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Understanding of Random Process, Random Variable and Probability Density Function

when we observe the Random Process at a specific time $t_k$, that is the value at $X(s_1,t_k), X(s_2,t_k),\ldots,X(s_n,t_k)$, if we denote them by $(a_1,a_2,\ldots, a_n)$. Now the mapping between ...
AlexTP's user avatar
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7 votes
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Correlation of independent random processes

No. Quoting Wikipedia's article Independence (probability theory): If $X$ and $Y$ are independent random variables, then the expectation operator $\operatorname{E}$ has the property $$\...
5 votes
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Random signals as power signals

Note that the condition $$\int_{-\infty}^{\infty}|f(t)|^2dt<\infty\tag{1}$$ (i.e., that the signal $f(t)$ has finite energy) is very restrictive when we try to model signals, even though ...
Matt L.'s user avatar
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5 votes

Rayleigh Fading Simulation

Rephrasing OP's Post for Clarity It wasn't stated, but I assume the OP is attempting to create a one-tap channel filter with: ...
Dan Boschen's user avatar
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5 votes

Random sampling vs uniform sampling

The key idea is that the random sampling approach enforces more constraints on the resulting signal than the uniform sampling approach does. The POCS (projections onto convex sets) algorithm used for ...
Peter K.'s user avatar
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4 votes

Addition of signal variance and noise variance

If the signal and noise are uncorrelated then the variance of the sum of the two equals the sum of their variances. So, yes, you can add variances (of uncorrelated signals). However, you cannot add dB-...
Matt L.'s user avatar
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3 votes

Random signals as power signals

I think simple. We want to model a random physical phenomenon for analysis purpose. One way is to model it by a stochastic process $X(t)$, i.e. a time series of random variables $\left\lbrace X(t_k) =...
AlexTP's user avatar
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3 votes

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

As you correctly say, if $\{X(t)\colon t \in \mathbb T\}$ is a Gaussian process, then for every possible choice of $n$ random variables $X(t_1), X(t_2), \ldots, X(t_n),$ where $t_k \in \mathbb T$ for $...
Dilip Sarwate's user avatar
3 votes
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Doubts and some confusion on variance for complex rv

If the imaginary and real components each has a variance of 0.5, then what would be the total variance? The variance of the complex RV in that case would be $0.5 + 0.5 = 1$. If on the other hand, ...
MBaz's user avatar
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3 votes
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Understanding of Random Process/Random Variable

Let me explain it in another way. Consider you have 6 different function of time. You only throw your dice once and regarding the outcome you choose on of six functions and the chose one is one ...
Mohammad M's user avatar
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3 votes
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Random process $X(t)$ with autocorrelation function given find the mean and the variance

The limit $\lim_{\tau\to\infty} R_x(\tau)$, if it exists, equals $E^2[X(t)]$ and so $E[X(t)]=0$ in this case. More generally, the mean of a WSS process is nonzero only if the power spectral density ...
Dilip Sarwate's user avatar
2 votes

What is the value of the standard deviation $\sigma$?

randn gives you Gaussian noise with std. dev. of 1. (x/255)*(randn(size(image))) gives you a noisy image of std. dev. ...
geometrikal's user avatar
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2 votes
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Detection of sine signals with random amplitudes

Since both the null hypothesis $\mathcal H_0$ and the alternative hypothesis $\mathcal H_1$ are signal + noise, this is not only the detection of a random signal in WGN, but also a discrimination ...
Gilles's user avatar
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2 votes

Random signals as power signals

In addition to Marcus Müller comment, If a signal has finite energy then the signal value must reach zero after long enough time, but for random signals your signals generally don't have such ...
Mohammad M's user avatar
  • 1,327
2 votes

Pink ($1/f$) pseudo-random noise generation

I have been using Corsini and Saletti's algorithm since 1990: G. Corsini, R. Saletti, "A 1/f^gamma Power Spectrum Noise Sequence Generator", IEEE Transactions on Instrumentation and Measurement, 37(4),...
Ed V's user avatar
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2 votes

Independence of Functions of random Variable

It happens that if $X$ and $Y$ are independent then so will their functions $g(X)$ and $h(Y)$ be; but not $g(X,Y)$ and $h(X,Y)$.
Fat32's user avatar
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2 votes
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Variance of function of random variable

Assume $Y = g(X)$ be the function of RV $X$, then by using the following $$E\{ g(X) \} = \int g(x) f_X(x) dx $$ variance of $Y$ can be computed without the computation of pdf $f_Y(y)$ as: $$ \begin{...
Fat32's user avatar
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2 votes
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Understanding Asymptotic Equipartition Property

The sentences you've mentioned in your question all have a quite different meaning. Let me try to explain them one by one: Clearly, it is not true that all $2^n$ sequences of length $n$ have the ...
Matt L.'s user avatar
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2 votes

If $X$ and $Y$ are zero mean independent Gaussian random variables with different variances, what is the density of $\sqrt{X^2+Y^2}$

According to your description, $x = Z \cdot \cos(W),\ y = Z \cdot \sin(W)$. Follow this answer for the derivation of joint PDF of $(Z, W)$ : Complex Gaussian Magnitude and Phase Joint PDF Derivation ...
DSP Rookie's user avatar
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2 votes
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How can I generate time-variant Rayleigh channel in MATLAB

Yes, you can have time variant channel in MATLAB either by using comm.RayleighChannel() , you can read about it in help of matlab. or the other way you use the ...
Zeyad_Zeyad's user avatar
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2 votes
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Autocorrelation of random walk

Let's try to figure this out from first principles. Let's start with $x$ our zero-mean, Gaussian, independent, identically distributed noise sequence: $$ x[n] \sim N(0,\sigma^2_x) $$ Then our random ...
Peter K.'s user avatar
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1 vote
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Determining the average probability of error

Note that $$ \mathbb{P}(n_m+i_m<-1\cap i_m = \frac{-1}{2}) $$ is equal to $$ \mathbb{P}(n_m < -0.5)\mathbb{P}(i_m = -0.5) = 0.25\mathbb{P}(n_m < -0.5) $$ since $n_m$ and $i_m$ are independent....
MBaz's user avatar
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1 vote
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Why the requirement of the GCD of the lengths of all circuits in the graph being one?

This is my understanding: the statistics of the source described in the paper depend on which character is produced first. If the first character is $a$, then one of the source properties is that ...
MBaz's user avatar
  • 15.3k
1 vote

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

You have the definition of the random process as : $$ X(t) = a \sin(\omega_0 t + \Theta) \tag{1} $$ where $a$ and $\omega_0$ are deterministic constants and $\Theta$ is a continuous R.V. uniformly ...
Fat32's user avatar
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1 vote

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

You have a single random variable $\Theta$ in this example, so taking the expectation with respect to that random variable results in a single integral. The formula for $R_X(\tau)$ in your question is ...
Matt L.'s user avatar
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1 vote
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Power contained in a random process $X(t)$

Assuming that the process is wide-sense-stationary, the total power in a random process with autocorrelation function $R(\tau)$ and power spectral density $S(f) = \mathcal F\{R(\tau)\}$ is given by $$\...
Dilip Sarwate's user avatar
1 vote
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Independence of Functions of random Variable

The canonical definition of independence of two random variables $X$ and $Y$ is $X$ and $Y$ are called independent random variables if for every choice of Borel sets $B_1, B_2$, the events $\{X ...
Dilip Sarwate's user avatar
1 vote
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Random Process at a particular time instance

You misunderstand what's going on here. The random process $X(t)$ is of the form $$X(t)=\sin(\Omega t)\tag{1}$$ where $\Omega$ is a random variable with PDF $$p_{\Omega}(\omega)=P[H]\cdot \delta(\...
Matt L.'s user avatar
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1 vote

Random Process at a particular time instance

Your equation (3) and (4) are wrong. When you say Also, the sum of probabilities of a random variable should be 1 you actually mean the total sum of the values of probability mass function $P_{...
Fat32's user avatar
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1 vote

How can I calculate the entropy of a signal that's not independent from itself?

I think that the answer is that there is no correct answer. Whilst this is a simple question, the answer is kinda open. You're asking about the entropy of correlated data with respect to random ...
Paul Uszak's user avatar

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