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Fourier Transforms

Date: 09/17/98 at 11:10:58
From: Kristjan
Subject: Fourier Transform

What is a Fourier Transform and what is it used for?


Date: 09/17/98 at 11:53:45
From: Doctor Mitteldorf
Subject: Re: Fourier Transform

A Fourier transform takes any function and converts it to an equivalent 
set of sine waves. For example, if you plot the electrical signal going 
into a loudspeaker that's playing music, you'll see a wavy line, where 
the waves are sometimes big and close together, other times smaller, 
other times farther apart. Your ear hears notes, not vibrations, and 
the collection of notes that you hear is the Fourier transform of the 
signal going into the loudspeaker.

A remarkable and wonderful theorem says that the same mathematical 
procedure that takes you from the function to its Fourier transform 
can be used again to take you from the Fourier transform back to the 
function. In other words, just do the operation twice and you're back 
where you started from.

It turns out that Fourier transforms are enormously useful in physics, 
sometimes in expected ways and sometimes in completely unexpected ways. 
Often the easiest way to solve a differential equation is to Fourier 
transform the whole equation, solve it more easily, then Fourier 
transform it back again. In quantum mechanics, a particle isn't in a 
given place at a given time, but must be described by a "wave function" 
that gives the amplitude for its being in a given place at a given 
time. If you know the wave function for the position, you can apply a 
Fourier transform to get the wave function for the particle's momentum.

- Doctor Mitteldorf, The Math Forum   

Date: 05/22/2004 at 01:41:22
From: Charles
Subject: Use of Fourier transforms

I understand that Fourier transforms are used in differential equations.  
Is there anywhere else they might be used?  And what's the difference 
between that and fast Fourier transforms?  Are they used for the same 
kinds of problems?  Thanks.

Date: 06/09/2004 at 18:55:43
From: Doctor Douglas
Subject: Re: Use of Fourier transforms

Hi Charles.

Thanks for writing to the Math Forum.  Fourier transforms are used for 
many applications where the "spectral content" of a function, or a 
waveform, is important.  For example, you might have an equalizer on 
your stereo that attenuates (filters) frequencies outside the audio 
passband, or perhaps "shapes" the spectrum to your liking ("more bass!").  
The equalizer/filter operates on the sound in the "frequency-domain", 
rather than in the "time-domain" (which might take the form of the
time-dependent voltage in your preamp).  Fourier transforms establish 
the mathematical connection between these two domains.

Fourier transform methods are important in audio applications, 
quantum mechanics, optics, and all sorts of wave phenomena.

Fast fourier transforms (FFTs) are essentially a special discretized 
version of Fourier Transforms.  They are defined for functions that are 
equally-sampled in time and usually have a number of points of the form 
2^k for some integer k.  FFTs are useful in computer implementations, 
because of these two properties.  And the "fast" is related to the speed 
and efficiency of the algorithm that is used to do the transform.  
These efficiencies depend critically on the fact that the samples are 
equally-spaced in time and that the number of points is a power of 2.

- Doctor Douglas, The Math Forum

Date: 06/09/2004 at 19:37:51
From: Charles
Subject: Thank you (Use of Fourier transforms)

Thank you very much for your help.  That clarifies things 
quite a bit.  
Associated Topics:
High School About Math
High School Physics/Chemistry

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