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 nth-order 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.

Examples

Frequency Response of Elliptic Lowpass Filter

Design a 6th-order elliptic analog lowpass filter with 5 dB of ripple in the passband and 50 dB of stopband attenuation.

[z,p,k] = ellipap(6,5,50);

Convert the zero-pole-gain filter parameters to transfer function form and display the frequency response of the filter.

[b,a] = zp2tf(z,p,k);
freqs(b,a)

Input Arguments

n — Filter order

positive integer scalar

Filter order, specified as a positive integer scalar.

Data Types: double

Rp — Passband ripple

positive scalar

Passband ripple, specified as a positive scalar in decibels.

Data Types: double

Rs — Stopband attenuation

positive scalar

Stopband attenuation down from the peak passband value, specified as a positive scalar in decibels.

Data Types: double

Output Arguments

z — Zeros

column vector

Zeros of the filter, returned as a column vector of length n. If n is odd, then z has a length equal to n – 1.

p — Poles

column vector

Poles of the filter, returned as a column vector of length n.

k — Gain

scalar

Gain of the filter, returned as a scalar.

Algorithms

The ellipap function 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. The function 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.

The transfer function in factored zero-pole form is

$$H(s)=\frac{z(s)}{p(s)}=k\frac{(s-z_1)(s-z_2)\dots (s-z_N)}{(s-p_1)(s-p_2)\dots (s-p_M)}$$

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.

References

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• All inputs must be constants. Expressions or variables are allowed if their values do not change.

Version History

Introduced before R2006a