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Allpass filter for lowpass to complex N-point transformation

`[AllpassNum,AllpassDen] = allpasslp2xc(Wo,Wt)`

`[AllpassNum,AllpassDen] = allpasslp2xc(Wo,Wt)` returns
the numerator, `AllpassNum`, and the denominator, `AllpassDen`,
of the `N`th-order allpass mapping filter, where `N` is
the allpass filter order, for performing a real lowpass to complex
multipoint frequency transformation. 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 the, 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.

Design the allpass filter moving four features of an original
complex filter given in `W`_{o} to
the new independent frequency locations `W`_{t}.
Please note that the transformation creates `N` replicas
of an original filter around the unit circle, where `N` is
the order of the allpass mapping filter:

```
Wo = [-0.2, 0.3, -0.7, 0.4]; Wt = [0.3, 0.5, 0.7, 0.9];
[AllpassNum, AllpassDen] = allpasslp2xc(Wo, Wt);
[h, f] = freqz(AllpassNum, AllpassDen, 'whole');
```

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

Wo | Frequency values to be transformed from the prototype filter |

Wt | Desired frequency locations in the transformed target filter |

AllpassNum | Numerator of the mapping filter |

AllpassDen | Denominator of the mapping filter |

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

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