of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 455
Innermost in the nest of loops is the useful calculation for a pair of
nodes. The complex numbers that form the node inputs, outputs, and W
function are handled one part at a time according to the rules of complex
arithmetic. Note that only the lower argument value (principal value) ofW is
explicitly evaluated, since the upper argument value gives the same result
with opposite sign. A count of useful operations for the node pair gives four
multiplications and six additions. These statements are evaluated (NlogzN)/2
times, which gives a total of 2NlogzN multiplications and 3Nlog2N
additions.
So far the FFT has been described in terms of transforming a set of time
samples into a frequency spectrum. Because of the symmetry of the W
coefficient array, transforming a spectrum into a sample record requires only a
trivial change in the FFT subroutine. All that is necessary is to place a minus
sign in front of N3 in statement 3213. When this is done, the spectrum to be
transformed is stored in the D arrays in the same format that it appears after a
forward transform (be sure to store both the normal and the conjugate
spectrum) and the modified FFT subroutine is called. The time samples will
then appear in the D1 array in natural sequence with the first one at the zero
subscript position. No amplitude adjustment is necessary after the inverse
transform.
Modification for Real Data
As was noted earlier, the complex formulation of the FFT tends to be
wasteful when the time samples are real. The multiplication count turns out
to be twice the promised value, which is due to the assumption of complex
data and the redundant set of computed harmonics. Fortunately, it is possible
to eliminate the wasted storage and redundant spectrum by the addition
of a real/complex conversion subroutine.
For the forward FFT, the idea is to store the even-numbered time
samples in the real part of the data array and the odd-numbered samples in
the imaginary part, thus doubling the number of samples stored. Next, the
FFT is performed as described previously. The spectrum that emerges is no
longer redundant and typically looks like a list of random numbers. At this
point, a complex-to-real conversion is performed on the spectrum, which
results in a complete, conventional spectrum of N + 1 harmonics for the 2N
data points. For convenience, the Nyquist component is stored as the sine
part of the dc component which would otherwise always be zero.
For an inverse FFT, the spectrum is first entered into the data array
normally with no duplication required. Next a real-to-complex conversion is
performed that transforms the set of real harmonics into a complex set.
Finally, the inverse FFT is executed to generate a list of 2N time sample1:
stored in even-odd order as before.
Figure 13-18 is a diagram of the computations involved in the
complex-to-real conversion. First the dc and Nyquist components are both