Questions tagged [stochastic]
A "stochastic process" is another term for "random process".
109
questions
2
votes
1
answer
69
views
Relation and differences between correlation sequence and correlation matrix
Suddenly I am stuck at a question while studying Statistical signal processing:
What are the differences between correlation sequence and matrix (can be auto/cross), and how are the two related? When ...
- 33
1
vote
0
answers
31
views
Parameter value in bias instability modeling
In IMU error modeling, bias instability/flicker noise is often modeled by a first order Gauss-Markov process whose standard deviation is equal to the given BI coefficient [1, p.185][2, p.31]. Because ...
- 11
1
vote
1
answer
73
views
Minimum Number of Base Accelerations Needed to Simulate Gaussian White Noise in Structural Dynamics?
I am currently working on a scientific paper where I subject a structure to base accelerations modeled as Gaussian white noise. I am relatively new to signal processing and would appreciate some ...
- 11
0
votes
0
answers
22
views
Two meanings for "innovation" in Wiener filter are the same?
This is related question to A question about Wiener filter based on Linear Estimation by Kailath, based on the textbook Linear Estimation by Kailath. In that link I talk about how I first learned what ...
- 1
0
votes
0
answers
21
views
A question about Wiener filter based on Linear Estimation by Kailath
In my linear estimation class based on the textbook Linear Estimation by Kailath, we went through the process of finding LLSE of $\hat{x}(t+\lambda)$ for fixed $\lambda$ given $\{y(\tau)|-\infty<\...
- 1
1
vote
1
answer
54
views
Ergodicity clarification
On page 201 of https://stanford.edu/~dntse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter5.pdf,
it is mentioned that
This observation suggests that the capacity result (5.89) holds for a ...
- 125
2
votes
1
answer
61
views
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(...
- 63
1
vote
1
answer
74
views
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 ...
- 111
1
vote
1
answer
96
views
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 ...
- 111
3
votes
1
answer
399
views
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^{-...
- 95
1
vote
1
answer
58
views
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 ...
- 13
0
votes
0
answers
167
views
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.
...
- 173
3
votes
1
answer
135
views
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 ...
- 141
5
votes
4
answers
356
views
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, ...
- 53.8k
0
votes
0
answers
31
views
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 ...
4
votes
1
answer
104
views
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 ...
- 41
1
vote
1
answer
343
views
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 ...
- 123
2
votes
0
answers
53
views
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 ...
- 149
2
votes
0
answers
81
views
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 $T$ ...
2
votes
1
answer
106
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 ...
1
vote
1
answer
182
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 ...
- 113
-1
votes
1
answer
131
views
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.
0
votes
1
answer
139
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?
- 377
1
vote
1
answer
2k
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 = ...
- 11
-1
votes
1
answer
833
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 ...
- 99
0
votes
0
answers
98
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} \...
- 125
0
votes
0
answers
41
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 ...
- 1
0
votes
1
answer
80
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 ...
- 303
0
votes
0
answers
242
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 ...
- 101
2
votes
3
answers
537
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) ...
- 31
0
votes
0
answers
49
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, ...
- 1
5
votes
1
answer
79
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 ...
- 53
1
vote
0
answers
80
views
Auto-correlation of absolute squared stochastic process
Consider the stochastic process $a(t) \in \mathbb{C}$. Its autocorrelation function is given as
$$
\phi_{aa}(\tau)=\left(a(t)\star a(t)\right)(\tau)=\int_{-\infty}^{\infty}a^*(t)\cdot a(t+\tau) \...
- 11
1
vote
3
answers
114
views
Noise added to a Random Process
if we have a discrete random process
\begin{equation}
x\left(n\right)\:=\:0.2x\left(n-1\right)+w\left(n\right)+w\left(n-1\right)
\end{equation}
where $ w\left(n\right)$ is a noise with a mean
$
m_w=0....
- 303
4
votes
2
answers
112
views
Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)
Assume we have the following system (coming from control systems theory, hence in s-domain)
$ Y(s) = H_A (s) \cdot A(s) - H_B (s) \cdot B(s) $
I now wish to consider $a(t)$ and $b(t)$ as white noise ...
user53750
0
votes
0
answers
57
views
How to characterize the randomness of an event using it's PSD?
I have the power spectral density function of a stochastic phenomenon.
how can I generate a signal (time series) representing the randomness of this event over time?
How can I draw the probability ...
- 1,347
2
votes
0
answers
37
views
Signifance of statistical information in a signal
I am learning control engineering for some time and I work with a lot of transfer functions and frequency domain design. Reading from textbook, to me everything seems deterministic. Whenever I come ...
- 27
-1
votes
2
answers
106
views
wide sense stationary of dynamic process
I am trying to understand the definition of wide sense stationary on my own and probably have some silly questions. Wikipedia says, wide sense stationary is a process with constant mean and ...
- 27
1
vote
1
answer
37
views
different between MVG and joint MVG?
Distribution for "joint multi-variate gaussian distribution" (joint MVG):
$$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}^...
- 203
0
votes
1
answer
150
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 ...
- 1
1
vote
1
answer
242
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 ...
- 111
0
votes
1
answer
110
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, \...
- 201
2
votes
1
answer
267
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 ...
- 289
3
votes
1
answer
1k
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] \...
- 39
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 ...
- 1
1
vote
2
answers
638
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 ...
- 691
1
vote
1
answer
70
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
...
- 125
1
vote
1
answer
466
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 ...
- 177
0
votes
0
answers
41
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}{||...
- 39
0
votes
1
answer
40
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 ...
- 1,117