Fixed-Point Blockset

Division

Division of values is an operation that should be avoided in fixed-point embedded systems, but it can occur in places. Therefore, consider the division of two real-world values:

These values are represented by the general [Slope Bias] encoding scheme described in Scaling:

In a fixed-point system, the division of values results in finding the variable Q_a:

This formula shows

• In general, Q_a is not computed through a simple division of Q_b by Q_c.
• In general, there are two multiplies of a constant and a variable, two additions, one division of a variable by a variable, one division of a constant by a variable, and some additional bit shifting.

Inherited Scaling for Speed

The number of arithmetic operations can be reduced with these choices:

• Set B_a = 0. This eliminates one addition operation.
• If B_c = 0, then set the fractional slope F_a = F_b/F_c. This eliminates one constant times variable multiplication.

The resulting formula is

If , then no clear recommendation can be made.

Inherited Scaling for Maximum Precision

You can determine the maximum precision scaling if the range of the variable is known. Example: Maximizing Precision shows that you can determine the range of a fixed-point operation from and . For division, you can determine the range from

where for nonzero denominators

Radix Point-Only Scaling

For radix point-only scaling, finding Q_a results in this simple expression:

Note    For the last two formulas involving Qa, a divide by zero, and zero divided by zero are possible. In these cases, the hardware will give some default behavior but you must make sure that these default responses give meaningful results for the embedded system.

Gain

Summary