GATE EC Random Processes questions

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

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 …

An information source generates a binary sequence $\left \{ \alpha _{n} \right \}$. $\alpha _{n}$ can take one of the two possible values $-1$ and $+1$ with equal probability and are statistically independent and identically distributed. This sequence is precoded to obtain another sequence $\left \{ \beta _{n} \right \}$, as …

Consider a random process $X\left ( t \right )=3V(t)-8,$ where $V(t)$ is a zero mean stationary random process with autocorrelation $R_{v}\left ( \tau \right )=4e^{-5\mid \tau \mid}.$ The power in $X(t)$ is _________

The input-output relationship of a causal stable LTI system is given as $$y[n] = \alpha \: y[n-1] + \beta \: x[n]$$. If the impulse response $h[n]$ of this system satisfies the condition $\sum_{n=0}^{\infty}h[n]= 2$, the relationship between $\alpha$ and $\beta$ is

The power spectral density of a real stationary random process $X(t)$ is given by $$S_X (f) = \begin{cases} \frac{1}{W}, & \mid f \mid \leq W \\ 0, & \mid f \mid > W \end{cases}$$ The value of the expectation $E \left [ \pi X(t)X\left ( t-\frac{1}{4W} \right ) \right ]$ is __________.

Let $X(t)$ be a wide sense stationary $(WSS)$ random process with power spectral density $S_{X}(f).$ If $Y(t)$ is the process defined as $Y(t)= X(2t-1)$, the power spectral density $S_{Y}( f )$ is

A real band-limited random process $X(t)$ has two-sided power spectral density $$S_{X}(f)= \begin{cases} 10^{-6} (3000-\mid f \mid) \text{Watts/Hz} & \text{for } \mid f \mid \leq 3 kHz \\ 0 & \text{otherwise} \end{cases}$$ where $f$ is the frequency expressed in Hz.The signal $X(t)$ modulates a carrier $\cos16000\pi t$ and the resultant …

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

A zero- mean white Gaussian noise is passed through an ideal lowpass filter of ban width $10 \; \mathrm{kHz}$. The output is the uniformly sampled with sampling period $t_{s}=0.03 \; \mathrm{msec}$. The samples so obtained would be

Statement for Linked Answer Questions 78 and 79 The following two questions refer to wide sense stationary stochastic processes It is desired to generate a stochastic process (as voltage process) with power spectral density $$S(\omega)=\frac{16}{16+\omega^{2}}$$ by driving a Linear-Time-Invariant system by zero mean white noise (as voltage process) with …

Statement for Linked Answer Questions 78 and 79 The following two questions refer to wide sense stationary stochastic processes The parameters of the system obtained in $\text{Q.78}$ would be

Noise with uniform power spectral density of $\text{N}_{0}$ $\mathrm{W / Hz}$ is passed through a filter $\mathrm{H}(\omega)=2$ exp $\left(-j \omega t_{d}\right)$ followed by an ideal low pass filter of bandwidth $\text{B Hz.}$ The output noise power in Watts is

The PSD and the power of a signal $g(t)$ are, respectively, $\mathrm{S}_{\text{g}}(\omega)$ and $\text{P}_{\text{g}}$. The PSD and the power of the signal $a \text{g}(t)$ are, respectively,

The spectral density of a real valued random process has