Question

324. Consider the random process X(t) = A + Bt2 for - <t < oo, where A and B are two statistically independent Gaussian random variables, each with zero mean and variance o?. a) Plot two sample functions of X(t). b) Find E{X(0)} c) Find the autocorrelation function Rx(t,t +T). d) Find the pdf of the random variable Y = X(1). e) Is X(t) a Gaussian process? Prove your result.

Answer 1

1 clear all; 2 clear variables; 3 clc; 4 t=-40:40 ; 5 N=length(t); 6 mean=0; 7 variance=1; 8 A=mean+sqrt(variance)*randn(1,N); 9 B=mean+sqrt(variance) *randn(1,N); 10 X=A+B. *t.^2; 11 plot(t, x) 12 xlabel('t') 13 ylabel ('X') 14 title('variance=1') 15 grid on 16

variance=1 2000 1500 1000 500 HH хон -500 -1000 -1500 -2000 -40 - 20 0 20 40

1 clear all; 2 clear variables; 3 clc; 4 t=-40:40; 5 N=length(t); 6 mean=0; 7 variance=2; 8 A=mean+sqrt(variance) *randn(1,N); 9 B=mean+sqrt(variance) *randn(1,1); 10 X=A+B. *t.^2; 11 plot(t, x) 12 xlabel('t') 13 ylabel('X') 14 title('variance=2'). 15 grid on 16

variance=2 3000 2000 1000 X-1000 -2000 -3000 -4000 -5000 -40 -20 0 20 40

b) X(t) = A + B +2 E[X(t)) - E[+ E[B]+ = 0 + 0 = 0. c) Rylt,t+t) - E{ x1)x(++)] = E( (+b+°) ( +8 64+0%)] + ABLt + 2)² + ABt² + B2+²/+103) In u E[A²] + E [AB] (t+23² + E[AB] +2 + E (B2) Łltre) d) = 0 + 02 (t+z)? 4 = X(1) - A+B. Elyde ELA) + E(B) = 0 E(42) - E((A+B)2) = E(A²+ B² + 2AB) = 2to² = 202 Ya N10,20).

e) E(X[4)= E(A) + Ell) +2 Elx*(t)) = E [C+ + 6+)?] = f[A2+B+++ + 2AB+] = 8²7 82tt E(x41)3= 8+(1++4) mean o & vanane ² (1+t) X(t) is gauman with can with x~ N(0, 041++)}