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:

images

More specifically, we can propose images. The square of the amplitude is then in the form:

images

where Ln (x2) is the generator polynomial of optimum filters and images is a parameter determined by the required attenuation for x = 1.

When images = 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:

images

The polynomials Ln (x2) verify the relations:

The filters do not obtain the same function ...

Get Digital Filters Design for Signal and Image Processing now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.