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Zero-pole-gain lowpass to complex bandstop frequency transformation

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2bsc(Z,P,K,Wo,Wt)`

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklp2bsc(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 a first-order real lowpass to complex bandstop 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`.

This transformation effectively places one feature of an original
filter, located at frequency -W_{o}, at the required
target frequency location, W_{t1}, and the second
feature, originally at `+`W_{o},
at the new location, W_{t2}. It is assumed that
W_{t2} is greater than W_{t1}.
Additionally the transformation swaps passbands with stopbands in
the target filter.

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.

Choice of the feature subject to the lowpass to bandstop transformation is not restricted only 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.

Lowpass to bandpass transformation can also be used for transforming other types of filters; e.g., real notch filters or resonators can be doubled and positioned at two distinct desired frequencies at any place around the unit circle forming a pair of complex notches/resonators.

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] = zpklp2bsc(z, p, k, 0.5, [0.2, 0.3]);

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

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

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 value to be transformed from the prototype filter. It 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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