Fast Fourier Transforms on Motorola's Digital Signal Processors

F( k)
a real sequence with N points, can be found according
to Eqn. 6-9. Combining Eqn. 6-8 and Eqn. 6-9,
we obtain the final equation Eqn. 6-10.
[ Z( k)
+ Z∗( N⁄ 2–
k)
] Z( k)
– Z∗ N⁄ 2 k
---------------------------------------------------- j
2
–
[ ( ) ]
=
– --------------------------------------------------W rk
2
N
where: k = 0,1,...(N/2)-1,
N = Number of real inputs
Notice that:
• Only 0 to N/2-1 points are saved by the
algorithm.
Eqn. 6-10
• The values F(0) and F(N/2) are real and
independent, to obtain entire spectrum, F(N/2)
in the imaginary part of F(0).
• The twiddle factors in the DFT and split complex
multiplication have different resolutions. In the
DFT, the period of W is N/2; in the split complex
multiplication, the period of W is N, though the
same number of points (N/2) are needed in both
cases. This means the algorithm may use more
memory space for twiddle factors.
Eqn. 6-10 can be decomposed further to a real multiplication
format that can be implemented on
DSPs.
6-12 MOTOROLA