Documentation |
Zero-pole-gain lowpass to complex bandpass frequency transformation
[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2bpc(Z,P,K,Wo,Wt)
[Z2,P2,K2,AllpassNum,AllpassDen] = zpklp2bpc(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 real lowpass to complex bandpass frequency transformation.
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 gain factor, K.
This transformation effectively places one feature of an original filter, located at frequency -W_{o}, at the required target frequency location, W_{t1}, and the second feature, originally at +W_{o}, at the new location, W_{t2}. It is assumed that W_{t2} is greater than W_{t1}.
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 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.
Lowpass to bandpass transformation can also be used for transforming other types of filters; e.g., real notch filters or resonators can be doubled and positioned at two distinct desired frequencies at any place around the unit circle forming a pair of complex notches/resonators. This transformation can be used for designing bandpass filters for radio receivers from the high-quality prototype lowpass filter.
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] = zpklp2bpc(z, p, k, 0.5, [0.2 0.3]);
Verify the result by comparing the prototype filter with the target filter:
fvtool(b, a, k2*poly(z2), poly(p2));
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. It should be normalized to be between -1 and 1, with 1 corresponding to half the sample rate. |
Wt | Desired frequency locations in the transformed target filter. They should be normalized to be between 0 and 1, with 1 corresponding to half the sample rate. |
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 |