What Is Fixed-Point Data?

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Before You Begin Fixed-Point Numbers Fixed-Point Operations

Before You Begin

Fixed-point numbers use integers and integer arithmetic to approximate real numbers. They are an efficient means for performing computations involving real numbers without requiring floating-point support in underlying system hardware.

See Tips for Using Fixed-Point Data.

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Fixed-Point Numbers

Fixed-point numbers use integers and integer arithmetic to represent real numbers and arithmetic with the following encoding scheme:

where

• V is a precise real-world value that you want to approximate with a fixed-point number.

• is the approximate real-world value that results from fixed-point representation.

• Q is an integer that encodes This value is the quantized integer.

  Q is the actual stored integer value used in representing the fixed-point number. If a fixed-point number changes, its quantized integer, Q, changes but S and B remain unchanged.

• S is a coefficient of Q, or the slope.

• B is an additive correction, or the bias.

Fixed-point numbers encode real quantities (for example, 15.375) using the stored integer Q. You set the value of Q by solving the equation

for Q and rounding the result to an integer value as follows:

Q = round((V – B)/S)

For example, suppose you want to represent the number 15.375 in a fixed-point type with the slope S = 0.5 and the bias B = 0.1. This means that

Q = round((15.375 – 0.1)/0.5) = 30

However, because Q is rounded to an integer, you lose some precision in representing the number 15.375. If you calculate the number that Q actually represents, you now get a slightly different answer.

Using fixed-point numbers to represent real numbers with integers involves the loss of some precision. However, if you choose S and B correctly, you can minimize this loss to acceptable levels.

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Fixed-Point Operations

Now that you can express fixed-point numbers as you can define operations between two fixed-point numbers.

The general equation for an operation between fixed-point operands is as follows:

c = a <op> b

where a, b, and c are all fixed-point numbers, and <op> refers to a binary operation: addition, subtraction, multiplication, or division.

The general form for a fixed-point number x is S_xQ_x + B_x (see Fixed-Point Numbers). Substituting this form for the result and operands in the preceding equation yields this expression:

(S_cQ_c + B_c) = (S_aQ_a + B_a) <op> (S_bQ_b + B_b)

The values for S_c and B_c are chosen by Stateflow software for each operation (see Promotion Rules for Fixed-Point Operations) and are based on the values for S_a, S_b, B_a and B_b that you enter for each fixed-point data (see Specifying Fixed-Point Data).

Note   You can be more precise in choosing the values for Sc and Bc when you use the := assignment operator (that is, c := a <op> b). See Assignment (=, :=) Operations.

Using the values for S_a, S_b, S_c, B_a, B_b, and B_c, you can solve the preceding equation for Q_c for each binary operation as follows:

• The operation c=a+b implies that

  Q_c = ((S_a/S_c)Q_a + (S_b/S_c)Q_b + (B_a + B_b – B_c)/S_c)

• The operation c=a-b implies that

  Q_c = ((S_a/S_c)Q_a – (S_b/S_c)Q_b – (B_a – B_b – B_c)/S_c)

• The operation c=a*b implies that

  Q_c = ((S_aS_b/S_c)Q_aQ_b + (B_aS_b/S_c)Q_a + (B_bS_a/S_c)Q_b + (B_aB_b – B_c)/S_c)

• The operation c=a/b implies that

  Q_c = ((S_aQ_a + B_a)/(S_c(S_bQ_b + B_b)) – (B_c/S_c))

The fixed-point approximations of the real number result of the operation c = a <op> b are given by the preceding solutions for the value Q_c. In this way, all fixed-point operations are performed using only the stored integer Q for each fixed-point number and integer operation.

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Using Fixed-Point Data in Stateflow Charts

How Fixed-Point Data Works in Stateflow Charts

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