Question

e let X(+) and Y) be independent faussian random process wth zedo means and the same co-variance function cx (o), whese T is the difference between of two distinct points define the amplitude. modulated signal by z(t) = X(t) cob wt + y(t) sin wt. Find the PDF of ZCt). Determine if Z(+) is a SSS 0r/ and WSS

Answer 1

solution: H OGY stept: let zlt) = alt) cosut & YLt) sinut where xlt) and ylt) are the Gaussian random zero mean and the same conviene process with function. z(t) = X(t) casut & Y(A) showta apply expectations on both side E[ zit) = EC x1) cowt + yle) sowot] E[xct) coswt+E(YLt)] Sinot r. zero mean ] z 0 x Swt + Ox sinwr E[z(] = 0 stepa; complete thee auto coveriance of zit) as Czltus ta) - E[ z(t)= (t)) - E[z(t) ] E [z_ty] Store zlt)z xlt) cowt + y(t) sinct 2 I and E{xto] - E[410] -0

2 Ext.) coswt, tylto) Sinuto) (x(+2) coswtz tiltasinut) – [0] are z fecxlto) x (ty) CowtocoButy + E[4Lt) Yltz))sincot, Sinusty 1 + E(XLtoY[ty] But, sinute + E[ylt) xtz]siswt , outy = f ( x[to) x+] Cebut, cos usty + €[ylt) yltz)] Siouote Shady { [[X14] = { [ 414) -0 2 el tort2) Coswt, coswtz + c(t it) sinust, sinuta covariance 1 fince xlt) and yet has common function cltosta)} 2 altute (coswto Coswtz + siswt, sinuta) 2 c. (toute) cos (ti- tz) ş i cos(&-B) a cosa cosp) + sina sing

step3 Here , common covariance teanctim is Call). terefore, Joe Ca (tostz) Gal T]cos cölte-to) a Caltortz) cosltorta) Hence, zlt) is wide sense stationaly rance poiccere. Stepq: Here we calculate the probability density function (pdf) of z(t) let Xt) be Gaussian sandem proceed with mean in and variance of the part of xlt) is fray (n) - - The poobability density feronction of Gaurian dès tribution z(+) cottn mian zero and covariance function cltista) = alto, tz) (CT) = yen erobakined veryone