Main Content

zpkshiftc

Zero-pole-gain complex shift frequency transformation

Syntax

[Z2,P2,K2,AllpassNum,AllpassDen] = zpkshiftc(Z,P,K,Wo,Wt)
[Num,Den,AllpassNum,AllpassDen] = zpkshiftc(Z,P,K,0,0.5)
[Num,Den,AllpassNum,AllpassDen] = zpkshiftc(Z,P,K,0,-0.5)

Description

[Z2,P2,K2,AllpassNum,AllpassDen] = zpkshiftc(Z,P,K,Wo,Wt) returns zeros, Z2, poles, P2, and gain factor, K2, of the target filter transformed from the real lowpass prototype by applying a first-order complex frequency shift transformation. This transformation rotates all the features of an original filter by the same amount specified by the location of the selected feature of the prototype filter, originally at Wo, placed at Wt in the target filter.

It also returns the numerator, AllpassNum, and the denominator, AllpassDen, of the allpass mapping filter. The prototype lowpass filter is given with zeros, Z, poles, P, and the gain factor, K.

[Num,Den,AllpassNum,AllpassDen] = zpkshiftc(Z,P,K,0,0.5) performs the Hilbert transformation, i.e. a 90 degree counterclockwise rotation of an original filter in the frequency domain.

[Num,Den,AllpassNum,AllpassDen] = zpkshiftc(Z,P,K,0,-0.5) performs the inverse Hilbert transformation, i.e. a 90 degree clockwise rotation of an original filter in the frequency domain.

Examples

collapse all

Design a prototype real IIR halfband filter using a standard elliptic approach.

[b, a] = ellip(3,0.1,30,0.409);
z = roots(b);
p = roots(a);
k = b(1);

Rotation by pi/4 Radians/Sample

Rotation by -0.25:

[z2,p2,k2] = zpkshiftc(z, p, k, 0.5, 0.25);
filterAnalyzer(b,a,k2*poly(z2),poly(p2),...
    FrequencyRange="centered",...
    FilterNames=["PrototypeFilter","TargetFilterAfterRotation"]);

Rotation by pi/2 Radians/Sample

[z2,p2,k2] = zpkshiftc(z, p, k, 0, 0.5);
filterAnalyzer(b,a,k2*poly(z2),poly(p2),...
    FrequencyRange="centered",...
    FilterNames=["PrototypeFilter","TargetFilterAfterHilbertTransformation"]);

Rotation by -pi/2 Radians/Sample

[z2,p2,k2] = zpkshiftc(z, p, k, 0, -0.5);
filterAnalyzer(b,a,k2*poly(z2),poly(p2),...
    FrequencyRange="centered",...
    FilterNames=["PrototypeFilter","FilterFromInverseHilbertTransformation"]);

Arguments

VariableDescription
Z

Zeros of the prototype lowpass filter

P

Poles of the prototype lowpass filter

K

Gain factor of the prototype lowpass filter

Wo

Frequency value to be transformed from the prototype filter

Wt

Desired frequency location in the transformed target filter

Z2

Zeros of the target filter

P2

Poles of the target filter

K2

Gain factor of the target filter

AllpassDen

Numerator of the mapping filter

AllpassDen

Denominator of the mapping filter

Frequencies must be normalized to be between -1 and 1, with 1 corresponding to half the sample rate.

References

Oppenheim, A.V., R.W. Schafer and J.R. Buck, Discrete-Time Signal Processing, Prentice-Hall International Inc., 1989.

Dutta-Roy, S.C. and B. Kumar, “On digital differentiators, Hilbert transformers, and half-band low-pass filters,” IEEE® Transactions on Education, vol. 32, pp. 314-318, August 1989.

Version History

Introduced in R2011a