FAST FOURIER TRANSFORM ALGORITHMS UNIT-II FAST FOURIER TRANSFORM ALGORITHMS

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

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+p2x2+ ... +pn-1xn-1

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

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+p4w4+ ... +pn-1wn-1 Even/Odd Separation P(w)= P1(w)+P2(w) P1(w)=p0+p2w2+p4w4+ ... +pn-2wn-2 P1(w)=Pe (w2)=p0+p2w+p4w1+...+pn-2w(n-2)/2

Even/Odd Separation Contd. P(w)= P1(w)+P2(w) P1(w)=p0+p2w2+p4w4+ ... +pn-2wn-2 P1(w)=Pe (w2)=p0+p2w+p4w1+...+pn-2w(n-2)/2 P2(w)=p1w+p3w3+p5w5+ ... +pn-1wn-1 P2(w)= w P3(w)=p1+p3w2+... +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

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

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)

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], …

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

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

FFT Algorithms - 8 point FFT