Chapter 2. Boolean Algebra and Logic Gates - Tong In Oh · Chapter 2. Boolean Algebra and Logic Gates ... • Application of Boolean algebra to gate -type circuits. ... Digital Design,

Digital Design, Kyung Hee Univ.

1

Chapter 2. Boolean Algebra and Logic Gates

Tong In Oh


2

Basic Definitions


3


4

2.3 Axiomatic Definition of Boolean Algebra• Boolean algebra:

• Algebraic structure defined by a set of elements, B, together with two binary operators, + and ·

• Closure/Identity element/Commutative law/Distributive law/Complement• Exist at least two elements

George Boole (1854)


5

2.3 Axiomatic Definition of Boolean Algebra• Difference with arithmetic and ordinary algebra (the field of real

numbers)• Not include the associative law• Distributive law of ‘+’ over ‘·’ is valid (Boolean algebra)• No additive or multiplicative inverses (no subtraction or division operations)• Complement (not available in ordinary algebra)• Ordinary algebra deals with the infinite set of elements vs. Boolean algebra

deals with the undefined set of elements, B

• Boolean algebra• Elements of the set B• Rules of operation for the two binary operators• Satisfy the six Huntington postulates

• We deal only with a two-valued Boolean algebra (0 and 1)• Application of Boolean algebra to gate-type circuits


6

Two-values Boolean Algebra


7

Boolean Algebra (switching algebra, binary logic)


8

Theorems & Properties of Boolean Algebra• Duality (valid if the operators and identity elements are interchanged)• Basic theorems• Operator Precedence

• Parentheses → NOT → AND → OR


9


10


11


12

DeMorgan’s Theorem


13

Operator Precedence


14

2.5 Boolean Functions

• Truth table• Evaluated by determining the binary value of

the expression for all possible values of the variables


15

Circuit Diagram• Transform from an algebraic expression into a circuit diagram

composed of logic gates connected in a particular structure• Dictate the interconnection of gates in the logic-circuit diagram

FIGURE 2.1 Gate implementation of F1 = x + y’z


16FIGURE 2.2 Implementation of Boolean function F2 with gates


17

Algebraic Manipulation• Use computer minimization programs• Algebraic manipulation of Boolean algebra


18

DeMorgan’s Theorems


19

2.6 Canonical and Standard Forms


20

Boolean Function Expression


21

Boolean Function Expression• Complement of a Boolean function• Boolean function : expressed as a product of maxterms


22

Sum of Minterms (Ex 2.4)


23

Product of Maxterms


24

Conversion between Canonical Forms


25

Standard Forms• Contain any number of literals• Sum of products

• AND terms = product terms• Sum = ORing of these terms

• Products of sums• OR terms = sum terms• Product = ANDing of these terms

• Two-level structure of gates


26

Nonstandard Form

FIGURE 2.4 Three‐ and two‐level implementation


27

2.7 Other Logic Operations


28


29

2.8 Digital Logic Gates• Considering the construction

• Feasibility and economy of producing the gate with physical components• Possibility of extending the gate to more than two inputs• Basic properties of the binary operator (Commutativity, associativity)• Ability of the gate to implement Boolean functions alone or in conjunction with

other gates

• Inhibition/Implication: not commutative or associative → impractical to use as standard logic gates

• Complement, transfer, AND, OR, NAND, NOR, exclusive-OR, equivalence

• Small circle = bubble = logic complement


30


31

• Gates can be more than two inputs• Consider commutative and associative• AND/OR … satisfied• NAND/NOR … commutative but not associative (fig 2.6)

• Multiple NOR:complemented OR • x↓y↓z=(x+y+z)’

Extension to Multiple Inputs

FIGURE 2.6 Demonstrating the nonassociativity of the NOR operator:

x ↓ y( )↓z ≠ x ↓ y ↓z( )


32

Extension to Multiple Inputs• Multiple NOR : x↓y↓z=(x+y+z)’• Multiple NAND: x↑y↑z=(xyz)’ (fig 2.7)• Expression in sum of products form with NAND and NOR gates

FIGURE 2.7 Multiple‐input and cascaded NOR and NAND gates


33

• XOR/XNOR … satisfied commutative and associative but not common• XOR = odd function (fig 2.8)

FIGURE 2.8 Three‐input exclusive‐OR gate


34

Positive and Negative Logic• Gates have one of two values, except

during transition• Assign signal values to two logic values• Higher signal level: H, lower signal level:

L• Positive logic system: H → logic 1• Negative logic system: L → logic 1• Polarity indicator (fig.2.10) – same

physical gate can operate either as a positive-logic AND gate or as a negative-logic OR gate

• Dual of a function


35

2.9 Integrated Circuits• IC: fabricated on a die of a silicon semiconductor crystal (a

chip) for constructing digital gates• Levels of integration (complexity, number of logic gates)

• Small-scale integration (SSI)• Medium-scale integration (MSI)• Large-scale integration (LSI)• Very large-scale integration (VLSI)

• Digital logic families (technology)• TTL: transistor-transistor logic• ECL: emitter-coupled logic (high speed)• MOS: metal-oxide semiconductor (high density)• CMOS: complementary metal-oxide semiconductor (low power)


36

• Important parameters• Fan-out• Fan-in• Power dissipation• Propagation delay• Noise margin

• Computer-Aided Design of VLSI Circuits• Electronic design automation (EDA)• ASIC/FPGA/PLD/full-custom IC• HDL-based synthesis tools and methodologies

• Verilog• VHDL

Documents

Chapter 2. Boolean Algebra and Logic Gates - Tong In Oh · Chapter 2. Boolean Algebra and Logic Gates ... • Application of Boolean algebra to gate -type circuits. ... Digital Design,