# dsp.FIRHalfbandInterpolator

Halfband interpolator

## Description

The `dsp.FIRHalfbandInterpolator`

System object™ performs efficient polyphase interpolation of the input signal using an
upsampling factor of two. You can use `dsp.FIRHalfbandInterpolator`

to
implement the synthesis portion of a two-band filter bank to synthesize a signal from
lowpass and highpass subbands. `dsp.FIRHalfbandInterpolator`

uses an FIR
equiripple design or a Kaiser window design to construct the halfband filters and a
polyphase implementation to filter the input. For more details, see Algorithms.

To upsample and interpolate your data:

Create the

`dsp.FIRHalfbandInterpolator`

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?

This object supports C/C++ code generation and SIMD code generation under certain
conditions. This object also supports code generation for ARM^{®}
Cortex^{®}-M and ARM
Cortex-A processors. For more information, see
Code Generation.

## Creation

### Syntax

### Description

returns a FIR halfband interpolation filter,
`firhalfbandinterp`

= dsp.FIRHalfbandInterpolator`firhalfbandinterp`

, with the default settings. Under the
default settings, the System object upsamples and interpolates the input data using a halfband
frequency of `11025`

Hz, a transition width of
`4.1`

kHz, and a stopband attenuation of
`80`

dB. The design method is set to
`"Auto"`

.

returns a halfband interpolator, with additional properties specified by one or
more `firhalfbandinterp`

= dsp.FIRHalfbandInterpolator(`Name=Value`

)`Name-Value`

pair arguments.

**Example: **```
firhalfbandinterp =
dsp.FIRHalfbandInterpolator(Specification="Filter order and stopband
attenuation")
```

creates an FIR halfband interpolator object with
filter order set to 52 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 following, you choose
two of the three available design parameters to design the FIR Halfband
filter.

`"Transition width and stopband attenuation"`

–– Transition width and stopband attenuation are the design parameters.`"Filter order and stopband attenuation"`

–– Filter order and stopband attenuation are the design parameters.`"Filter order and transition width"`

–– Filter order and transition width are the design parameters.

The filter is designed using optimal equiripple filter design method.

When you set `Specification`

to
`"Coefficients"`

, you specify the halfband filter
coefficients directly through the `Numerator`

property.

`FilterOrder`

— Filter order

`52`

(default) | even positive integer

Filter order, specified as an even positive integer.

#### 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`

— Stopband attenuation

`80`

(default) | positive real scalar

Stopband attenuation in dB, specified as a positive real scalar.

#### Dependencies

To enable this property, set `Specification`

to
`"Filter order and stopband attenuation"`

or
```
"Transition width and stopband
attenuation"
```

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`TransitionWidth`

— Transition width

`4100`

(default) | positive real scalar

Transition width in Hz, 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 output sample rate (2 ×`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 default transition width is automatically set to normalized frequency units using the default sample rate at which the filter operates (2 × 44100) Hz. The filter in the halfband interpolator effectively runs at twice the sample rate of the input signal.When you set the

`NormalizedFrequency`

property to`true`

after you create the object, the transition width must be specified 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*/(2×_{Hz}*Fs*/2).firhalfbandinterp = dsp.FIRHalfbandInterpolator; firhalfbandinterp.NormalizedFrequency = true; firhalfbandinterp.TransitionWidth = 4100/((2x44100)/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`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`Numerator`

— FIR halfband filter coefficients

`2*firhalfband("minorder",0.407,1e-4)`

(default) | row vector

FIR halfband filter coefficients, specified as a row vector. The
coefficients must comply with the FIR halfband impulse response format. For
details on this format, see Halfband Filters and FIR Halfband Filter Design. If
half the order of the filter, `(length(Numerator) - 1)/2`

is even, every other coefficient starting from the first coefficient must be
a zero except for the center coefficient which must be a 1.0. If half the
order of the filter is odd, the sequence of alternating zeros with a 1.0 at
the center starts at the second coefficient.

#### Dependencies

To enable this property, set `Specification`

to
`"Coefficients"`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`DesignMethod`

— Filter design method

`"Auto"`

(default) | `"Equiripple"`

| `"Kaiser"`

Specify the filter design method as one of the following:

`"Auto"`

–– The algorithm automatically chooses the filter design method depending on the filter design parameters. The algorithm uses the equiripple or the Kaiser window method to design the filter.If the design constraints are very tight, such as very high stopband attenuation or very narrow transition width, then the algorithm automatically chooses the Kaiser method, as this method is optimal for designing filters with very tight specifications. However, if the design constraints are not tight, then the algorithm chooses the equiripple method.

When you set the

`DesignMethod`

property to`"Auto"`

, you can determine the method used by the algorithm by examining the passband and stopband ripple characteristics of the designed filter. If the object used the equiripple method, the passband and stopband ripples of the designed filter have a constant amplitude in the frequency response. If the filter design method the object chooses in the`"Auto"`

mode is not suitable for your application, manually specify the`DesignMethod`

as`"Equiripple"`

or`"Kaiser"`

.`"Equiripple"`

–– The algorithm uses the equiripple method.`"Kaiser"`

–– The algorithm uses the Kaiser window method.

For more details on these two methods, see Algorithms.

#### Dependencies

To enable this property, set `Specification`

to
`"Transition width and stopband attenuation"`

,
`"Filter order and stopband attenuation"`

, or
`"Filter order and transition width"`

.

**Data Types: **`char`

| `string`

`FilterBankInputPort`

— Synthesis filter bank

`false`

(default) | `true`

Synthesis filter bank, specified as either `false`

or
`true`

. If this property is `false`

,
`dsp.FIRHalfbandInterpolator`

is an
interpolation filter for a single vector- or matrix-valued input when you
call the algorithm. If this property is `true`

, `dsp.FIRHalfbandInterpolator`

is a synthesis
filter bank and the algorithm accepts two inputs, the lowpass and highpass
subbands to synthesize.

`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 default transition width is automatically set to normalized frequency units using the default sample rate at which the filter operates (2 × 44100) Hz. The filter in the halfband interpolator effectively runs at twice the sample rate of the input signal.When you set the

`NormalizedFrequency`

property to`true`

after you create the object, the transition width must be specified 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*/(2×_{Hz}*Fs*/2).firhalfbandinterp = dsp.FIRHalfbandInterpolator; firhalfbandinterp.NormalizedFrequency = true; firhalfbandinterp.TransitionWidth = 4100/((2x44100)/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. The input
sample rate defaults to 44100 Hz. If you specify a transition width as one
of your filter design parameters, the transition width cannot exceed 1/2 the
output sample rate (2 × `SampleRate`

property)
value.

#### Dependency

To enable this property, set:

`Specification`

to any accepted value except`"Coefficients"`

.`NormalizedFrequency`

to`false`

.*(since R2023b)*

**Data Types: **`single`

| `double`

### Fixed-Point Properties

`CoefficientsDataType`

— Word and fraction lengths of coefficients

`numerictype(1,16)`

(default) | `numerictype`

object

Word and fraction lengths of coefficients, specified as a signed or
unsigned `numerictype`

object. The default,
`numerictype(1,16)`

corresponds to a signed numeric
type object with 16-bit coefficients and a fraction length determined
based on the coefficient values, to give the best possible
precision.

This property is not tunable.

Word length of the output is same as the word length of the input. Fraction length of the output is computed such that the entire dynamic range of the output can be represented without overflow. For details on how the fraction length of the output is computed, see Fixed-Point Precision Rules for Avoiding Overflow in FIR Filters.

`RoundingMethod`

— Rounding method for output fixed-point operations

`"Floor"`

(default) | `"Ceiling"`

| `"Convergent"`

| `"Nearest"`

| `"Round"`

| `"Simplest"`

| `"Zero"`

Rounding method for output fixed-point operations, specified as a character vector. For more information on the rounding modes, see Precision and Range.

## Usage

### Description

implements a halfband synthesis filter bank for the inputs
`y`

= firhalfbandinterp(`x1`

,`x2`

)`x1`

and `x2`

. `x1`

is the lowpass output of a halfband analysis filter bank and
`x2`

is the highpass output of a halfband analysis filter
bank. `dsp.FIRHalfbandInterpolator`

implements a synthesis filter
bank only when the `FilterBankInputPort`

property is set to
`true`

.

### Input Arguments

`x1`

— Data input

column vector | matrix

Data input to the FIR halfband interpolator, specified as a column vector or a matrix. This signal is the lowpass output of a halfband analysis filter bank. If the input signal is a matrix, each column of the matrix is treated as an independent channel.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `fi`

**Complex Number Support: **Yes

`x2`

— Second data input

column vector | matrix

Second data input to the synthesis filter bank, specified as a column vector or a matrix. This signal is the highpass output of a halfband analysis filter bank. If the input signal is a matrix, each column of the matrix is treated as an independent channel.

The size, data type, and complexity of both the inputs must be the same.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `fi`

**Complex Number Support: **Yes

### Output Arguments

`y`

— Output of interpolator

column vector | matrix

Output of the interpolator, returned as a column vector or a matrix. The number of rows in the interpolator output is twice the number of rows in the input signal.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `fi`

**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.FIRHalfbandInterpolator`

`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 |

`coeffs` | Returns the filter System object coefficients in a structure |

`polyphase` | Polyphase decomposition of multirate filter |

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

## Examples

### Impulse and Frequency Response of Halfband Interpolation Filter

Create two lowpass halfband interpolation filters. The design method in the first filter is set to `"Equiripple"`

and in the second filter is set to `"Kaiser"`

.

Specify a filter order of 52 and a transition width of 0.0930 in normalized frequency units.

Order = 52; TW = 0.0930; filterspec = "Filter order and transition width"; firhalfbandinterpEqui = dsp.FIRHalfbandInterpolator(... NormalizedFrequency=true,... Specification=filterspec,... FilterOrder=Order,... TransitionWidth=0.0930,... DesignMethod="Equiripple"); firhalfbandinterpKaiser = dsp.FIRHalfbandInterpolator(... NormalizedFrequency=true,... Specification=filterspec,... FilterOrder=Order,... TransitionWidth=TW,... DesignMethod="Kaiser");

Plot the impulse response of both the filters. The zeroth-order coefficient is delayed by 26 samples, which is equal to the group delay of the filter. This yields a causal halfband filter.

hfvt = fvtool(firhalfbandinterpEqui,firhalfbandinterpKaiser,... Analysis="Impulse"); legend(hfvt,{'Equiripple','Kaiser'})

Plot the magnitude and phase response.

If the filter specifications are tight, say a very high filter order with a very narrow transition width, the filter designed using the `"Kaiser"`

method converges more effectively.

hvftMag = fvtool(firhalfbandinterpEqui,firhalfbandinterpKaiser,... Analysis="Magnitude"); legend(hvftMag,{'Equiripple','Kaiser'})

hvftPhase = fvtool(firhalfbandinterpEqui,firhalfbandinterpKaiser,... Analysis="Phase"); legend(hvftPhase,{'Equiripple','Kaiser'})

### Design and Implement FIR Halfband Interpolator

Design an equiripple FIR halfband interpolator object of order 48 using the `designHalfbandFIR`

function. Set the `Verbose`

argument to `true`

.

hbFIR = designHalfbandFIR(FilterOrder=48,SystemObject=true,... Structure='interp',Verbose=true)

designHalfbandFIR(FilterOrder=48, TransitionWidth=0.1, DesignMethod="equiripple", Passband="lowpass", Structure="interp", SystemObject=true)

hbFIR = dsp.FIRHalfbandInterpolator with properties: Specification: 'Coefficients' Numerator: [0 -0.0082 0 0.0079 0 -0.0116 0 0.0165 0 -0.0227 0 0.0309 0 -0.0419 0 0.0571 0 -0.0800 0 0.1193 0 -0.2073 0 0.6350 1 0.6350 0 -0.2073 0 0.1193 0 -0.0800 0 0.0571 0 -0.0419 0 0.0309 0 -0.0227 0 0.0165 0 -0.0116 ... ] (1x49 double) FilterBankInputPort: 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); dfv(hbFIR);

The input is a cosine wave with an angular frequency of $\frac{\pi}{4}$ radians/sample.

input = cos(pi/4*(0:39)');

Interpolate the cosine signal using the FIR halfband interpolator.

output = hbFIR(input);

Plot the original and interpolated signals. In order to plot the two signals on the same plot, you must account for the output delay introduced by the FIR halfband interpolator and the scaling introduced by the filter. Use the `outputDelay`

function to compute the `delay`

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

Visualize the input and the resampled signals. The input and output values coincide every other sample due to the interpolation factor of 2.

[delay,FsOut] = outputDelay(hbFIR,FsIn=1)

delay = 12

FsOut = 2

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([-5,20]) legend('Interpolated by 2','Input signal','Location','best');

### 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, audio device writer, and interpolation filter. The sample rate of the audio data is 22050 Hz. The order of the halfband filter is 52, with a transition width of 2 kHz.

Set the design method to `"Auto"`

. This mode chooses one of the filter design methods, equiripple or Kaiser, based on the design parameters of the filter.

afr = dsp.AudioFileReader(... "speech_dft.mp3",... SamplesPerFrame=1024); filtSpec = "Filter order and transition width"; Order = 52; TW = 2000; firhalfbanddecim = dsp.FIRHalfbandDecimator(... Specification=filtSpec,... FilterOrder=Order,... TransitionWidth=TW,... DesignMethod="Auto",... SampleRate=afr.SampleRate); firhalfbandinterp = dsp.FIRHalfbandInterpolator(... Specification=filtSpec,... FilterOrder=Order,... TransitionWidth=TW,... DesignMethod="Auto",... SampleRate=afr.SampleRate/2); adw = audioDeviceWriter(SampleRate=afr.SampleRate);

View the magnitude response of the halfband filter.

fvtool(firhalfbanddecim)

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 = firhalfbanddecim(audioframe); ylow = firhalfbandinterp(xlo); adw(ylow); end

Wait until the audio file is played to the end, then close the input file and release the audio output resource.

release(afr); release(adw);

### 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 device writer. Construct the FIR 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 = 52; TW = 2000; firhalfbanddecim = dsp.FIRHalfbandDecimator(... Specification=filterspec,FilterOrder=Order,... TransitionWidth=TW,DesignMethod="Auto",... SampleRate=AF.SampleRate); firhalfbandinterp = dsp.FIRHalfbandInterpolator(... Specification=filterspec,FilterOrder=Order,... TransitionWidth=TW,SampleRate=AF.SampleRate/2,... DesignMethod="Auto",... 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] = firhalfbanddecim(audioInput); audioOutput = firhalfbandinterp(xlo,xhigh); spectrumInput = [audioInput audioOutput]; SpecAna(spectrumInput); AP(audioOutput); end release(AF); release(AP); release(SpecAna);

### Upsample and Interpolate Multichannel Input Using FIR Halfband Interpolator

Create a half-band interpolation filter. The filter order is 52 with a transition width of 0.0930 in normalized frequency units. Use the filter to upsample and interpolate a multichannel input.

Set the design method to `"Auto"`

. This mode chooses one of the filter design methods, equiripple or Kaiser, based on the specified filter design parameters.

Fs = 44.1e3; filterspec = "Filter order and transition width"; Order = 52; TW = 0.0930; firhalfbandinterp = dsp.FIRHalfbandInterpolator(... NormalizedFrequency=true,... Specification=filterspec,... FilterOrder=Order,... TransitionWidth=TW,... DesignMethod="Auto");

x = randn(1024,4); y = firhalfbandinterp(x);

## More About

### Halfband Filters

An ideal lowpass halfband filter is given by

$$h(n)=\frac{1}{2\pi}{\displaystyle {\int}_{-\pi /2}^{\pi /2}{e}^{j\omega n}}d\omega =\frac{\mathrm{sin}({\scriptscriptstyle \frac{\pi}{2}}n)}{\pi n}.$$

An ideal filter is not realizable because the impulse response is noncausal and not absolutely summable. However, the impulse response of an ideal lowpass filter possesses some important properties that are required in a realizable approximation. The impulse response of an ideal lowpass halfband filter is:

Equal to 0 for all even-indexed samples

Equal to 1/2 at n=0 as shown by L'Hôpital's rule on the continuous-valued equivalent of the discrete-time impulse response

The ideal highpass halfband filter is given by

$$g(n)=\frac{1}{2\pi}{\displaystyle {\int}_{-\pi}^{-\pi /2}{e}^{j\omega n}}d\omega +\frac{1}{2\pi}{\displaystyle {\int}_{\pi /2}^{\pi}{e}^{j\omega n}}d\omega .$$

Evaluating the preceding integral gives the following impulse response

$$g(n)=\frac{\mathrm{sin}(\pi n)}{\pi n}-\frac{\mathrm{sin}({\scriptscriptstyle \frac{\pi}{2}}n)}{\pi n}.$$

The impulse response of an ideal highpass halfband filter is:

Equal to 0 for all even-indexed samples

Equal to 1/2 at n=0

The FIR halfband interpolator uses a causal FIR approximation to the ideal halfband response, which is based on minimizing the $${\ell}^{\infty}$$ norm of the error (minimax). See Algorithms for more information.

### Kaiser Window

The coefficients of a Kaiser window are computed from this equation:

$$w(n)=\frac{{I}_{0}\left(\beta \sqrt{1-{\left(\frac{n-N/2}{N/2}\right)}^{2}}\right)}{{I}_{0}(\beta )},\text{\hspace{1em}}\text{\hspace{1em}}0\le n\le N,$$

where *I*_{0} is the zeroth-order modified Bessel function of the first kind.

To obtain a Kaiser window that represents an FIR filter with stopband attenuation of
*α* dB, use this *β*.

$$\beta =\{\begin{array}{ll}0.1102(\alpha -8.7),\hfill & \alpha >50\hfill \\ 0.5842{(\alpha -21)}^{0.4}+0.07886(\alpha -21),\hfill & 50\ge \alpha \ge 21\hfill \\ 0,\hfill & \alpha <21\hfill \end{array}$$

The filter order *n* is given by:

$$n=\frac{\alpha -7.95}{2.285(\Delta \omega )},$$

where Δ*ω* is the transition width.

## Algorithms

### Filter Design Method

The FIR halfband interpolator algorithm uses the equiripple or the Kaiser window method to
design the FIR halfband filter. When the design constraints are tight, such as very high
stopband attenuation or very narrow transition width, use the Kaiser window method. When
the design constraints are not tight, use the equiripple method. If you are not sure of
which method to use, set the design method to `Auto`

. In this mode, the
algorithm automatically chooses a design method that optimally meets the specified
filter constraints.

**Halfband Equiripple Design**

In the equiripple method, the algorithm uses a minimax (minimize the maximum error) FIR design to design a fullband linear phase filter with the desired specifications. The algorithm upsamples a fullband filter to replace the even-indexed samples of the filter with zeros and creates a halfband filter. It then sets the filter tap corresponding to the group delay of the filter in samples to 1/2. This yields a causal linear-phase FIR filter approximation to the ideal halfband filter defined in Halfband Filters. See [1] for a description of this filter design method using the Remez exchange algorithm. Since you can design a filter using this approximation method with a constant ripple both in the passband and stopband, the filter is also known as the equiripple filter.

**Kaiser Window Design**

In the Kaiser window method, the algorithm first truncates the ideal halfband filter defined in Halfband Filters, then it applies a Kaiser window defined in Kaiser Window. This yields a causal linear-phase FIR filter approximation to the ideal halfband filter.

The coefficients of the designed halfband interpolation filter are scaled by the interpolation factor, two, to preserve the output power of the signal.

For more information on designing FIR halfband filters, see FIR Halfband Filter Design.

### Polyphase Implementation with Halfband Filters

The FIR halfband interpolator uses an efficient polyphase implementation for halfband filters when you filter the input signal. You can use a polyphase implementation to move the upsampling operation after filtering. This allows you to filter at the lower sampling rate.

Splitting a filter’s impulse response *h*(*n*) into two
polyphase components results in an even polyphase component with
*z*-transform of

$${H}_{0}(z)={\displaystyle \sum _{n}h}(2n){z}^{-n},$$

and an odd polyphase component with *z*-transform of

$${H}_{1}(z)={\displaystyle \sum _{n}h}(2n+1){z}^{-n}.$$

The *z*-transform of the filter can be written in terms of the even and odd
polyphase components as

$$H(z)={H}_{0}({z}^{2})+{z}^{-1}{H}_{1}({z}^{2}).$$

You can represent the upsampling by 2 and then filtering the signal using this figure.

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

For a halfband filter, the only nonzero coefficient in the even polyphase component is the
coefficient corresponding to *z ^{0}*.
Implementing the halfband filter as a causal FIR filter shifts the nonzero coefficient
to approximately

*z*, where

^{-N/4}*N*is the number of filter taps. This process is shown in the following figure. The top plot shows a halfband filter of order 52. The bottom plot shows the even polyphase component. Both filters are noncausal. Delaying the even polyphase component by 13 samples creates a causal FIR filter.

To efficiently implement the halfband interpolator, the algorithm replaces the upsampling operator, delay block, and adder with a commutator switch. The commutator switch starts on the even branch and takes input samples from the two branches alternately, one sample at a time. This doubles the sampling rate of the input signal. Which polyphase component reduces to a simple delay depends on whether the half order of the filter is even or odd.

Here is the implementation when the filter half order is even. In this diagram,
*H*_{0}(z) becomes the gain followed by the
delay.

When the filter half order is odd, *H*_{1}(z)
becomes the gain followed by the delay and the switch. This is because the delay
required to make the even polyphase component causal can be odd or even depending on the
filter half order.

To confirm this behavior, run the following code in the MATLAB^{®} command prompt and inspect the polyphase components of the following
filters.

filterspec = "Filter order and stopband attenuation"; halfOrderEven = dsp.FIRHalfbandInterpolator(Specification=filterspec,... FilterOrder=64,StopbandAttenuation=80); halfOrderOdd = dsp.FIRHalfbandInterpolator(Specification=filterspec,... FilterOrder=54,StopbandAttenuation=80); polyphase(halfOrderEven) polyphase(halfOrderOdd)

One of the polyphase components has a single nonzero coefficient indicating that it is a simple delay. To preserve the output power of the signal, the coefficients are scaled by the interpolation factor, two. To see this scaling, compare the polyphase components of a halfband interpolator with the coefficients of a halfband decimator.

hfirinterp = dsp.FIRHalfbandInterpolator; hfirdecim = dsp.FIRHalfbandDecimator; polyphase(hfirdecim) polyphase(hfirinterp)

**Synthesis Filter Bank**

The FIR halfband interpolator implements the synthesis portion of a two-band filter bank to synthesize a signal from lowpass and highpass subbands.

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

To summarize, the FIR halfband interpolator:

Filters the input before upsampling with the even and odd polyphase components of the filter.

Exploits the fact that one filter polyphase component is a simple delay for a halfband filter.

Acts as a synthesis filter bank.

## References

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

## 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. To learn more about ARM Cortex code generation, see Code Generation for ARM Cortex-M and ARM Cortex-A Processors.

This object also supports SIMD code generation using Intel^{®} AVX2 code replacement library when the input signal has a data type of
`single`

or `double`

.

The SIMD technology significantly improves the performance of the generated code. For more information, see SIMD Code Generation. To generate SIMD code from this object, see Use Intel AVX2 Code Replacement Library to Generate SIMD Code from MATLAB Algorithms.

## Version History

**Introduced in R2014b**

### 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 default transition width is automatically set to normalized
frequency units using the default sample rate at which the filter operates (2 ×
44100) Hz. The filter in the halfband interpolator effectively runs at twice the
sample rate of the input signal.

firhalfbandinterp = dsp.FIRHalfbandInterpolator(NormalizedFrequency=true)

firhalfbandinterp = dsp.FIRHalfbandInterpolator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 0.0930 StopbandAttenuation: 80 DesignMethod: 'Auto' FilterBankInputPort: false NormalizedFrequency: true Show all properties

When you set the `NormalizedFrequency`

property to
`true`

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

firhalfbandinterp = dsp.FIRHalfbandInterpolator

firhalfbandinterp = dsp.FIRHalfbandInterpolator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 4100 StopbandAttenuation: 80 DesignMethod: 'Auto' FilterBankInputPort: false NormalizedFrequency: false SampleRate: 44100 Show all properties

`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, the corresponding values
in normalized units are computed using these equations.firhalfbandinterp.NormalizedFrequency = true; firhalfbandinterp.TransitionWidth = 4100/((2x44100)/2)

firhalfbandinterp = dsp.FIRHalfbandInterpolator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 0.0930 StopbandAttenuation: 80 DesignMethod: 'Auto' FilterBankInputPort: false NormalizedFrequency: true Show all properties

## See Also

### Functions

`designHalfbandFIR`

|`freqz`

|`fvtool`

|`info`

|`cost`

|`coeffs`

|`polyphase`

|`outputDelay`

### Objects

`dsp.FIRHalfbandDecimator`

|`dsp.IIRHalfbandInterpolator`

|`dsp.DyadicSynthesisFilterBank`

|`dsp.ChannelSynthesizer`

### Blocks

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