GATE EC 2012 Set 1 Q2 — Random Processes

Consider a real-valued random process $$f(t)=\sum_{n=1}^{N} a_{n} \: p(t-n T),$$ where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and $0$ otherwise. The coefficients $a_{n}$ are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the …

​​​​A white Gaussian noise $w(t)$ with zero mean and power spectral density $\frac{N_{0}}{2}$, when applied to a first-order RC low pass filter produces an output $n(t)$. At a particular time $t=t_{k}$, the variance of the random variable $n\left(t_{k}\right)$ is $\_\_\_\_\_\_\_$.

Let $X(t)$ be a white Gaussian noise with power spectral density $\frac{1}{2} \mathrm{~W} / \mathrm{Hz}$. If $X(t)$ is input to an LTI system with impulse response $e^{-t} u(t)$. The average power of the system output is _____________ $\mathrm{W}$ (rounded off to two decimal places).

The autocorrelation function $R_{X}\left ( \tau \right )$ of a wide-sense stationary random process $X(t)$ is shown in the figure. The average power of $X(t)$ is ________________

Let a random process $Y(t)$ be described as $Y(t)=h(t) \ast X(t)+Z(t),$ 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 $\mid H(f) \mid =0.5$ for $-5\leq f\leq5,$ and zero elsewhere. $Z(t)$ is a stationary random process, uncorrelated with $X(t)$, with …

Consider a white Gaussian noise process $\text{N(t)}$ with two-sided power spectral density $S_{N}\left ( f \right )=0.5\:W/Hz$ as input to a filter with impulse response $0.5e^{-t^{2}/2}$ (where $t$ is in seconds) resulting in output $Y(t)$. The power in $Y(t)$ in watts is