Fixed-Point Blockset

Multiplication

Consider the multiplication 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 multiplication of values results in finding the variable Q_a:

This formula shows

• In general, Q_a is not computed through a simple multiplication of Q_b and Q_c.
• In general, there is one multiply of a constant and two variables, two multiplies of a constant and a variable, three additions, and some additional bit shifting.

Inherited Scaling for Speed

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

• Set B_a = B_bB_c. This eliminates one addition operation.
• Set F_a = F_bF_c. This simplifies the triple multiplication - certainly the most difficult part of the equation to implement.
• Set E_a = E_b + E_c. This eliminates some of the bit-shifting.

The resulting formula is

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 multiplication, you can determine the range from

where

Radix Point-Only Scaling

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

Accumulation

Gain