Questions tagged [random-process]

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39 views

Conceptual Questions on Colored Noise Process

I am having a tough time finding answers to some specific questions and finding references where there is information regarding Brownian noise or Red Noise. I'm referring to white and colored noises ...
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3answers
304 views

Is white noise WSS by nature or not?

I want to know what is the difference between white noise and WSS white noise. is there any difference between them or they're equal? and what about white Gaussian Noise?
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1answer
52 views

Wide sense stationary that is not strict sense stationary? [duplicate]

A "wide-sense stationary process" (WSS) means that its mean is a constant, and its auto-correlation is time-invariant, that is: $$\begin{aligned}E[x(t)] &= c \\ R_X(t_1, t_2) &= R_X(...
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2answers
55 views

WSS vs SSS vs ergodic

Is this the correct "venn diagram" that related WSS, SSS, and Ergodic process types? $$\text{all process types}\begin{cases}\text{WSS} \begin{cases}SSS \begin{cases}\text{ergodic} \\ \text{...
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2answers
58 views

Practical implementation of Expected Value?

If i compute the average power of my input signal random variable $X(t)$ as $$R = E[X^2(t)]$$ i.e. as the expected value of a random process, is this really just an estimate of average power? More ...
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1answer
55 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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1answer
33 views

Complex gaussian random variable [closed]

In my work i need to generate circularly symmetric complex gaussian random variables with non zero mean and certain variance in matlab. I know the command for generating in case of zero mean , but ...
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2answers
82 views

Autocorrelation function and correlation integral

I am confused by the definition of autocorrelation function. It is originally defined as the expected value $$R_{XX}(\tau) = E[(X(t)X(t+\tau)] = \langle X(t)X(t+\tau)\rangle\tag{1}$$ where $\langle\...
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0answers
28 views

How to measure Noise

If I have a model that determines the current flowing through a membrane - how can I find out how big the random fluctuations in the current are? so how big is the noise level?
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1answer
93 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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2answers
35 views

Filter Percentage Uniform Noise from a DC-signal?

I'm not good with signal processing but i've looked around and have got no clue how to approach this. My Question is, If there is a Static value present - that is being corrupted by percentage ...
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0answers
23 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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0answers
37 views

Understanding white Gaussian noise variance

Referring to the bellow figure, assume the WGN has a constant PSD equal to N0. When filtered at $B_1 = 100 Hz$, the variance of the time domain noise is $σ_1^2$, and when filtered at $B_2 = 5 Hz$, the ...
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2answers
59 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
143 views

Blind Estimation of Signal Parameter and Noise Variance

Let $y[n]= h*x[n] + w[n]$, where $h$ is an unknown but deterministic parameter, $x[n]$ is a BPSK random variable with equal probability of +1 and -1, $w[n]$ are i.i.d. Gaussian with zero mean and ...
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1answer
70 views

Power spectrum of uniform white noise

Given a white noise image $W_{i,j} \sim U[a,b]$ where each pixel is distributed uniformly in $[a,b]$, how would I go about computing its power spectral density? That is, I want to find $E[|\hat{W}_{i,...
1
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1answer
21 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
48 views

Cyclostationary signal intuition

Images show a discussion I picked up from a PhD thesis about a cyclostationary process and need help interpreting it. "In the time domain the upsampling process creates a signal whose distribution of ...
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1answer
46 views

How to create a wide-sense stationary time series with a frequency of 40 Hz?

I want to create a time series in MATLAB which has a peak frequency of 40 Hz but is also a wide-sense stationary random process. I then want to use power spectral density estimation to recover the ...
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1answer
26 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
30 views

Example of Entropy and Channel Capacity Computation

Can you help me on verifying if this computation of entropy is correct and on understanding its meaning? I am not sure of the result especially because it is equal to 0: it means that we cannot ...
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0answers
44 views

Randomly Generate Synthetic Noise in an Image Text Document

I'm working on denoising dirty image document. I want to create a dataset wherein synthetic noise will be added to simulate real-world, messy artifacts. Simulated dirt may include coffee stains, faded ...
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1answer
68 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
246 views

Integral over power spectral density

The wikipedia entry on PSD has one confusing line: Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process) But ...
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0answers
18 views

Local noise intensity in an image

Noise can be assessed in uniform regions of an image, by subtracting a lowpass-filtered version of it. Then from the histogram of intensities, a global measure can be obtained (such as the average ...
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3answers
3k views

How can a signal be both periodic and random?

Do any examples of such signals exist where the signal is both periodic and random? Because as I see it, if a signal is periodic then the randomness kinda goes away because you know what the signal ...
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0answers
22 views

How to compute the energy of a NON-STATIONARY (transient) random discrete-time signal

When computing the energy of a NON-STATIONARY (transient) random discrete-time signal $x(n)$, does it make more sense to compute the energy as $ E=\sum_1^N{x^2(n)}$ over all the $N$ samples or does ...
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0answers
6 views

Clutter Model in MOT: joint distribution of a random matrix and its column size variable

Suppose $C_k$ is a random matrix contains columns of measurement vectors that are random variables: $$C_k=[c_k^1,...,c_k^{m_k^c}]$$ $m^c_k$ is the number of columns as well as a random variable. All ...
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1answer
538 views

Autocorrelation of a uniform random process

i am currently learning the basics of signal processing. As you may know the definition of the autocorrelation is different if you look at a random process or for example a deterministic signal My ...
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1answer
81 views

Questions about the stability (and stationarity) of a system and state space representations

i'm pretty new to the topic and I'm trying to understand how to determine the stability of a process. I'm giving this discrete-time stochastic system: $$ \cases{ s_t = 2s_{t-2} + 3w_{t-2} \\ y_t = ...
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2answers
111 views

Cross-correlation of filtered random processes

I have a wide-sense-stationary (WSS) process $\{x(t)\}$ and two linear filters with impulse functions $h_1$ and $h_2$. Let $\delta(\omega)$ be the power spectrum of $\{x(t)\}$ and $$H_1:\omega\...
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1answer
468 views

PSD from autocorrelation in MATLAB

I am trying to simulate a simple stochastic process defined by the equation: \begin{equation} \frac{1}{v}\frac{db}{dt} +\Gamma_0 b= \sqrt{\sigma}R(t), \end{equation} where $R(t)$ is a zero-mean white ...
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2answers
115 views

Why is there only one integration in the solution if there is two integral in the formula?

In this problem the random variable is theta and according to the formula there should be two integrations but in the solution there is only one . Nor am i able to understand the meaning of x1 and x2 ...
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0answers
61 views

Proof of weak stationary random process autocovariance always goes to zero?

Professor told me that if a random process is weak stationary, and it does not feature any periodic component, then its autocovariance always goes to zero. I can intuitively understand it, however, ...
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1answer
58 views

Question regarding AC power of ergodic process

We know Ergodic process is the subset of Weakly stationary process which permits us to substitute time average for ensemble Average My teacher said If $X(t)$ is Ergodic random process then following ...
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1answer
144 views

Correlation of independent random processes

Suppose $X(t)$ and $Y(t)$ be two independent random processes. Is $E(X(t_1)Y(t_2))$ necessarily zero?
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1answer
325 views

Power contained in a random process $X(t)$

How do we calculate the AC and DC power of random process $X(t)$ , provided we have $R_x (\tau)$, and $S_x(f)$ ?
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0answers
28 views

Is the expectation of a random process $X(t)$ with zero DC component necessarily zero?

Is the expectation of a random process $X(t)$ with zero DC component necessarily zero? Or can it be non-zero depending upon the process?
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1answer
44 views

Variance of function of random variable

Is their an easier way to find variance of function of random variable? Till now what I am doing is first find probability density function of (function of random variable) then integrate over range.
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2answers
51 views

Independence of Functions of random Variable

Consider I am given two functions of one random variable each for example x=cos(at),y=rect(bt) where a and b are random variables.And I am given Probability density function for a and b then if I am ...
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4answers
105 views

What is definition of independent random variable

I wan't to ask that if E{X}=0 E{Y}=0 and E{XY}=0 then how can I verify if the two random variables are independent or not. X , Y are both continuous random variables {I am not able to recall ...
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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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0answers
18 views

Early estimate the sign of the drift in a generalized Wiener process

I posting here my problem, perhaps somebody can point me how to proceed further :) [The challenge] I have an electronic system that can be modeled as a Wiener process with a drift $\mu$: $ X_t = \...
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1answer
101 views

Power Spectrum Estimation of three sinusoids in white noise

Let's assume we have a random process consisting of three sinusoids in white noise: $$x[n] = 3 \cdot \sin(ω_1 \cdot n + ϕ_1) + 5 \cdot \cos(ω_2 \cdot n + ϕ_2) + 2 \cdot \sin(ω_3 \cdot n + ϕ_3) + v[n]$$...
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1answer
88 views

An Interesting Model with Unknown Orthogonal Design Matrix

Suppose the multivariate one-way anova model for the raw data , i.e. $$ \label{Example_model_1} \mathbf{y}_{ij}=\mathbf{\mu}+\mathbf{z}_i+\mathbf{e}_{ij}, ~~ i=1,\ldots,m,~~j=1,\ldots,n_i,~~~~~~~~~~...
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0answers
35 views

Relation between power spectral density and mean absolute value

The root mean square $$\sigma_{x} = \sqrt{\frac{1}{T}\int_0^T x^2(t) \, \mathrm{d}t}$$ of a finite zero-mean random signal $x(t)$ in the range $0 < t < T$ is related to the signal's power ...
3
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2answers
139 views

How to find a variance of sample sequence

I have a sequence such as $$r[n] = y[n]v[n]$$ $y[n]$ and $v[n]$ are zero-mean and statistically independent. I need to find a variance of $r[n]$ and show that it is white and equal to $\sigma ^2_y\...
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1answer
46 views

Particular Correlation formula

I'm reading a book where the autocorrelation of white noise is expressed as: What is the term $Q(k)$ and why is is expressed as an average value of a dot product ?
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1answer
78 views

Simulate time series given temporal auto-correlation functions

Given a random process $x[n] \in \mathbb{R}$ (say of length $N$) and all correlation functions such as: \begin{align} \langle x[i]\rangle\\ \langle x[i]x[j]\rangle\\ \langle x[i]x[j]x[k]\rangle\\ \...
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1answer
74 views

response of LTI system to a Random Input Signal

what is LTI filter? what is the output when x(t) is input? let x(t) be the input signal to the system and y(t) denote the output signal. The output of the system may be expressed in terms of ...