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# ellipap

Elliptic analog lowpass filter prototype

## Syntax

[z,p,k] = ellipap(n,Rp,Rs)

## Description

[z,p,k] = ellipap(n,Rp,Rs) returns the zeros, poles, and gain of an order n elliptic analog lowpass filter prototype, with Rp dB of ripple in the passband, and a stopband Rs dB down from the peak value in the passband. The zeros and poles are returned in length n column vectors z and p and the gain in scalar k. If n is odd, z is length n - 1. The transfer function in factored zero-pole form is

$H\left(s\right)=\frac{z\left(s\right)}{p\left(s\right)}=k\frac{\left(s-{z}_{1}\right)\left(s-{z}_{2}\right)\dots \left(s-{z}_{N}\right)}{\left(s-{p}_{1}\right)\left(s-{p}_{2}\right)\dots \left(s-{p}_{M}\right)}$

Elliptic filters offer steeper rolloff characteristics than Butterworth and Chebyshev filters, but they are equiripple in both the passband and the stopband. Of the four classical filter types, elliptic filters usually meet a given set of filter performance specifications with the lowest filter order.

ellipap sets the passband edge angular frequency ω0 of the elliptic filter to 1 for a normalized result. The passband edge angular frequency is the frequency at which the passband ends and the filter has a magnitude response of 10-Rp/20.

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### Algorithms

ellipap uses the algorithm outlined in [1]. It employs ellipke to calculate the complete elliptic integral of the first kind and ellipj to calculate Jacobi elliptic functions.

## References

[1] Parks, T. W., and C. S. Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, chap. 7.