Allpass filter for lowpass to highpass transformation
[AllpassNum,AllpassDen] = allpasslp2hp(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2hp(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the first-order allpass mapping filter for performing a real lowpass to real highpass frequency transformation. This transformation effectively places one feature of an original filter, located originally at frequency, Wo, at the required target frequency location, Wt, at the same time rotating the whole frequency response by half of the sampling frequency. Result is that the DC and Nyquist features swap places.
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 highpass 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.
Lowpass to highpass transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can change their position in a simple way by using the lowpass to highpass transformation.
Design the allpass filter changing the lowpass filter to the highpass filter with its cutoff frequency moved from Wo=0.5 to Wt=0.25.
Plot the phase response normalized to π, which is in effect the mapping function Wo(Wt). Please note that the transformation works in the same way for both positive and negative frequencies:
Wo = 0.5; Wt = 0.25; [AllpassNum, AllpassDen] = allpasslp2hp(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');
Frequency value to be transformed from the prototype filter
Desired frequency location 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.