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

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

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

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

In general, the filter designs that contain more than L+2 alternations or ripples are called as Extra ripple filters.

|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. Since the new set of L+2 extremal frequencies are selected to increase in each iteration until it converges to the upper bound, this implies that when |E(ω)|≤δ for all frequencies on the dense set, the optimal solution has been found in terms of the polynomial H(ω).

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

We know that we can have at most L-1 local maxima and minima in the open interval 0<ω<π. In addition, ω=0 and π are also usually extrema. It is also maximum at ω for pass band and stop band frequencies. Thus the error function of a low pass filter has at most L+3 extremal frequencies.

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

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