Random Processes

Consider a real-valued random process $f(t) = \sum_{n=1}^{N} a_n p(t - nT)$, where $T > 0$ and $N$ is a positive integer. Here, $p(t) = 1$ for $t \in [0, 0.5T]$ and $0$ otherwise. The coefficients $a_n$ are pairwise inde...

Consider a real-valued random process $$ f(t)=\sum\limits_{n=1}^N a_n p(t-n T), $$ 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 indep...

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 ran...

For a real signal, which of the following is/are valid power spectral density/densities?

Let X(t) be a white Gaussian noise with power spectral density $$\frac{1}{2}$$W/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 ____________ W (rounde...

The random variable $$ Y=\int_{-\infty}^{\infty} W(t) \phi(t) d t, \quad \text { where } \phi(t)=\left\{\begin{array}{cc} 1, & 5 \leq t \leq 7 \\ 0, & \text { otherwise } \end{array}\right. $$ and $W(t)$ is a real white...

Consider the random process x(t) = U + Vt. Where U is a zero mean Gaussian random variable and V is a random variable uniformly distributed between 0 and 2. Assume that U and V are statistically independent. The mean val...

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...

Consider random process $$X(t) = 3V(t) - 8$$, where $$V$$ $$(t)$$ is a zero mean stationary random process with autocorrelation $${R_v}\left( \tau \right) = 4{e^{ - 5\left| \tau \right|}}$$. The power of $$X(t)$$ is ____...

$$\mathop {\left\{ {{X_n}} \right\}}\nolimits_{n = - \infty }^{n = \infty } $$ is an independent and identically distributed (i.i.d) random process with $${X_n}$$ equally likely to be $$+1$$ or $$-1$$. $$\mathop {\left\{...

A random binary wave $$y(t)$$ is given by $$$y\left( t \right) = \sum\limits_{n = - \infty }^\infty {{X_n}p\left( {t - nT - \phi } \right)} $$$ where $$p(t) = u(t) - u(t - T)$$, $$u(t)$$ is the unit step function and $$\...

Consider a random process $$X\left( t \right) = \sqrt 2 \sin \left( {2\pi t + \varphi } \right),$$ where the random phase $$\varphi $$ is uniformly distributed in the interval $$\left[ {0,\,\,2\pi } \right].$$ The auto -...

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

A real band-limited random process $$X( t )$$ has two -sided power spectral density $$${S_x}\left( f \right) = \left\{ {\matrix{ {{{10}^{ - 6}}\left( {3000 - \left| f \right|} \right)Watts/Hz} & {for\left| f \right| \le...

The power spectral density of a real stationary random process X(t) is given by $$${S_x}\left( f \right) = \left\{ {\matrix{ {{1 \over W},\left| f \right| \le W} \cr {0,\left| f \right| > W} \cr } } \right.$$$ The value...

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

The following question refer to wide sense stationary stochastic process: It is desired to generate a stochastic process (as voltage process) with power spectral density $$$S\left( \omega \right) = {{16} \over {16 + {\om...

The following question refer to wide sense stationary stochastic process: The parameters of the system obtained in Q. 12 would be

A zero-mean white Gaussian noise is passed through an ideal low-pass filter of bandwidth 10 kHz. The output is then uniformly sampled with sampling period t s = 0.03 msec. The samples so obtained would be

Noise with uniform power spectral density of N 0 W/Hz is passed through a filter H(ω ) = 2exp (-jωt d ) followed by an ideal low pass filter of bandwidth B Hz. The output noise power in Watts is

If the variance $$\sigma _d^2$$ of d(n) = x(n - 1) is one-tenth the variance $$\sigma _x^2$$ of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation function $${R_{xx}}\,(k)\,/\,\,\sigma...

The probability density function of the envelope of narrow band Gaussian noise is

The ACF of a rectangular pulse of duration T is

For a narrow band noise with Gaussian quadrature and inphase components, the probability density function of its enveolope will be

Zero mean Gaussian noise of variance N is applied to a half wave rectifier. The mean squared value of the rectifier output will be:

White Gaussian noise with zero mean and double - sided power spectral density $$\eta /2$$ is the input $$x(t)$$ to a linear system with impulse response $$h(t)$$ $$ = exp\left( { - t/RC} \right)\,\,\,\,\,\,\,\,u\left( t...

White Gaussian noise is passed through a linear narrow band filter. The probability density function of the envelope of the noise at the filter output is: