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iirlp2bp

Transform IIR lowpass filter to IIR bandpass filter

Description

example

[Num,Den,AllpassNum,AllpassDen] = iirlp2bp(B,A,Wo,Wt) transform IIR lowpass filter to IIR bandpass filter.

The iirlp2bp function returns the numerator and denominator vectors, Num and Den, respectively, of the target filter transformed from the real lowpass prototype by applying a second-order real lowpass to real bandpass frequency mapping. For more details, see IIR Lowpass to IIR Bandpass Transformation.

The function also returns the numerator, AllpassNum, and the denominator, AllpassDen, of the allpass mapping filter. The prototype lowpass filter is specified with the numerator B and denominator A.

Note

Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

Examples

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Design a prototype real IIR lowpass elliptic filter with a gain of about –3 dB at 0.5π rad/sample.

[b,a] = ellip(3,0.1,30,0.409);

Create a real bandpass filter by placing the cutoff frequencies of the prototype filter at 0.25π and 0.75π.

[num,den] = iirlp2bp(b,a,0.5,[0.25 0.75]);

Compare the magnitude responses of the filters using FVTool.

hvft = fvtool(b,a,num,den);
legend(hvft,'Prototype','Target')

Figure Filter Visualization Tool - Magnitude Response (dB) contains an axes and other objects of type uitoolbar, uimenu. The axes with title Magnitude Response (dB) contains 2 objects of type line. These objects represent Prototype, Target.

Input Arguments

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Numerator of the prototype lowpass filter, specified as a row vector.

Data Types: single | double
Complex Number Support: Yes

Denominator of the prototype lowpass filter, specified as a row vector.

Data Types: single | double
Complex Number Support: Yes

Frequency value to be transformed from the prototype filter, specified as a scalar. Frequency Wo must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

Data Types: single | double

Desired frequency locations in the transformed target filter, specified as a two-element vector. Frequencies in Wt must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

Data Types: single | double

Output Arguments

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Numerator of the target filter, returned as a row vector.

Data Types: single | double
Complex Number Support: Yes

Denominator of the target filter, returned as a row vector.

Data Types: single | double

Numerator of the mapping filter, returned as a row vector.

Data Types: single | double

Denominator of the mapping filter, returned as a row vector.

Data Types: single | double

More About

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IIR Lowpass to IIR Bandpass Transformation

IIR lowpass to IIR bandpass transformation effectively places one feature of the 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,” meaning that the Nyquist feature stays at Nyquist, but the DC feature moves to a location dependent on the selection of Wts.

Relative positions of other features of the 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. You can choose to transform any feature of the original filter like stopband edge, DC, deep minimum in the stopband, or others.

Real lowpass to bandpass transformation can also be used to transform other types of filters, for example, real notch filters or resonators can be doubled and positioned 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. “Design of Bandpass Digital Filters.” Proceedings of the IEEE 57, no. 6 (1969): 1229–31. https://doi.org/10.1109/PROC.1969.7216.

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

See Also

Functions

Introduced in R2011a