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

The maximum precision scaling can be determined if the range of the variable is known. As shown in Example: Maximizing Precision, the range of a fixed-point operation can be determined from and .

For multiplication, the range can be determined from

where

Radix Point-Only Scaling

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

Accumulation

Gain