Fourier

6 Pages 1508 Words

Linear transforms, especially Fourier and Laplace transforms, are widely used in solving problems in science and engineering. The Fourier transform is used in linear systems analysis, antenna studies, optics, random process modeling, probability theory, quantum physics, and boundary-value problems (Brigham, 2-3) and has been very successfully applied to restoration of astronomical data (Brault and White). The Fourier transform, a pervasive and versatile tool, is used in many fields of science as a mathematical or physical tool to alter a problem into one that can be more easily solved. Some scientists understand Fourier theory as a physical phenomenon, not simply as a mathematical tool. In some branches of science, the Fourier transform of one function may yield another physical function (Bracewell, 1-2).


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The Fourier Transform
The Fourier transform, in essence, decomposes or separates a waveform or function into sinusoids of different frequency which sum to the original waveform. It identifies or distinguishes the different frequency sinusoids and their respective amplitudes (Brigham, 4). The Fourier transform of f(x) is defined as
F(s) = f(x) exp(-i 2xs) dx.

Applying the same transform to F(s) gives

f(w) = F(s) exp(-i 2ws) ds.

If f(x) is an even function of x, that is f(x) = f(-x), then f(w) = f(x). If f(x) is an odd function of x, that is f(x) = -f(-x), then f(w) = f(-x). When f(x) is neither even nor odd, it can often be split into even or odd parts.

To avoid confusion, it is customary to write the Fourier transform and its inverse so that they exhibit reversibility:

F(s) = f(x) exp(-i 2xs) dx

f(x) = F(s) exp(i 2xs) ds

so that

f(x) = f(x) exp(-i 2xs) dx exp(i 2xs) ds

as long as the integral exists and any discontinuities, usually represented by multiple integrals of the form ½[f(x+) + f(x-)], are finite. The t...