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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.

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Important Application of the FFT

Exercise 1

How many complex multiplies and adds are required to convolute tow N-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. So the total cost = (2N1)(N+N1) 2 N 1 N N 1 ON2 O N 2

  • Now 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
  • Multiply DFTs XkHk X k H k The cost is O2N1=ON O 2 N 1 O N
  • 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 N-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 (figure1.png)

Summary of DFT

  • xn x n is an N-point signal.
Figure 2
Figure 2 (figure2.png)
  • 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 "twiddle factor".
  • 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 DTFT of xn x n and n=0N1xne(i2πFn) n 0 N 1 x n 2 F n is DTFT of xn x n at digital frequency F. 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

    XnhnXk·Hk 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 = length(x) + length(h) - 1
    • Zero padding increases frequency resolution in DFT domain
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 N-pt DFT by cleverly combining shorter DFTs
    • N-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
Figure 4
Figure 4 (figure4.png)

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