of Microprocessors

440 MUSICAL ApPLICATIONS OF MICROPROCESSORS
will be exactly twice the number of harmonics in the input record. The other
modification is for the sine and cosine form of the harmonic data. Thus,
when filling a waveform table of 256 entries, 128 harmonics must be
specified, although many if not most of the higher ones may be zero to avoid
aliasing problems when the table is used for synthesis.
The program segment below.illustrates the inverse slow Fourier transform.
The "C" and "5" arrays hold the cosine and sine harmonic data,
respectively. The de component uses subscript 0, the fundamental uses subscript
1, etc. Thus, the cosine component of the eighth harmonic would be
stored in C(8) and the sine component would be in 5(8). During execution,
the time samples will be stored in the T array, which also starts at subscript 0
and is assumed to initially contain zeroes. The constant N is simply the
number of samples in the record and should be even.
1000 FOR 1=0 TO N/2-1
1001 FORJ=O TO N-l
1002 LET T0)=T0)+C(I)*COS(6.283l8*I*J/N)
1003 NEXT J
1004 NEXT I
The outer loop steps through the harmonics starting at the zeroth and ending
one short of N /2. The inner loop steps through all of the time samples for
each harmonic adding the specified cosine and sine components to the current
content of the samples.
The "forward" transform, which converts time samples into harmonic
components, however, is what most people have in mind when they speak
about a Fourier transform. Unfortunately, its computation is not intuitively
obvious, although it turns out to be no more complex than the inverse
transform. The basic approach for determining the amplitude ofa component
with a particular frequency and phase is to generate samples of the sought
component, multiply them by corresponding samples of the signal to be
analyzed, and then add up the products. The sum, after being divided by
one-half the record size for averaging and mathematical anomalies is the
amplitude of the signal component having that frequency and phase! (The de
component will be twice its correct amplitude, however.)
The truth of this can be comprehended by remembering that multiplying
two sinusoidal signals is equivalent to balanced modulation and produces
a signal having only sum and difference frequency components. The reader
should also recall that any sinusoidal wave averaged over an integral number
of cycles is zero. It is easily seen then if the two signals being multiplied have
different frequencies, that the product consists of two sinusoidal waves that,
when averaged over a duration having an integral number ofperiods for'both,
gives zero. If the frequencies being multiplied are the same, the difference
frequency is zero, which is de and results in an average value equal to the
product of the signal amplitudes if they are perfectly in phase. When not in