## 5.3. Optimal approach of equal ripple in the stop-band and passband

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:

We have seen that in equations (5.30) and (5.31), the related transfer function satisfies the formula:

Taking into account equation (5.56), equation (5.57) is written as follows:

To simplify matters, we only consider the quantity *H*_{r}(*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.

**Figure 5.18.** *Frequency response ...*