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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Why are convolutions between those functions equivalent (signal processing for theoretical neuroscience)?
I'm reading a book on theoretical neuroscience [1], in which the following definitions are given:
$\rho(t)=\sum_{i=1}^n \delta(t-t_i)$
where $\delta$ is Dirac's delta and the $t_i$ are timestamps at ...
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Weiner Filter - why does this computation explain that the necessary filter is a weiner filter?
$X_1(t), X_2(t)$ are random WSS processes with expectation 0, and correlation functions $R_{X_1}(\tau), R_{X_2}(\tau), R_{X_1,X_2}(\tau)$ $n(t)$ is a white noise with SPD $S_n(f) = \frac{N_0}{2}$ ...
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Struggling with visualizing (drawing) a sample of a random process
I've had this question I don't really know how to answer.
let $t \ge 0$, $N_t$ is a possionian random process with parameter 1. let $-\infty < t < \infty$, $X_t$ is a random process that is ...
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random signals through LTI systems, why are these two signals joint wide sense stationary?
I’m trying to solve this problem but I don’t understand an assumption the solution makes:
The question:
let $\hat{W}$ be the best linear approximation of $W_t$ out of $Y_t$, find $\text{CoV}(W_4, \...
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Incorrect Power Calculation in MATLAB Convolution
I have been working on a MATLAB code to perform convolution and calculate the power of the resulting signal. However, I have encountered an issue with the power calculation in my code.
I am convolving ...
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Three dimensional Wiener-Khinchin formula for incompressible isotropic random field
When I am reading Uriel Frisch's book "Turbulence", on page 55 he claimed that the Wiener-Khinchin formula for an incompressible isotropic random field, such as the velocity field of ...
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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 is the difference between the general white process and the white process of order 𝑝?
Antoni uses the following definition of white process of order $p$:
a process whose all cumulants up to order p are such that
$$\text{Cum}\left[X(t),X(t-\tau_1)\cdots,X(t-\tau_{r-1})\right]=C_{rX}\...
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Understanding homework solution - why are $\{X_t\}$ and $\{Y_t\}$ joint WSS, and finding Wiener filter + error
I'm walking through the published solutions of my homework and I'm struggling interpreting them. In them I was given a random Gaussian process $\{X_t\}$ and a random variable
$$\{Y_t\} = X_t\cos(2\pi ...
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What is the jitter effect on the spectrum of impulse train?
How to derive the sepctrum of $$x(t)=\sum_{n=-\infty}^{\infty}\delta\left(t-nT-\tau_n\right)$$
where $\tau_n\sim N\left(0, \sigma^2\right)$
I assume that the randomness effect should behave as a low ...
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Is this signal an FRI (finite rate of innovation) signal?
I am trying to model the subsurface as a randomly layered medium, with stochastic acoustic attentuation that gives rise to a reflected waveform from an active source as the following signal:
$$
x(t) = ...
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Autocorrelation of a random process
I have some doubt about the following exercise.
Let's consider the signal $X(t)= \operatorname{rect}\left(\frac{t}{2A}\right) $ , where $A$ is a
discrete random variable which can assume one value ...
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Power of filtered Bernoulli process
I have some doubt about this exercise.
The Bernoulli random process $X(n)$ with means $p=0.5$ is sent in input to a LTI system with impulse response $h(n)= \cos(\frac{\pi n}{3}) R_3(n+1)$ , where $$...
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Symmetric Autocorrelation Function vs Asymmetric Autocorrelation Function
I am trying to work through the Cyclostationary Blog to create a cyclic autocorrelation function:
I have been given the above equation to determine the cyclic autocorrelation function where tau is ...
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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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positivity of the spectrum of quasi-stationary signals
I am working on the "System identification : theory for the user" by Lennart Ljung (freely available here) and it is one of these books which contains exercises but no answers... My exercise ...
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Expectation and autocorrelation for modulated sinusoid
Given
$$ Y(t) = A X(t) \cos(\omega t + \phi) $$
with $X(t)$ is zero-mean WSS (wide-sense stationary) process, $\phi$ ~ Unif$(0,2\pi)$. Suppose $X(t)$ and $\phi$ are independent random variables. I ...
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adding subtracting PSD vibration data
We performed some vibration tests on a shaker, with two accelerometers on our setup.
One was on the shaker table, to measure the input signal. And the other one was on a flexible part of our test ...
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Generating a random signal for an autocorrelation that is not square integrable
Suppose I have a function that is not square integrable such as the zeroth Bessel function of the first kind. How do I generate a random sequence, with mean zero of course, such that its ...
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If $X(t)$ is a WSS process with mean 5, what is the mean of $X(2t)$? [closed]
I know mean is constant for a WSS process but I am still confused about the mean for this process. My process was by integrating $X(2t)$ from $0$ to $T$, then substituting $t′=2t$. So the limits ...
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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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Why is a random process strictly stationary when its joint Probability density function is time invariant?
I don't understand what stationarity of random process mean.
I know they're statistical properties that are time invariant but I don't have intuition for it and I don't get what that has to do with ...
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The frequency function for $Y_t-18=0.4X_t+0.9X_{t-1}+e_t$
I am having trouble finding the frequency function that takes me from $X_t$ to $Y_t$ in the system stated in the title. $X_t$ and $Y_t$ are stationary stochastic processes and $e_t$ is zero mean white ...
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If add white noise to the signal, how much does the std of the noise affect the thd of the signal?
I'm an electrical department student studying signals and systems.
The RMS value of the general AC electrical signal is 6.31V, and the first harmonic value is 6.30932V(rms). (THD is approximately 1.5% ...
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Random telegraphic noise and Lorentzian noise power spectral density
Following the example of the Lorentzian noise power spectral density shown above (ref), I would like to clarify the following:
In the first figure (labeled by (c)), May I please know why the constant ...
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What is the difference between Yule Walker and Modified Yule Walker Equation that used in Stochastic Signal Modeling?
Our Professor couldn't explain the clear difference between the Yule Walker equation and the Modified version of it that is used in Stochastic models. Please explain both the equations and why we ...
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Why is Autocorrelation between a Zero-mean Random process and a finite deterministic sequence zero?
The Solution is given above:
The Question is, how did the $\mathbb{E}{[x(k)f(l)]}$ and $\mathbb{E}{[x(l)f(k)]}$ become zero? is there some rule that correlation between Random Process and ...
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Can FFT2 be used as a randomness test for image encryption?
Based on this link
Is it possible to consider a model to introduce FFT2 as a parameter to determine how random the image is? (randomness test like Entropy)
consider Lena's image and its FFT2:
If we ...
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Any Relationship Between the Entropy of an Image and Its Spectrum?
Is there a relationship between the Shannon entropy of image and the output of the 2D Fourier transform (DFT) of the image?
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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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Does bandlimited power spectral density correspond to original WSS random process being bandlimited almost surely?
If it is given that PSD of a random process is bandlimited to frequency $f_B$, then can we claim that any sample path of the random process is also bandlimited to $f_B$?
Intuitively, I always thought ...
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Condition for these sequences to be stationary correlated (tipp for integration of exponential functions)
I have two sequences $s$ and $r$ defined as :
$s = \{s_n\}_{n \in \mathbb{Z}}$ where $s_n(t) = (M_{\beta}^n s)(t) = s(t) e^{int}$ with arbitrary $s \in L^2(\mathbb{R})$ and $\beta > 0$
$r = \{r_n\}...
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Probability of the rate of change of a filtered random process
I am given the following problem: I have a filter with impulse response $h(t) = e^{-10t}, t \leq 0$, and autocorrelation function of the input signal, which is WDS and Gaussian with median equal to 0, ...
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Computing the mean of a random process with varying phase due to a random variable
Cheers, I have I am given the following signal $$A \cos ( 2 \pi f_o t + \Theta)$$ and $\Theta$ a random variable with pdf of $\frac{1}{2 \pi } ,0 \leq \theta \leq 2 \pi$ and 0 elsewhere and I am asked ...
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Power Spectral Density and Wiener–Khinchin theorem for 2 different stochastic processes
I know the famous Wiener-Khinchin theorem for stationary random processes: the Fourier transform of the autocorrelation function of a stationary random process is equal to the Power Spectral Density ...
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Why there is a difference from amplitude calculated from PSD and the real one? [duplicate]
I make a script in python to extract amplitude from power spectral density.
I try to verif it but I have a difference between the real amplitude and the amplitude extracted from PSD.
To calculate the ...
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How to draw the PSD from a time series
I try to draw the spectral density of a time series in order to compare it with the theoretical one.
Please can any one help me to do this.
This is the time series of all the data.
Thanks Dan Boschen ...
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Is maximum cross-correlation achieved at the origin?
Let $x[n]$ and $y[n]$ be two DT random signals with $x[n]\xrightarrow{\mathcal{H}}y[n]$ through some system $\mathcal{H}$ that is deterministic yet unknown.\
Define both the autocorrelation and cross-...
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Sampling rates for uncorrelated samples
I'm given the autocorrelation of a WSS random process
and the question asks to find the sampling rate that yields uncorrelated samples.
As far as I understand where looking for the $\tau$'s where $...
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Hard time figuring out whether the following random process is wide sense stationary
I'm dealing with a random process that's simply a square wave with pulse period T, where:
Each pulse takes either $A$ or $-A$ depending on a coin toss.
The wave is shifted by a random $t_d$ where $...
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Effect of sampling a cont. stochastic process on the variance
I am trying to understand the estimation of the power spectral density of a continuous time stochastic process from it's samples.
Consider a normal wide-sense stationary white noise process with ...
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Predicting distribution of integral of random process from power spectral density?
Suppose I have a random process $X(t)$ and I know the power spectral density of $X(t)$, $S_{XX}(f)$.
What can be said about the distribution of $Y(t) = \int_{t'=0}^T X(t') dt'$?
Bear in mind I have a ...
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Limiting value of autocorrelation of wide-sense stationary process
Let random process $X$ is wide-sense stationary process. Where could I find the source or verification of the statement that, when the limiting value of autocorrelation $\lim_{\tau\rightarrow\infty}...
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Is there relationship between autocorrelations Rx(2) and Rx(1)?
There is a random process, $X:\mathbb{Z}\rightarrow\mathbb{R}$ in discrete-time domain, which is wide-sense stationary with zero mean and autocorrelation function $R_X(\tau)$.
a) What range of values ...
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Specify a random process such that $R_Y[0]=3+u$, $R_Y[1]=-2+u$, and $R_Y[k]=u$ otherwise
there is a problem that I should specify a real-valued random process $Y[n]$ such that the autocorrelation function $R_Y[k]$ satisfies
$$R_Y[0]=3+u,\ R_Y[1]=R_Y[-1]=-2+u,\ \text{and}\ R_Y[k]=u, |k|>...
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Confusion In DC power,Average Power,Ac Power and total Power
These are Things I studied.
$ E[X]:-$ THis gives us Average value or Dc value
$E[X^2]:-$Total power
$\sigma^2=E[X^2]-E[X]^2$:-Ac power
From Autocorrelation
$R_x(0)=E[X^2]$--Total power
And $ \lim_{x\...
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Random Signal Energy
Why the energy of random signal(random process) is infinite? and why random signal can not be zero at infinity?
I know that there is a relation between the tow questions but I'am still confused.
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Generating violet noise with a specific PSD coefficient
I am trying to generate a time-domain violet noise signal with the following power spectral density (PSD):
$$ S_n(f) = A^2f^2 $$
Unfortunately, I am having trouble finding the right amplitude ...
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How to generate a power-law / pink noise signal?
Suppose I need to generate a time series where the intervals will be about 120 seconds every time, but with a small variation (e.g. 125, 130, 119, 118, 121, 129, etc)
I want this variation not to be ...
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Is "Introduction to Statistical Signal Processing" by RM Gray good for starting?
I am working on noise processes in electronic devices for my studies, by now Ive been doing a fairly large amount of processing of time measurements, like calculate PSD, estimate thermal, flicker ...
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