Questions tagged [random-process]

A random process (in time), also called "stochastic process" is a signal that, when sampled at any given time, is a random variable.

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Variance of White Gaussian Noise

It could seem an easy question and without any doubts it is but I'm trying to calculate the variance of white Gaussian noise without any result. The power spectral density (PSD) of additive white ...
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Power Spectrum: Definition

I am new to the study of time series. Recently I have asked a question about the covariance of real and imaginary part of a real(in time domain) stochastic time series and I have received an answer ...
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PSD of complex white gaussian noise

It may be a really simple question, but I'm not sure about this one: Given a complex white Gaussian noise process with iid real and imaginary parts and a double sided power spectral density of $N_0/2$...
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If the mean of a random process is constant, does it imply the process is first order stationary?

If a random process is first order stationary, its mean is constant. However, if a random process has a constant mean say $3$ and an autocorrelation equal to $9 + 15e^{|-\tau|}$. The process is ...
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What is a good example of an ergodic process?

I'm trying to find simple examples of an ergodic process. What process comes to your mind as a good illustration of its properties? A quick research (Wikipedia, another answer) mainly gives examples ...
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Is white noise WSS by nature or not?

I want to know what is the difference between white noise and WSS white noise. is there any difference between them or they're equal? and what about white Gaussian Noise?
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What is the practical meaning of the variance, covariance, mean value?

What is the practical meaning of variance, covariance and mean for a signal?
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Why is $A\cos(2\pi f_ct)$ a non-stationary process?

I am studying analog communication and having Communication system - Simon Hykin as one of the reference. There is a question Consider the sinusoidal process$$X(t) = A\cos(2\pi f_ct)$$where the ...
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The normalization of the autocorrelation function and how it changes the definitions you've learned about signal analysis in communication systems

Since this question is book-oriented, I will kindly ask you to accompany it with a book that is considered by many researchers the bible of Digital Communication: Proakis - Digital Communications, one ...
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Random process $X(t)$ with autocorrelation function given find the mean and the variance

Autocorrelation function is $$R_{xx}(\tau)=\frac{20}{1+2\tau^2}$$ So at $\tau=0$$$R_{xx}(0)=20=E[X(t)X(t)]=E[X^2(t)]$$ The variance is $$\mathrm{Var}[X(t)]=E[X^2(t)]-E^2[X(t)]=20-E^2[X(t)]$$ As $X(t)$...
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Why is $\sin(t)$ a stationary process?

I am trying to understand the meaning of the term Stationary Process. For example, I was told that $\sin(t)$ is a stationary process. Could someone try to explain, in simple words, why is $\sin(t)$ (...
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What are the statistics of the discrete Fourier transform of white Gaussian noise?

Consider a white Gaussian noise signal $ x \left( t \right) $. If we sample this signal and compute the discrete Fourier transform, what are the statistics of the resulting Fourier amplitudes?
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Understanding of Random Process, Random Variable and Probability Density Function

I just wanted to confirm my understanding of a Random Process, Random Variable and the its Probability density Function. Here is the way that I looked a Random Process/Random Variable: If we ...
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Relation between frequency spectrum and PDF of a random variable

I have a random variable that is being generated according to some probability distribution function (e.g. a Gaussian PDF). When looking at the frequency spectrum of the generated data does the ...
Television's user avatar
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What Is Continuous White Noise in The Context of Signal Processing and Broadly

How can one define Continuous White Noise in a coherent way? Is there a way to derive it Mathematically? Specifically, is there a way to define it which will works as the model in Signal Processing ...
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What's the meaning of ergodicity? [duplicate]

I just read the topic about Ergodicity but I have ambiguity about its meaning (by intuition). What does mean: (for mean) Statistical average = Time average. Could you please explain it in detail. ...
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PSD from autocorrelation in MATLAB

I am trying to simulate a simple stochastic process defined by the equation: \begin{equation} \frac{1}{v}\frac{db}{dt} +\Gamma_0 b= \sqrt{\sigma}R(t), \end{equation} where $R(t)$ is a zero-mean white ...
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variance in the time domain versus variance in frequency domain

Hi All: I'm trying to better understand the connection between variance of a time series and the integral of the spectral density over all frequencies. Rather than going through all of the relations, ...
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moving average rounding error analysis

I have implemented a moving average, similar to the Hogenauer Filter, with a reduced number of computation operations. I expect the expected error to behave as the random walk and its STD to be of ...
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Energy Detection in Presence of Colored Gaussian Noise

Before asking my question, let me introduce the context: For spectrum sensing based on energy detection, which has been widely studied in presence of AWGN, the optimal detection threshold is computed ...
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Average Power Spectral Density of PAM signals

I am reading through the PAM transmission scheme and about the power spectral density of the signals. Given that the Average Power Spectral Density of PAM Signals is: $$ \Phi_{ss}(f)=\Phi_{aa}\left(e^...
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What is the distribution of it?

If $\theta$ is uniformly distributed in $(0, 2\pi),$ then what is the distribution of $e^{i\theta},$ where $i = \sqrt{-1}?$ And what are the statistical properties of $\left[e^{i0\theta}\, e^{i1\theta}...
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explanation of correlation of stationary stochastic processes

I have some doubts about correlation in stationary stochastic processes. I know that the expectation of a random variable is $$E(x)=\int_{-\infty}^{+\infty} a f_x(...
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Band-limited random signal with arbitrary distribution?

I'd like to generate a random discrete-time signal that is band-limited to some bandwidth B (by means of a digital filter, ie in MATLAB). The catch is that I'd like this signal to have an arbitrary ...
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Proving the upper bound of cross correlation

I am reading about cross-correlation from this document and equation (5) states that The maximum value of the crosscorrelation is not always when the shift equals zero; however, we can prove the ...
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Is the sum of white noise and shifted white noise white noise again?

Let $W[k]$ be a stationary white noise with variance = 1 Question: Is $X[k] = W[k] + c \cdot W[k-1]$ white noise? $c$ is a real number.
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What's the FFT2 of white noise image?

I try to compute the FFT2 of the white noise image. I use this image with Entropy 7.995 and pixel scatter plot: based on this python code: ...
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2 answers
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How do random processes add in Rayleigh fading to get a Guassian random process?

I was studying Rayleigh channels from Wireless Communications, Second Edition by Andreas F. Molisch. The book states that by adding different in-phase and quadrature components multipath components ...
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1 answer
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Wide sense stationary that is not strict sense stationary? [duplicate]

A "wide-sense stationary process" (WSS) means that its mean is a constant, and its auto-correlation is time-invariant, that is: $$\begin{aligned}E[x(t)] &= c \\ R_X(t_1, t_2) &= R_X(...
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