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# iirlp2bs

Transform IIR lowpass filter to IIR bandstop filter

## Syntax

```[Num,Den,AllpassNum,AllpassDen] = iirlp2bs(B,A,Wo,Wt)[G,AllpassNum,AllpassDen] = iirlp2bs(Hd,Wo,Wt)```

where `Hd` is a `dfilt` object

## Description

```[Num,Den,AllpassNum,AllpassDen] = iirlp2bs(B,A,Wo,Wt)``` returns the numerator and denominator vectors, `Num` and `Den` of the bandstop digital filter. `AllpassNum` and `AllpassDen` are the vectors of numerator and denominator coefficients of the allpass mapping filter. The prototype lowpass filter is given with a numerator specified by `B` and a denominator specified by `A`.

This transformation effectively places one feature of an original filter, located at frequency -Wo, at the required target frequency location, Wt1, and the second feature, originally at `+`Wo, at the new location, Wt2. 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. It is assumed that Wt2 is greater than Wt1. Frequencies must be normalized to be between 0 and 1, with 1 corresponding to half the sample rate.

This transformation implements the "Nyquist Mobility," which means that the DC feature stays at DC, but the Nyquist feature moves to a location dependent on the selection of Wo and Wts.

Relative positions of other features of an original filter 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. After the transformation feature F2 will precede F1 in the target filter. However, the distance between F1 and F2 will not be the same before and after the transformation.

For more details on the lowpass to bandstop frequency transformation, see Digital Frequency Transformations.

`[G,AllpassNum,AllpassDen] = iirlp2bs(Hd,Wo,Wt)` returns transformed `dfilt` object `G` with a bandstop magnitude response. The coefficients `AllpassNum` and `AllpassDen` represent the allpass mapping filter for mapping the prototype filter frequency `Wo` and the target frequencies vector `Wt`. Note that in this syntax `Hd` is a `dfilt` object with a lowpass magnitude response.

## Examples

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

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

Create the real bandstop filter by placing the cutoff frequencies of the prototype filter at the band edge frequencies `W`t1`=0.25` and `W`t2`=0.75`:

`[num, den] = iirlp2bs(b, a, 0.5, [0.25, 0.75]);`

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

`fvtool(b, a, num, den);`

With both filters plotted in the figure, you see clearly the results of the transformation.

## Arguments

VariableDescription
`B`

Numerator of the prototype lowpass filter

`A`

Denominator of the prototype lowpass filter

`Wo`

Frequency value 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 0 and 1, with 1 corresponding to half the sample rate.

## References

Constantinides, A.G., "Spectral transformations for digital filters," IEEE® Proceedings, vol. 117, no. 8, pp. 1585-1590, August 1970.

Nowrouzian, B. and A.G. Constantinides, "Prototype reference transfer function parameters in the discrete-time frequency transformations," Proceedings 33rd Midwest Symposium on Circuits and Systems, Calgary, Canada, vol. 2, pp. 1078-1082, August 1990.

Nowrouzian, B. and L.T. Bruton, "Closed-form solutions for discrete-time elliptic transfer functions," Proceedings of the 35th Midwest Symposium on Circuits and Systems, vol. 2, pp. 784-787, 1992.

Constantinides, A.G., "Design of bandpass digital filters," IEEE Proceedings, vol. 1, pp. 1129-1231, June 1969.