iirftransf

IIR frequency transformation of digital filter

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Syntax

[num,den] = iirftransf(b,a,allpassNum,allpassDen)

Description

[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

Transform Lowpass Filter Using Allpass Mapping Filter

Open Live Script

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);
freqz(b,a)

$Figure contains 2 axes objects. Axes object 1 with title Phase, xlabel Normalized Frequency (\times\pi rad/sample), ylabel Phase (degrees) contains an object of type line. Axes object 2 with title Magnitude, xlabel Normalized Frequency (\times\pi rad/sample), ylabel Magnitude (dB) contains an object of type line.$

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.

filterAnalyzer(b,a,num,den,FilterNames=["PrototypeFilterTFForm","TransformedFilterFromTFForm"]);

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

Compare the magnitude response of the filters.

filterAnalyzer(ss(:,1:3),ss(:,4:6),num2,den2,FilterNames=["PrototypeFilterSOSForm",...
    "TransformedFilterFromSOSForm"]);

Input Arguments

`b` — Numerator coefficients of prototype lowpass IIR filter
row vector | matrix

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

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

$H (z) = \frac{B (z)}{A (z)} = \frac{b_{0} + b_{1} z^{- 1} + \dots + b_{n} z^{- n}}{a_{0} + a_{1} z^{- 1} + \dots + a_{n} z^{- n}},$
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.

$b = [\begin{matrix} b_{01} & b_{11} & b_{21} & ... & b_{Q 1} \\ b_{02} & b_{12} & b_{22} & ... & b_{Q 2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ b_{0 P} & b_{1 P} & b_{2 P} & \dots & b_{Q P} \end{matrix}]$
In the transfer function form, the numerator coefficient matrix b_ik of the IIR filter can be represented using the following equation:

$H (z) = \prod_{k = 1}^{P} H_{k} (z) = \prod_{k = 1}^{P} \frac{b_{0 k} + b_{1 k} z^{- 1} + b_{2 k} z^{- 2} + \dots + b_{Q k} z^{- Q}}{a_{0 k} + a_{1 k} z^{- 1} + a_{2 k} z^{- 2} + \dots + a_{Q k} z^{- Q}},$
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

`a` — Denominator coefficients of prototype lowpass IIR filter
row vector | matrix

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

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

$H (z) = \frac{B (z)}{A (z)} = \frac{b_{0} + b_{1} z^{- 1} + \dots + b_{n} z^{- n}}{a_{0} + a_{1} z^{- 1} + \dots + a_{n} z^{- n}},$
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.

$a = [\begin{matrix} a_{01} & a_{11} & a_{21} & \dots & a_{Q 1} \\ a_{02} & a_{12} & a_{22} & \dots & a_{Q 2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ a_{0 P} & a_{1 P} & a_{2 P} & \dots & a_{Q P} \end{matrix}]$
In the transfer function form, the denominator coefficient matrix a_ik of the IIR filter can be represented using the following equation:

$H (z) = \prod_{k = 1}^{P} H_{k} (z) = \prod_{k = 1}^{P} \frac{b_{0 k} + b_{1 k} z^{- 1} + b_{2 k} z^{- 2} + \dots + b_{Q k} z^{- Q}}{a_{0 k} + a_{1 k} z^{- 1} + a_{2 k} z^{- 2} + \dots + a_{Q k} z^{- Q}},$
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

`allpassNum` — Numerator coefficients of mapping filter
row vector

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

Data Types: single | double
Complex Number Support: Yes

`allpassDen` — Denominator coefficients of mapping filter
row vector

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

Data Types: single | double
Complex Number Support: Yes

Output Arguments

`num` — Numerator coefficients of transformed filter
row vector | matrix

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

`den` — Denominator coefficients of transformed filter
row vector | matrix

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

zpkftransf