Lowpass IIR

Design lowpass infinite impulse response filter

Description

A lowpass filter attenuates the energy of an input signal above a specified frequency threshold. An infinite impulse response (IIR) filter computes an output signal as a running weighted average of input samples and previous output values.

Signal Processing Toolbox™ provides different ways to design lowpass IIR filters:

• designfilt function — Design a lowpass IIR filter at the command line or in a script. Use "lowpassiir" as the first argument when you call the function.

  For an example, see Design Lowpass IIR Filter using designfilt.

• Design Filter Live Editor task — Design a lowpass IIR filter as part of a live script. The task displays code that you can paste into other MATLAB^® programs.

  For an example, see Design Lowpass IIR Filter Using Live Editor Task.

• Filter Designer app — Design a lowpass IIR filter interactively. Export your design to the MATLAB workspace, to Simulink^®, or to a file.

  For an example, see Design Lowpass IIR Filter Using Filter Designer.

All of these methods return digitalFilter objects. If you have a DSP System Toolbox™ license, you can generate your design as a filter System object™ and include additional Design Method Options.

For other ways to design lowpass IIR filters using MATLAB, see Other Lowpass IIR Filter Design Functions.

Examples

Design Lowpass IIR Filter using designfilt

To design the filter at the command line or in a script, use the designfilt function with "lowpassiir" as the first argument. Use name-value arguments to specify your design further.

d = designfilt("lowpassiir", ...   % Response type
    SampleRate=2000, ...           % Sample rate
    PassbandFrequency=400, ...     % Frequency constraints
    StopbandFrequency=550, ...
    PassbandRipple=5, ...          % Magnitude constraints
    StopbandAttenuation=25, ...
    DesignMethod="ellip", ...      % Design method
    MatchExactly="passband", ...   % Design method options
    ScaleSOS=true);
filterAnalyzer(d)

Design Lowpass IIR Filter Using Live Editor Task

To design the filter using the Design Filter Live Editor task:

1. Specify the filter:

   • Under Select filter response, select Lowpass IIR.

   • Under Sample Rate, specify Frequency units as Hz and Input sample rate (Hz) as 2000.

   • Under Specify frequency parameters, specify a passband frequency of 400 Hz and a stopband frequency of 550 Hz.

   • Under Specify magnitude parameters, specify a passband ripple of 5 dB and a stopband attenuation of 25 dB.

   • Under Specify algorithm, select Elliptic. Under Specify algorithm options, specify that the design match the Passband exactly.

2. The Live Editor task updates the filter automatically by default.

   

Design Lowpass IIR Filter Using Filter Designer

To design the filter using the Filter Designer app:

1. In the Response gallery of the Designer tab in the app toolstrip, select Lowpass IIR.

2. Specify the filter using the options in the Filter Parameters table:

   • In Design by, select Specification.

   • In Sample Rate, specify Frequency units as Hz and Input sample rate (Hz) as 2000.

   • Under Frequency Specifications, specify a passband frequency of 400 Hz and a stopband frequency of 550 Hz.

   • Under Magnitude Specifications, specify a passband ripple of 5 dB and a stopband attenuation of 25 Hz.

   • Under Algorithm, select Elliptic. Under Algorithm Options, specify that the design match the Passband exactly.

3. In the Filter section of the Designer tab, click Update Filter.

   

4. Export code to create your digital filter. On the toolstrip, click Export and select Generate MATLAB function > Digital Filter Object. The code appears in the editor.

   function designedFilter = lowpassiir1filt
   
   designedFilter = designfilt('lowpassiir', ...
       'PassbandFrequency',400,'StopbandFrequency',550, ...
       'PassbandRipple',5,'StopbandAttenuation',25, ...
       'SampleRate',2000,'DesignMethod','ellip', ...
       'MatchExactly','passband','ScaleSOS',true);
   
   end

Parameters

Design Method

Design Method — Filter design algorithm

Butterworth | Chebyshev type I | Chebyshev type II | elliptic | least Pth-norm

Specify the algorithm to design the filter. The available design methods correlate with the set of design specifications that you choose.

Butterworth — Butterworth algorithm

design method

Design the filter using the Butterworth algorithm. Butterworth filters have a smooth monotonic frequency response that is maximally flat in the passband. They trade rolloff steepness for flatness.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Specify DesignMethod="butter".	In the Specify algorithm section of the task, set Design method to Butterworth.	In the Algorithm section of the filter parameters panel, set Design method to Butterworth.

The Butterworth design method supports these design parameter combinations.

Filter Order	Frequency Specifications	Magnitude Specifications	Design Method Options	DSP System Toolbox License Required
N/A	Passband Frequency Stopband Frequency	Passband Ripple Stopband Attenuation	Match Exactly
Filter Order	Half-Power Frequency	N/A	N/A
Numerator Order Denominator Order	Half-Power Frequency	N/A	N/A

Chebyshev Type I — Chebyshev type I algorithm

design method

Design the filter using the Chebyshev type I algorithm. Chebyshev type I filters have a frequency response that is equiripple in the passband and maximally flat in the stopband. Their passband ripple increases with increasing rolloff steepness.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Specify DesignMethod="cheby1".	In the Specify algorithm section of the task, set Design method to Chebyshev type I.	In the Algorithm section of the filter parameters panel, set Design method to Chebyshev type I.

The Chebyshev type I design method supports these design parameter combinations.

Filter Order	Frequency Specifications	Magnitude Specifications	Design Method Options	DSP System Toolbox License Required
N/A	Passband Frequency Stopband Frequency	Passband Ripple Stopband Attenuation	Match Exactly
Filter Order	Passband Frequency	Passband Ripple	N/A
Filter Order	Passband Frequency Half-Power Frequency	N/A	N/A	✓
Filter Order	Half-Power Frequency	Passband Ripple	N/A	✓

Chebyshev Type II — Chebyshev type II algorithm

design method

Design the filter using the Chebyshev type II algorithm. Chebyshev type II filters have a frequency response that is maximally flat in the passband and equiripple in the stopband.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Specify DesignMethod="cheby2".	In the Specify algorithm section of the task, set Design method to Chebyshev type II.	In the Algorithm section of the filter parameters panel, set Design method to Chebyshev type II.

The Chebyshev type II design method supports these design parameter combinations.

Filter Order	Frequency Specifications	Magnitude Specifications	Design Method Options	DSP System Toolbox License Required
N/A	Passband Frequency Stopband Frequency	Passband Ripple Stopband Attenuation	Match Exactly
Filter Order	Stopband Frequency	Stopband Attenuation	N/A
Filter Order	Half-Power Frequency Stopband Frequency	N/A	N/A	✓
Filter Order	Half-Power Frequency	Stopband Attenuation	N/A	✓

Elliptic — Elliptic algorithm

design method

Design the filter using the elliptic algorithm. Elliptic filters have a frequency response that is equiripple in both passband and stopband.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Specify DesignMethod="ellip".	In the Specify algorithm section of the task, set Design method to Elliptic.	In the Algorithm section of the filter parameters panel, set Design method to Elliptic.

The elliptic design method supports these design parameter combinations.

Filter Order	Frequency Specifications	Magnitude Specifications	Design Method Options	DSP System Toolbox License Required
N/A	Passband Frequency Stopband Frequency	Passband Ripple Stopband Attenuation	Match Exactly
Filter Order	Passband Frequency	Passband Ripple Stopband Attenuation	N/A
Filter Order	Passband Frequency Stopband Frequency	Passband Ripple	N/A	✓
Filter Order	Half-Power Frequency	Passband Ripple Stopband Attenuation	N/A	✓

Least Pth-Norm — Least Pth-norm algorithm

design method

Design the filter using the least Pth-norm unconstrained optimization algorithm.

You must have a DSP System Toolbox license to use this design method.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Specify DesignMethod="lpnorm".	In the Specify algorithm section of the task, set Design method to IIR least p-norm.	In the Algorithm section of the filter parameters panel, set Design method to IIR least p-norm.

The least Pth-norm method supports these design parameter combinations.

Filter Order	Frequency Specifications	Magnitude Specifications	Design Method Options	DSP System Toolbox License Required
Filter Order	Passband Frequency Stopband Frequency	N/A	Norm Density Factor Initial Norm Initial Numerator Initial Denominator Maximum Pole Radius Passband Weight Stopband Weight	✓
Numerator Order Denominator Order	Passband Frequency Stopband Frequency	N/A	Norm Passband Weight Stopband Weight	✓

Sample Rate

Sample Rate — Filter sample rate

positive scalar

Specify the filter sample rate as a positive scalar expressed in hertz.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
To specify a sample rate, use the SampleRate name-value argument and input a value in hertz. To work with normalized frequencies, omit the sample rate.	In the Sample Rate section of the task: To specify a sample rate, set Frequency units to Hz and type a value for Input sample rate (Hz). To work with normalized frequencies, set Frequency units to Normalized.	In the Sample Rate section of the filter parameters panel: To specify a sample rate, set Frequency units to Hz and type a value for Input sample rate (Hz). To work with normalized frequencies, set Frequency units to Normalized.

Filter Order

Filter Order — Highest power in transfer function

positive integer

Specify the filter order N as a positive integer. Some design methods let you specify the order. Others generate minimum-order designs, which are the shortest filters that satisfy the specified constraints.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
To specify a filter order, use the FilterOrder name-value argument and input a positive integer. To use a minimum-order design, specify frequency and magnitude constraints accordingly.	In the Specify Filter Order section of the task: To specify a filter order, set Order mode to Specify and enter a value for Order. To use a minimum-order design, set Order mode to Minimum.	In the Filter Order section of the filter parameters panel: To specify a filter order, set Order mode to Specify and enter a value for Order. To use a minimum-order design, set Order mode to Minimum.

Numerator Order — Highest power in IIR transfer function numerator

positive integer

Specify the numerator order as a positive integer.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the NumeratorOrder name-value argument. Input the value as a positive integer.	In the Specify Filter Order section of the task: To specify a numerator order, set Order mode to Specify, select the Denominator order box, and enter a value for Numerator order.	In the Filter Order section of the filter parameters panel: To specify a numerator order, set Order mode to Specify, select the Denominator order box, and enter a value for Numerator order.

Denominator Order — Highest power in IIR transfer function denominator

positive integer

Specify the denominator order as a positive integer.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the DenominatorOrder name-value argument. Input the value as a positive integer.	In the Specify Filter Order section of the task: To specify a denominator order, set Order mode to Specify, select the Denominator order box, and enter a value for Denominator order.	In the Filter Order section of the filter parameters panel: To specify a denominator order, set Order mode to Specify, select the Denominator order box, and enter a value for Denominator order.

Frequency Specifications

Passband Frequency — Passband end frequency

positive scalar

Specify the passband frequency as a positive scalar smaller than the Nyquist frequency. If you work with normalized frequencies, the Nyquist frequency is 1. If you specify a sample rate F_s, the Nyquist frequency is F_s/2.

The passband frequency is the highest frequency for which the magnitude response of the filter is within an interval of width Passband Ripple from the reference value, usually 0 dB.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the PassbandFrequency name-value argument. Input the value as a positive scalar.	In the Specify Frequency Parameters section of the task, select a set of Frequency Constraints containing Passband frequency and enter a positive scalar as the value.	In the Frequency Specifications section of the filter parameters panel, select a set of Frequency Constraints containing Passband frequency and enter a positive scalar as the value.

Stopband Frequency — Stopband end frequency

positive scalar

Specify the stopband frequency as a positive scalar smaller than the Nyquist frequency. If you work with normalized frequencies, the Nyquist frequency is 1. If you specify a sample rate F_s, the Nyquist frequency is F_s/2.

The stopband frequency is the lowest frequency for which the magnitude response of the filter is at least Stopband Attenuation dB lower than the reference value, usually 0 dB.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the StopbandFrequency name-value argument. Input the value as a positive scalar.	In the Specify Frequency Parameters section of the task, select a set of Frequency Constraints containing Stopband frequency and enter a positive scalar as the value.	In the Frequency Specifications section of the filter parameters panel, select a set of Frequency Constraints containing Stopband frequency and enter a positive scalar as the value.

Half-Power Frequency — 3 dB frequency

positive scalar

Specify the half-power frequency as a positive scalar smaller than the Nyquist frequency. If you work with normalized frequencies, the Nyquist frequency is 1. If you specify a sample rate F_s, the Nyquist frequency is F_s/2.

The half-power frequency is the frequency at which the magnitude response of the filter is pow2db(1/2) = 10 log₁₀½ = –3.01 dB below the reference value, usually 0 dB.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the HalfPowerFrequency name-value argument. Input the value as a positive scalar.	In the Specify Frequency Parameters section of the task, set Frequency Constraints to Half power (3dB) frequency and enter a positive scalar as the value.	In the Frequency Specifications section of the filter parameters panel, set Frequency Constraints to Half power (3dB) frequency and enter a positive scalar as the value.

Magnitude Specifications

Passband Ripple — Gain fluctuation in passband

positive scalar

Specify the passband ripple as a positive scalar in decibels.

The passband ripple measures the fluctuation of the passband gain of the filter about the reference value, usually 0 dB.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the PassbandRipple name-value argument. Input the value as a positive scalar in decibels.	In the Specify Magnitude Parameters section of the task, select a set of Magnitude Constraints containing Passband ripple (dB) and enter a positive scalar in decibels as the value.	In the Magnitude Specifications section of the filter parameters panel, select a set of Magnitude Constraints containing Passband ripple (dB) and enter a positive scalar in decibels as the value.

Stopband Attenuation — Gain decrease in stopband

positive scalar

Specify the stopband attenuation as a positive scalar in decibels.

The stopband attenuation measures the extent by which the filter gain in the stopband is decreased in comparison to the reference value, usually 0 dB.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the StopbandAttenuation name-value argument. Input the value as a positive scalar in decibels.	In the Specify Magnitude Parameters section of the task, select a set of Magnitude Constraints containing Stopband attenuation (dB) and enter a positive scalar in decibels as the value.	In the Magnitude Specifications section of the filter parameters panel, select a set of Magnitude Constraints containing Stopband attenuation (dB) and enter a positive scalar in decibels as the value.

Design Method Options

Basic Options

Match Exactly — Band to match exactly

stopband | passband | both

Specify the band that the design must match exactly.

• By default, the Butterworth and Chebyshev type II algorithms match the stopband exactly.

• By default, the Chebyshev type I algorithm matches the passband exactly.

• By default, the elliptic algorithm matches both the passband and the stopband exactly. This option is available only for the elliptic design method.

Advanced Options

Density Factor — Density of frequency grid

positive scalar ≥ 10

Density of the frequency grid used by the Least Pth-Norm algorithm, specified as a positive scalar ≥ 10. The frequency grid has roughly (Density Factor × Filter Order)/(2 × Passband Frequency) frequency points. Increasing the density factor results in filters that more closely approximate an equiripple filter but take longer to compute.

You must have a DSP System Toolbox license to use this option. This parameter is available only in the designfilt function.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the DensityFactor name-value argument. Input the value as a positive scalar greater than 10.	N/A	N/A

Initial Denominator — Initial estimate of filter denominator coefficients

vector of size (N+1)-by-1

Initial estimate of the filter denominator coefficients for the Least Pth-Norm algorithm, specified as a vector of size (N+1)-by-1, where N is the Filter Order that you specify.

You must have a DSP System Toolbox license to use this option. This parameter is available only in the designfilt function.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the InitialDenominator name-value argument. Input the value as a vector of size (N+1)-by-1, where N is the filter order.	N/A	N/A

Initial Norm — Initial Pth norm

positive scalar

Initial Pth norm used by the Least Pth-Norm algorithm, specified as a positive scalar. Starting the optimization with a smaller initial value aids in the convergence of the algorithm. For more information, see Least Pth-Norm Optimal IIR Filter Design.

You must have a DSP System Toolbox license to use this option. This parameter is available only in the designfilt function.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the InitialNorm name-value argument. Input the value as a positive scalar.	N/A	N/A

Initial Numerator — Initial estimate of filter numerator coefficients

vector of size (N+1)-by-1

Initial estimate of the filter numerator coefficients for the Least Pth-Norm algorithm, specified as a vector of size (N+1)-by-1, where N is the Filter Order that you specify.

You must have a DSP System Toolbox license to use this option. This parameter is available only in the designfilt function.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the InitialNumerator name-value argument. Input the value as a vector of size (N+1)-by-1, where N is the filter order.	N/A	N/A

Maximum Pole Radius — Maximum pole radius

scalar in the range (0,1]

Maximum pole radius used by the Least Pth-Norm algorithm, specified as a scalar in the range (0,1]. This value indicates the maximum radius of each pole in the pole/zero plot of the designed filter.

You must have a DSP System Toolbox license to use this option. This parameter is available only in the designfilt function.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the MaxPoleRadius name-value argument. Input the value as a positive scalar less than or equal to 1	In the Specify Algorithm Options section of the task, specify Max pole radius as a positive scalar less than or equal to 1.	In the Algorithm Options section of the filter parameters panel, specify Max pole radius as a positive scalar less than or equal to 1.

Norm — L-infinity norm

positive scalar

L-infinity norm used by the Least Pth-Norm algorithm, specified as a positive scalar.

You must have a DSP System Toolbox license to use this option.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the Norm name-value argument. Input the value as a positive scalar.	In the Specify Algorithm Options section of the task, specify Norm as a positive scalar.	In the Algorithm Options section of the filter parameters panel, specify Norm as a positive scalar.

Passband Weight — Passband optimization weight

positive scalar

Passband optimization weight for a Least Pth-Norm design, specified as a positive scalar.

Weights are a way of specifying the relative importance of the passband ripple and the stopband attenuation in a filter design. By default, the passband and the stopband are equally weighted, each with unit weight. By increasing the passband weight, you can increase the passband ripple at the expense of increasing the stopband attenuation.

You must have a DSP System Toolbox license to use this option with some syntaxes.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the PassbandWeight name-value argument. Input the value as a positive scalar.	In the Specify Algorithm Options section of the task, specify Passband Weight as a positive scalar.	In the Algorithm Options section of the filter parameters panel, specify Passband Weight as a positive scalar.

Stopband Weight — Stopband optimization weight

positive scalar

Stopband optimization weight for a Least Pth-Norm design, specified as a positive scalar.

Weights are a way of specifying the relative importance of the passband ripple and the stopband attenuation in a filter design. By default, the passband and the stopband are equally weighted, each with unit weight. By increasing the stopband weight, you can increase the stopband attenuation at the expense of increasing the passband ripple.

You must have a DSP System Toolbox license to use this option with some syntaxes.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the StopbandWeight name-value argument. Input the value as a positive scalar.	In the Specify Algorithm Options section of the task, specify Stopband Weight as a positive scalar.	In the Algorithm Options section of the filter parameters panel, specify Stopband Weight as a positive scalar.

Scale SOS Coefficients — Option to scale filter coefficients representing second-order sections

true | false

Option to scale the filter coefficients representing second-order sections (SOS), specified as a logical scalar. When you set this parameter to true, the SOS coefficients are scaled using L-infinity norm scaling, so that the filter has unit gain at zero frequency.

You must have a DSP System Toolbox license to use this option.

designfilt Function	Design Filter Live Editor Task	Filter Designer App
Use the ScaleSOS name-value argument. Input the value as true or false.	In the Specify Algorithm Options section of the task, select the Scale SOS coefficients box.	In the Algorithm Options section of the filter parameters panel, select the Scale SOS coefficients box.

More About

Other Lowpass IIR Filter Design Functions

Use one of these functions to design a lowpass IIR filter:

• butter — Butterworth filter design

• cheby1 — Chebyshev type I filter design

• cheby2 — Chebyshev type II filter design

• ellip — Elliptic filter design

• iirlpnorm — Least Pth-norm optimal IIR filter

• iirlpnormc — Constrained least Pth-norm optimal IIR filter

• designLowpassIIR — Design and implement lowpass IIR filter

Use this function to lowpass-filter a signal:

• lowpass — Lowpass-filter signals