**2. Boolean Algebra**

In a discrete signals the two distinct values such as HIGH and LOW has equivalent voltage levels such as 5 volts and 0 volts respectively. This two distinct levels are represented as value 1 and value 0 respectively. Any algebraic functions performed with respect to this discrete values are defined as *Boolean algebra* developed by George Boole. He also developed various suitable theorems associated with this boolean for manipulation and simplification. There are set of basic definitions which are assumed to be true which defines all the information about the system. The following are the basic definition used in boolean algebra.

• *NOT:* The NOT of a variable is 1 if and only if, the variable itself is 0 and vice versa

• The logical diagram using logic gates is realized for the simplified expression

• In practical design and real time implementation one should consider to use

A combinational circuit that performs the addition of two bits is called *halfadder*. When the augend and addend numbers contain more significant digits, the carry obtained from the addition of two bits is added to the next higher order pair of significant bits. The combinational circuit that performs the addition of three bits is called a *full-adder*. The full adder can also be obtained by using two half adder

A half adder is a combination logic circuit that uses two inputs (A and B) and

**Figure 2** shows the design and implementation of half adder circuit in LabVIEW environment, where the front panel that two inputs Input A and Input B, the outputs are Sum and Carry [3]. The block diagram in LabVIEW environment shows

A full adder is a combination logic circuit that uses three inputs (A, B and Cin) and two outputs (Sum S and Carry C). **Table 2** shows the truth table the various combinations of inputs and its corresponding outputs. The output Sum S and Carry

*Sum S* ¼ *A*⨁*B* (1) *Carry C* ¼ *AB* (2)

*Sum S* ¼ *A*⨁*B*⨁*Cin* (3)

two outputs (Sum S and Carry C). **Table 1** shows the truth table the various combinations of inputs and its corresponding outputs. The output Sum S and Carry

C is obtained and the k-map is used to get the logical equation. The Boolean

the logic gate implementation for the above obtained expression.

C is obtained and the k-map is used to get the logical equation.

*Input Variables Output Variables*

**A B Sum S Carry C** 0 00 0 0 11 0 1 01 0 1 10 1

obtained in the previous step

*DOI: http://dx.doi.org/10.5772/intechopen.97611*

minimum number of gates.

**3.2 Arithmetic circuits**

*(i) Design of half-adders*

*(ii) Design of full-adders*

**Table 1.**

**159**

*Truth Table – Half Adder.*

expressions are

*3.2.1 Adder circuits*

*Digital System Design*

circuits.

