Questions tagged [random-process]

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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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27 views

Affine transformation on circularly symmetric Gaussian distribution

Assume the complex random vector $\mathbf{y} \in \mathbb{C}^N$ is circularly symmetric distributed i.e. $\mathbf{y} \sim \mathcal{CN}(\mathbf{0},\mathbf{R})$ $$ p_{\mathbf{y}}(\mathbf{y})=\pi^{-N}\...
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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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36 views

How to take amplitude and frequency from the power spectral density?

I try to take the amplitude and the frequency from the power spectral density. So I make a time series with a 5 component waves, then I draw the PSD with the welch method. Now I need to take the ...
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55 views

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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Relationship between quantum non - causal order and control/feedback

Recently, in the field of quantum information, the causal structure of the events in a quantum mechanical system has been questioned. I'm thinking of the results in the following paper Quantum ...
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1 answer
57 views

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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2 answers
182 views

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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1 answer
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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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1 answer
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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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1 answer
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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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2D space and 1D time evolution of a random field

I also asked this on math stack-exchange, but it is also relevant for the signal processing community. I want to develop a 2D random field and its change with time with constant velocity. My process: ...
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1 answer
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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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1 vote
1 answer
105 views

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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1 answer
44 views

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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The definition of amplitude probability density

I'm trying to figure out the formal definition of "amplitude probability density"(APD). First of all I didn't find a textbook which defines APD but there are some sources that explain it ...
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1 answer
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related to random nature of wireless channel

I am reading An Approximate BER Analysis for Ambient Backscatter Communication Systems With Tag Selection wherein it is mentioned that "when the distance between two nodes is very small and line ...
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Frequency response terminology: $N(i\omega)$, $Q(i\omega)$

I am reading through an analysis of a system $$ Y(t) = \frac{d}{dt}X(t) + X(t) $$ and the frequency response $H(i\omega)$ is defined as $$ H(i\omega) = \frac{N(i\omega)}{Q(i\omega)} = \frac{1+i\omega}{...
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1 answer
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How to Combine white gaussian random noises from different Seeds?

I have generated two different white gaussian random noises in MATLAB using two different seeds. For example: ...
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1 vote
2 answers
83 views

What's a Normalized function?

I'm studing Representations of Random Processes and the book talks about Orthonormal functions, but doesn't make it clear what is it. I was able to realize that a set of functions are orthonormal if ...
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1 answer
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RMS Amplitudes of voltage

I am looking here at the course of current fluctuations and have calculated the RMS values, but somehow it seems strange to me: with the black line there are small fluctuations, but the RMS value is ...
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Why is Power spectral density of random walk noise defined despite it being non-stationary? [duplicate]

While reading up on oscillator stability, I noticed that authors characterize random walk noise (Brownian noise) as having a PSD of $S_y(f) = h_{-2} f^{-2} $ where $h_{-2}$ is some constant. This is ...
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1 vote
1 answer
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Expected Value of a sequence with two random variables

If I have a signal of the form $x\left(n\right)=Acos\left(nω+ϕ\right)$ where $\omega \in \left[\omega -\lambda ,\omega \:+\lambda \:\right]$ is a uniform random variable and $\phi $ is also a uniform ...
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Global variability index for group of signals

Suppose I have a method that I can use to generate $n_p$ signals (we can intend them as realizations of an unknown not stationary discrete-time stochastic process). Modifying the method, I can obtain ...
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Signal-to-Noise ratio of multivariate stochastic process from Correlation Matrix

I'm not in signal processing, I'm from an another discipline. I've derived a simple result which I presume must be well known in SP and I'd like to know whether there's a paper or textbook that has it ...
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Response of an unstable LTI system to random signals

A convenient approach for studying the response of a stable LTI system with impulse response $h(t)$ to a WSS stochastic input $X(t)$ is to look at the power spectral density (PSD) of the output $Y(t)$ ...
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mathematical expression to detect modulated data within a vector

Assume I have such modulated data which is, for example, $x=0.7+0.7i$. That modulated data is encapsulated in a vector as below: OR where $c$ is any constant number, let’s say, for example: $c=0.7$ ...
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4 votes
3 answers
171 views

Under what conditions is there a one-to-one mapping between continuous-time and discrete-time signals?

As the sampling theorem dictates that the uniform sampling frequency must be at least twice the maximum frequency present in the bandlimited signal (Nyquist rate), a question arises about the ...
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Transformation of random variables vs shift of functions

I am a beginner to random variables and I am understanding the concept of the transformations of a random variable. Consider a random variable $X$ to be Gaussian distributed with $a_x = 1.6$ and $\...
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2 answers
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Intuitive definition of ergodicity for random signal

Is it possible to define the ergodicity of the random signal in an intuitive sense without using any statistical reference?
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3 answers
75 views

voltage current analysis in time

I am looking at the transmembrane voltage of a neuron model and would like to investigate this in more detail. The applied current is divided into a deterministic and a noisy component - if I set the ...
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3 votes
3 answers
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Signal variance and power connection

For a random signal $x(n)$, why is $E(x(n)^2)$ called signal power? Is it really power? Any proof?
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1 answer
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How to describe voltage fluctuations

I am looking for a way to describe the properties of voltage fluctuations - is the RMS amplitude a suitable measure? Attached is a screenshot showing the random current fluctuations (in the case of ...
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1 answer
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What is the relation between input and output PSDs given system transfer function $H(s)$

If I have the system transfer function $H(s)$ in the complex frequency domain, how would I relate the input/output power spectral densities? I have come across the relation $P_{out}(f) = |H(f)|^2P_{in}...
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1 vote
1 answer
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Variance of filtered white noise

I was asked a question, as posted here, and the answer given is (A) i.e. $\frac{3}{2} A^2 N_0$. My solution steps was: Finding the mean of the output process: Since input is gaussian the output will ...
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3 votes
1 answer
176 views

Variance of Integral of a real white Gaussian Noise Process

In this question, is the answer not equal to infinity ? Answer is mentioned as 6. But my doubt is cant we think of it like a linear combination of many independent random variables each having ...
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4 votes
2 answers
77 views

Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)

Assume we have the following system (coming from control systems theory, hence in s-domain) $ Y(s) = H_A (s) \cdot A(s) - H_B (s) \cdot B(s) $ I now wish to consider $a(t)$ and $b(t)$ as white noise ...
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3 votes
1 answer
388 views

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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0 votes
0 answers
51 views

How to characterize the randomness of an event using it's PSD?

I have the power spectral density function of a stochastic phenomenon. how can I generate a signal (time series) representing the randomness of this event over time? How can I draw the probability ...
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1 vote
1 answer
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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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2 votes
0 answers
28 views

Signifance of statistical information in a signal

I am learning control engineering for some time and I work with a lot of transfer functions and frequency domain design. Reading from textbook, to me everything seems deterministic. Whenever I come ...
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5 votes
1 answer
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Conceptual Questions on Colored Noise Process

I am having a tough time finding answers to some specific questions and finding references where there is information regarding Brownian noise or Red Noise. I'm referring to white and colored noises ...
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4 votes
3 answers
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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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-1 votes
1 answer
266 views

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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-2 votes
2 answers
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WSS vs SSS vs ergodic

Is this the correct "venn diagram" that related WSS, SSS, and Ergodic process types? $$\text{all process types}\begin{cases}\text{WSS} \begin{cases}SSS \begin{cases}\text{ergodic} \\ \text{...
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