The Multiplication Process

The multiplication of an n-bit binary number with an m-bit binary number results in a product that is up to m + n bits in length for both signed and unsigned words.

Suppose you want to multiply three numbers. Each of these numbers is represented by a 5-bit word, and each has a different binary-point-only scaling. Additionally, the output is restricted to a 10-bit word with binary-point-only scaling of 2^-4. The multiplication is shown in the following model for the input values 5.75, 2.375, and 1.8125.

Applying the rules from the previous section, the multiplication follows these steps:

The first two numbers (5.75 and 2.375) are multiplied:
$\begin{matrix} Q_{R a w P r o d u c t} = 1 01.11 \\ \frac{\times 10.011}{1 01.11 \cdot 2^{- 3}} \\ \frac{\begin{array}{l} 1 01.11 \cdot 2^{- 2} \\ + 1 01.11 \cdot 2^{1} \end{array}}{011 01.10101} \begin{matrix} = 13.65625. \end{matrix} \end{matrix}$
Note that the binary point of the product is given by the sum of the binary points of the multiplied numbers.
The result of step 1 is converted to the output data type:
$\begin{matrix} Q_{T e m p} = c o n v e r t (Q_{R a w P r o d u c t}) \\ = 001101.1010 = 13.6250. \end{matrix}$
Signal Conversions discusses conversions. Note that a loss in precision of one bit occurs, with the resulting value of Q_Temp determined by the rounding mode. For this example, round-to-floor is used. Furthermore, overflow did not occur but is possible for this operation.
The result of step 2 and the third number (1.8125) are multiplied:
$\begin{matrix} Q_{R a w P r o d u c t} = 011 01.1010 \\ \frac{\times 1.1101}{11 01.1010 \cdot 2^{- 4}} \\ \frac{\begin{array}{l} 11 01.1010 \cdot 2^{- 2} \\ 11 01.1010 \cdot 2^{- 1} \\ + 11 01.1010 \cdot 2^{0} \end{array}}{0011000 .10110010} \begin{matrix} = 24.6953125. \end{matrix} \end{matrix}$
Note that the binary point of the product is given by the sum of the binary points of the multiplied numbers.
The product is converted to the output data type:
$\begin{matrix} Q_{a} = c o n v e r t (Q_{R a w P r o d u c t}) \\ = 011000.1011 = 24.6875. \end{matrix}$
Signal Conversions discusses conversions. Note that a loss in precision of 4 bits occurred, with the resulting value of Q_Temp determined by the rounding mode. For this example, round-to-floor is used. Furthermore, overflow did not occur but is possible for this operation.

Blocks that perform multiplication include the Product, Discrete FIR Filter, and Gain blocks.