# 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:

Create the

`dsp.IIRHalfbandDecimator`

object and set its properties.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

### Description

returns a halfband decimator, `iirhalfbanddecim`

= dsp.IIRHalfbandDecimator`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.

returns an IIR halfband decimator, with additional properties specified by one or more
`iirhalfbanddecim`

= dsp.IIRHalfbandDecimator(`Name=Value`

)`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.

### Main Properties

`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

To enable this property, 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

To enable this property, 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
or in normalized frequency units* (since R2023b)*.

If you set the
`NormalizedFrequency`

property to:

`false`

–– The value of the transition width is in Hz and must be less than half the`SampleRate`

property value.`true`

–– The value of the transition width is in normalized frequency units. The value must be a positive scalar less than`1.0`

.When you set the

`NormalizedFrequency`

property to`true`

while creating the object and you do not set the transition width, the object sets the default transition width to normalized frequency units using the default sample rate of 44100 Hz.When you set the

`NormalizedFrequency`

property to`true`

after you create the object, you must specify the transition width in normalized units before you run the object algorithm. To specify the normalized frequency value, set`NormalizedFrequency`

to`true`

and manually convert the frequency value in Hz to the normalized value using the input sample rate in Hz. For example, if the input sample rate*Fs*is 44100 Hz, the corresponding transition width value in normalized units is*TW*/(_{Hz}*Fs*/2).iirhalfbanddecim = dsp.IIRHalfbandDecimator; iirhalfbanddecim.NormalizedFrequency = true; iirhalfbanddecim.TransitionWidth = 4100/(44100/2)

* (since R2023b)*

#### Dependencies

To enable this property, 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 you
set this property to `"Quasi-linear phase"`

, the first branch of the
polyphase structure is a pure delay, which results in an approximately linear phase
response.

#### Dependencies

To enable this property, set `Specification`

to any accepted
value except `"Coefficients"`

.

`NormalizedFrequency`

— Flag to set frequencies in normalized units

`false`

(default) | `true`

*Since R2023b*

Flag to set frequencies in normalized units, specified as one of these values:

`true`

–– The transition width must be in the normalized frequency units and less than`1.0`

.When you set the

`NormalizedFrequency`

property to`true`

while creating the object and you do not set the transition width, the object sets the default transition width to normalized frequency units using the default sample rate of 44100 Hz.When you set the

`NormalizedFrequency`

property to`true`

after you create the object, you must specify the transition width in normalized units before you run the object algorithm. To specify the normalized frequency value, set`NormalizedFrequency`

to`true`

and manually convert the frequency value in Hz to the normalized value using the input sample rate in Hz. For example, if the input sample rate*Fs*is 44100 Hz, the corresponding transition width value in normalized units is*TW*/(_{Hz}*Fs*/2).iirhalfbanddecim = dsp.IIRHalfbandDecimator; iirhalfbanddecim.NormalizedFrequency = true; iirhalfbanddecim.TransitionWidth = 4100/(44100/2)

`false`

–– The transition width is in Hz. You can specify the input sample rate through the`SampleRate`

property.

#### Dependency

To enable this property, set `Specification`

to any accepted
value except `"Coefficients"`

.

**Data Types: **`logical`

`SampleRate`

— Input sample rate

44100 (default) | positive real scalar

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

#### Dependency

To enable this property, set:

`Specification`

to any accepted value except`"Coefficients"`

.`NormalizedFrequency`

to`false`

.*(since R2023b)*

**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

To enable this property, 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

To enable this property, 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

To enable this property, 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

To enable this property, 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

To enable this property, 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

To enable this property, 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

To enable this property, 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

To enable this property, set `Specification`

to
`"Coefficients"`

.

### Code Generation Properties

`AllowArbitraryInputLength`

— Allow arbitrary input length in generated code

`false`

(default) | `true`

Allow arbitrary frame length for fixed-size input signals in the generated code,
specified as `true`

or `false`

. When you specify:

`true`

–– The input frame length does not have to be a multiple of the decimation factor 2. The output of the object in the generated code is a variable-size array.`false`

–– The input frame length must be a multiple of the decimation factor 2.

When you specify variable-size signals, the input frame length can be arbitrary
and the object ignores this property in the generated code. When you run this object
in MATLAB^{®}, the object supports arbitrary input frame lengths for fixed-size and
variable-size signals and this property does not affect the object behavior.

**Data Types: **`logical`

## Usage

### Description

`[`

computes the `ylow`

,`yhigh`

] = iirhalfbanddecim(`x`

)`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 and data type.

### Input Arguments

`x`

— Data input

column vector | matrix

Data input, specified as a column vector or a matrix. If the input is a matrix, each column is treated as an independent channel.

The number of rows in the input signal
*Ki* can be arbitrary and does not have to be a multiple of
2.* (since R2023b)*

This object supports variable-size input signal, that is, the frame length (number of rows) of the signal can change even when the object is locked. However, the number of channels (columns) must remain constant.

**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.

When the input is of size
*K _{i}*-by-

*N*, and

*K*is not a multiple of 2, the lowpass subband has an upper bound size of

_{i}`ceil`

(*K*/2)-by-

_{i}*N*.

*(since R2023b)*

If *K _{i}* is a multiple of 2, then the
lowpass subband is of size
(

*K*/2)-by-

_{i}*N*. The number of channels (columns) does not change.

**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.

When the input is of size
*K _{i}*-by-

*N*, and

*K*is not a multiple of 2, the highpass subband has an upper bound size of

_{i}`ceil`

(*K*/2)-by-

_{i}*N*.

*(since R2023b)*

If *K _{i}* is a multiple of 2, then the
highpass subband is of size
(

*K*/2)-by-

_{i}*N*. The number of channels (columns) does not change.

**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 |

`freqzmr` | Compute DTFT approximation of the impulse response of a multirate or a single-rate filter |

`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 |

`outputDelay` | Determine output delay of single-rate or multirate filter |

## Examples

### Frequency Response of Quasi-Linear Phase IIR Halfband Decimator

Create a minimum-order lowpass IIR halfband decimation filter. The filter has a transition width of 0.1859 in normalized frequency units and a stopband attenuation of 80 dB.

IIRHalfbandDecim = dsp.IIRHalfbandDecimator(... NormalizedFrequency=true,... TransitionWidth=0.1859,... DesignMethod='Quasi-linear phase')

IIRHalfbandDecim = dsp.IIRHalfbandDecimator with properties: Main Specification: 'Transition width and stopband attenuation' TransitionWidth: 0.1859 StopbandAttenuation: 80 DesignMethod: 'Quasi-linear phase' NormalizedFrequency: true Use get to show all properties

Obtain the filter coefficients.

c = coeffs(IIRHalfbandDecim);

Plot the magnitude and phase response.

freqz(IIRHalfbandDecim)

### 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.

filterAnalyzer(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);

### Design and Implement IIR Halfband Decimator

Design an elliptic IIR halfband decimator object of order 31 and a transition width of 0.1 using the `designHalfbandIIR`

function. Set the `Verbose`

argument to `true`

.

hbIIR = designHalfbandIIR(FilterOrder=31,SystemObject=true,... Structure='decim',Verbose=true)

designHalfbandIIR(FilterOrder=31, DesignMethod="butter", Structure="decim", SystemObject=true, Passband="lowpass")

hbIIR = dsp.IIRHalfbandDecimator with properties: Main Specification: 'Coefficients' Structure: 'Minimum multiplier' HasPureDelayBranch: false AllpassCoefficients1: [8x1 double] AllpassCoefficients2: [7x1 double] HasTrailingFirstOrderSection: false Use get to show all properties

Create a `dsp.DynamicFilterVisualizer`

object and visualize the magnitude response of the filter.

dfv = dsp.DynamicFilterVisualizer(NormalizedFrequency=true,YLimits=[-400 200]); dfv(hbIIR);

The input is a cosine wave.

Fs = 1; Fc = 0.03; input = cos(2*pi*Fc*(0:39)'/Fs);

Decimate the cosine signal using the IIR halfband decimator.

output = hbIIR(input);

Plot the original and decimated signals. In order to plot the two signals in the same plot, you must account for the output delay introduced by the IIR halfband decimator and the scaling introduced by the filter. Use the `outputDelay`

function to compute the `delay`

introduced by the decimator. Shift the output by this delay value.

Visualize the input and the resampled signals. Due to the decimation factor of 2, the output samples coincide with every other input sample.

[delay,FsOut] = outputDelay(hbIIR,FsIn=Fs,Fc=Fc)

delay = 9.9900

FsOut = 0.5000

nInput = (0:length(input)-1); tOutput = (0:length(output)-1)/FsOut-delay; stem(tOutput,output,'filled',MarkerSize=4); hold on; stem(nInput,input); hold off; xlim([-10,25]) legend('Decimated by 2','Input signal','Location','best');

### 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. Use a minimum-order design with a transition width of 0.0952 in normalized frequency units and a stopband attenuation of 60 dB.

IIRHalfbanddecim = dsp.IIRHalfbandDecimator(... NormalizedFrequency=true,... Specification='Transition width and stopband attenuation',... TransitionWidth=0.0952,... StopbandAttenuation=60);

Filter a two-channel input into lowpass and highpass subbands. The input signal can be of arbitrary frame size, that is, the number of input rows does not have to be a multiple of 2.

x = randn(1025,2); [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. When the first input sample is delivered, the commutator switch feeds this input to the first branch and the halfband decimator computes the first output value. As more input samples come in, the switch delivers one sample at a time to each branch alternatively. The decimator generates output every time the first branch generates an output. This halves the sampling rate of the input signal.

**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.

For more information on filter banks, see Overview of Filter Banks.

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**

### R2023b: Support for normalized frequencies

When you set the `NormalizedFrequency`

property to
`true`

, you must specify the transition width in normalized frequency
units (0 to 1).

When you set the `NormalizedFrequency`

property to
`true`

while creating the object and you do not set the transition width,
the object automatically sets the default transition width to normalized frequency units
using the default sample rate of 44100 Hz.

iirhalfbanddecim = dsp.IIRHalfbandDecimator(NormalizedFrequency=true)

iirhalfbanddecim = dsp.IIRHalfbandDecimator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 0.1859 StopbandAttenuation: 80 DesignMethod: 'Elliptic' NormalizedFrequency: true

When you set the `NormalizedFrequency`

property to
`true`

after you create the object, you must specify the transition width
in normalized units before you run the object
algorithm.

iirhalfbanddecim = dsp.IIRHalfbandDecimator

iirhalfbanddecim = dsp.IIRHalfbandDecimator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 4100 StopbandAttenuation: 80 DesignMethod: 'Elliptic' NormalizedFrequency: false SampleRate: 44100

`NormalizedFrequency`

to `true`

and manually convert
the frequency values in Hz to normalized values using the input sample rate in Hz. For
example, if the input sample rate is 44100 Hz, you can compute the corresponding values in
normalized units using these equations.iirhalfbanddecim.NormalizedFrequency = true; iirhalfbanddecim.TransitionWidth = 4100/(44100/2)

iirhalfbanddecim = dsp.IIRHalfbandDecimator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 0.1859 StopbandAttenuation: 80 DesignMethod: 'Elliptic' NormalizedFrequency: true

### R2023b: Support for arbitrary input frame length

This object supports an input signal with an arbitrary frame length, so the input frame length does not have to be a multiple of the decimation factor 2.

When you generate code, to support arbitrary frame length for fixed-size signals, you
must set the `AllowArbitraryInputLength`

property to
`true`

while generating code.

## See Also

### Functions

`freqz`

|`freqzmr`

|`fvtool`

|`info`

|`cost`

|`polyphase`

|`outputDelay`

|`designHalfbandIIR`

### Objects

### Blocks

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