Transform lowpass analog filters to highpass

`[bt,at]`

` = `

lp2hp(b,a,Wo)

[At,Bt,Ct,Dt]` = `

lp2hp(A,B,C,D,Wo)

`lp2hp`

transforms analog lowpass filter prototypes
with a cutoff angular frequency of 1 rad/s into highpass
filters with desired cutoff angular frequency. The transformation
is one step in the digital filter design process for the `butter`

, `cheby1`

, `cheby2`

, and `ellip`

functions.

The `lp2hp`

function 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.

`[bt,at]`

transforms an analog
lowpass filter prototype given by polynomial coefficients into a highpass
filter with cutoff angular frequency ` = `

lp2hp(b,a,Wo)`Wo`

. Row vectors `b`

and `a`

specify
the coefficients of the numerator and denominator of the prototype
in descending powers of *s*.

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

Scalar `Wo`

specifies the cutoff angular frequency
in units of radians/second. The frequency transformed filter is returned
in row vectors `bt`

and `at`

.

`[At,Bt,Ct,Dt]`

converts the continuous-time
state-space lowpass filter prototype in matrices ` = `

lp2hp(A,B,C,D,Wo)`A`

, `B`

, `C`

, `D`

below

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

into a highpass filter with cutoff angular frequency `Wo`

.
The highpass filter is returned in matrices `At`

, `Bt`

, `Ct`

, `Dt`

.

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