# iirlp2hp

Transform lowpass IIR filter to highpass filter

## Syntax

``[num,den] = iirlp2hp(b,a,wo,wt)``
``````[num,den,allpassNum,allpassDen] = iirbpc2bpc(b,a,wo,wt)``````

## Description

example

````[num,den] = iirlp2hp(b,a,wo,wt)`transforms the lowpass IIR filter to highpass filter.The `iirlp2hp` function defined by the input coefficients `b` and `a`, to a highpass IIR filter defined by the coefficients `num` and `den`. The function accepts `w0`, frequency value to be transformed from the prototype filter, and `wt`, desired frequency in the transformed highpass filter, and applies the lowpass to highpass frequency transformation. For more details, see IIR Lowpass to Complex Highpass Frequency Transformation. ```
``````[num,den,allpassNum,allpassDen] = iirbpc2bpc(b,a,wo,wt)```in addition returns the numerator and the denominator coefficients of the mapping filter.```

## Examples

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This example transforms an IIR filter from lowpass to highpass by moving the magnitude response at one frequency in the source filter to a new location in the transformed filter.

Generate a least P-norm optimal IIR lowpass filter with varying attenuation levels in the stopband. Specify a numerator order of 10 and a denominator order of 6. Visualize the magnitude response of the filter.

```[b,a] = iirlpnorm(10,6,[0 0.0175 0.02 0.0215 0.025 1], ... [0 0.0175 0.02 0.0215 0.025 1],[1 1 0 0 0 0], ... [1 1 1 1 20 20]); fvtool(b,a)```

To generate a highpass filter whose passband flattens out at 0.4π rad/sample, select the frequency in the lowpass filter at 0.0175π, the frequency where the passband starts to roll off, and move it to the new location. Compare the magnitude responses of the filters using FVTool.

```wc = 0.0175; wd = 0.4; [num,den] = iirlp2hp(b,a,wc,wd); hvft = fvtool(b,a,num,den); legend(hvft,'Prototype','Target')```

The transition band for the highpass filter is essentially the mirror image of the transition for the lowpass filter from 0.0175π to 0.025π, stretched out over a wider frequency range. In the passbands, the filter share common ripple characteristics and magnitude.

## Input Arguments

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Numerator coefficients of the prototype lowpass IIR filter, specified as a row vector.

Data Types: `single` | `double`
Complex Number Support: Yes

Denominator coefficients of the prototype lowpass IIR filter, specified as a row vector.

Data Types: `single` | `double`
Complex Number Support: Yes

Frequency value to be transformed from the prototype filter, specified as a real positive scalar in the range `(0,1)`.

Data Types: `single` | `double`

Desired frequency location in the transformed highpass filter, specified as a real positive scalar in the range `(0,1)`.

Data Types: `single` | `double`

## Output Arguments

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Numerator coefficients of the transformed highpass filter, returned as a row vector.

Data Types: `single` | `double`
Complex Number Support: Yes

Denominator coefficients of the transformed highpass filter, returned as a row vector.

Data Types: `single` | `double`

Numerator coefficients of the mapping filter, returned as a row vector.

Data Types: `single` | `double`

Denominator coefficients of the mapping filter, returned as a row vector.

Data Types: `single` | `double`

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### IIR Lowpass to Complex Highpass Frequency Transformation

IIR lowpass to complex highpass frequency transformation takes a selected frequency from the lowpass filter, `wo`, and maps the corresponding magnitude response value onto the desired frequency location in the highpass filter, `wt`. Note that all frequencies are normalized between zero and one and that the filter order does not change when you transform to a highpass filter.

When you select `wo` and designate `wt`, the transformation algorithm sets the magnitude response at the `wt` values of your bandstop filter to be the same as the magnitude response of your lowpass filter at `wo`. Filter performance between the values in `wt` is not specified, except that the stopband retains the ripple nature of your original lowpass filter and the magnitude response in the stopband is equal to the peak response of your lowpass filter. To accurately specify the filter magnitude response across the stopband of your bandpass filter, use a frequency value from within the stopband of your lowpass filter as `wo`. Then your bandstop filter response is the same magnitude and ripple as your lowpass filter stopband magnitude and ripple.

The transformation retains the shape of the original filter is what makes this function useful. If you have a lowpass filter whose characteristics, such as rolloff or passband ripple, particularly meet your needs, the transformation function lets you create a new filter with the same characteristic performance features, but in a highpass version without designing the highpass filter from the beginning.

In some cases transforming your filter may cause numerical problems, resulting in incorrect conversion to the highpass filter. Use `fvtool` to verify the response of your converted filter.

## References

[1] Nowrouzian, B., and A.G. Constantinides. “Prototype Reference Transfer Function Parameters in the Discrete-Time Frequency Transformations.” In Proceedings of the 33rd Midwest Symposium on Circuits and Systems, 1078–82. Calgary, Alta., Canada: IEEE, 1991. https://doi.org/10.1109/MWSCAS.1990.140912.

[2] Nowrouzian, B., and L.T. Bruton. “Closed-Form Solutions for Discrete-Time Elliptic Transfer Functions.” In [1992] Proceedings of the 35th Midwest Symposium on Circuits and Systems , 784–87. Washington, DC, USA: IEEE, 1992. https://doi.org/10.1109/MWSCAS.1992.271206.

[3] Constantinides, A.G.“Spectral transformations for digital filters.” Proceedings of the IEEE, vol. 117, no. 8: 1585-1590. August 1970.