GATE ECE 2014 Set 1 | Question: 50

Question

Consider a random process $X(t) = \sqrt{2}\sin(2\pi t + \varphi),$ where the random phase $\varphi$ is uniformly distributed in the interval $[0,2\pi].$ The auto-correlation $E[X(t_{1})X(t_{2})]$ is

• A. $\cos(2\pi(t_{1} + t_{2}))$
• B. $\sin(2\pi(t_{1} - t_{2}))$
• C. $\sin(2\pi(t_{1} + t_{2}))$
• D. $\cos(2\pi(t_{1} - t_{2}))$

Answer 1

• $R_X(t_1, t_2) = E[\sqrt{2} \sin(2\pi t_1 + \phi) \cdot \sqrt{2} \sin(2\pi t_2 + \phi)]$

  $= 2 E[\sin(2\pi t_1 + \phi) \sin(2\pi t_2 + \phi)]$

  Using the identity $2 \sin A \sin B = \cos(A - B) - \cos(A + B)$:

  $= E[\cos(2\pi(t_1 - t_2)) - \cos(2\pi(t_1 + t_2) + 2\phi)]$

• Applying Expectation: Since $\phi$ is uniform over a full period $[0, 2\pi]$, the expectation of the second term $\cos(2\pi(t_1 + t_2) + 2\phi)$ is zero.

• Result: $E[X(t_1)X(t_2)] = \cos(2\pi(t_1 - t_2))$.

Answer: D. $\cos(2\pi(t_1 - t_2))$