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

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 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

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.

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.

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

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

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

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