October 2006
Intermediate to advanced
369 pages
8h 11m
English
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4.8. Papoulis filters (optimum (On))
4.8.1. General characteristics
Compared to Butterworth filters of equivalent orders, these filters do not present ripple in their passband. Papoulis filters combine the advantages of Butterworth and Chebyshev filters.
These filters are obtained using a method that imposes a maximum value at the integral of the square of the attenuation A2(jx) for −1 ≤ x ≤ 1 or also to A2 (jx)−1 for −1 ≤ x ≤ 1:
More specifically, we can propose 
. The square of the amplitude is then in the form:
where Ln (x2) is the generator polynomial of optimum filters and 
is a parameter determined by the required attenuation for x = 1.
When 
= 1 and if we say the specification is symmetrical, determining Ln (jx) then leads to determining the polynomials of norm 1 linked to the scalar product:
The polynomials Ln (x2) verify the relations:
The filters do not obtain the same function ...
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