of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 457
derived from the de term in the complex spectrum according to the "/'
function. Next the pairs of harmonics that would have been conjugates if the
even-odd storage arrangement were not done are combined according to the
"g" function to produce a pair of real harmonics. Finally, the "h" function is
applied to the remaining N/2th spectral component. Note that cosines and
sines of angle increments half the size of the smallest increment used in the
FFT are utilized in evaluating g much like an extra step in the FFT. In fact,
the whole procedure can be regarded as an extra column of nodes in the FFT
algorithm.
Figures 13-19 and 13-20 are BASIC subroutines for complex-to-real
and real-to-complex conversion, respectively. The complex-to-real routine
also scales the spectrum output so that all of the harmonics, including the de
component, are of the correct amplitude. Except for scaling, the real-tocomplex
routine undoes what the complex-to-real routine does.
The conversion adds essentially 2N multiplications to the FFT, which
gives a total of 2N+2NlogzN for 2N data points. IfM = 2N = the number
of real data points, this is equal to MlogzM multiplications, which is the
4000 REM COMPLEX TO REAL TRANSFORMATION FOR FOURIER TRANSFORM
4001 REM IF REAL DATA POINTS ARE ALTERNATELY STORED IN D1 AND 02
4002 REM ARRAYS, I.E. TO->01(0), T1->02(0), T2->01(I), T3->02(1),
4003 REM THEN THIS ROUTINE CONVERTS THE COMPLEX SPECTRUM INTO A
4004 REM REAL COSINE-SINE SPECTRUM.
4005 REM THE ROUTINE ALSO DOES AMPLITUDE CORRECTION ON THE DC
4006 REM COMPONENT AND THE HARMONICS SO THAT THE FINAL SPECTRUM OUTPUT
4007 REM IS THE TRUE SPECTRUM OF THE WAVE.
4008 REM THE SINE PART OF THE DC COMPONENT IS SET EQUAL TO THE NYQUIST
4009 REM COMPONENT AND SHOULD BE NEAR ZERO IF THE SAMPLE RATE OF THE
4010 REM DATA WAS ADEQUATE.
4011 REM THIS ROUTINE USES THE SAME INPUT AS THE FFT ROUTINE
4100 N=2**K
4200 REM COMPUTE DC AND FOLDOVER COMP6NENTS
4201 Tl=D1(0)
4202 T2=D2(0)
4203 D1(0)=(T1+T2)/(2*N)
4204 D2(0)=(T1-T2)/(2*N)
4300 REM COMPUTE REMAINDER OF FREQUENCY COMPONENTS
4301 FOR N1=1 TO N/2
4302 N2=N-N1
4303 C=COS(-3.1415927*N1/N)
4304 S=SIN(-3.1415927*N1/N)
4305 T1=(D1(N1)+01(N2))/2
4306 T2=(D1(N1)-D1(N2))/2
4307 T3=(D2(N1)+D2(N2))/2
4308 T4=(D2(N1)-D2(N2))/2
4309 T5=T2*S-T3*C
4310 T6=T2*C+T3*S
4311 D1(N1)=(T1-T5)/N
4312 D1(N2)=(T1+T5)/N
4313 D2(N1)=(T4-T6)/N
4314 D2(N2)=(-T4-T6)/N
4315 NEXT N1
4316 RETURN
Fig. 13-19. Complex-to-real spectrum transformation routine in BASIC