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

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

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

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

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 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.

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.

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

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

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