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From: Nicholas Kinar on 9 Mar 2010 15:53
Hello--

I am reading a very interesting paper on acoustics which describes a
numerical procedure used to construct a discrete vector of values from a
signal. This vector is then used along with some mathematics to
calculate the attenuation of a sound pulse through a medium.
Essentially, the procedure has the following steps:

(1) Compute the frequency spectrum U[tau, omega] using the Gabor
transform of a discrete signal s[t]. The time is tau, and the angular
frequency is omega. Thus U[tau, omega] is a 2D matrix with each element
having real and complex parts.

(2) From U[tau, omega], compute the amplitude A[tau, omega] from the
real and imaginary parts. Thus A[tau, omega] = sqrt( (U_r^2) + (U_i^2)
), where U_r is the real part of U[tau, omega], and U_i is the imaginary
part of U[tau, omega].

(3) Next, define chi = (tau * omega), and then construct a 1D vector
A[chi] = A[tau * omega] from A[tau, omega] by integrating along constant
chi.

So what I need to do is take the 2D matrix A[tau, omega], and convert it
to A[tau * omega] by performing numerical integration along constant chi.

What is the most efficient way to perform numerical integration along
constant chi so that I can obtain A[tau * omega]? I've been running
around in circles for a while trying to do this, and I can't see a very
clear-cut or efficient solution.

Thanks,

Nicholas


From: Nicholas Kinar on 11 Mar 2010 18:23

> What is the most efficient way to perform numerical integration along
> constant chi so that I can obtain A[tau * omega]? I've been running
> around in circles for a while trying to do this, and I can't see a very
> clear-cut or efficient solution.
>


I've found a paper that deals with the numerical integration of line
integrals [1], but I'm still puzzled about performing the integration
along constant chi.

[1] K. Atkinson and E. Venturino, �Numerical evaluation of line
integrals,� SIAM Journal on Numerical Analysis, 1993, pp. 882�888.

Hmm....

Nicholas

From: Nicholas Kinar on 12 Mar 2010 10:31
>
>> What is the most efficient way to perform numerical integration along
>> constant chi so that I can obtain A[tau * omega]? I've been running
>> around in circles for a while trying to do this, and I can't see a very
>> clear-cut or efficient solution.
>>

Perhaps the integration is nothing more than an "average" along constant
chi. I'm going to run with this idea for a while and see where it leads
me, since this appears to be the most logical.

Nicholas
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