 [dB]

 K

Projected approaches with 4T=488 samples

MMSE−JD4 active usersVehicular AL = 1L

assumed = 1

1 antenna2000 time slots

WLP with dim(P) = 1 SS with dim(P) = 1 Training−based estimationWLP with dim(P) = d* SS with dim(P) = d* Known channel

Figure 3.13: Bit error rates versus signal-to-noise ratio; this figure is similar to figure 3.12 but 4T = 488blind observations are used; the choice of d is not as crucial as in the figure above; but if the bestdimension could be known, the performance would be significantly improved (0.8 dB for the semi-blindprojected subspace approach while the maximum gain is 1 dB (at BER=1%)

3.4 Simulations 61

17 18 19 20 21 22 23 24 25

10−2

10−1

Eb/N

0 [dB]

<||

g −

ges

t ||2 >

Projected approach with dim(P) = 1 for high SNRs

T = 122 samplesL = 1L

a = 1

Training−based channel estimationProjected semi−blind subspace

Figure 3.14: Mean square estimation error versus signal-to-noise ratio; simulation conditions: see fig-ure 3.12; for d = 1 and T = 122, the semi-blind projected subspace approach beats the training-basedapproach if SNR> 21 dB


3.5 Conclusions

Semi-blind channel estimation methods for synchronized uplink CDMA systems have been consideredin this chapter. The central idea of this approach was to minimize composite criteria and to tune the un-derlying regularizing constants by evaluating the corresponding asymptotic covariances of the estimationerror. We provided these covariance matrices for the subspace and a weighted linear prediction basedsemi-blind schemes. We distinguished two semi-blind approaches:

• the regularized approach for which the time slot and the midamble sizes (T and m) tend bothtowards the infinity while keeping constant the ratio T/m;

• the projected approach for which the time slot and the midamble sizes (T and m) tend both towardsthe infinity but m = o(T N). In this case, there is no regularizing constants to be tuned because theblind part dominates the trained one. The trained channel estimate is merely projected onto thenull space of the matrix Q, which is the matrix associated with the blind cost function.

We focused more on the regularized approach because the corresponding asymptotic assumption is theleast restrictive. We have identified the conditions under which the considered multi-variable minimiza-tions can be split into several one-variable minimizations, which considerably facilitates the research ofthe optimum values of the regularizing constants. Moreover, it has been seen how beneficial it can be toweight the linear prediction criteria: the proposed weighting both improves the statistical performanceof the blind part and makes the estimation of the covariance matrix of the error easier, which is neededto tune the regularizing constants.

At last, the potential of the presented approaches has been evaluated in a realistic context. In this respect,at least three points are worth being highlighted:

• the projected approach is very simple conceptually since it consists in projecting the training-basedvector channel estimate on a certain subspace and the parameter to be tuned is discrete (dimensionof the projection space). However, it is not easy to exploit in a realistic context: high signal-to-noise ratio (typically greater than 20 dB) or large number of observations (typically greater than400) is required;

• semi-blind approaches perform quite well but they are especially useful when training sequencesare short (5% to 10% of the time slot duration typically). In this case, they work dramatically betterthan the pure trained approach. In the context of the UMTS-TDD Mode, significant improvementshave been achieved in terms of data rate, quality of service or power consumption. For instance,semi-blind approaches allow us to achieve the same performance as the classical trained estimatebut with a training sequence which is three times shorter. This corresponds to increase the datathroughput by 15%;

• as for the comparison of the semi-blind schemes between themselves, it has been seen that thesemi-blind subspace approach generally outperforms the weighted linear prediction one in thecases of interest. This fact was not obvious since we showed (see also [?]) that the blind linearprediction approach dominates the blind subspace method in the cases where channel lengths areoverestimated.

As a conclusion, semi-blind estimation schemes based on second-order blind techniques are very promis-ing for systems pre-designed for this kind of channel estimation method. This means that midamble

3.5 Conclusions 63

lengths and time slot sizes should be chosen in order to fully exploit semi-blind estimators capabilities.On the other hand, the most critical point was the additional complexity induced by the proposed semi-blind techniques. For future systems, it should not be that critical. For UMTS-TDD systems however,semi-blind techniques are not envisageable regarding the receiver complexity limitation. This is why wehave considered other ways of improving channel estimation in chapter 4. This time, we will see thatif transmissions are continuous, it is possible to design simple and efficient channel estimators withoutusing the "blind" part of the time slot. The method consists also in looking for a projection matrix butwe will see that the corresponding performance is less sensitive to the number of available observations,which is the number of time slots available for the estimation procedure.

Chapter 4

Low complexity channel estimationschemes based on continuous time slottransmissions

4.1 Introduction

This chapter is also devoted to the channel estimation problem. In chapter 2 we studied the most efficientsemi-blind channel estimation scheme, which is the Maximum Likelihood estimation. EM algorithmcomplexity is the limiting factor so that it cannot be implemented if there is more than 2 active usersper time slot and ICI length has to be less than N−1 = 15 chip durations. This is why in chapter 3 weaddressed suboptimum semi-blind channel estimation schemes namely semi-blind techniques based onsecond-order blind methods. This time presented algorithms are implementable but channel estimationcomplexity is increased by a factor typically ranging from 10 to 20 [?] [?]. This means that this kind ofschemes cannot be implemented in current UMTS-TDD systems. Therefore, in this chapter we considerin first place the complexity constraint.

One critical point in the UMTS-TDD mode is that there are few degrees of freedom to improve channelestimation accuracy of the standardized procedure without compromising complexity. Indeed, UMTS-TDD training sequences have been designed in such a way that the training sequences matrix S (equation1.9) is right circulant (−P3−). This considerably reduces computational cost of its inversion because itcan be performed in the frequency-domain (via the Fast Fourier Transform algorithm). The conditionto be met to get this circularity property is to assume a 57-chip channel length for type 1 time slotsand 64-chip channel length for type 2 time slots [?], [?]. As a consequence, if we want to design moreefficient channel estimators with marginal complexity increase for the UMTS-TDD mode, it is preferableto derive them from the standard procedure.

We already know the major drawback of the special feature of the standard UMTS-TDD channel estima-tor, which is a consequent performance degradation. Indeed, the comparison of the performance achievedin the trained case with those achieved in the case where the channel is known from the receiver shows a3 dB loss on the signal to noise ratio (MMSE-BL-JD, BER target = 1%, 4 active users, Vehicular A en-vironment, single antenna, no channel coding). The performance loss is even more important if multipleaccess interference is higher (figure 2.4), channel coding is implemented [?] or multiple antennas is used.

66 Low complexity channel estimation schemes based on continuous time slot transmissions

This performance degradation is due to the presence of noise in the transmission chain. When channelsare "too overestimated", which is our case (−P5−), detector performance is very sensitive to noise. Infact, physical channel length corresponding to the Vehicular A environment is 19 chips while assumedchannel length is 57 chips (only type 1 time slot has been considered so far). This equates to 67% of theestimated coefficients being pure noise. Then, two questions arise. Are we able to efficiently eliminate1

non-significant channel coefficients, the pure noisy coefficients in particular? Does this approach lead toimplementable solutions? [?], [?] [?]

In order to go further with the raised questions, we first justify why it can be clever to replace an estimatedchannel coefficient with zero. To this end, let us consider the vector channel g as a deterministic vector.From equation (1.8), it is straigthforwardly found that the estimation error variance for the training-basedestimation is:

E[‖g−g‖2] = σ2Trace((SHS)−1). (4.1)

The one-dimensional counterpart of (4.1) is

∀i ∈ [0,K(`+1)−1],E[|g(i)−g(i)|2] = σ2uTi (SHS)−1ui , σ2

est(i) (4.2)

with ui = (0 . . .010 . . .0)T ; the one is on position (i + 1). Consequently, if instead of estimating g(i) byg(i), it was estimated by 0, then the estimation error would be E[|g(i)|2] = g2(i) (since g is assumed tobe deterministic). This means that if this energy is less than the estimation noise variance σ2

i , it is betterto replace the trained estimate of the considered coefficient with 0. Of course, as g2(i) is unknown, wehave to make use of g2(i) to know if g(i) has to be replaced or not. One possible way is to introduce athreshold η with which the estimated coefficients g(i) are compared. The selection rule is: if g2(i) > ηthen the coefficient is kept and it is removed otherwise. This method is called hard thresholding [?], [?].

Let us comment more on the considered method. With regard to complexity, the proposed strategy isnot computationally demanding. Indeed, fixing a threshold value is a free operation from the complexitystandpoint. From a performance point of view, the problem is to find a suitable threshold value. In thecase of sporadic transmissions, this problem is a "de-noising" problem and the corresponding estimator iscalled "hard thresholder" [?], [?], [?]. For each time slot, we have an estimated channel impulse responsefrom which we want to discard certain coefficients. The de-noising problem has drawn the attention ofmany researchers worldwide. In particular, D. Donoho et al. from Stanford University have had a hugeimpact on this area. We tried to exploit their results in order to adapt them to a radio communicationcontext. We still faced two major problems:

• many results are based on asympotical approaches [?], [?], [?] which means that channel lengthhas to be very high. Some preliminary simulation results showed us that this assumption is notsuited to our context namely the UMTS-TDD mode;

• many studies have been devoted to the wavelet framework. However, to our comprehension of theproblem, the wavelet approach [?], [?], [?], [?], [?] does not seem to be a powerful tool to improvethe de-noising process for radio communications.

As a result, finding a threshold value, both derived from theoretical and efficient results, does not seem tobe an easy task for sporadic transmissions. But if transmissions are continuous, several channel estimatesare available so that the channel coefficients energies can be estimated.

That is why in this chapter we propose considering this more restrictive framework, which is the con-tinuous transmissions case. In this framework, we consider the channel as a random process. In this

1By "eliminate" we mean that an estimated channel coefficient is replaced by a null coefficient.

4.2 The strongest-paths method 67

context, the hard thresholding method becomes the well-known strongest-paths method, which is widelyused in real radio systems. This method is easy to implement but does not fully exploit channel statisticsthat can be estimated from the different time slots. On the other hand, if no more prior information isavailable, applying a Wiener filtering on the current channel estimate is optimum statistically but leads toa complex solution. The main goal of this chapter is to design estimators based on the Wiener approachwhile avoiding important complexity increase. For this purpose, we first review major drawbacks ofthe strongest-paths method. Next, we present the different assumptions we intend to exploit to designboth efficient and simple estimation schemes. After describing the corresponding estimation schemes,we evaluate their performances in the UMTS-TDD mode context and compare them with the classicalstrongest-paths method.

4.2 The strongest-paths method

4.2.1 Principle

The strongest-paths method consists of selecting only the coefficients of the channel estimate that are themost significant in terms of energy. This method can be viewed as a hard thresholding [?], [?] approachas well. Indeed, for hard thresholding-based estimation, any channel coefficient whose estimated energyis found to be below a fixed energy threshold is then replaced by a null coefficient. Consequently, for bothmethods the selection is made according to an energy criterion: only the strongest channel coefficientsare kept. In order to estimate the energies of the coefficients of the current channel, use is made of thechannel estimates provided by other time slots.

4.2.2 Major drawbacks

Although the strongest-paths approach is used in the real life because it is easy to implement, it has atleast four major drawbacks:

(D1) it has the disadvantage that there is no ground, both theoretical and efficient, to select the thresholdvalue or the number of significant paths. As the threshold is fixed, it does not reflect or dependupon the propagation environment and the noise level;

(D2) the estimator is generally inconsistent. Indeed, the channel coefficients whose energy is foundto be beneath the threshold are systematically discarded from the channel impulse response, evenunder high signal to noise conditions;

(D3) a further problem is apparent in that, after selection, the coefficients remain noisy as estimationnoise is not removed.

(D4) a yet further problem is that the strongest-paths method does not fully exploit the statistics of thechannel.


Assumptions Scenario ReferenceTrue = 1 / False = 0

(a) (b) (c)0 0 0 S01 0 0 S10 1 0 S20 0 1 S31 1 0 S41 0 1 S50 1 1 S61 1 1 S7

Table 4.1: Possible scenarios when assumptions (a), (b) and (c) are considered

4.3 A revisited Wiener approach

4.3.1 Assumptions

As mentioned above the strongest-paths-like methods are simple but they are not able to fully exploit thestatistics of the channel. Therefore, in order to design more efficient strategies, we will exploit one orseveral of the following assumptions:

(a) there is basis where the channel can be described by only a few components;

(b) channel coefficients are not correlated;

(c) training sequences have ideal auto-correlation and inter-correlation properties (RSS = mI).

Depending on the desired performance level, the complexity constraint imposed by the system and thechannel nature, we will show how to exploit the Wiener approach to design the most adapted scheme.

4.3.2 The different scenarios

Depending on validity of assumptions (a), (b) and (c), different estimators can be designed. In orderto find the desired trade-off between performance and complexity, we tackle the estimation problem intwo steps. Firstly, we focus on performance improvements. Secondly, from the designed estimators wepropose several ways of reducing complexity. Eight scenarios are possible, as we see in table 4.1:

4.3.3 The optimum weighting procedure

As the strongest-paths method based procedure can be viewed as a binary weighting procedure of theestimated channel coefficients, it is very likely that estimation performance can be improved by extending

4.3 A revisited Wiener approach 69

the weights set 0,1 to the continuous interval [0,1]. In order to be more general, we can even extendthe method by looking for a weighting matrix W minimizing the channel estimation error. However,as the least-squares estimate is also the maximum likelihood estimate (1.10), the minimization of theestimation error leads to W 6= I only if the channel is considered as a random process.

By considering the channel as a random process, it makes sense to look for a weighting matrix W inorder to minimize E

[‖g−g‖2]

where g is the new channel estimate defined by g = Wg. Therefore, Wis evaluated as follows:

W = argminW

E[‖g−g‖2] (4.3)

= argminW

E[‖Wg−g‖2] (4.4)

= argminW

TraceW(Rg +σ2(SHS)−1)WH −WRg−RgWH +Rg (4.5)

= Rg(Rg +σ2(SHS)−1)−1

(4.6)

= Rg

(Rg +

σ2

mR−1

SS

)−1

(4.7)

where Rg = E[ggH ].

On the contrary to the Bayesian approach [?], [?] , which consists of maximizing the A Posteriori Prob-ability (APP) P(g/Y tr) with respect to the channel estimate g, the Wiener filtering approach does notneed the Gaussian assumption on the channel and the invertibility of Rg. If g is a Gaussian randomvector (Rayleigh fadings) and if Det(Rg) 6= 0, the Wiener filtering approach coincides with the Bayesianapproach. Indeed if g is Gaussian then

P(g/Y tr) =P(Y tr/g)P(g)

P(Y tr)(4.8)

= (2πσ2)−mexp(−σ2‖Y tr−Sg‖2)(2πDet(Rg))−K(`+1)exp(−gHR−1g g) (4.9)

∼ exp(−σ2‖Y tr−Sg‖2−gHR−1g g) (4.10)

where Y tr is the observation vector generated by the training sequence (1.8). Hence,

g = (σ−2SHS+R−1g )−1SHY tr (4.11)

= Rg(Rg +σ2

mR−1

SS )−1. (4.12)

This means that minimizing the estimation variance is equivalent to maximizing the APP in the Gaussiancase.

Our approach, which is based on the Wiener filtering, differs from the work done in [?] for at leastfour reasons. First, our framework is rather a "de-noising context" and the Gaussian assumption isnot necessary. Moreover, targeted systems are CDMA systems. Next, we show how lacunarity of thephysical channel (−P4−) can be integrated into the estimation procedure and at last show how it can boildown to a one-dimensional weighting procedure. In order to propose different "de-noising" schemes, weexploit one or several of the assumptions (a), (b) and (c).


4.3.4 The different schemes

4.3.4.1 The low rank approximation (S1)

The low rank assumptionIn this section, we assume that assumption (a) is verified. In this case, the performance of the con-ventional Wiener filtering-based estimator can be improved by reducing the number of parameters toestimate. If (a) holds, g is a linear combination (γ) of a low number (say p) of p column vectors from thematrix Π so that:

g = Πγ. (4.13)

For each time slot, only γ needs to be estimated whereas Π is a constant matrix to be determined once and

for all. As we did for (4.3), we can show that the estimate of γ, say γ = Wγ, minimizing E[‖Wγ− γ‖2

]

is given by:

γ = Rγ(Rγ +σ2

mΠHRSSΠ)−1 (4.14)

where Rγ = E[γγH ] and γ = (ΠHΠ)−1ΠH g = ΠH g. If g is Gaussian, this is equivalent to maximizeP(γ/Y ) with respect to γ.

The low rank approximationIn the real life, relation (4.13) is generally not perfectly true but we can approximate the received signalby:

Y tr ' SΠγ+V (4.15)

which is called the low rank approximation.

However, in order to implement this procedure we have to determine the matrix Π. In fact, the bestbasis to represent a random vector by a minimum number of components is the Karhunen-Loeve basis[?]. Therefore we can construct Π from the most significant eigenvectors of Rg. As for choosingp, which is the number of significant eigenvectors, one possibility is to keep the eigenvectors so thatλ(i)−σ2

est(i) ≥ 0, where λ(i) and σ2est(i) (i ∈ [0,K(`+ 1)− 1]) are the eigenvalues of Rg and the noise

variances respectively.

4.3.4.2 How to exploit the ideal correlation properties

In section 4.3.3, we have seen that the proposed channel estimators fully exploit the statistics of thechannel. However, a high additional computational cost has to be paid. Indeed, according to (4.7)and (4.14), in the proposed estimation procedures there is a need to perform large dimension matricesinversions and eigen decompositions. Typically, in the uplink of the UMTS-TDD mode, for 8 activeusers the size of the channel is 8×57 = 456 chips, which is the dimension of Rg.

But complexity of the two proposed channel estimation procedures can be significantly reduced in con-texts where assumptions (b) and (c) are well verified. For estimation strategies corresponding to S2, S3and S5, complexity reduction is still very small because inversions and eigen decompositions have to beperformed.

The most interesting estimation schemes for reducing complexity while maintaining good performanceare obtained in the scenarios S4, S6 and S7. For S4, Rg is diagonal, which means that no eigen decom-

4.3 A revisited Wiener approach 71

position is needed (the canonical basis is the Karhunen-Loeve basis we are looking for). For S6, whichis the most favorable case, Rg and RSS are diagonal. Consequently, the matrix weighting procedure as-sociated with equation (4.7) boils down to K(`+1) one-dimensional scalar weighting procedures, whichmeans that each channel coefficient is multiplied by a reliability factor. As for S7, it is easy to showthat the corresponding estimation strategy is equivalent to a hard thresholding estimation strategy wherethe selected channel coefficients are weighted by the "Wiener reliabilities". Let us detail more the mostrelevant cases, which are scenarios S6 and S7.

4.3.4.3 One-dimensional hybrid weighting (S6)

Assume that assumptions (a), (b) and (c) hold. From (4.7), it is easily found that each coefficient of thechannel impulse response estimate is weighted by:

∀k ∈ [1,K],∀i ∈ [0,K(`+1)−1],δ(k)(i) =Ek(i)

Ek(i)+σ2est(i)

(4.16)

whereEk(i) = E[|gk(i)|2] (4.17)

andσ2

est(i) = σ2uTi (SHS)−1ui. (4.18)

Of course, in real situations, both the energies of the channel coefficients and the noise variance haveto be estimated. The way of estimating the energies and the noise variance has a strong influence onthe performance of the proposed estimator. Let us see how to estimate them in an efficient way. Letδk(i), Ek(i) and σ2

i (i) be the estimates of δk(i), Ek(i) and σ2i (i) respectively. The most natural way of

implementing the proposed method is to choose:

δk(i) =Ek(i)

Ek(i)+ σ2est(i)

, (4.19)

Ek(i) =1

W

m=W

∑m=1

|gk(i,m)|2 (4.20)

where W is the number of available time slots and m is the time slot index. As for the noise estimation,one idea is to make use of the less significant paths. This is especially efficient when channel lengthsare overestimated, which is often the case in real life. Using the Q less significant paths of each channelgives:

σ2est(i) =

1KQ

k=K

∑k=1

q=Q

∑q=1

Ek(ik,q) (4.21)

where K is the number of active users and Q is the number of channel coefficients considered as purenoise and i(k,q) are the corresponding indices. However, the major drawback of (4.20) is that it isbased on the noisy channel coefficients. In particular, even if W →∞, pure noisy coefficients are weightedby limW→∞(δk(i)) = 1/2. For this reason we propose estimating the channel coefficients energies byEk(i) which is defined by:

Ek(i) =1

W

m=W

∑m=1

|gk(i,m)|2− σ2est(i). (4.22)

In this case, the new dimensional weight δk(i) are replaced with:

δk(i) =Ek(i)

Ek(i)+ σ2est(i)

(4.23)

=Ek(i)− σ2

est(i)Ek(i)

. (4.24)

At last, if we want to guarantee that weights δk(i) stay in the interval [0,1] for i ∈ [0,K(`+ 1)− 1], wecan modify these weights as follows:

δk(i) =∣∣∣∣1−

σ2est

Ek(i)

∣∣∣∣×max(

1− σ2est

Ek(i),0

)(4.25)

which is equivalent to apply a hard thresholding for estimated coefficients having an estimated signal-to-noise ratio less than 1. As a consequence, the proposed weighting procedure may be called an hybridone.

As a conclusion, we see that the proposed estimator

• tends to eliminate coefficients which are pure noise (limW→+∞ δk(i) = 0 for the pure noisy coeffi-cients);

• tries to remove estimation noise from the channel coefficients (δk(i) < 1);

• is consistent (limσ→0 δk(i) = 0).

4.3.4.4 Low rank approximation and ideal correlation properties (S7)

Assume that assumptions (a), (b) and (c) hold. Assumption (b) and (4.13) implies that Rγ is also diagonal.It is then possible to rewrite (4.14) as follows:

γ = Dγ

(Dγ +

σ2

mIp

)−1

ΠH g. (4.26)

Finally, the channel estimate is given by:

g = ΠΛΠH g (4.27)

where

Λ = Diag(λ0, . . . ,λp−1) = Diag

(Eγ(0)

Eγ(0)+ σ2

m

, . . . ,Eγ(p−1)

Eγ(p−1)+ σ2

m

)(4.28)

and ∀i ∈ [0, p−1],Eγ(i) = E[γ(i)γ∗(i)]. As Dγ has to be estimated in practice, we have to estimate Λ byΛ = Diag(λ0, . . . , λp−1) where:

λ(i) =Eγ(i)− σ2

m

Eγ(i)(4.29)

and

Eγ(i) =1

W

m=W

∑m=1

|γk(i,m)|2. (4.30)

4.4 Simulation results 73

At last, notice that γ is given by γ = Πg.

The matrix Rg being diagonal implies that Π is a simple selection matrix. Indeed, since Rg is alreadydiagonal, Π is constructed from the canonical basis, which is a Karhunen-Loeve basis. This is why thisprocedure is equivalent to a hard thresholding procedure in which selected coefficients are selected bythe matrix Π and weighted by the "Wiener reliabilities". Therefore, this estimator allows us to (partially)eliminate drawbacks (D3) and (D4). In terms of complexity, this postprocessing is not very demandingsince no eigen decomposition is needed and the inversion of the p-dimensional matrix in equation (4.26)is replaced by p scalar divisions.

4.4 Simulation results

4.4.1 Determination of the scenario corresponding to the UMTS-TDD context

First of all, we want to know how beneficial it is to make use of the Karhunen-Loeve basis instead of thecanonical basis. For this purpose, we analyze the power-delay profile of the channels under considerationin the two bases. As figures 4.1, 4.2 and 4.3 show, even in the canonical basis, the considered channelsare described by a few number of coefficients. Therefore, there is no use of considering a new basis inour context.

0 2 4 6 8 10 12 14

−102

−101

Power−delay profile

Chip index

Pow

er [d

B]

Canonical basis Karhunen−Loeve basis_____0 dB____

____−2 dB____

____−12 dB ____

____−30 dB____

PHYSICAL CHANNEL 1. Indoor B + 2. Raised−cosine

Figure 4.1: Power-delay profile of Indoor B channel; physical channel length = 12 chips; this figure rep-resents the channel coefficient energies in the canonical basis (top curve) and the Karhunen-Loeve basis(bottom curve); the KL basis is obtained by performing an eigenvalue decomposition on the covariancematrix of the channel; in both bases, the channel energy is concentrated on three coefficients; as a result,considering a new basis is useless


0 5 10 15

−102

−101


Chip index

Pow

er [d

B]

Canonical basis Karhunen−Loeve basis

____0 dB____

____−6 dB____

____−15 dB____

____−19 dB____

____−26 dB____ ____−29 dB____

0 5 10 15

−102

−101


Chip index

Pow

er [d

B]

Canonical basis Karhunen−Loeve basis

____0 dB____

____−6 dB____

____−15 dB____

____−19 dB____

____−26 dB____ ____−29 dB____

PHYSICAL CHANNEL 1. Vehicular A + 2. Raised−cosine

PHYSICAL CHANNEL 1. Vehicular A + 2. Raised−cosine

Figure 4.2: Power-delay profile of Vehicular A channel; physical channel length = 19 chips; this figurerepresents the channel coefficient energies in the canonical basis (top curve) and the Karhunen-Loevebasis (bottom curve); the KL basis is obtained by performing an eigenvalue decomposition on the co-variance matrix of the channel; in both bases, the channel energy is concentrated on six coefficients; as aresult, considering a new basis is useless

0 5 10 15

−102

−101


Chip index

Pow

er [d

B]

Canonical basis Karhunen−Loeve basis____0 dB____

____−5 dB____

____−14 dB____ ____−16 dB____

____−22 dB____ ____−25 dB____

____−35 dB____

PHYSICAL CHANNEL 1. Pedestrian B + 2. Raised−cosine

Figure 4.3: Power-delay profile of Pedestrian B channel; physical channel length = 24 chips; this figurerepresents the channel coefficient energies in the canonical basis (top curve) and the Karhunen-Loevebasis (bottom curve); the KL basis is obtained by performing an eigenvalue decomposition on the co-variance matrix of the channel; in both bases, the channel energy is concentrated on nine coefficients; asa result, considering a new basis is useless

Figure 4.4 represents a three-dimensional view of the matrix Rg for a given user. We see that it is almostdiagonal. As we know the channel model, we could have forecasted this observation. Indeed, coefficientsof propagation channels are decorrelated by construction. Secondly, we know the emission and reception


filters, which are root raised cosine filters. Therefore, Rg is not perfectly diagonal because of these filters.But assuming that assumption (b) is verified is still reasonable with regard to results from figure 4.4.

0

5

10

15

0

5

10

15

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Row index

3−D view of matrix Rg of the user k

Column index

Mag

nitu

de

Vehicular AEstimation over 1000 time slots

Figure 4.4: This figure depicts a three-dimensional view of the covariance matrix of the channel (Vehic-ular A) for a given user; we see that it is "almost" diagonal

For this purpose, assuming 8 active users and a 57-chips channel, we split the matrix S into 64 57×57blocks and consider the top left block and its right neighbor. The top left block (figure 4.5), which isdiagonal block represents the correlation properties of midamble for a given user. The other consideredblock (figure 4.6) shows the correlation between midambles of users #1 and #2. From these figures, wesee that the energy is essentially concentrated on the main diagonal. The diagonal coefficients magnitudeequal 456 (figure 4.5) while magnitude of the other coefficients do not go beyond 20 (figures 4.5 and 4.6).Therefore we consider RSS as a diagonal matrix.


010

2030

4050

60

010

2030

4050

600

50

100

150

200

250

300

350

400

450

500

Row index

3−D view of matrix RSS,kk

where k=1,..,K

Column index

Mag

nitu

de

Figure 4.5: This figure depicts a three-dimensional view of a diagonal block of the matrix of trainingsequences RSS; it represents the autocorrelation properties of the training sequence of a given user

010

2030

4050

60

0

10

20

30

40

50

600

5

10

15

20

25

Row index

3−D view of matrix RSS,kl

where k=1,..,K l=1,...,8

Column index

Mag

nitu

de

Figure 4.6: This figure depicts a three-dimensional view of a non-diagonal block of the matrix of trainingsequences RSS; this figure has to be compared with the figure above; it represents the intercorrelationproperties between two given users; we see that the maximum magnitude of the block entries underconsideration is less than 25 while the diagonal coefficients of the matrix depicted in figure 4.5 equal456, which means that RSS is "almost" diagonal

As a conclusion, assumptions (a), (b) and (c) are verified in the simulated context. We will still not takeadvantage of assumption (a). Indeed, one of the drawbacks of the strongest-paths method is that we haveto assume a certain number of significant paths. In the performance evaluation, we do not consider theschemes taking into account a priori information (a), since we want to avoid introducing a parameter tobe tuned (see (D3)). With this respect, the one-dimensional weighting based on the Wiener approachseems the most attractive method to us.


4.4.2 Comparison between the strongest-paths method and the one-dimensional weight-ing method based on the Wiener approach

4.4.2.1 Influence of the assumed number of significant paths

First of all, we consider the strongest-paths method and show its sensivity to the choice of the numberof significant paths (p). The BER target is 3%. The number of time slots used for channel coefficientsestimation is M = 11 and 4 active users per time slot are assumed. In this scenario, we plot the gainin terms of signal-to-noise ratio provided by the strongest-paths method as a function of the numberof selected channel coefficients. The best value of the number of significant paths is 3 for Indoor B(figure 4.7), 7 or 8 for Vehicular A (figure 4.8) and about 9 for Pedestrian B (figure 4.9). What isimportant to notice is that if this number is not properly chosen, it follows an important degradation ofthe performance. For instance, if p = 3, we obtain the optimum gain from the strongest-paths method inIndoor B whereas in Vehicular A or Pedetrian B the conventional training-based estimation works betterwith no post-processing, which correponds to negative gains (see figures 4.8 and 4.9).

3 4 5 6 7 8 9 10 11 120

0.5

1

1.5

2

2.5

3

3.5

4Influence of the number of selected coefficients in Indoor B

Number of selected coefficients

Gai

n on

Eb/

No

[dB

]

Maximum gain (Known channel)TS + Strongest paths method

Indoor BBER target = 3 %K = 4 active usersM = 11 time slotsNo channel codingv = 3 km/h

Figure 4.7: For a given BER target (3%): gain in terms of signal-to-noise ratio as a function of thenumber of selected channel coefficients (p) for the strongest-paths method; simulation conditions: IndoorB channel, MMSE block linear joint detection, 4 active users, M = 11 time slots are available, mobilestation speed = 3 km/h, no channel coding; the optimum number of paths to select is 3


3 4 5 6 7 8 9 10 11 12−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4


Gai

n on

Eb/

No

[dB

]

Influence of the number of selected coefficients in Vehicular A


Vehicular ABER target = 3%K = 4 active usersM = 11 time slotsNo channel codingv = 3 km/h

Figure 4.8: For a given BER target (3%): gain in terms of signal-to-noise ratio as a function of the numberof selected channel coefficients (p) for the strongest-paths method; simulation conditions: Vehicular Achannel, MMSE block linear joint detection, 4 active users, M = 11 time slots are available, mobilestation speed = 3 km/h, no channel coding; the optimum number of paths to select is about 7 or 8; if pis not properly chosen, say p = 3 the training-based estimation works better without the strongest-pathsmethod based selection

3 4 5 6 7 8 9 10 11 12−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4


Gai

n on

Eb/

No

[dB

]

Influence of the number of selected coefficients in Pedestrian B


Pedestrian BBER target = 3%K = 4 active usersM = 11 time slotsNo channel codingv = 3 km/h

Figure 4.9: For a given BER target (3%): gain in terms of signal-to-noise ratio as a function of the numberof selected channel coefficients (p) for the strongest-paths method; simulation conditions: Pedestrian Bchannel, MMSE block linear joint detection, 4 active users, M = 11 time slots are available, mobilestation speed = 3 km/h, no channel coding; the optimum number of paths to select is about 9; if p isnot properly chosen, say p = 3 the training-based estimation works better without the strongest-pathsmethod based selection


4.4.2.2 Comparison between the strongest-paths method and the one-dimensional weighting basedon the Wiener approach in typical situations

In figures 4.10, 4.11 and 4.12 we have plotted the MMSE-BL-JD performance with:

• the conventional training-based channel estimation;

• the training-based channel estimation equipped with the strongest-paths method for p = 3;

• the training-based channel estimation equipped with the strongest-paths method for p = 6;

• the training-based channel estimation followed by the one-dimensional weighting based on theWiener approach;

• and the perfect channel knowledge,

in Indoor B, Vehicular A and Pedestrian B respectively.

For the three figures, we assume K = 4 users, M = 61 time slots and that noise variance is known fromthe receiver. From these figures, several remarks are worth being made. Firstly, for typical choices of p (3and 6), we observe the lack of consistency of the estimator based on the paths selection. Secondly, atten-tion of the reader is drawn upon the fact that performances of the estimator based on paths selection arechannel-dependent. On the other hand, behavior of the hybrid weighting based on the Wiener approachis very regular and provides significant improvements over the conventional training-based estimationprocedure for all tested SNR and propagation channel. A minimum 2 dB gain (w.r.t the trained estima-tion performance) is obtained for Eb/No ranging from 5 dB to 15 dB and in three tested environments.

4.4.2.3 Influence of the number of active users

In figure 4.13 we consider the typical Vehicular A environment and there are M = 61 time slots availablefor improving the conventional channel estimation procedure. It represents the SNR gain provided bythe hybrid weighting for different numbers of users (up to 8). The top curve represents the maximumgain that would be achieved if the channels were known from the receiver. Corresponding simulationsimply show that channel knowledge is a more crucial point as multiple access interference increase andthat the training-based estimation equipped with the proposed hybrid procedure gives very good gainswhatever the number of users is. Typically, this gain reaches 75% of the available margin, which is veryremarkable.

4.4.2.4 Influence of the number of available time slots

In this subsection, we want to know whether a large number of time slots is needed in order to achievegood performance gain with the one-dimensional hybrid weighting. For the strongest-paths method, thecorresponding simulations are not included in this report because we saw that for a number of time slotsranging from 11 to 121 the performance gains were constant and this holds for the three propagationchannels under consideration in this chapter. As for the 1-D hybrid weighting procedure, the samekind of results are obtained as shown in figure 4.14. Beside performances provided by the conventional

5 6 7 8 9 10 11 12 13 14 1510

−3

10−2

10−1

100

Eb/No [dB]



Performance evaluation of MMSE−JD in Indoor B

Conventional TS TS + Strongest paths method (p=3)TS + Strongest paths method (p=6)TS + 1D hybrid weighting Known channel

Indoor BK = 4 active usersM = 61 time slotsKnown noise varianceNo channel codingv = 3 km/h

Figure 4.10: Bit error rates versus signal-to-noise ratio; the performance of the MMSE joint detectoris evaluated in Indoor B with 5 different channel estimates: the conventional training-based estimate,the training-based estimate followed by the strongest-paths selection with p = 3, the training-based es-timate followed by the strongest-paths selection with p = 6, the training-based estimate followed by theproposed one-dimensional hybrid weighting and the true channel; simulation conditions: 4 active users,M = 61 time slots are available, known noise variance, mobile station speed = 3 km/h; in any cases,the post-processing operation is beneficial to performance: the hard and the hybrid weighting operationsprovide a 2 dB gain for any Eb/No

trained (top curve) and the perfect channel knowledge (bottom curve) cases, it represents the MMSE-JD perfomances obtained with the 1-D hybrid weighting procedure for M = 11 and M = 121. Eventhough this estimator is more sensitive to the number of observations than the strongest-paths methods,the difference is still very small say between 0.1 to 0.2 dB. This is a very attractive feature because itmeans that the "continuous transmissions" assumption may not be that restrictive. Indeed, in one framethere are 15 time slots (subsection 1.2.2.2). If time slot allocation is symmetric, one frame comprises 7or 8 time slots. So, this means only two frames are needed to achieve the claimed gain, which is a verypositive point.

4.4.2.5 Influence of noise variance estimation accuracy

As we have so far assumed the noise variance to be knwon we question this assumption. Intentionally,we consider a bad case where there is only one user (K = 1) and eleven time slots (M = 11). For the noiseestimation purpose, use is made of the 6 weakest paths (Q = 6). Then we compare the performance of theMMSE joint detection with the hybrid weighting procedure when the noise variance is known and whenit is estimated with KMQ = 66 samples. As a result, the difference is negligible although it correspondsto a 10% error on the noise variance estimation as showed in table 4.2.

4.5 Conclusion 81

5 6 7 8 9 10 11 12 13 14 1510

−4

10−3

10−2

10−1

100

Eb/No [dB]



Performance evaluation of MMSE−JD in Vehicular A


Vehicular AK = 4 active usersM = 61 time slotsKnown noise varianceNo channel codingv = 3 km/h

Figure 4.11: Bit error rates versus signal-to-noise ratio; the performance of the MMSE joint detector isevaluated in Vehicular A with 5 different channel estimates: the conventional training-based estimate,the training-based estimate followed by the strongest-paths selection with p = 3, the training-based es-timate followed by the strongest-paths selection with p = 6, the training-based estimate followed by theproposed one-dimensional hybrid weighting and the true channel; simulation conditions: 4 active users,M = 61 time slots are available, known noise variance, mobile station speed = 3 km/h; the Vehicular Apropagation environment allows us to highlight one of the major defaults of the strongest-paths method,which is its consistency; behavior of the one-dimensional hybrid weighting method is very regular forany tested SNR

K 1 2 4 4 4M 11 11 11 21 61Q 6 6 6 6 6

Number of samples(KMQ =) 66 132 264 504 1664| σ−σ

σ |×100 10% 7% 5% 3.5% 2%

Table 4.2: Noise variance estimation accuracy as a function of the number of samples

4.5 Conclusion

In this chapter, we proposed both efficient and implementable schemes for removing noise from thetraining-based channel estimates. This is based on the continuous transmission assumption.

In particular, we showed how each estimated coefficient can be efficiently weighted by reliabilities basedon the Wiener approach, which is called one-dimensional hybrid weighting procedure. Using this post-processing after the trained procedure allows for SNR gains ranging from 2 dB to 3 dB; these gainscorrespond to 75% of the gain margin between the trained estimation case and the case where the channelis known from the receiver. On the contrary to the strongest-paths method, these gains are available for alarge range of Eb/No (at least from 5 dB to 15 dB) and different test propagation environments (indoor,pedestrian and vehicular environments). Applying this post-processing technique allows us to (at leastin part) eliminate the major drawbacks of the paths selection method (channel statistics are not fullyexploited; lack of consistency; noise is not removed from selected coefficients).

5 6 7 8 9 10 11 12 13 14 1510

−4

10−3

10−2

10−1

100

Eb/No [dB]



Performance evaluation of MMSE−JD in Pedestrian B


Pedestrian BK = 4 active usersM = 61 time slotsKnown noise varianceNo channel codingv = 3 km/h

Figure 4.12: Bit error rates versus signal-to-noise ratio; the performance of the MMSE joint detector isevaluated in Pedestrian B with 5 different channel estimates: the conventional training-based estimate,the training-based estimate followed by the strongest-paths selection with p = 3, the training-based es-timate followed by the strongest-paths selection with p = 6, the training-based estimate followed by theproposed one-dimensional hybrid weighting and the true channel; simulation conditions: 4 active users,M = 61 time slots are available, known noise variance, mobile station speed = 3 km/h; the observationdone for Vehicular A is more apparent; strongest-paths method behavior is related both to the SNR andthe number of selected paths whereas the hybrid weighting is "always" efficient

2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Influence of MAI on Eb/No gain

Number of active users

Gai

n on

Eb/

No

[dB

]

Known channel TS + 1D hybrid weighting

Vehicular ABER target = 3%MMSE −BL−JDM = 61 time slotsNo channel codingv = 3 km/h

80%

77%

76%

72%

Figure 4.13: For a given BER target (3%): gain in terms of signal-to-noise ratio versus number of activeusers; simulation conditions: see figure 4.11; the top curve is the maximum gain that would be achievedif the channel was known; the bottom curve represents the gain provided by the one-dimensional hybridweighting; the relative gain is quite stable with regard to the number of users

Regarding implementability issues, the 1-D hybrid weighting procedure is a very good candidate forimplementation because it is not computationally demanding (scalar operations) and a simple way of

4.5 Conclusion 83

5 6 7 8 9 10 11 12 13

10−2

10−1

Eb/No [dB]



Influence of the number of time slots

Conventional TS TS + 1D hybrid weighting (M = 11) TS + 1D hybrid weighting (M = 121)Known channel

Vehicular AK = 4 active usersMMSE−BL−JDKnown noise varianceNo channel codingv = 3 km/h

Figure 4.14: Bit error rates versus signal-to-noise ratio; simulation conditions are the same as in figure4.11 except for the number of time slots; the top curve is the performance of the trained estimationwhile the bottom curve corresponds to the case where the channel is assumed to be known from thereceiver; the two other curves are the performance of the hybrid weighting with M = 11 and M = 121; asa conclusion, we do not need a large number of time slots to take profit of the proposed channel estimatepost-processing

5 6 7 8 9 10 11 12 13

10−2

10−1

Eb/No [dB]



Performance evaluation of MMSE−JD in Vehicular A

Conventional TS TS + 1D hybrid weighting + Estimated noise varianceTS + 1D hybrid weighting + Known noise variance Known channel

Vehicular A K = 1 active userM = 11 time slotsNoise estimation based onthe 6 weakest pathsNo channel codingv = 3 km/h

Figure 4.15: Bit error rates versus signal-to-noise ratio; the influence of estimation noise variance accu-racy on performance is studied; simulation conditions: Vehicular A channel, 1 active user, M = 11 timeslots are available, noise estimation is based on the 6 weakest paths, mobile station speed = 3 km/h, nochannel coding; the difference between the curves corresponding to the case where the noise variance isassumed to be known and the case where it is estimated by using MQ = 66 samples is negligible

estimating noise variance is available. This point is very important because the "brute force" approachwould consist in inverting the matrix Rg, which is very computationally demanding. Indeed, in theUMTS-TDD uplink, if no additional assumptions are made, we have to invert a 456×456 for the 8-user


case with type 1 time slots (assumed channel length is 57 chips).

Additionnally, we saw that, in typical situations, using a ten of UMTS-TDD time slots suffices to achievesignificant improvements. This means that the "continuous transmissions" assumption is not that restric-tive.

As for the low-rank approximation, we could not exploit it. It would be particularly useful in situationswhere propagation channels are really correlated, which was not the case in our study.

Chapter 5

Implementable receiver structures

5.1 Uplink and downlink cases

The aim of this chapter is twofold. We want to propose implementable receiver structures both foruplink and downlink. For uplink, we merely summarize the results obtained so far, which is done in thisintroduction. Except for this section (5.1), this chapter is devoted to the downlink case.

In uplink, joint detection is thought to be the most complex detection scheme that a UMTS-TDD basestation can support (3GPP recommendations). To date, the best strategy would consist in trading aslight performance degradation for a considerable complexity reduction by exploiting the structure ofthe equalization matrices (1.21, 1.22). For instance, [?] and [?] exploit an approximate Cholesky tech-nique, [?] exploits the asymptotical equivalence between finite-order Toeplitz and circulant matricesand [?] proposes a fast joint detection with cyclic reduction exploiting displacement structures. In anycase, performance of the non-approximated joint detection is considered as the most efficient scheme.This is why we do not envisage more complex detection solutions in this study. With regard to channelestimation, semi-blind estimators based on second-order blind techniques are also too complex to be im-plemented in UMTS-TDD systems [?] [?]. For typical situations such as those described in chapter 3, thechannel estimation complexity is increased by 10 to 20 [?], [?] , which is a high cost to be paid regardingthe performance gain (1 dB typically). But if several time slots are available for the channel estimationpurpose, which corresponds to the continuous transmission case (e.g. the speech service), a very goodcandidate for implementation is the one-dimensional hybrid weighting-based trained estimation (section4.3.4.3). As a consequence, a receiver based on an approximated joint detection fed with trained chan-nel estimates based on an efficient "de-noising" seems to be a viable solution for a UMTS-TDD basestation operating in the uplink, which has been studied in chapter 4.

In downlink, one would like to avoid implementing joint detection in UMTS-TDD user equipments. Inorder to achieve satisfying user equipment reception performance, the idea is to share the equalizationtask between the mobile station and the base station. This consists in performing a pre-equalization stepat the base station side. Assuming a pre-equalization strategy, we want to know how necessary it is toimplement a multiuser scheme in a UMTS-TDD user equipment. To answer this question, we comparethe performance of MMSE joint detection with those of the matched filter-based detection in a multipleantenna context [?]. As for channel estimation, no specific algorithm is derived.

First, we envisage two pre-equalization schemes namely the joint pre-distortion and Transmit Adaptive

86 Implementable receiver structures

Antennas schemes (TxAA). We briefly justify why joint-predistortion does not seem a good strategy tobe selected. Next, we review the TxAA principle applied in the UMTS-TDD mode. In the simulationsection, performances of the MF and MMSE-BL-JD receivers are evaluated when TxAA is implementedwith 2, 4 or 8 antennas.

This is done in nearly flat fading and high frequency selective propagation channels. From simulationresults, we conclude about the possibility of reducing the complexity of conventional mobile stationreceivers through the implementation of a single-user detection stage (matched filter).

5.2 Downlink solutions

5.2.1 The two a priori candidates

The detection problem is that we do not directly have access to the transmitted data because they havebeen filtered by the channel and corrupted by noise. To compensate for the effect of the channel on datawe perform an equalization step, which roughly consists in inverting the filtering matrix (1.21), (1.22).Computational cost of user equipment joint detection essentially comes from the inversion of the matricesassociated with equations (1.21) or (1.22). One simple and quite natural idea is to perform this inversionat the base station side. This is the Joint pre-distortion principle. This solution has originally beenpresented by Bosch to 3GPP TSG-RAN Working Group 1 [?]. Like joint detection, joint pre-distortionis performed at the symbol rate. This is only performed on the two transmitted data parts of each UMTS-TDD time slots and not on the midamble. Additionally, perfect channel compensation is assumed, whichmeans that pre-equalized downlink channels are considered as single-path channels so that downlinkchannels are not estimated. This is an excellent strategy in static propagation environments. However, intime-varying propagation environments, it becomes more questionable since the base station is alone forcompensating downlink channel degradation from uplink channel estimates. In this respect, [?] showsthat joint pre-distortion performance is very sensitive to MS velocity. This is why in this chapter wefocus on a more simple BS pre-processing method, which is Transmit Adaptive Antennas (TxAA).

5.2.2 Transmit adaptive antennas

5.2.2.1 The TxAA idea

TxAA is a transmit spatial diversity technique, which means that several antennas are used in the basestation for the downlink. The idea is to transmit the same data from several antennas (figure 5.1) insteadof one as it is done in most of second generation cellular systems. As for UMTS systems, TxAA wasoriginally specified for the FDD mode and then adapted to the UMTS-TDD mode [?], [?]. Let q be thenumber of antennas of the base station. The TxAA algorithm consists in calculating the scalar weights(w1, . . . ,wq) to be applied to each antenna in order to maximize the power received by the mobile station(figure 5.1). In order to compute these weights, use of the downlink channels is made.

5.2 Downlink solutions 87

Figure 5.1: TxAA transmission; there are q antennas at the base station; the data sequences of the Kactive users are transmitted with the q antennas; each antenna is weighted by wi with i = 1, . . . ,q; thedata transmitted from the antenna #i experience the equivalent channel hi(z); the antenna weights arecalculated in order to maximize the useful power of the received signal

5.2.2.2 Weights calculation

When a single antenna is assumed at the base station, the chip-rate version of the signal received by themobile station of user k is expressed by:

yk(n) =j=K

∑j=1

[gk(z)]s j(n)+ v(n) (5.1)

where gk(z) represents the physical channel (including the emission, the propagation and the receptionfilters) of user k and s j(.) is the spread data sequence of user j. When TxAA is used, equation (5.1)becomes:

yk(n) =

(i=q

∑i=1

j=K

∑j=1

w(i)j [g(i)

k (z)]s j(n)

)+ v(n). (5.2)

where g(i)k (z) is the z-transform of the discrete-time impulse response of the physical channel for antenna

#i and user #k. In order to introduce the unknown weighting vector wk = (w(1)k , . . . ,w(q)

k )T for user k, itis convenient to rewrite equation (5.2) as:

yk(n) = (sk(n) . . . sk(n− `))[

g(1)k . . .g(q)

k

]

︸︷︷︸Gk

wk + ∑j 6=k

(s j(n) . . . s j(n− `))[

g(1)j . . .g(q)

j

]

︸︷︷︸G j

+v(n) (5.3)

where for all (k, i) ∈ [1,K]× [1,q], g(i)k is the vector impulse response of the overall channel for antenna

#i and user #k.


Now, let us define the useful received power as:

Pyk = E[yk(n)y∗k(n)] = wHk GH

k Gkwk. (5.4)

This is the power that would be received by user k in the situation where:

• the noise would be negligible,

• the MAI would be negligible and

• the spread data sequence (chip-rate) is white.

The TxAA algorithm is equivalent to the maximization of the useful received power defined by equation(5.4). In fact, the useful power Pyk coincides with the power of the signal in the output of an ideal(noiseless case) max ratio Rake receiver [?] for user k.

Additionally, we assume an unitary norm constraint on the weights vector, which is mandatory to keepthe transmitted power constant (constant power budget constraint). Finally, the optimum weights vectorΩk for user k is the solution of the following Lagrangian maximization (λ is the Lagrange multiplier):

Ωk = argmaxwk

[wHk GH

k Gkwk +λ(wHk wk−1)] (5.5)

that is to say that Ωk is the eigenvector associated with the largest eigenvalue of the matrix GHk Gk.

As mentioned above, calculating the weights vector from equation (5.5) requires the downlink channelsknowledge. For the UMTS-TDD mode, the uplink and downlink channel reciprocity (−P6−) is generallyassumed. This is justified by the fact that duplex division is made in time, which means that uplink anddownlink frequency bands coincide. Under this assumption, the most simple strategy is to use the uplinkchannel estimate to estimate the matrix GH

k Gk.

5.3 Simulations

5.3.1 Simulation conditions

Simulations assumptions detailed in this section correspond to a typical uncoded speech service. Down-link transmissions from the base station to the mobile station are considered. There is 1 receiving antennaat the mobile station and 2, 4 or 8 antennas are assumed at the base station. There are K = 8 active usersper time slot, each of them using a single channelization code with a spreading factor of N = 16.

Two propagation environments are considered: the Indoor A, which is nearly a flat fading channel, andthe Pedestrian B, which is characterized by a high frequency selectivity. For both channels, the mobilespeed is fixed at 3 km/h, which corresponds to a pedestrian speed. In uplink, channel estimations areperformed at a fixed signal-to-noise ratio (Eb/No = 6 dB). In downlink, channels are normally estimated,which means that estimation accuracy depends on the noise level.

A multi-switching points frame structure is assumed (symmetrical traffic assumption) where an uplinktime slot is immediately followed by a downlink time slot which is followed by an other downlink time

5.3 Simulations 89

slot and so on (figure 5.3.1). This means that the time difference between uplink channel estimationsused for downlink transmissions is 666.67 µs, that is the best configuration for using closed-loop transmitdiversity schemes such as TxAA.

There is no correlation between signals in downlink. Spacing between two adjacent antennas (sensors)is assumed to be large enough to ensure this non-correlation property.

666.67µs

10ms

Figure 5.2: Symmetrical allocation: the traffic between uplink and downlink is alternated


5 6 7 8 9 10 11 12 13 14 1510

−3

10−2

10−1

100

Eb/N

o [dB]



Comparison between MF and MMSE−JD detections

Matched Filter detection + Training−based estimationMMSE−JD + Training−based estimation

Indoor AK = 8 active usersq = 2 antennasMS speed = 3 km/hUL channel estimation @ Eb/No=6dB

Figure 5.3: Bit error rates versus signal-to-noise ratio; performances of the matched filter and the MMSEjoint detection are compared in Indoor A; simulation conditions: the channel estimation strategy is fixed(training-based estimation), 8 active users, 2 transmit antennas, mobile speed = 3 km/h, uplink channelestimates are performed at Eb/No = 6 dB, symmetrical allocation, no channel coding; the joint detectioncapability is not fully exploited over Indoor A-like channels because they are nearly flat fading channels;as a consequence, with 2 antennas, the joint detection provides a 0.8 dB gain at BER = 2% and 1.5 dBat BER = 1%

5 6 7 8 9 10 11 12 13 14 1510

−3

10−2

10−1

100

Eb/N

o [dB]





Pedestrian BK = 8 active usersq = 2 antennasMS speed = 3 km/hUl channel estimation @ Eb/No=6dB

Figure 5.4: Bit error rates versus signal-to-noise ratio; performances of the matched filter and the MMSEjoint detection are compared in Pedestrian B; simulation conditions are the same as for figure 5.3; inhigh-selective channels such as Pedestrian B, the use of joint detection is mandatory; the matched filterdetection performance reaches an error floor; for a 3% BER target, more than 4 dB are lost because ofthe use of the matched filter

2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

Eb/N

o [dB]





Indoor AK = 8 active usersq = 4 antennasMS speed = 3 km/hUL channel estimation @ Eb/No=6db

Figure 5.5: Bit error rates versus signal-to-noise ratio; performances of the matched filter and the MMSEjoint detection are compared in Indoor A; simulation conditions: the channel estimation strategy is fixed(training-based estimation), 8 active users, 4 transmit antennas, mobile speed = 3 km/h, uplink channelestimates are performed at Eb/No = 6 dB, symmetrical allocation, no channel coding; observationsfrom figure 5.3 are confirmed with 4 transmit antennas; at BER = 1% the joint detection outperforms thematched filter by only 0.3 dB

5.3 Simulations 91

2 3 4 5 6 7 8 9 1010

−2

10−1

100

Eb/N

o [dB]





Pedestrian BK = 8 active usersq = 4 antennasMS speed = 3 km/hUL channel estimation @ Eb/No=6dB

Figure 5.6: Bit error rates versus signal-to-noise ratio; performances of the matched filter and the MMSEjoint detection are compared in Pedestrian B; simulation conditions are the same as for figure 5.5; evenwith 4 transmit antennas, the joint detection is justified in high frequency selective propagation environ-ment

0 1 2 3 4 5 610

−3

10−2

10−1

Eb/N

o [dB]





Indoor AK = 8 active usersq = 8 antennasMS speed = 3 km/hUL channel estimation @ Eb/No=6dB

Figure 5.7: Bit error rates versus signal-to-noise ratio; performances of the matched filter and the MMSEjoint detection are compared in Indoor A; simulation conditions: the channel estimation strategy is fixed(training-based estimation), 8 active users, 8 transmit antennas, mobile speed = 3 km/h, uplink channelestimates are performed at Eb/No = 6 dB, symmetrical allocation, no channel coding; the curves corre-sponding to the matched filter and the MMSE joint detector are very close; in indoor environment, when8 transmit antennas are used, joint detection is not worth being implemented; notice also that the 1%BER target is achieved for Eb/No = 3 dB, which is due to the diversity gain

0 1 2 3 4 5 610

−2

10−1

100

Eb/N

o [dB]





Pedestrian BK = 8 active usersq = 8 antennasMS speed = 3 km/hUL channel estimation @ Eb/No=6dB

Figure 5.8: Bit error rates versus signal-to-noise ratio; performances of the matched filter and the MMSEjoint detection are compared in Pedestrian B; simulation conditions: the channel estimation strategy isfixed (training-based estimation), 8 active users, 8 transmit antennas, mobile speed = 3 km/h, uplinkchannel estimates are performed at Eb/No = 6 dB, symmetrical allocation, no channel coding; when 8antennas are used at the base station, the use of joint detection schemes in the mobile station is question-able; this means that the detection stage complexity can be significantly reduced (by using a matchedfilter based detection)

2 3 4 5 6 7 80

1

2

3

4

5

6

Number of antennas

Gai

n on

Eb/

No

[dB

]

Comparison between MF and MMSE−JD (with TxAA)

Difference between MF and MMSE−JD in Pedestrian BDifference between MF and MMSE−JD in Indoor A

Target BER = 2%K = 8 active usersUL channel estimation @ 6dB

Figure 5.9: For a given BER target (2%) joint detection is compared with the matched filter detection ina nearly flat fading channel and a high frequency selective channel; 2, 4 and 8 antennas are used

5.3.3 Interpretation

From the provided simulation results, we see that when a 8-sensor antenna is used the MF detectionperformance is close to that of the MMSE detection even in frequency selective channels such as the

5.3 Simulations 93

Pedestrian B channel. This is one of the reasons why we considered TxAA as a "pre-equalization"scheme (section 5.1). But this can also justified by theoretical considerations. In this subsection wetheoretically justify the fact that if the number of antennas increases, the equivalent channel for user knamely g

k, [g(1)

k . . .g(q)k ]Ωk becomes "more flat" than non pre-equalized channels g(1)

k , . . . ,g(q)k .

Let us consider the channel gk= [g(1)

k . . .g(q)k ]wk = GkΩk. First, it is important to notice that g

kis also

the eigenvector associated with the largest eigenvalue of the matrix GkGHk since

GHk GkΩk = λmaxΩk ⇒GkGH

k (GkΩk) = λmax(GkΩk). (5.6)

Conversely, if Ok is the eigenvector associated with the largest eigenvalue of the matrix GkGHk , then

GHk Ok is the eigenvector associated with the largest eigenvalue of the matrix GH

k Gk. It turns out that:

GkGHk =

∑i=qi=1 g(i)

k (0)g(i),∗k (0) . . . ∑i=q

i=1 g(i)k (0)g(i),∗

k (`)...

∑i=qi=1 g(i)

k (`)g(i),∗k (0) . . . ∑i=q

i=1 g(i)k (`)g(i),∗

k (`)

. (5.7)

As channel coefficients are assumed to be decorrelated, E[gk(m)g∗k(n)] = 0 for m 6= n. Therefore thematrix GkGH

k becomes diagonal as the number of antennas increases. If q tends towards the infinity, thevector Ok is a vector of the canonical basis. In the latter case, the way how TxAA works is particularilyeasy to understand. Indeed, if q → ∞, Ok = up = (0 . . .010 . . .0)T where the one is on the position

p. Hence, Ωk = GHk up = (g(1),∗

k (p) . . .g(q),∗k (p))T . In summary, the TxAA algorithm first selects the

delay for which the estimated energy (average is made over the number of antennas) and then weightscoherently the selected paths. This means that if there is a time window in which the most significantcoefficients of the channels g(1)

k , . . . ,g(q)k are confined, the power of the equivalent channel g

kwill also

be concentrated in this window. If the window is narrow, the equivalent channel tends to become flat inthe frequency domain. This is illustrated in figure 5.10 where 8 antennas are assumed. For a given timeslot, the 8 first waveforms represent the impulse responses for each antenna and the last one representsthe equivalent channel.


0 20 400

0.1

0.2

0.3

0.4

0 20 400

0.2

0.4

0.6

0 20 400

0.2

0.4

0.6

0.8

0 20 400

0.1

0.2

0.3

0.4

0 20 400

0.2

0.4

0.6

0.8

0 20 400

0.2

0.4

0.6

0 20 400

0.2

0.4

0.6

0.8

0 20 400

0.2

0.4

0.6

0 20 400

0.5

1

1.5

Figure 5.10: For a given time slot, the impulse responses of the physical channels of 8 antennas arerepresented. The last waveform (bottom right) shows the equivalent physical channel provided by theTxAA algorithm. The latter channel tends to become flat as the number of antennas increses

5.4 Conclusion

In this chapter we selected the TxAA algorithm as transmit diversity scheme. The TxAA algorithmperformance has been evaluated in Indoor A channel (nearly flat fading) and in Pedestrian B channel(high frequency selectivity) with 2, 4 and 8 transmit antennas.

As a result, if the mobile station is in an indoor propagation environment, even with only 2 antennas, theuse of joint detection in the user equipment is not justified. However, if it is in a high frequency selectivepropagation environment, joint detection is mandatory unless 8 or more antennas are used at the basestation.

This means that the detection stage complexity of the mobile station can be reduced if the number ofantennas is increased at the base station or if UMTS-TDD mobile stations avoid severe propagationenvironments, which seems to be unthinkable.

Conclusion

Summary

In this thesis, two problems have been investigated: the channel estimation and the symbol detection. Thesystems under consideration are the TD-CDMA systems and more particularly the UMTS-TDD modebased systems. The main aim of this study was to improve the UMTS-TDD receiver performance andpropose implementable receiver structures both for uplink and downlink. In this study, we focused moreon the channel estimation problem. Let us recall the two reasons motivating this choice:

• first, the channel estimation issue was more open than the symbol detection one because of thetraining sequences structure and the limited expectations from UMTS-TDD systems. The UMTS-TDD training sequences have been designed in order to minimize the inversion complexity of thetraining sequences matrix S. But this is possible only if the channel length is assumed to be 57chips and 64 chips for type 1 and type 2 time slots respectively. As a consequence, channel lengthsare "too overestimated". The implication is an important performance degradation of the symboldetector. Typically, the flawed channel knowledge is responsible for a 3 to 4 dB degradationof the signal-to-noise ratio. This means that there is a huge gap to bridge. Next, UMTS-TDDsystems should operate with no more than K = 8 active users per time slot while the correspondingspreading factor is N = 16. In [?], M. Honig and M. Tsatsanis show that for low user loads, sayK/N ≤ 1/2, the spectral efficiency (bits/chip) of the MMSE and the MMSE-DFE are close to themaximum spectral efficiency, which would be achieed in the ideal case where the orthogonalitywould be recovered;

• second, in chapter 2, we separated the influence of channel estimation accuracy from that of symboldetection reliability. It turns out that the performances of the MMSE and MAP joint detectors arevery close (at least for K/N = 0.125). On the other hand, significant gains were obtained when thechannel was assumed known from the receiver.

In order to improve channel estimation accuracy, two directions have been envisaged. First, in chapters2 and 3 we propose exploiting the observations generated by the whole time slot while the conventionalchannel estimation procedure uses only the observations generated by the midamble (training sequence).One of the advantages of this approach, which is called the semi-blind approach, is that no assumption ismade on the nature of the transmissions. This means that it works for sporadic time slot transmissions.Second, in chapter 4, we assume continuous time slot transmissions. Under this assumption, severaltime slots can be used in order to enhance the current channel estimate. Additionally, this leads to lowcomplexity solutions.

96 Conclusion

In chapter 2, we studied the optimum semi-blind receiver in order to obtain performance lower bounds.The study was conducted for 2 active users because the expectation maximization algorithm complexityis exponential in the number of users and the ISI duration. Moreover, impact of the flawed channel lengthknowledge on the receivers performance has been emphasized, which is paricularly strong in the UMTS-TDD mode; typically there is a 2 dB loss in terms of signal-to-noise ration because of overmodelling.

In chapter 3 suboptimum semi-blind schemes were considered. The proposed schemes involve compositecriteria in which certain regularizing constants have to be tuned. Indeed, the problem is to know how toweight the blind cost function with respect to the training-based cost function. It turns out that, underasymptotic assumptions, it is possible to tune properly the balance between the blind and the trained part.Many simulations performed in the UMTS-TDD mode context show that the proposed tuning is efficienteven though time slot and midamble sizes are not designed for semi-blind channel estimation schemes.We also showed how to deal with the underlying multidimensional optimization problem; indeed thereare K (K is the number of active users per time slot) regularizing constants to be tuned in the uplink.Under reasonable assumption on the matrix S, we saw how this problem boils down to K one-dimensionalminimization problems. In typical situations, the semi-blind approaches outperformed the training-basedapproach by 1 dB. It is important to mention that these results were achieved in a overmodelling context.Moreover, several blind techniques were envisaged: the subspace method, the linear prediction anda weighted linear prediction. In a blind context where channel lengths are overestimated, the linearprediction approaches are more efficient than the subspace method. However, in semi-blind scenarios,behaviors are different: the semi-blind was the most efficient one in all the configurations we tested(different signal-to-noise ratios, different midamble sizes and different number of blind observations).We also presented a weighted linear prediction approach that was more efficient than the conventionallinear prediction approach both from a statistical and pratical standpoint. We also saw that the semi-blindestimators capability is fully exploited when midamble lengths are short. Significant improvements interms of data and bit error rates were achieved (data throughputs increased by 15%, bit error rates dividedby 10). The main drawback of these methods stems from their computational complexity (multipliedby 10 to 20 typically). For current UMTS-TDD systems, such channel estimation schemes are notenvisageable.

In order to propose implementable channel estimators, we consider a more restrictive framework, whichwere the continuous transmissions (chapter 4). In this case, we saw that it is possible to remove in partestimation noise from the training-based channel estimates. We propose a simple and efficient way ofindividually weighting the estimated channel coefficients. The proposed method avoids the pitfalls ofthe strongest-paths method. As a consequence, it works for many propagation environments (indoor,vehicular, pedestrian) and signal-to-noise ratio conditions. The main simulation result is that 75% of theSNR gap between the trained case and the case where the channel is assumed to be known is bridged(for a MMSE joint detection).

So far, the study was devoted to the uplink case but is also applicable to the downlink case. In chapter5, we focus on the downlink case. We show through simulation results that joint detection is not manda-tory in the mobile station if transmit adptative antennas (multiple antennas) are used in the base station.Indeed, if an 8-sensor base station antenna is used, even in high frequency selective propagation environ-ments, the matched filter detection performance is very close to the MMSE joint detection performance.We also justified why TxAA can be viewed as a pre-equalization scheme.

Based on these results, we propose several receiver structures which are indicated in table 5.1.

97

The proposed receiver Channel estimation Symbolstuctures Sporadic transmission Continuous transmission detection

UL Limited Training Training + Joint detectioncomplexity 1D-Wiener approach

No complexity Semi-blind subspace Semi-blind subspace + Joint detectionlimitation Wiener approach

DL Nb of antennas < 8 Training Training + Joint detection1D-Wiener approach

Nb of antennas ≥ 8 Training Training + Matched filter1D-Wiener approach

Table 5.1: The proposed receiver structures based on the results from this study

Perspectives

From chapter 3

With regard to semi-blind channel estimation techniques applied to the UMTS-TDD mode, an importantissue remains to be addressed. Indeed, we studied the case where the spreading factor is fixed. Thiscase corresponds to the voice service. But for data services more blind observations are available for theestimation of the covariance matrix of the observations RM . More performance evaluations are neededto know more about the potential of semi-blind techniques in realistic frameworks.

As for the projected approach, an efficient way of tuning the assumed dim(null(Q)) has to be proposed.

From chapters 3 and 4: semi-blind estimation + multislot

An other extension of the semi-blind schemes would be the continuous time slot transmissions. The ideais to combine the semi-blind criteria derived in chapter 3 with the Wiener cost function derived in chapter4. This should allow us to both remove the noisy channel coefficients and refine the selected coefficients.

From chapters 3 and 5: semi-blind estimation + multiple antennas

In this study, we considered the semi-blind approach in a single antenna context (one antenna at the basestation and one antenna at the mobile station). However, more and more people are considering MultipleTransmit Multiple Receive systems (MTMR). In this case, the terminal is not necesarily a mobile stationbut it can be a laptop or other devices. This should be in favor of blind methods and therefore semi-blindmethods. First, the identifiability issue needs to be addressed. Second, results from chapter 3 may beextended.

98 Conclusion

From chapters 4 and 5: adaptative antennas + multislot

In TxAA, the weights calculation is based only on the uplink channel estimates from the last uplink timeslot. However, assuming continuous transmissions, it could be beneficial to make use of several timeslots to refine the TxAA weights or to implement a more efficient pre-equalization structure.

From chapter 5

At the beginning of chapter 5, the joint-predistortion scheme was mentioned. The problem of how toshare the equalization task between the mobile and the base stations remains an open issue. A jointoptimization of the transmitter and the receiver could be a viable solution to find the desired tradeoffbetween performance and complexity.

Appendix A

Multiple access methods: an analogy(Motorola SPS)

To illustrate the conceptual differences among the multiple access technologies, an analogy will be ap-plied. Picture a large room and a number of people, in pairs, who would like to hold conversations. Thepeople in each pair only want to talk and listen to each other, and have no interest in what is being saidby the other pairs. In order for these conversations to take place, however, it is necessary to define theenvironment for each conversation.

First, let us apply this analogy to an FDMA system. An FDMA environment would be simulated bybuilding walls in the single large room, creating a larger number of small rooms. A single pair of peoplewould enter each small room and hold their conversation. When that conversation is complete, the pairof people would leave and another pair would be able to enter that small room.

In a TDMA environment, each of these small rooms would be able to accommodate multiple conversa-tions "simultaneously." For example, with a 3-slot system such as IS-54, each "room" would contain upto 3 pairs of people, with each pair taking turns talking. Each pair has the right to speak for 20 secondsduring each minute. Even if there are fewer than three pairs in the small room, each pair is still limitedto its 20 seconds per minute.

Now, for CDMA, get rid of all of the little rooms. Pairs of people will enter the single large room.However, if every pair uses a different language, they can all use the air in the room as a carrier for theirvoices and experience little interference from the other pairs. The analogy here is that the air in the roomis a wideband "carrier" and the languages are represented by the "codes" assigned by the CDMA system.In addition, language "filters" are incorporated, people speaking German will hear virtually nothing fromthose speaking Spanish, etc. We can continue to add pairs, each speaking a unique language (as definedby the unique code) until the overall "background noise" (interference from other users) makes it toodifficult for some of the people to understand the other in their pair (frame erasure rates get too high). Bycontrolling the voice volume (signal strength) of all users to no more than necessary, we maximize thenumber of conversations which can take place in the room (maximize the number of users per carrier).Therefore, the maximum number of users, or effective traffic channels, per carrier depends on the amountof activity that is going on in each channel, and is therefore not precise. It is a "soft overload" conceptwhere an additional user (or conversation, in our analogy) can usually be accommodated if necessary, atthe "cost" of a bit more interference to the other users.

100 Multiple access methods: an analogy (Motorola SPS)

Appendix B

Review of polyphase components

For a FIR, IIR, causal or noncausal filter, it is possible to decompose its transfer function as [?] :

h(z) =p=P−1

∑p=0

z−p h(p)(zP) (B.1)

whereh(p)(z) = ∑

ih(iP+ p) z−i (B.2)

and P is the order of the decomposition.

We note that the inverse z-transform of the pth component is:

ZT−1[h(p)(z)] = h(iP+ p) (B.3)

where i is the index of the series (h)i and p the order of the component can be considered as a parameterof that series.

102 Review of polyphase components

Appendix C

Properties of the EM algorithm

The likelihood function does not decrease

Let x, θ, θ(p) and y be the hidden data, the unknown parameter, the estimate of the unknown parameter atiteration pth and the observations respectively. x, which is a random sequence, belongs to a finite discretealphabet of symbols. θ is independent of the hidden data.

We want to show that the likelihood function P(y/θ) does not decrease as the number of iterationsincreases. The auxiliary function Q is defined by:

Q(θ;θ(p)) , E[logP(x,y/θ)/ y,θ(p)]. (C.1)

In order to introduce the log-likelihood logP(y/θ), we use the Bayes’s formula. As the symbols alphabetis finite and θ being independent of x, we have :

Q(θ;θ(p)) = ∑x

P(x/y,θ(p)) log[P(x/y,θ) P(y/θ)]. (C.2)

To obtain the desired inequality, we introduce the Kullback divergence

K(P(x/y,θ(p)) ‖ P(x/y,θ)) = ∑x

P(x/y,θ(p)) logP(x/y,θ(p))

P(x/y,θ). (C.3)

Hence,

Q(θ;θ(p)) = ∑x

P(x/y,θ(p)) logP(x/y,θ)+ logP(y/θ)

= ∑x

P(x/y,θ(p)) logP(x/y,θ)

P(x/y,θ(p))+∑

xP(x/y,θ(p)) logP(x/y,θ(p))

+logP(y/θ),

θ(p) being defined by the maximization-step θ(p+1) = argmaxθ Q(θ;θ(p)) , we have:

∀ θ, Q(θ(p+1);θ(p)) > Q(θ;θ(p)). (C.4)

104 Properties of the EM algorithm

For θ = θ(p)

Q(θ(p+1);θ(p)) > Q(θ(p);θ(p)). (C.5)

The implication is that

logP(y/θ(p+1))− logP(y/θ(p)) > K(P(x/y,θ(p+1)) ‖ P(x/y,θ(p))) > 0. (C.6)

Positiveness of the Kullback divergence shows that the likelihood increases after each iteration of thealgorithm.

Stable points and stationary points

We want to show that if the derivative of the auxiliary function vanishes in a point then the log-likelihoodalso vanishes in this point. For this purpose, we just need to expand ∂Q(θ,θ(p))

∂θ as follows:[

∂Q(θ,θ(p))∂θ

]

θ=θ(p)

=

[∂E[logP(x/y,θ) /θ(p),y]

∂θ

]

θ=θ(p)

+[

∂logP(y/θ)θ

]

θ=θ(p)

=[

∂θ ∑

xP(x/y,θ(p))logP(x/y,θ)

]

θ=θ(p)

+[

∂logP(y/θ)θ

]

θ=θ(p)

(∗)=

[∑x

P(x/y,θ(p))∂P(x/y,θ)

∂θ1

P(x/y,θ)

]

θ=θ(p)

+[

∂logP(y/θ)θ

]

θ=θ(p)

=[∑x

∂P(x/y,θ)∂θ

]

θ=θ(p)

+[

∂logP(y/θ)θ

]

θ=θ(p)

=

∂∂θ ∑

xP(x/y,θ)

︸︷︷︸1

θ=θ(p)

+[

∂logP(y/θ)θ

]

θ=θ(p).

Note that in the equality (*), we assumed P(x/y,θ(p)) regular enough in order to enable switching be-tween sum and derivation operators.

At last, we see that every stable points of the auxiliary function is a stationary point for the log-likelihood.

Appendix D

Convergence in probability

Definition 1

Let Xnn∈N be a random serie. Xnn∈N converges towards 0 in probability if:

∀ε > 0, limn→+∞

P(|Xn|> ε) = 0.

This is also denoted by: Xn = oP(1).

Definition 2

Let Xnn∈N be a random serie. Xnn∈N is bound in probability if: ∀ε > 0,∃δ(ε) > 0 and n0 such that∀n≥ n0,P(|Xn|> δ(ε)) < ε. This is also denoted by: Xn = OP(1).

Definition 3

Let Xnn∈N be a random serie. Let X be a random variable. Let unn∈N be serie of strictly positivenumbers. Then we have:

1. Xnn∈N converges towards X if and only if Xn−X = oP(1);

2. Xn = oP(un) if and only if u−1n Xn = oP(1);

3. Xn = OP(un) if and only if u−1n Xn = OP(1).

Proposition

Let Xnn∈N and Ynn∈N be two random series. Let unn∈N and vnn∈N be two series of strictlypositive numbers. Then we have:

106 Convergence in probability

1.

Xn = oP(un),Yn = oP(vn)⇒

XnYn = oP(unvn)Xn +Yn = oP(max(un,vn))|Xn|r = oP(ur

n)

2.Xn = oP(un),Yn = OP(vn)⇒ XnYn = oP(unvn)

3.

Xn = OP(un),Yn = OP(vn)⇒

XnYn = OP(unvn)Xn +Yn = OP(max(un,vn))|Xn|r = OP(ur

n)

Appendix E

Kronecker product

Definition of Kronecker product

Let M d= m×n and M′ d= m′×n′ be two matrices. Then the Kronecker product (⊗) of M by M′ is:

M⊗M′ =

M1,1M′ . . . M1,nM′...

Mm,1M′ . . . Mm,nM′

d= mm′×nn′. (E.1)

Definition of vec operator

Let M d= m×n be a matrice such that M = [m1 . . .mn]. Then the vec of M is defined by:

vec(M) =

m1...

mn

. (E.2)

Proposition 1

Let M, M′, N and N′ be four matrices. Recall that superscipts H and † stand for the Hermitian operatorand the pseudo-inverse respectively. Then we have:

1. (M+N)⊗ (M′N′) = M⊗M′+M⊗N′+N⊗M′+N⊗N′;

2. (M⊗N)(M′⊗N′) = MM′⊗NN′ if dimensions of the matrices are compatible;

3. (M⊗N)⊗ (M′⊗N′) = M⊗N⊗M′⊗N′;

4. (M⊗N)H = MH ⊗NH ;

5. (M⊗N)† = M†⊗N†.

108 Kronecker product

Proposition 2

Let M, N and P be three matrices. Recall that superscipt T stands for the transpose operator. Then,

vec(MNP) = (PT ⊗M)vec(N). (E.3)

Appendix F

Proofs of chapter 3

F.1 Regularized semi-blind case (proposition 1)

Our goal is to evaluate the covariance matrix of estimation error for the regularized semi-blind case(regularized semi-blind subspace, linear prediction and weighted linear prediction). To this end, we firstevaluate the error δg defined by g = g+δg where g is the true vector channel and g is semi-blind estimateof the channel. From this, we deduce the expression of Cov(δg).

The proof is made according to an asymptotic analysis. The corresponding assumptions for the regular-ized semi-blind case are:

• T → ∞;

• m→ ∞;

• the ratio Tm = ρ is fixed.

First, for simplicity, we denote Q(α) and ∆(α) by Q and ∆ respectively. In addition, we define:

Q , R−1/2SS QR−1/2

SS , δQ , R−1/2SS (Q−Q)R−1/2

SS ,./Q, R−1/2

SS QR−1/2SS

RSY , R−1/2SS RSY , RSV , R−1/2

SS RSV

g , R1/2SS g,

./g, R1/2

SS g.

We start from the general expression of the semi-blind channel estimates which is (3.30):

g =[RSS +ρQ(α)

]−1RSY . (F.1)

The semi-blind estimator (F.1) left multiplied by R1/2SS writes:

110 Proofs of chapter 3

g = (RSS +ρQ)−1RSY (F.2)./g = (IK(l+1) +ρ

./Q)−1RSY (F.3)

= (IK(l+1) +ρQ+ρ δQ)−1RSY (F.4)

= (IK(l+1) +ρQ)−1[IK(l+1) +ρ δQ(IK(l+1) +ρQ)−1]−1RSY (F.5)

Using the facts that δQ = OP(T−1/2) and g = (IK(l+1) +αρQ)−1g (because g lies in the null space of Q)

we can re-expressed./g as:

./g = (IK(l+1) +ρQ)−1[IK(l+1)−ρ δQ(IK(l+1) +ρQ)−1]RSY +OP(T−1) (F.6)

= (IK(l+1) +ρQ)−1(g+ RSV −ρ δQg−ρ δQRSV )+OP(T−1) (F.7)

= (IK(l+1) +ρQ)−1[g+(IK(l+1)−ρ δQ)RSV −ρ δQg]+OP(T−1) (F.8)

= (IK(l+1) +ρQ)−1g+[IK(l+1)−ρ δQ(IK(l+1) +ρQ)−1]RSV −ρ δQg (F.9)

+OP(T−1)

Based on RSV = O(m−1/2)= O(T−1/2) and the fact that ρ is constant, we can neglect the term ρ δQRSV (IK(l+1)+

ρQ)−1 in the latter expression of./g. The channel estimation error can then be expressed as:

δg = (RSS +ρ Q)−1(RSY −ρ δQg)+OP(T−1) (F.10)

Calling independency between the training-based and blind estimators leads to:

Cov(δg) , limT→+∞,T/m=ρ

T E[δgδgH ] (F.11)

= (R(∞)SS +ρ Q)−1× lim

T→+∞,T/m=ρT E[

SHV trVHtrS

m2 +ρ2 δQggHδQH ] (F.12)

×(R(∞)SS +ρQ)−1 (F.13)

= (R(∞)SS +ρ Q)−1[ρσ2R(∞)

SS +ρ2Cov(δQg)] × (R(∞)SS +ρ Q)−1. (F.14)

Therefore, introducing W and Σ in the latter gives the closed-form of the estimation error covariancematrix:

Cov(δg) = (R(∞)SS +ρQ)−1[ρσ2R(∞)

SS +ρ2∆HWΣW∆] × (R(∞)SS +ρ Q)−1. (F.15)

F.2 Regularized semi-blind subspace (proposition 2)

We want to explicit Σ = Cov(δ∆g) in the regularized semi-blind case based on the subspace method.For this purpose, we firts express δ∆g. From this, we evaluate Cov(δ∆g).

Here again we ignore argument α of matrices Q(α) and ∆(α) for ease of reading.

F.2 Regularized semi-blind subspace (proposition 2) 111

Expression of δ∆g

By using (2.14) and (3.6) we readily obtain:

δ∆g =

D(δπ)C1g1...

D(δπ)CKgK

=

D(δπ)h1...

D(δπ)hK

=

vec(δπTM(h1))...

vec(δπTM(hK))

(F.16)

δπ is given by standard perturbation formulae [?]:

δπ =−(TM(H)THM(H))#δRπ−πδR(TM(H)TH

M(H))#. (F.17)

The latter right multiplied by TM(hk) writes

δπTM(hk) = πδR(TM(H)THM(H))#TM(hk)

= πδR(TM(H)THM(H))#TM(H)(IM+L+1⊗uk).

(F.18)

Hence,vec(D(δπ)hk) =−[(IM+L+1⊗uT

k )TTM(H)(TM(H)TH

M(H))#,T ]⊗πvec(δR). (F.19)

Expression of Cov(δ∆g)

Making use of equation (F.19) and the following relation [?]

Cov[vec(δRM)] =∫ π

−π[DM(ω)DH

M(ω)⊗SN(eiω)]∗⊗ [DM(ω)DHM(ω)⊗SN(eiω)]dω (F.20)

withSN(eiω) , H(eiω)HH(eiω)+σ2IN and DM(ω) , (1e−iω . . .e−i(M−1)ω)T (F.21)

and

we obtain that

Cov(δ∆g) =∫ π

−π

JT1...

JTK

T

#,∗M (H)

[DM(ω)DH

M(ω)⊗SN(eiω)]∗

T#M (J1 . . .JK) (F.22)

⊗π [DM(ω)DH

M(ω)⊗SN(eiω)]πHdω. (F.23)

Therefore, by the use of the two following identities

π(eiω)SN(eiω)πH(eiω) = σ2π(eiω)πH(eiω)T(eiω)H(eiω) = DH

M+L(ω)⊗ IK(F.24)

it is ready to show that

Cov(δ∆g) = σ2IK ⊗∫ π

−π

[DM(ω)DH

M(ω)⊗ [π(eiω)πH(eiω)

]]∗dω (F.25)

+σ4∫ π

−π

[JT

k T(ω)TH(ω)Jl]∗⊗ [

π(eiω)πH(eiω)]

dω.


F.3 Regularized semi-blind linear prediction (proposition 3)

We want to evaluate Σ = Cov(δ∆g) in the regularized semi-blind case based on the conventional andweighted linear prediction approaches. We mimic the steps followed to prove Propositions 1 and 2, thatis to say we first calculate δ∆g and then Cov(δ∆g). For sake of clarity, matrices Q(α) and ∆(α) arealways denoted by Q and ∆.

First of all, let split matrix Σ into K2 (l +1)× (l +1) blocks defined as:

∀(k, l) ∈ [1,K]2,Σ(k, l) , limT→+∞

T E[δ∆kgk(δ∆lgl

)H ]. (F.26)

In order to evaluate δ∆k recall ∆k , ΠS(A)Ck from equation (3.21). Differentiating the latter we obtainthat:

δ∆k = δΠ S(A)Ck +ΠS(δA)Ck (F.27)

where

δΠ =

δπD⊥ 00

. . .0 0

(F.28)

and

S(δA) =

0 0

δA(1). . .. . . 0

... δA(1)

δA(M)...

. . .0 δA(M)

. (F.29)

Plugging equations (F.29) and (F.28) into equation (F.27) we have:

δ∆k =

δπD⊥ 0 . . . 0δA(1) 0

. . . . . ....

.... . . 0

δA(1)δA(M)

. . ....

. . .0 δA(M)

×Ck (F.30)

In order to calculate matrix δ∆k in two steps, we split δ∆k into two blocks as follows

δ∆k =[

δPkδAk

](F.31)

F.3 Regularized semi-blind linear prediction (proposition 3) 113

where δPk , [δπD⊥ 0N,NL]Ckd= N× (l +1).

First step: expression of δPk , δPkgk

The idea is to introduce vector vec(δRM) since we know how to express its covariance matrix [?] [?].Using equation (2.14) we have

δPk = [δπD⊥ 0N,NL]Ckgk

(F.32)

= [δπD⊥ 0N,NL]hk. (F.33)

Using vector h(0)k comprising the first N entries of hk and standard perturbation formulae it follows that

δPk = δπD⊥h(0)k (F.34)

= (−D#δDπD⊥−πD⊥δDD#) h(0)

k (F.35)

= −D#δDπD⊥H(0)uk−πD⊥δDD#h(0)k (F.36)

where we have introduced uk = (0 . . .0 1 0 . . .0)T d= K×1 the one is on the position kth.

Since πD⊥ is the orthogonal projection matrix on the orthogonal complement of null(D) (equation (3.21))we have πD⊥H(0) = 0. Additionally, we know that δD = AδRMAH and for any matrices M1,M2,M3vec(M1M2M3) = (MT

3 ⊗M1)vec(M2) (Annex E). Hence,

δPk = −πD⊥δDD#h(0)k (F.37)

= −πD⊥AδRMAHD#h(0)k (F.38)

= −[AHD#h(0)k ]T ⊗πD⊥Avec(δRM). (F.39)

Finally, defining ηk

as ηk, AHD#h(0)

k = AHH(0)[HH(0)H(0)]−1uk gives:

δPk = −ηTk⊗πD⊥Avec(δRM). (F.40)

Second step: expression of δAk , δAkgk

From the definition of δAk it is ready to show that

δAk = vec[δA(1) . . .δA(M)] TM−1(hk). (F.41)

Since RM−1 = TM−1(H)THM−1(H)+σ2INM and TM−1(hk) = TM−1(H)(IM+L⊗uk) equation (F.41) can be

rewritten as

δAk = −vecAδRM[0TN,NM (RM−1−σ2INM)#,T ]T TM−1(hk) (F.42)

= −vecAδRM[0TN,NM (RM−1−σ2INM)#,T ]T TM−1(H)(IM+L⊗uk) (F.43)

= −vecAδRM[0TN,NM T

#,∗M−1(H)]T (IM+L⊗uk). (F.44)


In order to introduce vec(δRM) we once again make use of the classical property of the vec operator,which is M1,M2,M3 vec(M1M2M3) = (MT

3 ⊗M1)vec(M2). So,

δAk = −(IM+L⊗uk)T [0NM,N T

#,∗M−1(H)]⊗Avec(δRM) (F.45)

= −[0NM,N JTk T

#,∗M−1(H)]⊗Avec(δRM) (F.46)

with Jk = (IM+L⊗uk).

Finally, ∀ k ∈ [1,K]N,

δ∆k gk=

[−ηT

k⊗πD⊥A

−[0NM,N JTk T

#,∗M−1(H)]⊗A

]×vec(δRM). (F.47)

To conclude the proof we just need to replace δ∆k gk

with (F.47) and Cov(vec(δRM)) with (F.20) in(F.26). After simple manipulations, one can establish that:

Σ11(k, l) = σ2∫ 2π

0πD⊥A(eiω)AH(eiω)πD⊥dω δk,l

+ σ4∫ 2π

0uT

k [H#(0)A(eiω)AH(eiω)H#,H(0)]∗ul πD⊥A(eiω)AH(eiω)πD⊥dω

d= N×N

Σ22(k, l) = IM+L⊗D δk,l

+ σ2∫ 2π

0[JkT(eiω)TH(eiω)]∗dω Jl]∗⊗D

+ σ2∫ 2π

0[DM+L−1(ω)DH

M+L−1(ω)]∗⊗A(eiω)AH(eiω)dω δk,l

+ σ4∫ 2π

0[JkT(eiω)TH(eiω)Jl]∗⊗A(eiω)AH(eiω)dω

d= N(M +L)×N(M +L)

Σ12(k, l) = σ2∫ 2π

0e−iωDT

M+L−1(ω)⊗πD⊥A(eiω)AH(eiω)dω δk,l

= + σ4∫ 2π

0e−iω[uT

k H#A(eiω)TH(eiω)Jl]∗⊗πD⊥A(eiω)AH(eiω)dω

d= N×N(M +L)

Σ21(k, l) = ΣH12(k, l)

where [T(0) . . . T(M−1)] = T#M−1(H) .

F.4 Projected semi-blind case (proposition 4) 115

F.4 Projected semi-blind case (proposition 4)

The purpose of this proof is twofold. We want to prove that the estimation covariance matrix Γ inthe projected case is given by Γ = σ2

m PR−1SS P + O(σ4) and that the semi-blind channel estimate is the

projection of the trained estimate onto the null space of matrix Q that is g = Pgtr

+OP(m−1 +T−1/2).

The proof is made under asymptotic assumptions. The corresponding assumptions for the regularizedsemi-blind case are:

• T → ∞;

• m→ ∞;

• the ratio ρ is such that m = o(T N).

For clarity„ we denote Q(α) and ∆(α) by Q and ∆ respectively. In addition, we define:

Q , R−1/2SS QR−1/2

SS , δQ , R−1/2SS (Q−Q)R−1/2

SS ,./Q, R−1/2

SS QR−1/2SS

RSY , R−1/2SS RSY , RSV , R−1/2

SS RSV

g , R1/2SS g,

./g, R1/2

SS g.

First, recall the general expression of the semi-blind channel estimates from (3.30)

g =[RSS +ρQ

]−1RSY . (F.48)

From (F.48), the semi-blind estimate left multiplied by R1/2SS is written as

./g= (IK(l+1) +ρ

./Q)−1RSY . (F.49)

As we want to introduce P the orthogonal projector on subspace null(Q) we decompose IK(l+1) +ρ./Q as

follows

IK(l+1) +ρ./Q = IK(l+1) +ρQ+ρδQ (F.50)

=[

U V][

A11 A12A21 A22

][UH

VH

](F.51)

(F.52)

where U represents an orthogonal basis of the null space of Q so that P = UUH . Matrix V represents anorthogonal basis of the ortogonal complement of Q, which is null(Q). One can readily verify that

A11 = IK(l+1) +ρUHδQUA22 = IK(l+1) +ρVH(Q+δQ)VA12 = ρUHδQVA21 = AH

12

(F.53)


Since we have to express IK(l+1) +ρ./Q from (F.50) we call the block matrix inversion lemma ??

[A11 A12A21 A22

]−1

=[

M11 M12M21 M22

](F.54)

with

M11 = (A11−A12A−122 A21)−1

M22 = (A22−A21A−111 A12)−1

M12 = −(A11−A12A−122 A21)−1A12A−1

22M21 = MH

12

(F.55)

Then, to prove the claimed results, we follow three steps:

• we calculate the asympotical expression of matrix Ai j for all (i, j) in [1,2]2;

• we calculate the asympotical expression of matrix Mi j for all (i, j) in [1,2]2;

• from the new expression of IK(l+1) +ρ./Q we rewrite

./g.

Step 1: asymptotical expression of Ai, j / (i, j) ∈ 1,22

For this step, we only consider the case of the linear prediction approaches. In the subspace method casewe just need to replace ∆(Π,A) with ∆(π) and to choose W = I.

Denote ∆(Π,A) =∆(Π,A)R−1/2SS . Assume that either W = Wσ = IK(L+M+1)⊗(πD⊥+σ2D#) or W = I,

which case corresponds to the non-weighted linear prediction.

Looking at matrices A11, A12A21 and A22 we see that we need to evaluate the rate of covergence of

δQ =./Q−Q. To this end, let express

./Q

./Q = ∆H(Π, A)W(Π, D)∆(Π, A)

= [∆H(Π,A)+∆H(δΠ,A)+∆H(Π,δA)+∆H(δΠ,δA))]×[W(Π,D)+W(δΠ,D)+W(Π,δD)+W(δΠ,δD)]×[∆(Π,A)+∆(δΠ,A)+∆(Π,δA)+∆(δΠ,δA))]

Taking into account that

•

∆(Π,A) =

ΠS(A)C1 0. . .

0 ΠS(A)CK

(F.56)

•∆(δΠ,A) = OP(T−1/2), (Π,δA) = OP(T−1/2) and (δΠ,δA) = OP(T−1) (F.57)


•W(Π,D) = IK(L+M+1)⊗ (πD⊥ +σ2D#) (F.58)

•W(δΠ,D) = OP(T−1/2), W(Π,δD) = OP(T−1/2) and W(δΠ,δD) = OP(T−1). (F.59)

one can check that: UHδQU = OP(T−1), calling admissibilty of W; for all W, VHδQV = OP(T−1/2)and UHδQV = OP(T−1/2). From these considerations, we find that

A11 = IK(l+1) +ρUHδQU = IK(l+1) +OP(m−1)A22 = IK(l+1) +ρVH(Q+δQ)V = ρVHQV[IK(l+1) +OP(T−1/2)+OP(T−1m)]A12 = ρUHδQV = OP(T 1/2m−1)A21 = AH

12 = OP(T 1/2m−1)

(F.60)

Step2: asymptotical expression of Mi, j / (i, j) ∈ 1,22

Plugging new expressions of matrices Ai j from equations (F.60) into (F.55) we obtain

M11 = (A11−A12A−122 A21)−1 (F.61)

= (IK(l+1)−A12A−122 A21)−1A−1

11 (F.62)

= A−111 +A−1

11 A12A−122 A21A−1

11 + . . . (F.63)

= (IK(l+1) +ρUHδQU)−1 (F.64)

+(IK(l+1) +ρUHδQU)−1×ρUHδQV (F.65)

×ρVHQV[IK(l+1) +OP(T−1/2)+OP(T−1m)]−1

×ρVHδQU× (IK(l+1) +ρUHδQU)−1

+ . . .

= IK(l+1)−ρUHδQU

(IK(l+1)−ρUHδQU)×ρUHδQV (F.66)

×ρ−1VHQ#V[IK(l+1) +OP(T−1/2)+OP(T−1m)] (F.67)

×ρVHδQU× (IK(l+1)−ρUHδQU)+ . . .

M12 = −M11A12A−122

= −(IK(l+1) +ρUHδQU)A12A−122 + . . .

= −[IK(l+1) +OP(m−1)]A12A−122

= −[IK(l+1) +OP(m−1)]UHδQP⊥Q#V


M22 = (A22−A21A−111 A12)−1

= (IK(l+1)−A21A−111 A12)−1A−1

22

= A−122 +A−1

22 A21A−111 A12A−1

22 + . . .

= ρVHQV[IK(l+1) +OP(T−1/2)+OP(T−1m)]−1

+ρVHQV[IK(l+1) +OP(T−1/2)+OP(T−1m)]−1×ρVHδQU

×(IK(l+1) +ρUHδQU)−1×ρUHδQV

×ρVHQV[IK(l+1) +OP(T−1/2)+OP(T−1m)]−1

= ρ−1VHQ#V[IK(l+1) +OP(T−1/2)+OP(T−1m)]+. . .OP(T m−2). . .+ . . .

= ρ−1VHQ#V[IK(l+1) +OP(T−1/2)+OP(T−1m)]IK(l+1) +OP(T m−2). . .

By multiplying M11, M12 and M22 by U and/or V we obtain

UM11UH = P−ρPδQP+ρPδQQ#δQP+OP(T 1/2m−1)OP(T−1m)[OP(T−1/2)+OP(T−1m)]OP(T 1/2m−1

+OP(m−1)OP(T 1/2m−1)OP(T−1m)OP(T 1/2m−1)+ . . .

= P[IK(l+1)−ρδQ(IK(l+1)− Q#δQ)P]+OP(T−1/2m−1)+OP(T−1)+

+OP(m−2)

UM12VH = −PδQQ#[IK(l+1) +OP(T−1/2)+OP(T−1m)+OP(m−1)]

VM22VH = ρ−1P⊥Q#P⊥[IK(l+1) +OP(T−1/2)+OP(T−1m)+OP(m−1)]

= ρ−1Q#[IK(l+1) +OP(T−1/2)+OP(T−1m)+OP(m−1)]

and of course VM21UH = (UM12VH)H .

Step 3: new expression of./g

From (F.68), (F.68), F.68) we find that

./g = (IK(l+1) +ρ

./Q)−1RSY (F.68)

= [UM11UHUM12VHVM21UHVM22VH ]RSY (F.69)

= [P−ρPδQ+ρPδQQ#δQP+(. . .)Q# + . . .]RSY (F.70)

= [P+OP(m−1)]RSY (F.71)

= [P+OP(m−1)](g+ RSV ) (F.72)./g −g = PRSV +OP(m−1)+OP(T−1/2) (F.73)

= PRSV +OP(m−1)+OP(T−1/2). (F.74)

With standard notations we have

δg = R−1/2SS PR−1/2

SS RSV (F.75)

= R−1/2SS PR1/2

SS R−1SS RSV (F.76)

, PR−1SS RSV . (F.77)


Hence, the approximated error covariance matrix is:

E[δg δgH ] = σ2P(SHS)−1P+O(σ4) (F.78)

and the projected semi-blind estimate is:

g = Pgtr

+OP(m−1 +T−1/2). (F.79)

Samson Lasaulce - SupélecChannel Estimation and Multiuser Detection for TD-CDMA Systems Soutenue le...

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Transcript of Samson Lasaulce - SupélecChannel Estimation and Multiuser Detection for TD-CDMA Systems Soutenue le...