# iirlp2mbc

Transform IIR lowpass filter to IIR complex M-band filter

## Syntax

``````[Num,Den,AllpassNum,AllpassDen] = iirlp2mbc(B,A,Wo,Wt)``````

## Description

example

``` ```[Num,Den,AllpassNum,AllpassDen] = iirlp2mbc(B,A,Wo,Wt)``` transform IIR lowpass filter to IIR complex M-band filter.The `iirlp2mbc` returns the numerator and denominator vectors, `Num` and `Den`, respectively of the target filter transformed from the real lowpass prototype by applying an `Mth`-order real lowpass to complex multi-bandpass frequency transformation. For more details, see IIR Lowpass Filter to IIR Complex M-Band Filter Transformation.The function also 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`.```

## 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);`

Create a complex multiband filter with two passbands.

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

Compare the magnitude responses of the filters using FVTool. `iirlp2mbc` replicates the desired feature at 0.5 in the lowpass filter at four locations in the multiband filter.

```hvft = fvtool(b,a,num,den); legend(hvft,'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 value to be transformed 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`

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.

Data Types: `single` | `double`

## 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`
Complex Number Support: Yes

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

Data Types: `single` | `double`
Complex Number Support: Yes

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

Data Types: `single` | `double`
Complex Number Support: Yes

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### IIR Lowpass Filter to IIR Complex M-Band Filter Transformation

IIR lowpass filter to IIR complex M-band filter transformation effectively places one feature of the original filter, located at frequency Wo, at the required target frequency locations, Wt1,...,WtM.

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, 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.

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 M-band 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.