Wiley signals and systems e book TLFe BO 446

17.6. Describinr Discrete

Ittilldoat

Signals

43 I

wc will forego exaniiriirrg them in clclail. However. we need to hc w a r e of tbe
definitions of stat ionary and ergodicity as they apply to randorn secpienccs.
Stationaiy means in this ease that the second-order exp
~ ~ f ( z ~ ~ orily
l ] ~drlwnd
~ [ on
k . integer
~ ] ~differences
~
K = k.1 - k.2
discrete t i m e variables kl and k ~ .
The time-average, necessarv for the tlefiiiitioit of ergodicity, i s givcii for sample
r;mdoiri process by
sequenccs of a cIiser
(17.84)

From here, the a u ~ o - ~ o r ~ e l a ~cifo~n ), s ~ - ~ o r r and
~ l ~covariiknces
~tio~
can bc considcrcd in the s m i P way as for continuous random processes. Instead of autocorrelatitm aiid cross-correlation fimctions, we will be using auto-correlation and
cross-correlation sequertces.
For the cross-power density spectrum of two discrete weak s~,a~ioiia~y
faridam

processes, we obtain from the ctiscret,e-timc Fourier transform the spectrum
(17.85f

periodic in 52. The power density spectrim of a discrete randorn process corresponds to the case where y = LT.

Example 19.12
A weak statioriary continuous randorn proccss Ic(-t> is charactcv-ised by ail exponentially dccayiag ACF
). zzz e-wol r1 ,
Samples arc taken at iritervals T

z [ k ]= Li,.(kT) k t IIZ
The a i ~ t ~ - c o r r e l ~ sequence
t i ~ ) n is

The power density spectrum of the discrete raiidorn process i s

and i s slnown in Figure 17.12. 1%is gositive, real arid even, like the power density
spectrum of a continuom random process with real wfues. hi addition, it is 2n
periodic. As the auto-correlation sequences vS.[ K ] is yielded simply by sampling