of Microprocessors

668 MUSICAL ApPLICATIONS OF MICROPROCESSORS
insure that the initial input data values are small enough so that after
transformation they will still fit into the chosen data wordlength. This is
perfectly feasible for a dedicated function single block size FFT routine (or
hardware implementation of such a routine) but could get out of hand in a
routine designed for a wide range of possible block sizes. Another possibility
is to scan the data array after each iteration and divide all values by two.
Unfortunately, this sacrifices small signal accuracy, when freedom from
processing noise is needed most, in exchange for large signal-handling
capability. Yet a third is to scale the calculations so that overflow is a distinct
possibility and test for it during each iteration. Ifoverflow does occur, divide
the data array by two, recording the number of times this is done, and
continue. This technique is called conditional array scaling and has the best
tradeoff between accuracy and word size but suffers from additional
complexity and an indefinite execution time as well as burdening successive
processing steps with data having a variable scale factor.
Since the 68000 readily handles 32-bit data values, the brute-force
technique mentioned first above will be used. The intial input data,
intermediate results stored in the data arrays, and the output data will be 32­
bit signed integers. The SIN and COS multipliers will be obtained from a
table as 16-bit unsigned binary fractions. When multiplied by the 32-bit data
values, a 48-bit product is obtained. The intermediate calculations within a
node are perfotmed using these 48-bit values, but the final node result that is
stored back into the data array is just the most significant 32 bits. The sign of
the COS multiplier is handled by conditionally adding or subtracting the
product according to the angle. The SIN multiplier is always positive, since
the angle is always between 0 and 180°. For maximum accuracy, SIN or COS
multipliers of unity (1. 0) are represented by a special value in the table, and
multiplication is skipped when they are seen. Zero multipliers are also
skipped, which gives a noticeable speed increase for the smaller record sizes,
where zero and unity multipliers are proportionately abundant.
Figure 18-8 shows the assembled 68000 FFT subroutine listing.
Arguments and instructions for its use are given in the first few lines and
should be self-explanatory. When filling the data arrays with 32-bit values, it
is desirable to shift the data left as far as allowable to maximize accuracy. For
example, if the original data are signed 16-bit integers and the transform size
is 512 complex points, the 16-bit data can be sign extended to 32 bits and
then shifted left seven positions before being stored into the data arrays. This
leaves 9-bit positions available for the nine doublings incurred by the nine
FFT iterations needed for 512 points. However, if real data is being
transformed and this routine is followed by a real transformation routine,
only six shifts should be done because the real transformation also tends to
double the array values. The subroutine is completely self-contained except
for an external cosine table (not shown but instructions for its generation are
included), which can be shared with other synthesis routines. Since all