of Microprocessors

446 MUSICAL ApPLICATIONS OF MICROPROCESSORS
mere 28 instead of the original 256! Although it is not immediately apparent
how the redundancies can be identified in a general algorithm, one can use
matrix theory and trigonometric identities and derive the FFT algorithm
from this rype of table.
A completely different way to look at the redundancy involves the
concept of decimation. As an example, consider a sample record with N
samples where N is even. Using the SFT, one could perform N2 operations
and wind up with the spectrum. Now consider splitting the record into two
smaller records with NI2 samples in each such that the even numbered
samples go to one record and the odd numbered ones to go the other. If an
SFT is performed on each record, it will require N 2 /4 operations for a total of
only N 2 /2 operations. The trick is to combine the two resulting spectrums
into one representing the true spectrum of the original record. This can be
accomplished by duplicating the even sample spectrum ofN14 harmonics and
adding with a progressive phase shift the odd sample spectrum, also duplicated,
to yield N12 harmonics total. The combination requires N extra
multiplications and additions.
If N is divisible by 4, the decimation operation could be repeated to
give (our records of N14 samples each. The SFT can be performed on each,
requiring only N 2 /4 operations, and the four resulting spectrums combined
in two stages to form the final spectrum. In fact, if N is a power of two,
decimation can be repeated until there are NI2 subrecords, each having only
two samples as in Fig. 13-12. Note the resulting scrambled order of the
samples. This is called bit-reversed order because ifthe binary representation of
the scrambled sample numbers is observed in a mirror, it will appear to be a
simple ascending binary sequence.
Since the discrete Fourier transform of two samples is trivial (the cosine
component of the single harmonic is the sum of the sample values, while the
sine component is the difference), most of the computation is combining the
subspectra together in stages to form the final spectrum. The essence of the
FFT, then, is complete decimation of the time samples followed by recombination
of the frequency specrra to form the exact spectrum of the original
sample set.
Complex Arithmetic
At this point in the description of the FFT we will be forced to commit
a mathematical "copout" that is customary in such a discussion. In any case,
it will not only simplify the discussion but will facilitate the development of
an FFT program. The first step is to treat each harmonic, which has a cosine
and a sine component, as a single complex number. Although the cosine
component is normally called the "real" part and the sine component the
"imaginary" part, both are quite real as far as a physical waveform is concerned.
Therefore, it is best to forget about "real" and "imaginary" and