Documentation 
Discretetime, SOS, directform II filter
Refer to dfilt.df2sos in Signal Processing Toolbox™ documentation.
hd = dfilt.df2sos(s) returns a discretetime, secondorder section, directform II filter object hd, with coefficients given in the s matrix.
Make this filter a fixedpoint or singleprecision filter by changing the value of the Arithmetic property for the filter hd as follows:
To change to singleprecision filtering, enter
set(hd,'arithmetic','single');
To change to fixedpoint filtering, enter
set(hd,'arithmetic','fixed');
For more information about the property Arithmetic, refer to Arithmetic.
hd = dfilt.df2sos(b1,a1,b2,a2,...) returns a discretetime, secondorder section, directform II object, hd, with coefficients for the first section given in the b1 and a1 vectors, for the second section given in the b2 and a2 vectors, etc.
hd = dfilt.df2sos(...,g) includes a gain vector g. The elements of g are the gains for each section. The maximum length of g is the number of sections plus one. If g is not specified, all gains default to one.
hd = dfilt.df2sos returns a default, discretetime, secondorder section, directform II filter object, hd. This filter passes the input through to the output unchanged.
The figure below shows the signal flow for the directform II filter implemented with secondorder sections by dfilt.df2sos. 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.
To help you understand where and how the filter performs fixedpoint 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 frmt 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, sosMatrix 
DenProdFrmt  CoeffWordLength  DenProdFracLength  ProductMode, ProductWordLength, sosMatrix 
InputFrmt  InputWordLength  InputFracLength  None 
NumAccumFrmt  AccumWordLength  NumAccumFracLength  AccumMode, CastBeforeSum 
NumFrmt  CoeffWordLength  NumFracLength  CoeffAutoScale, Signed, sosMatrix 
NumProdFrmt  CoeffWordLength  NumProdFracLength  ProductWordLength, ProductMode 
OutputFrmt  OutputWordLength  OutputFracLength  OutputMode 
ScaleValueFrmat  CoeffWordLength  ScaleValueFracLength  CoeffAutoScale, ScaleValues 
SectionInputFormt  SectionInputWordLength  SectionInputFracLength  SectionInputAutoScale 
SectionOutputFrmt  SectionOutputWordLength  SectionOutputFracLength  SectionOutputAutoScale 
StateFrmt  StateWordLength  StateFracLength  CastBeforeSum, 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.
In this table you see the properties associated with secondorder section implementation of directform II 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 FixedPoint 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 readonly until you set CoeffAutoScale to false. 
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. 
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. 
OptimizeScaleValues  When true, the filter skips multiplicationbyone scaling. When false, the filter performs multiplicationbyone scaling. 
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:

OutputWordLength  Determines the word length used for the output data. 
OverflowMode  Sets the mode used to respond to overflow conditions in fixedpoint 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. 
ProductWordLength  Specifies the word length to use for multiplication operation results. This property becomes writable (you can change the value) when 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).
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. 
ScaleValueFracLength  Scale values work with SOS filters. Setting this property controls how your filter interprets the scale values by setting the fraction length. Only available when you disable AutoScaleMode by setting it to false. 
ScaleValues  Scaling for the filter objects in SOS filters. 
SectionInputAutoScale  Tells the filter whether to set the stage input data format to minimize the occurrence of overflow conditions. 
SectionInputFracLength  Lets you set the fraction length for section inputs in SOS filters, if you set SectionInputAutoScale to false. 
SectionInputWordLength  Lets you set the word length for section inputs in SOS filters, if you set SectionInputAutoScale to false. 
SectionOutputAutoScale  Tells the filter whether to set the section output data format to minimize the occurrence of overflow conditions. 
SectionOutputFracLength  Lets you set the fraction length for section outputs in SOS filters, if you set SectionOutputAutoScale to off. 
SectionOutputWordLength  Lets you set the word length for section outputs in SOS filters, if you set SectionOutputAutoScale to false. 
Signed  Specifies whether the filter uses signed or unsigned fixedpoint coefficients. Only coefficients reflect this property setting. 
SosMatrix  Holds the filter coefficients as property values. Displays the matrix in the format [sections x coefficients/section datatype]. A [15x6 double] SOS matrix represents a filter with 6 coefficients per section and 15 sections, using data type double to represent the coefficients. 
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. 
Specify a secondorder section, directform II dfilt object for a Butterworth filter converted to secondorder sections, with the following code:
[z,p,k] = butter(30,0.5); [s,g] = zp2sos(z,p,k); hd = dfilt.df2sos(s,g); % Convert filter to fixedpoint hd.arithmetic='fixed';