cheb2ap

Chebyshev Type II analog lowpass filter prototype

Syntax

[z,p,k] = cheb2ap(n,Rs)

Description

[z,p,k] = cheb2ap(n,Rs) returns the zeros, poles, and gain of an order n Chebyshev Type II analog lowpass filter prototype with Rs dB of ripple down from the passband peak value in the stopband.

Examples

Frequency Response of an Analog Chebyshev Type II Filter

Design a 6th-order Chebyshev Type II analog lowpass filter with 70 dB of ripple in the stopband. Display its magnitude and phase responses.

[z,p,k] = cheb2ap(6,70);      % Lowpass filter prototype
[num,den] = zp2tf(z,p,k);     % Convert to transfer function form
freqs(num,den)                % Frequency response of analog filter

Input Arguments

n — Filter order

integer

Filter order, specified as an integer.

Data Types: single | double

Rs — Stopband ripple

scalar

Stopband ripple, specified as a scalar in decibels.

Data Types: single | double

Output Arguments

z — Zeros

n-length column vector

Zeros of the filter, returned as an n-length column vector. If n is odd, z has length n–1.

p — Poles

n-length column vector

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

k — Gain

scalar

Gain of the filter, returned as a scalar.

Algorithms

Chebyshev Type II filters are monotonic in the passband and equiripple in the stopband. The pole locations are the inverse of the pole locations of the cheb1ap function, whose poles are evenly spaced about an ellipse in the left half plane. The Chebyshev Type II stopband edge angular frequency ω₀ is set to 1 for a normalized result. This value is the frequency at which the stopband begins. The filter has a magnitude response of 10^–Rs/20.

Chebyshev Type II filters are sometimes called inverse Chebyshev filters because of their relationship to Chebyshev Type I filters. The cheb2ap function is a modification of the Chebyshev Type I prototype algorithm:

1. cheb2ap replaces the frequency variable ω with 1/ω, turning the lowpass filter into a highpass filter while preserving the performance at ω = 1.

2. cheb2ap subtracts the filter transfer function from unity.

The transfer function is given by

$$H(s)=\frac{z(s)}{p(s)}=k\frac{(s-z(1))(s-z(2))\mathrm{...}(s-z(n))}{(s-p(1))(s-p(2))\mathrm{...}(s-p(n))}.$$

References

[1] Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987.

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