of Microprocessors

448 MUSICAL ApPLICATIONS OF MICROPROCESSORS
Fourier Multipliers Expressed as
Arguments of the Function
W(I) ~ COS(21TI/N), SIN(21TIIN)
N = 16
Time
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
3 0 3 6 9 12 15 2 5 8 11 14 1 4 7 10 13
4 0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12
F
5 0 5 10 15 4 9 14 3 8 13 2 7 12 1 6 11 r
6 0 6 12 2 8 14 4 10 0 6 12 2 8 14 4 10 e
7 0 7 14 5 12 3 10 1 8 15 6 13 4 11 2 9 q
8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 u
9 0 9 2 11 4 13 6 15 8 1 10 3 12 5 14 7 e
10 0 10 4 14 8 2 12 6 0 10 4 14 8 2 12 6 n
11 0 11 6 1 12 7 2 13 8 3 14 9 4 15 10 5 c
12 0 12 8 4 0 12 8 4 0 12 8 4 0 12 8 4 Y
13 0 13 10 7 4 1 14 11 8 5 2 15 12 9 6 3
14 0 14 12 10 8 6 4 2 0 14 12 10 8 6 4 2
15 0 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Fig. 13-13. Fourier multipliers expressed as arguments of the function W(I) =
cos(21TJIN),sin(21TIIN). N = 16.
consideration which is constant throughout the FFT computation. If I ever
exceeds the range of a to N -1, it is customary to use the principal value,
which is simply I mod N. Finally, as if nothing is sacred, even the time
samples are considered to be complex numbers with a real part and an
imaginary part.
As a result of going complex, the number of harmonics is equal to the
number of complex samples. Thus, the forward FFT takes a set of N complex
time samples and generates a set of N complex harmonics. As we will see
later, all of the'intetmediate arithmetic in the FFT algorithm is also complex.
In the real world, which includes audio, the time samples are, of
course, real, which means that the imaginary parts are zero. The corresponding
spectrum has only (N/2)+ 1 unique harmonics; the remaining (N/2)-1
are simply the complex conjugate (the signs of the sine components are reversed)
of the others less de and Nyquist frequency components. Thus, in actual use
with audio samples, one has N samples and (N/2)+ 1 harmonics, just as
before. Although the preceding implies considerable waste in the computation,
a method of eliminating it will be described later.