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Zero-pole-gain lowpass to complex N-point frequency transformation

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2xc(Z,P,K,Wo,Wt)`

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2xc(Z,P,K,Wo,Wt)` returns zeros, `Z`_{2},
poles, `P`_{2}, and gain factor, `K`_{2},
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 zeros, `Z`,
poles, `P`, and gain factor, `K`.

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.

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

[b, a] = ellip(3,0.1,30,0.409); z = roots(b); p = roots(a); k = b(1); [z2,p2,k2] = zpklp2xc(z, p, k, [-0.5 0.5], [-0.25 0.25]);

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

fvtool(b, a, k2*poly(z2), poly(p2));

Plotting the filters on the same axes lets you compare the results graphically, shown here.

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

Z | Zeros of the prototype lowpass filter |

P | Poles of the prototype lowpass filter |

K | Gain factor 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. |

Z2 | Zeros of the target filter |

P2 | Poles of the target filter |

K2 | Gain factor of the target filter |

AllpassNum | Numerator of the mapping filter |

AllpassDen | Denominator of the mapping filter |

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