iirlp2hp

Transform IIR lowpass filter to IIR highpass filter

Syntax

[num,den] = iirlp2hp(b,a,wo,wt)

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

[tfiltObj,allpassNum,allpassDen] =
iirlp2hp(pfiltObj,wo,wt)

Description

[num,den] = iirlp2hp(b,a,wo,wt) transforms an IIR lowpass filter to an IIR highpass filter.

The iirlp2hp function returns the numerator and denominator coefficients of the transformed highpass filter. The function accepts wo, 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 on the input prototype filter. The input prototype lowpass filter is specified with the numerator and denominator coefficients, b and a respectively. For more details on the transformation, see IIR Lowpass to Highpass Frequency Transformation.

example

[num,den,allpassNum,allpassDen] = iirlp2hp(b,a,wo,wt) additionally returns the numerator and the denominator coefficients of the mapping filter.

[tfiltObj,allpassNum,allpassDen] = iirlp2hp(pfiltObj,wo,wt) specifies the input prototype lowpass IIR filter as a filter object. The function returns the target highpass IIR filter as a filter object. (since R2026a)

example

Examples

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Transform IIR Lowpass Filter to IIR Highpass Filter

Open Live Script

Design an IIR lowpass filter using the iirlpnorm function. Transform the filter to highpass by moving the magnitude response at one frequency in the source filter to a new location in the transformed filter.

Input IIR lowpass 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. The function returns the coefficients both in the vector form and in the second-order sections (SOS) form. The output argument g specifies the overall gain of the filter when expressed in the second-order sections form.

[b,a,~,sos,g] = 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]);

Visualize the magnitude response of the filter.

filterAnalyzer(b,a)

Transform Filter Using iirlp2hp

Transform the IIR lowpass filter using the iirlp2hp function. Specify the filter as a vector of numerator and denominator coefficients.

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.

wc = 0.0175;
wd = 0.4;
[num,den] = iirlp2hp(b,a,wc,wd);

Compare the magnitude responses of the filters. 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 filters share common ripple characteristics and magnitude.

filterAnalyzer(b,a,num,den,...
    FilterNames=["PrototypeFilter_TFForm",...
    "TransformedHighpassFilter"])

Alternatively, you can also specify the input IIR lowpass filter as a matrix of coefficients. Pass the scaled second order section coefficient matrices as inputs. Apply the scaling factor g to the first section of the filter.

sosg = sos;
sosg(1,1:3) = g*sosg(1,1:3);
[num2,den2] = iirlp2hp(sosg(:,1:3),sosg(:,4:6),wc,wd);

Use the sos2ctf function to convert the second-order section matrices to the cascaded transfer function form.

[b_ctf,a_ctf] = sos2ctf(sosg);

Compare the magnitude response of the filters.

filterAnalyzer(b_ctf,a_ctf,num2,den2,...
    FilterNames=["PrototypeFilterFromSOSMatrix",...
    "TransformedHighpassFilterFromSOSMatrix"])

Transform IIR Lowpass Filter Object to IIR Highpass Filter Object

Since R2026a

Open Live Script

Design and implement an IIR lowpass filter object using the designLowpassIIR function. The function returns a dsp.SOSFilter object when you set the SystemObject argument to true.

psosFilt = designLowpassIIR(FilterOrder=30,HalfPowerFrequency=0.5,DesignMethod="butter",...
        SystemObject=true)

psosFilt = 
  dsp.SOSFilter with properties:

            Structure: 'Direct form II transposed'
    CoefficientSource: 'Property'
            Numerator: [15×3 double]
          Denominator: [15×3 double]
       HasScaleValues: false

  Show all properties

Create a dsp.DynamicFilterVisualizer object to visualize the magnitude response of the filter.

filterViz = dsp.DynamicFilterVisualizer(NormalizedFrequency=true,YLimits=[-80 20]);
filterViz(psosFilt)

Transform the IIR lowpass filter to an IIR highpass filter using the iirlp2hp function. Specify the function to transform 0.3 $π$ rad/sample in the prototype filter to 0.7 $π$ rad/sample in the target filter. The function implements the highpass filter as a dsp.SOSFilter object.

tsosFilt = iirlp2hp(psosFilt,0.3,0.7)

tsosFilt = 
  dsp.SOSFilter with properties:

            Structure: 'Direct form II transposed'
    CoefficientSource: 'Property'
            Numerator: [15×3 double]
          Denominator: [15×3 double]
       HasScaleValues: false

  Show all properties

Visualize the magnitude response of the transformed IIR highpass filter.

filterViz(tsosFilt)

Input Arguments

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`b` — Numerator coefficients of prototype lowpass IIR filter
row vector | matrix

Numerator coefficients of the prototype lowpass IIR filter, specified as either:

Row vector –– Specifies the values of [b₀, b₁, …, b_n], given this transfer function form:

$H (z) = \frac{B (z)}{A (z)} = \frac{b_{0} + b_{1} z^{- 1} + \dots + b_{n} z^{- n}}{a_{0} + a_{1} z^{- 1} + \dots + a_{n} z^{- n}},$
where n is the order of the filter.
Matrix –– Specifies the numerator coefficients in the form of an P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each filter section. If Q = 2, the filter is a second-order section filter. For higher-order sections, make Q > 2.

$b = [\begin{matrix} b_{01} & b_{11} & b_{21} & ... & b_{Q 1} \\ b_{02} & b_{12} & b_{22} & ... & b_{Q 2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ b_{0 P} & b_{1 P} & b_{2 P} & \dots & b_{Q P} \end{matrix}]$
In the transfer function form, the numerator coefficient matrix b_ik of the IIR filter can be represented using the following equation:

$H (z) = \prod_{k = 1}^{P} H_{k} (z) = \prod_{k = 1}^{P} \frac{b_{0 k} + b_{1 k} z^{- 1} + b_{2 k} z^{- 2} + \dots + b_{Q k} z^{- Q}}{a_{0 k} + a_{1 k} z^{- 1} + a_{2 k} z^{- 2} + \dots + a_{Q k} z^{- Q}},$
where,
- a –– Denominator coefficients matrix. For more information on how to specify this matrix, see a.
- k –– Row index.
- i –– Column index.
When specified in the matrix form, b and a matrices must have the same number of rows (filter sections) Q.

Data Types: single | double
Complex Number Support: Yes

`a` — Denominator coefficients of prototype lowpass IIR filter
row vector | matrix

Denominator coefficients for a prototype lowpass IIR filter, specified as one of these options:

Row vector –– Specifies the values of [a₀, a₁, …, a_n], given this transfer function form:

$H (z) = \frac{B (z)}{A (z)} = \frac{b_{0} + b_{1} z^{- 1} + \dots + b_{n} z^{- n}}{a_{0} + a_{1} z^{- 1} + \dots + a_{n} z^{- n}},$
where n is the order of the filter.
Matrix –– Specifies the denominator coefficients in the form of an P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each filter section. If Q = 2, the filter is a second-order section filter. For higher-order sections, make Q > 2.

$a = [\begin{matrix} a_{01} & a_{11} & a_{21} & \dots & a_{Q 1} \\ a_{02} & a_{12} & a_{22} & \dots & a_{Q 2} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ a_{0 P} & a_{1 P} & a_{2 P} & \dots & a_{Q P} \end{matrix}]$
In the transfer function form, the denominator coefficient matrix a_ik of the IIR filter can be represented using the following equation:

$H (z) = \prod_{k = 1}^{P} H_{k} (z) = \prod_{k = 1}^{P} \frac{b_{0 k} + b_{1 k} z^{- 1} + b_{2 k} z^{- 2} + \dots + b_{Q k} z^{- Q}}{a_{0 k} + a_{1 k} z^{- 1} + a_{2 k} z^{- 2} + \dots + a_{Q k} z^{- Q}},$
where,
- b –– Numerator coefficients matrix. For more information on how to specify this matrix, see b.
- k –– Row index.
- i –– Column index.
When specified in the matrix form, a and b matrices must have the same number of rows (filter sections) P.

Data Types: single | double
Complex Number Support: Yes

`pfiltObj` — Prototype lowpass IIR filter
filter object

Since R2026a

Prototype lowpass IIR filter, specified as one of these filter objects:

digitalFilter
dsp.IIRFilter
dsp.SOSFilter
dsp.FourthOrderSectionFilter
dsp.FilterCascade — Each filter stage in the filter cascade must be one of the supported filter objects or a scalar.
dsp.ParallelFilter — Each filter branch in the parallel filter must be one of the supported filter objects or a scalar.

`wo` — Frequency value to be transformed from prototype filter
positive scalar in (`0`, `1`)

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

Data Types: single | double

`wt` — Desired frequency location in transformed highpass filter
positive scalar in (`0`, `1`)

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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`num` — Numerator coefficients of transformed highpass filter
row vector | matrix

Numerator coefficients of the transformed highpass filter, returned as one of the following:

Row vector of length n+1, where n is the order of the input filter. The num output is a row vector when the input coefficients b and a are row vectors.
P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each section of the transformed filter. The num output is a matrix when the input coefficients b and a are matrices.

Data Types: single | double
Complex Number Support: Yes

`den` — Denominator coefficients of transformed highpass filter
row vector | matrix

Denominator coefficients of the transformed highpass filter, returned as one of the following:

Row vector of length n+1, where n is the order of the input filter. The den output is a row vector when the input coefficients b and a are row vectors.
P-by-(Q+1) matrix, where P is the number of filter sections and Q is the order of each section of the transformed filter. The den output is a matrix when the input coefficients b and a are matrices.

Data Types: single | double

`tfiltObj` — Target highpass IIR filter
filter object

Since R2026a

Target highpass IIR filter, returned as one of these filter objects:

`allpassNum` — Numerator coefficients of mapping filter
row vector

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

Data Types: single | double

`allpassDen` — Denominator coefficients of mapping filter
row vector

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

Data Types: single | double

More About

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IIR Lowpass to Highpass Frequency Transformation

IIR lowpass to 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 0 and 1 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 filterAnalyzer 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.

Extended Capabilities

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C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2011a

expand all

R2026a: Support for filter objects

You can now specify the input lowpass IIR filter as one of these filter objects:

digitalFilter
dsp.IIRFilter
dsp.SOSFilter
dsp.FourthOrderSectionFilter
dsp.FilterCascade — Each filter stage in the filter cascade must be one of the supported filter objects or a scalar.
dsp.ParallelFilter — Each filter branch in the parallel filter must be one of the supported filter objects or a scalar.

iirlp2hp

Syntax

Description

Examples

Transform IIR Lowpass Filter to IIR Highpass Filter

Transform IIR Lowpass Filter Object to IIR Highpass Filter Object

Input Arguments

b — Numerator coefficients of prototype lowpass IIR filter row vector | matrix

a — Denominator coefficients of prototype lowpass IIR filter row vector | matrix

pfiltObj — Prototype lowpass IIR filter filter object

wo — Frequency value to be transformed from prototype filter positive scalar in (0, 1)

wt — Desired frequency location in transformed highpass filter positive scalar in (0, 1)

Output Arguments

num — Numerator coefficients of transformed highpass filter row vector | matrix

den — Denominator coefficients of transformed highpass filter row vector | matrix

tfiltObj — Target highpass IIR filter filter object

allpassNum — Numerator coefficients of mapping filter row vector

allpassDen — Denominator coefficients of mapping filter row vector

More About

IIR Lowpass to Highpass Frequency Transformation

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

Version History

R2026a: Support for filter objects

See Also

Functions

`b` — Numerator coefficients of prototype lowpass IIR filter
row vector | matrix

`a` — Denominator coefficients of prototype lowpass IIR filter
row vector | matrix

`pfiltObj` — Prototype lowpass IIR filter
filter object

`wo` — Frequency value to be transformed from prototype filter
positive scalar in (`0`, `1`)

`wt` — Desired frequency location in transformed highpass filter
positive scalar in (`0`, `1`)

`num` — Numerator coefficients of transformed highpass filter
row vector | matrix

`den` — Denominator coefficients of transformed highpass filter
row vector | matrix

`tfiltObj` — Target highpass IIR filter
filter object

`allpassNum` — Numerator coefficients of mapping filter
row vector

`allpassDen` — Denominator coefficients of mapping filter
row vector

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.