X(t)=2(2 t+), where random phase is uniform on [0,2]. Find E[X(t 1)X(t 2)].

GATE 2014 EC (EC1) – Question 60

Communications · Random Processes · 2 marks · Multiple choice

$X(t)=\sqrt2\sin(2\pi t+\varphi)$, where random phase $\varphi$ is uniform on $[0,2\pi]$. Find $E[X(t_1)X(t_2)]$.

$\cos(2\pi(t_1+t_2))$
$\sin(2\pi(t_1-t_2))$
$\sin(2\pi(t_1+t_2))$
$\cos(2\pi(t_1-t_2))$

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Correct answer: (D) $\cos(2\pi(t_1-t_2))$

Explanation

Use2sinA sinB=cos(A−B)−cos(A+B). With amplitude√2 the product is cos[2π(t1−t2)]−cos[2π(t1+t2)+2φ]. Averaging the second term over a uniform phase gives zero. Thus **E[X(t1)X(t2)]=cos[2π(t1−t2)],D**.