Filter Design Toolbox™

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. This 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 W_t.

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 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.

Examples

Design the allpass filter changing the lowpass filter with cutoff frequency at W_o=0.5 to the real bandpass filter with cutoff frequencies at W_t1=0.25 and W_t2=0.375:

Wo = 0.5;
Wt = [0.25, 0.375];
[AllpassNum, AllpassDen] = allpasslp2bp(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 W_o(W_t). 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');

Shown in the figure, with the x-axis as the new frequency, you see the mapping filter for the example.

Arguments

Variable	Description
`Wo`	Frequency value to be transformed from the prototype filter
`Wt`	Desired frequency locations 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.

References

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.

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