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Introduction to DSP - filtering: FIR Filter design by window II

FIRfilter coefficients can be calculated by taking the inverse Fourier transform of the desired frequency response and throwing away the small values:

  • pretend we don't mind lots of filter coefficients
  • specify the desired frequency response using lots of samples
  • calculate the inverse Fourier transform
  • this gives us a lot of filter coefficients
  • so truncate the filter coefficients to give us less
  • then calculate the Fourier transform of the truncated set of coefficients to see if it still matches our requirement

BUT...

Truncating the filter coefficients means we have a truncated signal. And a truncated signal has a broad frequency spectrum:

truncated signal

So truncating the filter coefficients means the filter's frequency response can only be defined coarsely.

Luckily, we already know a way to sharpen up the frequency spectrum of a truncated signal, by applying a window function. So after truncation, we can apply a window function to sharpen up the filter's frequency response:

Window applied to a truncated signal

So here is an even better recipe for calculating FIR filter coefficients:

  • pretend we don't mind lots of filter coefficients
  • specify the desired frequency response using lots of samples
  • calculate the inverse Fourier transform
  • this gives us a lot of filter coefficients
  • so truncate the filter coefficients to give us less
  • apply a window function to sharpen up the filter's frequency response
  • then calculate the Fourier transform of the truncated set of coefficients to see if it still matches our requirement

This is called the window method of FIR filter design.

BUT...

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