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Intro to Digital Signal Processing

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Fast Convolution Using the FFT

Module by: Robert Nowak

Summary: This module describes FFT, convolution, filtering, LTI systems, digital filters and circular convolution.

Important Application of the FFT

Exercise 1

How many complex multiplies and adds are required to convolve two NN-pt sequences? yn=∑m=0N-1xmhn-m y n m 0 N 1 x m h n m

Solution 1

There are 2N-1 2 N 1 non-zero output points and each will be computed using NN complex mults and N-1 N 1 complex adds. Therefore, Total Cost=2N-1N+N-1≈ON2 Total Cost 2 N 1 N N 1 O N 2

Zero-pad these two sequences to length 2N-1 2 N 1 , take DFTs using the FFT algorithm xn→Xk x n X k hn→Hk h n H k The cost is O2N-1log2N-1=ONlogN O 2 N 1 2 N 1 O N N
Multiply DFTs XkHk X k H k The cost is O2N-1=ON O 2 N 1 O N
Inverse DFT using FFT XkHk→yn X k H k y n The cost is O2N-1log2N-1=ONlogN O 2 N 1 2 N 1 O N N

So the total cost for direct convolution of two NN-point sequences is ON2 O N 2 . Total cost for convolution using FFT algorithm is ONlogN O N N . That is a huge savings (Figure 1).

Figure 1

Summary of DFT

xn x n is an NN-point signal (Figure 2).

Figure 2

Xk=∑n=0N-1xnⅇ-ⅈ2πNkn=∑n=0N-1xnWNkn X k n 0 N 1 x n 2 N k n n 0 N 1 x n WN k n where WN=ⅇ-ⅈ2πN WN 2 N is a "twiddle factor" and the first part is the basic DFT.

What is the DFT

Xk=XF=kN=∑n=0N-1xnⅇ-ⅈ2πFn X k X F k N n 0 N 1 x n 2 F n where XF=kN X F k N is the DTFT of xn x n and ∑n=0N-1xnⅇ-ⅈ2πFn n 0 N 1 x n 2 F n is the DTFT of xn x n at digital frequency FF. This is a sample of the DTFT. We can do frequency domain analysis on a computer!

Inverse DFT (IDFT)

xn=1N∑n=0N-1Xkⅇⅈ2πNkn x n 1 N n 0 N 1 X k 2 N k n

Build xn x n using Simple complex sinusoidal building block signals
Amplitude of each complex sinusoidal building block in xn x n is 1NXk 1 N X k

Circular Convolution

DFT

xn⊕hn↔XkHk

↔ x n h n X k H k

(1)

Regular Convolution from Circular Convolution

Zero pad xn x n and hn h n to length=lengthx+lengthh-1 length length x length h 1
Zero padding increases frequency resolution in DFT domain (Figure 3)

Figure 3

Subfigure 3.1: 8-pt DFT of 8-pt signal	Subfigure 3.2: 16-pt DFT of same signal padded with 8 additional zeros

The Fast Fourier Transform (FFT)

Efficient computational algorithm for calculating the DFT
"Divide and conquer"
Break signal into even and odd samples keep taking shorter and shorter DFTs, then build NN-pt DFT by cleverly combining shorter DFTs
NN-pt DFT: ON2↓ONlog2N O N 2 O N 2 N