# iirlp2mbc

Transform IIR lowpass filter to IIR complex multiband filter

## Description

`[`

transform an IIR lowpass filter to an IIR complex multiband filter.`num`

,`den`

,`allpassNum`

,`allpassDen`

] =
iirlp2mbc(`b`

,`a`

,`wo`

,`wt`

)

The `iirlp2mbc`

returns the numerator and denominator
coefficients of the transformed complex IIR multiband filter. The function also
returns the numerator, `allpassNum`

, and the denominator,
`allpassDen`

, of the
*M*^{th} order allpass mapping filter.
The prototype lowpass filter is specified with the numerator `b`

and the denominator `a`

.

The function transforms a real lowpass prototype filter to a multiband filter by
applying an M^{th}-order real lowpass to complex multiple
bandpass frequency mapping. Parameter *M* is the number of times an
original feature is replicated in the transformed filter. For more details, see
IIR Lowpass Filter to IIR Complex Multiband Filter Transformation.

## Examples

### Transform Lowpass Filter to Complex Multiband Filter

Transform a real lowpass IIR filter to a complex multiband IIR filter using the `iirlp2mbc`

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 iirlp2mbc**

Transform the real prototype lowpass filter into a complex multiband filter with two passbands.

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

and `a`

respectively.

[num,den] = iirlp2mbc(b,a,0.5,[-7 -5 6 8]/10);

Compare the magnitude response of the filters. `iirlp2mbc`

replicates the desired feature at 0.5 in the lowpass filter at four locations in the multiband filter.

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

Alternatively, you can also specify the input lowpass IIR filter as a matrix of coefficients. Pass the second-order section matrices as inputs. The transformed filter is a fourth-order section filter given by the numerator and the denominator coefficients `num2`

and `den2`

.

ss = tf2sos(b,a); [num2,den2] = iirlp2mbc(ss(:,1:3),ss(:,4:6),0.5,[-7 -5 6 8]/10);

To visualize the magnitude response of the fourth-order section filter, first pass the filter coefficients to the `dsp.FourthOrderSectionFilter`

object. Then, use this object as an input to the filter analyzer.

```
fos = dsp.FourthOrderSectionFilter(Numerator=num2,...
Denominator=den2);
```

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,fos,FrequencyRange="centered",... FilterNames=["PrototypeFilterFromSOSMatrix",... "TransformedMultibandFilterFromSOSMatrix"]);

*Copyright 2012–2024 The MathWorks, Inc.*

## 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 [

*b*_{0},*b*_{1}, …,*b*_{n}], given this transfer function form:$$H(z)=\frac{B(z)}{A(z)}=\frac{{b}_{0}+{b}_{1}{z}^{-1}+\cdots +{b}_{n}{z}^{-n}}{{a}_{0}+{a}_{1}{z}^{-1}+\cdots +{a}_{n}{z}^{-n}},$$

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.$$b=\left[\begin{array}{ccccc}{b}_{01}& {b}_{11}& {b}_{21}& \mathrm{...}& {b}_{Q1}\\ {b}_{02}& {b}_{12}& {b}_{22}& \mathrm{...}& {b}_{Q2}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {b}_{0P}& {b}_{1P}& {b}_{2P}& \cdots & {b}_{QP}\end{array}\right]$$

In the transfer function form, the numerator coefficient matrix

*b*of the IIR filter can be represented using the following equation:_{ik}$$H(z)={\displaystyle \prod _{k=1}^{P}{H}_{k}}(z)={\displaystyle \prod _{k=1}^{P}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}+\cdots +{b}_{Qk}{z}^{-Q}}{{a}_{0k}+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}+\cdots +{a}_{Qk}{z}^{-Q}}},$$

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 [

*a*_{0},*a*_{1}, …,*a*_{n}], given this transfer function form:$$H(z)=\frac{B(z)}{A(z)}=\frac{{b}_{0}+{b}_{1}{z}^{-1}+\cdots +{b}_{n}{z}^{-n}}{{a}_{0}+{a}_{1}{z}^{-1}+\cdots +{a}_{n}{z}^{-n}},$$

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.$$a=\left[\begin{array}{ccccc}{a}_{01}& {a}_{11}& {a}_{21}& \cdots & {a}_{Q1}\\ {a}_{02}& {a}_{12}& {a}_{22}& \cdots & {a}_{Q2}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {a}_{0P}& {a}_{1P}& {a}_{2P}& \cdots & {a}_{QP}\end{array}\right]$$

In the transfer function form, the denominator coefficient matrix

*a*of the IIR filter can be represented using the following equation:_{ik}$$H(z)={\displaystyle \prod _{k=1}^{P}{H}_{k}}(z)={\displaystyle \prod _{k=1}^{P}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}+\cdots +{b}_{Qk}{z}^{-Q}}{{a}_{0k}+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}+\cdots +{a}_{Qk}{z}^{-Q}}},$$

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

`wo`

— Frequency value to transform from prototype filter

positive scalar

Frequency value to transform from the prototype filter, specified as a
positive scalar. Frequency `wo`

should be normalized to be
between `0`

and `1`

, with
`1`

corresponding to half the sample rate.

**Data Types: **`single`

| `double`

`wt`

— Desired frequency locations in transformed target filter

row vector

Desired frequency locations in the transformed target filter, specified as
a row vector. Frequencies in `wt`

should be normalized to
be between `-1`

and `1`

, with
`1`

corresponding to half the sample rate. The value of
*M* which is equal to the number of times an original
feature is replicated in the transformed filter equals the length of the
`wt`

vector.

**Data Types: **`single`

| `double`

## Output Arguments

`num`

— Numerator coefficients of transformed multiband filter

row vector | matrix

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

Row vector of length

*Mn*/2+1, where*M*is the number of times an original feature is replicated in the transformed filter and*n*is the order of the input filter. The value of*M*equals the length of the`wt`

vector.The

`num`

output is a row vector when the input coefficients`b`

and`a`

are row vectors.*P*-by-(*QM*/2+1) matrix, where*P*is the number of filter sections,*Q*is the order of each section of the transformed filter, and*M*is the number of times an original feature is replicated in 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 multiband filter

row vector | matrix

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

Row vector of length

*Mn*/2+1, where*M*is the number of times an original feature is replicated in the transformed filter and*n*is the order of the input filter. The value of*M*equals the length of the`wt`

vector.The

`den`

output is a row vector when the input coefficients`b`

and`a`

are row vectors.*P*-by-(*QM*/2+1) matrix, where*P*is the number of filter sections,*Q*is the order of each section of the transformed filter, and*M*is the number of times an original feature is replicated in 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

`allpassNum`

— Numerator coefficients of mapping filter

row vector

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

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

## More About

### IIR Lowpass Filter to IIR Complex Multiband Filter Transformation

IIR lowpass filter to IIR complex multiband filter transformation
effectively places one feature of the original filter, located at frequency
`w`

, at the required target
frequency locations, _{o}`w`

, …
,_{t1}`w`

._{tM}

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, F_{1} and F_{2}, with
F_{1} preceding F_{2}. Feature
F_{1} will still precede F_{2} after the
transformation. However, the distance between F_{1} and
F_{2} will not be the same before and after the
transformation.

Choice of the feature subject to this transformation is not restricted 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.

The IIR lowpass filter to IIR complex multiband filter transformation can also be used for 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., and I. Kale.
“High-Order Complex Frequency Transformations,” *Internal report
No. 27/2001*. Applied DSP and VLSI Research Group, University of
Westminster.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

## Version History

**Introduced in R2011a**

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