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Fast Convolution

Module by: Richard Baraniuk

Summary: This module is an overview of concepts behind fast convolution of LSI systems.

  • Cost of convolution y=x*h y x h for all ∈ ℝ N N . y=Hx y H x If support ( x=N x N ) and support ( h=L≤N h L N ) then the number of adds and the number of multiplies =OLN O L N with the worst case being ON2 O N 2
  • Cost of x → 𝔽 H X x → 𝔽 H X h→H h H by 2 FFTS, then Y=HX Y H X then _ → 𝔽 y _ → 𝔽 y by inverse fft. Number of adds and number of multiplies is =ONlogN O N N
  • So if L L is large, it is faster via FFT! In matlab:
    y=ifftfftxffth y ifft fft x fft h (1)

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