Create Fixed-Point Data

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This example shows the basics of how to use the fixed-point numeric object fi.

Notation

The fixed-point numeric object is called fi because J.H. Wilkinson used fi to denote fixed-point computations in his classic texts Rounding Errors in Algebraic Processes (1963), and The Algebraic Eigenvalue Problem (1965).

Setup

This example may use display settings or preferences that are different from what you are currently using. To ensure that your current display settings and preferences are not changed by running this example, the example automatically saves and restores them. The following code captures the current states for any display settings or properties that the example changes.

originalFormat = get(0, 'format');
format loose
format long g
% Capture the current state of and reset the fi display and logging
% preferences to the factory settings.
fiprefAtStartOfThisExample = get(fipref);
reset(fipref);

Default Fixed-Point Attributes

To assign a fixed-point data type to a number or variable with the default fixed-point parameters, use the fi constructor. The resulting fixed-point value is called a fi object.

For example, the following creates fi objects a and b with attributes shown in the display, all of which we can specify when the variables are constructed. Note that when the FractionLength property is not specified, it is set automatically to "best precision" for the given word length, keeping the most-significant bits of the value. When the WordLength property is not specified it defaults to 16 bits.

a = fi(pi)

a = 

              3.1416015625

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

b = fi(0.1)

b = 

        0.0999984741210938

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 18

Specifying Signed and WordLength Properties

The second and third numeric arguments specify Signed (true or 1 = signed, false or 0 = unsigned), and WordLength in bits, respectively.

% Signed 8-bit
a = fi(pi, 1, 8)

a = 

                   3.15625

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 8
        FractionLength: 5

The sfi constructor may also be used to construct a signed fi object

a1 = sfi(pi,8)

a1 = 

                   3.15625

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 8
        FractionLength: 5

% Unsigned 20-bit
b = fi(exp(1), 0, 20)

b = 

          2.71828079223633

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 20
        FractionLength: 18

The ufi constructor may be used to construct an unsigned fi object

b1 = ufi(exp(1), 20)

b1 = 

          2.71828079223633

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 20
        FractionLength: 18

Precision

The data is stored internally with as much precision as is specified. However, it is important to be aware that initializing high precision fixed-point variables with double-precision floating-point variables may not give you the resolution that you might expect at first glance. For example, let's initialize an unsigned 100-bit fixed-point variable with 0.1, and then examine its binary expansion:

a = ufi(0.1, 100);

bin(a)

ans =

    '1100110011001100110011001100110011001100110011001101000000000000000000000000000000000000000000000000'

Note that the infinite repeating binary expansion of 0.1 gets cut off at the 52nd bit (in fact, the 53rd bit is significant and it is rounded up into the 52nd bit). This is because double-precision floating-point variables (the default MATLAB® data type), are stored in 64-bit floating-point format, with 1 bit for the sign, 11 bits for the exponent, and 52 bits for the mantissa plus one "hidden" bit for an effective 53 bits of precision. Even though double-precision floating-point has a very large range, its precision is limited to 53 bits. For more information on floating-point arithmetic, refer to Chapter 1 of Cleve Moler's book, Numerical Computing with MATLAB. The pdf version can be found here: https://www.mathworks.com/company/aboutus/founders/clevemoler.html

So, why have more precision than floating-point? Because most fixed-point processors have data stored in a smaller precision, and then compute with larger precisions. For example, let's initialize a 40-bit unsigned fi and multiply using full-precision for products.

Note that the full-precision product of 40-bit operands is 80 bits, which is greater precision than standard double-precision floating-point.

a = fi(0.1, 0, 40);
bin(a)

ans =

    '1100110011001100110011001100110011001101'


b = a*a

b = 

        0.0100000000000045

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 80
        FractionLength: 86

bin(b)

ans =

    '10100011110101110000101000111101011100001111010111000010100011110101110000101001'

Access to Data

The data can be accessed in a number of ways which map to built-in data types and binary strings. For example,

DOUBLE(A)

a = fi(pi);
double(a)

ans =

              3.1416015625

returns the double-precision floating-point "real-world" value of a, quantized to the precision of a.

A.DOUBLE = ...

We can also set the real-world value in a double.

a.double = exp(1)

a = 

             2.71826171875

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

sets the real-world value of a to e, quantized to a's numeric type.

STOREDINTEGER(A)

storedInteger(a)

ans =

  int16

   22268

returns the "stored integer" in the smallest built-in integer type available, up to 64 bits.

Relationship Between Stored Integer Value and Real-World Value

In BinaryPoint scaling, the relationship between the stored integer value and the real-world value is

$$ \mbox{Real-world value} = (\mbox{Stored integer})\cdot
2^{-\mbox{Fraction length}}.$$

There is also SlopeBias scaling, which has the relationship

$$ \mbox{Real-world value} = (\mbox{Stored integer})\cdot
\mbox{Slope}+ \mbox{Bias}$$

where

$$ \mbox{Slope} = (\mbox{Slope adjustment factor})\cdot
2^{\mbox{Fixed exponent}}.$$

and

$$\mbox{Fixed exponent} = -\mbox{Fraction length}.$$

The math operators of fi work with BinaryPoint scaling and real-valued SlopeBias scaled fi objects.

BIN(A), OCT(A), DEC(A), HEX(A)

return the stored integer in binary, octal, unsigned decimal, and hexadecimal strings, respectively.

bin(a)

ans =

    '0101011011111100'


oct(a)

ans =

    '053374'


dec(a)

ans =

    '22268'


hex(a)

ans =

    '56fc'

A.BIN = ..., A.OCT = ..., A.DEC = ..., A.HEX = ...

set the stored integer from binary, octal, unsigned decimal, and hexadecimal strings, respectively.

$$\mbox{\texttt{fi}}(\pi)$$

a.bin = '0110010010001000'

a = 

              3.1416015625

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

$$\mbox{\texttt{fi}}(\phi)$$

a.oct = '031707'

a = 

           1.6180419921875

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

$$\mbox{\texttt{fi}}(e)$$

a.dec = '22268'

a = 

             2.71826171875

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

$$\mbox{\texttt{fi}}(0.1)$$

a.hex = '0333'

a = 

           0.0999755859375

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

Specifying FractionLength

When the FractionLength property is not specified, it is computed to be the best precision for the magnitude of the value and given word length. You may also specify the fraction length directly as the fourth numeric argument in the fi constructor or the third numeric argument in the sfi or ufi constructor. In the following, compare the fraction length of a, which was explicitly set to 0, to the fraction length of b, which was set to best precision for the magnitude of the value.

a = sfi(10,16,0)

a = 

    10

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 0

b = sfi(10,16)

b = 

    10

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 11

Note that the stored integer values of a and b are different, even though their real-world values are the same. This is because the real-world value of a is the stored integer scaled by 2^0 = 1, while the real-world value of b is the stored integer scaled by 2^-11 = 0.00048828125.

storedInteger(a)

ans =

  int16

   10


storedInteger(b)

ans =

  int16

   20480

Specifying Properties with Parameter/Value Pairs

Thus far, we have been specifying the numeric type properties by passing numeric arguments to the fi constructor. We can also specify properties by giving the name of the property as a string followed by the value of the property:

a = fi(pi,'WordLength',20)

a = 

          3.14159393310547

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 20
        FractionLength: 17

For more information on fi properties, type

help fi

or

doc fi

at the MATLAB command line.

Numeric Type Properties

All of the numeric type properties of fi are encapsulated in an object named numerictype:

T = numerictype

T =


          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 15

The numeric type properties can be modified either when the object is created by passing in parameter/value arguments

T = numerictype('WordLength',40,'FractionLength',37)

T =


          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 40
        FractionLength: 37

or they may be assigned by using the dot notation

T.Signed = false

T =


          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 40
        FractionLength: 37

All of the numeric type properties of a fi may be set at once by passing in the numerictype object. This is handy, for example, when creating more than one fi object that share the same numeric type.

a = fi(pi,'numerictype',T)

a = 

          3.14159265359194

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 40
        FractionLength: 37

b = fi(exp(1),'numerictype',T)

b = 

          2.71828182845638

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 40
        FractionLength: 37

The numerictype object may also be passed directly to the fi constructor

a1 = fi(pi,T)

a1 = 

          3.14159265359194

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 40
        FractionLength: 37

For more information on numerictype properties, type

help numerictype

or

doc numerictype

at the MATLAB command line.

Display Preferences

The display preferences for fi can be set with the fipref object. They can be saved between MATLAB sessions with the savefipref command.

Display of Real-World Values

When displaying real-world values, the closest double-precision floating-point value is displayed. As we have seen, double-precision floating-point may not always be able to represent the exact value of high-precision fixed-point number. For example, an 8-bit fractional number can be represented exactly in doubles

a = sfi(1,8,7)

a = 

                 0.9921875

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 8
        FractionLength: 7

bin(a)

ans =

    '01111111'

while a 100-bit fractional number cannot (1 is displayed, when the exact value is 1 - 2^-99):

b = sfi(1,100,99)

b = 

     1

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 100
        FractionLength: 99

Note, however, that the full precision is preserved in the internal representation of fi

bin(b)

ans =

    '0111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111'

The display of the fi object is also affected by MATLAB's format command. In particular, when displaying real-world values, it is handy to use

format long g

so that as much precision as is possible will be displayed.

There are also other display options to make a more shorthand display of the numeric type properties, and options to control the display of the value (as real-world value, binary, octal, decimal integer, or hex).

For more information on display preferences, type

help fipref
help savefipref
help format

or

doc fipref
doc savefipref
doc format

at the MATLAB command line.

Cleanup

The following code sets any display settings or preferences that the example changed back to their original states.

% Reset the fi display and logging preferences
fipref(fiprefAtStartOfThisExample);
set(0, 'format', originalFormat);
%#ok<*NOPTS,*NASGU>

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