Random processes - Signal Processing and Optimization for Transceiver Systems - page 781

and is a positive semidefinite matrix. As in the scalar case, the covariance

sequence − m x , which is a zero-mean

process. The autocorrelation of a WSS process satisfies the property

C xx ( k ) is the autocorrelation of x ( n )

R xx ( −k )= R xx ( k ) .

(E . 9)

The Fourier transform of R xx ( k )iscalledthepowerspectrumor PSD matrix :

∞

S xx ( e jω )= R xx ( k ) e −jωk . (E . 10)

k = −∞

This is an M M matrix function of ω. It can be shown that this is a positive

semidefinite matrix for all ω.

×

E.2.2 Joint stationarity and cross power spectra

Two random processes x ( n )and y ( n ) are said to be jointly WSS if each of them
is WSS and, in addition, the cross correlation defined by E [ x ( n ) y † ( n − k )] is

independent of n. In this case we denote the cross correlation by

R xy ( k )= E [ x ( n ) y † ( n

− k )] . (E . 11)

Note that this is an N × M matrix sequence, where x ( n )is N × 1and y ( n )is
M 1. The cross power spectrum of x ( n )with y ( n )isgivenbytheFourier

transform of R xy ( k ):

×

∞

S xy ( e jω )= R xy ( k ) e −jωk . (E . 12)

k = −∞

It is readily verified that

R yx ( k )= R xy ( −k ) (E . 13)

and

S yx ( e jω )= S xy ( e jω ) .
For example, in the scalar case

R yx ( k )= R xy ( yx ( e jω )= S xy ( e jω ) . − k ) , (E . 14)

It can be shown that the joint WSS condition is equivalent to the condition that
the vector process

v ( n )= x ( n )
y ( n )

(E . 15)

be WSS. In this case the power spectrum of v ( n )is

⎡ ⎤

S xx ( e jω ) S xy ( e jω )

S vv ( e jω )=

⎣ ⎦ .

(E . 16)

S yx ( e jω ) S yy ( e jω )