Random Processes - Communications - Electronics & Communication Engineering Previous Year Questions

Noise with double-sided power spectral density of K over all frequencies is passed through a RC low pass filter with 3 dB cut-off frequency of f_c. The noise power at the filter output is

X(t) is a stationary random process with autocorrelation function $R_X(\tau) = \exp(-\pi\tau^2)$. This process is passed through the system shown below. The power spectral density of the output process Y(t) is

The power spectral density of a real process \(X(t)\) for positive frequencies is shown below. The values of \(E[X^2(t)]\) and \(|E[X(t)]|\), respectively, are

A random variable \(X\) takes values \(-0.5\) and \(0.5\) with probabilities \(\frac{1}{4}\) and \(\frac{3}{4}\), respectively. The noisy observation of \(X\) is \(Y = X + Z\), where \(Z\) has uniform probability density over the interval \((-1, 1)\). \(X\) and \(Z\) are independent. If the MAP rule based detector outputs \(\hat{X}\) as

\[ \hat{X} = \begin{cases} -0.5, & Y < \alpha \\ 0.5, & Y \ge \alpha, \end{cases} \]

then the value of \(\alpha\) (accurate to two decimal places) is ________.

Let a random process Y(t) be described as Y(t) = h(t) * X(t) + Z(t), where X(t) is a white noise process with power spectral density S_X(f) = 5 W/Hz. The filter h(t) has a magnitude response given by |H(f)| = 0.5 for -5 ≤ f ≤ 5, and zero elsewhere. Z(t) is a stationary random process, uncorrelated with X(t), with power spectral density as shown in the figure. The power in Y(t), in watts, is equal to ___________ W (rounded off to two decimal places).