Filter Design Toolbox

Optimal Filter Design Solutions

We have stated that the optimal filter design problem is to find the filter whose magnitude response, |H()|, minimizes

for a given , p, W() and D(). You can use both FIR and IIR filters to meet this requirement.

For the FIR case, with p equal to , and the additional constraint that the filter must have linear phase, you can use a very efficient design method, based on the Remez exchange algorithm to determine the optimal solution.

Function gremez in the toolbox implements this method. Additionally, gremez provides optional calling syntaxes that enable variations and enhancements to the filter design problem.

To design optimal FIR solutions in the general case where p is not necessarily equal to infinity, the toolbox includes the function firlpnorm. You may find this useful in cases where minimax solutions lead to abrupt time-domain responses. firlpnorm does not use the Remez exchange algorithm and generally takes longer to design a filter than gremez and other filter design functions. Moreover, firlpnorm is not constrained to linear phase filters.

Note that Signal Processing Toolbox provides the function firls, an efficient FIR linear phase solution to the optimal filter design problem in the least-squares sense, that is, when p equals 2.

IIR solutions to the optimal filter design problem are more involved than their FIR counterparts. Filter Design Toolbox offers two functions that design IIR filters that are optimal in the least-p norm sense: iirlpnorm and iirlpnormc.

iirplnorm uses a somewhat faster, unconstrained algorithm, while iirplnormc uses a constrained algorithm that designs an optimal filter that meets the specifications while restricting the maximum radius of its poles to a specified value less than one.

Elliptic filters, such as those you use the function ellip (in Signal Processing Toolbox) to design, are optimal IIR filters for the case p equals infinity, when the desired magnitude response is piecewise constant, and the filter numerator and denominator orders are the same.

The Parks-McClellan method, which implements the Remez exchange algorithm, produces a filter design that just meets your design requirements, but does not exceed them. In many instances, when you use the window method to design a filter, the result is a filter that performs too well in the stopband. This wastes performance and taxes computational power by using more filter coefficients than necessary. When you use a rectangular window in the window design method, the resulting filter can be shown to be the optimal, unweighted least squares solution to the filter design problem. In summary, the optimal solution is not always a good solution to the filter design problem.

Filters designed using the Parks-McClellan method have equal ripple in their passbands and stopbands. For this reason, they are often called equiripple filters. They represent the most efficient filter designs for a given specification, meeting your frequency response specification with the lowest order filter.

Optimal Filter Design Theory

Advanced FIR Filter Designs