Zero-pole-gain lowpass to bandpass frequency transformation
= zpklp2bp(Z,P,K,Wo,Wt) returns zeros,
P2, and gain factor,
of the target filter transformed from the real lowpass prototype by
applying a second-order real lowpass to real bandpass frequency mapping.
It also returns the numerator,
and the denominator
AllpassDen, of the allpass
mapping filter. The prototype lowpass filter is given with zeros,
P, and gain factor,
This transformation effectively places one feature of an original
filter, located at frequency -Wo, at the required
target frequency location, Wt1, and the second
feature, originally at
at the new location, Wt2. It is assumed that
Wt2 is greater than Wt1.
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
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.
Choice of the feature subject to the lowpass to bandpass transformation is not restricted only 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.
Real lowpass to bandpass transformation can also be used for transforming other types of filters; e.g., real notch filters or resonators can be easily doubled and positioned at two distinct, desired frequencies.
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); [Z2,P2,K2] = zpklp2bp(Z,P,K, 0.5, [0.2 0.3]); hfvt = fvtool(B,A,K2*poly(Z2),poly(P2)); legend(hfvt,'Prototype Lowpass Filter', 'Bandpass Filter'); axis([0 1 -70 10]);
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
Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
Constantinides, A.G., “Spectral transformations for digital filters,” IEE Proceedings, vol. 117, no. 8, pp. 1585-1590, August 1970.
Nowrouzian, B. and A.G. Constantinides, “Prototype reference transfer function parameters in the discrete-time frequency transformations,” Proceedings 33rd Midwest Symposium on Circuits and Systems, Calgary, Canada, vol. 2, pp. 1078-1082, August 1990.
Nowrouzian, B. and L.T. Bruton, “Closed-form solutions for discrete-time elliptic transfer functions,” Proceedings of the 35th Midwest Symposium on Circuits and Systems, vol. 2, pp. 784-787, 1992.
Constantinides, A.G., “Design of bandpass digital filters,” IEEE® Proceedings, vol. 1, pp. 1129-1231, June 1969.