Allpass filter for lowpass to complex N-point transformation
[AllpassNum,AllpassDen] = allpasslp2xc(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2xc(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the Nth-order allpass mapping filter, where N is the allpass filter order, for performing a real lowpass to complex multipoint frequency transformation. Parameter N also specifies the number of replicas of the prototype filter created around the unit circle after the transformation. This transformation effectively places N features of the, original filter located at frequencies Wo1,...,WoN, at the required target frequency locations, Wt1,...,WtM.
Relative positions of other features of an original filter are the same in the target filter for the Nyquist mobility and are reversed for the DC mobility. For the Nyquist mobility 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. For DC mobility feature F2 will precede F1 after the transformation.
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; e.g., the stopband edge, the DC, the deep minimum in the stopband, or other ones. The only condition is that the features must be selected in such a way that when creating N bands around the unit circle, there will be no band overlap.
This transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can be easily replicated at a number of required frequency locations. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.
Design the allpass filter moving four features of an original complex filter given in Wo to the new independent frequency locations Wt. Please note that the transformation creates N replicas of an original filter around the unit circle, where N is the order of the allpass mapping filter:
Wo = [-0.2, 0.3, -0.7, 0.4]; Wt = [0.3, 0.5, 0.7, 0.9]; [AllpassNum, AllpassDen] = allpasslp2xc(Wo, Wt); [h, f] = freqz(AllpassNum, AllpassDen, 'whole');
Frequency values to be transformed from the prototype filter
Desired frequency locations in the transformed target filter
Numerator of the mapping filter
Denominator of the mapping filter
Frequencies must be normalized to be between -1 and 1, with 1 corresponding to half the sample rate.