Questions tagged [complex-random-variable]

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7
votes
3answers
225 views

Distribution of $e^{j\theta}$

Is there a canonical or analytic expression for the probability distribution for the circularly-symmetric complex random variable $Z$: $$ Z = e^{j\theta}, $$ where $\theta \sim \mathcal U(0, 2\pi)$? ...
3
votes
2answers
123 views

Signal with Complex Gaussian noise

If I have: $$ y = x_r+jx_i + n_r +j n_i$$ with $n_r$ and $n_i$ Gaussian with mean 0 and variance $\sigma^2$, what is the pdf of the envelope |y| and phase(y)? Is it still Rayleigh-distributed and ...
2
votes
2answers
931 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 $N_0/2$...
1
vote
1answer
48 views

Characteristic function of a random Gaussian variable

I have to find the characteristic function of a random Gaussian variable of $$ \sigma_z (w) = E e^{i w z } $$. This is the variable and I know , from the theory that the characteristic function of ...
1
vote
1answer
44 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.
1
vote
1answer
36 views

(For c-FastICA) On covariance and pseudocovariance matrix of a complex random vector

I am currently studying complex FastICA and the paper says that Suppose $\mathbf{s}$ is a $n\times1$ complex random vector. If $\mathbf{s}$ has zero mean, unit variance, and uncorrelated real and ...
1
vote
1answer
133 views

Energy definition for Autocorrelation lag 0 and lag 1 for complex signals

I am studying the role of an auto-correlation matrix for random signals and the difference of energy between a lag 0 and lag 1 matrix. Consider a complex input signal $x(k)=[x1,x2]^T$ and $x(k-1)=[x0,...
0
votes
4answers
101 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 ...
0
votes
1answer
45 views

Clarifying matrix notation from an ICA-CMN paper

In the paper I am referring (and here from citeseer), complex vectors $\mathbf{z}$ and matrix $\mathbf{M}$ were defined as follows \begin{align} {{\bf z}} &= \left[z_{1},z_{2},\ldots,z_{N}\...
0
votes
0answers
24 views

Calculating average power of a discrete random signal with complex values

I have a Rayleigh channel that transmits discrete values according to the following: $$y(n) = h(n) * s_n + \omega_n$$ Where $y$ is the received message, $s_n$ is the symbol sent (constellation 2-PSM), ...
0
votes
0answers
26 views

I want to invert my fourier transform components to waves again

Hi I am using R to analyze some data I basically did fft(data) and got a vector of complex numbers but from that now I want to remove certain harmonics from my actual wave but how do I convert one of ...
0
votes
2answers
107 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
37 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
102 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{...