of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 441
phase, the average is proportional to the cosine of the phase difference.
Fortunately, this procedure works even when one of the signals has many
frequency components; if one component matches it will contribute a de
component to the product samples. Thus, by using two "probe" waves 90°
apart in phase at all of the possible harmonic frequencies of the data record,
one may determine its complete harmonic amplitude and phase makeup.
The program segment below, which is remarkably similar to the inverse
transform segment, performs discrete Fourier analysis. The C, S, and T
arrays and N are as before and C and S are assumed to be initially zeroes.
1000 FOR 1=0 TO N-l
1001 FOR ]=0 TO N/2-1
1002 LET CO)=CO)+T(l)*COS(3.14159*I*]/N)/(N/2)
1003 LET SO)=SO)+T(I)*SIN(3.14159*I*]/N)/(N/2)
1004 NEXT]
1005 NEXT I
The outer loop in this case scans the time waveform, while the inner loop
accumulates running averages of cosine and sine components of each harmonic.
Note that the Nyquist frequency component is not computed. Therefore,
if one transforms arbitrary random data and then transforms it back, the
result will not be exactly equivalent. However, practical, unaliased data will
be returned with only a slight roundoff error. Note also that in both of these
program segments that the inner and outer loops may be interchanged with
no real effect on the results.
Fast Fourier Transform
Examination of the preceding two program segments reveals that N2
useful multiplications and additions are required for the transformation. Useful
is emphasized because a well-thought-out assembly language implementation
of the programs could eliminate multiplications within the cosine and
sine arguments by proper indexing through a sine table. Likewise, subscript
calculations can be eliminated by use of index registers. The division by N/2
in the forward transform may still be required, but it can be deferred until
the computation is completed and then need only be a shift if the record size
is a power of two. In the case of the inverse transform, the number of
multiplications may be cut in half by using the amplitude-phase form for the
harmonics rather than cosine-sine.
Even with the world's most efficient assembly language program, a
tremendous amount of computation is needed for even a moderate number of
samples. For example, 20-msec blocks taken at a 25 ks/s sample rate would
be 500 samples that, when squared, implies about a quarter of a million
operations for the transform. Upping the sample rate to 50 ks/s quadruples
the work ro a million operations. An efficient assembly language program on