Question

A(t) is a wide-sense stationary random process and is a random variable distributed uniformly over [0, 211]. Furthermore, is independent of A(t). Three random processes X(t), Y(t), and Z(t) are given by X(t) = A(t) cos(20ft + 0) Y(t) = A(t) cos(507t + 0) z(t) = X(t) + y(t) a. Show that X(t) and Y(t) are stationary in the wide sense. b. Show that Z(t) is not stationary in the wide sense.

Answer 1

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in (2ext to 2 Sin Sin(20 xt) oxt + 2x) - 2 = 2x sin 25 x cos(40 nt + 2x) 20 E ( x)) =0 tot. -- Cow ( x 62, X(h)) = E(X(t) * X (h)) - E (ALU) A (H) 60s (20x7 +0) les (200h+o). = E(At) A (h)) « Ellos (203t to) Cos/203.hto) = f[t-h) NJE Cos (1000 + 20nh + 20 ) 2 1 + los (20 x (th))). = f(t-h) & Balcos (conlt-h))) o [ E(los (rozeth) + 20 )) =0

" X (t ) is wide-sense stationary Similarly, using E ( ces (sont to)) = 27 sin(507t to) we have ELY(t)) = 0. t t. f CowCY (+ 2, Y(")) = E(Y(t ) x Y(b)) E CALE ) 4 (9) E (as (**t to)asbenh to = f(t-h) E cos (Soult th)+ 20 ) to 2 0 + cos ( 507 (t-h) = f(t-h) & cos (so ñ (t-h)) = a function of (th). - Y(t) is wide-sense stationary

Now z (t) = X (t) + Y (t ) n Elzit)) = o. t Cou(z(t), 28(h)) = cov( x (D ) +y (t ), X (h) + Y Cb)) e cow (X (t), X (h)) + Cow (e (t), Y (h)) + cowl X(+ ), 4(4)) + (ou (Y Lt ), x(h)). Now Coulx (t) Xh) & Covey (t), Y (²)) are functions of (th). so lets see Coul X(t), ĭ (6)). = E(X(t) Y b)) (= E(X(t))= 0 = E(46) = El Act) A (h) cos forth toy cos (sortato) = f(t-h) & E (os / 20x t + 50th+20) - Cos/50th - 20 *t)

Now, Ellos ( 205 t t sokh +20) ) Sin 200 t +50th +20 4 a 2 = a Junction of tfh & not of (t-h) so 2 (t) is not stationary.