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Transform lowpass analog filters to bandstop

`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
ω_{0} and bandwidth
*B*_{w}, the standard *s*-domain
transformation is

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

where *Q* =
ω_{0}/*B*_{w} and
*p* = *s*/ω_{0}. The
state-space version of this transformation is

$$Q=\frac{{\omega}_{0}}{{B}_{w}}$$

$$At=\left[\frac{{\omega}_{0}}{Q\cdot {A}^{-1}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega}_{0}\cdot \text{eye}(ma);-{\omega}_{0}\cdot \text{eye}(ma)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{zeros}(ma)\right]$$

$$Bt=-\left[\frac{{\omega}_{0}}{Q\left(A\backslash B\right)};\text{zeros}(ma,n)\right]$$

$$Ct=\left[\frac{C}{A}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{zeros}(mc,ma)\right]$$

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

`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.