Zero-pole-gain lowpass to N-point frequency transformation


[Z2,P2,K2,AllpassNum,AllpassDen] = zpklp2xn(Z,P,K,Wo,Wt)
[Z2,P2,K2,AllpassNum,AllpassDen] = zpklp2xn(Z,P,K,Wo,Wt,Pass)


[Z2,P2,K2,AllpassNum,AllpassDen] = zpklp2xn(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 an Nth-order real lowpass to real multipoint frequency transformation, where N is the number of features being mapped. By default the DC feature is kept at its original location.

[Z2,P2,K2,AllpassNum,AllpassDen] = zpklp2xn(Z,P,K,Wo,Wt,Pass) allows you to specify an additional parameter, Pass, which chooses between using the "DC Mobility" and the "Nyquist Mobility". In the first case the Nyquist feature stays at its original location and the DC feature is free to move. In the second case the DC feature is kept at an original frequency and the Nyquist feature is allowed to move.





This transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can be easily replicated at a number of required frequency locations. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.



[b, a] = ellip(3,0.1,30,0.409);
z = roots(b);
p = roots(a);
k = b(1);
[z2,p2,k2] = zpklp2xn(z, p, k, [-0.5 0.5], [0 0.25], 'pass');
hfvt = fvtool(b, a, k2*poly(z2), poly(p2));
legend(hfvt,'Original Filter','Half-band Filter');

As demonstrated by the figure, the target filter has the desired response shape and values replicated from the prototype.



Zeros of the prototype lowpass filter


Poles of the prototype lowpass filter


Gain factor of the prototype lowpass filter


Frequency value to be transformed from the prototype filter


Desired frequency location in the transformed target filter




Zeros of the target filter


Poles of the target filter


Gain factor of the target filter


Numerator of the mapping filter


Denominator of the mapping filter



Cain, G.D., A. Krukowski and I. Kale, "High Order Transformations for Flexible IIR Filter Design," VII European Signal Processing Conference (EUSIPCO'94), vol. 3, pp. 1582-1585, Edinburgh, United Kingdom, September 1994.

Krukowski, A., G.D. Cain and I. Kale, "Custom designed high-order frequency transformations for IIR filters," 38th Midwest Symposium on Circuits and Systems (MWSCAS'95), Rio de Janeiro, Brazil, August 1995.

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