Name: FFTR2B.ASM
         Type: Assembler Macro
      Version: 1.1
  Last Change:  2-Oct-86

  Description: Radix 2, In-place, Decimation-in-time Complex FFT Macro

 This macro performs a complete Fast Fourier Transform (FFT) on complex
 data.  The basic algorithm is the Decimation-in-time (DIT), Radix 2
 FFT algorithm using 24 bit fixed-point arithmetic.  The algorithm uses
 a sine-cosine lookup table for the FFT coefficients (twiddle factors).
 The macro can be called to perform any FFT from 4-32768 points.  Simply
 call it with the arguments of number of FFT points, location of the
 data array and location of the sine-cosine table.  All register
 initialization is performed by this macro.  However, the macro assumes
 that registers which should not be altered by the FFT have already been
 saved by the main program.  This allows the user to fit the FFT macro
 into his application and thus control the context switching overhead.
 No data scaling is performed and no overflow detection is done.
 Modifications to this routine could allow it to be used with the
 scaling modes and thus allow dynamic scaling for each FFT pass.

 All data and coefficients are complex, with the real part in X Data
 memory and the imaginary part in Y Data memory.  For an N point FFT,
 the data buffer requires N X Data and N Y Data memory locations.
 The algorithm is performed "in-place", meaning that only one data
 buffer is required for both input and output data.  The input
 data is assumed to be in normal (time-sequential) order and the
 output is in bit-reversed order.  By using the reverse-carry
 address modifier and a separate output data buffer, the output
 data may be easily unscrambled.  Other methods also exist to
 unscramble the output data without a separate output data buffer. 
 The FFTR2B macro uses "twiddle factors" (-cosine and -sine tables)
 stored in data memory.  For maximum speed, the FFT macro performs
 a lookup table operation to get new sine and cosine values for
 each group of butterflies.  A SINCOS macro is available to
 generate these tables.  For an N point FFT, N/2 X Data and N/2
 Y Data locations are required.  Sine and cosine values could be
 calculated in real-time to save data memory at the expense of
 execution time.

 The FFTR2B macro is slightly faster than the FFTR2A library macro.
 The speed increase is obtained by splitting the last pass out from
 the triple nested DO loop and giving it a separate DO loop.  The
 reason this is faster is that the FFTR2A inner loop is started
 with a loop count of 1 on the last pass.  Note that the separate last
 pass DO loop uses different addressing modes to increment through the
 butterflies, thus avoiding outer loops.  Additional details are
 included in the source file; however, more algorithm description
 would be required for complete understanding by typical users.  The
 FFTR2B macro can directly replace the FFTR2A macro using the
 calling procedure demonstrated in the FFTR2AT test program.  A
 summary of performance using a 20.5 MHz clock is given below.

 Complex
 Points     Int.P,X,Y            Int.P,Ext.X,Y            Ext.P,X,Y
 ------     ---------            -------------            ----------
    16      0.032 msec           0.048 msec               0.072 msec
    64      0.148                0.238                    0.369
   256      0.712                1.175                    1.849
  1024     (3.413)               5.661                    8.958
  4096     (16.01)               26.59                    42.18
 16384     (73.55)               122.3                    194.2

where ( ) indicates not possible with internal DSP56000/1 data memory.