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Introduction to DSP - filtering: FIR Filter design by window method I

So the filter coefficients for an FIR filter can be calculated simply by taking the inverse Fourier transform of the desired frequency response.

BUT...

  • The inverse Fourier transformhas to take samples of the continuous desired frequency response.
  • to define a sharp filter needs closely spaced frequency samples - so a lot of them
  • so the inverse Fourier transform will give us a lot of filter coefficients
  • but we don't want a lot of filter coefficients

We can do a better job by noting that:

  • the filter coefficients for an FIR filterare also the impulse response of the filter
  • the impulse response of an FIR filter dies away to zero
  • so many of the filter coefficients for an FIR filter are small
  • and perhaps we can throw away these small values as being less important
FIR filter

Here is a better recipe for calculating FIR filter coefficients based on throwing away the small ones:

  • 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...

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