Question

Consider two random processes X(t) and Y(t) defined as

X(t)=Acos(w_ot+z), Y(t)=Bsin(w_o+z)

where A and B and w_o are constants and z is a random variable that is uniformly distributed between 0 and 2pi. find the cross-correlation function of X(t) and Y(t). If both X(t) and Y(t) were wide sense stationary , could they also be jointly wide sense stationary?

Answer 1

XCH) - Acos (wot+2) y(t) = B sin wot +2) A, B, wo constants Z - uniform R.U Ifz(z) 1 Yax 2717 Rxylte tz) = cross cusselation of x Gt) and you Rx y Ltr 1) = € (x+) YCb] je = Ef Acos@oti +x). B sân @o tz+z)] El AB COS (woti + 2) sin(wo t2+2)] - AB el cos @oti + 2). sin 60 t3+2)] = ABEL'sincwortit 2+ wotz + 3) 7 + sin (wo ty + Z-600+1+2)) = {i 2 sina Comb = sin 6t 6) + Sin C4–6)] El sin wo fit t) + 22) + sin wo (ty to

Abde ( sin (wa6to the ) +22] + e ( sin wo (tar ::* = AB, Sin women constant [v=ta-ti] Rxy (tit) = Rxy(r) Independent of time So, x,y are jointly jointly wiss stationary Cuidesenve station culy process] C. Bonus .. E[xy) = s[x].ECY] Indipendent R.v] 0.0, ЕСху. XIY are alss LE CRY] =) : xiy are ws7 inclipendelly