Chapter 2. Boolean Algebra and Logic Gates


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1 Chapter 2. Boolean Algebra and Logic Gates Tong In Oh 1
2 Basic Definitions 2
3 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 George Boole (1854) Closure/Identity element/commutative law/distributive law/complement Exist at least two elements 4
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 twovalued Boolean algebra (0 and 1) Application of Boolean algebra to gatetype circuits 5
6 Twovalues Boolean Algebra 6
7 Boolean Algebra (switching algebra, binary logic) 7
8 Theorems & Properties of Boolean Algebra Duality (valid if the operators and identity elements are interchanged) Basic theorems Operator Precedence Parentheses NOT AND OR 8
9 9
10 10
11 11
12 DeMorgan s Theorem 12
13 Operator Precedence 13
14 2.5 Boolean Functions Truth table Evaluated by determining the binary value of the expression for all possible values of the variables 14
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 logiccircuit diagram FIGURE 2.1 Gate implementation of F 1 = x + y z 15
16 FIGURE 2.2 Implementation of Boolean function F 2 with gates 16
17 Algebraic Manipulation Use computer minimization programs Algebraic manipulation of Boolean algebra 17
18 DeMorgan s Theorems 18
19 2.6 Canonical and Standard Forms 19
20 Boolean Function Expression 20
21 Boolean Function Expression Complement of a Boolean function Boolean function : expressed as a product of maxterms 21
22 Sum of Minterms (Ex 2.4) 22
23 Product of Maxterms 23
24 Conversion between Canonical Forms 24
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 Twolevel structure of gates 25
26 Nonstandard Form FIGURE 2.4 Three and two level implementation 26
27 2.7 Other Logic Operations 27
28 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, exclusiveor, equivalence Small circle = bubble = logic complement 29
30 30
31 Extension to Multiple Inputs Gates can be more than two inputs Consider commutative and associative AND/OR satisfied NAND/NOR commutative but not associative (fig 2.6) ( x y) z x y z ( ) Multiple NOR:complemented OR x y z=(x+y+z) FIGURE 2.6 Demonstrating the nonassociativity of the NOR operator: 31
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 32 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) 33 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 positivelogic AND gate or as a negativelogic OR gate Dual of a function 34
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) Smallscale integration (SSI) Mediumscale integration (MSI) Largescale integration (LSI) Very largescale integration (VLSI) Digital logic families (technology) TTL: transistortransistor logic ECL: emittercoupled logic (high speed) MOS: metaloxide semiconductor (high density) CMOS: complementary metaloxide semiconductor (low power) 35
36 Important parameters Fanout Fanin Power dissipation Propagation delay Noise margin ComputerAided Design of VLSI Circuits Electronic design automation (EDA) ASIC/FPGA/PLD/fullcustom IC HDLbased synthesis tools and methodologies Verilog VHDL 36
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