zpkbpc2bpc

Zero-pole-gain complex bandpass frequency transformation

Syntax

[Z2,P2,K2,AllpassNum,AllpassDen] = zpkbpc2bpc(Z,P,K,Wo,Wt)


Description

[Z2,P2,K2,AllpassNum,AllpassDen] = zpkbpc2bpc(Z,P,K,Wo,Wt) returns zeros, Z₂, poles, P₂, and gain factor, K₂, of the target filter transformed from the complex bandpass prototype by applying a first-order complex bandpass to complex bandpass frequency transformation.

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

This transformation effectively places two features of an original filter, located at frequencies W_o1 and W_o2, at the required target frequency locations, W_t1, and W_t2 respectively. It is assumed that W_t2 is greater than W_t1. In most of the cases the features selected for the transformation are the band edges of the filter passbands. In general it is possible to select any feature; e.g., the stopband edge, the DC, the deep minimum in the stopband, or other ones.

Relative positions of other features of an original filter do not change in the target filter. This means that it is possible to select two features of an original filter, F₁ and F₂, with F₁ preceding F₂. Feature F₁ will still precede F₂ after the transformation. However, the distance between F₁ and F₂ will not be the same before and after the transformation.

This transformation can also be used for transforming other types of filters; e.g., complex notch filters or resonators can be repositioned at two distinct desired frequencies at any place around the unit circle; e.g., in the adaptive system.

Examples

Apply Complex Bandpass Frequency Transformation

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

[b,a] = ellip(3,0.1,30,0.409);

Create a complex passband from 0.25 to 0.75.

[b, a] = iirlp2bpc(b,a,0.5,[0.25,0.75]);
z = roots(b);
p = roots(a);
k = b(1);
[z2,p2,k2] = zpkbpc2bpc(z,p,k,[0.25, 0.75],[-0.75, -0.25]);

Verify the result by comparing the prototype filter with the target filter.

filterAnalyzer(b,a,k2*poly(z2),poly(p2),...
    FrequencyRange="centered",...
    FilterNames=["PrototypeFilter","TargetFilterWithComplexPassband"]);

Arguments

Variable	Description
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
AllpassNum	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.

Version History

Introduced in R2011a