lp2hp (Signal Processing Toolbox)

Lowpass to highpass analog filter transformation.

Syntax

[bt,at] = lp2hp(b,a,Wo)
[At,Bt,Ct,Dt] = lp2hp(A,B,C,D,Wo)

Description

lp2hp transforms analog lowpass filter prototypes with a cutoff frequency of 1 rad/sec into highpass filters with desired cutoff 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.

Transfer Function Form (Polynomial)

[bt,at] = lp2hp(b,a,Wo)

transforms an analog lowpass filter prototype given by polynomial coefficients into a highpass filter with cutoff frequency Wo. Row vectors b and a specify the coefficients of the numerator and denominator of the prototype in descending powers of s:

Scalar Wo specifies the cutoff frequency in units of radians/second. The frequency transformed filter is returned in row vectors bt and at.

State-Space Form

[At,Bt,Ct,Dt] = lp2hp(A,B,C,D,Wo)

converts the continuous-time state-space lowpass filter prototype in matrices A, B, C, D:

into a highpass filter with cutoff frequency Wo. The highpass filter is returned in matrices At, Bt, Ct, Dt.

Algorithm

lp2hp is a highly accurate state-space formulation of the classic analog filter frequency transformation. If a highpass filter is to have cutoff frequency

₀, the standard s-domain transformation is

The state-space version of this transformation is

At = Wo*inv(A);
Bt = -Wo*(A\B);
Ct = C/A;
Dt = D - C/A*B;

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

`bilinear`	Bilinear transformation method of analog-to-digital filter conversion.
`impinvar`	Impulse invariance method of analog-to-digital filter conversion.
`lp2bp`	Lowpass to bandpass analog filter transformation.
`lp2bs`	Lowpass to bandstop analog filter transformation.
`lp2lp`	Lowpass to lowpass analog filter transformation.