of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 449
With these conventions for representing everything as complex numbers,
the SFT coefficient matrix can be rewritten as in Fig. 13-13. The
numbers listed in the matrix are arguments to the W function mentioned
earlier needed to generate the proper complex multiplier. The operations
involved in going from time to frequency and back are the same as before,
except all arithmetic is now complex. In case one has forgotten (or never
knew), the basic arithmetic operations on two complex numbers, A,B and
C,D, are given below:
A,B + C,D = A+C,B+D
A,B C,D = A-D,B-D
A,B X Cp = AC-BD,AD+BC
A,B _ AC+BD
C,D C 2 +D 2 '
The nice thing about the complex form of the SFT is that it is completely
symmetrical. There are no special cases for dc or Nyquist components,
and the form of the output is exactly the same as the form of the input.
The FFT Algorithm
At this point, we are ready to look at the FFT algorithm itself. Actually,
there are many different algorithms, each with an edge over the others
in certain applications. They all, within roundoff error, give the same result,
however. The one that will be described here is one of the most straightforward
and has the desirable property that sines and cosines (the W function)
are used in ascending order. In technical terms, it is a radix 2 FFT using time
decomposition and input bit reversal. Radix 2 means that the record size
must be a power of two. Other radices are possible, but two is the most
efficient, particularly in a binary computer.
The first step in the FFT is to completely decompose the input samples,
which means that they are scrambled in bit-reversed order as in Fig. 13-12.
Fortunately, the scrambling can be accomplished without need for a second
array. One simply programs two counters, one normal and one bit-reversed,
and iterates them from 0 to N -1. Whenever the bit-reversed counter is
numerically greater than the normal counter, the corresponding samples are
swapped. This is illustrated in Fig. 13-14 for N= 16, but the procedure
applies for any record size as long as it is a power of two. In assembly
language on a microprocessor, on~ could use just one counter and write a
bit-reversal function, which would take a binary number and return its
bit-reversed equivalent.