Allpass filter for lowpass to bandpass transformation
returns the numerator,
AllpassDen] = allpasslp2bp(
AllpassNum, and the denominator,
AllpassDen, of the second-order allpass mapping filter for performing
a real lowpass to real bandpass frequency transformation. For more information, see Real Lowpass to Real Bandpass Frequency Transformation.
Design Allpass Mapping Filter from Lowpass Filter
Design the allpass mapping filter changing the lowpass filter with cutoff frequency
0.5 to the real–valued bandpass filter with cutoff frequencies
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');
Wo — Frequency value to be transformed from prototype filter
real scalar in the range
Frequency value to be transformed from the prototype filter, specified as a real
scalar in the range
Wt — Desired frequency locations in target filter
real vector with values in the range
Desired frequency locations in the transformed target filter, specified as a real
vector with values in the range
AllpassNum — Numerator of mapping filter
Numerator of the mapping filter, returned as a real-valued vector.
AllpassDen — Denominator of mapping filter
Denominator of the mapping filter, returned as a real-valued vector.
Real Lowpass to Real Bandpass Frequency Transformation
Real lowpass to real bandpass frequency 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
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.
Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.
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, for example, the stopband edge, the DC, the deep minimum in the stopband, or other features.
Lowpass to bandpass transformation can also be used for transforming other types of filters, for example, real notch filters or resonators can be doubled and repositioned at two distinct desired frequencies.
 Nowrouzian, B., and A.G. Constantinides. “Prototype Reference Transfer Function Parameters in the Discrete-Time Frequency Transformations.” In Proceedings of the 33rd Midwest Symposium on Circuits and Systems, 1078–82. Calgary, Alta., Canada: IEEE, 1991. https://doi.org/10.1109/MWSCAS.1990.140912.
 Nowrouzian, B., and L.T. Bruton. “Closed-Form Solutions for Discrete-Time Elliptic Transfer Functions.” In  Proceedings of the 35th Midwest Symposium on Circuits and Systems , 784–87. Washington, DC, USA: IEEE, 1992. https://doi.org/10.1109/MWSCAS.1992.271206.
 Constantinides, A.G.“Spectral transformations for digital filters.” Proceedings of the IEEE, vol. 117, no. 8: 1585-1590. August 1970.
Introduced in R2011a