547 Appendix A Solutions to the Exercises - - E{s(t)} = z i ( t ) ,because of (17.21) Applies to cnseniblcs 1, and 4: see Exercise l a - j E { : c ( t l ) ~ ( ) )= z , ( t l ) x , ( ) , hewuse of (17.20) In Exercise 17.lb thc special case t l = t is investigated The condition only applies to eriseiiible Becaiise t.his ra.ndom process is not, weak stn.tionary, however, it cannot be weak ergodic none of’t,he ramloin proc:esses are weak ergodic Solution 17.3 a) Linear t,ime-averag.e :c;(t)= O! because t,liere is iio tic cornporient The expected value at a c.ert,ain point t o F:(z(to)},is spread across many poirii,s of the CD signal (at,most by as many samples its there are on the CD), because 111 our experiment h sample function conies from it ritiidom time Since music signals in general not h a w dc components, E{z(t)) = h) With first order expectred vrtliies wc can only discuss the condition for stationnrity given in (17.14): In a) we xerta,ined that p, = E{s(l)} = constant Wit,li thc same notation we can assunie E(x2(t)} = constant, because it is determined hrrl rnaiiy va,liies of the output signal 2( t ) + the ra,iidom process could be stationary c) For ergodic:itjy:thc first and second order eiiseinblc nieaiis rnusl agree with t,he corresponding tsiine-aver;igcsof any sample funct,ion, e.g., K(.r2 ( t ) }= x,2 ( t ) This does not, apply because to form thc time-average sqiiared xi2(t)only a ten second section of the CD i s considered, which for different sample fiiiictions 7, is trrlce~nfrom different points on the CD The a\iera,ges are generally different, lbllowing t,o the loudness of the music =+ t,lie random process is not ergodic S o l u t i o n 17.4 ir, ;= - P.,Q i- P r = + S ( t ) ) ) = E ( r ? ( t ) }+ E{21(l):x2(t)}+ E { r ; ( l ) }= fiW(t)} = E((:x:,( t ) 0;: = E{y”t)} -/I,; =3 =O S o l u t i o n 17.5 + IT = ~ { ( v (t )P , $ ) ) ~ } E ( [ s ( i j y ( t )- (LL&) + L ~ L l y ( t ) ) ] ) Since y ( t ) is deterministalc,p u ( t )= ;y(t),and therefore: CT; = E { ( z ( t )- / l & ) ) } Adding ;t = D: = 10 deterininistic sigiial docs not change the variance