# dsp.AllpassFilter

Single section or cascaded allpass filter

## Description

The `dsp.AllpassFilter`

object filters each channel of the input using
allpass filter implementations. To import this object into Simulink^{®}, use the MATLAB^{®} System block.

To filter each channel of the input:

Create the

`dsp.AllpassFilter`

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

### Description

returns an
allpass filter System object™, `Allpass`

= dsp.AllpassFilter`Allpass`

, that filters each channel of the input signal
independently using an allpass filter, with the default structure and coefficients.

returns an allpass filter System object, `Allpass`

= dsp.AllpassFilter(`Name=Value`

)`Allpass`

, with each property set to the specified value
by one or more `Name-Value`

pair arguments. `Name`

is the property name and `Value`

is the corresponding value. For
example, `Structure='Lattice'`

sets the filter structure to
`'Lattice'`

.

## 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.

`Structure`

— Internal allpass filter structure

`'Minimum multiplier'`

(default) | `'Lattice'`

| `'Wave Digital Filter'`

Internal allpass filter implementation structure, specified as one of these:

`'Minimum multiplier'`

`'Lattice'`

`'Wave Digital Filter'`

Each structure uses a different set of coefficients, independently stored in the corresponding object property.

`AllpassCoefficients`

— Allpass polynomial coefficients

`[-2^(-1/2) 0.5]`

(default) | *N*-by-1 | *N*-by-2 | *N*-by-4

Real allpass polynomial filter coefficients, specified as one of these:

*N*-by-1 matrix ––*N*first-order allpass sections.*N*-by-2 matrix ––*N*second-order allpass sections.*N*-by-4 matrix ––*N*fourth-order allpass sections.*(since R2024a)*

The default value of `[-2^(-1/2) 0.5]`

defines a stable
second-order allpass filter with poles and zeros located at ±π/3 in the
*z*-plane.

**Tunable: **Yes

#### Dependencies

To enable this property, set the `Structure`

property to
`'Minimum multiplier'`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`WDFCoefficients`

— Wave Digital Filter allpass coefficients

[`1/2, -2^(1/2)/3`

] (default) | *N*-by-1 | *N*-by-2

Real allpass coefficients in the Wave Digital Filter form, specified as an
*N*-by-`1`

or
*N*-by-`2`

matrix of *N*
first-order or second-order allpass sections. All elements must have absolute values
less than or equal to `1`

. This value is a transformed version of the
default value of `AllpassCoefficients`

, computed using
`allpass2wdf(AllpassCoefficients)`

. These coefficients define the
same stable second-order allpass filter as when `Structure`

is set to
`'Minimum multiplier'`

.

**Tunable: **Yes

#### Dependencies

To enable this property, set the `Structure`

property to
`'Wave Digital Filter'`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`LatticeCoefficients`

— Lattice allpass coefficients

`[-2^(1/2)/3, 1/2]`

(default) | vector

Real or complex allpass coefficients as lattice reflection coefficients, specified
as a row vector (single-section configuration) or a column vector. This value is a
transformed and transposed version of the default value of
`AllpassCoefficients`

, computed using ```
transpose(tf2latc([1
h.AllpassCoefficients]))
```

. These coefficients define the same stable
second-order allpass filter as when `Structure`

is set to
`'Lattice'`

.

**Tunable: **Yes

#### Dependencies

To enable this property, set the `Structure`

property to
`'Lattice'`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

**Complex Number Support: **Yes

`TrailingFirstOrderSection`

— Indicate if last section is first order

`false`

(default) | `true`

Indicate if last section is first order. When you set the
`TrailingFirstOrderSection`

property to `true`

,
the last section is considered to be first-order, and the second order term in the last
row of the *N*-by-2 matrix is ignored. In case of the minimum
multiplier structure, if you specify an *N*-by-4 matrix, this property
has no effect on the filter coefficients.

#### Dependencies

To enable this property, set the `Structure`

property to
`'Minimum multiplier'`

or ```
'Wave Digital
Filter'
```

.

## Usage

### Syntax

### Description

### Input Arguments

`x`

— Data input

vector | matrix

Data input, specified as a vector or a matrix. This object also accepts variable-size inputs. Once the object is locked, you can change the size of each input channel, but you cannot change the number of channels.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

### Output Arguments

`y`

— Filtered output

vector | matrix

Filtered output, returned as a vector or a matrix. The size, data type, and complexity of the output signal matches that of the input signal.

**Data Types: **`double`

| `single`

**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.AllpassFilter`

`freqz` | Frequency response of discrete-time filter System object |

`filterAnalyzer` | Analyze filters with Filter Analyzer app |

`impz` | Impulse response of discrete-time filter System object |

`info` | Information about filter System object |

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

`cost` | Estimate cost of implementing filter System object |

`grpdelay` | Group delay response of discrete-time filter System object |

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

## Examples

### Lowpass Filtering using Two Allpass Filters

Construct the two `dsp.AllpassFilter`

objects.

Fs = 48000; % in Hz FL = 1024; APF1 = dsp.AllpassFilter(AllpassCoefficients=[-0.710525516540603 0.208818210000029]); APF2 = dsp.AllpassFilter(AllpassCoefficients=[-0.940456403667957 0.6;... -0.324919696232907 0],... TrailingFirstOrderSection=true);

Construct the Transfer Function Estimator to estimate the transfer function between the random input and the Allpass filtered output.

TFE = dsp.TransferFunctionEstimator(FrequencyRange='onesided',... SpectralAverages=2);

Construct the `dsp.ArrayPlot`

object to plot the magnitude response.

AP = dsp.ArrayPlot(PlotType='Line',YLimits=[-80 5],... YLabel='Magnitude (dB)',SampleIncrement=Fs/FL,... XLabel='Frequency (Hz)',Title='Magnitude Response',... ShowLegend=true,ChannelNames={'Magnitude Response'});

Filter the Input and show the magnitude response of the estimated transfer function between the input and the filtered output.

tic; while toc < 5 in = randn(FL,1); out = 0.5.*(APF1(in) + APF2(in)); A = TFE(in, out); AP(db(A)); end

### Design and Implement Quasilinear IIR Halfband Filter

*Since R2024a*

Design a quasi-linear IIR halfband filter with the order of 32 using the `designHalfbandIIR`

function. Assign the filter coefficients to a coupled allpass filter.

`[a0,a1] = designHalfbandIIR(FilterOrder=32,DesignMethod="quasilinphase",Verbose=true)`

designHalfbandIIR(FilterOrder=32, TransitionWidth=0.1, DesignMethod="quasilinphase", Structure="single-rate", Datatype="double", SystemObject=false)

`a0 = `*4×4*
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

`a1 = `*4×4*
0 0.8085 0 0.3051
0 0.0387 0 0.2695
0 -0.7054 0 0.2604
0 0.3546 0 -0.4386

Construct the corresponding coupled allpass filter using the `cascade`

and `parallel`

functions. Alternatively, if you set the `SystemObject`

argument of the `designHalfbandIIR`

function to `true`

, the function designs the same object.

B0 = dsp.AllpassFilter(AllpassCoefficients=a0); B1 = dsp.AllpassFilter(AllpassCoefficients=a1); filtObj = cascade(parallel(B0,cascade(dsp.Delay, B1)),0.5)

filtObj = dsp.FilterCascade with properties: Stage1: [1x1 dsp.ParallelFilter] Stage2: 0.5000 CloneStages: true

Create a `dsp.DynamicFilterVisualizer`

object and visualize the magnitude response of the filter.

dfv = dsp.DynamicFilterVisualizer(NormalizedFrequency=true); dfv(filtObj);

A quasi-linear IIR filter has a fairly constant group delay (hence almost linear phase) in the passband region of the filter.

grpdelay(filtObj)

phasez(filtObj)

Create a `spectrumAnalyzer`

object to visualize the spectra of the input and output signals.

scope = spectrumAnalyzer(SampleRate=2, ... PlotAsTwoSidedSpectrum=false,... ChannelNames=["Input Signal","Filtered Signal"]);

Stream in a noisy sinusoidal signal and filter the signal using the IIR halfband filter. The sinusoidal tone falls in the passband frequency of the filter and is therefore unaffected.

sine = dsp.SineWave(Frequency=1000,SampleRate=44100,SamplesPerFrame=1024)

sine = dsp.SineWave with properties: Amplitude: 1 Frequency: 1000 PhaseOffset: 0 ComplexOutput: false Method: 'Trigonometric function' SamplesPerFrame: 1024 SampleRate: 44100 OutputDataType: 'double'

for i = 1:1000 x = sine()+0.005*randn(1024,1); y = filtObj(x); scope(x,y); end

### Design and Implement Butterworth IIR Halfband Filter

Design a Butterworth IIR halfband filter with the order of 13 and construct the corresponding coupled allpass filter using the `designHalfbandIIR`

function. Set the `SystemObject`

argument of the function to `true`

.

filtObj = designHalfbandIIR(FilterOrder=13,Verbose=true,SystemObject=true)

designHalfbandIIR(FilterOrder=13, DesignMethod="butter", Structure="single-rate", Datatype="double", SystemObject=true, Passband="lowpass")

filtObj = dsp.FilterCascade with properties: Stage1: [1x1 dsp.ParallelFilter] Stage2: 0.5000 CloneStages: true

Create a `dsp.DynamicFilterVisualizer`

object and visualize the magnitude response of the filter.

dfv = dsp.DynamicFilterVisualizer(NormalizedFrequency=true); dfv(filtObj);

Create a `spectrumAnalyzer`

object to visualize the spectra of the input and output signals.

scope = spectrumAnalyzer(SampleRate=2, ... PlotAsTwoSidedSpectrum=false,... ChannelNames=["Input Signal","Filtered Signal"]);

Stream in random data and filter the signal using the IIR halfband filter.

for i = 1:1000 x = randn(1024, 1); y = filtObj(x); scope(x,y); end

## Algorithms

The transfer function of an allpass filter is given by

$$H(z)=\frac{c(n)+c(n-1){z}^{-1}+\mathrm{...}+{z}^{-n}}{1+c(1){z}^{-1}+\mathrm{...}+c(n){z}^{-n}}$$.

*c* is allpass polynomial coefficients vector.
The order, *n*, of the transfer function is the length
of vector *c*.

In the minimum multiplier form and wave digital form, the allpass
filter is implemented as a cascade of either second-order (biquad)
sections or first-order sections. When the coefficients are specified
as an *N*-by-2 matrix, each row of the matrix specifies
the coefficients of a second-order filter. The last element of the
last row can be ignored based on the trailing first-order setting.
When the coefficients are specified as an *N*-by-1
matrix, each element in the matrix specifies the coefficient of a
first-order filter. The cascade of all the filter sections forms the
allpass filter.

In the lattice form, the coefficients are specified as a vector.

These structures are computationally more economical and structurally more stable compared to the generic IIR filters, such as df1, df1t, df2, df2t. For all structures, the allpass filter can be a single-section or a multiple-section (cascaded) filter. The different sections can have different orders, but they are all implemented according to the same structure.

### Minimum Multiplier

This structure realizes the allpass filter with the minimum number of required multipliers,
equal to the order `n`

. It also uses `2n`

delay units
and `2n`

adders. The multipliers uses the specified coefficients, which
are equal to the polynomial vector *c* in the allpass transfer
function.

In this second-order section of the minimum multiplier structure, the coefficients
vector, *c*, is equal to [`0.1 -0.7`

].

In this fourth-order section of the minimum multiplier structure, the coefficients
vector, *c*, is equal to [```
-0.7071 0.5 0.4
0.25
```

].

### Wave Digital Filter

This structure uses `n`

multipliers, but only `n`

delay
units, at the expense of requiring `3n`

adders. To
use this structure, specify the coefficients in wave digital filter
(WDF) form. Obtain the WDF equivalent of the conventional allpass
coefficients using `allpass2wdf(allpass_coefficients)`

.
To convert WDF coefficients into the equivalent allpass polynomial
form, use `wdf2allpass(WDF coefficients)`

. In this
second-order section of the WDF structure, the coefficients vector *w* is
equal to `allpass2wdf([0.1 -0.7])`

.

### Lattice

This lattice structure uses `2n`

multipliers, `n`

delay
units, and `2n`

adders. To use this structure, specify
the coefficients as a vector.

You can obtain the lattice equivalent of the conventional allpass
coefficients using `transpose(tf2latc(1, [1 allpass_coefficients]))`

.
In the following second-order section of the lattice structure, the
coefficients vector is computed using ```
transpose(tf2latc(1,
[1 0.1 -0.7]))
```

. Use these coefficients for a filter that
is functionally equivalent to the minimum multiplier structure with
coefficients [0.1 -0.7].

## References

[1] Regalia, Philip A. and Mitra Sanjit K. and Vaidyanathan, P. P.
(1988) “The Digital All-Pass Filter: A Versatile Signal Processing Building
Block.” *Proceedings of the IEEE*, Vol. 76, No. 1, 1988, pp.
19–37

[2] M. Lutovac, D. Tosic, B. Evans, *Filter Design for
Signal Processing Using MATLAB and Mathematica.* Upper Saddle River, NJ: Prentice
Hall, 2001.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The System object supports code generation only when the

`Structure`

property is set to`Minimum multiplier`

or`Lattice`

.See System Objects in MATLAB Code Generation (MATLAB Coder).

## Version History

**Introduced in R2013a**

### R2024a: Minimum multiplier structure supports fourth-order allpass sections

When you set the `Structure`

property to ```
'Minimum
multipler'
```

, you can specify an *N*-by-4 matrix of filter
coefficients in the `AllpassCoefficients`

property, where
*N* is the number of filter sections.

## See Also

### Functions

### Objects

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