of Microprocessors

458 MUSICAL ApPLICATIONS OF MICROPROCESSORS
5000 REM REAL TO COMPLEX TRANSFORMATION FOR INVERSE FOURIER TRANSFORM.
5001 REM THIS ROUTINE CONVERTS A REAL COSINE-SINE SPECTRUM INTO A
5002 REM COMPLEX SPECTRUM THAT WHEN INVERSE FOURIER TRANSFORMED WILL
5003 REM PRODUCE REAL DATA POINTS STORED SUCH THAT EVEN NUMBERED
5004 REM POINTS ARE IN THE REAL PART OF THE DATA ARRAY AND ODD
5005 REM NUMBERED POINTS ARE IN THE IMAGINARY PART OF THE ARRAY.
5006 REM THIS ROUTINE IS THE INVERSE OF THE COMPLEX TO REAL ROUTINE
5007 REM EXCEPT FOR SCALING WHICH HAS BEEN SET FOR THE PROPER OUTPUT.
5008 REM THIS ROUTINE FOLLOWED BY AN INVERSE FOURIER TRANSFORM IS THE
5009 REM EXACT INVERSE OF A FORWARD FOURIER TRANSFORM FOLLOWED BY THE
5010 REM COMPLEX TO REAL ROUTINE.
5011 REM THIS ROUTINE USES THE SAME INPUT AS THE FfT ROUTINE
5100 N=2**K
5200 REM RESTORE DC AND FOLDOVER COMPONENTS
5201 Tl=D1(0)
5202 T2=02(0}
5203 01(O}=T1+T2
5204 D2(O}=T1-T.2
5300 REM COMPUTE REMAINDER OF FREQUENCY COMPONENTS
5301 FOR N1=1 TO N/2
5302 N2=N-N1
5303 C1=COS(-3.1415927*Nl/N}
5304 Sl=SIN(-3.1415927*N1/N}
5305 T1=(Ol(N1)+D1(N2}}/2
53Q6 T4=(D2(N1}-02(N2})/2
5307 T5=(D1(N2)-D1(N1}}/2
5308 T6=( -02 (N1) -02 (N2) )/2
5309 T2=T5*Sl+T6*C1
5310 T3=T6*Sl-T5*C1
5311 D1(N1}=T1+T2
5312 01(N2}=T1-T2
5313 D2(N1}=T3+T4
5314 D2(N2)=T3-T4
5315 NEXT N1
5316 RETURN
Fig. 13-20. Real-to-complex spectrum conversion routine in BASIC
promised value. With clever programming to avoid multiplication by zero
and one in the FFT and real conversion routines (especially in the first
column of the FFT), one can ultimately reach a lower limit of
M[(logzM)-2] multiplications, which is a significant reduction when Mis
small.
Using the FFT for Synthesis
Although the greatest value of the FFT is in sound analysis and modification,
it is covered in this chapter because it is also a powerful "synthesisfrom-scratch"
technique. Unfortunately, its record-oriented properties complicate
application to changing spectra with arbitrary frequency components.
However, if the required spectral detail is great, the FFT can be much more
efficient in computing samples than a direct Fourier series evaluation in spite
of these complications.
In all FFT-based synthesis procedures, the general idea is to compute a
sequence of sample records using the FFT and then combine them sequen-