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# Fast Convolution Using the FFT

Module by: Robert Nowak. E-mail the author

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=0N1xmhnm y n m 0 N 1 x m h n m

#### Solution

There are 2N1 2 N 1 non-zero output points and each will be computed using NN complex mults and N1 N 1 complex adds. Therefore, Total Cost=(2N1)(N+N1)ON2 Total Cost 2 N 1 N N 1 O N 2

1. Zero-pad these two sequences to length 2N1 2 N 1 , take DFTs using the FFT algorithm xnXk x n X k hnHk h n H k The cost is O(2N1)log(2N1)=ONlogN O 2 N 1 2 N 1 O N N
2. Multiply DFTs XkHk X k H k The cost is O2N1=ON O 2 N 1 O N
3. Inverse DFT using FFT XkHkyn X k H k y n The cost is O(2N1)log(2N1)=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).

## Summary of DFT

• xn x n is an NN-point signal (Figure 2).
• Xk=n=0N1xne(i2πNkn)=n=0N1xn WN kn 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=0N1xne(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=0N1xne(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=1Nn=0N1Xkei2πNkn 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

xnhnXkHk 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+lengthh1 length length x length h 1
• Zero padding increases frequency resolution in DFT domain (Figure 3)

### 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: ON2ONlog 2 N O N 2 O N 2 N

### Fast Convolution

• Use FFT's to compute circular convolution of zero-padded signals
• Much faster than regular convolution if signal lengths are long
• ON2ONlog 2 N O N 2 O N 2 N

See Figure 4.

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