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Transform IIR complex bandpass filter to IIR complex bandpass filter with different characteristics

`[Num,Den,AllpassNum,AllpassDen] =
iirbpc2bpc(B,A,Wo,Wt)`

`[Num,Den,AllpassNum,AllpassDen] =
iirbpc2bpc(B,A,Wo,Wt)` returns the numerator and denominator
vectors, `Num` and `Den` respectively,
of the target filter transformed from the complex bandpass prototype
by applying a first-order complex bandpass to complex bandpass frequency
transformation.

It also returns the numerator, `AllpassNum`,
and the denominator, `AllpassDen`, of the allpass
mapping filter. The prototype lowpass filter is given with the numerator
specified by `B` and the denominator specified by `A`.

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} respectively. 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; e.g., in the adaptive system.

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

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

Create a complex passband from 0.25 to 0.75:

[b, a] = iirlp2bpc (b, a, 0.5, [0.25,0.75]); [num, den] = iirbpc2bpc(b, a, [0.25, 0.75], [-0.5, 0.5]);

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

fvtool(b, a, num, den);

Using FVTool to plot the filters shows you the comparison, presented in this figure.

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 |

Wt | Desired frequency locations in the transformed target filter |

Num | Numerator of the target filter |

Den | Denominator of the 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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