Two Iterative Channel Estimation Algorithms in Single-Input Multiple-output (SIMO) LTE Systems_2012...

TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIESTrans. Emerging Tel. Tech. 2013; 24:59–68

Published online 24 October 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.2582

SPECIAL ISSUE - LTE-A

Two iterative channel estimation algorithms insingle-input multiple-output (SIMO) LTE systemsYang Liu and Serdar Sezginer*

Sequans Communications, 19 Le Parvis de la Défense, 92073 Paris, France

ABSTRACT

Accurate channel estimation is a challenging problem in a long term evolution systems especially in highly selectivechannels. In this paper, two iterative channel estimation algorithms are proposed to efficiently estimate channels overthe orthogonal frequency-division multiplexing (OFDM) symbols containing pilot symbols. The first one is proposedbased on a simple interpolation algorithm, named as simple iterative frequency interpolation (SIFI), and improves thesystem performances with low implementation complexity. The second one is derived from the traditional expectationmaximisation (EM) algorithm in OFDM systems. In order to deal with the ‘null’ subcarriers in the guard band andmake the algorithm converge, the truncated singular value decomposition method is integrated to truncated singular valuedecomposition expectation-maximisation (TSVD-EM). Both channel estimation algorithms are followed by a simple linearinterpolation in time domain to obtain channel estimates over the whole subframe. Complexities of the proposed algorithmsare also analysed and compared. Simulation results show that the SIFI algorithm outperforms the traditional interpolationmethod with a low complexity, and the TSVD-EM algorithm approaches the best achievable performance with a reducedcomplexity. Copyright © 2012 John Wiley & Sons, Ltd.

KEY WORDS

iterative channel estimation; long term evolution; expectation maximisation; orthogonal frequency-division multiplexing

*Correspondence

Serdar Sezginer, Sequans Communications, 19 Le Parvis de la Défense, 92073 Paris, France.

E-mail: [email protected]

Received 12 March 2012; Revised 18 June 2012; Accepted 12 July 2012

1. INTRODUCTION

In long term evolution (LTE) systems, as in other wirelesscommunication systems, the accuracy of channel estimatesseriously affects the system performance. In order to obtainprecise channel estimates with appropriate complexity,many studies have been devoted to channel estimation inorthogonal frequency-division multiplexing (OFDM) sys-tems [1–4]. For time-varying channels, channel estimationmay be implemented on both frequency and time domains.In terms of mean square error (MSE), the optimal chan-nel estimator is two-dimensional (2D) Wiener filter-basedinterpolation, which is derived from minimum MSE(MMSE) criterion. However, because of its high complex-ity, it is not feasible to use 2D Wiener filter in a practicalsystem. Various methods have been studied to reduce thecomplexity of MMSE estimator such as linear mismatchedMMSE estimator [5], exponential mismatched MMSE esti-mator [3], 2�1D MMSE estimator [6] and so on. As can

be expected, these estimators reduce the complexity oftraditional 2D MMSE estimator at the expense of perfor-mance degradation. In practical systems, thanks to its lowcomplexity, the 2�1D interpolation method is preferredwhere frequency and time interpolations are performedindependently [7]. This approach provides a good trade-offbetween complexity and performance.

In recent years, iterative receivers have become moreand more popular because of their attractive performanceespecially after the appearance of ‘turbo principle’ [8]. Dif-ferent iterative detection mechanisms have already beenproposed and studied in detail (see, e.g., [9] and refer-ences therein). However, these iterative mechanisms areseriously affected by channel estimator performance. Thus,in iterative receivers, one needs more accurate channelestimates in order to improve system performances. More-over, for future standards, one of the main objectives is tobuild more efficient transmission systems. In this manner,decreasing the power level or the number of pilot symbols

Copyright © 2012 John Wiley & Sons, Ltd. 59

Y. Liu and S. Sezginer

are the possible ways to improve the spectral efficiency.In such systems, the estimation algorithms used in cur-rent systems may result in less accuracy and more robustestimation algorithms will be needed. In such situations,iterative channel estimation that uses the ‘soft’ informationof data can be considered in order to improve the accuracy.

Different iterative channel estimators have already beenproposed for OFDM systems [10, 11] by using the ‘soft’information from the channel decoder. However, some ofthese proposed iterative algorithms have extremely highcomplexity and most of them do not consider the ‘null’subcarrier in practical systems. In this paper, in order tohave low complexity iterative channel estimator, we pro-pose two iterative channel estimation algorithms based ona simple 2� 1D interpolation method: the simple itera-tive frequency interpolation (SIFI) algorithm has a lowcomplexity and the truncated singular value decomposi-tion expectation-maximisation (TSVD-EM) algorithm per-forms as well as the traditional EM algorithm but with alower complexity. Both the SIFI and the TSVD-EM aresuitable to systems with ‘null’ subcarriers.

The rest of the paper is organised as follows. InSection 2, the interpolation-based channel estimationmethod in LTE systems is briefly explained. Next,Section 3 describes the proposed iterative channel estima-tion algorithms. In Section 4, complexities of the proposedalgorithms are analysed and compared. Simulation resultsare shown in Section 5. Finally, conclusions are drawnin Section 6.

2. CONVENTIONAL CHANNELESTIMATION IN LTE

In LTE systems, the channel estimates can be obtainedusing conventional linear interpolation algorithms basedon reference symbols (RS). In Figure 1, the cell-specificRS locations corresponding to normal cyclic prefix con-figuration up to four transmit antennas are shown, forone subframe and two adjacent resource blocks, wherePn .0 6 n 6 NT � 1/ represents the RSs for the nth trans-mit antenna port [12]. From Figure 1, the RSs for transmitantenna 0 and 1 have the same distribution: every sixth car-rier in frequency domain and the first and fifth symbolsin time domain. In this paper, we focus on one transmitantenna case.

In practical systems, in order to have simple channelestimation, usually, the conventional 2x1D interpolationmethods [7] are performed. According to the orders ofinterpolations in time domain and frequency domain, wecan have different strategies to implement the channel esti-mation. In this paper, we focus on the method that performsthe frequency domain interpolation first, which is depictedin Figure 2.

The channel estimates on pilot positions are noted asOhPk;l

, where k stands for the subcarrier index in one OFDMsymbol and l denotes the index of OFDM symbols in onesubframe. If the position .k; l/ does not correspond to a

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

PP

P

P

P

P

P

P

P

Figure 1. Reference symbols in long term evolution system fornormal cyclic prefix [12].

Frequency domain interpolation

Time domain interpolation

Channel estimates on RS

Figure 2. Conventional channel estimation in an long termevolution subframe.

pilot position, the channel estimate on this position is null,that is, OhP

k;lD 0. On the basis of channel estimates on

pilot positions, frequency domain channel estimation isimplemented with a filter. The filter has a coefficient vectorc D .c0; � � � ; cLFLTR�1/, where LFLTR stands for the filterlength. With this filter, the channel estimate on the resourceelement (RE) .k; l/ is expressed as

Ohk;l D

LFLTR�1XnD0

cn OhPk�

LFLTR2 Cn;l

(1)

Finally, interpolation in time domain is performed to getall the channel estimates. For time domain interpolation,we consider a simple linear interpolation in order to keepthe complexity as low as possible. When the previous andthe next RSs exist, the channel estimate on the RE .k; l/

can be obtained by

Ohk;l DOhRS, prevdnext C OhRS, nextdprev

dprevC dnext(2)

where OhRS, prev and OhRS, next stand for the channel estimateson the previous and next pilot positions; dprev and dnext

60 Trans. Emerging Tel. Tech. 24:59–68 (2013) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett


CHE Mapping

Detection&

Demapping

CP FFT

CP FFT

DecoderEqualizer

soft info.

Figure 3. Receiver block diagram with iterative channel estimator.

are the distances from .k; l/ to the previous and next RSs,respectively. In the sequel, we will discuss the channel esti-mation over one OFDM symbol and the subscript l willbe omitted.

In this paper, iterative channel estimation algorithms willbe studied based on the iterative receiver shown in Figure 3.After cyclic prefix removal and fast Fourier transform(FFT) for each received antenna, the received symbolvector on the r th received antenna yr D

�yr0 ; � � � ; y

rN�1

�Tcan be written as

yr D Xhr C nr (3)

where the matrix X stands for a diagonal matrix contain-ing transmitted symbol Xkk at the kth diagonal entry; thevector hr represents the channel coefficients in the fre-quency domain; n is a complex Gaussian noise vector withzero mean and variance 2�2. Then, the received symbolsyr are fed to the equaliser (EQU) and the equaliser com-bines multiple streams (two receive antennas as shown inFigure 3) into a single one yEQU and passes it to the detec-tor. In the detector, likelihood of each symbol is calculated.Then, on the basis of likelihoods of symbols, informa-tion on coded bits is obtained, deinterleaved and providedto the channel decoder. From channel decoder, some softinformation (soft info.) on coded bits is generated. Afterinterleaving and mapping, the soft information is used toimplement iterative channel estimation. With new chan-nel estimates, the whole process will be performed again.Through successive iterations, system performance canbe improved.

3. ITERATIVE CHANNELESTIMATION

In this section, two iterative channel estimation algo-rithms for LTE systems will be proposed: SIFI and TSVD-EM algorithm.

3.1. Simple iterative frequencyinterpolation algorithm

Because the 2 � 1D interpolation method is widelyused in practical systems, it is natural to develop iter-ative channel estimation methods based on this simpleinterpolation algorithm.

For the initial (first) iteration, the channel estimates onpilot positions can be obtained by

Oh0

P;n DX�P;nnyP;n

jXP;nnj2(4)

where XP;nn and yP;n stand for the nth transmitted andreceived pilot symbol. Then, the FI is implemented based

on the channel estimates Oh0

P D�Oh0

P;0; � � � ;Oh0

P;n; � � ��T

. In

an LTE subframe, the interval between two pilot symbolsis 6 for antenna ports 0 and 1. Thus, the interpolation isperformed with a spacing of 6 symbols

Oh.0/ D FLTR�Oh0

P

�6

(5)

where FLTR.�/ stands for a simple frequency domain filterwith a spacing of 6 symbols; the vector Oh.0/ representschannel estimates in the initial iteration, which includes thechannel estimates on pilot positions, denoted as Oh.0/P;n, and

the channel estimates on data positions, denoted as Oh.0/k

, asshown in Figure 4.

From the second iteration, we propose to implement theFI with a smaller spacing (SSP) to improve channel esti-mation. In order to implement interpolation with SSP, some

other channel estimates Oh.0/;Prodk

between two channel esti-mates from actual pilot symbols have to be built. We canuse soft information from channel decoder to build thesenew channel estimates. However, using only the soft infor-mation is not very reliable to guarantee the convergence ofthe algorithm through iterations. Therefore, a new estimateOh.0/;Prodk

is proposed that is built by using both the channel

estimate from the initial iteration Oh.0/k

and the channel esti-mate based on soft information from the channel decoderOh.0/;Sk

.

Trans. Emerging Tel. Tech. 24:59–68 (2013) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett

61


Channel estimate based on pilots

Soft channel estimate

Channel estimate after filter

Iter 2Iter 1

Pilot

Pilot

"Soft Info"

Figure 4. Iterative frequency interpolation.

For pilot symbol XP;nn, the received pilot symbol is

yP;n DXP;nnhP;nC nP;n (6)

where nP;n is an additive white Gaussian noise componentwith zero mean and variance 2�2. Together with (4),we have

p�Oh0

P;n

�� CN

hP;n;

2�2

jXP;nnj2

!(7)

where the subscript P represents pilots, the distributionCN is a complex normal distribution and hP;n standsfor the actual channel coefficient for the nth subcarrier.Because Oh.0/

kare obtained from Oh

0

P;n through the linearfilter FLTR.�/ as in (5), it is assumed that the probabilitydensity function (pdf) of channel coefficient in the initial

iteration h.0/k

has the same variance as the pdf of Oh0

P;n, butthe mean value is different as shown in (10).

With the soft information from the channel decoder, thecorresponding channel estimate can be calculated as

Oh.0/;SkDQX.0/�kk

yk

B

jX.0/kkj2

(8)

where QX .0/kk

represents a soft symbol containing a posteri-ori information (APP) as

QX.0/kkDXm

smAPP.0/.Xkk D sm/ (9)

B

jX.0/kkj2 D

Xm

jsmj2APP.0/.Xkk D sm/, and sm stands

for a constellation point. Here, it is assumed that theobtained APP for channel estimation is perfect. Then, the

soft symbol QX .0/kk

can be viewed as a ‘virtual’ pilot sym-bol and the pdf of the channel estimate based on soft

information p�h.0/;Sk

�also has Gaussian distribution as

the pdf based on pilots. Finally, it can be proved that both

the distribution of channel estimate Oh.0/k

and that of Oh.0/;Sk

are subject to complex Gaussian distribution, that is,

p�h.0/k

�� CN

Oh.0/k;

2�2

jXP;nnj2

!(10)

and

p�h.0/;Sk

�� CN

0B@ Oh.0/;Sk;2�2

B

jX.0/kkj2

1CA (11)

In order to get the channel estimate Oh.0/;Prodk

, we proposeto multiply the two Gaussian distributions in (10) and (11)and get a new distribution

p�h.0/;Prodk

�� p

�h.0/k

�� p

�h.0/;Sk

�(12)

On the basis of Appendix A, the new distribution is stilla Gaussian distribution with mean value and variance,respectively, as

jXP;nnj2 Oh.0/kCB

jX.0/kkj2 Oh

.0/;Sk

jXP;nnj2CB

jX.0/kkj2

(13)

and

2�2

jXP;nnj2CB

jX.0/kkj2

(14)

We take the mean value as the value of Oh.0/;Prodk

and imple-ment the second FI with SSP to get new channel estimatesover all subcarriers,

Oh.1/ D FLTR��Oh.0/P ; Oh.0/;Prod

��SSP

(15)

The procedure is depicted in Figure 4 with SSP D 3 as

an example, where the vector�Oh.0/P ; Oh.0/;Prod

�represents

all channel estimates over the whole OFDM symbol withappropriate ordering.



For the subsequent iterations, two methods are proposed

to get Oh.i/;Prodk

:

� Method A: Combine channel estimates from softinformation Oh.i/;S

kwith the estimates from filtering

based on pilot symbols Oh.0/k

. With this method,Oh.i/;Prodk

can be easily obtained from (13)

Oh.i/;Prodk

DjXP;nnj

2 Oh.0/kCB

jX.i/kkj2 Oh

.i/;Sk

jXP;nnj2CB

jX.i/kkj2

(16)

� Method B: Combine channel estimates from softinformation Oh.i/;S

kwith those from the previous

iteration Oh.i�1/k

and the new distribution can beexpressed as

p�h.i/;Prodk

�� p

�h.i�1/k

�� p

�h.i/;Sk

�(17)

Then, after some derivations, the mean value of thisnew distribution becomes

Oh.i/;Prodk D

jXP;nnj

2C

i�1XtD1

AjX .t/

kk j2

!Oh.i�1/k CAjX .i/

kk j2 Oh.i/;Sk

jXP;nj2C

iXtD1

AjX .t/

kk j2

:

(18)

However, it is complex to use eachB

jX.t/kkj2 with

0 6 t 6 i � 1. In order to reduce complexity, thefollowing approximation is used:

i�1XtD1

B

jX.t/kkj2 � .i � 1/E

njXkk j

2o

(19)

Denoting E˚jXkk j

2�

as E and using (19), (18)reduces to

Oh.i/;Prodk �

ŒjXP;nnj2C .i � 1/E� Oh.i�1/k CAjX .i/

kk j2 Oh.i/;Sk

jXP;nj2C .i � 1/E CAjX .i/

kk j2

(20)

Then, together with the channel estimates on pilot posi-

tions from the previous iteration Oh.i/P;n, these values willbe used to implement the frequency domain interpola-tion with an SSP to obtain the channel estimates in the.i C 1/th iteration.

3.2. TSVD-EM algorithm

In this section, a simplified EM channel estimation algo-rithm is presented that particularly focuses again on theLTE subframe structure. However, the approach can begeneralised for any kind of practical systems.

3.2.1. Traditional EM in OFDM systems.

The EM algorithm provides a recursive solution to MLestimation [13], and it performs a two-step procedureas follows:

(1) E-step: compute the auxiliary functionQ�

gjOg.i/�D

E�

hlog p .� jg/ jy; g.i/

i;

(2) M-step: update the parameters Og.iC1/ D arg maxgQ�

gjOg.i/�

,

where g stands for the vector containing the time-domainchannel parameters to be estimated, Og.i/ represents the esti-mated parameters vector at the i th iteration, y denotes theobserved data vector and � is the so-called complete data.In particular, � contains observed data and some misseddata. The EM channel estimation in coded OFDM systemsis performed as [14]

Oh.iC1/EM D�L

��L

R.i/N�N

�L

��1��LeX.i/�y (21)

where�L is a matrix containing the first L columns of theFFT matrix, L is the value of channel delay spread, eX rep-resents a diagonal matrix consisting of soft symbols thatcontain the APPs of the transmitted symbol matrix X at thei th iteration as defined in (9) and

R.i/N�N

DX

C2QAPPi .XD C/C�C (22)

Here, Q represents the set of all possible codeword matri-ces of X, and the matrix C stands for one realisationfrom Q.

3.2.2. TSVD-EM in LTE.

In (21), all subcarriers are considered in EM channelestimation and small condition numbers [15] are guar-anteed that is defined as the ratio between the greatestand the smallest singular values. Such a small conditionnumber makes the EM channel estimation perform per-fectly. However, in most practical multi-carrier commu-nication systems, some null subcarriers are kept to actas ‘guard band’. With these null subcarriers, (21) can beexpressed as

Oh.iC1/EM D�LDP

��LDP

R.i/NDP�NDP

�LDP

��1��LDP

eX.i/�y(23)

where NDP represents the number of modulated subcarri-ers and�LDP contains only the rows of�L correspondingto the modulated subcarriers, including data and pilot sym-bols. In order to reduce the complexity, the SVD method isused to decompose the matrix�LDP

�LDP D U†V� (24)


63


where the matrix U is an NDP � NDP unitary matrix,the matrix † is NDP � LDP diagonal matrix with non-negative real numbers on the diagonal and V denotes anLDP � LDP unitary matrix. Using (24), the EM channelestimate becomes

Oh.iC1/SVD-EM D�LDP V†�1U��

R.i/NDP�NDP

��1eX.i/�y(25)

In (25), because the matrix † is not square, the inversionoperation is performed over the non-zero part of †. Eventhough the SVD method can reduce the complexity of theEM algorithm, with the matrix�LDP , some very small sin-gular values may exist in the matrix †. This can lead tolarge ‘condition numbers’ [15] that result in the so-called‘border effect’. In order to deal with the null subcarriers,in [15], a threshold for singular values is set to guarantee asmall condition number for channel estimation with pilots.In SVD-EM, we also consider only the S singular valuesthat are greater than the threshold and the correspondingvectors from matrices U and V. This will be named trun-cated SVD-EM (TSVD-EM). The corresponding matricesare denoted as NDP�S matrix Us, LDP�S matrix Vs andS�S matrix†s. Thus, the TSVD-EM can be expressed as

Oh.iC1/TSVD-EM D�LDP Vs†�1s U�s

�R.i/NDP�NDP

��1eX.i/ �y(26)

With proper selection of the threshold, the ill-conditioncases can always be eliminated.

In [16], the MSE property has been analysed based on(26) and the efficiency of the TSVD-EM algorithm hasbeen shown by some simulation results. In this paper, theperformance of the TSVD-EM will be compared with thatof the SIFI algorithm.

4. COMPLEXITY COMPARISON

In this section, complexities of different algorithms willbe analysed and compared, including conventional non-iterative method and the proposed iterative algorithms. Inorder to compare complexities of different algorithms, thenumber of complex multiplications at each iteration foreach algorithm is checked, because complex multiplicationoperation is more complicated than other operations and isthe most important factor in evaluating the complexity ofan algorithm.

4.1. Frequency interpolation

To obtain channel estimate of one subcarrier using (1),the non-iterative FI algorithm needs dLFLTR=6e complexmultiplications. Hence, for all modulated subcarriers overone OFDM symbol,NDP�dLFLTR=6e complex multiplica-tions are needed. For example, if the filter length is 21 and

the modulated subcarriers are 600, i.e. 50 resource blocks(RB), the number of complex multiplications needed toobtain all 600 channel estimates is 600� 4D 2400.

4.2. SIFI algorithm

In order to perform frequency interpolation with SSP, newchannel estimates based on soft information have to bebuilt between two channel estimates based on pilot sym-bols. From (16) and (18), we see that, for each new chan-nel estimate, only one complex multiplication is needed.Then, over one OFDM symbol, the needed complex multi-plication number is NDP=SSP � NDP=6. Then, for FI, thenumber is NDP � dLFLTR=SSPe. Consequently, the totalnumber of complex multiplications becomes NDP=SSP �NDP=6CNDP � dLFLTR=SSPe. For the filter length of 21,the modulated subcarriers are 600 (50 RB), and SSP is3, the total number of complex multiplications is 4300 ateach iteration.

4.3. TSVD-EM algorithm

With traditional EM in OFDM systems, the matrixRNDP�NDP is always in the matrix inversion. Because thematrix RNDP�NDP contains soft information from decoder,it has to be recalculated and inverted at each iteration. Thecomplexity of this matrix inversion is O

�L3�

[17], and itis logical to assume that the number of complex multipli-cations needed for matrix inversion is L3. Furthermore, inorder to obtain the matrix to be inverted, NDP � L

2 com-plex multiplications are needed at each iteration. There-fore, for each inversion, the complexity is NDP�L

2CL3.After matrix inversion, N 2DP complex multiplications arerequired to get all channel estimates over all subcarriers.Therefore, the overall complexity isN 2DPCNDP�L

2CL3.In (26), the matrix inversion is not needed anymore.

With TSVD-EM, the matrix computation�LDP Vs†�1s U�s

can be carried out offline once the threshold of singularvalues is set. For a given iteration, we only need to cal-culate soft information on symbols and multiply with thepre-calculated matrix. Accordingly, the complexity forTSVD-EM becomes N 2DP complex multiplications at eachiteration. Thus, the complexity is reduced by NDP �L

2 C

L3 compared with the traditional EM, a reduction that canbe significant considering long delay spread channels expe-rienced in wireless systems. For example, the delay spreadvalueL of typical LTE extended typical urban (ETU) chan-nel model is 80. If the number of modulated subcarriers is60 (50 RB), the TSVD-EM needs 3:6� 105 complex mul-tiplications at each iteration and the difference of complexmultiplications between traditional EM and the TSVD-EM becomes 4:35 � 106, which is large and important toLTE systems.

All algorithms analysed earlier are summarised andcompared in Table I. For the SIFI, TSVD-EM and Tradi-tional EM algorithms, all equations and numbers indicatecomplexities in one iteration.



Table I. Complexity comparison.

Algorithm Complexity per iterationExample

NDP D 600, LD 80, LFLTR D 21, SSPD 3

FI NDP � dLFLTR=6e 2400SIFI NDP=SSP�NDP=6CNDP � dLFLTR=SSPe 4300TSVD-EM N2

DP 3:6� 105

Traditional EM N2DPCNDP � L2C L3 4:71� 106

FI, frequency interpolation; SIFI, simple iterative frequency interpolation; TSVD-EM, truncated singular value decomposi-tion expectation-maximisation; EM, expectation-maximisation.

Table II. Simulation parameters.

Parameter Value

FFT size 1024Number of modulated subcarriers 600Allocated RB 50Number of OFDM symbols in one subframe 14Cyclic prefix (samples) 80Transmission mode SIMO (1� 2)Modulation scheme 16-QAMChannel coding rate 1=2Channel coding type Duo-binary turbo codeChannel type ETU70/EVA5 (modified Jake’s Doppler spectrum [19])

OFDM, orthogonal frequency-division multiplexing.

5. SIMULATION RESULTS

In this section, simulation results are presented to showthe benefits of the proposed methods. To assess theperformance of the proposed algorithms, simulations areconducted over ETU channel model with 70-Hz Dopplerfrequency and extended vehicular A (EVA) channel modelwith 5-Hz Doppler frequency that are defined in [18]. Themain parameters are summarised in Table II.

In the simulations, the zero forcing equaliser is used.For the channel estimation on the whole subframe, lineartime interpolation is implemented after frequency domainchannel estimation at each iteration. For the SIFI algo-rithm, from the second iteration, in order to check theperformance of Methods A and B and the impact of SSPvalues, both Methods A and B are simulated with twodifferent spacing values: SSP D 1 and SSP D 3. Forthe TSVD-EM algorithm, initial channel estimation is per-formed with the FI algorithm. Furthermore, because per-formances of channel estimators are affected by the a prioriknowledge of channels, simulations with and without per-fect a priori knowledge are conducted in Sections 5.1 and5.2, respectively.

5.1. Perfect a priori knowledge of channels

In this section, all results are obtained by assuming thatthe perfect a priori knowledge of channels are available.In Figures 5 and 6, packet error rate (PER) performancecurves are shown over ETU70 and EVA5 channel models.

10-3

10-2

10-1

100

4.0 6.0 8.0 10.0

PE

R

SNR (dB)

ETU70 PerCSIFI

SIFI A SSP=3 Iter 5SIFI A SSP=1 Iter 5SIFI B SSP=3 Iter 5SIFI B SSP=1 Iter 5

TSVD-EM Iter 5

Figure 5. Packet error rate performances with 16QAM 1/2 overETU70 channel model.

For the proposed iterative algorithms, performances with 5iterations are shown.

5.1.1. SIFI algorithm.

For the SIFI algorithm, first, both Methods A and Bconverge through iterations with different SSP values. Sec-ond, as expected, performances with SSP D 1 are betterthan those with SSP D 3 for both Methods A and B.Over ETU70 channel, with Method A, the performancewith SSP D 1 is about 0.5 dB better than that withSSP D 3; with Method B, the difference is about 0.3 dB.


65


10-3

10-2

10-1

100

2.0 4.0 6.0 8.0 10.0 12.0

PE

R

SNR (dB)

EVA5 PerCSIFI

SIFI A SSP=3 Iter 5SIFI A SSP=1 Iter 5SIFI B SSP=3 Iter 5SIFI B SSP=1 Iter 5

TSVD-EM Iter 5

Figure 6. Packet error rate performances with 16QAM 1/2 overEVA5 channel model.

The improvement comes from smaller SSP value that intro-duces more soft information from the channel decoder toiterative channel estimation. The more soft informationresults in better channel estimates and better system per-formances. Furthermore, with Figures 5 and 6, we compareperformances of Methods A and B. Over ETU70, MethodsA and B have approximately the same performance withSSP D 3; however, with SSP D 1, Method A outperformsMethod B about 0.3 dB. Over EVA5, we observe the samerelation as we do over ETU70.

Even though the proposed SIFI algorithm improves thePER performance, the degradation compared with the per-formance with perfect channel state information is stilllarge, about 2 dB at 10�3 PER over ETU70 channel modeland about 1dB over EVA5 channel model.

5.1.2. TSVD-EM algorithm.

In Figures 5 and 6, we see that the TSVD-EM alwaysoutperforms the SIFI, and the improvement is on the orderof 1 dB. Furthermore, the PER performances of TSVD-EM approach those with perfect channel state informationby five iterations.

However, from the analysis in Section 4, the TSVD-EMhas a higher complexity than the SIFI.

5.2. Without a priori knowledgeof channels

As presented in Section 3.1, the proposed SIFI algorithmdoes not need any a priori knowledge of channels. How-ever, the TSVD-EM algorithm needs the channel delayspread L to decide the size of the matrix �LDP and toperform the corresponding SVD. In Figure 7, the perfor-mances of TSVD-EM without knowing the exact delayspread are compared with those of SIFI. For the TSVD-EM, different delay spread values are considered in simu-lations, including L, LC 5, L� 15 and L=2.

From Figure 7, we see that the PER performancedegrades when considered delay spread values are smaller

10-3

10-2

10-1

10 0

2.0 4.0 6.0 8.0 10.0

PE

R

SNR (dB)

ETU70 PerCSIFI

SIFI A SSP=1 Iter 5TSVD-EM ( L) Iter 5TSVD-EM ( L+5) Iter 5TSVD-EM (L-15) Iter 5TSVD-EM ( L/2) Iter 5

Figure 7. Packet error rate performances with 16QAM 1/2 overETU70 channel model.

than the exact value. At 10�2 PER, the degradation isabout 2 dB with the value L� 15 and about 1 dB with thevalue L=2. However, if the considered value is greater thanthe exact one, for example, LC5 as shown in Figure 7, theperformance of the TSVD-EM is very close to that with theexact value of delay spread. Compared with the SIFI algo-rithm, the TSVD-EM has some degradation with smallerdelay spread values. For example, at 10�2 PER, the TSVD-EM with the delay spread value L=2 has almost the sameperformance as the SIFI; with the value L�15, the TSVD-EM performs even worse than the SIFI. However, when thevalue is greater than the exact value, the TSVD-EM alwaysoutperforms the SIFI algorithm, about 1 dB improvementwith the value LC 5 at 10�2 PER.

In practice, it is not possible to obtain the exact valueof delay spread but possible to estimate it. Because theTSVD-EM is sensitive to the value of channel delay spread,the accuracy of delay spread estimation is important tothe TSVD-EM algorithm. A large constant value can alsobe set for the TSVD-EM to avoid the delay spread esti-mation. For OFDM-based systems, the value can be thelength of cyclic prefix that is normally greater than channeldelay spread.

6. CONCLUSION

In this paper, two iterative channel estimation algorithmsare proposed for LTE systems. Both of them are derivedfrom the non-iterative conventional interpolation algorithmthat performs frequency domain estimation first followedby a linear time domain interpolation. The first proposedalgorithm is named SIFI that used both the channel esti-mates from pilot symbols and the soft information fromchannel decoders to perform frequency domain interpola-tion with an SSP starting from the second iteration. Thesecond algorithm is TSVD-EM that integrates the tra-ditional EM channel estimation in OFDM systems withthe truncated singular value decomposition. Complexi-ties of the two algorithms are analysed and compared:



the SIFI algorithm has a low complexity and the TSVD-EM reduces the complexity from the traditional EM algo-rithm. Simulation results show that both the SIFI andthe TSVD-EM algorithms improve system performancesthrough iterations.

ACKNOWLEDGEMENT

The research leading to these results has received fundingfrom the European Commission’s seventh framework pro-gramme FP7-ICT-2009 under grant agreement no. 247223also referred to as ARTIST4G.

APPENDIX A: DERIVATION OFGAUSSIAN DISTRIBUTION

From (10), (11) and (12), the distribution of h.0/;Prodk

canbe expressed as

p�h.0/;Prodk

�� CN

�Oh.0/k; �2 .0/

��CN

�Oh.0/;Sk

; �2 .0/;S�

(A.1)

where �2 .0/ D 2�2

jXP;nnj2and �2 .0/;S D 2�2

A

jX.0/

kkj2

,

and where �2 .0/ D 2�2=jXP;nnj2 and �2 .0/;S D

2�2=B

jX.0/kkj2. Using the definition of Gaussian distribu-

tion, (A.1) can be expended as

p�h.0/;Prodk

�� exp

��

1

2�2 .0/

ˇ̌̌h.0/;Prodk

� Oh.0/k

ˇ̌̌2�� exp

��

1

2�2 .0/;S

ˇ̌̌h.0/;Prodk

� Oh.0/;Sk

ˇ̌̌2�

� exp

8<:��2 .0/C �2 .0/;S2�2 .0/�2 .0/;S

24ˇ̌̌h.0/;Prodk

ˇ̌̌2� 2<e

8<:�2 .0/ Oh.0/;SkC �2 .0/;S Oh

.0/k

�2 .0/C �2 .0/;Sh.0/;Prod�k

9=;C�2 .0/

ˇ̌̌Oh.0/;Sk

ˇ̌̌2C �2 .0/;S

ˇ̌̌Oh.0/k

ˇ̌̌2�2 .0/ C �2 .0/;S

3759>=>;

� exp

8̂̂̂̂ˆ̂̂<̂ˆ̂̂̂̂̂:��2 .0/C �2 .0/;S

2�2 .0/�2 .0/;S

266666664ˇ̌̌̌ˇ̌h.0/;Prodk

��2 .0/ Oh

.0/;SkC �2 .0/;S Oh

.0/k

�2 .0/C �2 .0/;S

ˇ̌̌̌ˇ̌2

C�2 .0/

ˇ̌̌Oh.0/;Sk

ˇ̌̌2C �2 .0/;S

ˇ̌̌Oh.0/k

ˇ̌̌2�2 .0/C �2 .0/;S

�

0@�2 .0/ Oh.0/;SkC �2 .0/;S Oh

.0/k

�2 .0/C �2 .0/;S

1A2„ ƒ‚ …

irrelevant to h.0/;Prodk

377777775

9>>>>>>>=>>>>>>>;� exp

8̂<̂:��

2 .0/C �2 .0/;S

2�2 .0/�2 .0/;S

ˇ̌̌̌ˇ̌h.0/;Prodk

��2 .0/ Oh

.0/;SkC �2 .0/;S Oh

.0/k

�2 .0/C �2 .0/;S

ˇ̌̌̌ˇ̌29>=>; (A.2)

The term at the right-hand side of (A.2) is irrelevant to the

variable h.0/;Prodk

and all the parameters in this item arefrom previous iteration that keep constant in the currentiteration. Thus, considering the normalisation of a pdf, thisirrelevant item can be neglected. Then, the mean value andvariance of the new distribution are obtained as in (13).

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