Optimum "L" filter

The Optimum "L" filter (also known as a Legendre filter) is a kind of electronic filter. The Optimum "L" filter was proposed by Athanasios Papoulis in 1958. Compared to other linear filters, 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".

Like Butterworth and Bessel filters, the Optimum "L" filter is a "pole-only" filter.

A Optimum "L" low-pass filter is characterized by its transfer function:

$H(s)={\frac {1}{...[FIXME]...}}$

In particular, a second order Legendre filter has a transfer function

$H(s)={\frac {3}{x^{2}+2^{1/2}x+1}}$.

A third order Legendre filter has a transfer function

$H(s)={\frac {3^{1/2}}{x^{3}+1.31x^{2}+1.36x+3^{1/2}}}$.

See also

• Bessel filter
• Butterworth filter
• Chebyshev filter
• Elliptic filter

External links and references

• Kuo, Franklin F. (1966). Network Analysis and Synthesis. Wiley. ISBN 0-471-51118-8. Second Edition.
• "Optimum “L” Filters: Polynomials, Poles and Circuit Elements" by C. Bond 2004
• "Design and Analysis of Analog Filters: A Signal Processing Perspective" by L. D. Paarmann, Chapter 8.3: Legendre filters.