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iirlp2bpc

Transform IIR lowpass filter to complex bandpass filter

Description

[num,den,allpassNum,allpassDen] = iirlp2bpc(b,a,wo,wt) transforms an IIR lowpass filter to a complex bandpass filter.

The function transforms a real lowpass prototype filter, specified as the numerator and denominator coefficients b and a respectively, to a complex bandpass filter by applying a first-order real lowpass to complex bandpass frequency transformation.

The function returns the numerator and denominator coefficients of the transformed complex bandpass filter. The function also returns the numerator and denominator coefficients of the allpass mapping filter, allpassNum and allpassDen respectively.

For more details on the transformation, see IIR Lowpass to Complex Bandpass Transformation.

example

Examples

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Transform a lowpass IIR filter to a complex bandpass filter using the iirlp2bpc function.

Input Lowpass IIR Filter

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);
filterAnalyzer(b,a)

Transform Filter Using iirlp2bpc

Transform the prototype lowpass filter into a complex bandpass filter by placing the cutoff frequencies of the prototype filter at 0.25π and 0.75π.

Specify the prototype filter as a vector of numerator and denominator coefficients, b and a respectively.

[num,den] = iirlp2bpc(b,a,0.5,[0.25 0.75]);

Compare the magnitude response of the filters.

filterAnalyzer(b,a,num,den,FrequencyRange="centered",...
    FilterNames=["PrototypeFilter_TFForm",...
    "TransformedFilter"]);

Alternatively, you can also specify the input lowpass IIR filter as a matrix of coefficients. Pass the second order section coefficient matrices as inputs.

ss = tf2sos(b,a);
[num2,den2] = iirlp2bpc(ss(:,1:3),ss(:,4:6),0.5,[0.25 0.75]);

Use the sos2ctf function to convert the second-order section matrices to the cascaded transfer function form.

[b_ctf,a_ctf] = sos2ctf(ss);

Compare the magnitude response of the filters.

filterAnalyzer(b_ctf,a_ctf,num2,den2,FrequencyRange="centered",...
    FilterNames=["PrototypeFilterFromSOSMatrix", ...
    "TransformedFilterFromSOSForm"]);

Copyright 2012–2024 The MathWorks, Inc.

Input Arguments

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Numerator coefficients of the prototype lowpass IIR filter, specified as either:

  • Row vector –– Specifies the values of [b0, b1, …, bn], given this transfer function form:

    H(z)=B(z)A(z)=b0+b1z1++bnzna0+a1z1++anzn,

    where n is the order of the filter.

  • Matrix –– Specifies the numerator coefficients in the form of an P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each filter section. If Q = 2, the filter is a second-order section filter. For higher-order sections, make Q > 2.

    b=[b01b11b21...bQ1b02b12b22...bQ2b0Pb1Pb2PbQP]

    In the transfer function form, the numerator coefficient matrix bik of the IIR filter can be represented using the following equation:

    H(z)=k=1PHk(z)=k=1Pb0k+b1kz1+b2kz2++bQkzQa0k+a1kz1+a2kz2++aQkzQ,

    where,

    • a –– Denominator coefficients matrix. For more information on how to specify this matrix, see a.

    • k –– Row index.

    • i –– Column index.

    When specified in the matrix form, b and a matrices must have the same number of rows (filter sections) Q.

Data Types: single | double
Complex Number Support: Yes

Denominator coefficients for a prototype lowpass IIR filter, specified as one of these options:

  • Row vector –– Specifies the values of [a0, a1, …, an], given this transfer function form:

    H(z)=B(z)A(z)=b0+b1z1++bnzna0+a1z1++anzn,

    where n is the order of the filter.

  • Matrix –– Specifies the denominator coefficients in the form of an P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each filter section. If Q = 2, the filter is a second-order section filter. For higher-order sections, make Q > 2.

    a=[a01a11a21aQ1a02a12a22aQ2a0Pa1Pa2PaQP]

    In the transfer function form, the denominator coefficient matrix aik of the IIR filter can be represented using the following equation:

    H(z)=k=1PHk(z)=k=1Pb0k+b1kz1+b2kz2++bQkzQa0k+a1kz1+a2kz2++aQkzQ,

    where,

    • b –– Numerator coefficients matrix. For more information on how to specify this matrix, see b.

    • k –– Row index.

    • i –– Column index.

    When specified in the matrix form, a and b matrices must have the same number of rows (filter sections) P.

Data Types: single | double
Complex Number Support: Yes

Frequency value to transform from the prototype filter, specified as a real scalar. Frequency wo should 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 should be normalized to be between -1 and 1, with 1 corresponding to half the sample rate.

Data Types: single | double

Output Arguments

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Numerator coefficients of the transformed complex bandpass filter, returned as one of the following:

  • Row vector of length n+1, where n is the order of the input filter. The num output is a row vector when the input coefficients b and a are row vectors.

  • P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each section of the transformed filter. The num output is a matrix when the input coefficients b and a are matrices.

Data Types: single | double
Complex Number Support: Yes

Denominator coefficients of the transformed complex bandpass filter, returned as one of the following:

  • Row vector of length n+1, where n is the order of the input filter. The den output is a row vector when the input coefficients b and a are row vectors.

  • P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each section of the transformed filter. The den output is a matrix when the input coefficients b and a are matrices.

Data Types: single | double
Complex Number Support: Yes

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

Data Types: single | double
Complex Number Support: Yes

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

Data Types: single | double
Complex Number Support: Yes

More About

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Extended Capabilities

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C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2011a

See Also

Functions