Radix-2 decimation-in-time FFT (Cooley-Tukey)
Procedure and important points
1. The number of input samples is N = 2ν here ν is an integer.
2. The input sequence is shuffled through the bit-reversal. The index n of sequence x(n) can be expressed in binary and then reversed.
3. The number of stages in flow graph can be given by ν =log2N .
4. Every stage consists of N/2 butterflies.
5. Inputs/outputs for each of the butterfly are separated as follows:
Separation = 2m-1 samples where m = stage index, stages being numbered from left to right (i.e., m = 1 for stage 1, m = 2 for stage 2 etc.).
This amounts to the separation increasing from left to right in order 1, 2, 4, ..., N/2.
6. The number of complex additions
and the number of complex multiplications is 
7. The elementary computation block in flow graph, called as butterfly, is shown here.
This is an in-place calculation in which the outputs
can be calculated and stored in same locations as A and B.
Example: Radix-2, 8-point, decimation-in-time FFT for sequence
Solution
The twiddle factors are given below
One of the elementary calculations is shown below:
Solution The twiddle factors are given below
One of the elementary computations is shown as follows:
The signal flow graph is 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 program:
x = [1, 2, 3, 4, -4, -3, -2, -1], X = fft(x)
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