Transform IIR lowpass filter to IIR M-band filter

```
[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)

```
[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 `M`

th-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 W_{o}, at the required
target frequency locations, W_{t1},...,W_{tM}.

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, 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. 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.

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

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

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.

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.

Variable | Description |
---|---|

`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 ( |

`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.

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.

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