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# 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)
[G,AllpassNum,AllpassDen] = iirlp2mb(Hd,Wo,Wt)
[G,AllpassNum,AllpassDen] = iirlp2mb(...,Pass)

## Description

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

[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" and the "Nyquist Mobility." In the first case the Nyquist feature stays at its original location and the DC feature is free to move. In the second case the DC feature is kept at an original frequency and the Nyquist feature is movable.

It also returns the numerator, AllpassNum, and the denominator, AllpassDen, of the allpass mapping filter. The prototype lowpass filter is given with a numerator specified by B and a denominator specified by A.

This transformation effectively places one feature of an original filter, located at frequency Wo, at the required target frequency locations, Wt1,...,WtM.

Relative positions of other features of an 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. In general it is possible to select any feature; e.g., the stopband edge, the DC, the deep minimum in the stopband, or other ones.

This transformation can also be used for transforming other types of filters; e.g., 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.

[G,AllpassNum,AllpassDen] = iirlp2mb(Hd,Wo,Wt) returns transformed dfilt object G with an IIR real M-band filter frequency response. The coefficients AllpassNum and AllpassDen represent the allpass mapping filter for mapping the prototype filter frequency Wo and the target frequencies vector Wt. Note that in this syntax Hd is a dfilt object with a lowpass magnitude response.

[G,AllpassNum,AllpassDen] = iirlp2mb(...,Pass) returns transformed dfilt object G with an IIR real M-band filter frequency response. This syntax allows you to specify an additional parameter, Pass, which chooses between using the "DC Mobility" and the "Nyquist Mobility." In the first case the Nyquist feature stays at its original location and the DC feature is free to move. In the second case the DC feature is kept at an original frequency and the Nyquist feature is allowed to move.

The coefficients AllpassNum and AllpassDen represent the allpass mapping filter for mapping the prototype filter frequency Wo and the target frequencies vector Wt. Note that in this syntax Hd is a dfilt object with a lowpass magnitude response.

## Examples

Design a prototype real IIR halfband filter using a standard elliptic approach:

`[b, a] = ellip(3, 0.1, 30, 0.409);`

### Example 1

Create the real multiband filter with two passbands:

```[num1, den1] = iirlp2mb(b, a, 0.5, [2 4 6 8]/10);
[num2, den2] = iirlp2mb(b, a, 0.5, [2 4 6 8]/10, 'pass');```

The second code snippet uses the pass option to select the Nyquist mobility option. In this case the resulting filter is the same.

### Example 2

Create the real multiband filter with two stopbands:

`[num3, den3] = iirlp2mb(b, a, 0.5, [2 4 6 8]/10, 'stop');`

Verify the result by comparing the prototype filter with target filters:

`fvtool(b, a, num1, den1, num2, den2, num3, den3);`

Combining all of the filters, prototypes and targets, on one figure makes comparing them straightforward. Passbands for the filters in example 1 appear separately in the figure, although they overlap to a degree that makes them hard to identify — they have identical coefficients.

## Arguments

VariableDescription
B

Numerator of the prototype lowpass filter

A

Denominator of the prototype lowpass filter

Wo

Frequency value to be transformed from the prototype filter

Wt

Desired frequency locations in the transformed target filter

Pass

Choice ('pass'/'stop') of passband/stopband at DC, 'pass' being the default

Num

Numerator of the target filter

Den

Denominator of the target filter

AllpassNum

Numerator of the mapping filter

AllpassDen

Denominator of the mapping filter

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

## References

Franchitti, J.C., "All-pass filter interpolation and frequency transformation problems," MSc Thesis, Dept. of Electrical and Computer Engineering, University of Colorado, 1985.

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.

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

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.