The Fast Fourier Transform (FFT) is a collection of techniques that exploits symmetries in the Discrete Fourier Transform (DFT) calculation to significantly reduce the computational complexity from O(N^2) to O(NlogN). It divides the DFT calculation into smaller pieces by splitting the input sequence into even and odd parts, recursively applying this splitting to obtain a reduction in computation time. A graphical representation shows how the direct DFT calculation becomes more efficient using the FFT approach.

1. Fast Fourier Transform

2. What is it?
• A collection of “tricks” that exploit the symmetry of the DFT
calculation to make its execution much faster .
• Speedup increases with DFT size.

3. Computational Complexity of Directly Calculating
the DFT
• The N-point DFT equation for a finite-duration sequence, x(n), is given
by
• A single complex multiplication itself requires 4 real multiplications and 2 real
additions.

4. Here’s a graphical representation of what happens
when we stick with the direct DFT calculation

6. The working principle of FFT

7. Here goes the generalised approach

8. ……continued

9. Letting
• G(k) is the even-part H(k)
is the odd part as stated in the
previous slide.
- Here we roughly divided our
time by a factor of 2.

11. Further Splitting…….

12. Ex : x(n) = {1,2,3,4} (A 4-point Decimation)