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# iirbpc2bpc

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

## Syntax

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

## Description

[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 Wo1 and Wo2, at the required target frequency locations, Wt1, and Wt2 respectively. It is assumed that Wt2 is greater than Wt1. 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, F1 and F2, with F1 preceding F2. Feature F1 will still precede F2 after the transformation. However, the distance between F1 and F2 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.

## Examples

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.

## Arguments

VariableDescription
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.