# iirlp2mb

Transform IIR lowpass filter to IIR M-band filter

## Syntax

``````[Num,Den,AllpassNum,AllpassDen] = iirlp2mb(B,A,Wo,Wt)``````
``````[Num,Den,AllpassNum,AllpassDen] = iirlp2mb(B,A,Wo,Wt,Pass)``````

## Description

example

``` ```[Num,Den,AllpassNum,AllpassDen] = iirlp2mb(B,A,Wo,Wt)``` transform IIR lowpass filter to IIR M-band filter.The `iirlp2mb` function 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 real multiple bandpass frequency mapping. By default the DC feature is kept at its original location. For more details, see IIR Lowpass Filter to IIR M-Band Filter Transformation.```
``` ```[Num,Den,AllpassNum,AllpassDen] = iirlp2mb(B,A,Wo,Wt,Pass)``` allows you to specify an additional parameter, `Pass`, which chooses between using the “DC Mobility”, the Nyquist feature stays at its original location and the DC feature is free to move and the “Nyquist Mobility.”, the DC feature is kept at an original frequency and the Nyquist feature is movable.```

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`.

Note

Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

## 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 real multiband filter with two passbands.

`[num1,den1] = iirlp2mb(b,a,0.5,[2 4 6 8]/10);`

Create a real multiband filter with two stopbands.

`[num2,den2] = iirlp2mb(b,a,0.5,[2 4 6 8]/10, 'stop');`

Compare the magnitude responses of the filters using FVTool.

```hvft = fvtool(b,a,num1,den1,num2,den2); legend(hvft,'Prototype','Two passbands','Two stopbands')```

## 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 real scalar. Frequency `Wo` must 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` must be normalized to be between `0` and `1`, with `1` corresponding to half the sample rate.

Data Types: `single` | `double`

Choice of passband or stopband at DC, specified as a `'pass'` or `'stop'` .

## 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`

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

Data Types: `single` | `double`

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

Data Types: `single` | `double`

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

IIR lowpass filter to IIR 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. 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 M-band 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] Franchitti, J.C., “All-pass filter interpolation and frequency transformation problems.” MSc Thesis, Dept. of Electrical and Computer Engineering, University of Colorado, 1985.

[2] Feyh, G., J.C. Franchitti and C.T. Mullis.“All-pass filter interpolation and frequency transformation problem.” Proceedings 20th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, California, pp. 164-168, November 1986.

[3] Mullis, C.T. and R. A. Roberts. Digital Signal Processing, section 6.7, Reading, Mass., Addison-Wesley, 1987.

[4] Feyh, G., W.B. Jones and C.T. Mullis. “An extension of the Schur Algorithm for frequency transformations.” Linear Circuits, Systems and Signal Processing: Theory and Application. C. J. Byrnes et al Eds, Amsterdam: Elsevier, 1988.