lp2bs

Transform lowpass analog filters to bandstop

Syntax

[bt,at] = lp2bs(b,a,Wo,Bw)

[At,Bt,Ct,Dt] = lp2bs(A,B,C,D,Wo,Bw)

Description

[bt,at] = lp2bs(b,a,Wo,Bw) transforms an analog lowpass filter prototype given by polynomial coefficients (specified by row vectors b and a) into a bandstop filter with center frequency Wo and bandwidth Bw. The input system must be an analog filter prototype.

[At,Bt,Ct,Dt] = lp2bs(A,B,C,D,Wo,Bw) converts the continuous-time state-space lowpass filter prototype (specified by matrices A, B, C, and D) to a bandstop filter with center frequency Wo and bandwidth Bw. The input system must be an analog filter prototype.

Examples

Lowpass to Bandstop Transformation

Design a 10th-order lowpass Butterworth analog filter prototype.

n = 10;
[z,p,k] = buttap(n);

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

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

Transform the prototype to a bandstop filter with a stopband from 20 Hz to 60 Hz. Specify the center frequency and bandwidth in rad/s.

fl = 20;
fh = 60;

Wo = 2*pi*sqrt(fl*fh); % center frequency
Bw = 2*pi*(fh-fl); % bandwidth

[bt,at] = lp2bs(b,a,Wo,Bw);

Display the magnitude and frequency responses of the transformed filter.

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 — Center frequency

scalar

Center frequency, specified as a scalar. For a filter with lower band edge w1 and upper band edge w2, use Wo = sqrt(w1*w2). Express Wo in units of rad/s.

Data Types: single | double

Bw — Bandwidth

scalar

Bandwidth, specified as a scalar. For a filter with lower band edge w1 and upper band edge w2, use Bw = w2–w1. Express Bw 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

lp2bs transforms analog lowpass filter prototypes with a cutoff angular frequency of 1 rad/s into bandstop filters with the desired bandwidth and center frequency. The transformation is one step in the digital filter design process for the butter, cheby1, cheby2, and ellip functions.

lp2bs is a highly accurate state-space formulation of the classic analog filter frequency transformation. If a bandstop filter has center frequency ω₀ and bandwidth B_w, the standard s-domain transformation is

$$s=\frac{p}{Q(p^2+1)}$$

where Q = ω₀/B_w and p = s/ω₀. The state-space version of this transformation is

$$Q=\frac{{\omega }_0}{B_w}$$

$$At=\left[\frac{{\omega }_0}{Q\cdot A^{-1}}{\omega }_0\cdot \text{eye}(ma);-{\omega }_0\cdot \text{eye}(ma)\text{zeros}(ma)\right]$$

$$Bt=-\left[\frac{{\omega }_0}{Q\left(A\backslash B\right)};\text{zeros}(ma,n)\right]$$

$$Ct=\left[\frac{C}{A}\text{zeros}(mc,ma)\right]$$

$$Dt=D-C/A\cdot B$$

lp2bs 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.