Parks-Mc Clellan Filter Design Algorithm - Additional Notes

Additional Notes

Before applying the Chebyshev approximation, a set of steps were necessary:

1. Define the set of basis function for the approximation, and
2. Exploit the fact that the pass and stop bands of bandpass filters would always be separated by transition regions.

Since FIR filters could be reduced to the sum of cosines case, the same core program could be used to perform all possible linear-phase FIR filters. In contrast to the Maximum Ripple approach, the band edges could now be specified ahead of time.

To achieve an efficient implementation of the optimal filter design using the Park-McClellan algorithm, two difficulties have to be overcome:

1. Defining a flexible exchange strategy, and
2. Implementing a robust interpolation method.

In some sense, the programming involved the implementation and adaptation of a known algorithm for use in FIR filter design. Two faces of the exchange strategy were taken to make the program more efficient:

1. Allocating the extremal frequencies between the pass and stop bands, and
2. Enabling movement of the extremals between the bands as the program iterated.

At initialization, the number of extremals in the pass and stop band could be assigned by using the ratio of the sizes of the bands. Furthermore, the pass and stop band edge would always be placed in the extremal set, and the program's logic kept those edge frequencies in the extremal set. The movement between bands was controlled by comparing the size of the errors at all the candidate extremal frequencies and taking the largest. The second element of the algorithm was the interpolation step needed to evaluate the error function. They used a method called the Barycentric form of Lagrange interpolation, which was very robust.

All conditions for the Parks-McClellan algorithm are based on Chebyshev's Alternation Theorem. The Alternation Theorem states that the polynomial of degree L that minimizes the maximum error will have at least L+2 extrema. The optimal frequency response will barely reach the maximum ripple bounds. The extrema must occur at the pass and stop band edges and at either ω=0 or ω=π or both. Now the derivative of a polynomial of degree L is a polynomial of degree L-1, which can be zero at most at L-1 places. So the maximum number of local extrema is the L-1 local extrema plus the 4 band edges, giving a total of L+3 extrema.