E401: Advanced Communication Theory

E401: Advanced Communication Theory

Professor A. ManikasChair of Communications and Array Processing

Imperial College London

Multi-Antenna Wireless CommunicationsPart-C: Array Receivers for SIMO and MIMO

Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.16.18.c1 1 / 104

Table of Contents7 Antenna Array Receivers for SIMO and MIMO

General Problem Formulation M < NArray Covariance MatrixTheoretical Covariance MatrixPractical Covariance MatrixGenerating L Snapshots having given Cov. Matrix

Summary - General Problem FormulationThe Detection ProblemDetection Criteria: Infinite Observation Interval (L=∞ snapshots))Detection Criteria: Finite Observation Interval (L < ∞ snapshots) - AIC & MDLDetection Problem - Summary

The Estimation ProblemThe Maximum Likelihood (ML) approachSpace Channel Parameter Estimation:Subspace-typeThe Concept of the “Signal Subspace”The Concept of the “Manifold”AmbiguitiesIntersections of Signal Subspace with the Array ManifoldThe MUSIC AlgorithmEstimation of Signal Powers, Cross-correlation etc

The Reception Problem: Array Pattern & BeamformingMain Categories of BeamformersDefinitions - Array PatternSome Popular BeamformersExamples of Array Patterns (Beamformers)Beamformers in Mobile Communications

Array Performance Criteria and BoundsOutput SNIR CriterionOutage Probability CriterionCRB (Estimation Accuracy Bound)Detection and Resolution Bounds

Overall Summary


Antenna Array Receivers for SIMO and MIMO

General Objective

By observing a vector-signal x(t) = Sm(t)+n(t) using an arraysystem the aim is to obtain information about a signal environment.

There are three general problems to solve.

1. The Detection problem: M =?(i.e. to detect the presence of M co-channel emitting sources)

2. The Estimation problem:to estimate various signal and channel parameterse.g. DOAs=?∀i ; Pmi = εm2i (t) =?∀i ; Pn = σ2n =?;ρij = εmi (t)m∗j (t) =?∀i , j , with i 6= jpolarization parameters, fading coeffi cients, signal spread.

3. The Reception problem:to receive one signal (desired signal) and suppress the remaining M − 1as unwanted cochannel interference


Antenna Array Receivers for SIMO and MIMO

These problems are highlighted in the following block structure:


Antenna Array Receivers for SIMO and MIMO General Problem Formulation M < N

General Problem Formulation M<NConsiner an observed (N × 1) complex signal-vector x(t) that is modelledas follows

x(t) ,N×M︷︸︸︷S(p)·

M×1︷︸︸︷m(t) +

N×1︷︸︸︷n(t) (1)

Note that by observing x(t), its 2nd order statistics become known, i.e.the covariance matrix Rxx is known, where

Rxx = Ex(t).x(t)H (2)

Estimate M, p1, p2, ..., pM ,Rmm , σ2n, etc.

where

S4= S(p) =

[S(p1), S(p2), ..., S(pM )

]- (unknown)

m(t) : message signal-vector - (unknown)

Rmm : 2nd order statistics of m(t) - (unknown)

n(t): AWGN vector - (power σ2n unknown)

with

p = the vector of generic (unknown) parameters p1, p2, ..., pMN = known (this is a system parameter)

M = unknown (this is a channel parameter - number of signals)

with M < N (later this condition will be removed)


Antenna Array Receivers for SIMO and MIMO Array Covariance Matrix

Theoretical Covariance MatrixIf the (N × 1) vector-signal x(t) = Sm(t) + n(t) is observed over infiniteobservation interval then its 2nd order statistics can be calculated. Theseare given by the theoretical covariance matrix Rxx which is an(N ×N) complex matrix - always Hermitian. That is,

Rxx , Ex(t)x(t)H

(3)

=

E x1(t)x1(t)∗ , E x1(t)x2(t)∗ , ..., E x1(t)xN (t)∗E x2(t)x1(t)∗ , E x2(t)x2(t)∗ , ..., E x2(t)xN (t)∗

..., ..., ..., ...E xN (t)x1(t)∗ , E xN (t)x2(t)∗ , ..., E xN (t)xN (t)∗

= E

(Sm(t) + n(t)) . (Sm(t) + n(t))H

= E

Sm(t)m(t)HSH + n(t)n(t)H + Sm(t)n(t)H + n(t)m(t)HSH

= S.E

m(t)m(t)H

︸︷︷︸

,Rmm

.SH + En(t)n(t)H

︸︷︷︸

,Rnn

+ SEm(t)n(t)H

︸︷︷︸

=OM ,N

+ En(t)m(t)H

︸︷︷︸ SH

= S .Rmm . SH +Rnn (4)



i.e.

x(t) Rxx , Ex(t)x(t)H

= S .Rmm . SH +Rnn

where

Rmm , Em(t).m(t)H

= 2nd order statistics of m(t) (unknown)

=

E m1(t).m1(t)∗︸︷︷︸Em1(t)2=P1

, E m1(t).m2(t)∗ , ..., E m1(t).mM (t)∗

E m2(t).m1(t)∗ , E m2(t).m2(t)∗︸︷︷︸Em2(t)2=P2

, ..., E m2(t).mM (t)∗

......

...,...

E mM (t).m1(t)∗ , E mM (t).m2(t)∗ , ..., E mM (t).mM (t)∗︸︷︷︸EmM (t)2=PM

= an (M ×M) complex matrix (always Hermitian) - unknown



Rnn , En(t).n(t)H

is 2nd order statistics of n(t)

=

E n1(t).n1(t)∗︸︷︷︸En1(t)2=Pn1

, E n1(t).n2(t)∗︸︷︷︸0

, ..., E n1(t).nN (t)∗︸︷︷︸0

E n2(t).n1(t)∗︸︷︷︸0

, E n2(t).n2(t)∗︸︷︷︸En2(t)2=Pn2

, ..., E n2(t).nN (t)∗︸︷︷︸0

..., ..., ..., ...E nN (t).n1(t)∗︸︷︷︸

0

, E nN (t).n2(t)∗︸︷︷︸0

, ..., E nN (t).nN (t)∗︸︷︷︸EnN (t)2=PnN

= σ2nIN (5)

= an (N ×N) complex matrix (always Hermitian) - unknown

Note that, because we have assumed isotropic AWGN noise,

Pn1 = Pn2 = ... = PnN = σ2n (6)



Practical Covariance MatrixConsider that the signal x(t) = Sm(t) + n(t) is observed over finiteobservation interval equivalent to L snapshots.These L observations (snapshots) at times t1, t2, ..., tL (i.e. finiteobservation interval) are denoted as [x(t1), x(t2), ..., x(tL)] andrepresented by the N × L complex matrix X

i.e.

X , [x(t1), x(t2), ..., x(tL)] (7a)

=[S.m(t1) + n(t1), S.m(t2) + n(t2), ..., S.m(tL) + n(tL)

]= S.M+N (7b)

with

S = [S1, S2, ..., SM ] (N ×M)M = [m(t1), m(t2), ..., m(tL)] (M × L)N = [n(t1), n(t2), ..., n(tL)] (N × L)

(8)

where the matrices S, M and N (as well as the dimension M) areunknown



In this case the 2nd order statistics of x(t) are estimated by thepractical covariance matrix Rxx

Practical Model:

x(t) Rxx =1L

L

∑l=1

x(tl )x(tl )H

i.e. Rxx =1L

XXH

= S1L

MMH︸︷︷︸=Rmm

SH +1L

NNH︸︷︷︸=Rnn

= S .Rmm . SH +Rnn

N.B.: In an array system the matrix Rxx (theoretical or practical)contains all the geometrical information about the various sourcesrelative to the array.



Generating L Snapshots having a given Covariance MatrixTo generate L snapshots of x(t) having a predefined covariancematrix Rxx the vectors x(tl ) for l = 1, 2, .., L should be generatedusing the following expression

x(tl ) = E D12 z(tl ) (9)

whereI E and D are the eigenvector-matrix and eigenvalue-matrix of Rxx , andI z(tl ) ∈ CN is a Gaussian random complex vector of N elements of zeromean and variance 1, i.e.

Ez(tl ).z(tl )H = IN (10)

That is,

X =[x(t1), x(t2), ..., x(tL)

]=

[E D

12 z(t1), E D

12 z(t2), ..., E D

12 z(tL)

](11)



Proof:

Let z ∈ CN such as Ez .zH = IN . Then

Rxx = Ex .xH (12)

However,

Rxx = EDEH = ED12 D

12 EH = ED

12 IND

12 EH

= ED12 Ez .zH︸︷︷︸

=IN

D12 EH

= EED12 z︸︷︷︸

x

.zHD12 EH︸︷︷︸

xH

(13)

By comparing Equation 13 and 12 we have

x = E D12 z (14)

Note: In a similar fashion L snapshots of m(t), n(t) or any othervector-signal can be generated.


Antenna Array Receivers for SIMO and MIMO Summary - General Problem Formulation

Summary - General Problem Formulation

Condition: M < N

Estimate M, p = [p1, p2, ..., pM ]T ,Rmm , σ2n, etcwhere pi is a parameter of interest associated with the i th source.


Antenna Array Receivers for SIMO and MIMO The Detection Problem

The ‘Detection’Problem

This is to determine the parameter M

i.e. to determine the number of signals and thus the dimensions ofthe vectors/matrices m(t), S, Rmm and M

In other words, to detect how many emitting sources/transmitters arepresent in an array environment

i.e. to detect the presence of M sources



Detection Criteria: Infinite Observation IntervalBased on x(t)we can form the matrix Rxx (representing the statisticsof x(t)):

Rxx , Ex(t)x(t)H

= S ·Rmm · SH︸︷︷︸

=Rsignals

+ Rnn︸︷︷︸=σ2nIN

(15)

When the number of sources M is smaller than the number of systemdimensions N (e.g. number of array-sensors) then the determinant ofthe Rsignals is equal to zeroi.e.

if M < N ⇒ det(Rsignals) = 0 (16)

This is due to the fact that the presence of an emitting sourceincreases the rank of the matrix Rsignals by one.i.e.

rank Rsignals = M (17)

⇒ rank

Rxx − σ2nIN= M (18)



However, using eigen-decomposition of Rxx we have

Rxx = E.D.EH (19)

where

D =

d1, 0, ..., 0, 0, 0, ..., 00, d2, ..., 0, 0, 0, ..., 0..., ..., ..., ..., ..., ..., ..., ...0, 0, ..., dM , 0, 0, ..., 00, 0, ..., 0, dM+1, 0, ..., 00, 0, ..., 0, 0, dM+2, ..., 0..., ..., ..., ..., ..., ..., ..., ...0, 0, ..., 0, 0, 0, ..., dN

(20)

= (N ×N) matrix



An alternative way to express Rxx as the addition of signals and noisecovariance matrices and then to eigen-decompose the signalcovariance matrix Rsignals . That is,

Rxx = Rsignals +Rnoise

= E.Λ.EH + σ2nIN = E.Λ.EH + σ2nE.EH︸︷︷︸

IN

= E.(Λ+ σ2nIN

)︸︷︷︸D

.EH (21)

where Λ =

λ1, 0, ..., 0, 0, 0, ..., 00, λ2, ..., 0, 0, 0, ..., 0..., ..., ..., ..., ..., ..., ..., ...0, 0, ..., λM , 0, 0, ..., 00, 0, ..., 0, 0, 0, ..., 00, 0, ..., 0, 0, 0, ..., 0..., ..., ..., ..., ..., ..., ..., ...0, 0, ..., 0, 0, 0, ..., 0

(22)

= (N ×N) matrixProf. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.16.18.c1 17 / 104


That is, from Equ 21 in conjunction with Equs 20 and 22 we have

D = Λ+ σ2nIN (23)

D =

λ1 + σ2n︸︷︷︸=d1

0 ... 0 0 0 ... 0

0 λ2 + σ2n︸︷︷︸=d2

... 0 0 0 ... 0

... ... ... ... ... ... ... ...

0 0 ... λM + σ2n︸︷︷︸=dM

0 0 ... 0

0 0 ... 0 σ2n↑

=dM+1

0 ... 0

0 0 ... 0 0 σ2n↑

=dM+2

... 0

... ... ... ... ... ... ... ...0 0 ... 0 0 0 ... σ2n

↑=dN



This implies that the eigenvalues of the data covariance matrixRxx (i.e. the diagonal elements of D) are related to the eigenvaluesof the emitting signals covariance matrix Rsignals (i.e. diagonalelements of Λ) as follows:

eigi Rxx = eigi Rsignals+ σ2n

di = λi + σ2n (24)

Now, since the smallest eigenvalue of Rsignals is zero

eigmin Rsignals = 0 (25)

with multiplicity N −M, that meanseigmin Rxx = σ2n (26)

with multiplicity also N −M.Therefore, theoretically, the number of emitting sources M can bedetermined by the eigenvalues of the covariance matrix Rxx of the Rxsignal-vector x(t), and more specifically by the following expression

M = N−(multiplicity of minimum eigenvalue of Rxx ) (27)Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.16.18.c1 19 / 104


Note: another useful expression is

Rxx = E.D.EH = [Es,En].

[Ds O

O Dn

][Es,En]

H

= Es.Ds.EHs +EnDnEH

n (28)

where



Detection Criteria: Finite Observation Interval (L=finite)AIC and MDL criteria

D. =

λ1 + σ21︸︷︷︸=d1

0 ... 0 0 0 ... 0

0 λ2 + σ22︸︷︷︸=d2

... 0 0 0 ... 0

... ... ... ... ... ... ... ...

0 0 ... λM + σ2M︸︷︷︸=dM

0 0 ... 0

0 0 ... 0 σ2M+1↑

=dM+1

0 ... 0

0 0 ... 0 0 σ2M+2↑

=dM+2

... 0

... ... ... ... ... ... ... ...0 0 ... 0 0 0 ... σ2N

↑=dN

(29)



In theory,

σ21 = σ22 = ... = σ2M = σ2M+1 = .... = σ2N ,σ2n (30)

However, in practice

σ21 6= σ22 6= ... 6= σ2M 6= σ2M+1 6= .... 6= σ2N (31)

butσ2n ≈ σ21 ≈ σ22 ≈ ... ≈ σ2M ≈ σ2M+1 ≈ .... ≈ σ2N (32)

if M is known then .

σ2n = the average of the N −M smallest eigenvalues

=1

N −M(σ2M+1 + σ2M+2 + ....+ σ2N

)(33)



AIC - Detection Criterion

AIC = −2L

ln

∏Ni=1 di...

d1d2d3d1d2d1

+

NN − 1...321

ln

NN − 1...321

− ln

∑Ni=1 di....

d1 + d2 + d3d1 + d2d1

+2

01...

N − 2N − 1

2N2N − 1...

N + 2N + 1

an (N × 1) real vector (34)



Minimum Description Length (MDL) Detection Criterion

MDL = −L

ln

∏Ni=1 di...

d1d2d3d1d2d1

+

NN − 1N − 2...21

ln

NN − 1N − 2...21

− ln

∑Ni=1 di....

d1 + d2 + d3d1 + d2d1

+12ln L

01...

N − 2N − 1

2N2N − 1...

N + 2N + 1

an (N × 1) real vector (35)



remember: di , (for i = 1 to N with d1 < d2 < . . . < dN ) denotes thei-th eigenvalue of Rxx

Notes:I if the first element of the vector AIC or MDL is minimumthen M = 0

I if the second element of the vector AIC or MDL is minimumthen M = 1

I if the third element of the vector AIC or MDL is minimumthen M = 2

I etc.

Reference: M. Wax and T. Kailath, "Detection of Signals byInformation Theoretic Criteria", IEEE Transactions on ASSP, vol. 33,pp. 387-392, Apr. 1985.



Detection Problem - SummaryIf L = ∞ , i.e. Theoretical Rxx = E

x(t).x(t)H

,

then

M = N − (multiplicity of min. eigenvalue of Rxx ) (36)

noise power σ2n = min.eigenvalue of Rxx (37)

If L =finite, i.e. Practical Rxx =1L

L

∑l=1

x(tl ).x(tl )H = 1LX.XH

then

M = can be found using AIC or MDL (38)

σ2n = the average of the N −M smallest eigenvalues

= 1N−M

(σ2M+1 + σ2M+2 + ....+ σ2N

)(39)

Remember:

I N =number of array elementsI M = number of signals/sources


Antenna Array Receivers for SIMO and MIMO The Estimation Problem

The Estimation ProblemThe Maximum Likelihood (ML) approach

Consider an observed (N × 1) complex signal-vector x(t) modelled asfollows

x(t)4= S(p).m(t) + n(t) (40)

In this case the L observations at times t1, t2,...,tL (i.e. finiteobservation interval) are

[x(t1), x(t2), . . . , x(tL)] defined as the N × L complex matrix X

since the noise is modelled as a zero mean complex Gaussian random

process, with a covariance matrix Rnn4= σ2IN then the observed

array signal x(t) has a mean vector and covariance matrix which aregiven as follows:

Ex(t) = S(p).m(tl )

E(x(t)− Ex(t)) . (x(t)− Ex(t))H︸︷︷︸Rnn

= σ2nIN



This implies that if there are L observations, which are independent,then the conditional probability density function (likelihood function -LF)

LFx4= pdfx

(x(t1), x(t2), . . . , x(tL)

∣∣p,M, σ2n)

(41)

ML Solution:

M = (S(p)HS(p))−1S(p)HX (42)

m(ti ) = (S(p)HS(p))−1S(p)H︸︷︷︸,WH

ML

x(ti ) (43)

pML

= argmaxpLFx

= argmaxpTr (PS.Rxx ) (44)

wherePS = S(p)

(SH (p)S(p)

)−1S(p)H (45)



Space Channel Parameter Estimation:Subspace-type

In this type of algorithms the parameter M is assumed known(M < N) and involves in some way, or another, two concepts:

i) the concept of the "signal-subspace" associated with the observedsignal-vector x(t) and its properties

F This is an unknown linear subspace of dimensionality equal M -embedded in an N-dimensional (M < N)

ii) the concept of the "manifold" associated with the system/problem’scharacteristicsin the case of arrays known as "array manifold". It isindependent of the noisy observed signal-vector x(t) and its properties.

F This is a non-linear subspace (e.g. a curve, surface etc) - embedded inan N-dimensional observation space



Solution = M points

∈ system’s manifold∈ ‘signal subspace’ of x(t)

As a result the objective is firstly, from the data, to estimate thesignal subspace and then to search the manifold to find itsintersection with the estimated signal-subspace.



The Concept of the “Signal Subspace”

The first step is to utilize the observed (received) signal vector

x(t) =

,m1(t)︷︸︸︷β1m1(t)S1 +

,m2(t)︷︸︸︷β2m2(t)S2 + ....+

,mM (t)︷︸︸︷βMmM (t)SM

+n(t) (46)

= Sm(t) + n(t), ∀t (infinite observation interval) (47)

or, over L snapshots (finite observation interval)

to estimate the “signal subspace ”.The “signal subspace ”should have dimensionality (in most cases)equal to M (known - or estimated) and the signal term S.m(t)belongs always to this subspace.i.e.

dim(signal subspace ) = MSm(t) ∈ signal subspace (48)



The Subspace of a Manifold (Response) Vector

Let us consider the subspace spanned by only one signal-term ofEquation 46, for instance the first term m1(t)S1 at t = t1 and t = t2as well as the manifold vector S1 are shown as below:

It is clear from the above figures that

L[S1] = L[m(t)S1, ∀t] (49)



The Subspace of a Manifold (Response) Vector

For instance consider a SIMO multipath channel

Here the symbols/vectors S and a are equivalent and both denote anarray manifold vector



The Subspace of a Single Signal/Source

m(t)a⇒ L[βm(t)a] = L[a] where a , a(p)



The Subspace of two Signals/Sources

m1(t)S1 +m2(t)S2 ⇒ L[m1(t)S1 +m2(t)S2] = L[S1,S2]



The Subspace of the received signal-vector (t)

Furthermore, Equation 47 can be represented as follows(at a specific time t):

⇒

That is,

”subspace of x(t), ∀t” = L[x(t), ∀t] =whole obs. spacewith dim (L[x(t), ∀t]) = N (50)



As the dimensionality of this subspace is M and the number ofcolumns of S is equal to M (remember S = [S1, S2, ..., SM ]) thisimplies that

”signal subspace” = L[S]with dim (L[S]) = M (51)

That is, the signal subspace is spanned by the unknown M manifoldvectors associated with the M signals (one signal - one vector)

The complement subspace tothe signal subspace is known as“noise subspace”i.e.

”noise subspace” = L[S]⊥

with dim(L[S]⊥) = N −M



Signal-Subspace type techniques are based on partitioning theobservation space into

I the Signal Subspace L[S] andI the Noise Subspace L[S]⊥

However, as the matrix S

remains unknown, the signalsubspace and consequently thenoise subspace remainunknown.

Note:

observation-space , L[X]

= L[Rxx ]

= L[x(t), ∀t](52)



Estimation of the two subspaces: This is achieved by performing anEigenvector decomposition of the received data covariance matrixRxx,i.e.

Rxx = E.D.EH = [Es,En].

[Ds O

O Dn

][Es,En]

H (53)

Rxx = Es .Ds .EHs +En.Dn.E

Hn (54)

This implies that

signal subspace = L[S] = L[Es ] = L[En ]⊥

noise subspace = L[S]⊥ = L[Es ]⊥ = L[En ]remember: S 6= Es although L[S] = L[Es ]observation space = L[Rxx ] = L[x(t), ∀t] = L[X]

(55)



The Concept of the "Manifold"One-parameter Manifolds in Wireless Comms

Consider the manifold vector (or array response vector) of a singleparameter p representing for instance θ or φ or Fci.e.

a(p) ∈ CN

This is a single-parameter vector function.



Locus of the Manifold Vectors



By recording the locus of the manifold vectors as a function of theparameter p ( e.g. direction), a “continuum”(i.e. a geometricalobject such as a curve) is formed lying in an N-dimensional space.This geometrical object (locus of manifold vectors i.e. S(p),∀p) isknown as the array manifold.In an array system the manifold (array manifold) can be calculated(and stored) from only the knowledge of the locations and directionalcharacteristics of the sensors.

Let S(p) ∈ CN be the manifold vector of a system of N dimensions(e.g. of an array of N sensors) where p is a generic system parameter.This is a single-parameter vector function and as p varies the pointS(p) will trace out a curve A (see figure), embedded in anN-dimensional space CN .



Important Parameters of a Curve



Array of Changing Geometry

By changing the array geometry the corresponding curve will change



Length of a Curve

Example: important parameters:

I arclengths , s(p) =p∫0

∥∥S(p)∥∥ dpI rate-of-change ofarclengths , s(p) =

∥∥S(p)∥∥I length of manifoldlm

I curvaturesa set of real numbersκ1, κ2, κ2, etc(curve’s shape)

A curve may have "bad" areas (.s=small) and "good" areas (

.s=large)



"Bad" CurvesThere are "bad" and "good" curves (i.e. "bad" and "good" antennageometries)Example of a "bad" curve:



Ambiguities



Linear Antenna ArraysAll linear array geometries have manifolds of "hyperhelical" shapeembedded in N-dimensional complex space.

Visualisation of a

hyperhelix in 3D

C4=

0 −κ1 0 · · · 0 0κ1 0 −κ2 · · · 0 00 κ2 0 · · · 0 0...

......

. . ....

...0 0 0 · · · 0 −κd−10 0 0 · · · κd−1 0

I Curvatures: forming a matrix known as the Cartan Matrix C.I Hyperhelices: curvatures=constant for every s or pI As we have infinite number of linear arrays we have infinite number ofdifferent hyperhelical curves - different set of curvatures



Linear Antenna Array Design

The Frobenius norm of the Cartan matrix (i.e. ‖C‖F ) is related tothe array symmetricity as follows:

1) ‖C‖F = 1 if array = symmetric

2) ‖C‖F =√2 if array = fully asymmetric

3) 1 < ‖C‖F <√2 if array = partially symmetric

eig(C)⇒antennas location r x



Manifold Surfaces in Wireless CommsTwo-Parameter Manifolds: Visualisation



In a similar fashion if there are two (unknown) parameters (p,q) persignal then S(p, q) ∈ CN is a two-parameter manifold vector (avector function)and as (p,q) varies the point S(p, q) ∈ CN will form a surfaceM(see figure), formally defined as follows

Array Manifold: M 4= S(p, q) ∈ CN , ∀(p, q) : p, q ∈ Ω (56)

where Ω denotes the parameter space.

This surfaceM is the locus ofall manifold vectorsS(p,q); ∀p,q



For a point (p, q) on themanifold surface the mostimportant parameters are:

I The Gaussian curvature:KG (p, q)

I The manifold metric:G(p, q)

I The Christoffel matrices:Γ(p, q)

For a curve on the manifoldsurface, the parameters ofinterest are:

I The arc length: sI The geodesic curvature: κg(curve=geodesic⇒ κg = 0)



Some Important ResultsAll planar antenna geometries (2-Dim arrays) have:

KG = 0 (always)⇒ flat or parabolic of conoid shapewith the apex at point φ=90

Examples:

Note: ‘flatness’ doesnot imply that thereexist straight lines , asin the case of a surface inR3.It means that suchsurfaces can begenerated by rotating apassing geodesic curvearound an apex point.



Manifold Surfaces as Families of Curves



Manifold Surfaces: ParameterisationThere are many parameterisations of a hypersurface.We should always try to find a parameterization that makes solvingthe problem easier.

e.g.:"Cone"-angles: (α, β) where

cos α = cos φ cos(θ −Θ)cos β = cos φ sin(θ −Θ)

where Θ =const



Planar Antenna Array Design Based on Curves



Intersections of Signal Subspace with the Array Manifold

Both the manifold and L[Es] are embedded on the sameN-dimensional observation space

=⇒

Therefore, the intersection of the manifold with L[Es ] will providethe end-points of the columns of the matrix S, i.e. it will provide theparameters p1, p2, ..., pM .



Example:

Note:

S1,S2,E 1,E 2 ⊥ L[En ]

S1,S2,E 1,E 2 ∈ L[Es ]

L[Es] is a plane which intersects the array manifold in 2 points S1,S2Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.16.18.c1 63 / 104


The MUSIC Algorithm

It estimates the intersection L[Es] and the array manifold byemploying the following procedure:

I Let p denote a parameter value.I Form the associated S(p) and then project S(p) on to the subspaceL[En ]. This will give us the vector

z(p) = PEn .S(p) (57)



I The norm-squared of z(p) can be written as

ξ(p) = z(p)H z(p)

= S(p)H .PHEn︸︷︷︸=PEn

.PEn

︸︷︷︸=PEn

.S(p)

= S(p)H .PEn .S(p)

= S(p)H .En .(EHn En︸︷︷︸=IN×N

)−1.EHn .S(p)

= S(p)H .En .EHn .S(p) (58)

I It is obvious that ξ(p) = 0 iffp = p1 orp = p2

I Therefore, we search the array manifold, i.e. we evaluate the expression(58),∀p , and we select as our estimates the p’s which satisfy

ξ(p) = 0⇒ S(p)H .En .EHn .S(p) = 0, ∀p (59)



Equation-59 is known as the Multiple Signal Classification (MUSIC)algorithm and it is a Signal-Subspace type technique.MUSIC Algorithm in Step format:

step-0. assumptions: M and array geometry are knownstep-1. receive the analogue signal vector x(t) ∈ CN×1or its discrete version

x(tl ), for l = 1, 2, ..., Lstep-2. find the covariance matrix

Rxx =

Ex(t).x(t)H

∈ CN×N in theory

1L

L

∑l=1

x(tl ).x(tl )H ∈ CN×N in practice

step-3. find the Eigenvectors and Eigenvalues of Rxxstep-4. form the matrix Es with columns the eigevectors which correspond to

the M largest eigenvalues of Rxxstep-5. find the arg of the M minima of the function

ξ(p) = S(p)H .Pn .S(p), ∀pwhere Pn = IN −PEs and PEs is the projection operator onto LEs,

i.e. [p1, p2, .., pM ]T = argmin

pξ(p)



Example (see Experiment AM1)

Example of MUSIC used in conjunction with a Uniform Linear Arrayof 5 receiving elements.

The array operates in the presence of 3 unknown emitting sourceswith DOA’s (30, 0), (35, 0), (90, 0)

0 20 40 60 80 100 120 140 160 1805

0

5

10

15

20

25

30

35MuSIC spectrum (Theoretical) SNR=10dB

Azimuth Angle degrees

dB



MUSIC Limitations:I MUSIC breaks down if some incident signals are coherent, i.e. fullycorrelated, (e.g. multipath situation or ‘smart’jamming case)

I Then L[Es ] 6= L[S] or, to be more precise, L[Es ]∈ L[S]I Therefore the ‘intersection’argument cannot be used.I e.g. same environment as before but the (30,0) & (35, 0) sourcesare coherent (fully correlated)

0 20 40 60 80 100 120 140 160 1805

0

5

10

15

20

25

30

35MuSIC spectrum (coherent)


dB

∃ algorithms which can handle coherent signals in conjunction wothMUSIC: spatial smoothing (see AM1 experiment)



Estimation of Signal Powers, Cross-correlation etc

Firstly estimate the DOA’s and noise power and then use the conceptof ‘pseudo inverse’to estimate Rmmi.e.

step− 1 : Based on Rxx , estimate p and σ2n

step− 2 : form S

step− 3 : Rmm = S# · (Rxx − σ2nIN ) · S#H (60)

where S# = (SH · S)−1 · SH = pseudo-inverse of S (61)

Note that:σ2n =(min eigenvalue of Rxx or given by Equation-33)

=noise power



Proof of Equation-60 :

Rxx = S.Rmm .SH + σ2nIN (62)

Rxx − σ2nIN = S.Rmm .SH

By pre & post multiplying both sides of the previous equation withthe pseudo inverse of S we have

S# .(Rxx − σ2nIN

).S#H =

=S#︷︸︸︷(SHS)−1SH .S .Rmm .S

H .

=S#H︷︸︸︷S(SHS)−1

=⇒ S# · (Rxx − σ2nIN ) · S#H = Rmm


Antenna Array Receivers for SIMO and MIMO The Reception Problem: Array Pattern & Beamforming

Main Catergories of BeamformersCategory-1 : Single sensor with directional response.

Green Bank Telescope, National Radio Astronomy Observatory, WestVirginia.100 m clear aperture; Largest fully steerable antenna in the world.



Main Catergories of Beamformers

Horn- Antenna : Another example of Category-1 (Single sensorwith directional response).



Main Catergories of BeamformersCategory-2 : Array of Sensor

I Used in SONAR, RADAR, communications, medical imaging, radioastronomy, etc.

I Line array of directional sensors: Westerbork Synthesis Array RadioTelescope, (WSRT), the Neterlands.



Main Catergories of Beamformers (Array of Sensors)switched beamformer : there is a finite number of fixed arraypatterns and the system chooses one of them to maximise signalstrength (the one with main lobe closer to the desired user/signal)and switches from one to beam to another as the user/signal movesthroughout the sector).adaptive beamformer (or adaptive array): array patterns areadjusted automatically (main lobe extending towards a user/signalwith a null directed towards a cochannel user/signal)

switched beamformer adaptive beamformer Adaptive for 2 cochannel signals/users



Adaptive Beam (ULA, N=5, d=1)



Adaptive Beam (Two co-Channel Signals)

ULAN = 5d = 1



The ‘Reception’Problem: Array Pattern & BeamformingDefinitions

If the array elements are weighted by complex-weights then the arraypattern provides the gain of the array as a function of DOAse.g.

if θ 7−→ S (θ) then g(θ) = wHS(θ) (63)

where g(θ) denotes the gain of the array for a signal arriving fromdirection θThen,

g(θ), ∀θ : is known as the array pattern (64)

The array pattern is a function of the array manifold S(θ) (i.e. arraygeometry and channel parameter θ) and the Rx weight vector w



N.B.: default pattern:g(θ) = 1TNS (θ) (65)

i.e. w = 1N (i.e. no weights)e.g. Array Pattern of a Uniform Linear Array of 5 elements(w = 15, i.e. no weights)

0 20 40 60 80 100 120 140 160 18020

15

10

5

0

5

10initial pattern


gain

in d

B

BeamwidthThe array pattern has anumber of lobes.

I the largest lobe is called the‘main lobe’ while

I the remaining lobes areknown as ‘sidelobes’.

beamwidth = 2 sin−1(

λ

Nd

)× 180

π(66)



Note that d =inter-sensor-spacing, and

if d =λ

2⇒ beamwidth = 2 sin−1

(2N

)× 180

π(67)

To steer the main lobe towards a specific (known) direction θ, a‘spatial correction weight’ wmain-lobe can be used which should beequal to

wmain-lobe = exp(−jrT kmain-lobe) (68)

wmain-lobe = S(θmain-lobe) (69)



ARRAY PATTERN: for arrays of N = 1, 2, 3, 4 and 5 sensors(Mainlobe at 90, d = 1 half-wavelength)

r x = [0] =

[−0.50.5

]T=

−10+1

T =

−1.5−0.50.51.5

T

=

−1.5−0.500.51.5

T

w →= = [1,1]T = [1,1,1]T = [1,1,1,1]T = [1,1,1,1,1]T



ARRAY PATTERN for ULA arrays of N = 2, 3, 4, 5 sensors(mainlobe at 120, d = 1 half-wavelength)

r x = [0] =

[−0.50.5

]T=

−10+1

T =

−1.5−0.50.51.5

T

=

−1.5−0.500.51.5

T

w = S(120, 0) = exp(−j rT k(120, 0)) (70)

(simplified to) = exp(−jπr x cos(120)) (71)Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.16.18.c1 82 / 104


A beamformer is an array system which receives a ‘desired’signaland suppresses (according to a criterion) co-channel interferenceand noise effects, by synthesizing an array pattern with high-gaintowards the DOA of the desired signal and deep nulls towards theDOAs of the interfering signals (adaptive arrays).

The state of the art: Superresolution ‘blind’beamformersProf. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.16.18.c1 83 / 104


Some Popular Beamformers

WIENER-HOPF Beamformer:

w = c.R−1xx Sdesiredsignal

(72)

where c = a constant scalar

I Maximizes the SNIR at the array output.I It is optimum wrt SNIR criterionI It is a conventional beamformer (i.e. resolution is a function of theSNRin)

I No need to know the DOAs of the interfering signalsI (please try to prove Equation-72)



Modified WIENER-HOPF Beamformer:

w = c ·R−1n+JSdesiredsignal

(73)

where c = a constant scalar

I comments similar to Wiener-HopfI robust to ‘pointing’errors (i.e. robust to errors associated with thedirection of the desired signal)



Minimum Variance BeamformerI It is also known as Capon’s beamformerI It is the beamformer that solves the following optimisation problem

minw

(wHRxxw

)(74)

subject to wHS(θ) = 1 (75)

I Equation 74 aims to minimise the effect of the desired signal plus noise.However, the constraint wHS(θ) = 1 (Equ 75) prevents the gainreduction in the direction of the desired signal.

I Solution of Equations 74 and 75:

w = c .R−1xx Sdesiredsignal

(76)

where c = a constant scalar chosen such as wHS(θ) = 1



A Superresolution Beamformer based on DOA estimation:

w = P⊥SJSdesiredsignal(77)

where S = [Sdesiredsignal

, SJ ]

I Provides complete (asymptotically) interference cancellation.I Maximizes the SIR at the array output.I It is optimum wrt SIR criterionI It is a superresolution beamformer (i.e. resolution is not a function ofthe SNRin)

I Needs an estimation algorithm to provide the DOAs of all incidentsignals.



A Superresolution Beamformer not based on DOA estimation ofinterfering sources

w = P⊥Enj· Sdesired

signal(78)

where Enj =noise subspace of Rn+J

Note: Rn+J =covariance matrix where the effects of the

desired signal have been removed

Maximum Likelihood (ML) Beamformer

w = coldes(

S#)

(79)

where S# = S.(SHS)−1 (80)



Examples of Array Patterns (Beamformers)

Consider a uniform linear array of 5 elements operating in thepresence of three signals with directions (90, 0), (30, 0) and(35, 0). One of the signals is the ‘desired’signal and the other twoare unknown interferences.

Initially, the ‘desired’DOA is

(90, 0) = known

and the DOAs of interfering sources are:

(30, 0) = unknown

(35, 0) = unknown



WIENER-HOPF Beamformer (Equ. 72): Superresolution Beamformer (Equ. 78)

SNR=40dB (high) or 10dB (low) SNR=10dB and 40dB

0 20 40 60 80 100 120 140 160 18045

40

35

30

25

20

15

10

5

0

5WH array pattern SNR=40dB


gain

in d

B

0 20 40 60 80 100 120 140 160 180160

140

120

100

80

60

40

20

0

20array pattern (90 deg) Complete Interference Cancellation


gain

in d

B

0 20 40 60 80 100 120 140 160 18030

25

20

15

10

5

0WH array pattern SNR=10dB


gain

in d

B



Superresolution Beamformer (Equ.-77) (all DOAs known).

a) desired source=30 b) desired source=35

0 20 40 60 80 100 120 140 160 180160

140

120

100

80

60

40

20

0



gain

in d

B

0 20 40 60 80 100 120 140 160 180160

140

120

100

80

60

40

20

0



gain

in d

B



Beamformers in Mobile Communications

Some Applications of Beamformers in Communications:

1 analogue access methods FDMA (e.g. AMPS, TACS, NMT)2 digital access methods TDMA (e.g GSM, IS136), CDMA3 duplex methods FDD, TDD

Main properties of beamforming in Communications

Properties Advantages1 signal gain better range/coverage(see figure below)2 interference rejection increase capacity3 spatial diversity multipath rejection4 power effi ciency reduced expense



Coverage patterns for switched beam and adaptive array antenna


Antenna Array Receivers for SIMO and MIMO Array Performance Criteria and Bounds

Array Performance Criteria and BoundsIntroduction

Two popular performance evaluation criteria are:

I SNIRoutI Outage Probability

Three Array Bounds

I Detection BoundI Resolution BoundI Cramer-Rao Bound (estimation accuracy)



SNIRout Criterion

The signal at the output of the beamformer can be expressed as

y(t) = wHx(t) = wH (Sm(t) + n(t))

= wH (S1m1(t) + SJmJ (t) + n(t))

where

S = [S1,S2, . . . , SM︸︷︷︸,SJ

]

m(t) = [m1(t),m2(t), . . . ,mM (t)︸︷︷︸,mTJ

]T



Power of y(t) :

Py = Ey(t)2

=

= E y(t)y(t)∗ = EwHx(t)x(t)Hw

= wHE

x(t)x(t)H

︸︷︷︸

=Rxx

w

= wH

P1S1SH1︸︷︷︸4=Rdd

+ SJRmJmJSHJ︸︷︷︸

4=RJJ

+ σ2IN︸︷︷︸4=Rnn

w= wH (Rdd +RJJ +Rnn)w

(assuming desired, interfs & noise are uncorrelated)



i.e.Py = wHRddw︸︷︷︸

=Pd ,out

+ wHRJJw︸︷︷︸=PJ ,out

+ wHRnnw︸︷︷︸=Pn,out

where

Pd ,out = o/p desired term

= P1wHS1SH1 w = P1(w

HS1)2

PJ ,out = o/p interf. term

=M

∑i=2

M

∑j=2

ρijwHS iS

Hj w with ρij , corr.coeff

Pn,out = o/p noise term

= σ2nwHw



Therefore,

SNIRout =Pd ,out

PJ ,out + Pn,out=

P1(wHS1)2

M

∑i=2

M

∑j=2

ρijwHS iS

Hj w︸︷︷︸

wHSJRmJmJ SHJ w

+ σ2nwHw

(81)

Note:I An alternative equivalent expression to Equ 81 is given below

SNIRout =wHRddw

wH (Rxx −Rdd ) w(82)

I Both Equations 81 and 82 are general expressions for any beamformer.However, for different beamformers (i.e. different weights) theseequations give different values.



Outage Probability Criterion

outage probability (OP) is defined as follows:

OP = Pr(SNIRout < SNIRpr ) (83)

or

OP = Pr(SIRout < SIRpr ) (84)

It is a performance evaluation criterion.

An example of an array-CDMA system’s Outage Probability withN = 1, 8, 15 receiving elements is shown below clearly showing thatby employing an antenna array and using, for instance, thebeamformer of Equation 77 (complete interference cancellationbeamformer) at the base station, the system capacity is increased.



Outage Probability Examples/Graphs

For example, for 0.001 outage probability, the system capacity per cellincreases from 20 mobiles, for a single antenna case (i.e. N = 1), toabout 40 and 80 mobiles for N equal to 8 and 15, respectively.



Uncertainty Hyperspheres and CRBModel the uncertainty remaining in the system after L snapshots as anUncertainty Hypersphere of effective radius σe

σe =√CRB[s ] (85)

=

√1

2 (SNR× L)CCRB = Cramer-Rao Bound

[see chapter-8 of my book]: The uncertainty sphere represents the smallest achievable

uncertainty (optimal accuracy) due to the presence of noise after L snapshots, when all

the effects of the presence of other sources have been eliminated by an “ideal” parameter

estimation algorithm

The Parameter C (0 < C ≤ 1): models any additional uncertainty introduced by apractical parameter estimation algorithm. N.B.: Ideal algorithm: C = 1



Detection and Resolution Bounds



Detection and Resolution BoundsAngular Separation: Resolution & Detection Laws

Note: σe ∝ −2√SNR × L; where L =number of snapshotsSquare-root Law :

Detection: ∆p = f.s, σe (86)

Fourth-root Law :

Resolution: ∆pth = f.s, 4√

κ1,√

σe (87)

Remember - Frequency Selective Channels :I number of resolvable paths=

⌊Delay -Spread

channel-symbol period

⌋+ 1

⇓I two paths with a relative delay < channel-symbol-period

cannot be resolved

for more info see Chapter 8 of my bookProf. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.16.18.c1 103 / 104

Antenna Array Receivers for SIMO and MIMO Overall Summary

Summary


E401: Advanced Communication Theory

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