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# 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

lp2bs transforms analog lowpass filter prototypes with a cutoff angular frequency of 1 rad/s into bandstop filters with 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 can perform the transformation on two different linear system representations: transfer function form and state-space form. In both cases, the input system must be an analog filter prototype.

### Transfer Function Form (Polynomial)

[bt,at] = lp2bs(b,a,Wo,Bw) transforms an analog lowpass filter prototype given by polynomial coefficients into a bandstop filter with center frequency Wo and bandwidth Bw. Row vectors b and a specify the coefficients of the numerator and denominator of the prototype in descending powers of s.

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

Scalars Wo and Bw specify the center frequency and bandwidth in units of radians/second. For a filter with lower band edge w1 and upper band edge w2, use Wo = sqrt(w1*w2) and Bw = w2-w1.

lp2bs returns the frequency transformed filter in row vectors bt and at.

### State-Space Form

[At,Bt,Ct,Dt] = lp2bs(A,B,C,D,Wo,Bw) converts the continuous-time state-space lowpass filter prototype in matrices A, B, C, D shown below

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

into a bandstop filter with center frequency Wo and bandwidth Bw. For a filter with lower band edge w1 and upper band edge w2, use Wo = sqrt(w1*w2) and Bw = w2-w1.

The bandstop filter is returned in matrices At, Bt, Ct, Dt.

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### Algorithms

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

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

where Q = ω0/Bw and p = s0. The state-space version of this transformation is

```Q = Wo/Bw;
At = [Wo/Q*inv(A) Wo*eye(ma);-Wo*eye(ma) zeros(ma)];
Bt = -[Wo/Q*(A\B); zeros(ma,n)];
Ct = [C/A zeros(mc,ma)];
Dt = D - C/A*B;
```

See lp2bp for a derivation of the bandpass version of this transformation.