autocorrelation

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...

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

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...

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 samp...

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 autocorrelatio...

The ACF of a rectangular pulse of duration T is

The ACF of a rectangular pulse of duration T is

The autocorrelation function of an energy signal has

Sketch the waveform (with properly marked axes) at the output of a matched filter matched for a signal s(t), of duration T, given by $$s(t) = \left\{ {\matrix{ {A\,\,\,\,for} & {0...