Questions tagged [complex-random-variable]

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What's the distribution of the DFT of a real-valued, zero-mean, normally distributed random vector?

Suppose $X$ is a real-valued N-dimensional Gaussian vector, $X \sim \mathcal{N}(\mathbf{0}, C_X)$. The discrete Fourier transform can be obtained by left-multiplying with the unitary DFT matrix, i.e. $...
DangerousTim's user avatar
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How do I determine stationarity from a set of 50 complex values collected every 10 minutes?

I am trying to determine stationarity from a somewhat stochastic process. Every 10 minutes, I collect a set of 50 FFTs, i.e., 1 trial over $50$ seconds, so an FFT occurs every time second. I ...
Cold Atom's user avatar
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Improving Modeling of Thermal Noise Behavior at Antenna

Background I recently asked this question over on Electrical Engineering Stack Exchange. On the advice of some commenters there, I've broken off those pieces which are appropriate for asking as ...
MomentumEigenstate's user avatar
5 votes
4 answers
324 views

Bandpass Stationary Stochastic Process

I was following this interesting post by a new user Rubem Pacelli and got stuck at Proakis' referenced definition (see Section 4-1-4 starting on page 159 here). The math, all repeated further below, ...
Dan Boschen's user avatar
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Question on Signal to noise ratio (SNR)

I am trying to form the SNR expression for the system model but I am getting stucked due to presence of multiple antennas at receiver. System model: It consists of one single antenna Transmitter, $L$ ...
paru's user avatar
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Pdf of product of two i.i.d random variable distributed as circular symmetric gaussian

What is a distribution of the product of two circular symmetric Gaussian random variables? i.e $X ~ CN(0,1)$ and $Y ~ CN(0,1)$ then distribution of $XY.$?
Jaimin Shah's user avatar
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1 answer
2k views

Exact formula for 8-PSK BER

Is there an exact formula for the probability of bit error (or bit error rate, BER) for 8-PSK (in the literature, course slides, etc.)? I am not referring to SEP (Symbol Error Probability) but BER. ...
Loran's user avatar
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How can I add complex WGN to a complex damped signal for a specified SNR?

I want to add complex wgn to a signal for a specified SNR. I have been using the following lines: ...
Neuling's user avatar
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nature of output of the addition of white noise with colord noise

I want to know that the addition of a uncorrelated time series x with a correlated time series, say y generates a time series ...
Sm1's user avatar
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0 answers
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Detecting complex signal within complex AWGN using Neyman Pearson statistic

I need to construct a Neyman Pearson statistic for detecting a complex signal with known amplitude and unknown phase in additive complex Gaussian noise. My hypotheses are: \begin{align} \mathcal{H}_0 &...
Razor's user avatar
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2 answers
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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 ...
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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 ...
Elena Martini's user avatar
1 vote
1 answer
463 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,...
chaosmind's user avatar
1 vote
2 answers
340 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 ...
Vishal Poddar's user avatar
2 votes
1 answer
680 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.
Buzz bee's user avatar
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4 answers
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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 ...
Buzz bee's user avatar
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(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 ...
Karn Watcharasupat's user avatar
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
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1 answer
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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}\right]^...
Karn Watcharasupat's user avatar
7 votes
3 answers
421 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 $\theta \sim \mathcal U(0, 2\pi)$? ...
Robert L.'s user avatar
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