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

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.

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

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

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

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

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.

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.

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