# iirlp2xc

Transform IIR lowpass filter to IIR complex N-point filter

## Syntax

``````[Num,Den,AllpassNum,AllpassDen] = iirlp2xc(B,A,Wo,Wt)``````

## Description

example

``````[Num,Den,AllpassNum,AllpassDen] = iirlp2xc(B,A,Wo,Wt)```transform IIR lowpass filter to IIR complex N-point filter.The `iirlp2xc` function returns the numerator and denominator vectors, `Num` and `Den` respectively, of the target filter transformed from the real lowpass prototype by applying an `Nth`-order real lowpass to complex multipoint frequency transformation. For more details, see IIR Lowpass Filter to IIR Complex N-Point 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);`

Transform the lowpass filter to an IIR complex N-point filter.

`[num,den] = iirlp2xc(b,a,[-0.5 0.5],[-0.25 0.25])`
```num = 1×4 complex 0.0643 + 0.0000i 0.0464 - 0.0000i 0.0464 - 0.0000i 0.0643 - 0.0000i ```
```den = 1×4 complex 1.0000 + 0.0000i -1.6918 + 0.0000i 1.2340 - 0.0000i -0.3207 + 0.0000i ```

Compare the magnitude responses of the filters using FVTool.

```fvt = fvtool(b,a,num,den); legend(fvt,'Prototype','Target')```

The target filter has complex coefficients and is indeed a bandpass filter.

## 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 values to be transformed from the prototype filter, specified as a row vector with even number of elements. Frequencies in `Wo` should be normalized to be between `0` and `1`, with `1` corresponding to half the sample rate.

Note

Length of `Wo` and `Wt` vectors must be the same.

Data Types: `single` | `double`

Desired frequency locations in the transformed target filter, specified as a row vector with even number of elements. Frequencies in `Wt` should be normalized to be between `-1` and `1`, with `1` corresponding to half the sample rate.

Note

Length of `Wo` and `Wt` vectors must be the same.

Data Types: `single` | `double`

## Output Arguments

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Numerator 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 N-Point Filter Transformation

IIR lowpass filter to IIR complex N-point filter transformation effectively places `N` features of the original filter, located at frequencies Wo1,...,WoN, at the required target frequency locations, Wt1,...,WtM.

Relative positions of other features of the original filter are the same in the target filter for the Nyquist mobility and are reversed for the DC mobility. For the Nyquist mobility 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. For DC mobility feature F2 will precede F1 after the transformation.

Choice of the feature subject to this transformation is not restricted to the cutoff frequency of an original lowpass filter. In general it is possible to select any feature; e.g., a stopband edge, DC, the deep minimum in the stopband, or other ones. Select features such that there is no band overlap when creating N bands around the unit circle.

IIR lowpass filter to IIR complex N-point filter transformation can also be used to transform 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.