of Microprocessors

672 MUSICAL ApPLICATIONS OF MICROPROCESSORS
temporary storage is in the machine registers (a big advantage of the 68000)
and the stack, the routine could even be put into ROM if desired.
In general, the subroutine is a literal translation of the BASIC program
of Fig. 13-17. However, the first portion, in which the data arrays are
scrambled into bit-reversed order, has been redone. Although the algorithm
used in the BASIC version is reasonably efficient for a high-level language
implementation, using bit manipulation instructions is much faster in
assembly language. Essentially, two counters are incremented from 0 to N-l,
one normal and one bit-reversed, and the corresponding array elements are
swapped whenever the bit-reversed counter is larger.
The second portion is reasonably straightforward except when multiplication
by a sine or cosine is performed. What is actually needed is a 16-bit
unsigned X 32-bit signed multiply to yield a 48-bit signed product,
whereas the 68000 only provides 16 X 16-bit multiplies where both
operands are either signed or unsigned. The desired result is achieved in the
TRGMPY subroutine by treating the 32-bit signed multiplicand as an
unsigned value, performing a 32 X 16-bit unsigned multiply using two of
the 16 X 16-bit machine instructions, and then correcting the product by
subtracting the unsigned multiplier from the most significant 16 bits of the
48-bit product if the multiplicand was negative. This routine also bypasses
multiplication completely when the multiplier is zero or unity. If an inverse
transform is required, the instruction in line 106 can be changed to a BSET
instruction, which has the effect of negating the SIN multipliers.
For efficient transformation of real data, a complex-to-real transform
routine can be coded using these same techniques and the BASIC version in
Fig. 13-19 as a guide. Note that the complex-to-real transform uses cosine
and sine angle increments half the size of the corresponding complex FFT.
Thus, to acrually transform 4096 real points using a 2048-point complex
transform, either the cosine table will have to be made twice as large (and the
given subroutine slightly modified to accommodate it), or the complex-toreal
routine will have to interpolate in the given table. Of course, this
problem does not arise for smaller transforms.
As coded, this subroutine gives results as accurate as 16-bit input data
allow. For many analysis tasks, particularly those in which the final result is a
plot or display, less accuracy is needed. This routine (and a companion
complex-to-real routine) could in fact be rewritten for the 12-bir data and
multipliers, which would probably increase the speed two to three times for
transform sizes up to 256 points. Real-time analysis of speech bandwidth
data then would be possible even on an 8-MHz 68000.
NOTRAN Music System
After coming this far, it is about time to see how the various synthesis
and processing techniques can be put together to form a usable music