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
BUT...
