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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

$H(f)\left\{\begin{matrix} 1,\: \left | f \right |\leq \frac{1}{2}\, Hz\\ 0,\: \left | f \right |>\frac{1}{2}\, Hz \end{matrix}\right.$

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:

I.  $E(X(t))=E(Y(t))$

II. $E(X^{2}(t))=E(Y^{2}(t))$

III.$E(Y^{2}(t))=2$

Select the correct option:

  1. only I is true
  2. only II and III are true
  3. only I and II are true
  4. only I and III are true
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