LOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D.

Size: px
Start display at page:

Download "LOGIC GATES. Basic Experiment and Design of Electronics. Ho Kyung Kim, Ph.D."

Transcription

1 Basic Eperiment and Design of Electronics LOGIC GATES Ho Kyung Kim, Ph.D. School of Mechanical Engineering Pusan National University

2 Outline Boolean algebra Logic gates Karnaugh maps 2

3 Analog and digital signals Analog signal An electric signal whose value varies in analogy with a physical quantity (e.g., temp., force, or acceleration) Sensitive to noise Digital signal Immune to noise 3

4 Binary signal Characterized by transitions between two states (f /f, on/off, 5 V/ V ) Knowledge of the transition between one state to another is equivalent to knowledge of the state Digital logic circuits can operate by detecting transitions (edges) between voltage levels 4

5 Binary number system bits = binary digits LSB (least significant bit), MSB (most significant bit) 8 bits = byte 6 bits = 2 bytes = word 3.25 = = = = =? =? 2 MSB LSB MSB LSB 5

6 Addition Subtraction Multiplication Division 6

7 Negative binary numbers +5, -5 2 n N +(2 n ) for n-bit signed integer words X + Y instead of X Y in digital computers Complements Ones complement a = ones complement of a = 2 4 a = Twos complement (= ones complement + ) a = ones complement of a =2 4 a = (= + ) Headecimal system,, 2,, 9, A (= ), B (= ), C (= 2), D (= 3), E (= 4), (= 5) B5 6 = = 26 7

8 E) Perform the following subtractions using the twos complement arithmetic. X Y = 2. X Y = Sol.). X Y = X + Y = = + = Since X Y <, X Y = 2. X Y = X + Y = = + = Since X Y >, X Y = 8

9 Binary codes BCD (binary-coded decimal) 39 e.g., hand calculator Gray codes Any consecutive numbers differ only by bit Effective in encoding mechanical angular position BCD code Gray code 9

10 Encoders

11 Boolean algebra Mathematics associated with the binary number system (and with the more general field of logic) Logical algebra Positive logic Logic = true, logic = false Negative logic Logic = false, logic = true Analysis of logic functions, that is, functions of logical (Boolean) variables, can be carried out in terms of truth tables Logic gates Physical devices that can be used to implement logic functions

12 Logic gates: OR, AND, NOT Logical addition Logical multiplication Logical complement 2

13 V CC +V CC Input A B Output Input A B Output V BB RB R C V CE 3

14 E) Realize the following statement with the logic gates: The output Z shall be logic only when the condition (X = AND Y = ) OR (W = ) occurs, and shall be logic otherwise 4

15 E) Realize the following statement with the logic gates: The output Z shall be logic only when the condition (X = AND Y = ) OR (W = ) occurs, and shall be logic otherwise The output Z shall be logic only when the condition ( X = AND Y = ) OR (W = ) occurs, and shall be logic otherwise 5

16 Rules of Boolean algebra rom 2: A + B C + = rom 7: A + B A + C D = A + B C D Proof of rule 6 by perfect induction (truth table) = + = (duality) 6

17 De Morgan s theorems (X + Y) = X Y (X Y) = X + Y NOR NAND Any logic gates can be implemented by using only OR and NOT gates, or only AND and NOT gates 7

18 E) Simplify the following functions f A, B, C, D = A B D + A B D + B C D + A C D Sol.) f = A D ( B + B) + B C D + A C D f = A D + B C D + A C D f = ( A + A C) D + B C D f = ( A + C) D + B C D f = A D + C D + B C D f = A D + ( + B) C D f = A D + C D f = ( A + C) D Rule 4 Rule 8 Rule 2 8

19 E) Realize the logic function described by the truth table below Sol.) y = A B C + A B C + A B C + A B C + A B C + A B C y = A C B + B + A B C + C + A B ( C + C) y = A C + A B + A B y = A C + A ( B + B) y = A C + A y = A + C 9

20 Logic gates: NAND, NOR, XOR NAND = NOT AND NOR = NOT OR Just regard this bubble as the NOT gate 2

21 AND function with NAND gates 2

22 Realization of various logic gates with only NAND gates Inverter AND OR NOR 22

23 AND function with NOR gates f = f = A B = A + B 23

24 Realization of various logic gates with only NOR gates Inverter AND OR NOR 24

25 XOR (eclusive OR) gate When its inputs are all logic s, the output is eclusively a logic ; otherwise, identical to the OR gate Z = X Y = (X + Y) (X Y) 25

26 Summary NOT 또는 inverter = ' 입력이반전되어출력 게이트이름회로기호논리식진리표 비고 buffer = 게이트이름회로기호논리식진리표 입력이그대로출력 비고 AND y = y y 입력이모두 일때출력이 NAND y = = (y)' y 입력이모두 일때에만때출력이 OR y = +y y 입력중하나라도 이면출력이 NOR y = = (+y)' y 입력중입력이하나라도모두 일때에만 이면출력이 NOT 또는 inverter = ' 입력이반전되어출력 NOT XOR 또는 (eclusive-or) inverter y = y = ' = 'y + y' y 입력이입력에반전되어 이홀수개일때출력출력이 buffer = 입력이그대로출력 XNOR (eclusive-nor) buffer 또는 equivalence y = ( y)' = = 'y'+y y 입력이입력에그대로 이짝수출개일때력출력이 NAND y = (y)' y 입력이모두 일때에만출력이 NAND y = (y)' y 입력이모두 일때에만출력이 NOR y = (+y)' y 입력이모두 일때에만출력이 NOR y = (+y)' y 입력이 26모두 일때에만출력이

27 Standard forms Sum-of-products epression Product-of-sums epression Any logical epression can be reduced to one of these two forms! 27

28 Sum-of-products OR of minterms = X Y Z + XY Z + X YZ + XY Z Products-of-sums AND of materms = (X + Y + Z)(X + Y + Z)( X + Y + Z)( X + Y + Z) X Y Z Minterm Materm X Y Z XY Z X YZ XY Z X + Y + Z X + Y + Z X + Y + Z X + Y + Z 28

29 Karnaugh maps and logic designs Karnaugh map Describing all possible combinations of the N variables present in the logic function of interest Arranging variables in a -bit change between adjacent terms 2 N cells Minterm in each cell AND-combination of the N variables in either uncomplemented or complemented form Product of the variables appearing at the corresponding vertical and horizontal coordinates Define a subcube with logical value cell 2 cells = pair 4 cells = quad 8 cells = octet 29

30 cell f = W X Y Z + W X Y Z + W X Y Z + W X Y Z + W X Y Z + W X Y Z+W X Y Z 2 cells f = W X Y + W Y Z + X Y Z + W X Y Z + W X Y Z+W X Y Z 3

31 4 cells f = W X + W Z + X Z 8 cells f = Z 3

32 E) Simplify the following logic circuit X Y Z YZ X X Z f = X + Y Z + Y Z = X + Y + Y Z = X+ Z 32

33 Sum-of-products realizations E) Design a logic circuit that implements the following truth table y = A B D + B C + C D + A D 33

34 E) Derive the truth table and minimum sum-of-products epression for the following circuit f = y + y z 34

35 Product-of-sums realization. Solve for the s eactly as for the s in sum-of-products epressions 2. Complement the resulting epression z Product-of-sums f = y z + y = y z y f = ( + y + z) ( + y) Sum-of-products f = y + y z + y 35

36 Don t care conditions Use the don t care entry with whenever it does not matter whether a position in the K- map is filled by a or a Then, can be used as either a or a, depending on which results in a greater simplification (i.e., helps in forming the smallest number of maimal subcubes) If = in the right truth table; f = B D + B C + A C D 36

Digital Logic (2) Boolean Algebra

Digital Logic (2) Boolean Algebra Digital Logic (2) Boolean Algebra Boolean algebra is the mathematics of digital systems. It was developed in 1850 s by George Boole. We will use Boolean algebra to minimize logic expressions. Karnaugh

More information

Lecture 2 Review on Digital Logic (Part 1)

Lecture 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 information

Lecture 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 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 information

E&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 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 information

Unit 2 Session - 6 Combinational Logic Circuits

Unit 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 information

Combinational Logic. Review of Combinational Logic 1

Combinational 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 information

Boolean Algebra & Logic Gates. By : Ali Mustafa

Boolean 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 information

XI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL.

XI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL. 2017-18 XI STANDARD [ COMPUTER SCIENCE ] 5 MARKS STUDY MATERIAL HALF ADDER 1. The circuit that performs addition within the Arithmetic and Logic Unit of the CPU are called adders. 2. A unit that adds two

More information

Chapter 7 Logic Circuits

Chapter 7 Logic Circuits Chapter 7 Logic Circuits Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary

More information

E&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates

E&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 information

DIGITAL LOGIC CIRCUITS

DIGITAL LOGIC CIRCUITS DIGITAL LOGIC CIRCUITS Introduction Logic Gates Boolean Algebra Map Specification Combinational Circuits Flip-Flops Sequential Circuits Memory Components Integrated Circuits Digital Computers 2 LOGIC GATES

More information

Logic Design. Chapter 2: Introduction to Logic Circuits

Logic 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 information

ENG2410 Digital Design Combinational Logic Circuits

ENG2410 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 information

Systems I: Computer Organization and Architecture

Systems I: Computer Organization and Architecture Systems I: Computer Organization and Architecture Lecture 6 - Combinational Logic Introduction A combinational circuit consists of input variables, logic gates, and output variables. The logic gates accept

More information

Learning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr.

Learning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr. /7/ CE 4 Digital ystem Design Dr. Arshad Aziz Fundamental of ogic Design earning Objectives Review the basic concepts of logic circuits Variables and functions Boolean algebra Minterms and materms ogic

More information

Signals and Systems Digital Logic System

Signals 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 information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: 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 4 Following the slides of Dr. Ahmed H. Madian محرم 439 ه Winter 28

More information

Review. EECS Components and Design Techniques for Digital Systems. Lec 06 Minimizing Boolean Logic 9/ Review: Canonical Forms

Review. EECS Components and Design Techniques for Digital Systems. Lec 06 Minimizing Boolean Logic 9/ Review: Canonical Forms Review EECS 150 - Components and Design Techniques for Digital Systems Lec 06 Minimizing Boolean Logic 9/16-04 David Culler Electrical Engineering and Computer Sciences University of California, Berkeley

More information

CS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman

CS 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 information

Number System conversions

Number 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 information

Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions

Lecture 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 information

Ch 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1

Ch 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 information

Textbook: Digital Design, 3 rd. Edition M. Morris Mano

Textbook: Digital Design, 3 rd. Edition M. Morris Mano : 25/5/ P-/70 Tetbook: Digital Design, 3 rd. Edition M. Morris Mano Prentice-Hall, Inc. : INSTRUCTOR : CHING-LUNG SU E-mail: kevinsu@yuntech.edu.tw Chapter 3 25/5/ P-2/70 Chapter 3 Gate-Level Minimization

More information

Cs302 Quiz for MID TERM Exam Solved

Cs302 Quiz for MID TERM Exam Solved Question # 1 of 10 ( Start time: 01:30:33 PM ) Total Marks: 1 Caveman used a number system that has distinct shapes: 4 5 6 7 Question # 2 of 10 ( Start time: 01:31:25 PM ) Total Marks: 1 TTL based devices

More information

ECE 545 Digital System Design with VHDL Lecture 1. Digital Logic Refresher Part A Combinational Logic Building Blocks

ECE 545 Digital System Design with VHDL Lecture 1. Digital Logic Refresher Part A Combinational Logic Building Blocks ECE 545 Digital System Design with VHDL Lecture Digital Logic Refresher Part A Combinational Logic Building Blocks Lecture Roadmap Combinational Logic Basic Logic Review Basic Gates De Morgan s Law Combinational

More information

CHAPTER1: Digital Logic Circuits Combination Circuits

CHAPTER1: 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 information

Combinational Logic Fundamentals

Combinational Logic Fundamentals 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

More information

Combinatorial Logic Design Principles

Combinatorial 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 information

Digital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit

Digital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit Digital Techniques 1. Binary System The digital computer is the best example of a digital system. A main characteristic of digital system is its ability to manipulate discrete elements of information.

More information

Chapter 2. Boolean Algebra and Logic Gates

Chapter 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 information

Chapter 2: Princess Sumaya Univ. Computer Engineering Dept.

Chapter 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 information

Possible logic functions of two variables

Possible logic functions of two variables ombinational logic asic logic oolean algebra, proofs by re-writing, proofs by perfect induction logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,..., minimal set Logic realization

More information

Combinational Logic Design Principles

Combinational 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 information

COMBINATIONAL LOGIC FUNCTIONS

COMBINATIONAL LOGIC FUNCTIONS COMBINATIONAL LOGIC FUNCTIONS Digital logic circuits can be classified as either combinational or sequential circuits. A combinational circuit is one where the output at any time depends only on the present

More information

Fundamentals of Computer Systems

Fundamentals of Computer Systems Fundamentals of Computer Systems Boolean Logic Stephen A. Edwards Columbia University Summer 2015 Boolean Logic George Boole 1815 1864 Boole s Intuition Behind Boolean Logic Variables X,,... represent

More information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: 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 information

XOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure.

XOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure. XOR - XNOR Gates Lesson Objectives: In addition to AND, OR, NOT, NAND and NOR gates, exclusive-or (XOR) and exclusive-nor (XNOR) gates are also used in the design of digital circuits. These have special

More information

Logic and Boolean algebra

Logic and Boolean algebra Computer Mathematics Week 7 Logic and Boolean algebra College of Information Science and Engineering Ritsumeikan University last week coding theory channel coding information theory concept Hamming distance

More information

Minimization techniques

Minimization 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 information

Chapter 2: Switching Algebra and Logic Circuits

Chapter 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 information

EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive

EECS150 - 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 information

Binary logic consists of binary variables and logical operations. The variables are

Binary 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 information

Part 5: Digital Circuits

Part 5: Digital Circuits Characteristics of any number system are: Part 5: Digital Circuits 5.: Number Systems & Code Conversions. ase or radix is equal to the number of possible symbols in the system 2. The largest value of digit

More information

University of Technology

University of Technology University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year معالجات دقيقة المرحلة الرابعة ھندسة الليزر والبصريات االلكترونية Lecture 3 & 4 Boolean Algebra and Logic Gates

More information

DIGITAL LOGIC CIRCUITS

DIGITAL LOGIC CIRCUITS DIGITAL LOGIC CIRCUITS Introduction Logic Gates Boolean Algebra Map Specification Combinational Circuits Flip-Flops Sequential Circuits Memor Components Integrated Circuits BASIC LOGIC BLOCK - GATE - Logic

More information

Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps

Lecture 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 information

II. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):

II. 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 information

Chapter 2. Digital Logic Basics

Chapter 2. Digital Logic Basics Chapter 2 Digital Logic Basics 1 2 Chapter 2 2 1 Implementation using NND gates: We can write the XOR logical expression B + B using double negation as B+ B = B+B = B B From this logical expression, we

More information

Total Time = 90 Minutes, Total Marks = 100. Total /10 /25 /20 /10 /15 /20

Total Time = 90 Minutes, Total Marks = 100. Total /10 /25 /20 /10 /15 /20 University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Midterm Examination Instructor: M. Sachdev October 30th, 2006 Total Time = 90 Minutes, Total

More information

WORKBOOK. Try Yourself Questions. Electrical Engineering Digital Electronics. Detailed Explanations of

WORKBOOK. Try Yourself Questions. Electrical Engineering Digital Electronics. Detailed Explanations of 27 WORKBOOK Detailed Eplanations of Try Yourself Questions Electrical Engineering Digital Electronics Number Systems and Codes T : Solution Converting into decimal number system 2 + 3 + 5 + 8 2 + 4 8 +

More information

CPE100: Digital Logic Design I

CPE100: Digital Logic Design I Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu CPE100: Digital Logic Design I Midterm01 Review http://www.ee.unlv.edu/~b1morris/cpe100/ 2 Logistics Thursday Oct. 5 th In normal lecture (13:00-14:15)

More information

Fundamentals of Computer Systems

Fundamentals of Computer Systems Fundamentals of Computer Systems Boolean Logic Stephen A. Edwards Columbia University Fall 2011 Boolean Logic George Boole 1815 1864 Boole s Intuition Behind Boolean Logic Variables x, y,... represent

More information

Midterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes

Midterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes Page 1 of 10 School of Computer Science 60-265-01 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name

More information

ECEN 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 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 information

Chapter 2: Boolean Algebra and Logic Gates

Chapter 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 information

Combinational Logic. By : Ali Mustafa

Combinational Logic. By : Ali Mustafa Combinational Logic By : Ali Mustafa Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer How to Analyze any combinational circuit like this? Analysis Procedure To obtain the output

More information

MODULAR CIRCUITS CHAPTER 7

MODULAR CIRCUITS CHAPTER 7 CHAPTER 7 MODULAR CIRCUITS A modular circuit is a digital circuit that performs a specific function or has certain usage. The modular circuits to be introduced in this chapter are decoders, encoders, multiplexers,

More information

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION

More information

L2: Combinational Logic Design (Construction and Boolean Algebra)

L2: 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 information

Propositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173

Propositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173 Propositional Logic CSC 17 Propositional logic mathematical model (or algebra) for reasoning about the truth of logical expressions (propositions) Logical expressions propositional variables or logical

More information

for Digital Systems Simplification of logic functions Tajana Simunic Rosing Sources: TSR, Katz, Boriello & Vahid

for Digital Systems Simplification of logic functions Tajana Simunic Rosing Sources: TSR, Katz, Boriello & Vahid SE140: omponents and Design Techniques for Digital Systems Simplification of logic functions Tajana Simunic Rosing 1 What we covered thus far: Number representations Where we are now inary, Octal, Hex,

More information

Gate-Level Minimization

Gate-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 information

Why digital? Overview. Number Systems. Binary to Decimal conversion

Why digital? Overview. Number Systems. Binary to Decimal conversion Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not

More information

Department of Electrical & Electronics EE-333 DIGITAL SYSTEMS

Department of Electrical & Electronics EE-333 DIGITAL SYSTEMS Department of Electrical & Electronics EE-333 DIGITAL SYSTEMS 1) Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X -Y and (b) Y - X using 2's complements. a) X = 1010100

More information

UNIVERSITI TENAGA NASIONAL. College of Information Technology

UNIVERSITI TENAGA NASIONAL. College of Information Technology UNIVERSITI TENAGA NASIONAL College of Information Technology BACHELOR OF COMPUTER SCIENCE (HONS.) FINAL EXAMINATION SEMESTER 2 2012/2013 DIGITAL SYSTEMS DESIGN (CSNB163) January 2013 Time allowed: 3 hours

More information

Chapter 2 Boolean Algebra and Logic Gates

Chapter 2 Boolean Algebra and Logic Gates Chapter 2 Boolean Algebra and Logic Gates The most common postulates used to formulate various algebraic structures are: 1. Closure. N={1,2,3,4 }, for any a,b N we obtain a unique c N by the operation

More information

CHAPTER 7. Exercises 17/ / /2 2 0

CHAPTER 7. Exercises 17/ / /2 2 0 CHAPTER 7 Exercises E7. (a) For the whole part, we have: Quotient Remainders 23/2 /2 5 5/2 2 2/2 0 /2 0 Reading the remainders in reverse order, we obtain: 23 0 = 0 2 For the fractional part we have 2

More information

Working with Combinational Logic. Design example: 2x2-bit multiplier

Working with Combinational Logic. Design example: 2x2-bit multiplier Working with ombinational Logic Simplification two-level simplification exploiting don t cares algorithm for simplification Logic realization two-level logic and canonical forms realized with NNs and NORs

More information

Part 1: Digital Logic and Gates. Analog vs. Digital waveforms. The digital advantage. In real life...

Part 1: Digital Logic and Gates. Analog vs. Digital waveforms. The digital advantage. In real life... Part 1: Digital Logic and Gates Analog vs Digital waveforms An analog signal assumes a continuous range of values: v(t) ANALOG A digital signal assumes discrete (isolated, separate) values Usually there

More information

Digital Logic Design. Combinational Logic

Digital 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 information

10/14/2009. Reading: Hambley Chapters

10/14/2009. Reading: Hambley Chapters EE40 Lec 14 Digital Signal and Boolean Algebra Prof. Nathan Cheung 10/14/2009 Reading: Hambley Chapters 7.1-7.4 7.4 Slide 1 Analog Signals Analog: signal amplitude is continuous with time. Amplitude Modulated

More information

Chapter 5. Digital systems. 5.1 Boolean algebra Negation, conjunction and disjunction

Chapter 5. Digital systems. 5.1 Boolean algebra Negation, conjunction and disjunction Chapter 5 igital systems digital system is any machine that processes information encoded in the form of digits. Modern digital systems use binary digits, encoded as voltage levels. Two voltage levels,

More information

MC9211 Computer Organization

MC9211 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 information

Chapter 2 Combinational logic

Chapter 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 information

EECS Variable Logic Functions

EECS 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 information

Review for Test 1 : Ch1 5

Review for Test 1 : Ch1 5 Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8

More information

Logic Gate Level. Part 2

Logic 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 information

Logic Simplification. Boolean Simplification Example. Applying Boolean Identities F = A B C + A B C + A BC + ABC. Karnaugh Maps 2/10/2009 COMP370 1

Logic Simplification. Boolean Simplification Example. Applying Boolean Identities F = A B C + A B C + A BC + ABC. Karnaugh Maps 2/10/2009 COMP370 1 Digital Logic COMP370 Introduction to Computer Architecture Logic Simplification It is frequently possible to simplify a logical expression. This makes it easier to understand and requires fewer gates

More information

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.

Ex: 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 information

Digital Logic. Lecture 5 - Chapter 2. Outline. Other Logic Gates and their uses. Other Logic Operations. CS 2420 Husain Gholoom - lecturer Page 1

Digital Logic. Lecture 5 - Chapter 2. Outline. Other Logic Gates and their uses. Other Logic Operations. CS 2420 Husain Gholoom - lecturer Page 1 Lecture 5 - Chapter 2 Outline Other Logic Gates and their uses Other Logic Operations CS 2420 Husain Gholoom - lecturer Page 1 Digital logic gates CS 2420 Husain Gholoom - lecturer Page 2 Buffer A buffer

More information

Logic. Combinational. inputs. outputs. the result. system can

Logic. Combinational. inputs. outputs. the result. system can Digital Electronics Combinational Logic Functions Digital logic circuits can be classified as either combinational or sequential circuits. A combinational circuit is one where the output at any time depends

More information

Karnaugh Maps Objectives

Karnaugh 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 information

ECE 545 Digital System Design with VHDL Lecture 1A. Digital Logic Refresher Part A Combinational Logic Building Blocks

ECE 545 Digital System Design with VHDL Lecture 1A. Digital Logic Refresher Part A Combinational Logic Building Blocks ECE 545 Digital System Design with VHDL Lecture A Digital Logic Refresher Part A Combinational Logic Building Blocks Lecture Roadmap Combinational Logic Basic Logic Review Basic Gates De Morgan s Laws

More information

/ M Morris Mano Digital Design Ahmad_911@hotmailcom / / / / wwwuqucscom Binary Systems Introduction - Digital Systems - The Conversion Between Numbering Systems - From Binary To Decimal - Octet To Decimal

More information

Chap 2. Combinational Logic Circuits

Chap 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 information

EC-121 Digital Logic Design

EC-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 information

Digital Circuit And Logic Design I. Lecture 4

Digital 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 information

Review: Additional Boolean operations

Review: Additional Boolean operations Review: Additional Boolean operations Operation: NAND (NOT-AND) NOR (NOT-OR) XOR (exclusive OR) Expressions: (xy) = x + y (x + y) = x y x y = x y + xy Truth table: x y (xy) x y (x+y) x y x y 0 0 1 0 1

More information

Boolean Algebra, Gates and Circuits

Boolean Algebra, Gates and Circuits Boolean Algebra, Gates and Circuits Kasper Brink November 21, 2017 (Images taken from Tanenbaum, Structured Computer Organization, Fifth Edition, (c) 2006 Pearson Education, Inc.) Outline Last week: Von

More information

Combinational Logic. Mantıksal Tasarım BBM231. section instructor: Ufuk Çelikcan

Combinational Logic. Mantıksal Tasarım BBM231. section instructor: Ufuk Çelikcan Combinational Logic Mantıksal Tasarım BBM23 section instructor: Ufuk Çelikcan Classification. Combinational no memory outputs depends on only the present inputs expressed by Boolean functions 2. Sequential

More information

Circuits & Boolean algebra.

Circuits & Boolean algebra. Circuits & Boolean algebra http://xkcd.com/730/ CSCI 255: Introduction to Embedded Systems Keith Vertanen Copyright 2011 Digital circuits Overview How a switch works Building basic gates from switches

More information

Chapter 2 Boolean Algebra and Logic Gates

Chapter 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 information

Z = F(X) Combinational circuit. A combinational circuit can be specified either by a truth table. Truth Table

Z = F(X) Combinational circuit. A combinational circuit can be specified either by a truth table. Truth Table Lesson Objectives In this lesson, you will learn about What are combinational circuits Design procedure of combinational circuits Examples of combinational circuit design Combinational Circuits Logic circuit

More information

Combinational Logic. Course Instructor Mohammed Abdul kader

Combinational Logic. Course Instructor Mohammed Abdul kader Combinational Logic Contents: Combinational and Sequential digital circuits. Design Procedure of combinational circuit. Adders: Half adder and Full adder. Subtractors: Half Subtractor and Full Subtractor.

More information

UNIT 1. BOOLEAN ALGEBRA AND COMBINATIONAL CIRCUITS

UNIT 1. BOOLEAN ALGEBRA AND COMBINATIONAL CIRCUITS UNIT 1. BOOLEAN ALGEBRA AND COMBINATIONAL CIRCUITS Numerical Presentation: In science, technology, business, and, in fact, most other fields of endeavour, we are constantly dealing with quantities. Quantities

More information

Karnaugh Map & Boolean Expression Simplification

Karnaugh Map & Boolean Expression Simplification Karnaugh Map & Boolean Expression Simplification Mapping a Standard POS Expression For a Standard POS expression, a 0 is placed in the cell corresponding to the product term (maxterm) present in the expression.

More information

The Design Procedure. Output Equation Determination - Derive output equations from the state table

The Design Procedure. Output Equation Determination - Derive output equations from the state table The Design Procedure Specification Formulation - Obtain a state diagram or state table State Assignment - Assign binary codes to the states Flip-Flop Input Equation Determination - Select flipflop types

More information

CHAPTER III BOOLEAN ALGEBRA

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 information

211: Computer Architecture Summer 2016

211: 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 information