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Zero-pole-gain lowpass to complex M-band frequency transformation

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

`[Z2,P2,K2,AllpassNum,AllpassDen]
= zpklpmbc(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 `M`th-order real lowpass to complex multibandpass
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 locations, W_{t1},...,W_{tM}.

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, for example, 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., to replicate notch filters and resonators at any required location.

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); [z1,p1,k1] = zpklp2mbc(z, p, k, 0.5, [2 4 6 8]/10); [z2,p2,k2] = zpklp2mbc(z, p, k, 0.5, [2 4 6 8]/10);

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

fvtool(b, a, k1*poly(z1), poly(p1), k2*poly(z2), poly(p2));

You could review the coefficients to compare the filters, but the graphical comparison shown here is quicker and easier.

However, looking at the coefficients in FVTool shows the complex nature desired.

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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