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## Fourier transform as additive synthesis

Now consider an arbitrary signal that repeats every samples. (Previously we had assumed that could be obtained as a sum of sinusoids, and we haven't yet found out whether every periodic can be obtained that way.) Let denote the Fourier transform of for :

In the second version we rearranged the exponents to show that is a sum of complex sinusoids, with complex amplitudes and frequencies for . In other words, can be considered as a Fourier series in its own right, whose th component has strength . (The expression makes sense because is a periodic signal). We can also express the amplitude of the partials of in terms of its own Fourier transform. Equating the two gives:

This means in turn that can be obtained by summing sinusoids with amplitudes . Setting gives:

This shows that any periodic can indeed be obtained as a sum of sinusoids. Further, the formula explicitly shows how to reconstruct from its Fourier transform , if we know its value for the integers .

Next: Properties of Fourier transforms Up: Fourier analysis of periodic Previous: Periodicity of the Fourier   Contents   Index
Miller Puckette 2006-12-30