The weighting function on the approximation error allows to choose the relative size of the errors in the different frequency bands. In particular, it is convenient to normalize W(ω) to unity in the stop band and set W(ω)=δ2/δ1 in the pass band.

The real valued desired frequency response Hdr(ω) is simply defined to be unity in the pass band and zero in the 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.

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

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

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

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

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.

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

The weighted approximation error is defined as E(ω) which is given as