Digital Filters Design for Signal and Image Processing

4.8. Papoulis filters (optimum (O_n))

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 L_n (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 L_n (jx) then leads to determining the polynomials of norm 1 linked to the scalar product:

The polynomials L_n (x²) verify the relations:

The filters do not obtain the same function ...