History of Optimal FIR Filter Design
In the 1960s, researchers within the field of analog filter design were using the Chebyshev approximation for filter design. During this time, it was well known that the best filters contain an equiripple characteristic in their frequency response magnitude and the elliptic filter (or Cauer filter) was optimal with regards to the Chebyshev approximation. When the digital filter revolution began in the 1960s, researchers used a bilinear transform to produce infinite impulse response (IIR) digital elliptic filters. They also recognized the potential for designing FIR filters to accomplish the same filtering task and soon the search was on for the optimal FIR filter using the Chebyshev approximation.
It was well known in both mathematics and engineering that the optimal response would exhibit an equiripple behavior and that the number of ripples could be counted using the Chebyshev approximation. Several attempts to produce a design program for the optimal Chebyshev FIR filter were undertaken in the period between 1962 and 1971. Despite the numerous attempts, most did not succeed, usually due to problems in the algorithmic implementation or problem formulation. Otto Herrmann, for example, proposed a method for designing equiripple filters with restricted band edges. This method obtained an equiripple frequency response with the maximum number of ripples by solving a set of nonlinear equations. Another method introduced at the time implemented an optimal Chebyshev approximation, but the algorithm was limited to the design of relatively low-order filters.
Similar to Herrmann's method, Ed Hofstetter presented an algorithm that designed FIR filters with as many ripples as possible. This has become known as the Maximal Ripple algorithm. The Maximal Ripple algorithm imposed an alternating error condition via interpolation and then solved a set of equations that the alternating solution had to satisfy. One notable limitation of the Maximal Ripple algorithm was that the band edges were not specified as inputs to the design procedure. Rather, the initial frequency set {ωi} and the desired function D(ωi) defined the pass and stop band implicitly. Unlike previous attempts to design an optimal filter, the Maximal Ripple algorithm used an exchange method that tried to find the frequency set {ωi} where the best filter had its ripples. Thus, the Maximal Ripple algorithm was not an optimal filter design but it had quite a significant impact on how the Parks-McClellan algorithm would formulate.
Read more about this topic: Parks-Mc Clellan Filter Design Algorithm
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