Documentation |
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, W_{o}, at the required target frequency location, W_{t}, 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, F_{1} and F_{2}, with F_{1} preceding F_{2}. After the transformation feature F_{2} will precede F_{1} in the target filter. However, the distance between F_{1} and F_{2} 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 W_{o}=0.5 to W_{t}=0.25.
Plot the phase response normalized to π, which is in effect the mapping function W_{o}(W_{t}). 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');
Variable | Description |
---|---|
Wo | Frequency value to be transformed from the prototype filter |
Wt | Desired frequency location in the transformed target filter |
AllpassNum | Numerator of the mapping filter |
AllpassDen | Denominator of the mapping filter |
Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
Constantinides, A.G., "Spectral transformations for digital filters," IEE Proceedings, vol. 117, no. 8, pp. 1585-1590, August 1970.
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.
Nowrouzian, B. and L.T. Bruton, "Closed-form solutions for discrete-time elliptic transfer functions," Proceedings of the 35th Midwest Symposium on Circuits and Systems, vol. 2, pp. 784-787, 1992.
Constantinides, A.G., "Frequency transformations for digital filters," Electronics Letters, vol. 3, no. 11, pp. 487-489, November 1967.