1. FAST FOURIER TRANSFORM ALGORITHMS
UNIT-II
FAST FOURIER TRANSFORM ALGORITHMS

2. INTRODUCTION
All Periodic Waves Can be Generated by Combining Sin and Cos Waves of Different Frequencies
Number of Frequencies may not be finite
Fourier Transform Decomposes a Periodic Wave into its Component Frequencies

3. DFT Definition
Sample consists of n points, wave amplitude at fixed intervals of time: (p0,p1,p2, ..., pn-1) (n is a power of 2)
Result is a set of complex numbers giving frequency amplitudes for sin and cos components
Points are computed by polynomial: P(x)=p0+p1x+p2x pn-1xn-1

4. DFT Definition, continued
The complete DFT is given by P(1), P(w), P(w2), ... ,P(wn-1)
w Must be a Primitive nth Root of Unity
wn=1, if 0<i<n then wi ¹ 1

5. Divide and Conquer Method
Compute an n-point DFT using one or more n/2-point DFTs
Need to find Terms involving w2 in following polynomial
P(w)=p0+p1w+p2w2+p3w3+p4w pn-1wn-1
Even/Odd Separation
P(w)= P1(w)+P2(w)
P1(w)=p0+p2w2+p4w pn-2wn-2
P1(w)=Pe (w2)=p0+p2w+p4w1+...+pn-2w(n-2)/2

6. Even/Odd Separation Contd.
P(w)= P1(w)+P2(w)
P1(w)=p0+p2w2+p4w pn-2wn-2
P1(w)=Pe (w2)=p0+p2w+p4w1+...+pn-2w(n-2)/2
P2(w)=p1w+p3w3+p5w pn-1wn-1
P2(w)= w P3(w)=p1+p3w pn-1wn-2
P3(w)=Po(w2)= p1+p3w+... +pn-1w(n-2)/2
P(w)= Pe(w2)+ wPo(w2)
Pe & Po come from n/2 point FFTs

7. Fast Fourier Transform Algorithms
Consider DTFT
Basic idea is to split the sum into 2 subsequences of length N/2 and continue all the way down until you have N/2 subsequences of length 2
Log2(8) N

8. Radix-2 FFT Algorithms - Two point FFT
We assume N=2^m
This is called Radix-2 FFT Algorithms
Let’s take a simple example where only two points are given n=0, n=1; N=2
Butterfly FFT y0 y0 y1
Advantage: Less computationally intensive: N/2.log(N)

9. General FFT Algorithm First break x[n] into even and odd
Let n=2m for even and n=2m+1 for odd
Even and odd parts are both DFT of a N/2 point sequence
Break up the size N/2 subsequent in half by letting 2mm
The first subsequence here is the term x[0], x[4], …
The second subsequent is x[2], x[6], …

10. Example
Let’s take a simple example where only two points are given n=0, n=1; N=2
Same result

11. FFT Algorithms - Four point FFT
First find even and odd parts and then combine them:
The general form:

12. FFT Algorithms - 8 point FFT