Fourier TransformsDate: 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/ |
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