Allpass filter for lowpass to bandstop transformation
[AllpassNum,AllpassDen] = allpasslp2bs(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2bs(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the second-order allpass mapping filter for performing a real lowpass to real 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. This transformation implements the "Nyquist Mobility," which means that the DC feature stays at DC, but the Nyquist feature moves to a location dependent on the selection of Wo and Wt.
Relative positions of other features of an original filter 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. After the transformation feature F2 will precede F1 in the target filter. 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.
Design the allpass filter changing the lowpass filter with cutoff frequency at Wo=0.5 to the real bandstop filter with cutoff frequencies at Wt1=0.25 and Wt2=0.375:
Wo = 0.5; Wt = [0.25, 0.375]; [AllpassNum, AllpassDen] = allpasslp2bs(Wo, Wt); [h, f] = freqz(AllpassNum, AllpassDen, 'whole'); plot(f/pi, abs(angle(h))/pi, Wt, Wo, 'ro'); title('Mapping Function Wo(Wt)'); xlabel('New Frequency, Wt'); ylabel('Old Frequency, Wo');
In the figure, you find the mapping filter function as determined by the example. Note the response is normalized to π:
Frequency value 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 0 and 1, with 1 corresponding to half the sample rate.
Nowrouzian, B. and A.G. Constantinides, "Prototype reference transfer function parameters in the discrete-time frequency transformations," Proceedings 33rd Midwest Symposium on Circuits and Systems, Calgary, Canada, vol. 2, pp. 1078-1082, August 1990.