random process

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....

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. T...

Let X (t) = A cos(2π f₀t+θ) be a random process, where amplitude A and phase θ are independent of each other, and are uniformly distributed in the intervals [-2,2] and [0,2π], resp...

Let $X(t) = A\cos(2\pi f_0 t+\theta)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2...

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{arra...

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 Sx (f) = 5 W/Hz. The filter h(t) has a magnitud...

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...

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 \r...

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 st...

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

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 $$p(t) = u(t) - u(t - T)$$, $$...

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution fun...

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 $$\l...

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 \r...