Zero-pole-gain lowpass to complex M-band frequency transformation
[Z2,P2,K2,AllpassNum,AllpassDen] = zpklpmbc(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 Mth-order real lowpass to complex multibandpass 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 Wo, at the required target frequency locations, Wt1,...,WtM.
Choice of the feature subject to this transformation is not restricted to the cutoff frequency of an original lowpass filter. In general it is possible to select any feature, for example, the stopband edge, the DC, the deep minimum in the stopband, or other ones.
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.
This transformation can also be used for transforming other types of filters; e.g., to replicate notch filters and resonators at any required location.
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); [z1,p1,k1] = zpklp2mbc(z, p, k, 0.5, [2 4 6 8]/10); [z2,p2,k2] = zpklp2mbc(z, p, k, 0.5, [2 4 6 8]/10);
Verify the result by comparing the prototype filter with the target filter:
fvtool(b, a, k1*poly(z1), poly(p1), k2*poly(z2), poly(p2));
You could review the coefficients to compare the filters, but the graphical comparison shown here is quicker and easier.
However, looking at the coefficients in FVTool shows the complex nature desired.
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. It should be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
Desired frequency locations in the transformed target filter. They should be normalized to be between -1 and 1, with 1 corresponding to half the sample rate.
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