lp2lp

Change cutoff frequency for lowpass analog filter

Syntax

[bt,at] = lp2lp(b,a,Wo)

[At,Bt,Ct,Dt] = lp2lp(A,B,C,D,Wo)

Description

[bt,at] = lp2lp(b,a,Wo) transforms an analog lowpass filter prototype with unity cutoff frequency (1 rad/s) into a lowpass filter with cutoff angular frequency Wo rad/s. Specify the filter prototype with numerator coefficients b and denominator coefficients a as row vectors. The input system must be an analog filter prototype.

[At,Bt,Ct,Dt] = lp2lp(A,B,C,D,Wo) converts the continuous-time state-space lowpass filter prototype (specified by matrices A, B, C, and D) with unity cutoff frequency (1 rad/s) to a lowpass filter with cutoff angular frequency Wo rad/s. The input system must be an analog filter prototype.

Examples

Lowpass to Lowpass Transformation

Design an 8th-order Chebyshev Type I analog lowpass filter prototype with 3 dB of ripple in the passband.

[z,p,k] = cheb1ap(8,3);

Convert the prototype to transfer function form and display its magnitude and frequency responses.

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

Transform the prototype to a lowpass filter with a cutoff frequency of 30 Hz. Specify the cutoff frequency in rad/s. Display the magnitude and frequency responses of the transformed filter.

Wo = 2*pi*30;

[bt,at] = lp2lp(b,a,Wo);
freqs(bt,at)

Input Arguments

b, a — Prototype numerator and denominator coefficients

row vectors

Prototype numerator and denominator coefficients, specified as row vectors. b and a specify the coefficients of the numerator and denominator of the prototype in descending powers of s:

$$\frac{B(s)}{A(s)}=\frac{b(1)s^n+\cdots +b(n)s+b(n+1)}{a(1)s^m+\cdots +a(m)s+a(m+1)}$$

Data Types: single | double

A, B, C, D — Prototype state-space representation

matrices

Prototype state-space representation, specified as matrices. The state-space matrices relate the state vector x, the input u, and the output y through

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

Data Types: single | double

Wo — Cutoff angular frequency

scalar

Cutoff angular frequency, specified as a scalar. Express Wo in units of rad/s.

Data Types: single | double

Output Arguments

bt, at — Transformed numerator and denominator coefficients

row vectors

Transformed numerator and denominator coefficients, returned as row vectors.

At, Bt, Ct, Dt — Transformed state-space representation

matrices

Transformed state-space representation, returned as matrices.

Algorithms

lp2lp transforms an analog lowpass filter prototype with a cutoff angular frequency of 1 rad/s into a lowpass filter with any specified cutoff angular frequency. The transformation is one step in the digital filter design process for the butter, cheby1, cheby2, and ellip functions.

lp2lp is a highly accurate state-space formulation of the classic analog filter frequency transformation. If a lowpass filter has cutoff angular frequency ω₀, the standard s-domain transformation is

$$s=p/{\omega }_0$$

The state-space version of this transformation is

$$At={\omega }_0\cdot A$$

$$Bt={\omega }_0\cdot B$$

$$Ct=C$$

$$Dt=D$$

The lp2lp function can perform the transformation on two different linear system representations: transfer function form and state-space form. See lp2bp for a derivation of the bandpass version of this transformation.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

The input transfer function coefficients, num and den, must be real.

Version History

Introduced before R2006a