# Create Fixed-Point Data

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' 

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 There is also SlopeBias scaling, which has the relationship where and 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. a.bin = '0110010010001000' 
a = 3.1416015625 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 13 a.oct = '031707' 
a = 1.6180419921875 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 13 a.dec = '22268' 
a = 2.71826171875 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 13 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).

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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