Combinational Logic Fundamentals
|
|
- Jeffrey Adams
- 5 years ago
- Views:
Transcription
1 Topic 3: Combinational Logic Fundamentals In this note we will study combinational logic, which is the part of digital logic that uses Boolean algebra. All the concepts presented in combinational logic can be implemented using logic gates, which are identified to be basic elements of combinational logic. In topic 4, we will study some aspects of their physical implementation. Objectives: - Understand the theory of Boolean algebra and its application in combinational logic. - Understand the behavior of basic logic gates. - Understand the principle of minimization of combinational functions using Karnaugh Maps. Combinational logic networks only process information. Their output signal values depend entirely on input signal values at each instant of time; we usually think of then as acting instantaneously. Switching theory deals with binary variables {x i } whose two values are represented by bits (binary digits) 0 and 1. A combinational (switching) function is a mapping z: K n K, where K = {0,1} and K n denotes the set of 2 n binary n-tuples. Any combinatory function can, in principle, be defined by a truth table, which specifies for every input combination (x 1, x 2,.x n ) the corresponding z(x 1, x 2,.x n ). Logic diagrams, truth tables, timing diagrams are use to represent functions realize by combinational systems. Figure 3.1gives an example of a truth table that defines two-variable functions z(x 0,y 0, c 0 ) and c 1 (x 0,y 0,c 0 ). Inputs Outputs x 0 y 0 c 0 z 0 c Figure 3.1. Truth table for 3-variable combinational functions: z 0 (sum) and c 1 (carry). Figure 3.2 shows different mode of representation of a logic network. 1
2 Figure 3.2. An example of logic networks. Combinational functions do not involve time, in this manner; they can be used to represent the behavior of any logical operation or binary numerical operation Boolean Algebra. An alternative approach to truth table of a logic function is to the use of algebraic expressions in which variables represent binary logic signals and the operators of the algebra represent gate functions. The algebra, which models combinational circuits, is a type of Boolean algebra, which originated with the work of George Boole ( ). Boolean algebra, like any algebra, is defined by a set of elements K, a set of operators P over K, and a set of axioms or laws defining the properties of P and K. It is convenient to let P = { and, or, not} and K = {0,1}. 2
3 AND, OR, NOT are the standard gate operators. The more common and useful laws of Boolean algebra are summarized in figure 3.3. Given a, b of K, a AND b = ab; a OR b = a+b, NOT a = a'. No. Statement of axiom Name 1 There exits elements 0, 1 of K such that Existence of identity elements a + 0 = a a1 = a 2 a +b = b+a Commutative Laws ab = ba 3 a(b+c) = ab + ac Distributive Laws a +(bc) = (a+b)(a+c) 4 For every a K, there exists a' K such that Existence of Inverse aa' = 0 a + a' = 1 5 a + (b +c) = (a + b) + c Associative Laws a(bc) = (ab)c 6 a + a = a Idempotent Laws aa = a 7 (a+b)' = a'b' De Morgan s Laws (ab)' = a' + b' 8 (a')' = a Involution Figure 3.3. The basic laws of Boolean algebra (a, b and c are arbitrary elements of K). Notation: In this note, for a given element x of K complement ( x) = x' = x To enable us to deal with a number of variables, it is useful to define some two- and three-variable algebraic identities. For each identity, its dual version is also given. These identities are often referred to as properties. If a, b, and c are the variables in K, then the following properties hold: Absorption a) a +ab = a b) a(a+b) = a c) a + a'b = a+b d) a(a' + b) = ab Consensus a) ab + a'c + bc = ab + a'c b) (a+b)(a' + c)(b + c) = (a +b)(a' + c) Principle of Duality: Any theorem or identity in Boolean algebra remains true if 0 and 1 are swapped and. and + are swapped throughout. n-variable Theorems (Textbook pp ) The following table shows several n-variable important theorems: Statement of Theorem Name X + X + X..+X = X Generalized idempotency X.X.X.X = X (X1.X2.Xn) = X1 + X2 +.+ Xn (Demorgan s Theorems (X1+ X2+ +Xn) = X1.X2..Xn [F(X1,X2,,Xn, +,.)] = F(X1, X2,,Xn,., +) (Generalized Demorgan s theorem) F(X1,X2,X3,.Xn) = X1.F(1,X2,.Xn) + X1.F(0,X2,..Xn) Shannon s expansion theorems F(X1,X2, Xn) = [X1+F(0,X2,,Xn)].[X1 + F(1,X2, Xn)] 3
4 Most of these theorem can be proved using a two-step method called finite induction First proving that the theorem is true for n = 2 ( that basis step) and then proving that if the theorem is true for n = I, then it is also true for n = I +1 ( the induction step). Shannon s expansion theorems are demonstrated by considering the dual concept.: If F(X1, X2,.., Xn, +,., ) is a fully parenthesized logic expression involving the variable X1, X2, Xn and the operators +,., and, then the dual of F, written F D, is the same expression with +, and. swapped: F D (X1,X2,,Xn, +,., ) = F(X1, X2, X3,.,Xn,., +, ) The fundamental design problem for combinational circuits may be expressed as follows. Design a logic circuit to realize a given set of combinational functions using minimum number of gates. These techniques are based on the correspondence between two-level combinational circuits and Boolean equations of the form: N f ( x, x,..., x N = m i And 0 1 ) i= 0 f ( x, x o 1,..., xn ) i N = = i= 0 M i Where Σ and Π denote the logical sum (OR, SOP: sum-of-product) and product (AND, POS: product-of-sum) operations, and m i the minterm and M i the maxterm. For example for N = 2 n -1 and n =3 (x 0, x 1, x 2 ) we have the following expressions for the minterm and maxterm. Msb = x 0, Lsb = x 2 Minterm Maxterm m 0 = x 0 'x 1 'x 2 ' (equivalent code of 0) M 0 = x 0 + x 1 + x 2 m 1 = x 0 'x 1 'x 2 M 1 = x 0 + x 1 + x 2 ' m 2 = x' 0 x 1 x 2 ' M 2 = x 0 + x 1 ' + x 2 ' m 3 = x 0 'x 1 x 2 M 3 = x 0 + x 1 ' + x 2 ' m 4 = x 0 x 1 'x 2 ' M 4 =x 0 ' + x 1 + x 2 m 5 = x 0 x 1 'x 2 M 5 = x 0 ' + x 1 + x 2 ' m 6 = x 0 x 1 x 2 ' M 6 = x 0 ' + x 1 ' + x 2 m 7 = x 0 x 1 x 2 (equivalent code of 7) M 7 = x 0 ' +x 1 ' +x 2 ' A Boolean function F may be represented by a sum (ORed together) of its minterms. They represent the input combinations needed to yield F = 1. So minterms represent the 1 s in the truth table for F. A Boolean function F may be represented by a product (ANDed together) of its maxterms. They represent the input combinations needed to yield F = 0. So maxterms represent the 1 s in the truth table for F. We have m i = M i ' and M i = m i ' Using minterms or maxterms to represent a combinational function represent the canonical form in which the expressions are not minimized. The same function can be represented in a standard form with minimized expressions (sums and products). Some definitions: - A literal is a variable or the complement of a variable. Examples: X, Y, X, Y. - A product term is a single literal or logic product of two or more literals. Examples: Z, WXY, XY Z. - A sum-of-products expression is a logical sum of product terms. Example: Z + WXXY + XY Z. - A sum term is a single literal or logical sum of two or more literals. Examples: Z, W + X + Y, X + Y + Z. - A product-of-sums expression is a logical product of sum terms. Example: Z (W + X + Y)(X+Y +Z). - A normal term is a product or sum term in which no variable appears more than once. 4
5 - The canonical sum of a logic function is a sum of the minterms corresponding to truth-table rows (input combinations) for which the function produces a 1 output. Example: F = X, Y, Z ( 0,3,4,6,7) = X ' Y ' Z' + X ' YZ + XY ' Z' + XYZ' + XYZ. - The canonical product of a logic function is a product of the maxterms corresponding to input combinations for which the function produces a 0 output. Example: F = ( 1,2,5) = ( X + Y + Z')( X + Y ' + Z)( X ' + Y + Z'). X, Y, Z 3.2. Gates and Circuits. A physical realization of a combinatory function is called a combinatory circuit. Combinational circuits are connected from standard components called gates, which themselves realize very simple combinational functions. The most important gate types are listed in figure 3.4. Figure 3.4 Major logic gate types. 5
6 The function performed by each gate is defined by a truth table. Gates will be represented in logic diagrams by the special symbols shown in figure Gate Minimization An SOP expression E for a function f is said to be minimal if it contains the smallest possible number of product terms among all SOP expressions defining f, and if as a secondary condition, no literals can be deleted from any of the product terms without changing the function specified by E. A two-level circuit corresponding to E is also minimal in the sense that it contains the fewest gates among all AND-OR circuits realizing f, and the fan-in of none of the AND gates can be reduced. Minimal POS expressions and OR-AND circuits can be defined analogously. There are three common methods for minimizing combinational expression: a) Boolean algebra The basis for all methods Difficult to see the best path to take and to know which law to be used b) Karnaugh Maps Pictorial approach based on recognition of patterns Fast and easy for 2 6 variables Difficult for large numbers of variables c) Tabulation methods Tedious to perform by hand, but well suited to computer implementation Not limit on the number of variables In this paragraph, we will present K-map as method gate minimization. It was developed by Maurice Karnaugh at AT&T Bell Lab in The K-map is based on the principle of looping 2 i "1" adjacent. By doing, we eliminate variables that appear in complemented and uncomplemented forms Karnaugh Maps (K-maps) Essentially a 2D truth table arrange in a table or map so that adjacent cells in the map differ in only one bit position. Each cell in the map corresponds to a minterm Combining 1 s in the Kmap into groups of 1, 2, 4, 8, etc. can yield a quick minimization of the combinational function Configurations: 6
7 7
8 8
9 a) Minimizing Sums of Products Karnaugh minimization is based on the principle of the groupment of 2 i cells of adjacent 1, which itself lies on the fact that Y.term + Y.term = term, with: - Y = variable for two adjacent 1 cells, - Y = a product of two variables for four adjacent 1 cells, - Y = a product of three variables for eight adjacent 1 cells, - Y = a product of four variables for sixteen adjacent 1 cells. Term is also called a prime implicant as defined below. Different configurations of adjacent 1s will be given in class. 9
10 Some Definitions (Figure 3.5 = Fig Textbook): A minimal sum of a logic function F(X1,.., Xn) is a sum-of-products expression for F such that no sum-of-products for F such that no sum-of-products expression for F has fewer products terms, and any sum-of-products expression with the same number of product terms has at least as many literals. A logic function P(X1,., Xn) implies a logic functions F(X1, Xn) if for every input combination such that P = 1, then F = 1 also. A prime implicant of a logic function F(X1,.,Xn) is a normal product term P(X1,,Xn) that implies F, such that if any variable is removed from P, then the resulting product term does not imply F. Prime-Implicant theorem: A minimal sum is a sum of prime implicants. Figure 3.5: (a) Karnaugh map; (b) prime implicants. The sum of all prime implicants of a logic function is called the complete sum. Although the complete sum is always a legitimate way to realize a logic function, it s not always minimal (Figure 3.6 = Figure 4.32 Textbook). It s therefore important in the minimization process to know which prime implicants to leave out. This can be done by considering the following definitions : - A distinguished 1-cell of a logic function is an input combination that is covered by only one prime implicant. - An essential prime implicant of a logic function is a prime implicant that covers one or more distinguished 1-cells. The minimal sum is obtained by identifying the distinguished 1-cells and the corresponding prime implicants and include the essential prime implicants in the minimal sum (Figure 4.33 textbook) Figure 3.6: (a) Karnaugh map; (b) prime implicants and distinguished 1-cells. 10
11 In the example of figure 3.6, all of the 1-cells are covered by essentials prime implicants, so we need go no further. We have a different configuration in figure 3.7, the approach consist of removing the essential prime implicants and the 1-cells they cover, the remaining prime implants will be added to the minimum sum. Figure 3.7: (a) Karnaugh map; (b) prime implicants and distinguished 1-cells; (c) reduced map after removal of essential prime implicants and covered 1-cells. The following example shows a more difficult case: a logic function with no essential prime implicant. 11
12 Another useful way of describing the Karnaugh algorithms consists of steps that show how to use K-map method for simplifying a Boolean Expression: 1. Construct the K map and place 1s in those squares corresponding to the 1s in the truth table. Place 0s in the other squares. 2. Examine the map for adjacent 1s and loop those 1s, which are not adjacent to any other 1s. They are called isolated 1s. 3. Next, look for those 1s, which are adjacent to only one other 1. Loop any pair containing such a Loop any octet event it contains some 1s that have already been loop 5. Loop any quad that contains one or more 1s that have not already been looped, making sure to use the minimal number of loops. 6. Loop any pairs necessary to include any 1s that have not yet been looped, making sure to use the minimal number of loops. 7. Form the OR sum of all the terms generated by each loop. Examples (figures 3.8 & 3.9) Figure 3.8. Examples of looping pairs of adjacent 1s. 12
13 Figure 3.9. More K-maps 13
14 b) Simplifying Products of sums: the principle of duality is applied. We consider the 0s on a Karnaugh map and express F. c) Don t care combination (textbook, page 232) Design Example: Three-way Light Control Assume that a large room has tree doors and that a switch near each door controls a light in the room. It has to be possible to turn the light on or off by changing the state of any one of the switches. As a first step, let us turn this word statement into a formal specification using a truth table. Let x 1, x 2 and x 3 be the input variables that denote the state of each switch. Assume that the light is off if all switches are open. Closing any one of the switches will turn the light on. Then turning on a second switch will have to turn off the light. Thus the light will be on if exactly one switch is closed, and it will be off if two (or no) switches are closed. If the light is off when two switches are closed, then it must be possible to turn it on by closing the third switch. If f(x 1, x 2, x 3 ) represents the state of the light, then the required functional behavior can be specified as shown in the truth table in figure x 1 x 2 x 3 f Figure 3.10: Truth table for the three-way light control. The canonical sum-of-products expression for the specified function is f = m 1 + m 2 + m 4 + m 7 = x 1 'x 2 'x 3 + x 1 'x 2 x 3 ' + x 1 x 2 'x 3 ' + x 1 x 2 x 3 This expression cannot be simplified into a lower-cost sum-of-products expression. The resulting circuit is shown in Figure 3.11a. An alternative realization for this function is in the product-of-sums forms. The canonical expression of this type is f = M 0.M 3.M 5.M 6 = (x 1 + x 2 + x 3 ). (x 1 + x 2 ' + x 3 '). (x 1 ' + x 2 + x 3 '). (x 1 ' + x 2 ' + x 3 ) The resulting circuit is depicted in figure 3.8b. It has the same cost as the circuit in part (a) of the figure
15 Figure Implementation of the function in figure
CHAPTER III BOOLEAN ALGEBRA
CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationCombinational Logic Design Principles
Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems
More informationChapter 2: Switching Algebra and Logic Circuits
Chapter 2: Switching Algebra and Logic Circuits Formal Foundation of Digital Design In 1854 George Boole published An investigation into the Laws of Thoughts Algebraic system with two values 0 and 1 Used
More informationDigital Circuit And Logic Design I. Lecture 3
Digital Circuit And Logic Design I Lecture 3 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinational-circuit analysis 4. Combinational-circuit synthesis Panupong
More informationII. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):
ENGI 386 Digital Logic II. COMBINATIONAL LOGIC DESIGN Combinational Logic output of digital system is only dependent on current inputs (i.e., no memory) (a) Boolean Algebra - developed by George Boole
More informationUnit 2 Session - 6 Combinational Logic Circuits
Objectives Unit 2 Session - 6 Combinational Logic Circuits Draw 3- variable and 4- variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the Product-of-Sums
More informationCombinational Logic. Review of Combinational Logic 1
Combinational Logic! Switches -> Boolean algebra! Representation of Boolean functions! Logic circuit elements - logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational
More informationMidterm1 Review. Jan 24 Armita
Midterm1 Review Jan 24 Armita Outline Boolean Algebra Axioms closure, Identity elements, complements, commutativity, distributivity theorems Associativity, Duality, De Morgan, Consensus theorem Shannon
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active
More informationChapter 2 : Boolean Algebra and Logic Gates
Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University 1431-1432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean
More informationCHAPTER 2 BOOLEAN ALGEBRA
CHAPTER 2 BOOLEAN ALGEBRA This chapter in the book includes: Objectives Study Guide 2.1 Introduction 2.2 Basic Operations 2.3 Boolean Expressions and Truth Tables 2.4 Basic Theorems 2.5 Commutative, Associative,
More informationLogic Design. Chapter 2: Introduction to Logic Circuits
Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits perform operation on digital signal Digital signal: signal values are restricted to a few discrete values Binary logic
More information2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS
CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS What will we learn? 2 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0 Terminology 3 Digital
More informationThis form sometimes used in logic circuit, example:
Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (k-map). 1. Deriving of logical expression from
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev
E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean
More informationEECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive
EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150 - Lec19-cl1 Page 1 Boolean Algebra I (Representations of Combinational
More informationECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering
ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. Overview Part 1 Gate
More informationNumber System conversions
Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number
More information1. Name the person who developed Boolean algebra
MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 67 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR TH 2TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA
More informationFunctions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways:
Boolean Algebra (1) Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y)
More informationBinary logic consists of binary variables and logical operations. The variables are
1) Define binary logic? Binary logic consists of binary variables and logical operations. The variables are designated by the alphabets such as A, B, C, x, y, z, etc., with each variable having only two
More informationEx: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.
Boolean Expression Forms: Sum-of-products (SOP) Write an AND term for each input combination that produces a 1 output. Write the input variable if its value is 1; write its complement otherwise. OR the
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates
Digital Circuits & Systems Lecture Transparencies (Boolean lgebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean lgebra & Logic Gates Major topics Boolean algebra NND & NOR gates Boolean algebra
More informationLecture 2 Review on Digital Logic (Part 1)
Lecture 2 Review on Digital Logic (Part 1) Xuan Silvia Zhang Washington University in St. Louis http://classes.engineering.wustl.edu/ese461/ Grading Engagement 5% Review Quiz 10% Homework 10% Labs 40%
More informationChapter 2 Boolean Algebra and Logic Gates
Ch1: Digital Systems and Binary Numbers Ch2: Ch3: Gate-Level Minimization Ch4: Combinational Logic Ch5: Synchronous Sequential Logic Ch6: Registers and Counters Switching Theory & Logic Design Prof. Adnan
More informationSlide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary
Slide Set 3 for ENEL 353 Fall 2016 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2016 SN s ENEL 353 Fall 2016 Slide Set 3 slide
More informationWeek-I. Combinational Logic & Circuits
Week-I Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and
More informationBoolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE
Boolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE Today s Topics Boolean algebra applications in logic
More informationLecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University
Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Aby K George, ECE Department, Wayne State University Contents The Map method Two variable
More informationChapter-2 BOOLEAN ALGEBRA
Chapter-2 BOOLEAN ALGEBRA Introduction: An algebra that deals with binary number system is called Boolean Algebra. It is very power in designing logic circuits used by the processor of computer system.
More informationCS 226: Digital Logic Design
CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions
More informationDigital Circuit And Logic Design I. Lecture 4
Digital Circuit And Logic Design I Lecture 4 Outline Combinational Logic Design Principles (2) 1. Combinational-circuit minimization 2. Karnaugh maps 3. Quine-McCluskey procedure Panupong Sornkhom, 2005/2
More informationAdministrative Notes. Chapter 2 <9>
Administrative Notes Note: New homework instructions starting with HW03 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2
More informationEEE130 Digital Electronics I Lecture #4
EEE130 Digital Electronics I Lecture #4 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi Topics to be discussed 4-1 Boolean Operations and Expressions 4-2 Laws and Rules of Boolean
More informationBoolean Algebra & Logic Gates. By : Ali Mustafa
Boolean Algebra & Logic Gates By : Ali Mustafa Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions
More informationLecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps
EE210: Switching Systems Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps Prof. YingLi Tian Feb. 21/26, 2019 Department of Electrical Engineering The City College of New York
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Chapter 2 - Part 1 2 Chapter 2 - Part 1 3 Chapter 2 - Part 1 4 Chapter 2 - Part
More informationLecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions
EE210: Switching Systems Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions Prof. YingLi Tian Feb. 15, 2018 Department of Electrical Engineering The City College of New York The
More informationCHAPTER 12 Boolean Algebra
318 Chapter 12 Boolean Algebra CHAPTER 12 Boolean Algebra SECTION 12.1 Boolean Functions 2. a) Since x 1 = x, the only solution is x = 0. b) Since 0 + 0 = 0 and 1 + 1 = 1, the only solution is x = 0. c)
More informationBinary Logic and Gates. Our objective is to learn how to design digital circuits.
Binary Logic and Gates Introduction Our objective is to learn how to design digital circuits. These circuits use binary systems. Signals in such binary systems may represent only one of 2 possible values
More informationCS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman
CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions
More informationIf f = ABC + ABC + A B C then f = AB C + A BC + AB C + A BC + A B C
Examples: If f 5 = AB + AB then f 5 = A B + A B = f 10 If f = ABC + ABC + A B C then f = AB C + A BC + AB C + A BC + A B C In terms of a truth table, if f is the sum (OR) of all the minterms with a 1 in
More informationChapter 2: Boolean Algebra and Logic Gates
Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a
More informationDigital Logic Design. Combinational Logic
Digital Logic Design Combinational Logic Minterms A product term is a term where literals are ANDed. Example: x y, xz, xyz, A minterm is a product term in which all variables appear exactly once, in normal
More informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two- and three-variable Four-variable Determination of minimum expressions using essential prime implicants Five-variable
More informationENG2410 Digital Design Combinational Logic Circuits
ENG240 Digital Design Combinational Logic Circuits Fall 207 S. Areibi School of Engineering University of Guelph Binary variables Binary Logic Can be 0 or (T or F, low or high) Variables named with single
More informationDigital Logic Design. Malik Najmus Siraj
Digital Logic Design Malik Najmus Siraj siraj@case.edu.pkedu LECTURE 4 Today s Agenda Recap 2 s complement Binary Logic Boolean algebra Recap Computer Arithmetic Signed numbers Radix and diminished radix
More informationCPE100: Digital Logic Design I
Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris Combinational Logic Design Chapter
More informationCombinatorial Logic Design Principles
Combinatorial Logic Design Principles ECGR2181 Chapter 4 Notes Logic System Design I 4-1 Boolean algebra a.k.a. switching algebra deals with boolean values -- 0, 1 Positive-logic convention analog voltages
More informationCHAPTER1: Digital Logic Circuits Combination Circuits
CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.
More informationECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2
ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 Instructor: Andrew B. Kahng (lecture) Email: abk@ucsd.edu Telephone: 858-822-4884 office, 858-353-0550 cell Office: 3802
More informationSimplification of Boolean Functions. Dept. of CSE, IEM, Kolkata
Simplification of Boolean Functions Dept. of CSE, IEM, Kolkata 1 Simplification of Boolean Functions: An implementation of a Boolean Function requires the use of logic gates. A smaller number of gates,
More informationKarnaugh Maps Objectives
Karnaugh Maps Objectives For Karnaugh Maps of up to 5 variables Plot a function from algebraic, minterm or maxterm form Obtain minimum Sum of Products and Product of Sums Understand the relationship between
More informationBOOLEAN ALGEBRA TRUTH TABLE
BOOLEAN ALGEBRA TRUTH TABLE Truth table is a table which represents all the possible values of logical variables / statements along with all the possible results of the given combinations of values. Eg:
More informationELC224C. Karnaugh Maps
KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to K-maps How to use K-maps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on
More informationChap 2. Combinational Logic Circuits
Overview 2 Chap 2. Combinational Logic Circuits Spring 24 Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization Two-Level Optimization
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an
More informationGate-Level Minimization
Gate-Level Minimization Dr. Bassem A. Abdullah Computer and Systems Department Lectures Prepared by Dr.Mona Safar, Edited and Lectured by Dr.Bassem A. Abdullah Outline 1. The Map Method 2. Four-variable
More informationChapter 2 Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Huntington Postulates 1. (a) Closure w.r.t. +. (b) Closure w.r.t.. 2. (a) Identity element 0 w.r.t. +. x + 0 = 0 + x = x. (b) Identity element 1 w.r.t.. x 1 =
More informationCh 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1
Ch 2. Combinational Logic II - Combinational Logic Contemporary Logic Design 1 Combinational logic Define The kind of digital system whose output behavior depends only on the current inputs memoryless:
More informationChapter 2. Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Basic Definitions A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common
More informationDiscrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST
Discrete Mathematics CS204: Spring, 2008 Jong C. Park Computer Science Department KAIST Today s Topics Combinatorial Circuits Properties of Combinatorial Circuits Boolean Algebras Boolean Functions and
More informationLogic Gate Level. Part 2
Logic Gate Level Part 2 Constructing Boolean expression from First method: write nonparenthesized OR of ANDs Each AND is a 1 in the result column of the truth table Works best for table with relatively
More informationSlides for Lecture 10
Slides for Lecture 10 ENEL 353: Digital Circuits Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 30 September, 2013 ENEL 353
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal
More informationSignals and Systems Digital Logic System
Signals and Systems Digital Logic System Prof. Wonhee Kim Chapter 2 Design Process for Combinational Systems Step 1: Represent each of the inputs and outputs in binary Step 1.5: If necessary, break the
More informationStandard Expression Forms
ThisLecture will cover the following points: Canonical and Standard Forms MinTerms and MaxTerms Digital Logic Families 24 March 2010 Standard Expression Forms Two standard (canonical) expression forms
More informationChapter 2: Princess Sumaya Univ. Computer Engineering Dept.
hapter 2: Princess Sumaya Univ. omputer Engineering Dept. Basic Definitions Binary Operators AND z = x y = x y z=1 if x=1 AND y=1 OR z = x + y z=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 Boolean Algebra
More informationCprE 281: Digital Logic
CprE 281: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Boolean Algebra CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev
More information211: Computer Architecture Summer 2016
211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic - Storage: Recap - Review: cache hit rate - Project3 - Digital Logic: - truth table => SOP - simplification: Boolean
More informationEECS Variable Logic Functions
EECS150 Section 1 Introduction to Combinational Logic Fall 2001 2-Variable Logic Functions There are 16 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs X
More informationIntroduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps
Introduction to Digital Logic Missouri S&T University CPE 2210 Karnaugh Maps Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology
More informationComputer Organization I
Computer Organization I Lecture 6: Boolean Algebra /2/29 Wei Lu CS283 Overview Two Principles in Boolean Algebra () Duality Principle (2) Complement Principle Standard Form of Logic Expression () Sum of
More informationMC9211 Computer Organization
MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (29-2/ODD) (29 - / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra
More informationEECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits)
EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4-bool1 Page 1, 9/5 9am Outline Review of
More informationL2: Combinational Logic Design (Construction and Boolean Algebra)
L2: Combinational Logic Design (Construction and Boolean Algebra) Acknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. Borriello, Contemporary Logic Design (second edition), Pearson
More informationBinary Logic and Gates
1 COE 202- Digital Logic Binary Logic and Gates Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Introduction Boolean Algebra Elements of Boolean Algebra (Binary Logic) Logic Operations & Logic
More informationOutline. EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined
EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) January 30, 2003 John Wawrzynek Outline Review of three representations for combinational logic:
More informationMinimization techniques
Pune Vidyarthi Griha s COLLEGE OF ENGINEERING, NSIK - 4 Minimization techniques By Prof. nand N. Gharu ssistant Professor Computer Department Combinational Logic Circuits Introduction Standard representation
More informationChapter 2 Combinational logic
Chapter 2 Combinational logic Chapter 2 is very easy. I presume you already took discrete mathemtics. The major part of chapter 2 is boolean algebra. II - Combinational Logic Copyright 24, Gaetano Borriello
More informationCS/EE 181a 2010/11 Lecture 4
CS/EE 181a 21/11 Lecture 4 General topic of today s lecture: Logic Optimization Karnaugh maps. Quine-McCluskey tabulation method (not in detail). Non series-parallel networks (some care is required). Reference
More informationBOOLEAN ALGEBRA CLASS XII. Presented By : Dinesh Patel PGT CS KV IIT Powai
BOOLEAN ALGEBRA CLASS II Presented By : Dinesh Patel PGT CS KV IIT Powai Introduction Boolean Algebra is a set of rules and regulation which is suitable for Digital Circuits, whose answer is either True
More information2 Application of Boolean Algebra Theorems (15 Points - graded for completion only)
CSE140 HW1 Solution (100 Points) 1 Introduction The purpose of this assignment is three-fold. First, it aims to help you practice the application of Boolean Algebra theorems to transform and reduce Boolean
More informationLogic Design I (17.341) Fall Lecture Outline
Logic Design I (17.341) Fall 2011 Lecture Outline Class # 06 October 24, 2011 Dohn Bowden 1 Today s Lecture Administrative Main Logic Topic Homework 2 Course Admin 3 Administrative Admin for tonight Syllabus
More informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines Basic
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2
More informationEC-121 Digital Logic Design
EC-121 Digital Logic Design Lecture 2 [Updated on 02-04-18] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter
More informationDIGITAL ELECTRONICS & it0203 Semester 3
DIGITAL ELECTRONICS & it0203 Semester 3 P.Rajasekar & C.M.T.Karthigeyan Asst.Professor SRM University, Kattankulathur School of Computing, Department of IT 8/22/2011 1 Disclaimer The contents of the slides
More informationAdvanced Digital Design with the Verilog HDL, Second Edition Michael D. Ciletti Prentice Hall, Pearson Education, 2011
Problem 2-1 Recall that a minterm is a cube in which every variable appears. A Boolean expression in SOP form is canonical if every cube in the expression has a unique representation in which all of the
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 1. Developed by George Boole in 1847 2. Applied to the Design of Switching Circuit by Claude Shannon in 1939 Department of Communication Engineering, NCTU 1 2.1 Basic Operations
More informationCombinational Logic Circuits Part II -Theoretical Foundations
Combinational Logic Circuits Part II -Theoretical Foundations Overview Boolean Algebra Basic Logic Operations Basic Identities Basic Principles, Properties, and Theorems Boolean Function and Representations
More informationCombinational Logic. Fan-in/ Fan-out Timing. Copyright (c) 2012 Sean Key
Combinational Logic Fan-in/ Fan-out Timing Copyright (c) 2012 Sean Key Fan-in & Fan-out Fan-in The number of inputs to a logic gate Higher fan-in can lead to longer gate delays because of the higher input
More informationEEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系. September 2004
EEA051 - Digital Logic 數位邏輯 吳俊興高雄大學資訊工程學系 September 2004 Boolean Algebra (formulated by E.V. Huntington, 1904) A set of elements B={0,1} and two binary operators + and Huntington postulates 1. Closure
More informationLOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D.
Basic Eperiment and Design of Electronics LOGIC GATES Ho Kyung Kim, Ph.D. hokyung@pusan.ac.kr School of Mechanical Engineering Pusan National University Outline Boolean algebra Logic gates Karnaugh maps
More informationChapter 2 Combinational
Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations HOANG Trang Reference: 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean
More informationBoolean Algebra and Digital Logic
All modern digital computers are dependent on circuits that implement Boolean functions. We shall discuss two classes of such circuits: Combinational and Sequential. The difference between the two types
More informationDIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA
DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA 1 Learning Objectives Understand the basic operations and laws of Boolean algebra. Relate these operations and laws to circuits composed of AND gates, OR gates, INVERTERS
More information