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# dfilt.df2

Discrete-time, direct-form II filter

## Synopsis

Refer to `dfilt.df2` in Signal Processing Toolbox™ documentation.

## Description

`hd = dfilt.df2(b,a)` returns a discrete-time, direct-form II filter object `hd`, with numerator coefficients `b` and denominator coefficients `a`.

Make this filter a fixed-point or single-precision filter by changing the value of the `Arithmetic` property for the filter `hd` as follows:

• To change to single-precision filtering, enter

`set(hd,'arithmetic','single');`
• To change to fixed-point filtering, enter

`set(hd,'arithmetic','fixed');`

For more information about the property `Arithmetic`, refer to Arithmetic.

`hd = dfilt.df2` returns a default, discrete-time, direct-form II filter object `hd`, with `b` = 1 and `a` = 1. This filter passes the input through to the output unchanged.

### Note

The leading coefficient of the denominator `a(1)` cannot be 0. To allow you to change the arithmetic setting to `fixed` or `single`, `a(1)` must be equal to 1.

## Fixed-Point Filter Structure

The following figure shows the signal flow for the direct-form II filter implemented by `dfilt.df2`. To help you see how the filter processes the coefficients, input, and states of the filter, as well as numerical operations, the figure includes the locations of the formatting objects within the signal flow.

### Notes About the Signal Flow Diagram

To help you understand where and how the filter performs fixed-point arithmetic during filtering, the figure shows various labels associated with data and functional elements in the filter. The following table describes each label in the signal flow and relates the label to the filter properties that are associated with it.

The labels use a common format — a prefix followed by the letters“frmt” (format). In this use, “frmt” means the word length and fraction length associated with the filter part referred to by the prefix.

For example, the InputFrmt label refers to the word length and fraction length used to interpret the data input to the filter. The format properties `InputWordLength` and `InputFracLength` (as shown in the table) store the word length and the fraction length in bits. Or consider NumFrmt, which refers to the word and fraction lengths (`CoeffWordLength`, `NumFracLength`) associated with representing filter numerator coefficients.

Signal Flow Label

Corresponding Word Length Property

Corresponding Fraction Length Property

Related Properties

DenAccumFrmt

`AccumWordLength`

`DenAccumFracLength`

`AccumMode`, `CastBeforeSum`

DenFrmt

`CoeffWordLength`

`DenFracLength`

`CoeffAutoScale`, `Signed`, `Denominator`

DenProdFrmt

`CoeffWordLength`

`DenProdFracLength`

`ProductMode`, `ProductWordLength`

InputFrmt

`InputWordLength`

`InputFracLength`

None

NumAccumFrmt

`AccumWordLength`

`NumAccumFracLength`

`AccumMode`, `CastBeforeSum`

NumFrmt

`CoeffWordLength`

`NumFracLength`

`CoeffAutoScale`, `Signed`, `Numerator`

NumProdFrmt

`CoeffWordLength`

`NumProdFracLength`

`ProductWordLength`, `ProductMode`

OutputFrmt

`OutputWordLength`

`OutputFracLength`

`OutputMode`

StateFrmt

`StateWordLength`

`StateFracLength`

`States`

Most important is the label position in the diagram, which identifies where the format applies.

As one example, look at the label DenProdFrmt, which always follows a denominator coefficient multiplication element in the signal flow. The label indicates that denominator coefficients leave the multiplication element with the word length and fraction length associated with product operations that include denominator coefficients. From reviewing the table, you see that the DenProdFrmt refers to the properties `ProdWordLength`, `ProductMode` and `DenProdFracLength` that fully define the denominator format after multiply (or product) operations.

## Properties

In this table you see the properties associated with the df2 implementation of `dfilt` objects.

### Note

The table lists all the properties that a filter can have. Many of the properties are dynamic, meaning they exist only in response to the settings of other properties. You might not see all of the listed properties all the time. To view all the properties for a filter at any time, use

`get(hd)`

where `hd` is a filter.

For further information about the properties of this filter or any `dfilt` object, refer to Fixed-Point Filter Properties.

Property Name

Brief Description

`AccumMode`

Determines how the accumulator outputs stored values. Choose from full precision (`FullPrecision`), or whether to keep the most significant bits (`KeepMSB`) or least significant bits (`KeepLSB`) when output results need shorter word length than the accumulator supports. To let you set the word length and the precision (the fraction length) used by the output from the accumulator, set `AccumMode` to `SpecifyPrecision`.

`AccumWordLength`

Sets the word length used to store data in the accumulator/buffer.

`Arithmetic`

Defines the arithmetic the filter uses. Gives you the options `double`, `single`, and `fixed`. In short, this property defines the operating mode for your filter.

`CastBeforeSum`

Specifies whether to cast numeric data to the appropriate accumulator format (as shown in the signal flow diagrams) before performing sum operations.

`CoeffAutoScale`

Specifies whether the filter automatically chooses the proper fraction length to represent filter coefficients without overflowing. Turning this off by setting the value to `false` enables you to change the `NumFracLength` and `DenFracLength` properties to specify the precision used.

`CoeffWordLength`

Specifies the word length to apply to filter coefficients.

`DenAccumFracLength`

Specifies the fraction length used to interpret data in the accumulator used to hold the results of sum operations. You can change the value for this property when you set `AccumMode` to `SpecifyPrecision`.

`DenFracLength`

Set the fraction length the filter uses to interpret denominator coefficients. `DenFracLength` is always available, but it is read-only until you set `CoeffAutoScale` to `false`.

`Denominator`

Holds the denominator coefficients for IIR filters.

`DenProdFracLength`

Specifies how the filter algorithm interprets the results of product operations involving denominator coefficients. You can change this property value when you set `ProductMode` to `SpecifyPrecision`.

`FilterStructure`

Describes the signal flow for the filter object, including all of the active elements that perform operations during filtering — gains, delays, sums, products, and input/output.

`InputFracLength`

Specifies the fraction length the filter uses to interpret input data.

`InputWordLength`

Specifies the word length applied to interpret input data.

`NumAccumFracLength`

Specifies how the filter algorithm interprets the results of addition operations involving numerator coefficients. You can change the value of this property after you set `AccumMode` to `SpecifyPrecision`.

`Numerator`

Holds the numerator coefficient values for the filter.

`NumFracLength`

Sets the fraction length used to interpret the value of numerator coefficients.

`NumProdFracLength`

Specifies how the filter algorithm interprets the results of product operations involving numerator coefficients. Available to be changed when you set `ProductMode` to `SpecifyPrecision`.

`OutputFracLength`

Determines how the filter interprets the filter output data. You can change the value of `OutputFracLength` when you set `OutputMode` to `SpecifyPrecision`.

`OutputMode`

Sets the mode the filter uses to scale the filtered data for output. You have the following choices:

• `AvoidOverflow` — directs the filter to set the output data word length and fraction length to avoid causing the data to overflow.

• `BestPrecision` — directs the filter to set the output data word length and fraction length to maximize the precision in the output data.

• `SpecifyPrecision` — lets you set the word and fraction lengths used by the output data from filtering.

`OutputWordLength`

Determines the word length used for the output data.

`OverflowMode`

Sets the mode used to respond to overflow conditions in fixed-point arithmetic. Choose from either `saturate` (limit the output to the largest positive or negative representable value) or `wrap` (set overflowing values to the nearest representable value using modular arithmetic). The choice you make affects only the accumulator and output arithmetic. Coefficient and input arithmetic always saturates. Finally, products never overflow—they maintain full precision.

`ProductMode`

Determines how the filter handles the output of product operations. Choose from full precision (`FullPrecision`), or whether to keep the most significant bit (`KeepMSB`) or least significant bit (`KeepLSB`) in the result when you need to shorten the data words. For you to be able to set the precision (the fraction length) used by the output from the multiplies, you set `ProductMode` to `SpecifyPrecision`.

`PersistentMemory`

Specifies whether to reset the filter states and memory before each filtering operation. Lets you decide whether your filter retains states from previous filtering runs. `False` is the default setting.

`RoundMode`

Sets the mode the filter uses to quantize numeric values when the values lie between representable values for the data format (word and fraction lengths).

• `ceil` - Round toward positive infinity.

• `convergent` - Round to the closest representable integer. Ties round to the nearest even stored integer. This is the least biased of the methods available in this software.

• `fix` - Round toward zero.

• `floor` - Round toward negative infinity.

• `nearest` - Round toward nearest. Ties round toward positive infinity.

• `round` - Round toward nearest. Ties round toward negative infinity for negative numbers, and toward positive infinity for positive numbers.

The choice you make affects only the accumulator and output arithmetic. Coefficient and input arithmetic always round. Finally, products never overflow — they maintain full precision.

`Signed`

Specifies whether the filter uses signed or unsigned fixed-point coefficients. Only coefficients reflect this property setting.

`StateFracLength`

When you set `StateAutoScale` to `false`, you enable the `StateFracLength` property that lets you set the fraction length applied to interpret the filter states.

`States`

This property contains the filter states before, during, and after filter operations. States act as filter memory between filtering runs or sessions.

`StateWordLength`

Sets the word length used to represent the filter states.

## Examples

Specify a second-order direct-form II filter structure for a `dfilt` object, `hd`, with the following code:

```b = [0.3 0.6 0.3]; a = [1 0 0.2]; hd = dfilt.df2(b,a); % Change filter to fixed-point set(hd,'arithmetic','fixed')```