If the filter has a anti-symmetric unit sample response, then we know that

According to Alternation theorem, a necessary and sufficient condition for P(ω) to be unique, best weighted chebyshev approximation, is that the error function E(ω) must exhibit at least L+2 extremal frequencies in S.

Having the solution for P(ω), we can now compute the error function E(ω) from

If the filter has a anti-symmetric unit sample response, then we know that

The error function E(ω) alternates in sign between two successive extremal frequency, Hence the theorem is called as Alternative theorem.

Let us consider the design of a low pass filter with the pass band edge frequency ωP and the ripple in the pass band is δ1, then from the general specifications of the chebyshev filter, in the pass band the filter frequency response should satisfy the condition

FX denotes an array of maximum size 10 that specifies the weight function in each band.

Initially, we neither know the set of external frequencies nor the parameters. To solve for the parameters, we use an iterative algorithm called the Remez exchange algorithm, in which we begin by guessing at the set of extremal frequencies.

In general, the filter designs that contain maximum number of alternations or ripples are called as maximal ripple filters.

Let us consider the design of a low pass filter with the stop band edge frequency ωs and the ripple in the stop band is δ2, then from the general specifications of the chebyshev filter, in the stop band the filter frequency response should satisfy the condition