Allpass filter for lowpass to lowpass transformation
[AllpassNum,AllpassDen] = allpasslp2lp(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2lp(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the first-order allpass mapping filter for performing a real lowpass to real lowpass frequency transformation. This transformation effectively places one feature of an original filter, located originally at frequency Wo, at the required target frequency location, Wt.
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 lowpass 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 and so on.
Lowpass to lowpass 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 applying the lowpass to lowpass transformation.
Design the allpass filter changing the lowpass filter cutoff frequency originally at 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] = allpasslp2lp(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');
As shown in the figure, allpasslp2lp generates a mapping function that converts your prototype lowpass filter to a target lowpass filter with different passband specifications.
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.