Re: [Phys-l] Integral Help

["David Bowman" <David_Bowman@georgetowncollege.edu>]

Regarding Josh's multi-dimensional integral problem:
 
> 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. 
>
> The .doc file mentioned below can be downloaded at:  [
> http://www.wikiphys.org/1.doc ]http://www.wikiphys.org/1.doc
> ...
 
First off, I noticed that there is an error in eqn. (2) in the above
1.doc file.  The problem is that a factor of 4*[pi]^2 is missing from
the RHS, or equivalently, the reciprocal of that factor is missing
from the LHS.
 
Secondly, I was able to reduce the generic N-dimensional case to a
single 1-dimensional integral over an exponentially decaying function
times a product of N different modified Bessel Functions.  So far I
haven't been able to do that last remaining 1-dimensional integral in
closed form for a generic value of N, or even in the special case of
interest when N = 5.
 
In any event the result is given below.  But first I need to define a
few ASCII-notated versions of some mathematical operations:
 
INT{x=a,b|dx*f(x)} == the definite integral of the function f(x)
times dx over the domain of x running from the lower limit of a to
the upper limit of b.
 
Also the notation I use for a repeated product is
 
PI{j=1,N|f_j} == f_1*f_2*f_3*...*f_N  .
 
Also I_0(x) == the Modified Bessel Function of order 0 and argument
x.
 
The integral in eqn. (1) of the 1.doc document then becomes:
 
((2*[pi])^N)*INT{x=0,[infinity]|dx*exp(-x)*F(x,{a_j})}  where
 
F(x,{a_j}) == PI{j=1,N|I_0(2*a_j*sqrt(x))} .
 
Even though I haven't been able to do this last integral in the
generic N-value case, we can still evaluate it numerically for a
given known set {a_j|j = 1, N} using our favorite numerical
integration algorithm once the domain of integration is transformed
to something most suitable for the particular numerical integration
algorithm to be used.
 
I will leave the derivation of the above 1-d integral result as an
excercise for the reader.  But I will give three useful hints that
can be used to derive the result.
 
First, notice that the exponent in eqn. (1) of 1.doc is actually
the squared absolute value of a sum of N complex numbers where the
j-th complex number has an absolute value of a_j and a complex phase
of [theta]_j.  Or equivlently, the real part of each such complex
number is a_j*cos([theta]_j) and the imaginary part of the number is
a_j*sin([theta]_j).
 
Secondly, notice that the following integral identity:
 
exp(z^2)=INT{u=-[infinity],+[infinity]|du*exp(-u^2+2*u*z)}/sqrt([pi])
 
can be used to replace the square of a quantity in an exponential by
an integral over an exponential of something proportional to the
first power of that quantity instead.
 
Thirdly, since the exponent in eqn. (1) of 1.doc is the
squared absolute value of a sum of complex numbers that means that
we can think of it as the sum of the square of the sum of the real
parts of those numbers and the square of the sum of the imaginary
parts of those numbers.
 
David Bowman