Allpass filter for lowpass to complex bandstop transformation
[AllpassNum,AllpassDen] = allpasslp2bsc(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2bsc(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the first-order allpass mapping filter for performing a real lowpass to complex bandstop frequency transformation. 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 +Wo, at the new location, Wt2. It is assumed that Wt2 is greater than Wt1. Additionally the transformation swaps passbands with stopbands in the target filter.
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 bandstop 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 bandstop filters for band attenuation or frequency equalizers, from the high-quality prototype lowpass filter.
Design the allpass filter changing the real lowpass filter with the cutoff frequency of Wo=0.5 into a complex bandstop filter with band edges of Wt1=0.2 and Wt2=0.4 precisely defined:
Wo = 0.5; Wt = [0.2,0.4]; [AllpassNum, AllpassDen] = allpasslp2bsc(Wo, Wt); [h, f] = freqz(AllpassNum, AllpassDen, 'whole');
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.
Numerator of the mapping filter
Denominator of the mapping filter