The chebyshev approximation problem is viewed as an optimum design criterion on the sense that the weighted approximation error between the desired frequency response and the actual frequency response is spread evenly across the pass band and evenly across the stop band of the filter minimizing the maximum error. The resulting filter designs have ripples in both pass band and stop band.

The filter design method described in the design of optimum equi ripple linear phase FIR filters is formulated as a chebyshev approximation problem.

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

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

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.

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.

If |E(ω)|≥δ for some frequencies on the dense set, then a new set of frequencies corresponding to the L+2 largest peaks of |E(ω)| are selected and computation is repeated.

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

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