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 [2] and uses elliptic functions to compute the zeros and poles.

The function sets the passband edge angular frequency ω_p of the elliptic filter to 1 for a normalized result and finds a value for the stopband edge angular frequency ω_s. At these frequencies, the filter magnitude response has a passband ripple and stopband attenuation of R_p dB and R_s dB, respectively.

The transfer function in factored zero-pole form is

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

As part of the filter prototype design process, ellipap verifies that the filter order n, passband ripple Rp, and stopband attenuation Rs satisfy

$$n\ge \frac{K(m_0)K\left(1-m_1\right)}{K\left(1-m_0\right)K(m_1)}\text{,}$$

where $$m_0=\frac{1}{{\omega }_s^2}$$ must be less than 1, $$m_1=\frac{{10}^{R_p/10}-1}{{10}^{R_s/10}-1}$$, and K(m) is the complete elliptic integral of the first kind with parameter m such that

$$K(m)={\displaystyle {\int }_0^1\frac{dt}{\sqrt{(1-t^2)(1-m{t}^2)}}}.$$

Note

If you specify too high a value for n, ellipap can error out. To resolve this error, reduce n.

References

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

[2] Abramowitz, Milton, and Stegun, Irene A. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematical Series, 1972.

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.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Refer to the usage notes and limitations in the C/C++ Code Generation section. The same usage notes and limitations apply to GPU code generation.

Version History

Introduced before R2006a