Questions tagged [stochastic]

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Approximating a Gaussian Process

Suppose that $\theta_t$ is an exogenous variable with known Gaussian process. Next, suppose that for any index $i\in [0,1]$, $$ a_{i,t} = (1-\beta)\mathbb E[\theta_t|\mathcal I_{i,t}]+\beta \mathbb E[...
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18 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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31 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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1answer
30 views

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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44 views

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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105 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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1answer
51 views

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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2answers
566 views

What does the frequency axis of a Power Spectral Density mean?

I have never really understood what the frequency axis meant when we plot the Power Spectral Density(PSD). Does it correspond to frequency as we get after we take the Fourier Transform of a time ...
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68 views

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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21 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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44 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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38 views

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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3answers
102 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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2answers
61 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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1answer
3k views

Understanding Ergodicity and Ensemble Averaging

Literature says that a stationary signal is ergodic, if its ensemble average = time averages. Should it be the statistics computed by time averaging = statistics computed by ensemble averaging?The way ...
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47 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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26 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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1answer
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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2answers
767 views

What is the difference between the PSD of a deterministic and stochastic signal?

I am learning about stochastic processes and I don't get one thing: What is the advantage of calculating the PSD of a signal using the Wiener-Khinchin theorem $\Phi(\omega) =\mathcal{F}\{R_{xx}\}$ ...
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1answer
67 views

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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1answer
60 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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1answer
232 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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2answers
2k views

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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1answer
189 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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24 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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107 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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1answer
31 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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1answer
59 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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25 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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31 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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1answer
118 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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257 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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1answer
44 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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1answer
1k views

Power Spectral Density of Brownian Motion despite non-stationary

Note: I originally asked this on Physics Stack Exchange but haven't attracted any interest there so I'm asking here where it may be more relevant. A white noise process, $\xi(t)$ with delta ...
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2answers
64 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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1answer
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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1answer
199 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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2answers
270 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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217 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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28 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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2answers
138 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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1answer
405 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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1answer
75 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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1answer
124 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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1answer
49 views

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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1answer
881 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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2answers
14k views

generating white gaussian noise in matlab using two different functions

I want to know the difference between the two Gaussian noises generated below? Which one is white and how can i make the other one white? y=wgn(1,10000,0) and <...
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32 views

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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1answer
78 views

Null autocorrelation function and stationary

I can show that a process $X(t)$ is Wide Sense stationary (WSS) by showing that $E[X(t)]$ is constant and that its autocorrelation function is in function of $\tau=t_1-t_2$, that is, $R_X(t+\tau,t)=...
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2answers
626 views

Ornstein Uhlenbeck with drift

The Ornstein-Uhlenbeck (OU) process $dX_t = -\frac{1}{\mu} X_t + \sqrt{\frac{2\sigma^2}{\mu}} dW_t $ generates coloured noise with autocorrelation function $R(t) = \langle X_t,X_{t'}\rangle = \...