Wiley signals and systems e book TLFe BO 449

17. Describing Random Signals

434

a) g(t) = r(C) -i- li
b) p(f)
c)

=1

a(t}+ s h t

y(t) = s.(f)

d) p ( t ) = .(t)

+F ( f )
'

5E(f)

For e a d i part, say wlrethrr y ( t ) is crgodic. Assume in part d) that all sample
functions z t ( t )arc cvcn,
Exercise 17.7

-

Give p r ( C ) , E { x 2 ( f ) )ol(f)
.
and x L ( t for
) tlir deterministic sigm.1 x { t ) = e

" ltc(L)*

Exercise 17.8
Consider the discrete random process Lthrowinga die', where a ) r[k]is the nunibrr
t h o w n and h) z [ k ]is the square of the nurribcr thrown. Find p , 5 [ k ] ,u 5 [ k ] and
B ( z 2 [ k ] )Ale
, the processes crgoodir?

Exercise 17.9

Consider the disc
raritloin process 'throwing a. loaded die' where six always
appears at times
3 N , A T E 22, But the rxinibcv-s t,lirown at other tixms
~-wrr
d ~ s ~ r i equdIy,
b ~ ~ ~ and
~ d wherc ryik] is the riuniber thrown. Find
am$ y//k],
and also plJ[k.],crglJi] arid E(y"[kj). Is thc process stationary and/or ergodic.'
Exercise 17.10
Calculak the ACF of ~ ~ t r ~ signal
~ i i~ ~
( t ~)Fc.~ Kt EC
j ~with (17.56).

Exercise 3 7-13
What pc~uliaritic>s
does thc ACF have for the random process from Exercisc 1 7 3

Does it oiily depend oii t11e di€fert.iice of the avwaging poiiitb? C 3 w ps7(tn,f o
10s).

+

Exercise 17.12
Lct r ( t )and y(d) be two real raiidoiii signals.

-

a) how tjiat . ~ T j n ' ( f ) y ( t ) )= 1) eritails from E ( ( . E ( ~ )yit))')

W Y 2 (J 11

=

n(r2(t))
+

1

h) How can 37{x(t}ry(t)Jbe simplified if r(t)aid g ( t ) are UKlcorrelated?
cf

Give the conditions for uncorrelated ~ ( tancl
) ~ / ( 80
t ) that ~

{

~= 0. (

Exercise 17.13
What are thc formu1:i.e for power, DC: c ~ ~ r r ~ ~ ( ~ effeetivc
r i r r i ~ , value and
iiiriistic real signal d(t)'?

AC power

~