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

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

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.

If |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.

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.

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

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

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