Allpass filter for lowpass to bandpass transformation
[AllpassNum,AllpassDen] = allpasslp2bp(Wo,Wt)
[AllpassNum,AllpassDen] = allpasslp2bp(Wo,Wt) returns
AllpassNum, and the denominator,
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
at the required target frequency location, Wt1,
and the second feature, originally at
at the new location,
It is assumed that
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 will still precede
the transformation. However, the distance between
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
the real–valued bandpass filter with cutoff frequencies at
Compute the frequency response and plot the phase response normalized
to π, which is in effect the mapping function
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.
Constantinides, A.G., “Spectral transformations for digital filters,” IEEE® 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., “Design of bandpass digital filters,” IEEE Proceedings, vol. 1, pp. 1129-1231, June 1969.