TABLE OF CONTENTS Objective: This chapter describes the mathematical formalities behind binary functions. It includes Boolean algebra and its theorems, followed by standard function representation formats and corresponding standard circuit implementations. The study of Karnaugh maps and other function-simplification techniques is also included, along with a discussion on timing diagrams and glitch generation in combinational circuits.
Chapter Contents
| 5.1 | Boolean Algebra |
| 5.2 | Truth Tables |
| 5.3 | Minterms and SOP Equations |
| 5.4 | Maxterms and POS Equations |
| 5.5 | Standard Circuits for SOP and POS Equations |
| 5.6 | Karnaugh Maps |
| 5.7 | Large Karnaugh Maps |
| 5.8 | Other Function-Simplification Techniques |
| 5.9 | Propagation Delay and Glitches |
| 5.10 | Exercises |
Formal analysis of digital circuits is based on Boolean algebra, whose initial foundations were laid by G. Boole [Boole54] in the 1850s. It contains a set of mathematical rules that govern a two-valued (binary) system represented by zeros and ones. Such rules are discussed in this chapter, where several examples are also presented. To represent bit or bit vector values, VHDL syntax will again be employed whenever appropriate, which consists of a pair of single quotes for single bits or a pair of double quotes for bit vectors.
A Boolean function is a mathematical function involving binary variables, logical addition ("+", also called OR operation), logical multiplication (" ", also called AND operation), and logical inversion (" '"). In summary, Boolean functions can be implemented using only the three fundamental gates depicted in Figure 5.1, which were introduced in Chapter 4. (More precisely, because there...
