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
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
