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Zero-pole-gain real shift frequency transformation
[Z2,P2,K2,AllpassNum,AllpassDen]
= zpkshift(Z,P,K,Wo,Wt)
[Z2,P2,K2,AllpassNum,AllpassDen] = zpkshift(Z,P,K,Wo,Wt) returns the zeros,Z2 , poles, P2, and gain factor, K2, of the target filter transformed from the zeros, poles, and gain factor of real lowpass prototype by applying a second-order real shift frequency mapping. It also returns the numerator, AllpassNum, and the denominator, AllpassDen of the allpass mapping filter.
This transformation places one selected feature of an original filter, located at frequency W_{o}, at the required target frequency location, W_{t}. This transformation implements the "DC Mobility," which means that the Nyquist feature stays at Nyquist, but the DC feature moves to a location dependent on the selection of W_{o} and W_{t}.
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, F_{1} and F_{2}, with F_{1} preceding F_{2}. Feature F_{1} will still precede F_{2} after the transformation. However, the distance between F_{1} and F_{2} will not be the same before and after the transformation.
Choice of the feature subject to the real shift transformation is not restricted to the cutoff frequency of an original lowpass filter. 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.
This transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can change their position in a simple way without the need to design them again.
Rotate frequency response by π/2 radians/sample:
[B,A] = ellip(10,0.1,40,0.25); % Elliptic lowpass filter with passband frequency 0.25*pi rad/sample Z = roots(B); % get roots of numerator polynomial- filter zeros P = roots(A); % get roots of denominator polynomial- filter poles K = B(1); [Z2,P2,K2] = zpkshift(Z,P,K,0.25,0.75); % shift by 0.25*pi rad/sample Num = poly(Z2); Den = poly(P2); hfvt = fvtool(B,A,K2*Num,Den); legend(hfvt,'Original Filter','Rotate by \pi/2 radians/sample'); axis([0 1 -90 10]);