GATE ECE 2017 Set 1 | Question: 51

Question

Let $X(t)$ be a wide sense stationary random process with the power spectral density $S_{X}(f)$ as shown in Figure (a), where $f$ is in Hertz$(Hz)$. The random process $X(t)$ is input to an ideal lowpass filter with the frequency response

$$H(f)  = \begin{cases} 1, & \mid f \mid \leq \frac{1}{2} Hz \\ 0, &\mid f \mid > \frac{1}{2} Hz \end{cases}$$

as shown in Figure(b).The output of the lowpass filter is $Y(t)$.

Let $E$ be the expectation operator and consider the following statement:

1. $E(X(t))=E(Y(t))$
2. $E(X^{2}(t))=E(Y^{2}(t))$
3. $E(Y^{2}(t))=2$

Select the correct option:

• A. only I is true
• B. only II and III are true
• C. only I and II are true
• D. only I and III are true