1. An Iterative FFT We rewrite the loop to calculate nkyk[1] once
We refer to this calculation as a butterfly operation

2. Order of Evaluation Recursive call sequence for n = 8
The order 0,4,2,6,1,5,3,7 is a bit reversal
Start with 000,001,010,011,100,101,110,111
Reverse the bits of each binary number to obtain 000,100,010,110,001,101,011,111
We show how to calculate this shortly

3. Developing an Iterative Solution
Consider the tree structure on the prior slide
We computer the DFT for each pair of items first
We take the pairs and compute the DFT for 4 vectors
We continue until we reach the final n-element DFT
Expressing this strategy as an algorithm
We now expand the code in line 3 in more detail

4. An Intermediate Solution

5. The Final Solution

6. The Bit Reversal
Assume we have a routine rev(k) that reverses a binary numeral
We can reverse n-bits in O( lg n) time, so the overall complexity is O( n lg n)
An alternative is to store the reversed bits in a table so bit-reverse-copy runs in O( n ) time

7. Efficient FFT Complexity
The complexity is O( n lg n); to show this we must show the innermost loop (lines 9-12) runs in O( n lg n) time

8. A Parallel FFT Circuit

9. Design of the Circuit a bit reversal permutation is at the left
there are lg n stages, each stage has n/2 butterflies
the butterfly operations at each stage can be done in parallel
For each stage s from 1,2,…, lg n, there are n/2s butterfly groups with 2s-1 butterflies per group
at stage s we use where m = 2s