Questions tagged [stochastic]

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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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2 votes
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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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2 votes
1 answer
61 views

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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1 vote
1 answer
82 views

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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5 votes
1 answer
77 views

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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1 vote
1 answer
522 views

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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1 answer
209 views

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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0 answers
26 views

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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38 views

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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1 answer
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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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2 votes
3 answers
191 views

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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0 answers
32 views

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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6 votes
1 answer
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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) \...
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1 vote
3 answers
110 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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4 votes
2 answers
79 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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0 answers
52 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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2 votes
0 answers
29 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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-1 votes
2 answers
79 views

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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1 vote
1 answer
29 views

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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1 answer
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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 ...
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1 vote
1 answer
93 views

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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1 answer
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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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2 votes
1 answer
247 views

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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3 votes
1 answer
390 views

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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0 votes
0 answers
29 views

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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2 answers
257 views

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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1 vote
1 answer
45 views

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 ...
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1 vote
1 answer
118 views

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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35 views

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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0 votes
1 answer
33 views

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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1 vote
1 answer
215 views

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 ...
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0 votes
1 answer
57 views

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 ...
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4 votes
2 answers
71 views

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 ...
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2 votes
1 answer
278 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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29 views

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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1 vote
1 answer
383 views

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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1 vote
2 answers
198 views

Physical meaning of average values of random signals

This question might be a bit stupid, anyway, i'll risk it, since i want to get better understanding of this subject. Let's consider random signal x(t), and let's say that we know that it is ergodic ...
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3 votes
1 answer
493 views

Higher-order moment of output of LTI system

Assume a very simple LTI system. Assume $x$ is white Gaussian i.i.d. with variance $\sigma^2$. The output variance is straightforward to obtain. For example, for a continuous-time system: $$\mbox{...
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0 votes
1 answer
85 views

When a stochastic process would be a beneficial model in terms of noise

Let's say we have an image/signal with some noise in it. When would it be beneficial to model the signal as an outcome of a stochastic process? More specifically: How significant would noise have to ...
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1 vote
1 answer
206 views

Relationship between the autocorrelations of X(t) and X(nt)

Defining: $X(t)$ WSS random process with autocorrelation function $R_{X}(\tau) = \mathbb{E}[X(t)X(t+\tau)]$. $Y[n] = X(nT)$ (sampling of $X$ at a rate $\frac1T$) with autocorrelation function $R_Y(\...
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0 votes
1 answer
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Are there any signals with brickwall autocorrelation?

Are there any signals whose autocorrelation $R(\tau)$ has the following form? Assuming $\tau_c > 0$ and $R_0 > 0$ a constant, $$R(\tau) = \begin{cases}R_0, \text{ for $|\tau| < \tau_c$} \\ 0,...
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0 votes
1 answer
3k views

Autocorrelation: numpy versus FFT

I have a series a of values (0 and 1) coming from a Brownian process with drift for which I am studying the autocorrelation. I used two methods: 1) numpy autocorrelation: ...
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2 votes
2 answers
367 views

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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1 vote
2 answers
321 views

expected value of two LTI output signals multiplied (cross correlation)

I have an input signal x (assumed to be iid Gaussian with $\mu=0$, $\sigma^2$) which is fed into two linear systems: $y_1 = h_1 * x$ $y_2 = h_2 * x$ Now I would like to calculate $\mathbb{E}[y_1 y_2]...
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3 votes
0 answers
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Is it safe to call this WSSUS channel a Gaussian process?

BACKGROUND: Equation (3.6) of Wireless Communications by Goldsmith gives the baseband impulse response of a time-varying channel as: $$ c(\tau,t) = \sum_{n=0}^{N(t)}\alpha_n(t)e^{-j\phi_n(t)}\delta(\...
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2 votes
1 answer
1k views

What is an "innovation filter"?

I'm a math postgrad student working through a paper on eigenvalue decompositions of matrices of FIR filters (used for stuff like total decorrelation, convolutive mixing, MIMO). Towards the beginning, ...
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