Fourier Transforms

Date: 04/20/97 at 01:18:21
From: Tracey
Subject: Fourier Transforms

Dear Dr. Math,

I am having great trouble understanding the relationship between 
Fourier transforms, Laplacian equations, and why we are always 
converting them to Z tranforms.  Why is it useful to be in the 
frequency domain rather than in the time domain?

Yours faithfully,
Tracey (A confused 6th grader)

Date: 04/20/97 at 06:22:22
From: Doctor Mitteldorf
Subject: Re: Fourier Transforms

Dear Tracey,
    
It's not every day that we get a question about Fourier transforms 
from a 6th grade student, so I don't know where to begin.  I guess 
I'll just assume you know a whole lot and ask you to slow me down if 
I'm assuming too much.
    
Fourier transforms have a world of applications, some of them in 
expected ways and some in very unexpected ways. I'm a physicist, so 
the physics applications come most easily to mind for me.
   
An example of an "expected" application is to derive the musical 
pitches making up a sound from the sound signal itself. You can find 
the overtones in a musical note, or separate the notes in a chord 
played by an orchestra, or even analyze the sound of speech to 
distinguish various phonemes (consonant and vowel sounds.)
   
An example of a "less expected" application is the solution of 
differential equations. There are many differential equations that 
are difficult to solve in the time domain, but if you just transform 
them to the frequency domain they become very easy.
   
An example of a "completely unexpected" application is quantum 
mechanics. A "wave equation" is a standard quantum mechanical 
equation for the probability of a particle being at any location 
(x,y,z) in space. If you take the Fourier transform of the function 
that solves that equation, you end up with another function that 
relates to the probability that the same particle will have any given 
momentum!
    
This is just a tiny sampling of the applications of Fourier 
transforms.  Whole books have been written on the subject.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/