Questions tagged [stochastic]
93 questions
7
votes
1 answer
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 I understood from the book in the link is as…
Srishti M
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6
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2 answers
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 frequency $f_c$ is constant and amplitude $A$ is…
TIWARI
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5
votes
1 answer
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 seems to me a deterministic quantity and I cannot…
xhensa
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5
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3 answers
Power Spectral Density of Brownian Motion despite non-stationary
Note: I originally asked this on Physics Stack Exchange but haven't attracted any interest there so I'm asking here where it may be more relevant.
A white noise process, $\xi(t)$ with delta correlated two-correlation function $\langle…
Jagerber48
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5
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1 answer
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 consider a sample space $S$ consisting of $n$ outcomes…
sundar
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5
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1 answer
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 stationary or not?
Thomas Lee
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4
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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?
Thomas
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4
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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 of unit variance, and I'm interested in the Power…
user53750
4
votes
2 answers
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 input $v[k]$ at time $k$ and the output $x[k − 1]$ at…
Mr.Sh4nnon
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4
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2 answers
What's the meaning of ergodicity?
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. Thanks.
Amin
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4
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2 answers
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 domain signal. This doesn't make sense to me because…
3
votes
1 answer
Higher-order moment of output of LTI system
Assume a very simple LTI system. Assume $x$ is white Gaussian i.i.d. with variance $\sigma^2$.
The output variance is straightforward to obtain. For example, for a continuous-time system:
$$\mbox{Var}(y) = \mathbb{E}(y^2) = \sigma^2…
divB
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3
votes
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Mean Square Continuity of Random Process
Show that a stochastic process $X(t)$ is mean square continuous if and only if its autocorrelation function $R_X(t_1,t_2)$ is continous
$\Rightarrow$ Proof:
We have $E[(X(t)-X(t_0))^2]=R_X(t,t)-R_X(t_0,t)-R_X(t,t_0)+R_X(t_0,t_0)$, so if $t…
Numbermind
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3
votes
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Average Power Spectral Density of PAM signals
I am reading through the PAM transmission scheme and about the power spectral density of the signals. Given that the Average Power Spectral Density of PAM Signals is:
$$
\Phi_{ss}(f)=\Phi_{aa}\left(e^{j2\pi ft}\right)\frac{\lvert…
sundar
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3
votes
2 answers
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 measure this seems to measure the variance of the…
Brian P
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