It looks like I volunteered to help with an integral that's a little over
my head. A fellow ham is trying to write a program to autodecode Morse
code from an audio signal. This involves the Rayleigh fading channel and
some probability work, naming integrating over the phase to calculate
conditional probability. I think that I've verified his assertion of the
answer for N=2, but he wants higher orders - in particular N=5 - and I
don't think that I can help. I'd love to hear any responses, and will
forward them on to him.
His statement of the problem:
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I am working on the problem of CW
detection in the Rayleigh fading channel in the Bayesian framework. Since
CW
signals are incoherent, the probability density function must be
marginalized over the phase. This leads to the integral (1) in the attached
DOC file. The solution for N=2 is known (2), where I0 is the modified
Bessel
function of the first kind with order zero. I am interested in the solution
for higher N, in particular, for N=5, but I cannot find a way to take the
integral or to prove that a closed form solution does not exist. I will
appreciate your help with this.