of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 439
transform normally deals in sine and cosine components of the harmonic,
which can be negative as well as positive. Conversion between the two forms
is avally quite simple. The overall harmonic amplitude, A, is given by
A = S2+C 2 , where S is the sine component value and C is the cosine
component. The effective phase angle (in radians) is listed in the following
table:
c S Phase angle
+ + Tan- 1 (SjC)
+ Tan- 1 (Sj-C)
Tan- 1 (SjC)
+ Tan- 1 ( -SjC)
Note that these formulas always give a positive amplitude and phase
angle. Also, C will have to be tested for zero before calculating the phase
angle to avoid division overflow. When C is zero, the phase is 7T/2 if S is
positive and 37T/2 if S is negative. When translating from amplitude and
phase to sine and cosine the formulas are: C=Acos(P) and S=Asin(P).
Thus, conversion from one form to the other is fairly simple.
The dc and Nyquist frequency components ofthe wave are a special case,
however. The cosine part of the zeroth harmonic is the actual dc component,
while the sine part would be expected to be zero, since a zero frequency wave
cannot have a phase angle other than zero. For mathematical completeness, it
is also necessary to consider the harmonic at exactly one-half the sample rate.
As it turns out, the Nyquist frequency harmonic must have a zero sine part
also. Note that if this Nyquist frequency component is very strong, it is an
indication ofserious aliasing in the data. Thus, in a real audio signal application,
its existence can usually be ignored.
Now, if the harmonic components are counted up, we find that there is
exactly the same quantity of numbers forming the frequency domain representation
as are in the time domain representation. Actually, this is a requirement
if the transformation is to be precisely reversible for any arbitrary
sample set as was stated earlier. Fourier transformation is exactly what the
name implies, a data transformation. It alone is not a magic data compression
technique.
Slow Fourie,. Transform
Before delving into the fast Fourier transform, it is necessary to understand
how the straightforward but slow version works. A few pages ago there
was some discussion about filling a waveform table by evaluating a Fourier
series. By making a couple of modifications to the procedure, the inverse
(frequency domain to time domain) slow Fourier transform results. The first
modification is that the number of samples generated in the output record