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Topic: Distribution of the peak from a size-N complex vector
Replies: 1   Last Post: May 15, 2013 6:53 PM

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 David Jones Posts: 80 Registered: 2/9/12
Re: Distribution of the peak from a size-N complex vector
Posted: May 15, 2013 6:53 PM

wrote in message

Hope this is the right place to ask this question. I'm trying to determine
the distribution of the peak of a length-N complex vector whose real and
imaginary components are Gaussian distributed with zero mean and identical
variance.

This situation can be simulated in Matlab as follows:

test_mat = (randn(10000,8192)+1j*randn(10000,8192))/sqrt(2);
pks = max(abs(test_mat),[],2);
hist(pks,50)

So I'm generating 10,000 complex vectors of length N=8192. I'm then taking
the peak magnitude from each vector and constructing a histogram.

I think I have determined that the distribution of the variable 'pks' is
Generalized Extreme Value, and I think I've determined the "mu" parameter of
the distribution as sqrt(ln(N)), where N in this case is 8192. I kind of
"lucked" into this by looking at other vector lengths and seeing how this
value changed.

I'd like to be able to link the other parameters of the Generalized Extreme
Value distribution to the vector length and/or the variance of the original
real/imaginary components.

I'd appreciate any help anyone out there can provide.

Thanks,
Scott

========================================================

The GEV distribution is only an approximation. In this case it is easy to
get the exact distribution of the maximum, since it is the maximum of N
independent random variables whose distributions are known to be the square
root of an exponential random variable, and hence the distribution of the
maximum is known. So, either use this exact distribution for everything ...
or find an approximating GEV distribution by matching selected percentage
points.

David Jones

Date Subject Author
5/15/13 sbennett86@gmail.com
5/15/13 David Jones