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 ( 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 , [ k ] ,u [ k ] and B ( z [ k ] )Ale , the processes crgoodir? Exercise 17.9 Consider the disc raritloin process 'throwing a loaded die' where six always appears at times 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 7-13 What pc~uliaritic>s does thc ACF have for the random process from Exercisc Does it oiily depend oii t11e di€fert.iice of the avwaging poiiitb? C 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 (J 11 = n(r2(t)) + 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 ~ { ~= ( 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 ~