Transform IIR lowpass filter to IIR real N-point filter

```
[Num,Den,AllpassNum,AllpassDen] =
iirlp2xn(B,A,Wo,Wt)
```

[Num,Den,AllpassNum,AllpassDen]=
iirlp2xn(B,A,Wo,Wt,Pass)

[G,AllpassNum,AllpassDen] = iirlp2xn(Hd,Wo,Wt)

[G,AllpassNum,AllpassDen] = iirlp2xn(...,Pass)

```
[Num,Den,AllpassNum,AllpassDen] =
iirlp2xn(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 `N`

th-order real lowpass to real multipoint
frequency transformation, where `N`

is the number
of features being mapped. By default the DC feature is kept at its
original location.

```
[Num,Den,AllpassNum,AllpassDen]=
iirlp2xn(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 allowed to move.

It also returns the numerator, `AllpassNum`

,
and the denominator, `AllpassDen`

, of the allpass
mapping filter. The prototype lowpass filter is given with the numerator
specified by `B`

and the denominator specified by `A`

.

Parameter `N`

also specifies the number of
replicas of the prototype filter created around the unit circle after
the transformation. This transformation effectively places `N`

features
of an original filter, located at frequencies W_{o1},...,W_{oN},
at the required target frequency locations, W_{t1},...,W_{tM}.

Relative positions of other features of an 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, 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. For DC mobility
feature F_{2} will precede F_{1} 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. The
only condition is that the features must be selected in such a way
that when creating `N`

bands around the unit circle,
there will be no band overlap.

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] = iirlp2xn(Hd,Wo,Wt)`

returns
transformed `dfilt`

object `G`

with
an IIR real N-point 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] = iirlp2xn(...,Pass)`

returns
transformed `dfilt`

object `G`

with
an IIR real N-point 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);

Move the cutoffs of the prototype filter to the new locations `W`

_{t1}`=0.25`

and `W`

_{t2}`=0.75`

creating
a real bandpass filter:

[num, den] = iirlp2xn(b, a, [-0.5 0.5], [0.25 0.75], ... pass');

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

fvtool(b, a, num, den);

`iirlp2xn`

has created the desired bandpass
filter with the cutoff locations specified in the command.

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

`B` | Numerator of the prototype lowpass filter |

`A` | Denominator of the prototype lowpass filter |

`Wo` | Frequency values 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.

Cain, G.D., A. Krukowski and I. Kale, "High
Order Transformations for Flexible IIR Filter Design," *VII
European Signal Processing Conference (EUSIPCO'94)*, vol.
3, pp. 1582-1585, Edinburgh, United Kingdom, September 1994.

Krukowski, A., G.D. Cain and I. Kale, "Custom
designed high-order frequency transformations for IIR filters," *38th
Midwest Symposium on Circuits and Systems (MWSCAS'95)*,
Rio de Janeiro, Brazil, August 1995.

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