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Allpass filter for complex bandpass transformation

`[AllpassNum,AllpassDen] = allpassbpc2bpc(Wo,Wt)`

`[AllpassNum,AllpassDen] = allpassbpc2bpc(Wo,Wt)` returns
the numerator, `AllpassNum`, and the denominator, `AllpassDen`,
of the first-order allpass mapping filter for performing a complex
bandpass to complex bandpass frequency transformation. This transformation
effectively places two features of an original filter, located at
frequencies W_{o1} and W_{o2},
at the required target frequency locations W_{t1} and
W_{t2}. It is assumed that W_{t2} is
greater than W_{t1}. In most of the cases the
features selected for the transformation are the band edges of the
filter passbands. 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.

Relative positions of other features of an original filter do
not change in the target filter. 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.

This transformation can also be used for transforming other types of filters; e.g., complex notch filters or resonators can be repositioned at two distinct desired frequencies at any place around the unit circle. This is very attractive for adaptive systems.

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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