Digital Sound & Music: Concepts, Applications, & Science, Chapter 2, last updated 6/25/2013
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The Fourier transform is a mathematical process that can transform audio data from the
time to the frequency domain. Sometimes it's more convenient to represent sound data one way
as opposed to another because it's easier to manipulate it in a certain domain. For example, in
the time domain we can easily change the amplitude of the sound by multiplying each amplitude
by a number. On the other hand, it may be easier to eliminate certain frequencies or change the
relative magnitudes of frequencies if we have the data represented in the frequency domain.
2.3.7 The Discrete Fourier Transfer and its Inverse
The discrete Fourier transform is a version of the Fourier transform that can be applied to
digitized sound. The algorithm is given in Algorithm 2.1.
/*Input:
f, an array of digitized audio samples
N, the number of samples in the array
Note:

Output:
F, an array of complex numbers which give the amplitudes of the
frequency components of the sound given by f */
for ( )
(∑ )
Algorithm 2.1 Discrete Fourier transform
Each time through the loop, the
nth
frequency components is computed, . Each is a
complex number with a cosine and sine term, the sine term having the factor i in it.
We assume that you're familiar with complex numbers, but if not, a short introduction
should be enough so that you can work with the Fourier algorithm.
A complex number takes the form , where . Thus,
( )
is a complex number. In this case, a is replaced with and b with ( ).
Handling the complex numbers in an implementation of the Fourier transform is not difficult.
Although i is an imaginary number,

, and you might wonder how you‟re supposed to do
computation with it, you really don‟t have to do anything with it at all except assume it‟s there.
The summation in the formula can be replaced by a loop that goes from 0 through N-1. Each
time through that loop, you add another term from the summation into an accumulating total.
You can do this separately for the cosine and sine parts, setting aside i. This is explained in more
detail in the exercise associated with this section. Also, in object-oriented programming
languages, you may have a Complex number class to do complex number calculations for you.
The result of the Fourier transform is a list of complex numbers , each of the form
, where the magnitude of the frequency component is equal to

.
The inverse Fourier transform transforms audio data from the frequency domain to the
time domain. The inverse discrete Fourier transform is given in Algorithm 6.2.
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