legendre_filters calculating coefficients for "optimal-l" filters from legendre polynomials last updated: 27022011
not much information is available on the "optimal-l" filter. from wikipedia: "The Optimum "L" filter (also known as a Legendre filter) was proposed by Athanasios Papoulis in 1958. It has the maximum roll off rate for a given filter order while maintaining a monotonic frequency response. It provides a compromise between the Butterworth filter which is monotonic but has a slower roll off and the Chebyshev filter which has a faster roll off but has ripple in either the pass band or stop band. The filter design is based on Legendre polynomials which is the reason for its alternate name and the "L" in Optimum "L"." one of the few places where information could be found online was at c.r.bond's website here: http://www.crbond.com/papers/optf2.pdf. after contacting c.r.bond, vinnie falco (http://code.google.com/p/dspfilterscpp/) was provided with c++ code for calculating the filter coefficients. his dsp filter library now implements the "optimal-l" filter. the original papulis document can be bought from IEEE for 30$ (A. Papoulis, On Monotonic Response Filters, Proc. IRE, 47, Feb. 1959, 332-333), but also c.r.bond was kind enough to write a paper explaining how the "optimal-l" coefficients are calculated: the filter is formed by taking the legendre polynomial of first kind (http://en.wikipedia.org/wiki/Legendre_polynomials), integrating it (a modification) and then producing a transfer function. there are some complications since for odd/even orders of the optimal-l polynomials, different integrals have to be formed. with some guidance i was able to understand the procedures and input all the formulas in maxima (http://maxima.sourceforge.net) the maxima output bellow (an image) shows formulas for forming legendre polynomials (Pn) and applying papoulis's modifications (Ln). it also shows optimal-l transfer function for orders zero to six. there are no limitations on the order which can be produced from these calculations.  for more information see the links bellow. also please feel free to contact me for any questions you may have. i would be happy to answer such, if any. contact: lubomir i. ivanov neolit123 [at] gmail download: zip file containing the maxima files and html preview:
links on topic: discussion at the kvr forum: with robin schmidt and vinnie falco |