Radix-2 decimation in frequency FFT
Procedure and significant points
1. The number of input samples is N = 2ν
here ν is an integer.
2. The input sequence is in natural order; output is in the bit reversed order.
3. The number of stages in flow graph can be given by ν =log2N.
4. Each stage comprises of N/2 butterflies.
5. Inputs/outputs for each butterfly are separated in reverse order from that of DIT.
The separation decreases from left to right in order N/2, ... , 4, 2, 1.
6. The number of complex additions = 
and number of complex multiplications
can be given by
.
7. The basic calculation block in flow graph of the DIF FFT is butterfly shown
here. This is an in-place computation in that the 2 outputs (A + B) and (A - B) 
can be computed and stored in same locations as A and B.
Example 3.3.1 Radix-2, 8-point, decimation in frequency FFT for the
sequence
n→ 0 1 2 3 4 5 6 7
x(n) = {1, 2 3 4 -4 -3 -2 -1}
Solution The twiddle factors are same as in the DIT FFT done earlier :
One of the elementary computations is shown as follows:
The DFT is X(k) = {0, (5 - j12.07), (-4 + j4), (5 - j2.07), -4, (5 + j2.07), (-4 - j4), (5 + j12.07)}
The MATLAB progarm is same as shown in the Example 1.
(DIT Template)
The elementary calculation (Butterfly):
The signal flow graph:
(DIF Template)
The elementary computation (Butterfly):
The signal flow graph:
The signal flow graph:
16-point DIF FFT
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