An optimal filter can be obtained by using the Remez Exchange Algorithm to find the coefficients for the filter from a minimized Chebyshev approximation in the form:

 

(1)        

 

Where and the coefficients are chosen to yield an optimal .

 

The desired response for an Ideal Filter is given by:

 

(2)        

 

where is the set of bandpass frequencies and is the set of bandstop frequencies. Thus an optimal approach can be defined the as one that minimizes the maximum error given by:

 

 

However, this error treats bandpass and bandstop errors alike. For a more general approach, a weight function is included:

 

(3)        

 

Which allows bandstop errors to be more important than bandpass errors (or vice versa, if a larger weight is given to bandpass errors). Thus, the error function is defined as:

 

 

Then we use Chebyshev approximation to find the coefficients in (1) above that minimize . In turn, the Remez Exchange Algorithm is the method by which to arrive to the optimal .

 

The Alternation Theorem states that the impulse response given by equation (1) above will be a unique, best-weighted Chebyshev approximation to the desired frequency response if and only if, the error function exhibits at least extrema at frequencies in . The frequencies at which extrema occur are called Extremal Frequencies. So let denote the extremal frequency so that

 

 

Then it can be proven that

(4)        ,

(5)        

 

Together, equations (4) and (5) mean that the error is equal at all extremal frequencies. In addition, Equation (4) indicates that maxima and minima alternate, hence, the alternation theorem.

 

The alternation theorem tells us how to recognize an optimal set of for equation (1) when we have one, but it does not say how to obtain the . The Remez Exchange Algorithm provides an approach for finding the set of coefficients corresponding to this optimization.