dsp.IIRHalfbandDecimator

Decimate by factor of two using polyphase IIR

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Description

Thedsp.IIRHalfbandDecimatorSystem object™ performs efficient polyphase decimation of the input signal by a factor of two. To design the halfband filter, you can specify the object to use an elliptic design or a quasi-linear phase design. The object uses these design methods to compute the filter coefficients. To filter the inputs, the object uses a polyphase structure. The allpass filters in the polyphase structure are in a minimum multiplier form.

Elliptic design introduces nonlinear phase and creates the filter using fewer coefficients than quasi linear design. Quasi-linear phase design overcomes phase nonlinearity at the cost of additional coefficients.

Alternatively, instead of designing the halfband filter using a design method, you can specify the filter coefficients directly. When you choose this option, the allpass filters in the two branches of the polyphase implementation can be in a minimum multiplier form or in a wave digital form.

You can also use thedsp.IIRHalfbandDecimatorobject to implement the analysis portion of a two-band filter bank to filter a signal into lowpass and highpass subbands.

To filter and downsample your data:

Create thedsp.IIRHalfbandDecimatorobject and set its properties.
Call the object with arguments, as if it were a function.

To learn more about how System objects work, seeWhat Are System Objects?

Creation

Syntax

iirhalfbanddecim = dsp.IIRHalfbandDecimator

iirhalfbanddecim = dsp.IIRHalfbandDecimator(Name,Value)

Description

iirhalfbanddecim= dsp.IIRHalfbandDecimatorreturns a halfband decimator,iirhalfbanddecim, with the default settings. Under the default settings, the System object filters and downsamples the input data with a halfband frequency of22050Hz, a transition width of4100Hz, and a stopband attenuation of80dB.

example

iirhalfbanddecim= dsp.IIRHalfbandDecimator(Name,Value)returns an IIR halfband decimator, with additional properties specified by one or moreName,Valuepair arguments.

Example:iirhalfbanddecim = dsp.IIRHalfbandDecimator('Specification','Filter order and stopband attenuation')creates an IIR halfband decimator object with filter order set to9and stopband attenuation set to80dB.

Properties

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Unless otherwise indicated, properties arenontunable,这意味着你无法改变他们的价值观calling the object. Objects lock when you call them, and thereleasefunction unlocks them.

If a property istunable, you can change its value at any time.

For more information on changing property values, seeSystem Design in MATLAB Using System Objects.

`Specification`—Filter design parameters
`'Transition width and stopband attenuation'`(default) |`'Filter order and stopband attenuation'`|`'Filter order and transition width'`|`'Coefficients'`

Filter design parameters, specified as a character vector. When you setSpecificationto one of the filter design options, you can specify the filter design parameters using the correspondingFilterOrder,StopbandAttenuation, andTransitionWidthproperties. Also, you can specify the design method usingDesignMethod. When you setSpecificationto'Coefficients', you can specify the coefficients directly.

`FilterOrder`—Order of the IIR halfband filter
9(default) |positive scalar integer

Order of the IIR halfband filter, specified as a positive scalar integer. If you setDesignMethodto'Elliptic', thenFilterOrdermust be an odd integer greater than one. If you setDesignMethodto'Quasi-linear phase', thenFilterOrdermust be a multiple of four.

Dependencies

This property applies when you setSpecificationto'Filter order and stopband attenuation'or'Filter order and transition width'.

`StopbandAttenuation`—Minimum attenuation needed in stopband
80(default) |positive real scalar

Minimum attenuation needed in the stopband of the IIR halfband filter, specified as a positive real scalar. Units are in dB.

Dependencies

This property applies only when you setSpecificationto'Filter order and stopband attenuation'or'Transition width and stopband attenuation'.

Data Types:single|double

`TransitionWidth`—Transition width
4100(default) |positive real scalar

Transition width of the IIR halfband filter, specified as a positive real scalar. Units are in Hz. The value of the transition width must be less than half the input sample rate.

Dependencies

This property applies only when you setSpecificationto'Transition width and stopband attenuation'or'Filter order and transition width'.

Data Types:single|double

`DesignMethod`—Design method
`'Elliptic'`(default) |`'Quasi-linear phase'`

Design method for the IIR halfband filter, specified as'Elliptic'or'Quasi-linear phase'. When the property is set to'Quasi-linear phase', the first branch of the polyphase structure is a pure delay, which results in an approximately linear phase response.

Dependencies

This property applies only when you setSpecificationto any accepted value except'Coefficients'.

`SampleRate`—Input sample rate
44100(default) |positive real scalar

Input sample rate, specified as a positive real scalar. Units are in Hz.

Dependencies

This property applies only when you setSpecificationto any accepted value except'Coefficients'.

Data Types:single|double

`Structure`—Internal allpass filter implementation structure
`'Minimum multiplier'`(default) |`'Wave Digital Filter'`

Internal allpass filter implementation structure, specified as'Minimum multiplier'or'Wave Digital Filter'.

这个属性不是可调。

Dependencies

This property applies only when you set'Specification'to'Coefficients'. Each structure uses a different coefficients set, independently stored in the corresponding object property.

`AllpassCoefficients1`—Allpass polynomial filter coefficients of first branch
`[0.1284563; 0.7906755]`(default) |`[0.1284563 0.1534; 0.7906755 0.6745]`

Allpass polynomial filter coefficients of the first branch, specified as anN-by-1orN-by-2matrix.Nis the number of first-order or second-order allpass sections.

Tunable:Yes

Dependencies

This property applies only when you setSpecificationto'Coefficients'andStructureto'Minimum multiplier'.

`AllpassCoefficients2`—Allpass polynomial filter coefficients of second branch
`[0.4295667]`(default) |`[0.7906755 0.1534]`

Allpass polynomial filter coefficients of the second branch, specified as anN-by-1orN-by-2matrix.Nis the number of first-order or second-order allpass sections.

Tunable:Yes

Dependencies

This property applies only when you setSpecificationto'Coefficients'andStructureto'Minimum multiplier'.

`WDFCoefficients1`—Allpass filter coefficients of first branch in Wave Digital Filter form
`[0.1284563; 0.7906755]`(default) |`[0.1284563 0.1534; 0.7906755 0.6745]`

Allpass filter coefficients of the first branch in Wave Digital Filter form, specified as anN-by-1orN-by-2matrix.Nis the number of first-order or second-order allpass sections. Each element must have an absolute value less than or equal to1.

这个属性不是可调。

Dependencies

This property applies only when you setSpecificationto'Coefficients'andStructureto'Wave Digital Filter'.

`WDFCoefficients2`—Allpass filter coefficients of second branch in Wave Digital Filter form
`[0.4295667]`(default) |`[0.7906755 0.1534]`

Allpass filter coefficients of the second branch in Wave Digital Filter form, specified as the comma-separated pair consisting of'WDFCoefficients2'and aN-by-1orN-by-2matrix.Nis the number of first-order or second-order allpass sections. Each element must have an absolute value less than or equal to 1.

这个属性不是可调。

Dependencies

This property applies only when you set'Specification'to'Coefficients'and'Structure'to'Wave Digital Filter'.

`HasPureDelayBranch`—Make first branch a pure delay
`false`(default) |`true`

Flag to make the first allpass branch a delay, specified as a logical scalar. When this property is true, the first branch is treated as a pure delay and the propertiesAllpassCoefficients1andWDFCoefficients1do not apply.

这个属性不是可调。

Dependencies

This property applies only when you setSpecificationto'Coefficients'.

`Delay`—Length of delay
`1`(default) |finite positive scalar

Length of the first branch delay, specified as a finite positive scalar. The value of this property specifies the number of samples by which you can delay the input to the first branch.

这个属性不是可调。

Dependencies

This property applies only when you setSpecificationto'Coefficients'andHasPureDelayBranchto 1.

Data Types:single|double

`HasTrailingFirstOrderSection`—Treat the last section of the second branch as first order
`false`(default) |`true`

Option to treat the last section of the second branch as first order, specified as a logical scalar. When this property is 1 and the coefficients of the second branch are in anN-by-2 matrix, the object ignores the second element of the last row of the matrix. The last section of the second branch then becomes a first-order section. When this property is set to0, the last section of the second branch is a second-order section. When the coefficients of the second branch are in anN-by-1 matrix, this property is ignored.

这个属性不是可调。

Dependencies

This property applies only when you setSpecificationto'Coefficients'.

Usage

Syntax

ylow= iirhalfbanddecim(x)

[ylow,yhigh] = iirhalfbanddecim(x)

Description

example

ylow= iirhalfbanddecim(x)filters the input signal,x, using the IIR halfband filter,iirhalfbanddecim, and downsamples the output by a factor of 2.

example

[ylow,yhigh] = iirhalfbanddecim(x)computes theylowandyhigh, of the analysis filter bank,iirhalfbanddecimfor inputx. AKi-by-Ninput matrix is treated asNindependent channels. The System object generates two power-complementary output signals by adding and subtracting the two polyphase branch outputs respectively.ylowandyhighare of the same size (Ko-by-N) and data type.Ko=Ki/2, where2is the decimation factor.

Input Arguments

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`x`—Data input
column vector|matrix

Data input, specified as a column vector or a matrix. The number of rows in the input signal must be even since the decimation factor is always 2 for this object. If the input is a matrix, each column is treated as an independent channel.

Data Types:single|double
Complex Number Support:Yes

Output Arguments

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`ylow`— Lowpass subband of decimator output
column vector | matrix

Lowpass subband of decimator output, returned as a column vector or a matrix. The output,ylowis a lowpass halfband filtered and downsampled version of the inputx. Due to the halfband nature of the filter, the downsampling factor is always 2.

Data Types:single|double
Complex Number Support:Yes

`yhigh`— Highpass subband of decimator output
column vector | matrix

Highpass subband of decimator output, returned as a column vector or a matrix. The output,yhighis a highpass halfband filtered and downsampled version of the inputx. Due to the halfband nature of the filter, the downsampling factor is always 2.

Data Types:single|double
Complex Number Support:Yes

Object Functions

To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object namedobj, use this syntax:

release(obj)

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Specific to`dsp.IIRHalfbandDecimator`

`freqz`	Frequency response of discrete-time filterSystem object
`fvtool`	Visualize frequency response of DSP filters
`info`	Information about filterSystem object
`cost`	Estimate cost of implementing filterSystem object
`polyphase`	Polyphase decomposition of multirate filter

Common to All System Objects

`step`	RunSystem objectalgorithm
`release`	Release resources and allow changes toSystem objectproperty values and input characteristics
`reset`	Reset internal states ofSystem object

Examples

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Frequency Response of Quasi-Linear Phase IIR Halfband Decimator

Open Live Script

Create a minimum-order lowpass IIR halfband decimation filter for data sampled at 44.1 kHz. The filter has a transition width of 4.1 kHz, and a stopband attenuation of 80 dB.

IIRHalfbandDecim = dsp.IIRHalfbandDecimator(...'DesignMethod','Quasi-linear phase');

Obtain the filter coefficients.

c = coeffs(IIRHalfbandDecim);

Plot the magnitude and phase response.

fvtool(IIRHalfbandDecim,'Analysis','freq')

Figure Magnitude Response (dB) and Phase Response contains an axes object. The axes object with title Magnitude Response (dB) and Phase Response contains an object of type line.

Extract Low Frequency Subband from Speech

Open Live Script

Use a halfband analysis filter bank and interpolation filter to extract the low frequency subband from a speech signal.

Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalentstep语法。For example,obj(x)becomesstep(obj,x).

Note:TheaudioDeviceWriterSystem object™ is not supported inMATLAB Online.

Set up the audio file reader, the analysis filter bank, the audio device writer, and the interpolation filter. The sampling rate of the audio data is 22050 Hz. The halfband filter has an order of 21 and a transition width of 2 kHz.

afr = dsp.AudioFileReader('speech_dft.mp3','SamplesPerFrame',1024); filterspec ='Filter order and transition width'; Order = 21; TW = 2000; IIRHalfbandDecim = dsp.IIRHalfbandDecimator(...'Specification',filterspec,'FilterOrder',Order,...'TransitionWidth',TW,'SampleRate',afr.SampleRate); IIRHalfbandInterp = dsp.IIRHalfbandInterpolator(...'Specification',filterspec,'FilterOrder',Order,...'TransitionWidth',TW,'SampleRate',afr.SampleRate/2); ap = audioDeviceWriter('SampleRate',afr.SampleRate);

View the magnitude response of the halfband filter.

fvtool(IIRHalfbandDecim)

Figure Magnitude Response (dB) contains an axes object. The axes object with title Magnitude Response (dB) contains 2 objects of type line.

Read the speech signal from the audio file in frames of 1024 samples. Filter the speech signal into lowpass and highpass subbands with a halfband frequency of 5512.5 Hz. Reconstruct a lowpass approximation of the speech signal by interpolating the lowpass subband. Play the filtered output.

while~isDone(afr) audioframe = afr(); xlo = IIRHalfbandDecim(audioframe); ylow = IIRHalfbandInterp(xlo); ap(ylow);end

Wait until the audio file ends, and then close the input file and release the audio output resource.

release(afr); release(ap);

Two-Channel Filter Bank

Open Live Script

Use a halfband decimator and interpolator to implement a two-channel filter bank. This example uses an audio file input and shows that the power spectrum of the filter bank output does not differ significantly from the input.

Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalent step syntax. For example,obj(x)becomesstep(obj,x).

Note: TheaudioDeviceWriterSystem object™ is not supported in MATLAB Online.

Set up the audio file reader and audio device writer. Construct the IIR halfband decimator and interpolator. Finally, set up the spectrum analyzer to display the power spectra of the filter-bank input and output.

AF = dsp.AudioFileReader('speech_dft.mp3','SamplesPerFrame',1024); AP = audioDeviceWriter('SampleRate',AF.SampleRate); filterspec ='Filter order and transition width'; Order = 51; TW = 2000; IIRHalfbandDecim = dsp.IIRHalfbandDecimator(...'Specification',filterspec,'FilterOrder',Order,...'TransitionWidth',TW,'SampleRate',AF.SampleRate); IIRHalfbandInterp = dsp.IIRHalfbandInterpolator(...'Specification',filterspec,'FilterOrder',Order,...'TransitionWidth',TW,'SampleRate',AF.SampleRate/2,...'FilterBankInputPort',true); SpecAna = spectrumAnalyzer('SampleRate',AF.SampleRate,...'PlotAsTwoSidedSpectrum',false,...'ShowLegend',true,...'ChannelNames',{'Input signal','Filtered output signal'});

Read the audio 1024 samples at a time. Filter the input to obtain the lowpass and highpass subband signals decimated by a factor of two. This is the analysis filter bank. Use the halfband interpolator as the synthesis filter bank. Display the running power spectrum of the audio input and the output of the synthesis filter bank. Play the output.

while~isDone(AF) audioInput = AF(); [xlo,xhigh] = IIRHalfbandDecim(audioInput); audioOutput = IIRHalfbandInterp(xlo,xhigh); spectrumInput = [audioInput audioOutput]; SpecAna(spectrumInput); AP(audioOutput);endrelease(AF); release(AP); release(SpecAna);

Filter Input into Lowpass and Highpass Subbands

Open Live Script

Note: This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalentstep语法。例如,myObject (myOb (x)变成了一步ject,x).

Create a halfband decimator for data sampled at 44.1 kHz. Use a minimum-order design with a transition width of 2 kHz and a stopband attenuation of 60 dB.

IIRHalfbanddecim = dsp.IIRHalfbandDecimator(...'Specification','Transition width and stopband attenuation',...'TransitionWidth',2000,'StopbandAttenuation',60,'SampleRate',44.1e3);

Filter a two-channel input into lowpass and highpass subbands.

x = randn(1024,2); [ylow,yhigh] = IIRHalfbanddecim(x);

Algorithms

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Polyphase Implementation with Halfband Filters

When you filter your signal,dsp.IIRHalfbandDecimatoruses an efficient polyphase implementation for halfband filters. You can use the polyphase implementation to move the downsample operation before filtering. This change enables you to filter at the lower sampling rate.

IIR halfband filters are generally modeled using two parallel allpass filter branches.

$H (z) = 0.5 * [A_{1} (z^{2}) + z^{- 1} A_{2} (z^{2})]$

Elliptic Design

The allpass filters for elliptic IIR halfband filter are given as

$A_{1} (z) = \prod_{k = 1}^{K_{1}} \frac{a_{k}^{(1)} + z^{- 1}}{1 + a_{k}^{(1)} z^{- 1}}$

$A_{2} (z) = \prod_{k = 1}^{K_{2}} \frac{a_{k}^{(2)} + z^{- 1}}{1 + a_{k}^{(2)} z^{- 1}}$

Quasi-Linear Phase Design

To achieve a near-linear phase response for IIR halfband filters, make one of the branches a pure delay. In this design, the cost of the filter increases.

The allpass filters for the quasi-linear phase IIR halfband filter are

$A_{1} (z) = z^{- k}$

wherekis the length of the delay.

$A_{2} (z) = \prod_{K = 1}^{K_{2}^{(1)}} \frac{a_{k} + z^{- 1}}{1 + a_{k} z^{- 1}} \prod_{K = 1}^{K_{2}^{(2)}} \frac{c_{k} + b_{k} z^{- 1} + z^{- 2}}{1 + b_{k} z^{- 1} + c_{k} z^{- 2}}$

whereNis the order of the IIR halfband filter.

This figure represents filtering and downsampling the input by two.

Using the multirate noble identity for downsampling, you can move the downsampling operation before filtering. This change enables you to filter at the lower rate.

To implement the halfband decimator efficiently,dsp.IIRHalfbandDecimatorreplaces the delay block and downsampling operator with a commutator switch.

Analysis Filter Bank

传递函数的互补highpass filter branch of the analysis filter bank is given by

$G (z) = 0.5 * [A_{1} (z^{2}) - z^{- 1} A_{2} (z^{2})]$

Graphically, you can represent the analysis filter bank as follows.

dsp.IIRHalfbandDecimatorgenerates two power-complementary output signals by adding and subtracting the two polyphase branch outputs respectively.

To summarize,dsp.IIRHalfbandDecimator

Decimates the input prior to filtering.
Acts as an analysis filter bank.
Has non-linear phase response and uses few coefficients with elliptic design method.
Has near-linear phase response at the cost of additional coefficients with quasi-linear phase design method. One of the branches in this design is a pure delay.

References

[1] Lang, M.Allpass Filter Design and Applications.IEEE Transactions on Signal Processing. Vol. 46, No. 9, Sept 1998, pp. 2505–2514.

[2] Harris, F.J.Multirate Signal Processing for Communication Systems. Prentice Hall. 2004, pp. 208–209.

[3] Regalia, Phillip A., Sanjit K. Mitra, and P. P. Vaidyanathan. "The Digital All-Pass Filter: A Versatile Signal Processing Building Block."Proceedings of the IEEE.Vol. 76, Number 1, 1988, pp. 19-37.

Extended Capabilities

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

Usage notes and limitations:

SeeSystem Objects in MATLAB Code Generation(MATLAB Coder).

This object supports code generation for ARM^®Cortex^®-M and ARM Cortex-A processors.

Version History

Introduced in R2015b

dsp.IIRHalfbandDecimator

Description

Creation

Syntax

Description

Properties

Specification—Filter design parameters'Transition width and stopband attenuation'(default) |'Filter order and stopband attenuation'|'Filter order and transition width'|'Coefficients'

FilterOrder—Order of the IIR halfband filter9(default) |positive scalar integer

Dependencies

StopbandAttenuation—Minimum attenuation needed in stopband80(default) |positive real scalar

Dependencies

TransitionWidth—Transition width4100(default) |positive real scalar

Dependencies

DesignMethod—Design method'Elliptic'(default) |'Quasi-linear phase'

Dependencies

SampleRate—Input sample rate44100(default) |positive real scalar

Dependencies

Structure—Internal allpass filter implementation structure'Minimum multiplier'(default) |'Wave Digital Filter'

Dependencies

AllpassCoefficients1—Allpass polynomial filter coefficients of first branch[0.1284563; 0.7906755](default) |[0.1284563 0.1534; 0.7906755 0.6745]

Dependencies

AllpassCoefficients2—Allpass polynomial filter coefficients of second branch[0.4295667](default) |[0.7906755 0.1534]

Dependencies

WDFCoefficients1—Allpass filter coefficients of first branch in Wave Digital Filter form[0.1284563; 0.7906755](default) |[0.1284563 0.1534; 0.7906755 0.6745]

Dependencies

WDFCoefficients2—Allpass filter coefficients of second branch in Wave Digital Filter form[0.4295667](default) |[0.7906755 0.1534]

Dependencies

HasPureDelayBranch—Make first branch a pure delayfalse(default) |true

Dependencies

Delay—Length of delay1(default) |finite positive scalar

Dependencies

HasTrailingFirstOrderSection—Treat the last section of the second branch as first orderfalse(default) |true

Dependencies

Usage

Syntax

Description

Input Arguments

x—Data inputcolumn vector|matrix

Output Arguments

ylow— Lowpass subband of decimator outputcolumn vector | matrix

yhigh— Highpass subband of decimator outputcolumn vector | matrix

Object Functions

Specific todsp.IIRHalfbandDecimator

Common to All System Objects

Examples

Frequency Response of Quasi-Linear Phase IIR Halfband Decimator

Extract Low Frequency Subband from Speech

Two-Channel Filter Bank

Filter Input into Lowpass and Highpass Subbands

Algorithms

Polyphase Implementation with Halfband Filters

References

Extended Capabilities

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

Version History

See Also

Functions

Objects

Blocks

Topics

`Specification`—Filter design parameters
`'Transition width and stopband attenuation'`(default) |`'Filter order and stopband attenuation'`|`'Filter order and transition width'`|`'Coefficients'`

`FilterOrder`—Order of the IIR halfband filter
9(default) |positive scalar integer

`StopbandAttenuation`—Minimum attenuation needed in stopband
80(default) |positive real scalar

`TransitionWidth`—Transition width
4100(default) |positive real scalar

`DesignMethod`—Design method
`'Elliptic'`(default) |`'Quasi-linear phase'`

`SampleRate`—Input sample rate
44100(default) |positive real scalar

`Structure`—Internal allpass filter implementation structure
`'Minimum multiplier'`(default) |`'Wave Digital Filter'`

`AllpassCoefficients1`—Allpass polynomial filter coefficients of first branch
`[0.1284563; 0.7906755]`(default) |`[0.1284563 0.1534; 0.7906755 0.6745]`

`AllpassCoefficients2`—Allpass polynomial filter coefficients of second branch
`[0.4295667]`(default) |`[0.7906755 0.1534]`

`WDFCoefficients1`—Allpass filter coefficients of first branch in Wave Digital Filter form
`[0.1284563; 0.7906755]`(default) |`[0.1284563 0.1534; 0.7906755 0.6745]`

`WDFCoefficients2`—Allpass filter coefficients of second branch in Wave Digital Filter form
`[0.4295667]`(default) |`[0.7906755 0.1534]`

`HasPureDelayBranch`—Make first branch a pure delay
`false`(default) |`true`

`Delay`—Length of delay
`1`(default) |finite positive scalar

`HasTrailingFirstOrderSection`—Treat the last section of the second branch as first order
`false`(default) |`true`

`x`—Data input
column vector|matrix

`ylow`— Lowpass subband of decimator output
column vector | matrix

`yhigh`— Highpass subband of decimator output
column vector | matrix

Specific to`dsp.IIRHalfbandDecimator`

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