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Transform lowpass analog filters to highpass


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



[bt,at] = lp2hp(b,a,Wo) transforms an analog lowpass filter prototype given by polynomial coefficients (specified by row vectors b and a) into a highpass analog filter with cutoff angular frequency Wo. The input system must be an analog filter prototype.

[At,Bt,Ct,Dt] = lp2hp(A,B,C,D,Wo) converts the continuous-time state-space lowpass filter prototype (specified by matrices A, B, C, and D) to a highpass analog filter with cutoff angular frequency Wo. The input system must be an analog filter prototype.


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Design a 5th-order highpass elliptic filter with a cutoff frequency of 100 Hz, 3 dB of passband ripple, and 30 dB of stopband attenuation

Design the prototype. Convert the zero-pole-gain output to a transfer function.

f = 100;

[ze,pe,ke] = ellipap(5,3,30);
[be,ae] = zp2tf(ze,pe,ke);

Transform the prototype to a highpass filter. Specify the cutoff frequency in rad/s.

[bh,ah] = lp2hp(be,ae,2*pi*f);

Compute and plot the frequency response of the filter. Divide the normalized frequency by 2π so the x-axis of the plot is in Hz.

[hh,wh] = freqs(bh,ah,4096);

axis([10 400 -40 5])

Input Arguments

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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:


Data Types: single | double

Prototype state-space representation, specified as matrices. The state-space matrices relate the state vector x, the input u, and the output y through


Data Types: single | double

Cutoff angular frequency, specified as a scalar. Express the cutoff angular frequency in rad/s.

Data Types: single | double

Output Arguments

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Transformed numerator and denominator coefficients, returned as row vectors.

Transformed state-space representation, returned as matrices.


lp2hp transforms analog lowpass filter prototypes with a cutoff angular frequency of 1 rad/s into highpass filters with a desired cutoff angular frequency. The transformation is one step in the digital filter design process for the butter, cheby1, cheby2, and ellip functions.

lp2hp is a highly accurate state-space formulation of the classic analog filter frequency transformation. If a highpass filter is to have a cutoff angular frequency ω0, 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.

Introduced before R2006a