Allpass filter for lowpass to N-point transformation
[AllpassNum,AllpassDen] = allpasslp2xn(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2xn(Wo,Wt,Pass)
[AllpassNum,AllpassDen] = allpasslp2xn(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 real 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 an original filter, located at frequencies Wo1,...,WoN, at the required target frequency locations, Wt1,...,WtM. By default the DC feature is kept at its original location.
[AllpassNum,AllpassDen] = allpasslp2xn(Wo,Wt,Pass) allows you to specify an additional parameter, Pass, which chooses between using the "DC Mobility" and the "Nyquist Mobility." In the first case the Nyquist feature stays at its original location and the DC feature is free to move. In the second case the DC feature is kept at an original frequency and the Nyquist feature is movable.
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 without the need of designing them again. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.
Frequency values to be transformed from the prototype filter
Desired frequency locations in the transformed target filter
Choice ('pass'/'stop') of passband/stopband at DC, 'pass' being the default
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.
Cain, G.D., A. Krukowski and I. Kale, "High Order Transformations for Flexible IIR Filter Design," VII European Signal Processing Conference (EUSIPCO'94), vol. 3, pp. 1582-1585, Edinburgh, United Kingdom, September 1994.
Krukowski, A., G.D. Cain and I. Kale, "Custom designed high-order frequency transformations for IIR filters," 38th Midwest Symposium on Circuits and Systems (MWSCAS'95), Rio de Janeiro, Brazil, August 1995.