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Transform IIR lowpass filter to IIR complex N-point filter

`[Num,Den,AllpassNum,AllpassDen] =
iirlp2xc(B,A,Wo,Wt)[G,AllpassNum,AllpassDen] = iirlp2xc(Hd,Wo,Wt)`

where `Hd` is a `dfilt` object

`[Num,Den,AllpassNum,AllpassDen] =
iirlp2xc(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 complex multipoint
frequency transformation.

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`.

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., a stopband
edge, 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] = iirlp2xc(Hd,Wo,Wt)` returns
transformed `dfilt` object `G` with
an IIR complex 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.

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

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

Create the complex bandpass filter from the real lowpass filter:

[num, den] = iirlp2xc(b, a, [-0.5 0.5], [-0.25 0.25]);

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

fvtool(b, a, num, den);

Reviewing the coefficients and the figure produced by the example shows that the target filter has complex coefficients and is indeed a bandpass filter as expected.

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. They should be normalized to be between 0 and 1, with 1 corresponding to half the sample rate. |

Wt | Desired frequency locations in the transformed target filter. They should be normalized to be between -1 and 1, with 1 corresponding to half the sample rate. |

Num | Numerator of the target filter. |

Den | Denominator of the target filter. |

AllpassNum | Numerator of the mapping filter. |

AllpassDen | Denominator of the mapping filter. |

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