Questions tagged [random-process]

A random process (in time), also called "stochastic process" is a signal that, when sampled at any given time, is a random variable.

Filter by
Sorted by
Tagged with
30 votes
3 answers
86k views

Variance of White Gaussian Noise

It could seem an easy question and without any doubts it is but I'm trying to calculate the variance of white Gaussian noise without any result. The power spectral density (PSD) of additive white ...
Mazzy's user avatar
  • 545
23 votes
3 answers
27k views

What is a good example of an ergodic process?

I'm trying to find simple examples of an ergodic process. What process comes to your mind as a good illustration of its properties? A quick research (Wikipedia, another answer) mainly gives examples ...
bluenote10's user avatar
13 votes
2 answers
17k views

What are the statistics of the discrete Fourier transform of white Gaussian noise?

Consider a white Gaussian noise signal $ x \left( t \right) $. If we sample this signal and compute the discrete Fourier transform, what are the statistics of the resulting Fourier amplitudes?
DanielSank's user avatar
  • 1,026
9 votes
4 answers
10k views

Gaussian White Noise - Relation Between Distribution and Correlation

Im a beginner in signal processing so my question may be obvious. A white noise has the property to have its autocorrelation function that is equal to $$\mathbb{E}[f(t+\tau)f(t)]=\sigma^2 \delta(\...
StarBucK's user avatar
  • 279
9 votes
3 answers
4k 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 $N_0/2$...
Phobos's user avatar
  • 425
7 votes
3 answers
3k views

Random signals as power signals

Why are random signals considered as power signals (i.e. signals with infinite energy and finite average power)? Does this make any sense? What does it mean for random signals to have infinite energy ...
Likely's user avatar
  • 153
6 votes
3 answers
5k 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 ...
dydx's user avatar
  • 71
6 votes
2 answers
1k views

Why is $A\cos(2\pi f_ct)$ a non-stationary process?

I am studying analog communication and having Communication system - Simon Hykin as one of the reference. There is a question Consider the sinusoidal process$$X(t) = A\cos(2\pi f_ct)$$where the ...
TIWARI's user avatar
  • 163
6 votes
1 answer
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 ...
Srishti M's user avatar
  • 616
5 votes
1 answer
4k views

Understanding of Random Process, Random Variable and Probability Density Function

I just wanted to confirm my understanding of a Random Process, Random Variable and the its Probability density Function. Here is the way that I looked a Random Process/Random Variable: If we ...
sundar's user avatar
  • 343
4 votes
2 answers
10k views

What is the practical meaning of the variance, covariance, mean value?

What is the practical meaning of variance, covariance and mean for a signal?
ylcn's user avatar
  • 109
4 votes
4 answers
1k 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?
m-sh-shokouhi's user avatar
4 votes
4 answers
543 views

What Is Continuous White Noise in The Context of Signal Processing and Broadly

How can one define Continuous White Noise in a coherent way? Is there a way to derive it Mathematically? Specifically, is there a way to define it which will works as the model in Signal Processing ...
Royi's user avatar
  • 19.5k
4 votes
3 answers
249 views

Under what conditions is there a one-to-one mapping between continuous-time and discrete-time signals?

As the sampling theorem dictates that the uniform sampling frequency must be at least twice the maximum frequency present in the bandlimited signal (Nyquist rate), a question arises about the ...
HYMD's user avatar
  • 77
4 votes
2 answers
3k 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. ...
Amin's user avatar
  • 201
4 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] \...
user674907's user avatar
4 votes
2 answers
78 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 ...
Mr.Sh4nnon's user avatar
4 votes
2 answers
4k views

Deterministic / Non-deterministic Stochastic Process

Problem 6.1-6 of Probability, Random Variables, and Random Signal Principles, 4th Edition by Peebles asks If a process is defined by $X(t) = A$, where $A$ is a continuous random variable uniformly ...
spectre's user avatar
  • 555
4 votes
3 answers
958 views

The normalization of the autocorrelation function and how it changes the definitions you've learned about signal analysis in communication systems

Since this question is book-oriented, I will kindly ask you to accompany it with a book that is considered by many researchers the bible of Digital Communication: Proakis - Digital Communications, one ...
Rubem Pacelli's user avatar
4 votes
2 answers
108 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 ...
user avatar
4 votes
2 answers
1k 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 ...
Probably's user avatar
  • 107
4 votes
1 answer
229 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 ...
Sm1's user avatar
  • 291
4 votes
1 answer
138 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 ...
Robert L.'s user avatar
  • 2,212
4 votes
1 answer
1k 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(...
Numbermind's user avatar
4 votes
1 answer
2k views

Relation between frequency spectrum and PDF of a random variable

I have a random variable that is being generated according to some probability distribution function (e.g. a Gaussian PDF). When looking at the frequency spectrum of the generated data does the ...
Television's user avatar
3 votes
3 answers
8k views

Capacity of cascade binary symmetric channels

Let's imagine that we have interconnected in cascade $L$ binary symmetric channels each with the same transition probability $p(y|x) \in \{p, q=1-p\}$, where the output of each BSC is connected to the ...
CristoJV's user avatar
  • 131
3 votes
2 answers
3k views

Why is $\sin(t)$ a stationary process?

I am trying to understand the meaning of the term Stationary Process. For example, I was told that $\sin(t)$ is a stationary process. Could someone try to explain, in simple words, why is $\sin(t)$ (...
user135172's user avatar
3 votes
2 answers
322 views

Hard time figuring out whether the following random process is wide sense stationary

I'm dealing with a random process that's simply a square wave with pulse period T, where: Each pulse takes either $A$ or $-A$ depending on a coin toss. The wave is shifted by a random $t_d$ where $...
Essam's user avatar
  • 267
3 votes
2 answers
168 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\...
ViniLL's user avatar
  • 33
3 votes
1 answer
9k views

variance in the time domain versus variance in frequency domain

Hi All: I'm trying to better understand the connection between variance of a time series and the integral of the spectral density over all frequencies. Rather than going through all of the relations, ...
mark leeds's user avatar
  • 1,107
3 votes
1 answer
727 views

explanation of correlation of stationary stochastic processes

I have some doubts about correlation in stationary stochastic processes. I know that the expectation of a random variable is $$E(x)=\int_{-\infty}^{+\infty} a f_x(...
Andrea's user avatar
  • 539
3 votes
1 answer
124 views

Understanding homework solution - why are $\{X_t\}$ and $\{Y_t\}$ joint WSS, and finding Wiener filter + error

I'm walking through the published solutions of my homework and I'm struggling interpreting them. In them I was given a random Gaussian process $\{X_t\}$ and a random variable $$\{Y_t\} = X_t\cos(2\pi ...
Piratemetaldrinkingcrew's user avatar
3 votes
2 answers
5k views

Definition of average power?

There are two kind of average power I encountered in random signal class and textbook: definition 1: average power =$$E[|x(t)|^2]=R_{xx}(0)=\int^\infty_{-\infty} S_{xx}(f)\,df$$ definition 2: average ...
Rikeijin's user avatar
  • 189
3 votes
2 answers
78 views

What is the distribution of it?

If $\theta$ is uniformly distributed in $(0, 2\pi),$ then what is the distribution of $e^{i\theta},$ where $i = \sqrt{-1}?$ And what are the statistical properties of $\left[e^{i0\theta}\, e^{i1\theta}...
D Satya Ganesh's user avatar
3 votes
3 answers
911 views

Signal variance and power connection

For a random signal $x(n)$, why is $E(x(n)^2)$ called signal power? Is it really power? Any proof?
DSPinfinity's user avatar
3 votes
1 answer
370 views

Variance of Integral of a real white Gaussian Noise Process

In this question, is the answer not equal to infinity ? Answer is mentioned as 6. But my doubt is cant we think of it like a linear combination of many independent random variables each having ...
Sreejith's user avatar
3 votes
1 answer
695 views

moving average rounding error analysis

I have implemented a moving average, similar to the Hogenauer Filter, with a reduced number of computation operations. I expect the expected error to behave as the random walk and its STD to be of ...
Gideon Genadi Kogan's user avatar
3 votes
1 answer
2k 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) + ...
sugab's user avatar
  • 197
3 votes
1 answer
223 views

Does a collection of Gaussian random variables necessarily constitute a Gaussian Process?

If $\{X(t)\}$ is a Gaussian Process then the random variables $X(t_k)$ where $k = 1,2,3...n$, are jointly Gaussian. If each random variable $X(t)$ is a Gaussian variable, then will the random ...
user1825567's user avatar
3 votes
2 answers
1k views

Autocorrelation function $R_{yy}(t_1,t_2)$?

If $x(t)$ is a zero mean stationary Gaussian process and if $y(t)=x^2(t)$,then $\{y(t)\}$ is called a square law detector process. Now i want to find autocorrelation function $R_{yy}(t_1,t_2)$,that is ...
Rohit's user avatar
  • 578
3 votes
1 answer
8k views

Random process $X(t)$ with autocorrelation function given find the mean and the variance

Autocorrelation function is $$R_{xx}(\tau)=\frac{20}{1+2\tau^2}$$ So at $\tau=0$$$R_{xx}(0)=20=E[X(t)X(t)]=E[X^2(t)]$$ The variance is $$\mathrm{Var}[X(t)]=E[X^2(t)]-E^2[X(t)]=20-E^2[X(t)]$$ As $X(t)$...
Don's user avatar
  • 133
3 votes
2 answers
274 views

Any Relationship Between the Entropy of an Image and Its Spectrum?

Is there a relationship between the Shannon entropy of image and the output of the 2D Fourier transform (DFT) of the image?
user64854's user avatar
3 votes
1 answer
264 views

Generating violet noise with a specific PSD coefficient

I am trying to generate a time-domain violet noise signal with the following power spectral density (PSD): $$ S_n(f) = A^2f^2 $$ Unfortunately, I am having trouble finding the right amplitude ...
diemilio's user avatar
3 votes
1 answer
2k 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 ...
OscarNieves's user avatar
3 votes
1 answer
1k views

Average Power Spectral Density of PAM signals

I am reading through the PAM transmission scheme and about the power spectral density of the signals. Given that the Average Power Spectral Density of PAM Signals is: $$ \Phi_{ss}(f)=\Phi_{aa}\left(e^...
sundar's user avatar
  • 343
3 votes
1 answer
135 views

What is the jitter effect on the spectrum of impulse train?

How to derive the sepctrum of $$x(t)=\sum_{n=-\infty}^{\infty}\delta\left(t-nT-\tau_n\right)$$ where $\tau_n\sim N\left(0, \sigma^2\right)$ I assume that the randomness effect should behave as a low ...
Gideon Genadi Kogan's user avatar
3 votes
1 answer
152 views

Dimensional analysis of integrated white noise process

This question is somewhat related to this post. Let us consider we have a white noise current source $i_n(t)$, with a variance $\sigma_i^2$, and mean, $\mu_n=0$. Assume that this current is passed ...
Saqib Shah's user avatar
3 votes
1 answer
123 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 ...
Anmol Gupta's user avatar
3 votes
2 answers
306 views

Power Spectrum Density and Frequency

if i have some random signals (sampling rate = 10Hz, 0.1s per data) Using python library i transformed it to power spectral density power spectral density forms = f, psd (using mlab.psd) I'm really ...
이병철's user avatar
3 votes
0 answers
150 views

Energy Detection in Presence of Colored Gaussian Noise

Before asking my question, let me introduce the context: For spectrum sensing based on energy detection, which has been widely studied in presence of AWGN, the optimal detection threshold is computed ...
Luis M Gato's user avatar

1
2 3 4 5