416 17 Describing Randoiii Signals = - cosfw0tl- w&) 2 = - coswgI z mit z -= tl t I The integral averages all phase angles between arid 2n The process i s also crgodic because the time-average (17.19) agrees with the ensemble average We can verify this with the rxample of the ACT, where -1 cosw(-Jz If the random process tdso has amraadom peak vdue P in addition to the random phase p7so that ~ ( t=)2 sin(uoO%-t- 'p>I not ergoclic For cxanlpJer while E{z2(t)} iE{,22), the t,ime-avera~geof tlic square of a cert,ain sample function i is, however, zP(t) = it i s indeed statiotrary, brit ~ = l ~ Now that we are comtort&le with expected values and Iiavo learnt ways of c a l m lclting t h i n , we van return to the task from Section 17.i a t the bcginiiing of this chitpter We want to describe t,he propel ties of random signals with deterministic vaiiables* This will IWW be doric with first- and second-ordei expected W ~ U P S We start with ergodic random processrs if needed, so that, the expected values c m be with time avrragc~s The most irriportant c.xpectt.d valiie e the linear avm age, tlw auto-con.rlation f~iizctioiiand a generalisation derived froin it, tlre cross-correlatiuu function