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

Important application of the fft

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

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

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  • Zero-pad these two sequences to length 2 N 1 , take DFTs using the FFT algorithm x n X k h n H k The cost is O 2 N 1 2 N 1 O N N
  • Multiply DFTs X k H k The cost is O 2 N 1 O N
  • Inverse DFT using FFT X k H k y n The cost is O 2 N 1 2 N 1 O N N

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

Summary of dft

  • x n is an N -point signal ( ).
  • X k n 0 N 1 x n 2 N k n n 0 N 1 x n W N k n where W N 2 N is a "twiddle factor" and the first part is the basic DFT.

What is the dft

X k X F k N n 0 N 1 x n 2 F n where X F k N is the DTFT of x n and n 0 N 1 x n 2 F n is the DTFT of 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)

x n 1 N n 0 N 1 X k 2 N k n

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

Circular convolution

Dft

x n h n X k H k

Regular convolution from circular convolution

  • Zero pad x n and h n to length length x length h 1
  • Zero padding increases frequency resolution in DFT domain ( )

8-pt DFT of 8-pt signal
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 N -pt DFT by cleverly combining shorter DFTs
  • N -pt DFT: O N 2 O N 2 logbase --> 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 N 2 O N 2 logbase --> N

See .

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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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