besselap

Bessel analog lowpass filter prototype

Syntax

[z,p,k] = besselap(n)

Description

[z,p,k] = besselap(n) returns the poles and gain of an order-n Bessel analog lowpass filter prototype.

Examples

Frequency Response of an Analog Bessel Filter

Design a 6th-order Bessel analog lowpass filter. Display its magnitude and phase responses.

[z,p,k] = besselap(6);        % 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 scalar

Filter order, specified as an integer scalar. n must be less than or equal to 25.

Data Types: double

Output Arguments

z — Zeros

matrix

Zeros of the filter prototype, returned as an empty matrix. z is empty because there are no zeros.

p — Poles

column vector

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

k — Gain

scalar

Gain of the filter prototype, returned as a scalar.

Algorithms

The besselap function finds the filter roots from a lookup table constructed using Symbolic Math Toolbox™ software.

Analog Bessel filters are characterized by a group delay that is maximally flat at zero frequency and almost constant throughout the passband. The group delay at zero frequency is

$${\left(\frac{\left(2n\right)!}{2^{n}n!}\right)}^{1/n}.$$

The besselap function normalizes the poles and gain so that at low frequency and high frequency the Bessel prototype is asymptotically equivalent to the Butterworth prototype of the same order [1]. The magnitude of the filter is less than $$1/\sqrt{2}$$ at the unity cutoff frequency Ω_c = 1. The transfer function is expressed in terms of z, p, and k as

$$H(s)=\frac{k}{\left(s-p(1)\right)\left(s-p(2)\right)\cdots \left(s-p(n)\right)}.$$

References

[1] Rabiner, L. R., and B. Gold. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975, pp. 228–230.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• Filter order must be a constant. Expressions or variables are allowed if their values do not change.

Version History

Introduced before R2006a