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Zero-pole-gain complex shift frequency transformation
[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)
[Z2,P2,K2,AllpassNum,AllpassDen] = zpkshiftc(Z,P,K,Wo,Wt) returns zeros, Z_{2}, poles, P_{2}, and gain factor, K_{2}, 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 W_{o}, placed at W_{t }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.
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 -0.25:
[z2,p2,k2] = zpkshiftc(z, p, k, 0.5, 0.25); hfvt = fvtool(b,a,k2*poly(z2),poly(p2));
[z3,p3,k3] = zpkshiftc(z, p, k, 0, 0.5); addfilter(hfvt,k3*poly(z3),poly(p3)); legend(hfvt,'Original Filter','Rotation by -\pi/4 radians/sample',... 'Rotation by \pi/2 radians/sample');
[z2,p2,k2] = zpkshiftc(z, p, k, 0.5, -0.5); fvtool(b, a, k2*poly(z2), poly(p2));
After performing the rotation, the resulting filter shows the features desired.
Similar to the first example, performing the Hilbert transformation generates the desired target filter, shown here.
Finally, using the inverse Hilbert transformation creates yet a third filter, as the figure shows.
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 |
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.
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.