Questions tagged [stochastic]

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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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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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31 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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45 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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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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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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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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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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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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45 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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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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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
61 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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69 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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233 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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191 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
119 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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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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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
200 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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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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258 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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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
407 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
127 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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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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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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2answers
272 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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218 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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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
889 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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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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628 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 = \...
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1answer
784 views

Mean Square Continuity of Random Process

Show that a stochastic process $X(t)$ is mean square continuous if and only if its autocorrelation function $R_X(t_1,t_2)$ is continous $\Rightarrow$ Proof: We have $E[(X(t)-X(t_0))^2]=R_X(t,t)-R_X(...
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360 views

Applications of Power Spectral Density [closed]

I have a class covering Power Spectral Density but I have no idea why it matters. Could someone provide some examples of its use? Thanks
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628 views

Why look at power spectral density for stochastic processes?

I have been told that for deterministic signals, it makes sense to look at their respective Fourier transforms/spectra. For stochastic processes on the other hand, I am supposed to work with power ...
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1answer
160 views

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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1answer
243 views

Doubt about wide sense stationary random process

I have white Gaussian noise $F[n]$ with zero mean and autocorrelation $R_F[n_1,n_2]=\delta[n_1-n_2]$. If now I consider the random process defined as $$X[n]=u[n]e^{-kn}F[n]$$ Is $X[n]$ a wide-ense ...
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105 views

Is the output of function of two ergodic processes ergodic?

Let $\{\xi_k\}_{k\in \mathbf{Z}}$ and $\{\epsilon_k\}_{k\in \mathbf{Z}}$ be two independent zero-mean Gaussian processes (i.i.d.). Is the output of the function $f$ such that $y = f(\dots,\xi_{k-1},\...
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2answers
768 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 $\Phi(\omega) =\mathcal{F}\{R_{xx}\}$ ...