Finding an optimal solution to the problem of the amplitude approximation of specifications is obtained by minimizing a distance criterion between the theoretical frequency responses and those brought about by synthesis.
To present this approach, we will consider a low-pass linear phase FIR filter of type II (see equations (5.30) and (5.31)); that is, with a symmetrical impulse response and a choice of N odd response.
We then introduce the quantities:
To simplify matters, we only consider the quantity Hr(f), which determines the amplitude:
The problem is in estimating the coefficients b(n) so that the frequency response is optimal by distributing the approximation error in the passband and the attenuated band.