## Description

The `dsp.BiquadFilter` object implements a cascade of biquadratic sections, where the coefficients for each section are supplied by a separate row of an N-by-6 second-order sections (SOS) matrix. Each row of the SOS matrix contains the numerator and denominator coefficients of the corresponding section of the filter. The resulting filter can be applied to a vector or matrix input, where each column represents a channel of data that is processed independently.

To implement an IIR filter structure using biquadratic or SOS:

1. Create the `dsp.BiquadFilter` object and set its properties.

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

## Creation

### Syntax

``biquad = dsp.BiquadFilter``
``biquad = dsp.BiquadFilter(sosmatrix,scalevalues)``
``biquad = dsp.BiquadFilter(Name,Value)``

### Description

````biquad = dsp.BiquadFilter` returns a biquadratic IIR (SOS) filter System object™, `biquad`, which independently filters each channel (column) of the input over time using the SOS section ```[1 0.3 0.4 1 0.1 0.2]``` with a direct-form II transposed structure.```

example

````biquad = dsp.BiquadFilter(sosmatrix,scalevalues)` returns a biquadratic filter object, with the `SOSMatrix` property set to `sosmatrix` and the `ScaleValues` property set to `scalevalues`.```

example

````biquad = dsp.BiquadFilter(Name,Value)` returns a biquadratic filter object, `biquad`, with each property set to the specified value.```

## Properties

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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.

Specify the filter structure as `'Direct form I'`, `'Direct form I transposed'`, ```'Direct form II'```, `'Direct form II transposed'`.

Specify the source of the SOS matrix as `'Property'` or `'Input port'`.

Specify the second-order section (SOS) matrix as an N-by-6 matrix, where N is the number of sections in the filter. Each row of the SOS matrix contains the numerator and denominator coefficients of the corresponding section of the filter. The system function, H(z), of a biquad filter is:

`$H\left(z\right)=\frac{\sum _{k=0}^{2}{b}_{k}{z}^{-k}}{1-\sum _{l=1}^{2}{a}_{l}{z}^{-l}}$`

The coefficients are ordered in the rows of the SOS matrix as (b0, b1,b2,1, –a1, –a2). You can use coefficients of real or complex values. This property applies only when you set the `SOSMatrixSource` property to `Property`. The leading denominator coefficient of the biquad filter, a0, equals 1 for each filter section, regardless of the specified value.

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

Specify the scale values to apply before and after each section of a biquad filter. `ScaleValues` must be either a scalar or a vector of length `N+1`, where `N` is the number of sections. If you set this property to a scalar, the scalar value is used as the gain value only before the first filter section. The remaining gain values are set to `1`. If you set this property to a vector of `N+1`values, each value is used for a separate section of the filter.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property`.

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

Specify the initial conditions of the filter states when the `Structure` property is one of | ```Direct form II``` | `Direct form II transposed` |. The number of states or delay elements (zeros and poles) in a direct-form II biquad filter equals twice the number of filter sections. You can specify the initial conditions as a scalar, vector, or matrix.

When you specify a scalar value, the biquad filter initializes all delay elements in the filter to that value. When you specify a vector of length equal to the number of delay elements in the filter, each vector element specifies a unique initial condition for the corresponding delay element.

The biquad filter applies the same vector of initial conditions to each channel of the input signal. When you specify a vector of length equal to the product of the number of input channels and the number of delay elements in the filter, each element specifies a unique initial condition for the corresponding delay element in the corresponding channel. When you specify a matrix with the same number of rows as the number of delay elements in the filter, and one column for each channel of the input signal, each element specifies a unique initial condition for the corresponding delay element in the corresponding channel.

#### Dependencies

This property applies only when you set the `Structure` property to one of ```Direct form II``` or ```Direct form II transposed```.

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

Specify the initial conditions of the filter states on the side of the filter structure with the zeros. The number of states or delay elements in the numerator of a direct-form I biquad filter equals twice the number of filter sections. You can specify the initial conditions as a scalar, vector, or matrix. When you specify a scalar, the biquad filter initializes all delay elements on the zeros side in the filter to that value. When you specify a vector of length equal to the number of delay elements on the zeros side in the filter, each vector element specifies a unique initial condition for the corresponding delay element on the zeros side.

The biquad filter applies the same vector of initial conditions to each channel of the input signal. When you specify a vector of length equal to the product of the number of input channels and the number of delay elements on the zeros side in the filter, each element specifies a unique initial condition for the corresponding delay element on the zeros side in the corresponding channel. When you specify a matrix with the same number of rows as the number of delay elements on the zeros side in the filter, and one column for each channel of the input signal, each element specifies a unique initial condition for the corresponding delay element on the zeros side in the corresponding channel.

#### Dependencies

This property applies only when you set the `Structure` property to one of ```Direct form I``` or ```Direct form I transposed```.

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

Specify the initial conditions of the filter states on the side of the filter structure with the poles. The number of denominator states, or delay elements, in a direct-form I (noncanonic) biquad filter equals twice the number of filter sections. You can specify the initial conditions as a scalar, vector, or matrix. When you specify a scalar, the biquad filter initializes all delay elements on the poles side of the filter to that value. When you specify a vector of length equal to the number of delay elements on the poles side in the filter, each vector element specifies a unique initial condition for the corresponding delay element on the poles side.

The object applies the same vector of initial conditions to each channel of the input signal. When you specify a vector of length equal to the product of the number of input channels and the number of delay elements on the poles side in the filter, each element specifies a unique initial condition for the corresponding delay element on the poles side in the corresponding channel. When you specify a matrix with the same number of rows as the number of delay elements on the poles side in the filter, and one column for each channel of the input signal, each element specifies a unique initial condition for the corresponding delay element on the poles side in the corresponding channel.

#### Dependencies

This property only applies when you set the `Structure` property to one of ```Direct form I``` or ```Direct form I transposed```.

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

When this Boolean property is set to `true`, the biquad filter removes all unity scale gain computations. This reduces the number of computations and increases the fixed-point accuracy.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property`.

Select how to specify scale values. By default, this property is `true`, and the scale values are specified via the input port. When this property is `false`, all scale values are 1.

#### Dependencies

This property applies only when the `SOSMatrixSource` property is ```Input port```.

### Fixed-Point Properties

Specify the rounding method.

Specify the overflow action as one of `Wrap` or `Saturate`.

Specify the multiplicand fixed-point data type as one of ```Same as output``` or `Custom`.

#### Dependencies

This property applies only when you set the `Structure` property to ```Direct form I transposed```.

Specify the multiplicand fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies only when you set the `MultiplicandDataType` property to `Custom`.

Specify the section input fixed-point data type as either `Same as input` or `Custom`.

Specify the section input fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies only when you set the `SectionInputDataType` property to `Custom`.

Specify the section output fixed-point data type as either `Same as section input` or `Custom`.

Specify the section output fixed-point type as a signed, scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies only when you set the `SectionOutputDataType` property to `Custom`.

Specify the numerator coefficients fixed-point data type as `Same word length as input` or `Custom`. Setting this property also sets the `DenominatorCoefficientsDataType` and `ScaleValuesDataType` properties to the same value.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property`.

Specify the numerator coefficients fixed-point type as a `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The word length of the `CustomNumeratorCoefficientsDataType`, `CustomDenominatorCoefficientsDataType`, and `CustomScaleValuesDataType` properties must be the same.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property` and the `NumeratorCoefficientsDataType` property to `Custom`.

Specify the denominator coefficients fixed-point data type as `Same word length as input` or `Custom`. Setting this property also sets the `NumeratorCoefficientsDataType` and `ScaleValuesDataType` properties to the same value.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property`.

Specify the denominator coefficients fixed-point type as a `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorCoefficientsDataType`, `CustomDenominatorCoefficientsDataType`, and `CustomScaleValuesDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property` and the `DenominatorCoefficientsDataType` property to `Custom`.

Specify the scale values fixed-point data type as ```Same word length as input``` or `Custom`. Setting this property also sets the `NumeratorCoefficientsDataType` and `DenominatorCoefficientsDataType` properties to the same value.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property`.

Specify the scale values fixed-point type as a `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorCoefficientsDataType`, `CustomDenominatorCoefficientsDataType`, and `CustomScaleValuesDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `SOSMatrixSource` property to `Property` and the `ScaleValuesDataType` property to `Custom`.

Specify the mode to determine the numerator product fixed-point data type as:

• `Same as input` (default) — The numerator product word and fraction lengths are same as that of the input.

• `Custom` — Enables the `CustomNumeratorProductDataType` property, which you can use to specify the custom numerator product data type. Specify the data type as a `numerictype` object.

• `Full precision` — Use full-precision rules to specify the data type. These rules provide the most accurate fixed-point numerics. The rules prevent quantization from occurring within the object. Bits are added, as needed, so that no roundoff or overflow occurs. For more information, see Full Precision for Fixed-Point System Objects.

Setting this property also sets the `DenominatorProductDataType` property to the same value.

Specify the product fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorProductDataType` and `CustomDenominatorProductDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `NumeratorProductDataType` property to `Custom`.

Specify the mode to determine the denominator product fixed-point data type as:

• `Same as input` (default) — The denominator product word and fraction lengths are same as that of the input.

• `Custom` — Enables the `CustomDenominatorProductDataType` property, which you can use to specify the custom denominator product data type. Specify the data type as a `numerictype` object.

• `Full precision` — Use full-precision rules to specify the data type. These rules provide the most accurate fixed-point numerics. The rules prevent quantization from occurring within the object. Bits are added, as needed, so that no roundoff or overflow occurs. For more information, see Full Precision for Fixed-Point System Objects.

Setting this property also sets the `NumeratorProductDataType` property to the same value.

Specify the product fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorProductDataType` and `CustomDenominatorProductDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `DenominatorProductDataType` to `Custom`.

Specify the numerator accumulator fixed-point data type as `Same as input`, ```Same as product```, or `Custom`. Setting this property also sets the `DenominatorAccumulatorDataType` property to the same value.

Specify the numerator accumulator fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorAccumulatorDataType` and `CustomDenominatorAccumulatorDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `NumeratorAccumulatorDataType` property to `Custom`.

Specify the denominator accumulator fixed-point data type as `Same as input`, ```Same as product```, or `Custom`. Setting this property also sets the `NumeratorAccumulatorDataType` property to the same value.

Specify the denominator accumulator fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorAccumulatorDataType` and `CustomDenominatorAccumulatorDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `DenominatorAccumulatorDataType` property to `Custom`.

Specify the state fixed-point data type as ```Same as input```, `Same as accumulator`, or `Custom`.

#### Dependencies

This property applies when you set the Structure property to `Direct form II` or ```Direct form II transposed```.

Specify the state fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies only when you set the `StateDataType` property to `Custom`.

Specify the numerator state fixed-point data type as ```Same as input```, `Same as accumulator`, or `Custom`. Setting this property also sets the `DenominatorStateDataType` property to the same value.

#### Dependencies

This property applies only when you set the Structure property to `Direct form I transposed`.

Specify the numerator state fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorProductDataType` and `CustomDenominatorProductDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `StateDataType` property to `Custom`.

Specify the denominator state fixed-point data type as ```Same as input```, `Same as accumulator`, or `Custom`. Setting this property also sets the `NumeratorStateDataType` property to the same value.

#### Dependencies

This property applies only when you set the Structure property to `Direct form I transposed`.

Specify the denominator state fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`. The `CustomNumeratorStateDataType` and `CustomDenominatorStateDataType` properties must have the same word lengths.

#### Dependencies

This property applies only when you set the `StateDataType` property to `Custom`.

Specify the output fixed-point data type as ```Same as input```, `Same as accumulator`, or `Custom`.

Specify the output fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies only when you set the OutputDataType property to `Custom`.

## Usage

### Syntax

``y = biquad(x)``
``y = biquad(x,num,den)``
``y = biquad(x,num,den,g)``

### Description

example

````y = biquad(x)` filters the input signal `x` , and outputs the filtered values, `y`. The biquad filter object filters each channel of the input signal over successive calls to the algorithm.```
````y = biquad(x,num,den)` filters the input using `num` as the numerator coefficients, and `den` as the denominator coefficients of the biquad filter. This configuration applies when the `SOSMatrixSource` property is `Input port` and the `ScaleValuesInputPort` property is `false`. ```
````y = biquad(x,num,den,g)` specifies the scale values, `g`, of the biquad filter. This configuration applies when the `SOSMatrixSource` property is `Input Port` and the `ScaleValuesInputPort` property is `true`.```

### Input Arguments

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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.

The data type of all the inputs must be the same. If the input is fixed-point, it must be signed fixed point with power-of-two slope and zero bias.

The complexity of `x`, `num`, and `den` must be the same.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `fi`
Complex Number Support: Yes

Numerator coefficients, specified as a 3-by-N numeric matrix, where N is the number of biquad filter sections. The complexity of `x`, `num`, and `den` must be the same.

The data type of all the inputs must be the same. If `num` is fixed point, it must be signed fixed point with power-of-two slope and zero bias.

#### Dependencies

This input applies only when you set `SOSMatrixSource` property is ```Input port```.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `fi`
Complex Number Support: Yes

Denominator coefficients, specified as a 2-by-N numeric matrix, where N is the number of biquad filter sections. The object assumes that the first denominator coefficient of each section is 1.

The data type of all the inputs must be the same. If `den` is fixed point, it must be signed fixed point with power-of-two slope and zero bias.

The complexity of `x`, `num`, and `den` must be the same.

#### Dependencies

This input applies only when you set `SOSMatrixSource` property is ```Input port```.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `fi`
Complex Number Support: Yes

Scale values of the biquad filter, specified as a 1-by-(N +1) numeric vector, where `N` is the number of biquad filter sections.

The data type of all the inputs must be the same. If `g` is fixed point, it must be signed fixed point with power-of-two slope and zero bias.

#### Dependencies

This input applies when the `SOSMatrixSource` property is ```Input Port``` and the `ScaleValuesInputPort` property is `true`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `fi`

### Output Arguments

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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: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `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)`

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 `freqz` Frequency response of discrete-time filter System object `fvtool` Visualize frequency response of DSP filters `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 `scale` Scale second-order sections `scaleopts` Create an options object for second-order section scaling `scalecheck` Check scaling of biquadratic filter `cumsec` Cumulative second-order section of the biquadratic filter `generatehdl` Generate HDL code for quantized DSP filter (requires Filter Design HDL Coder) `tf` Convert discrete-time filter System object to transfer function `reorder` Reorder second-order sections of biquadratic filter System object
 `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

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

Use a fourth order, lowpass biquadratic filter object with a normalized cutoff frequency of 0.4 to filter high frequencies from an input signal. Display the result as a power spectrum using the Spectrum Analyzer.

```t = (0:1000)'/8e3; xin = sin(2*pi*0.3e3*t)+sin(2*pi*3e3*t); % Input is 0.3 &amp; % 3kHz sinusoids src = dsp.SignalSource(xin, 100); sink = dsp.SignalSink; [z,p,k] = ellip(4,1,60,.4); % Set up the filter [s,g] = zp2sos(z,p,k); biquad = dsp.BiquadFilter(s,g,'Structure','Direct form I'); sa = dsp.SpectrumAnalyzer('SampleRate',8e3,... 'PlotAsTwoSidedSpectrum',false,... 'OverlapPercent', 80,'PowerUnits','dBW',... 'YLimits', [-160 -10]); while ~isDone(src) input = src(); filteredOutput = biquad(input); sink(filteredOutput); sa(filteredOutput) end filteredResult = sink.Buffer; fvtool(biquad,'Fs',8000) ```

Design and apply a lowpass biquad filter System object using the `design` function.

```lpSpec = fdesign.lowpass('Fp,Fst,Ap,Ast',500,550,0.5,60,10000); lpfilter = design(lpSpec,'butter','systemobject',true) fvtool(lpfilter); ```
```lpfilter = dsp.BiquadFilter with properties: Structure: 'Direct form II' SOSMatrixSource: 'Property' SOSMatrix: [42x6 double] ScaleValues: [43x1 double] InitialConditions: 0 OptimizeUnityScaleValues: true Use get to show all properties ```

Demonstrate the Linf-norm scaling of a biquad filter using the `scale` function.

```Fs = 8000; Fcutoff = 2000; [z,p,k] = butter(10,Fcutoff/(Fs/2)); [s,g] = zp2sos(z,p,k); biquad = dsp.BiquadFilter('Structure', 'Direct form I', ... 'SOSMatrix', s,'ScaleValues', g); scale(biquad,'linf','scalevalueconstraint','none','maxscalevalue',2)```

Create an options scaling object that contains the scaling options settings you require.

`EllipI = design(fdesign.lowpass('N,Fp,Ap,Ast',10,0.5,0.5,20), 'ellip', 'FilterStructure', 'df1sos','SystemObject',true)`
```EllipI = dsp.BiquadFilter with properties: Structure: 'Direct form I' SOSMatrixSource: 'Property' SOSMatrix: [5x6 double] ScaleValues: [6x1 double] NumeratorInitialConditions: 0 DenominatorInitialConditions: 0 OptimizeUnityScaleValues: true Show all properties ```
`opts = scaleopts(EllipI)`
```opts = sosReorder: 'auto' MaxNumerator: 2 NumeratorConstraint: 'none' OverflowMode: 'wrap' ScaleValueConstraint: 'unit' MaxScaleValue: 'Not used' ```

## Algorithms

This object implements the algorithm, inputs, and outputs described on the Biquad Filter block reference page. The object properties correspond to the block parameters, except:

• Coefficient source

• Action when the a0 values of the SOS matrix are not one – the biquad filter object assumes the zero-th-order denominator coefficient equals 1 regardless of the specified value. The biquad filter object does not support the `Error` or `Warn` options found in the corresponding block.

Both this object and its corresponding block support variable-size input. When you call the object, it can handle an input argument which is changing in size.

## Extended Capabilities

Introduced in R2012a

Watch now