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iirlp2xn

Transform IIR lowpass filter to IIR real N-point filter

Description

example

[Num,Den,AllpassNum,AllpassDen] = iirlp2xn(B,A,Wo,Wt) transform IIR lowpass filter to IIR real N-point filter.

The iirlp2xn function returns the numerator and denominator vectors, Num and Den, respectively of the target filter transformed from the real lowpass prototype by applying an Nth-order real lowpass to real multipoint frequency transformation, where N is the number of features being mapped. By default the DC feature is kept at its original location. For more details, see IIR Lowpass to IIR Real N-Point Filter Transformation.

[Num,Den,AllpassNum,AllpassDen] = iirlp2xn(B,A,Wo,Wt,Pass) allows you to choose between using the "DC Mobility" and the "Nyquist Mobility". In the first case the Nyquist feature stays at its original location and the DC feature is free to move. In the second case the DC feature is kept at an original frequency and the Nyquist feature is allowed to move.

The function 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);

Transform the lowpass filter to an IIR real N-point filter.

[num,den] = iirlp2xn(b,a,[-0.5 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 object and other objects of type uitoolbar, uimenu. The axes object 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 values to be transformed from the prototype filter, specified as a row vector. Frequencies in Wo should be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

Length of vectors Wo and Wt must be the same.

Data Types: single | double

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

Length of vectors Wo and Wt must be the same.

Data Types: single | double

Choice of passband or stopband at DC, specified as a 'pass' or 'stop'.

Output Arguments

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Numerator coefficients of the transformed filter, returned as a row vector.

Data Types: single | double
Complex Number Support: Yes

Denominator coefficients of the transformed filter, returned as a row vector.

Data Types: single | double

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

Data Types: single | double

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

Data Types: single | double

More About

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IIR Lowpass to IIR Real N-Point Filter Transformation

IIR Lowpass to IIR Real N-Point Filter transformation effectively places N features of the original filter, located at frequencies Wo1,...,WoN, at the required target frequency locations, Wt1,...,WtM.

Relative positions of other features of the original filter are the same in the target filter for the Nyquist mobility and are reversed in DC mobility. For the Nyquist mobility 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. For DC mobility feature F2 will precede F1 after the transformation.

Choice of the feature subject to this transformation is not restricted 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. Select features such that there is no band overlap when creating N bands around the unit circle.

IIR Lowpass to IIR Real N-Point Filter transformation can also be used to transforming other types of filters, for example, notch filters or resonators can be easily replicated at a number of required frequency locations. A good application would be an adaptive tone cancellation circuit reacting to the changing number and location of tones.

References

[1] Krukowski, A., G.D. Cain, and I. Kale. “Custom Designed High-Order Frequency Transformations for IIR Filters.” In 38th Midwest Symposium on Circuits and Systems. Proceedings, 1:588–91. Rio de Janeiro, Brazil: IEEE, 1996.

[2] Cain, G.D., A. Krukowski and I. Kale, “High Order Transformations for Flexible IIR Filter Design,” VII European Signal Processing Conference (EUSIPCO'94), vol. 3, pp. 1582-1585, Edinburgh, United Kingdom, September 1994.

See Also

Functions

Introduced in R2011a