From: Jim Choate <ravage@ssz.com>
Date: Wed, 11 Feb 1998 03:37:16 +0800
To: cypherpunks@ssz.com (Cypherpunks Distributed Remailer)
Subject: Fourier & Signal Analysis...
Message-ID: <199802101939.NAA05111@einstein.ssz.com>
MIME-Version: 1.0
Content-Type: text



Hi,

There is one final point I would like to make regarding at least one comment
on Fourier Analysis of signals in a noisy environment.

Fourier signal analysis is not good for this sort of stuff, what you want is
a function called a Laplace Transform.

Fourier is a method to take a complex signal and break it down into
component trigonometric functions (eg sin) such that we can apply filtering
and other frequency dependant operations to the signal to improve the
signals characteristic signature or operate on a particular component (ie a
component that is resonating with something and decreasing the s/n ratio -
ringing on a pulse edge for example).

If you want to find a signal in the noise you want to apply a Laplace. What
it does is convert the amplitude-time variant signal (ie f(t)) into a
power-frequency variant signal (ie f(s)). What this does is take, for example,
a sample of a signal and describe how that signals power is divided between
the various frequencies in that signal (eg McLauren Series). In most
applications the grass of the signal will be evenly distributed across the
s-mapping while the signal (at least its carrier component) itself will show
up as a spike of noticeable amplitude. The reason you do this in signal
analysis is because that transormation function will often be a simpler
equation to solve numericaly that actualy trying to deal with f(t).

So, if you want to find a signal in a complex environment use a Laplace
Transform, once you've found the signal and want to know about its
components use a Fourier Transform.

Good hunting!


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