Allpass filter for real shift transformation
[AllpassNum,AllpassDen] = allpassshift(Wo,Wt)
[AllpassNum,AllpassDen] = allpassshift(Wo,Wt) returns the numerator, AllpassNum, and the denominator, AllpassDen, of the second-order allpass mapping filter for performing a real frequency shift transformation. This transformation places one selected feature of an original filter, located at frequency Wo, at the required target frequency location, Wt. 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 Wo and 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 real shift 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.
This transformation can also be used for transforming other types of filters; e.g., notch filters or resonators can be moved to a different frequency by applying a shift transformation. In such a way you can avoid designing the filter from the beginning.
Design the allpass filter precisely shifting one feature of the lowpass filter originally at Wo=0.5 to the new frequencies of Wt=0.25:
Wo = 0.5; Wt = 0.25; [AllpassNum, AllpassDen] = allpassshift(Wo, Wt);
Calculate the frequency response of the mapping filter in the full range:
[h, f] = freqz(AllpassNum, AllpassDen, 'whole');
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:
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.