Questions tagged [stochastic]

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PDF of a Shifted Rectangular Pulse

I wanted to determine the PDF of a Stochastic Process. I am familiar with the concept of PDF for a Random Variable which maps the outcomes to its probabilities but I am not able to find the PDF of a ...
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1answer
790 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^...
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1answer
1k views

Autocorrelation and Power Spectral Density (Discrete)

The Autocorrelation, $\phi_{aa}[\kappa]$, of a discrete time random process, $a[k]$, is defined as: $$ \phi_{aa}[\kappa] = \mathrm{E}\left\{ a[k+\kappa]a^*[k] \right\} $$ Taking its fourier ...
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2answers
78 views

Stochastic approximation algorithm

The goal is to find the FIR filter coefficients $\mathbf{h} = [5;3]$ with the help of the adaptive FIR filter $\mathbf{w}$ of order $p = 2$. I have implemented the Stochastic approximation algorithm ...
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2answers
1k views

Understanding the definition of mean/autocorrelation

I was studying about the definitions of mean, expected value and autocorrelation. I wanted to verify my understanding the evaluation of mean, expected value and autocorrelation. At the same time to ...
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1answer
191 views

Understanding of Random Process/Random Variable

At a simpler level to my previous question, I wanted to confirm my understanding on Random Process based on Random Variables using an example. So, I took this example: If we consider a dice, which ...
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1answer
3k 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 ...
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3answers
306 views

What really means stochastic in field of signal processing

I met two definitions of word stochastic, the first one (cited from wikipedia Stochastic) The word stochastic is an adjective in English that describes something that was randomly determined The ...
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1answer
227 views

LTI filtering for wide-sense stationary process

Why is it that if $U[n]$ is wide-sense stationary and it is convolved with $h[n]$ to produce $W[n]$, the autocorrelation becomes $R_{WW}[n] = R_{UU}[n]*h[n]*h[-n]$? I know that in general $R_{WW}[n_{...
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1answer
520 views

How to treat noise (in the acceleration) in a Kalman filter when tracking position?

I have an equation of motion for which I am wanting to track the position. The equation looks like this: $$\ddot{x}(t) = -a\dot{x}(t) - bx(t) + F(t)$$ where $F(t)$ is an external Stochastic force ...
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2answers
620 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 ...
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2answers
2k 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. ...
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1answer
157 views

Analyzing fluctuations within the Signal

I am trying to figure out how to analyze the signal shown below. It shows the fluctuation of number of review in for a product in Amazon, where positive means addition number of review while negative ...
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2answers
504 views

What does the frequency axis of a Power Spectral Density mean?

I have never really understood what the frequency axis meant when we plot the Power Spectral Density(PSD). Does it correspond to frequency as we get after we take the Fourier Transform of a time ...
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1answer
78 views

Is there any computational method to prove whether a series is stationary or not?

I have a discrete series $x[n]$. It is extracted from real life and I do not have probability distribution of each value $x[n]$. Is there any computational method to prove whether the series is ...
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1answer
182 views

Output of lowpass filter with damped sine wave input

The random process $$Y(t)=\cos(\omega_0t)\cos(\omega_0t+\pi N(t))$$where $N(t)$ is a Poisson process of parameter $\lambda$ enters a lowpass filter with transfer function $$H(j\omega) = \left\{ \...
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2answers
140 views

If noise is your signal, what is your noise?

Consider the following contrived situation. Imagine a Gaussian white noise process $x[t]$, with bandwidth $Δf$, with PSD equal to some quantity $A$ which you would like to measure. So the way to ...
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2answers
14k views

generating white gaussian noise in matlab using two different functions

I want to know the difference between the two Gaussian noises generated below? Which one is white and how can i make the other one white? y=wgn(1,10000,0) and <...
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1answer
212 views

Ergodicity of joint process

If we have two processes and both of them are ergodic. Does this mean that the joint proces is ergodic? Or other way around? If we have the dynamics for both components of the joint process what are ...
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1answer
396 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(...
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1answer
3k 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 ...
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1answer
325 views

AR model order selection for half second EEG fragments

I am using MATLAB to evaluate power spectral density estimates of half second EEG signals, using modified covariance method. Can anyone suggest me how to select the AR model order for this process? Is ...
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0answers
1k views

Fourier transforms of random processes

In the Wikipedia article on Brownian noise, the Fourier transform of Brownian noise is determined. How is that Fourier transform defined? It seems it is a non-random quantity there, so it is not ...
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1answer
151 views

How to represent multiple related signals via mathematical function?

I have three data sets from same source. I know they are interdependent and stochastic also. The datasets are of very high frequency (see the graph, X axis is time and Y axis is value). I want to ...
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1answer
428 views

Is ergodic in mean a property defined only for WSS stochastic processes?

I understand the definition of a random process $X(t)$ being ergodic in mean (first-order ergodic) is that the expectation of the sample mean $<u_X>_T=\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}...
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1answer
141 views

Properties of Stochastic Processes [closed]

I have a basic question about stochastic processes: When some informations such as wss, uncorraleted sampled, white about random signal (say x[n]) are given, what do we exactly have? For example ...
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0answers
3k views

Generate time-domain random signal from PSD

Given an analytical description of the PSD, for example (MATLAB "pseudocode"): ...
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1answer
115 views

How do we compute distrubtions of the value of a random process conditional on initial conditions?

Suppose I have a stationary process $\phi(t)$ with a known autocorrelation function $$ A(\tau) \equiv \langle \phi(0) \phi(\tau) \rangle$$ and suppose I also know that $\phi(t)$ is Gaussian ...
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1answer
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 ...

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