Allpass filter for lowpass to bandpass transformation
[AllpassNum,AllpassDen] = allpasslp2bp(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2bp(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the second-order allpass mapping filter for performing a real lowpass to real bandpass 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 "DC mobility," which means that the Nyquist feature stays at Nyquist, but the DC feature moves to a location dependent on the selection of 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 bandpass 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.
Lowpass to bandpass transformation can also be used for transforming other types of filters; e.g., real notch filters or resonators can be doubled and repositioned at two distinct desired frequencies.
Design the allpass mapping filter changing the lowpass filter with cutoff frequency at Wo=0.5 to the real–valued bandpass filter with cutoff frequencies at Wt10.25 and Wt2=0.375.
Compute the frequency response and 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, 0.375]; [AllpassNum, AllpassDen] = allpasslp2bp(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 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.