dsp.IIRHalfbandDecimator

Decimate by factor of two using polyphase IIR

Description

The dsp.IIRHalfbandDecimator System 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 the dsp.IIRHalfbandDecimator object 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:

1. Create the dsp.IIRHalfbandDecimator object and set its properties.

2. Call the object with arguments, as if it were a function.

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

Creation

Syntax

iirhalfbanddecim = dsp.IIRHalfbandDecimator

iirhalfbanddecim = dsp.IIRHalfbandDecimator(Name,Value)

Description

iirhalfbanddecim = dsp.IIRHalfbandDecimator returns 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 of 22050 Hz, a transition width of 4100 Hz, and a stopband attenuation of 80 dB.

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

Example: iirhalfbanddecim = dsp.IIRHalfbandDecimator('Specification','Filter order and stopband attenuation') creates an IIR halfband decimator object with filter order set to 9 and stopband attenuation set to 80 dB.

Properties

Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

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

For more information on changing property values, see System 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 set Specification to one of the filter design options, you can specify the filter design parameters using the corresponding FilterOrder, StopbandAttenuation, and TransitionWidth properties. Also, you can specify the design method using DesignMethod. When you set Specification to '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 set DesignMethod to 'Elliptic', then FilterOrder must be an odd integer greater than one. If you set DesignMethod to 'Quasi-linear phase', then FilterOrder must be a multiple of four.

Dependencies

This property applies when you set Specification to 'Filter order and stopband attenuation' or 'Filter order and transition width'.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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 set Specification to '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 set Specification to '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 set Specification to 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 set Specification to 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'.

This property is not tunable.

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 an N-by-1 or N-by-2 matrix. N is the number of first-order or second-order allpass sections.

Tunable: Yes

Dependencies

This property applies only when you set Specification to 'Coefficients' and Structure to 'Minimum multiplier'.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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 an N-by-1 or N-by-2 matrix. N is the number of first-order or second-order allpass sections.

Tunable: Yes

Dependencies

This property applies only when you set Specification to 'Coefficients' and Structure to 'Minimum multiplier'.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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 an N-by-1 or N-by-2 matrix. N is the number of first-order or second-order allpass sections. Each element must have an absolute value less than or equal to 1.

This property is not tunable.

Dependencies

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

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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 a N-by-1 or N-by-2 matrix. N is the number of first-order or second-order allpass sections. Each element must have an absolute value less than or equal to 1.

This property is not tunable.

Dependencies

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

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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 properties AllpassCoefficients1 and WDFCoefficients1 do not apply.

This property is not tunable.

Dependencies

This property applies only when you set Specification to '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.

This property is not tunable.

Dependencies

This property applies only when you set Specification to 'Coefficients' and HasPureDelayBranch to 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 an N-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 to 0, the last section of the second branch is a second-order section. When the coefficients of the second branch are in an N-by-1 matrix, this property is ignored.

This property is not tunable.

Dependencies

This property applies only when you set Specification to 'Coefficients'.

Usage

Syntax

ylow = iirhalfbanddecim(x)

[ylow,yhigh] = iirhalfbanddecim(x)

Description

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

[ylow,yhigh] = iirhalfbanddecim(x) computes the ylow and yhigh, of the analysis filter bank, iirhalfbanddecim for input x. A Ki-by-N input matrix is treated as N independent channels. The System object generates two power-complementary output signals by adding and subtracting the two polyphase branch outputs respectively. ylow and yhigh are of the same size (Ko-by-N) and data type. Ko = Ki/2, where 2 is the decimation factor.

Input Arguments

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

ylow — Lowpass subband of decimator output

column vector | matrix

Lowpass subband of decimator output, returned as a column vector or a matrix. The output, ylow is a lowpass halfband filtered and downsampled version of the input x. 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, yhigh is a highpass halfband filtered and downsampled version of the input x. 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 named obj, use this syntax:

release(obj)

Specific to dsp.IIRHalfbandDecimator

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

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
reset	Reset internal states of System object

Examples

Frequency Response of Quasi-Linear Phase IIR Halfband Decimator

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')

Extract Low Frequency Subband from Speech

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

Note: The audioDeviceWriter System object™ is not supported in MATLAB 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)

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

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: The audioDeviceWriter System 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);
end

release(AF);
release(AP);
release(SpecAna);

Filter Input into Lowpass and Highpass Subbands

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

Polyphase Implementation with Halfband Filters

When you filter your signal, the IIR halfband decimator uses 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 a 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

$$A_1(z)={\displaystyle \prod _{k=1}^{K_1}\frac{a_k^{(1)}+z^{-1}}{1+a_k^{(1)}{z}^{-1}}}$$

$$A_2(z)={\displaystyle \prod _{k=1}^{K_2}\frac{a_k^{(2)}+z^{-1}}{1+a_k^{(2)}{z}^{-1}}}$$

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

$$A_1(z)=z^{-k}$$

where k is the length of the delay.

$$A_2(z)={\displaystyle \prod _{K=1}^{K_2^{(1)}}\frac{a_k+z^{-1}}{1+a_k{z}^{-1}}}{\displaystyle \prod _{K=1}^{K_2^{(2)}}\frac{c_k+b_k{z}^{-1}+z^{-2}}{1+b_k{z}^{-1}+c_k{z}^{-2}}}$$

where N is the order of the IIR halfband filter.

You can represent filtering the input signal and then downsampling it by 2 using this figure.

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

To implement the halfband decimator efficiently, this algorithm replaces the delay block and downsampling operator with a commutator switch.

Analysis Filter Bank

The transfer function of the complementary highpass filter branch of the analysis filter bank is given by

$$G(z)=0.5*[A_1(z^2)-z^{-1}{A}_2(z^2)]$$

You can represent the analysis filter bank as in this diagram.

The IIR halfband decimator generates two power-complementary output signals by adding and subtracting the two polyphase branch outputs respectively.

To summarize, the IIR halfband decimator:

• Decimates the input prior to filtering.

• Acts as an analysis filter bank.

• Has a nonlinear phase response and uses few coefficients with the elliptic design method.

• Has near-linear phase response at the cost of additional coefficients with the quasi-linear phase design method, where one of the branches 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:

See System 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