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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? y⁢n=∑m=0N−1x⁢m⁢h⁢n−m y n m 0 N 1 x m h n m

Solution

There are 2⁢N−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=(2⁢N−1)⁢(N+N−1)≃O⁢N2 Total Cost 2 N 1 N N 1 O N 2

  1. Zero-pad these two sequences to length 2⁢N−1 2 N 1 , take DFTs using the FFT algorithm x⁢n→X⁢k x n X k h⁢n→H⁢k h n H k The cost is O⁢(2⁢N−1)⁢log(2⁢N−1)=O⁢N⁢logN O 2 N 1 2 N 1 O N N
  2. Multiply DFTs X⁢k⁢H⁢k X k H k The cost is O⁢2⁢N−1=O⁢N O 2 N 1 O N
  3. Inverse DFT using FFT X⁢k⁢H⁢k→y⁢n X k H k y n The cost is O⁢(2⁢N−1)⁢log(2⁢N−1)=O⁢N⁢logN O 2 N 1 2 N 1 O N N

So the total cost for direct convolution of two NN-point sequences is O⁢N2 O N 2 . Total cost for convolution using FFT algorithm is O⁢N⁢logN O N N . That is a huge savings (Figure 1).

Figure 1
Figure 1 (figure1.png)

Summary of DFT

  • x⁢n x n is an NN-point signal (Figure 2).
Figure 2
Figure 2 (figure2.png)
  • X⁢k=∑n=0N−1x⁢n⁢e−(i⁢2⁢πN⁢k⁢n)=∑n=0N−1x⁢n⁢ WN k⁢n X k n 0 N 1 x n 2 N k n n 0 N 1 x n WN k n where WN =e−(i⁢2⁢πN) WN 2 N is a "twiddle factor" and the first part is the basic DFT.

What is the DFT

X⁢k=X⁢F=kN=∑n=0N−1x⁢n⁢e−(i⁢2⁢π⁢F⁢n) X k X F k N n 0 N 1 x n 2 F n where X⁢F=kN X F k N is the DTFT of x⁢n x n and ∑n=0N−1x⁢n⁢e−(i⁢2⁢π⁢F⁢n) n 0 N 1 x n 2 F n is the DTFT of x⁢n 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)

x⁢n=1N⁢∑n=0N−1X⁢k⁢ei⁢2⁢πN⁢k⁢n x n 1 N n 0 N 1 X k 2 N k n

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

Circular Convolution

DFT

x⁢n⊕h⁢n↔X⁢k⁢H⁢k ↔ x n h n X k H k (1)

Regular Convolution from Circular Convolution

  • Zero pad x⁢n x n and h⁢n h n to length=length⁢x+length⁢h−1 length length x length h 1
  • Zero padding increases frequency resolution in DFT domain (Figure 3)

Figure 3
(a) 8-pt DFT of 8-pt signal(b) 16-pt DFT of same signal padded with 8 additional zeros
Figure 3(a) (figure3-1.png)Figure 3(b) (figure3-2.png)

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: O⁢N2→O⁢N⁢log 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
  • O⁢N2→O⁢N⁢log 2 N O N 2 O N 2 N

See Figure 4.

Figure 4
Figure 4 (figure4.png)

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