allpasslp2bp

Allpass filter for lowpass to bandpass transformation

Syntax

[AllpassNum,AllpassDen] = allpasslp2bp(Wo,Wt)

Description

[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. For more information, see Real Lowpass to Real Bandpass Frequency Transformation.

Examples

Design Allpass Mapping Filter from Lowpass Filter

Design the allpass mapping filter changing the lowpass filter with cutoff frequency Wo at 0.5 to the real–valued bandpass filter with cutoff frequencies Wt1 and Wt2 at 0.25 and 0.375, respectively.

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');

Input Arguments

Wo — Frequency value to be transformed from prototype filter

real scalar in the range (0,1)

Frequency value to be transformed from the prototype filter, specified as a real scalar in the range (0,1).

Data Types: single | double

Wt — Desired frequency locations in target filter

real vector with values in the range (0,1)

Desired frequency locations in the transformed target filter, specified as a real vector with values in the range (0,1).

Data Types: single | double

Output Arguments

AllpassNum — Numerator of mapping filter

real-valued vector

Numerator of the mapping filter, returned as a real-valued vector.

Data Types: double

AllpassDen — Denominator of mapping filter

real-valued vector

Denominator of the mapping filter, returned as a real-valued vector.

Data Types: double

More About

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 -W_o, at the required target frequency location, W_t1, and the second feature, originally at +W_o, at the new location, W_t2. It is assumed that W_t2 is greater than W_t1. 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.

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.

References

[1] 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.

[2] Nowrouzian, B., and L.T. Bruton. “Closed-Form Solutions for Discrete-Time Elliptic Transfer Functions.” In [1992] 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.

[3] Constantinides, A.G.“Spectral transformations for digital filters.” Proceedings of the IEEE, vol. 117, no. 8: 1585-1590. August 1970.

Version History

Introduced in R2011a