The decomposition of a periodic function as a fourier series is of great importance in many areas of physics such as vibration, acoustics and electromagnetics. It is also a fundamental technique of digital signal processing.
Fourier Series
A periodic function f(t) of period T can be expanded in the form
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Alternatively, we may use the complex form of the series,
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Fourier Transform of Discretely Sampled Data
Usually the periodic function f(t) will be realised only at discrete sample points. Suppose that f1, f2, ... fN is a sequence of discrete values of f(t); fk = f( [(k-1 )/N] T). By substitution of these values into the Fourier transform equation above and using the rectangle integration rule,
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The Fast Fourier Transform is an efficient algorithm for obtaining the cn from the fk.