Questions tagged [random-process]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
1answer
18 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 ...
1
vote
0answers
19 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 ...
1
vote
1answer
53 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 ...
1
vote
1answer
20 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 ...
1
vote
0answers
10 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 ...
6
votes
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 ...
0
votes
0answers
19 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 ...
0
votes
0answers
5 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 ...
2
votes
1answer
178 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 ...
1
vote
1answer
46 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 = ...
1
vote
2answers
62 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\...
3
votes
1answer
144 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 ...
0
votes
2answers
72 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 ...
0
votes
0answers
52 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, ...
1
vote
1answer
32 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 ...
2
votes
1answer
112 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?
0
votes
1answer
139 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 $S_x(f)$ ?
0
votes
0answers
26 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?
0
votes
0answers
33 views

Low pass representation of Bandpass Random process

Can A WSS random process with non-zero mean be also represented in such form.
0
votes
0answers
59 views

How to Derive Rayleigh distribution using transformation formula

Consider a complex random variable $Z=X+\jmath Y$, where the probability density function of $X$ and $Y$ are given by $$p(x) = \frac{1}{\sqrt{2\pi\sigma^2}} {\rm e}^{-\frac{x^2}{2\sigma^2}}\quad\mbox{...
1
vote
1answer
32 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.
0
votes
2answers
46 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 ...
0
votes
4answers
80 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 ...
4
votes
2answers
60 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 ...
0
votes
0answers
17 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 = \...
1
vote
1answer
71 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: $$x[n] = 3 \cdot \sin(ω_1 \cdot n + ϕ_1) + 5 \cdot \cos(ω_2 \cdot n + ϕ_2) + 2 \cdot \sin(ω_3 \cdot n + ϕ_3) + v[n]$$...
1
vote
0answers
30 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 ...
0
votes
0answers
57 views

Analytical spectral density of a On/Off modulation defined by a Bernoulli process

Consider a narrow band signal (laser) that I can modulate digitally with a on/off switch controlled by a digital pseudo random number generator. The resulting signal features a linewidth broadened by ...
0
votes
0answers
27 views

Images as Markov chains

I have seen literature on representing black and white images as probability distributions and then computing "distances" between them, for example, in optimal transport. I was wondering if there is ...
0
votes
0answers
74 views

On the spectral representation of deterministic and random signals

I went back to many references in order to fix some of the confusions that I have on many concepts in signal spectral representation. I concluded that: 1) Deterministic signals may be represented ...
3
votes
2answers
135 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\...
0
votes
0answers
16 views

Rate distortion function for a Gaussian process with a squared exponential kernel

This is probably a question whose answer should be available in some paper or textbook, but my searching for it hasn't helped me find a result that I could use. The question is basically just what ...
0
votes
1answer
37 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 ?
2
votes
1answer
76 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\\ \...
0
votes
1answer
53 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 ...
2
votes
2answers
156 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 ...
0
votes
1answer
58 views

Cramér-Rao lower bound

I have been trying to implement the Cramér-Rao lower bound from the paper - A reference-free time difference of arrival source localization using a passive sensor array (eq. 6 and eq. 7). $$ \...
2
votes
2answers
502 views

PSD of complex white gaussian noise

It may be a really simple question, but I'm not sure about this one: Given a complex white Gaussian noise process with iid real and imaginary parts and a double sided power spectral density of $N_0/2$...
0
votes
2answers
70 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]...
1
vote
2answers
60 views

Random Process at a particular time instance

I was studying Random Process and I thought I understood what it was all about until I came across this example. Consider a random experiment of tossing a coin with sample space S = {H, T} The sample ...
0
votes
1answer
74 views

Converting a non-stationary random process into a WSS process by adding a random phase

Here is an example where this method has been implemented. We were trying to calculate the spectrum of a transmitted signal(Random signal/weighted pulse) The auto correlation function of the pulse ...
1
vote
2answers
53 views

Fourier-Analysis of Stationary Random Signals

Let's say we have discrete-time stationary random signals with Gaussian PDF of mean value 0 and variance 1, whose individual signal values are uncorrelated. For such a signal, how can we determine ...
4
votes
1answer
88 views

Physical interpretation of 4th-order correlations

BACKGROUND: Let's say we have samples of a random process $X(t)$ at two different times, $t_1$ and $t_2$, denoted $X(t_1), X(t_2)$. The values of $X(t)$ represent some voltage-like quantity (i.e. a ...
1
vote
1answer
245 views

Random Signals - statistical properties are time dependant?

I'm taking a course on DSP and we're being introduced to the random signals, in particular continuous time and discrete time random signals. We're told that if we repeat a single random experiment at ...
1
vote
1answer
135 views

Are two jointly stationary white noise processes independent?

I am currently dealing with a problem concerning beamforming, where two "jointly stationary zero-mean white noise processes" form the input of an adaptive system. One of those processes resembles the ...
0
votes
1answer
213 views

Covariance matrix associated with random DC level in Gaussian noise

Given a signal $x[n] = A + w[n]$ where $A$ is a Gaussian random variable and $w[n]$ is Gaussian white noise, then the covariance matrix of the signal is given by $[C(\sigma^2_A)]_{ij}=E[x[i-1]x[j-...
1
vote
1answer
50 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)=...
0
votes
2answers
241 views

Understanding PSD: Why Does Power at High Frequencies Affect Low Frequencies?

I'm trying to wrap my head around power spectral density on a conceptual level, but I am having some difficulty. Suppose I have a communication system where I am receiving and sampling white Gaussian ...
0
votes
1answer
432 views

Autocorrelation of Addition of Two Independent Signals

Given a random signal $ Z \left( t \right) $ which is addition of two independent signals $ X \left( t \right) $ and $ Y \left( t \right) $ with constant parameters $ a $ and $ b $: $$ Z (t) = aX(t) +...
1
vote
2answers
90 views

Applying the CUSUM algorithm to a correlated random process

As far as I know, the CUSUM algorithm is meant for detecting change points on discrete-time uncorrelated random processes. For instance, to apply the CUSUM algorithm to a discrete Gaussian process, ...