E401: Advanced Communication Theory Receivers_SIMO_MIMO_2019.pdfE401: Advanced Communication Theory...

E401: Advanced Communication Theory

Professor A. ManikasChair of Communications and Array Processing

Imperial College London

Multi-Antenna Wireless CommunicationsArray Receivers for SIMO and MIMO

Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 1 / 128

Table of Contents

1 General Objectives 32 General Problem Formulation M < N 5

Array Covariance Matrix 6Theoretical Covariance MatrixPractical Covariance Matrix

Generating L Snapshots having given Cov. Matrix 11Summary - General Problem Formulation 13

3 The Detection Problem: Basic Detection Theory 14Hypothesis and Hypothesis Testing 15Terminology 17Hypothesis Testing re-Defined 19Antenna Array: Hypothesis Testing 20Infinite Observation Interval (Infinity Snapshots) 22Finite Observation Interval (Finite Snapshots) 28Information Theoretic Criteria 30Problem FormulationAIC CriterionMDL Criterion

Detection Problem - Summary 41

4 The Estimation Problem: Basic Estimation Theory 42The Maximum Likelihood (ML) approaches 42The Subspace-type Approach 44The Concept of the “Signal Subspace” 46The Concept of the “Manifold” 56Ambiguities 67Single-Parameter Manifolds: Linear Antenna Arrays 68Two-Parameter Manifolds 70Intersections of Signal Subspace & Array Manifold 77The MUSIC Algorithm 79Estimation of Signal Powers, Cross-correlation etc 84

5 The Reception Problem: Array Pattern & Beamforming 86Main Categories of Beamformers 87Definitions - Array Pattern 94Some Popular Beamformers 100Examples of Array Patterns (Beamformers) 105Beamformers in Mobile Communications 108

6 Array Performance Criteria and Bounds 110Output SNIR Criterion 111Outage Probability Criterion 115CRB (Estimation Accuracy Bound) 117Detection and Resolution Bounds 118

7 Overall Summary 1208 Appendix-A: Basic Decision Theory 121

Decision Criteria 121Decision Criteria: Mathematical Architectures 124

9 Appendix-B: Optimum M-ary Receivers and DecisionTheory 125


General Objectives

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


General Objectives

These problems are highlighted in the following block structure:


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)


General Problem Formulation M < N Array Covariance Matrix

Array Covariance MatrixTheoretical Covariance Matrix

If 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 (9)

i.e. Rxx =1L

XXH = S1L

MMH︸︷︷︸=Rmm

SH +1L

NNH︸︷︷︸=Rnn

= S .Rmm . SH +Rnn (10)

N.B.:In an array system the matrix Rxx (theoretical or practical) containsall the geometrical and other information about the varioussources relative to the array.

Remember that, sometimes, we will use Rxx to denote thepractical/estimated covariance matrix of the vector signal x(t).


General Problem Formulation M < N Generating L Snapshots having given Cov. Matrix

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 ) (11)

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 (12)

That is,

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

]=

[E D

12 z(t1), E D

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

12 z(tL)

](13)


General Problem Formulation M < N Generating L Snapshots having given Cov. Matrix

Proof.

Let z ∈ CN such as Ez .zH = IN . Then

Rxx = Ex .xH (14)

However,

Rxx = EDEH = ED12 D

12 EH = ED

12 IND

12 EH

= ED12 Ez .zH︸︷︷︸

=IN

D12 EH

= EED12 z︸︷︷︸

x

.zHD12 EH︸︷︷︸

xH

(15)

By comparing Equation 15 and 14 we have

x = E D12 z

N.B.:In a similar fashion we can generate L snapshots of m(t), n(t) or anyother vector-signal with known cov. matrix.


General Problem Formulation M < N 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.


The Detection Problem: Basic Detection Theory

The ‘Detection’Problem: Basic Detection TheoryProblem

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

SolutionTo use optimum Detection and Decision Theory

Note-1: The main optimum decision criteria are summarised inAppendix-A (at the end of this handout).

Note-2: The ML decision criterion will be employed in this section.


The Detection Problem: Basic Detection Theory Hypothesis and Hypothesis Testing

Hypothesis Testing

Definition (Hypothesis)

A Hypothesis , a statement of a possible condition

Definition (Hypethesis Testing)

To choose one from a number (two or more) hypotheses

Example (1: Detection of a signal s(t) in the presence of noise)

we have an observed signal r(t)

we define two hypotheses H1 and H2

Hypethesis Testing:

H1 : r(t) = s(t) + n(t) i.e. the signal is present

H2 : r(t) = n(t) i.e.the signal is not present


The Detection Problem: Basic Detection Theory Hypothesis and Hypothesis Testing

Example (2: Hypethesis Testing in an M-ary Comm System)In an M-ary Comm. System we have M hypotheses

The aim is to design a receiver which operates on an observed signalr(t) and chooses one of the following M hypotheses:

r t =s t +n t( ) ( ) ( )iDecision rule=?

(To determine whichsignal is present)

Di

Hypethesis Testing

H1 : r(t) = s1(t) + n(t),H2 : r(t) = s2(t) + n(t),· · · · · ·HM : r(t) = sM (t) + n(t),

(16)

where si (t)= one of M signals (channel symbols)


The Detection Problem: Basic Detection Theory Terminology

TerminologyConsider an observed signal r(t) and M hypotheses H1, H2,..., HM .

A priori probabilities ,

Pr(H1),Pr(H2), ...,Pr(HM )

(these are calculated BEFORE the experiment is performed)

A posterior probabilities ,

Pr(H1/r),Pr(H2/r), ...,Pr(HM/r)

That is, if r =observation variablethen we have M Conditional Probabilities

Pr(Hi/r), ∀i ∈ [1, . . . ,M ]

known as a POSTERIOR PROBABILITIES(since these are calculated AFTER the experiment is performed).


The Detection Problem: Basic Detection Theory Terminology

Pr(Hi/r) ∀i : diffi cult to find. A more natural approach is to findPr(r/Hi ), ∀i (17)

since in general pdfr/Hi , ∀iI are known orI can be found

Definition (Likelihood Functions (LF) )the M conditional probability density functions pdfr/Hi (r), ∀i , i.e.

pdfr/H1(r), pdfr/H2(r), ..., pdfr/HM (r) (18)

are known as "Likelihood Functions"

Definition (Likelihood Ratio (LR) )The ratio

pdfr/Hi (r)pdfr/Hj (r)

for i 6= j (19)

is known as "Likelihood Ratio"Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 18 / 128

The Detection Problem: Basic Detection Theory Hypothesis Testing re-Defined

Hypothesis Testing re-Defined

Note that the statistics of the observed signal r(t) are different fordifferent hypotheses.

Definition (Hypothesis Testing re-defined)

If the distributions of the observed signal r(t) for various hypotheses areknown

then the problem of choosing one of many (say M) hypotheses istranslated to make a decision about one of the M distributions afterhaving observed r(t).

This is called ‘Hypothesis Testing’


The Detection Problem: Basic Detection Theory Antenna Array: Hypothesis Testing

Antenna Array: Hypothesis TestingBased on the model of the signal received by an antenna array system(see Equ 1) we can define the following set of hypotheses:

H1 : x(t) = n(t); LF(0) = pdfx (t)|H1H2 : x(t) = S1m1(t) + n(t); LF(1) = pdfx (t)|H2

H3 : x(t) =2∑i=1S imi (t) + n(t); LF(2) = pdfx (t)|H3

· · · · · · · · ·

HN : x(t) =N−1∑i=1

S imi (t) + n(t); LF(N−1) = pdfx (t)|HN

(20)

Then we may use the ML decision rule, i.e.

Maximum Likehood (ML) = maxk

LF(k )


The Detection Problem: Basic Detection Theory Antenna Array: Hypothesis Testing

Equation 20, can be rewritten in an equivalent way as follows:

H1 : R(0)xx = Rnn; LF(0) = pdfx (t)|H1

H2 : R(1)xx = P1S1S

H1︸︷︷︸

Rsignals

+Rnn; LF(1) = pdfx (t)|H2

H3 : R(2)xx =

2

∑i=1PiS iS

Hi︸︷︷︸

Rsignals

+Rnn; LF(2) = pdfx (t)|H3

· · · · · · · · ·

HN : R(N−1)xx =

N−1∑i=1

PiS iSHi︸︷︷︸

Rsignals

+Rnn; LF(N−1) = pdfx (t)|HN

(21)

N.B.:

The LF(k ) for k = 0, 2, ...,N − 1 is a function of the eigenvalues of thematrices Rsignals and, consequently, of the eigenvalues of R

(k )xx

ML= maxk

LF(k )


The Detection Problem: Basic Detection Theory Infinite Observation Interval (Infinity Snapshots)

Infinite Observation Interval (Infinity snapshots)Based 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

(22)

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 (23)

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 (24)

⇒ rank

Rxx − σ2nIN= M (25)



However, using eigen-decomposition of Rxx we have

Rxx = E.D.EH (26)

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

(27)

= (N ×N) matrix

with d`, ` = 1, 2, ...N, denoting the `-th eigenvalue of Rxx and

d1 > d2 > ... > dN > 0 (28)



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 (29)

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

(30)

= (N ×N) matrixProf. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 24 / 128


That is, from Equ 29 in conjunction with Equs 27 and 30 we have

D = Λ+ σ2nIN (31)

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 (32)

Now, since the smallest eigenvalue of Rsignals is zero

eigmin Rsignals = 0 (33)

with multiplicity N −M, that meanseigmin Rxx = σ2n (34)

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 ) (35)Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 26 / 128


Note: another useful expression is

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

[Ds O

O Dn

][Es,En]

H

= Es.Ds.EHs +EnDnEH

n (36)

where


The Detection Problem: Basic Detection Theory Finite Observation Interval (Finite Snapshots)

Detection Problem: Finite Observation Interval (L=finite)Eigen-values

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

(37)


The Detection Problem: Basic Detection Theory Finite Observation Interval (Finite Snapshots)

N.B.:In theory,

σ21 = σ22 = ... = σ2M = σ2M+1 = .... = σ2N ,σ2n (38)

However, in practice

σ21 6= σ22 6= ... 6= σ2M 6= σ2M+1 6= .... 6= σ2N (39)

although

σ2n ≈ σ21 ≈ σ22 ≈ ... ≈ σ2M ≈ σ2M+1 ≈ .... ≈ σ2N (40)

if M is known then .

σ2n = the average of the N −M smallest eigenvalues

=1

N −M(σ2M+1 + σ2M+2 + ....+ σ2N

)(41)

If M is unknown then its estimation is not an easy task and a naiveapproach is likely to fail. Solution: see next section (Decision Theory).


The Detection Problem: Basic Detection Theory Information Theoretic Criteria

Information Theoretic CriteriaThe information theoretic criteria for model selection, introduced by

I Akaike (Information Theory Symposium 1973, Automatic Control 1974)I Schwartz (“Estimating the dimension of a model,” Ann. Stat, 1978), andI Rissanen (“Modeling by shortest datad escription,” Automatica, 1978)

address the following general problem.

ProblemGiven a set of L observations (L snapshots), i.e. data,

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

and a family of models , that is, a parameterised family of probabilitydensities pdf(X|Hi ),∀i , select the model that best fits the data (i.e.the model that fits the set of L observations).

Remember that the conditional probability densities pdf(X|Hi ) ,∀i ,are known as Likelihood Functions (LF)



Solution (AIC)Select the model which gives the minimum of AICa, i.e.

minkAIC(k) (42)

where

AIC(k) , −2 ln(maxkLF(k )

)︸︷︷︸ML estimator

+ 2k (43)

with k denoting the number of free adjusted parameters in the model.

aAIC - Akaike Information Criterion

N.B.:The first term in Equation-43 is the well-known log-likelihood of themaximum likelihood estimator of the parameters of the model.The second term is a bias correction term, inserted so as to make theAIC an unbiased estimate of the mean Kulback-Liebler distance betweenthe modeled density and the estimated density .



MDL Criterion

Inspired by Akaike’s pioneering work, Schwartz and Rissanenapproached the problem from quite different points of view.

I Schwartz’s approach is based on Bayesian arguments (Bayes criterion).He assumed that each competing model can be assigned a priorprobability, and proposed to select the model that yields the maximumposterior probability.

I Rissanen’s approach is based on information theoretic arguments. Sinceeach model can be used to encode the observed data, Rissanenproposed to select the model that yields the minimum code length.

It turns out that in the large-sample limit, both Schwartz’s andRissanen’s approaches yield the same criterion, (known as MDL2

criterion)

2MDL: Minimum Description LengthProf. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 32 / 128


Solution (MDL)Select the model which gives the minimum of MDL, i.e.

minkMDL(k) (44)

where

MDL(k) , − ln(maxkLF(k )

)︸︷︷︸ML estimator

+12k ln L (45)

with k denoting the number of free adjusted parameters in the model.

N.B.:Apart from a factor of 2, the first term in Equation 45 is identical tothe corresponding one in the AIC, while the second term has anextra factor of 12 ln L.



Solution (Likehood Function, LF)It can be proven that

LF(k ) = −L. ln(det

R(k )xx

)− Tr

R(k )xx

−1.Rxx (46)

= ln

N

∏`=k+1

d1/(N−k )`

1N−k

N

∑`=k+1

d`

(N−k )L

(47)

where k ∈ (0, 1, ...,N − 1)and d` = the `-th eigenvalue of Rxx (48)

Note that in Equation 47 the term in the brackets is the ratio

geometric mean of the smallest N − k eigenvalues of Rxx

arithmetic mean of the smallest N − k eigenvalues of Rxx

(49)



In Equation 46:I Rxx the practical/estimated covariance matrix, i.e,

Rxx =1L

L

∑l=1

x(tl ).xH (tl )

or= 1

LX.XH

I R(k )xx is the model (theoretical) cov matrix that is used in Equation 21

which defines the N hypotheses, i.e.,

H1 : R(0)xx = Rnn ; LF(0) = pdfx (t)|H1

H2 : R(1)xx = P1S1S

H1︸︷︷︸

Rsignals

+Rnn ; LF(1) = pdfx (t)|H2

H3 : R(2)xx =

2

∑i=1


Rsignals

+Rnn ; LF(2) = pdfx (t)|H3

· · · · · · · · ·

HN : R(N−1)xx =

N−1∑i=1


Rsignals

+Rnn ; LF(N−1) = pdfx (t)|HN



Solution (AIC and MDL: equivalent expressions)

It can be proven (based on Equ 47) that

AIC (k) = −2 ln

N

∏`=k+1

d1/(N−k )`

1N−k

N

∑`=k+1

d`

(N−k )L

+ 2k(2N − k) (50)

MDL(k) = − ln

N

∏`=k+1

d1/(N−k )`

1N−k

N

∑`=k+1

d`

(N−k )L

+12k(2N − k) ln L

(51)

Reference: M. Wax and T. Kailath, "Detection of Signals by InformationTheoretic Criteria", IEEE Transactions on ASSP, vol. 33, pp. 387-392, Apr.1985.Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 36 / 128


ExampleConsider a ULA of 7 antennas with halfwavelength spacing (i.e. N = 7)operating in the presence of two sources with directions-of-arrivals 20

and 25. The signal-to-noise ratio is 10 dB. Using L = 100 snapshots (i.e.samples), the resulted eigenvalues of the sample-covariance matrix are:21.2359, 2.1717, 1.4279, 1.0979, 1,0544, 0.9432, and 0.7324.Observing the gradual decrease of the eigenvalues it is clear that theseparation of the “smallest” eigenvalues from the “large”ones is adiffi cult task.However, the AIC and MDL provide the following values

(k) 0 1 2 3 4 5 6

AIC (k) 1180.8 100.5 71.4 75.5 86.8 93.2 96.0

MDL(k) 590.4 67.2 66.9 80.7 95.5 105.2 110.5

minimum M = 2

That is, the minimum of both the AIC and the MDL is obtained, asexpected, for the (k) equal to 2, i.e. M = 2



AIC Criterion in Vector Format

AIC =[AIC (0), AIC (1) ..., AIC (k), ..., AIC (N − 1)

]T

= −2L

ln

N∏`=1d`

N∏`=2d`

...N∏

`=N−2d`

N∏

`=N−1d`

dN

+

NN − 1

...

321

ln

NN − 1

...

321

− ln

N∑`=1d`

N∑`=2d`

...N∑

`=N−2d`

N∑

`=N−1d`

dN

+2

01

...

N − 3N − 2N − 1

2N2N − 1

...

N + 3N + 2N + 1

; an (N × 1) real vector (52)



MDL Criterion in Vector Format

MDL =[MDL(0), MDL(1) ..., MDL(k), ..., MDL(N − 1)

]T

= −L

ln

N∏`=1d`

N∏`=2d`

...N∏

`=N−2d`

N∏

`=N−1d`

dN

+

NN − 1

...

321

ln

NN − 1

...

321

− ln

N∑`=1d`

N∑`=2d`

...N∑

`=N−2d`

N∑

`=N−1d`

dN

+12

01

...

N − 3N − 2N − 1

2N2N − 1

...

N + 3N + 2N + 1

ln L; an (N × 1) real vector (53)



remember: d` , (for ` = 1 to N with d1 > d2 > . . . > dN ) denotesthe `-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.


The Detection Problem: Basic Detection Theory Detection Problem - Summary

Detection Problem - SummarySummary

If L = ∞ , i.e. Theoretical Rxx = Ex(t).x(t)H

, then

M = N − (multiplicity of min. eigenvalue of Rxx ) (54)

σ2n = min.eigenvalue of Rxx = noise power (55)

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 (56)

σ2n = the average of the N −M smallest eigenvalues

= 1N−M

(σ2M+1 + σ2M+2 + ....+ σ2N

)(57)

Remember:

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


The Estimation Problem: Basic Estimation Theory The Maximum Likelihood (ML) approaches

The Estimation Problem: Basic Estimation TheoryThe Maximum Likelihood (ML) approaches

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

x(t)4= S(p).m(t) + n(t) (58)

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


The Estimation Problem: Basic Estimation Theory The Maximum Likelihood (ML) approaches

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)

(59)

ML Solution:

M = (S(p)HS(p))−1S(p)HX (60)

m(ti ) = (S(p)HS(p))−1S(p)H︸︷︷︸,WH

ML

x(ti ) (61)

pML

= argmaxpLFx

= argmaxpTr (PS.Rxx ) (62)

wherePS = S(p)

(SH (p)S(p)

)−1S(p)H (63)


The Estimation Problem: Basic Estimation Theory The Subspace-type Approach

The Signal-Subspace Approaches

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


The Estimation Problem: Basic Estimation Theory The Subspace-type Approach

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 Estimation Problem: Basic Estimation Theory The Concept of the “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) (64)

= Sm(t) + n(t), ∀t (infinite observation interval) (65)

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 (66)



The Subspace of a Manifold (Response) Vector

Let us consider the subspace spanned by only one signal-term ofEquation 64, 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] (67)



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 65 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 (68)



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 (69)

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](70)



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 (71)

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

Hn (72)

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]

(73)


The Estimation Problem: Basic Estimation Theory The Concept of the “Manifold”

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:


The Estimation Problem: Basic Estimation Theory Ambiguities

Ambiguities


The Estimation Problem: Basic Estimation Theory Single-Parameter Manifolds: Linear Antenna Arrays

Single-Parameter Manifolds: 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


The Estimation Problem: Basic Estimation Theory Single-Parameter Manifolds: Linear Antenna Arrays

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


The Estimation Problem: Basic Estimation Theory Two-Parameter Manifolds

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 (74)

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


The Estimation Problem: Basic Estimation Theory Intersections of Signal Subspace & Array Manifold

Intersections of Signal Subspace and 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 .


The Estimation Problem: Basic Estimation Theory Intersections of Signal Subspace & Array Manifold

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.19c1 78 / 128

The Estimation Problem: Basic Estimation Theory The MUSIC Algorithm

The MUSIC Algorithm

The Multiple Signal Classification (MUSIC) algorithm belongs to thefamily of Signal-Subspace type technique .i.e.

MUSIC ∈ Signal-SubspaceIt 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) (75)



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) (76)

I With reference to the figures in slide-78, it is obvious that

ξ(p) = 0 iffp = p1 orp = p2

I Therefore, we search the array manifold, i.e. we evaluate the expression(76),∀p, and we select as our estimates the p’s which satisfy

ξ(p) = 0⇒ S(p)H .En .EHn .S(p) = 0, ∀p (77)



MUSIC Algorithm in Step-formatStep-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), ∀p

where 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)


The Estimation Problem: Basic Estimation Theory Estimation of Signal Powers, Cross-correlation etc

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 (78)

where S# = (SH · S)−1 · SH = pseudo-inverse of S (79)

Note that:σ2n =(min eigenvalue of Rxx or given by Equation-41)

=noise power


The Estimation Problem: Basic Estimation Theory Estimation of Signal Powers, Cross-correlation etc

Proof.Proof of Equation-78

Rxx = S.Rmm .SH + σ2nIN

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


The Reception Problem: Array Pattern & Beamforming

The Reception Problem: Array Pattern & Beamforming

1This figure is shown in the cover of Prof Manikas’book entitled "Beamforming: SensorSignal Processing For Defence Applications" .Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 86 / 128

The Reception Problem: Array Pattern & Beamforming Main Categories of Beamformers

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 & Beamforming Definitions - Array Pattern

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) (80)

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

g(θ), ∀θ : is known as the array pattern (81)

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) (82)

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(83)



Note that d =inter-sensor-spacing, and

if d =λ

2⇒ beamwidth = 2 sin−1

(2N

)× 180

π(84)

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) (85)

wmain-lobe = S(θmain-lobe) (86)



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)) (87)

(simplified to) = exp(−jπr x cos(120)) (88)Prof. A. Manikas (Imperial College) EE401: Array Rx’s for SIMO & MIMO v.19c1 98 / 128


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.19c1 99 / 128

The Reception Problem: Array Pattern & Beamforming Some Popular Beamformers

Some Popular Beamformers

WIENER-HOPF Beamformer:

w = c.R−1xx Sdesiredsignal

(89)

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-89)



Modified WIENER-HOPF Beamformer:

w = c ·R−1n+JSdesiredsignal

(90)

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

)(91)

subject to wHS(θ) = 1 (92)

I Equation 91 aims to minimise the effect of the desired signal plus noise.However, the constraint wHS(θ) = 1 (Equ 92) prevents the gainreduction in the direction of the desired signal.

I Solution of Equations 91 and 92:

w = c .R−1xx Sdesiredsignal

(93)

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



A Superresolution Beamformer based on DOA estimation:

w = P⊥SJSdesiredsignal(94)

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(95)

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#)

(96)

where S# = S.(SHS)−1 (97)


The Reception Problem: Array Pattern & Beamforming Examples of Array Patterns (Beamformers)

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. 89): Superresolution Beamformer (Equ. 95)

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.-94) (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


The Reception Problem: Array Pattern & Beamforming Beamformers in Mobile Communications

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


The Reception Problem: Array Pattern & Beamforming Beamformers in Mobile Communications

Coverage patterns for switched beam and adaptive array antenna


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)


Array Performance Criteria and Bounds Output SNIR Criterion

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

(98)

Note:I An alternative equivalent expression to Equ 98 is given below

SNIRout =wHRddw

wH (Rxx −Rdd ) w(99)

I Both Equations 98 and 99 are general expressions for any beamformer.However, for different beamformers (i.e. different weights) theseequations give different values.


Array Performance Criteria and Bounds Outage Probability Criterion

Outage Probability Criterion

outage probability (OP) is defined as follows:

OP = Pr(SNIRout < SNIRpr ) (100)

or

OP = Pr(SIRout < SIRpr ) (101)

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 94 (complete interference cancellationbeamformer) at the base station, the system capacity is increased.


Array Performance Criteria and Bounds Outage Probability Criterion

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.


Array Performance Criteria and Bounds CRB (Estimation Accuracy Bound)

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

σe =√CRB[s ] (102)

=

√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


Array Performance Criteria and Bounds Detection and Resolution Bounds

Detection and Resolution Bounds


Array Performance Criteria and Bounds 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 (103)

Fourth-root Law :

Resolution: ∆pth = f.s, 4√

κ1,√

σe (104)

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.19c1 119 / 128

Overall Summary

Summary


Appendix-A: Basic Decision Theory Decision Criteria

Appendix-A: Basic Decision TheoryDecision Criteria

Consider an observed signal r(t) and M hypotheses H1, H2,..., HM .

Define the sets of parameters P1 and P2 where:P1 denotes the set of parameters Pr(H1),Pr(H2), ..., Pr(HM )P2 represents the set of costs/weights Cij , ∀i , j

associated with the probabilities Pr(Di |Hj ),(where Di indicates "decision" by choosing hypothesis Hi )

Estimate/identify the likelihood functions

pdfr |H1(r)︸︷︷︸l

LF (1)

, pdfr |H2(r)︸︷︷︸l

LF (2)

, ..., pdfr |HM (r)︸︷︷︸l

LF (M )



Main Decision CriteriaDecision: choose the hypothesis Hi (i.e. Di )with the maximum Gi (r)where Gi (r) depends on the chosen criterion as follows:

BAYES CriterionMinimum Probability of Error (min(pe)) CriterionMAP CriterionMINIMAX CriterionNewman-Pearson (N-P ) CriterionMaximum Likelihood (ML ) Criterion

P1 P2 choose Hypothesis with max(Gj (r))

Bayes known known Gi (r) ,weight i × Pr(Hi )×pdfr |Himin(pe ) known unknown Gi (r) , Pr(Hi )× pdfr |Hior MAP

Minimax unknown known Gi (r) ,weight i× pdfr |HiN-P unknown unknown by solving a constraint optim. problem

ML don’t care don’t care Gi (r) , pdfr |Hi



N.B.:I Note-1:if an approximate/initial solution is required then any informationabout the sets of parameters P1 and/or P2 can be ignored. In thiscase the Maximum Likelihood (ML) Criterion should be used.

I Note-2:

weighti ,M

∑j=1j 6=i

(Cji − Cii

)(105)

or (since the term for i = j is equal to zero) simply,

weighti =M

∑j=1

(Cji − Cii

)(106)

I Note-3:Sometimes, for convenience, Gi will be used (i.e. Gi , Gi (r)) - i.e. theargument will be ignored.

I Note-4:for binary: weight0 , C10 − C00; weight1 , C01 − C11


Appendix-A: Basic Decision Theory Decision Criteria: Mathematical Architectures

Decision Criteria: Mathematical Architectures


Appendix-B: Optimum M-ary Receivers and Decision Theory

Appendix-B: Optimum M-ary Rx’s and Decision TheoryObjective : to design a receiver which operates on r(t) and choosesone of the following M hypotheses:

H1 : r(t) = s1(t) + n(t)H2 : r(t) = s2(t) + n(t)· · · · · ·HM : r(t) = sM (t) + n(t)

(107)

Corollary (LFs)It can be proven that:

pdfr/Hi(r(t)) = const · exp− 1N0

∫ Tcs

0(r(t)− si (t))2dt

(108)

= const · exp

− 1N0

∫ Tcs0 r(t)2dt

− 1N0

∫ Tcs0 si (t)2dt

+ 2N0

∫ Tcs0 r(t)si (t)dt

(109)



Optimum M-ary Architecture based on Decision TheoryIt can be easily proven that:

Di =

argmaxi pdfr/Hi(r(t))︸︷︷︸

=Gi (r )

ML

argmaxiPr(Hi )× pdfr/Hi (r(t))︸︷︷︸

=Gi (r )

MAP, ormin (pe)

argmaxiweighti × pdfr/Hi (r(t))︸︷︷︸

=Gi (r )

minmax

argmaxiweighti × Pr(Hi )× pdfr/Hi (r(t))︸︷︷︸

=Gi (r )

Bayes

(110)

for i = 1, . . . ,M

Based on Equ 109, the parameter Gi (r) shown in the previous equation, canbe simplified as follows:

Gi (r) =∫ Tcs

0r(t)s∗i (t)dt+DCi , ∀i , ∀criterion (111)



where DCi ∀i , depends on the decision criterion.

For all the criteria, DCi is given as follows:

DCi ,

− 12Ei ML

N02 ln(Pr(Hi ))−

12Ei MAP, or

min (pe)

N02 ln(weighti )−

12Ei minmax

N02 ln(weighti .Pr(Hi ))−

12Ei Bayes

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(112)

with Ei , energy of si (t) (113)



Based on 111 which is repeated below:

Gi (r) =∫ Tcs

0r(t)s∗i (t)dt+DCi , ∀i , ∀criterion,

Equ 110, can be implemented by the following optimum architecture:

s (t)1

0

Tcs

+DC1

s (t)2

0

Tcs

+DC2

s (t)M

0

Tcs

+DCM

G1

G2

GM

CompareDecisionVariables

and

choosemaximum

Di

If Gi=maxthen

Gi

Di

r t( )

t

Tcs

Tcs

Tcs

Tcs

OPTIMUM Mary RECEIVER: CORREL. RECEIVER


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