Introduction to Digital Systems: Modeling, Synthesis, and Simulation Using VHDL

3.10 SUM-OF-PRODUCTS DESIGN

A function can be represented by an expression that is a sum of minterms only, where each minterm is ANDed with every other minterm to represent the function when it is equal to 1. The resulting implementation is functionally correct and unique but not necessarily the lowest-cost realization of the function. The sum of products (SOP) is a logic expression consisting of product (AND) terms that are summed (ORed) with each other. If each product term is a minterm, the expression is called a canonical sum of products for the function. For example, consider the truth table in Figure 3.10 of a logic function f of three variables. Using the minterms for which the function is equal to 1, the function can be written explicitly as follows:

Figure 3.10 Truth Table for the Logic Function f

Through the use of Boolean algebra identities, this expression can be simplified algebraically as follows:

This is the minimum-cost sum-of-products expression for f. The cost of a logic circuit is the total number of gates plus the total number of inputs to all gates in the circuit. Minterms, given their row-number subscripts, can be used to specify a given function in a more ...