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iirftransf

IIR frequency transformation of digital filter

Description

example

[num,den] = iirftransf(b,a,allpassNum,allpassDen) returns the numerator and the denominator coefficients of the transformed filter.

The iirftransf function transforms a prototype filter, specified by the numerator b and denominator a, by using an allpass mapping filter, specified by the numerator allpassNum and the denominator allpassDen. If you do not specify an allpass mapping filter, then the function returns an original filter.

Examples

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Using the iirftransf function, extend the passband of a lowpass IIR filter by using an allpass mapping filter.

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);
fvtool(b,a)

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains an object of type line.

Transform Filter Using iirftransf

Extend the passband of the real prototype filter by specifying the allpass mapping filter as a vector of numerator and denominator coefficients, alpnum and alpden respectively. Use the allpasslp2lp function to generate the allpass mapping filter coefficients.

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

[b,a] = ellip(3,0.1,30,0.409);      
[alpnum,alpden] = allpasslp2lp(0.5,0.25);
[num,den] = iirftransf(b,a,alpnum,alpden);

Compare the magnitude response of the filters using FVTool.

fvt = fvtool(b,a,num,den);
legend(fvt,"Prototype Filter (TF Form)",...
    "Transformed Filter")

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 2 objects of type line. These objects represent Prototype Filter (TF Form), Transformed Filter.

You can also specify the input lowpass IIR filter as a matrix of coefficients. Pass the second-order section matrices as inputs. The numerator and the denominator coefficients of the transformed filter are given by num2 and den2, respectively.

ss = tf2sos(b,a);
[num2,den2] = iirftransf(ss(:,1:3),ss(:,4:6),...
    alpnum,alpden);

Compare the magnitude response of the filters using FVTool.

hvft = fvtool(ss,[num2 den2]);
legend(hvft,"Prototype Filter (SOS Form)",...
    "Transformed Filter")

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 2 objects of type line. These objects represent Prototype Filter (SOS Form), Transformed Filter.

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

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

Data Types: single | double
Complex Number Support: Yes

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

Data Types: single | double
Complex Number Support: Yes

Output Arguments

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Numerator coefficients of the transformed 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 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

Extended Capabilities

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

Version History

Introduced in R2011a

See Also

Functions