of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 453
each other. This is important because the actual number of multiplications
required to evaluate the node pair can then be cut in half.
After the first column of nodes has been processed, everything is back
in storage in a partially transformed state, and processing of the next column
of nodes can begin. When the rightmost column has been evaluated, the
array contains the N complex harmonics. Note that for the first column the
nodes that should be paired for calculation are adjacent. In the next column,
the pair members are two apart, in the third column four apart, and so forth.
Also note that the mass of crossing lines forms a group of two nodes in the
first column, four nodes in the next column and so forth until there is only
one mass in the final column. These and other symmetries discovered by
examining the diagram should make the progression sufficiently clear so that
generalization for other values ofN can be understood. Since the progression
is consistent from column to column, it is clear that a general FFT subroutine
can be written for any value of N that is a power of two.
Table 13-1 shows the actual sequence of intermediate results corresponding
to the diagram. The example time samples are of a seven-harmonic
approximation to a sawtooth wave with a few degrees of phase shift included
to make life interesting. Note that the spectrum output starts with the dc
component in the zero array position, fundamental in position 1, etc. up to
the seventh harmonic in position 7. The Nyquist component is in position 8
(the sine part is always zero), while higher array positions contain the first
through the seventh harmonics again but in descending sequence and with
the sign of the imaginary components (sines) flipped. Also note that the
correct amplitudes of the frequency components have been multiplied by N/2
except for the dc component, which is N times too large.
An FFT Subroutine in BASIC
Figure 13-17 is an FFT program written in BASIC. Although BASIC
and fast are contradictory terms, the program serves to illustrate the algorithm
and to allow the reader to become familiar with it. The array of
complex input samples is actually stored and handled as two arrays of normal
numbers; D 1 being the real part and D2 being the imaginary part. Note that
the array subscript starts at zero. When the transformation is complete, the
spectrum is stored in the same two arrays. To insure that N is a power of two,
the record size is specified by K, which is 10gzN and must be a positive
integer.
In the program itself, statements 3110 to 3125 scramble the naturally
ordered time samples into bit-reversed order using the method illustrated
earlier. N3 being the "for" loop index counts naturally from 1 to N. Nl in
conjunction with statements 3114 to 3117 acts like a bit-reversed counter
taking on successive values of 0, 8, 4, 12, 2, etc. (for the N= 16 case).
Whenever N 1 is greater than N3, the samples pointed to by N 1 and N3 are
swapped.