CORRELATION MATRIX - GUC · 2017-02-27 · is D × D source correlation matrix R nn =σ n 2 I is...

ADAPTIVE ANTENNAS. PROF. A.M.ALLAM

1 1

CORRELATION MATRIX

ADAPTIVE

ANTENNAS


1-Revision on stochastic process

Stochastic process

Are discrete and uniformly spaced instants of time for a certain statistical phenomena .

It is sampled uniformly with sampling rate chosen to be greater than twice the highest

frequency component of the process like radar signal, digital computer data…

-It is called discrete time series or time series

-A sequence represents a time series consists of

u(n) made at time n and m past observations of

the process made at times n-1, n-2, …n-M is

shown

The stochastic process is not a single function

of time, it represents an infinite number of

realizations of the process

u(n) u(n-1) u(n-M) n ≡ t

Ensemble

-It seems the last observation is controlled by the

memory storage of M+1 size


-The ensemble mean of stochastic

process is the ensemble average of the

values of all the sample functions

(realizations) at time instant t (n)

)]([)( nuEn

E is the statistical expectation operator

Expected value, Ensemble average or Mean

u(n) u(n-1) u(n-M) n ≡ t

Realization 1

u(n) u(n-1) u(n-M) n ≡ t

Realization 2

u(n) u(n-1) u(n-M) n ≡ t

Realization 3

u(n) u(n-1) u(n-M) n ≡ t

Realization 4

It is clear that the ensemble mean

μ(n) is function of time i.e., it has a

certain value depends on the instant

of its determination across all the

realizations (across the process)


Autocorrelation and Autocovariance

-The autocorrelation function of a stochastic process is the expectation of the product of the two random variables u(n), u(n-k) obtained by observing the process at times n, n-k respectively

The autocorrelation function measures the similarity between observations as a function of

two different time instants t1,t2 (n, n-k)

u(n) u(n-1) u(n-M) n ≡ t

Realization 1

u(n) u(n-1) u(n-M) n ≡ t

Realization 2

u(n) u(n-1) u(n-M) n ≡ t

Realization 3

u(n) u(n-1) u(n-M) n ≡ t

Realization 4

MkknunuEknnr ,...2,1,0)],....()([),( *

-It is an (M+1)x(M+1) matrix, each element is the

correlation of one observation with the others

across the process

MkknknunnuEknnc ,...2,1,0],....))(()())(()([(),( *

-They are interrelated as )()(),(),( * knnknnrknnc

i.e., the autocorrelation and the autocovariance

are the same for zero mean process for all n

-The same for the covariance

-It is also an (M+1)x(M+1) matrix, each element

is the correlation of one observation with the

others across the process

k=0 k=1 k=2


-For wide sense stationary or stationary to the second order process , the mean is

constant

,...2,1,0)....(),( kkrknnr

nallforconstn .........)(

,...2,1,0)....(),( kkcknnc

For k =0 ])([)0(2

nuEr the mean square value of u(n)

2)0( Uc the variance of u(n)

-For mean ergodic process, the mean square value of the error between the ensemble

average (μ, expectation) and the time average (estimated average) approaches zero

as the number of observations approaches infinity

while r and c are invariant with time shift, it depend on the difference between the

observation times n, n-k , that is k


0]))([( 2

lim

NN

)(1 1

0

nuN

N

n

mean ergodic process is The condition for -

-The time average is defined as

where N is the number of observations along one realization of the process which

supposed to be infinity

-The use of mean ergodic theorem, can be extended to autocorrelation function as,

the process is said to be correlated ergodic if the mean square value of the values of

r(k) and the estimated (time average ) r(k) approaches zero as the number of

observations approaches infinity

-In this case the autocorrelation will be calculated from a single realization of

observations for N tend to infinity

u(n) u(n-1) u(n-M) n ≡ t

-The time average is the mean over one

sample function (single realization) of the

stochastic process i.e., along the process


Correlation matrix

-Let Mx1 observation vector U(n) represents the sequence of a sampled elements

from a stochastic process

U(n)= [u(n), u(n-1), u(n-2,…,u(n-M+1)]T

u(n) u(n-1) u(n-M+1) n ≡ t

-The correlation matrix of a stationary discrete stochastic process represented by

the expectation of the product of the U(n) with its Hermitian transposition

)]()([ nUnUER H

uu

( transposition operation combined with complex conjugation)


-Using the condition of wide sense stationary ( depends only on the difference

between observation times ; k ) it takes expanded form as MxM matrix

The element r(0) in the main diagonal is always real valued, but

for complex valued data the remain elements are complex values


9 2/27/2017 LECTURES 9

Properties of correlation matrix

1-R of a stationary discrete-time stochastic process is Hermitian

RRR TH ][ *

*

* *

0 1 1

1 0 2

1 2 0

r r r M

r r r M

r M r M r

R

)(*)( krkr i.e.,

We only need M values of the autocorrelation function r(k) for

k=0,1,2,…,M-1 in order to completely define the correlation

matrix R

2-R for a stationary discrete-time stochastic process is Toeplitz

A square matrix is said to be Toeplitz (referred to Otto Toeplitz) if :

-all the elements on its main diagonal are equal , like r(0)

-and all the elements on any other diagonal are also equal, like r(1) , r*(1)


10 10

3-R for a stationary discrete-time stochastic process is positive definite

“Its determinant is greater than zero”

i.e., any correlation matrix is nonsingular

*

* *

0 1 2

1 0 1 0

2 1 0

r r r

r r r

r r r

4-When the elements of the observation vector of a stationary discrete-time

stochastic process are rearranged backward, the effect is equivalent to the

transposition of R

UBT(n)= [u(n-M+1), u(n-M+2),… , u(n-2), u(n-1), u(n)]

* *

*

0 1 1

1 0 2

1 2 0

u u RB BH T

r r r M

r r r ME n n

r M r M r

)]()([ nUnUE BHB

Then


11 2/27/2017 LECTURES 11

5- RM and RM+1 of a stationary discrete-time stochastic process, pertaining to M

and M+1 observations of the process respectively, are related by

*

* *

1

* * *

0 1 2

1 0 1 1

2 1 0 2

1 2 0

RM

r r r r M

r r r r M

r r r r M

r M r M r M r

*1

0R

R

T

M

M

r r

r

H


12

-Let us start with a description of the array, the received signal, and the additive noise

-They are received by an array of M elements with M potential weights

-The array output y can be given in the

following form

The array output can be determined for any snapshot of an incoming signal plus noise

once the weight vector is calculated

-Consider D signals arriving from D directions

)(

)(

.

.

)(

)(

)(..)()()(

2

1

21 tn

ts

ts

ts

aaatX

D

D

)()()( tntAStX

….is an M× D matrix of steering vectors A

)()( tXWtyT

TMwwwW ..21

S(t) … vector of incident complex monochromatic signals at a time

n(t) … noise vector at each array element m, zero mean, variance

σn2

a(θ) … M-element array steering vector for the θ direction of arrival

-Each array element will get a snapshot at a time

t; x(t); an element of the vector X(t) as

-Each incoming signal sd(k) has a correspondence

additive zero mean Gaussian noise

S1(t)

S2(t)

S3(t)

y(t)

2-Correlation matrix of adaptive antenna arrays


-It is understood that the arriving signals are time varying and thus our calculations

are based upon time snapshots of the incoming signal

-The M× M array correlation matrix Rxx can be written as

RXX= ARSSAH + Rnn

where

RSS is D × D source correlation matrix

Rnn =σn2 I is M× M noise correlation matrix

I = M× M identity matrix

- If we do not know the exact statistics for the noise and signals, but we can assume that

the process is ergodic, then we can approximate the correlation by use of a time

averaged correlation defined by

)()(1

1

kXkXK

R HK

k

XX

)()(1

1

kSkSK

R HK

k

SS

)()(1

1

knknK

R HK

k

nn

This time average is taken over different snapshots of number K, i.e., a sum of MxM

matrices each represents of a certain snapshot a time k, k=1,2,…,K )()( kXkX H

)]().([ kXkXER H

XX )]().([ kSkSER H

SS

RXX = E[X XH] = E[(AS + n)(SHAH + nH)]

= AE[S SH]AH + E[n nH]

= ARSS AH + Rnn


-For adaptive algorithms, a number of estimations of RXX are required before

convergence is obtained and convergence typically occurs after 50 iterations

Thus, the convergence time, TC can be estimated for S number of samples, L

number of iterations required for convergence and TS is the sampling period

TC=LSTS

-Often in the literature, the array correlation matrix is referred to as the covariance

matrix. This is only true if the mean values of the signals and noise are zero. The

arriving signal mean value must necessarily be zero because antennas cannot receive

d.c. signals. The noise inherent in the receiver may or may not have zero mean

depending on the source of the receiver noise


15 15 2/27/2017 LECTURES 15

CORRELATION MATRIX - GUC · 2017-02-27 · is D × D source correlation matrix R nn =σ n 2 I is...

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