of Microprocessors

MUSIC SYNTHESIS SOFTWARE 667
also possible to shift the baseline of the waveform t9 force posltlve and
negative symmetry and thus insure use of all 256 quantization levels in the
waveform table. Unfortunately, this may also introduce a significant de
component to the waveform, which may cause trouble, such as a "thump"
when a fast amplitude envelope is applied.
Fast Founer Transform for the 68000
For the last example, we will examine a translation of the complex FFT
subroutine described in Chapter 13 from BASIC into 68000 assembly
language. The main practical reason for being interested in an assembly
language FFT is its much greater speed relative to one in a high-level
language. For example, to transform 1024 complex data points (which could
be 2048 real data points if a real transformation routine was also written),
only 900 msec are required on a typical-speed 68000 compared to a minute
or more for the BASIC program running on most systems. With smaller
record sizes of 64 complex (128 real) points and a fast (l2-MHz) 68000, one
could approach real-time analysis of speech bandwidth (8 ks/s) data! For any
practical use with real-world data, the accuracy of the routine will be
equivalent to one using floating-point arithmetic.
Before describing the workings of the routine itself, let's examine the
scaling properties of the FFT algorithm. First refer back to Fig. 13-15, in
which the calculations involved in each "node" of the FFT were diagrammed.
These calculations are all of the form:
(NEWDATA) = (OLDDATA1) + (OLDDATA2) X COS(A) +
(OLDDATA3) X SIN(A)
where A is some angle, OLDDATA1, 2, and 3 are current values in the data
array, and NEWDATA is a new value stored back into the data array. Doing a
range analysis where the ranges of OLDDATA 1, 2, and 3 are all the same, we
find that the range of NEWDATA is at most 2.41 times larger than the
OLDDATA ranges, which could occur when the angle A is 45°. In actual use
inside an FFT, it turns out that the range of NEWDATA is only twice that of
the OLDDATA values. The point is that, since all of the array data
participates in calculations of this type for each column of nodes in the FFT
butterfly diagram, it follows thar" the range for the array doubles as each
column of nodes is evaluated. Since there are LOG2N node columns in an N­
point FFT calculation, the range of the final result array is simply N times
the range of the original data array. In actuality, this theoretical result range
will only be approached when the input signal contains a single-frequency
component near full-scale amplitude and it has not been windowed.
There are numerous ways to handle this tendency for data values to
double every iteration through the FFT algorithm. The first is to simply