Mathworks SIMULINK FIXED POINT 6 Reference

Simulink®Fixed Po

Reference

int™ 6

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Revision History

March 2009 Online only New for Version 6.1 (Release 2009a)
September 2009 Online only Revised for Version 6.2 (Release 2009b)
March 2010 Online only Revised for Version 6.3 (Release 2010a)

Fixed Point™ Reference

1

A

Contents

Function Reference

Selected Bibliography

Glossary

Index

iii

iv Contents

Function Reference

1

autofixexp

Purpose Automatically change scaling of fixed-point data types

Syntax autofixexp

Description The autofixexp script automatically changes the scaling for model

objects that specify fixed-point data types. However, if an object’s Lock
output data type setting against changes by the fixed-point tools

parameter is selected, the script refrains from scaling that object.

This script collects range data for model objects, either from design
minimum and maximum values that objects specify explicitly, or from
logged minimum and maximum values that occur during simulation.
Based on these values, the tool changes the scaling of fixed-point data
typesinamodelsoastomaximizeprecisionandcovertherange.

You can specify design minimum and maximum values for m odel objects
using parameters typically titled Output minimum and Output
maximum. See “Blocks That Allow Signal Range Specification” in

Simulink

specify these values. In the autoscaling procedure that the
script executes, design minimum and maximum values take precedence
over the simulation range.

®

User’s Guide for a list of Simulink blocks that permit you to

autofixexp

1-2

If yo u intend to scale fixed-point data t ypes using simulation minimum
and maximum values, the script yields meaningful results when
exercising the full range of values over which your design is meant
to run. Therefore, the simulation you run prior to using
must simulate your design over its full intended operating range. It is
especially important that you use simulation inputs with appropriate
speed and amplitude profiles for dynamic systems. The response of
a linear dynamic system is frequency dependent. For example, a
bandpass filter will show almost noresponsetoveryslowandvery
fast sinusoid inputs, whereas the signal of a sinusoid input with a
frequency in the passband will be passed or even significantly amplified.
The response of nonlinear dynamic systems can have complicated
dependence on both the signal speed and amplitude.

autofixexp

Note If you already know the simulation range you need to cover,
you can use an alternate autoscaling technique described in the

fixptbestprec reference page in the Simulink Reference.

To control the parameters associated with automatic scaling, such as
safety margins, use the Fixed-Point Tool.

For more information, see “Overview of the Fixed-Point Tool”.

To learn how to use the Fixed-Point Tool, refer to “Tutorial: Feedback
Controller”.

See Also fxptdlg

autofixexp

1-3

fixpt_instrument_purge

Purpose Remove corrupt fixed-point instrumentation from model

Note fixpt_instrument_purge will be removed in a future release.

Syntax fixpt_instrument_purge

fixpt_instrument_purge(modelName, interactive)

Description The fixpt_instrument_purge script finds and removes fixed-point

instrumentation from a model left by the Fixed-Point Tool and the
fixed-point autoscaling script. The Fixed-Point Tool and the fixed-point
autoscaling script each add callbacks to a model. For example, the
Fixed-Point Tool appends commands to model-level callbacks. These
callbacks make the Fixed-Point Tool respond to simulation events.
Similarly, the autoscaling script adds instrumentation to some
parameter values that gathers information required by the script.

Normally, these types of instrumentation are automatically removed
from a model. The Fixed-Point Tool removes its instrumentation when
the model is closed. The autoscaling script removes its instrumentation
shortly after it is added. However, there are cases where abnormal
termination of a model leaves fix ed -point instrumentation behind . The
purpose of
instrumentation left over from abnormal termination.

fixpt_instrument_purge is to find and remove fixed-point

fixpt_instrument_purge(modelName, interactive) removes

instrumentation from model
default, w hich prompts you to make each change. When
is set to false, all found instrumentation is automatically removed
from the model.

See Also autofixexp, fxptdlg

1-4

modelName. interactive is true by

interactive

showfixptsimerrors

Purpose Show overflows from most recent fixed-point simulation

Note showfixptsimerrors will be removed in a future release. Use

fxptdlg instead.

Syntax showfixptsimerrors

Description The showfixptsimerrors script displays any overflows from the most

recent fixed-point simulation. This information is also visible in the
Fixed-Point Tool.

See Also autofixexp, fxptdlg

1-5

showfixptsimranges

Purpose Show logged maximum values, minimum values, and overflow data

from fixed-point simulation

Note showfixptsimranges will be removed in a future release. Use

fxptdlg instead.

Syntax showfixptsimranges

showfixptsimranges(action)

Description showfixptsimranges displays the l ogged maximum values, minimum

values, and overflow data from the most recent fixed-point simulation
in the MATLAB

showfixptsimranges(action) stores the l ogge d maximum values,

minimum values, and overflow data from the most recent fixed-point
simulation in the workspace variable

'verbose', the logged data also appears in the MATLAB Command

Window. If

®

Command Window.

FixPtSimRanges.Ifaction is

action is 'quiet', no data appears.

See Also autofixexp, fxptdlg

1-6

Selected Bibliography

[1] Burrus,C.S.,J.H.McClellan,A.V.Oppenheim,T.W.Parks,R.W.Schafer,
and H.W. Schuessler, C omputer-Based Exercises for Signal Processing Using
MATLAB, Prentice Hall, Englewood Cliffs, New Jersey, 1994.

[2] Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of
Dynamic Systems, Second Edition, Addison-Wesley P ub li shing Company,
Reading, Massachusetts, 1990.

[3] Handbook For Digital Signal Processing,editedbyS.K.MitraandJ.F.
Kaiser, John Wiley & Sons, Inc., New York, 1993.

A

[4] Hanselmann, H., “Implementation of Digital Controllers — A Survey,”
Automatica, Vol. 23, No. 1, pp. 7-32, 1987.

[5] Jackson, L.B., Digital Filters and Signal Processing, Second Edition,
Kluwer Academic Publishers, Seventh Printing, Norwell, Massachusetts,

1993.

[6] Middleton, R. and G. Goodwin, Digital Control and Estimation — A
Unified Approach, Prentice Hall, Englewood Cliffs, New Jersey. 1990.

[7] Moler, C., “Floating points: IEEE Standard unifies arithmetic model,”
Cleve’s Corner, The MathWorks, Inc., 1996. You can find this article at

http://www.mathworks.com/company/newsletters/news_notes/clevescorner/index.html.

[8] Ogata, K., Discrete-Time Control Systems, Second Edition,PrenticeHall,
Englewood Cliffs, New Jersey, 1995.

[9] Roberts, R.A. and C.T. Mullis, Digital Signal Processing, Addison-Wesley
Publishing Company, Reading, Massachusetts, 1987.

A Selected Bibliography

A-2

Glossary

Glossary

This glossary defines terms related to fixed-point data types and numbers.
These terms may appear in some or all of the documents that describe
products from The MathWorks™ that have fixed-point support.

arithmetic shift

Shift of the bits of a binary word for which the sign bit is recycled for
each bit shift to the right. A zero is incorporated into the least significant
bit of the word for each bit shift to the left. In the absence of overflows,
each arithmetic sh if t to the right is equivalent to a division by 2, and
each arithmetic shift to the left is equivalent to a multiplication by 2.

See also binary point, binary word, bit, logical shift, most significant bit

bias

Part of the numerical representation used to interpret a fixed-point
number. Along with the slope, the bias forms the scaling of the number.
Fixed-point numbers can be represented as

real world value slope stored integer bias- =× +()

wheretheslopecanbeexpressedas

slope fractional slope

=×2

See also fixed-point representation, fractional slope, integer, scaling,
slope, [Slope Bias]

binary number

Value represented in a system of numbers that has two as its base and
that uses 1’s and 0’s (bits) for its notation.

See also bit

exponent

Glossary-1

Glossary

binary point

Symbol in the shape of a period that separates the integer and fractional
parts of a binary number. Bits to the left of the binary point are
integer bits and/or sign bits, and bits to the right of the binary point
are fractional bits.

See also binary number, bit, fraction, integer, radix point

binary point-only scaling

Scaling of a binary number that results from shifting the binary point of
the number right or left, and which therefore can only occur by powers
of two.

See also binary number, binary point, scaling

binary word

Fixed-length sequence of bits (1’s and 0’s). In digital hardware, numbers
are stored in binary words. The way in which hardware components or
software functions interpret this sequence of 1’s and 0’s is described
by a data type.

Glossary-2

See also bit, data type, word

bit

Smallest unit of information in computer software or hardware. A bit
canhavethevalue0or1.

ceiling (round toward)

Rounding mode that rounds to the closest representable number in the
direction of positive infini t y. This is equivalent to the

ceil mode in

Fixed-Point Toolbox™ software.

See also convergent rounding, floor (round toward), nearest (round
toward), rounding, truncation, zero (round toward)

contiguous binar y point

Binary point that occurs within the word length of a data type. For
example, if a data type has four bits, its contiguous binary point must
be understood to occur at one of the following five positions:

.

0000

.

0 000

.

00 00

.

000 0

.

0000

See also data type, noncontiguous binary point, word length

convergent rounding

Rounding mode that rounds to the nearest allowable quantized value.
Numbers that are exactly halfw ay between the two nearest allowable
quantized values are rounded up only if the least significant bit (after
rounding) would be set to

0.

Glossary

See also ceiling (round toward), floor (round toward), nearest (round
toward), rounding, truncation, zero (round toward)

data type

Set of characteristics that define a group of values. A fixed-point data
type is defined by its word length, its fraction length, and whether it
is signed or unsigned. A floating-point data type is defined by its word
length and whether it is signed or unsigned.

See also fixed-point representation, floating-point representation,
fraction length, signedness, word length

data type override

Parameter in the Fixed-Point Tool that allows you to set the output data
type and scaling of fixed-point blocks on a system or subsystem level.

See also data type, scaling

exponent

Part of the numerical representation used to express a floating-point or
fixed-point number.

Glossary-3

Glossary

1. Floating-point numbers are typically represented as

real - world value mantissa

=×2

exponent

2. Fixed-point numbers can be represented as

real world value slope stored integer bias- =× +()

wheretheslopecanbeexpressedas

slope fractional slope

=×2

exponent

The exponent of a fixed-point number is equal to the negative of the
fraction length:

exponent fraction length=− ×1

See also bias, fixed-point representation, floating-point representation,
fraction length, fraction al slope, integer, mantissa, slope

fixed-point representation

Method for representing numerical values and data types that have
a set range and precision.

1. Fixed-point numbers can be represented as

Glossary-4

real world value slope stored integer bias- =× +()

wheretheslopecanbeexpressedas

slope fractional slope

=×2

The slope and the bias together repres

exponent

ent the scaling of the fixed-point

number.

2. Fixed-point data types can be defined by their word length, their
fraction length, and whether they are signed or unsigned.

See also bias, data type, exponent, fraction length, fractional slope,
integer, precision, range, scaling, slope, word length

floating-point representa tion

Method for representing numerical values and data types that can have
changing range and precision.

1. Floating-point numbers can be represented as

Glossary

real - world value mantissa

=×2

exponent

2. Floating-point data types are defined by their word length.

See also datatype,exponent,mantissa,precision,range,wordlength

floor (round toward)

Rounding mode that rounds to the closest representable number in
the direction of negative infinity.

See also ceiling (round toward), convergent rounding, nearest (round
toward), rounding, truncation, zero (round toward)

fraction

Part o f a fixed-point number represented by the bits to the right of the
binary point. The fraction represents numbers that are less than one.

See also binary point, bit, fixed-point representation

fraction length

Number of bits to the right of the binary point in a fixed-point
representation of a number.

See also binary point, bit, fixed-point representation, fraction

Glossary-5

Glossary

fractional slope

Part of the numerical representation used to express a fixed-point
number. Fixed-point numbers can be represented as

real world value slope stored integer bias- =× +()

wheretheslopecanbeexpressedas

slope fractional slope

=×2

The term slope adjustment is sometimes u

exponent

sed as a synonym for

fractional slope.

See also bias, exponent, fixed-point representation, integer, slope

guard bits

Extra bits in either a hardware register or software simulation that are
added to the high end of a binary word to ensure that no information
is lost in case of overflow.

See also binary word, b it, overflow

integer

1. Part of a fixed-point number represented by the bits to the left of the
binary point. The integer represents numbers that are greater than
or equal to one.

2. Also called the "stored integer." The raw binary number, in which the
binary point is assumed to be at the far right of the word. The integer
is part of the numerical re pre sentation used to express a fixed-point
number. Fixed-point numbers can be represented as

Glossary-6

fraction length

real - world value stored integer

−

=×

2

or

real world value slope stored integer bias- =× +()

wheretheslopecanbeexpressedas

Glossary

slope fractional slope

=×2

exponent

See also bias, fixed-point representation, fractional slope, integer,
real-world value, slope

integer length

Number of bits to the left of the bina ry point in a fixed-point
representation of a number.

See also binary point, bit, fixed-point representation, fraction length,
integer

least significant bit (LSB)

Bit in a binary word that can represent the smallest value. The LSB is
the rightmost bit in a big-endian-ordered binary word. The weight of
the LSB is related to the fraction length according to

weight of LSB

fraction length

=−2

See also big-endian, binary word, bit, most significant bit

logical shift

Shift of the bits of a binary word, for which a zero is incorporated into
the m ost significant bit for each bit shift to the right and into the least
significant bit for eac h bit shift to the left.

See also arithmetic shift, binary point, binary word, bit, most significant
bit

mantissa

Part of the numerical re presentation used to express a floating-point
number. Floating-point numbers are typically represented as

real - world value mantissa

=×2

exponent

See also exponent, floating-point representation

Glossary-7

Glossary

most s ig n ificant bit (MSB)

Bit in a binary word that can represent the largest value. The MSB is
the leftmost bit in a big-endian-ordered binary word.

See also binary word, bit, least significant bit

nearest (round toward)

Rounding mode that rounds to the closest representable number, with
the exact midpoint rounded to the closest representable number in the
direction of posit iv e infinity. This is equiv al ent to the

nearest mode

in Fixed-Point Toolbox software.

See also ceiling (round toward), convergent rounding, floor (round
toward), rounding, truncation, zero (round toward)

noncontiguous binar y point

Binary point that is understood to fall outside the word length of a
data type. For example, the binary point for the following 4-bit word is
understood to occur two bits to the right of the word length,

Glossary-8

0000 .

thereby giv ing the bits of the word the following potential values:

5432

2222

.

See also binary point, data type, word length

one’s complement representation

Representation of signed fixed-point numbers. Negating a binary
number in one’s complement requires a bitwise complement. That is, all
0’sareflippedto1’sandall1’sareflippedto0’s. Inone’scomplement
notation there are two ways to represent zero. A binary word of all
0’srepresents"positive"zero,while a binary word of all 1’s represents
"negative" zero.

See also binary number, binary word, sign/magnitude representation,
signed fixed-point, two’s complement representation

overflow

Situation that occurs when the magnitude of a calculation result is too
large for the range of the data type being used. In many cases you can
choose to either saturate or wrap overflows.

See also saturation, wrapping

padding

Extending the least significant bit of a binary word with one or more
zeros.

See also least significant bit

precision

1. Measure of the smallest numerical interval that a fixed-point data
type and scaling can represent, determined by the value of the number’s
least significant bit. The precision is given by the slope, or the number
of fractional bits. The term resolution is sometimes used as a synonym
for this definition.

Glossary

2. Measure of the difference between a real-world numerical value and
the value of its quantized representation. This is sometimes called
quantization error or quantization noise.

See also data type, fraction, least significant bit, quantization,
quantization error, range, slope

Qformat

Representation used by Texas Instruments™ to encode signed two’s
complement fixed-point data types. This fixed-point notation takes the
form

Qm n.

where

• Q indicates that the number is in Q format.

• m is the number of bits used to designate the two’s complement

integer part of the number.

Glossary-9

Glossary

• n is the number of bits used to designate the two’s complement

fractional part of the number, or the number of bits to the right
of the binary point.

In Q format notation, the most significant bit is assumed to be the sign
bit.

See also binary point, bit, data type, fixed-point representation, fraction,
integer, two’s complement

quantization

Representation of a value by a data type that has too few bits to
representitexactly.

See also bit, data type, quantization error

quantization error

Error introduced when a value is represented by a data type that has
too few bits to represent it exactly, or when a value is converted from
one data type to a shorter data type. Quantization error is also called
quantization noise.

Glossary-10

See also bit, data type, quantization

radix point

Symbol in the shape of a period that separates the integer and fractional
parts of a number in any base system. Bits to the left of the radix point
are integer and/or sign bits, and bits to the right of the radix point are
fraction bits.

See also binary point, bit, fraction, integer, sign bit

range

Span of numbers that a certain data type can represent.

See also data type, precision

real-world value

Stored integer value with fixed-point scaling applied. Fixed-point
numbers can be represented as

fraction length

real - world value stored integer

−

=×

2

or

real world value slope stored integer bias- =× +()

wheretheslopecanbeexpressedas

Glossary

slope fractional slope

=×2

exponent

See also integer

resolution

See precision

rounding

Limiting the number of bits required to express a number. One or
more least significant bits are dropped, resulting in a loss of precision.
Rounding is necessary when a value cannot be expressed exactly by the
number of bits designated to represent it.

See also bit, ceiling (round toward), convergent rounding, floor (round
toward), least significant bit, nearest ( round toward), precision,
truncation, zero (round toward)

saturation

Method of handling numeric overflow that represents positive overflows
as the largest positive number in therangeofthedatatypebeingused,
and negative overflows as the largest negative number in the range.

See also overflow, wrapping

Glossary-11

Glossary

scaled double

A double data type that retains fixed-point scaling information. For
example, in Simulink and Fixed-Point Toolbox software you can use
data type override to convert your fixed-point data ty pe s to scaled
doubles. You can then simulate to determine the ideal floating-point
behavior of your system. After you gather that information you can
turn data type override off to return to fixed-point data types, and your
quantities still have their original scaling information because it was
held in the scaled double data types.

scaling

1. Format used for a fixed-point number of a given word length and
signedness. The slope and bias together form the scaling of a fixed-point
number.

2. Changing the slope and/or bias of a fixed-point number without
changing the stored integer.

See also bias, fixed-point representation, integer, slope

Glossary-12

shift

Movement of the bits of a binary word either towa rd the most significant
bit ("to the left") or toward the least significant bit ("to the right"). Shifts
to the right can be either logical, where the spaces emptied at the f ront
ofthewordwitheachshiftarefilledinwithzeros,orarithmetic,where
thewordissignextendedasitisshiftedtotheright.

See also arithmetic shift, logical shift, sign extension

sign bit

Bit (or bits) in a signed binary number that indicates whether the
number is pos itiv e or negative.

See also binary number, bit

sign extension

Addition of bits that have the value of the most significant bit to the
high end of a two’s complement number. Sign extension does not change
the value of the binary number.

See also binary number, guard bits, most significant bit, two’s
complement representation, word

sign/magnitude representation

Representation of signed fixed-point or floating-point numbers. In
sign/magnitude representation, one bit of a binary word is always
the dedicated sign bit, while the remaining bits of the word encode
the magnitude of the number. Negation using sign/magnitude
representation consists of flipping the sign bit from 0 (positive) to 1
(negative), or from 1 to 0.

See also binary word, bit, fixed-point representation, floating-point
representation, one’s complement representation, sign bit, signed
fixed-point, signedness, two’s complement representation

signed fixed-point

Fixed-point number or data type that can represent both positive and
negative numbers.

See also data type, fixed-point representation, signedness, unsigned
fixed-point

Glossary

signedness

The signedness of a number or data type can be signed or unsigned.
Signed numbers and data types can represent both positive and negative
values, whereas unsigned numbers and data types can only represent
values that are greater than or equal to zero.

See also data type, sign bit, sign/magnitude representation, signed
fixed-point, unsigned fixed-point

slope

Part of the numerical representation used to express a fixed-point
number. Along with the bias, the slope forms the scaling of a fixed-point
number. Fixed-point numbers can be represented as

real world value slope stored integer bias- =× +()

wheretheslopecanbeexpressedas

slope fractional slope

=×2

exponent

Glossary-13

Glossary

See also bias, fixed-point representation, fractional slope, integer,
scaling, [Slope Bias]

slope adjustment

See fractional slope

[Slope Bias]

Representation used to define the scali ng of a fixed-point number.

See also bias, scaling, slope

stored integer

See integer

trivial scaling

Scaling that results in the real-world value of a number being simply
equal to its stored integer value:

real - world value stored integer=

Glossary-14

In [Slope Bias] representation, fixed-point numbers can be represented
as

real world value slope stored integer bias- =× +()

In the trivial case, slope = 1 and bias = 0.

In terms of binary point-only scaling, the binary point is to the right o
the least significant bit for trivial scaling, meaning that the fracti

f

on

length is zero:

fraction length

real - world value stored integer stored i

=× =

−

2 nnteger × 2

Scaling is always trivial for pure integers, such as int8, and also for the
true floating-point types

single and double.

See also bias, binary point, binary point-only scaling, fixed-point
representation, fraction length, integer, least significant bit, scaling,
slope, [Slope Bias]

0

truncation

Roundingmodethatdropsoneormoreleastsignificantbitsfroma
number.

See also ceiling (round toward), convergent rounding, floor (round
toward), nearest (round toward), rounding, zero (round toward)

two’s complement repres en tation

Common representation of signed fixed-point numbers. Negation using
signed two’s complement representation consists of a translation into
one’s complement followed by the binary addition of a one.

See also binary word, one’s complement representation, sign/magnitude
representation, signed fixed-point

unsigned fixed-point

Fixed-point number or data type that can only represent numbers
greaterthanorequaltozero.

Glossary

See also data type, fixed-point representation, signed fixed-point,
signedness

word

Fixed-length sequence of binary digits (1’s and 0’s). In digital hardware,
numbers are stored in words. The way hardware components or
software functions interpret this sequence of 1’s and 0’s is described
by a data type.

See also binary word, data type

word length

Number of bits in a binary word or data type.

See also binary word, bit, data type

wrapping

Method of handling overflow. Wrapping uses modulo arithmet
a number that falls outside of the representable range the d
being used back into the representable range.

See also data type, overflow, range, saturation

ic to cast

ata type

Glossary-15

Glossary

zero (round toward)

Rounding mode that rounds to the closest representable number in the
direction of zero. This is equivalent to the

fix mode in Fixed-Point

Toolbox software.

See also ceiling (round toward), convergent rounding, floor (round
toward), nearest (round toward), rounding, truncation

Glossary-16

Index

IndexA

autofixexp function 1-2

automatic scaling

autofixexp 1-2
fixpt_instrument_purge 1-4

script 1-2

autoscaling

autofixexp 1-2
fixpt_instrument_purge 1-4

script 1-2

F

fixpt_instrument_purge function 1-4

functions

autofixexp 1-2
fixpt_instrument_purge 1-4
showfixptsimerrors 1-5
showfixptsimranges 1-6

S

showfixptsimerrors function 1-5
showfixptsimranges function 1-6

Index-1