** Next:** Properties of Fourier transforms
** Up:** Fourier analysis of periodic
** Previous:** Periodicity of the Fourier
** Contents**
** Index**

##

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