Wiley signals and systems e book TLFe BO 448

13 3

17.7. Exercises

Ensemble 3

Ensernble 4

t

Ensenihle 5: from Figwe 17.1

Exercise 17.2
Discuss which of L21e eiisenibks from Exercise 17.1 could be:
a) weak stationary,
1 J ) wcak ergodic.
Exercise 17.3
Corisiclrr a xandom process ~ ( ta t) the output of i - ~CD-player where (lie sample
fiinctioiis are prodiicrd 1-y someone coritiriiially rcpeating a t m serond section from
R rock C n . Assimic that it is tlrr idcd case in which this has been happciiiing fbr
RII irifiriite time aid t he CII-player will go on playiirg infinitely.
-

a) Ilow large are E(r(t)} and x z ( t ) iintler the assimiptiori that the outpiat, has
no DC c*omponent'!

b) TJsing the first-order expected values, discuss whet her the process could be
stalionwy.
r) TJsing the first-orrlrr expected valiies, discuss whether the proccm coiild be
ergodic.

Exercise 19.4
Take two uiicorrrlatbdraildoin processes z l ( t )R I I J~Z ( ~ ) . where p L 1= 2, p L 2 = 0,
E{.c;(t)) = 5 and E { z ; ( t ) }= 2. Calculate the averages p g ( t )n$(t).
,
E ( y L ( f ) }For
the rmdoin process y ( t ) = xl(i) .rZ(tf.

+

Exercise 17.5
Find (,hevariance of v ( t ) = r(t)i-g(t), wliere x ( t ) is a random signal with varianw
= 10 aiid y(t) is any delerriiiriistic signal.

Exercise 17.6
The ergotiic raritlom process x ( t ) has p, = 1 id nz = 1. Calrxlate pg(t),o;(t),
E { g 2 ( t ) and
} v,(t) for
~