Questions tagged [stochastic]

A "stochastic process" is another term for "random process".

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Signal Processing of Long Term Behavior in Stochastic Systems

I am quite new to techniques of signal processing. I have a fairly generic problem and wish to find information about topics and/or techniques that may help me address this problem. Let $x(t)$ and $y(...
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Probability distribution sampling from a trajectory

Disclaimer: this question was originally posted on Physics SE but with very limited replies and I am reproducing it here as a suggestion from one of the comments. Consider a sequence of random ...
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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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Proof of the Wiener-Khintchine theorem in time domain

In a proof for the Wiener-Khintchine theorem (See p. 572-573) I have seen the following operation being done: \begin{align*} S_{xx}(f) &= \lim_{T \rightarrow \infty} \int_{-2T}^{2T} Rxx(\tau) e^{-...
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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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What happens if you use the Fourier transform of the autocorrelation of a non-WSS process to compute power spectral density?

The Wiener–Khinchin theorem states that the power spectral density of a wide-sense stationary stochastic process can be obtained through the Fourier transform of the autocorrelation of the signal i.e. ...
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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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Bandpass Stationary Stochastic Process

I was following this interesting post by a new user Rubem Pacelli and got stuck at Proakis' referenced definition (see Section 4-1-4 starting on page 159 here). The math, all repeated further below, ...
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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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Approximate a Known System with Adaptive Filter and an Unknown System in a Series

I am using gradient descent on an adaptive IIR filter for the below system 1. At the moment I am just assuming the known system is not there and it works fine. However, occasionally when the known ...
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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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Deterministic, stochastic and chaotic components of a signal ; their combined effect

If a signal contains deterministic, stochastic as well as chaotic components, does this generally represent a problem in the analysis of the signal? By a problem, I mean that a signal containing all ...
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White noise does not contradicts Wide Sense Stationarity?

I am studying White Noise. But I am really beginner level, so I have a confusion with its construction. White Noise is usually defined as a Wide Sense Stationary process $N=\{N_t\}_{t\in T}$ (for $T$ ...
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Noise from irregular sampling pattern

In this paper the author says By using an irregular sampling pattern and filtering the irregular samples to create the pixels, featureless noise is produced from such high frequencies rather than ...
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Calculating error of estimation of signal from noisy data (Explanation of result)

Given random varaible X with distribution $$ \begin{cases} \mathbb{P}\left(X=1\right)=\alpha\\ \mathbb{P}\left(X=-1\right)=1-\alpha \end{cases} $$ Where $ \alpha $ is a given parameter($X$ is a binary ...
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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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How Could One Accelerate the Convergence of the Least Mean Squares (LMS) Filter?

How can the convergence of an LMS filter be accelerated? Can we do better than the Vanilla algorithm?
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Find coefficients of optimal Wiener filter of length 2

I need to find the coefficients (impulse response) of a FIR Wiener filter with length equal to 2. I have a gaussian white noise signal that is generated using the Standard Normal Distribution (mean = ...
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Sampling Noise Given Power Spectral Density

I am trying to write a Monte Carlo physics simulation which involves, given a power spectral density, sampling rate, and total number of samples, generating noise with such a power spectral density. I ...
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When is the Correlation Coefficient Ergodic

Given Wide Sense Stationary (WSS) processes X and Y that are ergodic to the mean and autocovariance. Under what conditions is the correlation coefficient ergodic to the mean? ie: $lim_{T->\infty} \...
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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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Variance Due to white noise input

I have the problem below. It sounds simple but for some reason I have been stuck on it for a long time and don't know what am doing wrong. Am trying to solve this using correlations. So we all know ...
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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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How to find the output mean and autocorrelation of a non-linear system

I need help with this question. I am sure this is the right StackExchange forum for this type of question. Consider a nonlinear device such that the output is $Y(t) = aX^2(t)$, where the input X(t) ...
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When is Markov a Martingale

I have two questions and I am very confused about the concepts Can a Markov process of order one also be a a Martingale? Is any Markov process of order one also a Martingale? For 1. I would say yes, ...
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Stochastic Methods for Image Deconvolution Problem

If we convolve an image with a point spread function and from the resulting image to find the input image, can we use any stochastic approaches? I feel like we will not be able to. A single image ...
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Auto-correlation of absolute squared stochastic process

Consider the stochastic process $a(t) \in \mathbb{C}$. Its autocorrelation function is given as $$ \phi_{aa}(\tau)=\left(a(t)\star a(t)\right)(\tau)=\int_{-\infty}^{\infty}a^*(t)\cdot a(t+\tau) \...
Torsten R's user avatar
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3 answers
114 views

Noise added to a Random Process

if we have a discrete random process \begin{equation} x\left(n\right)\:=\:0.2x\left(n-1\right)+w\left(n\right)+w\left(n-1\right) \end{equation} where $ w\left(n\right)$ is a noise with a mean $ m_w=0....
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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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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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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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wide sense stationary of dynamic process

I am trying to understand the definition of wide sense stationary on my own and probably have some silly questions. Wikipedia says, wide sense stationary is a process with constant mean and ...
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different between MVG and joint MVG?

Distribution for "joint multi-variate gaussian distribution" (joint MVG): $$f_{X}(x) = \frac{1}{(2\pi)^{n/2}\prod \limits_{i=0}^{n}\sigma_i} ~~\text{exp}\bigg[-\frac{1}{2} \sum \limits_{i=1}^...
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Showing that the sum of zero-mean noise is zero. Then computing the convolution of zero-mean noise with a given function

This is likely to be a quick fix for people with experience in stochastic processes. Let $ \eta[k] $ be a sequence of Uniform noise, $ \eta \sim U([-M,M]) $. I want to test if the following is correct ...
Dorian's user avatar
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Steady state variance of a stochastic differential equation - relation between the frequency and time domains

Consider a stochastic differential equation: $$ dx(t) = a x(t)dt + b y(t)dt \quad (1) $$ where $y(t)$ is a stochastic process satisfying $\langle y(t)y(t')\rangle = \delta(t-t')$. We will assume that ...
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Stationarity, discrete-translation operator, and the power spectral density matrix

Let $\mathbf{T}$ be the translation operator/matrix in discrete-time domain which can be written as $\mathbf{T} = \mathbf{\Phi} \mathbf{P} \mathbf{\Phi}^*$ where $\mathbf{P} = \exp(-i Diag([w_0, w_1, \...
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Super basic questions on statistical process

Before starting: I am really a beginner in statistical process in time. I mainly do quantum information and while learning aspect of quantum noise I realized that I am actually too weak on basics of ...
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Processes: Orthogonal, Uncorrelated, Statistically Independent

How are they all related? You can define them as: Orthogonal Processes: $E[XY] = 0$ Uncorrelated Processes: $E[XY] = E[(X - \mu_x)(Y - \mu_y)] = 0$ Statistically Independent Processes: $E[XY] = E[X] \...
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Proving that this process is weakly-stationary [duplicate]

Let $X(t) = Acos(2\pi f_c t)$ be a random process where $A$ is a uniform random variable within $(-1,1)$. I'm trying to prove this is a weakly(i.e. wide sense) stationary process. I need to show two ...
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When deriving the power spectral density of stochastic processes, why does taking an expectation allow the $T\rightarrow\infty$ limit to be taken?

I am following the arguments presented in the paper AN-255 Power Spectra Estimation, from Texas Instruments, to learn how to derive the power spectral density for a stationary stochastic process, and ...
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Why the requirement of the GCD of the lengths of all circuits in the graph being one?

I am reading A Mathematical Theory of Communication. The second requirement of an ergodic process confuses me (emphasis mine): All the examples of artificial languages given above are ergodic. This ...
nalzok's user avatar
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How do I find variance from the PSD of a stochastic process?

I have a time series that consists of noise and a signal, shown here windowed and Wiener filtered: and the PSD of just the noise (used in filtering): I want to find the variance of the noise using ...
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Channel Impulse Response is zero mean Gaussian random variable?

In the Paper "Key Generation From Wireless Channels" the channel estimation is given as: $\tilde{h}_{1,A} = \sigma_1^2 + \frac{\sigma^2}{||S_B||^2}$, $\tilde{h}_{1,A} = \sigma_1^2 + \frac{\sigma^2}{||...
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question related to something in karlin and taylor stochastic processes one text

This question is essentially a question about something in Karlin and Taylor's Stochastic Processes One text in the spectral chapter. Since this is a DSP list, Karlin and Taylor may not be so popular ...
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Autocorrelation for Stationary Signals

I'm having trouble grasping the autocorrelation function for stationary signals, both strict stationary and WSS. First for strict sense, we have $$\forall(\tau,t_1, \ldots, t_n) \in \mathbb{R} \land ...
Colin Hicks's user avatar
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Test for Equivalence of two time series

I wish to test whether two time series are equal. So I believe best way to define equivalence is that given two time series, say $\{x1_t\}$ and $\{x2_t\}$, we show that both the series come from the ...
Dayne's user avatar
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2 answers
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Intuition about independent signals

Given is this Wiener filter: From this we take \begin{equation} x[k]-a x[k-1]=v[k] \end{equation} $v(k)$ is assumed to be a white gaussian noise. In the textbook it is then stated that The ...
Mr.Sh4nnon's user avatar
3 votes
1 answer
340 views

Second moment ergodicity of gaussian random process

How can I prove that a WSS Gaussian stochastic process with mean 0 is mean-square ergodic in the second moment if and only if: $$\lim_{n \to \infty} \frac{1}{n}\sum_{k=0}^n r_{xx}^2(k) = 0$$ When $...
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The mean value of phase noise as a stochastic process

What is the mean value of phase noise as a stochastic process? Where can I get a theoretical analysis of this topic? PS: PLL produces cos(2*πfct+φ(t)). The phase noise refers to φ(t). The mean value ...
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ADC and Matched Filtering

A basic theorem in communications is the matched filter maximizes the SNR at sampling. I'm a little confused on how this relates to discrete time systems and sampling rate. Normally if you sample at ...
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