Simplification of Boolean Functions

Size: px
Start display at page:

Download "Simplification of Boolean Functions"

Transcription

1 COM111 Introduction to Computer Engineering (Fall ) NOTES 5 -- page 1 of 5 Introduction Simplification of Boolean Functions You already know one method for simplifying Boolean expressions: Boolean algebra. There are two serious limitations to using Boolean algebra though: Two Level Form of a Boolean Expression The two level form of an expression refers to the number of subexpressions in the Boolean equation or the number of gates in the longest path through the gate implementation of the expression. Inverter gates don't count as a separate level. There is no algorithm you can follow that is guaranteed to lead to the simplest form of the expression Given any intermediate result there is no way to tell if it is in fact the simplest form of the expression In this lecture you will learn an algorithmic procedure for finding the simplest two level form of a Boolean expression. This method doesn't give you the simplest form in any number of levels. However, the simplest two level form is usually what we want because as you add more levels the expression may get smaller but the propagational delay will also increase. Propagational delay determines the speed of the circuit. Motivation In essence the method we are about to discuss is a pictorial way to apply the distributive law to factor out common subexpressions. For example: F = AB + AB' F = A(B + B') F = A Or, F = ABC + ABC' + AB'C + AB'C' F = A (BC + BC' + B'C + B'C') F = A(1) F = A Intuitively, if you can find two terms that are equivalent except that one variable is the complement of the matching variable in the other term you can factor out this variable. The second example above shows that it works for multiple variable subsets-- as long as they are powers of two. The two examples above make it look easy to remove subexpressions with Boolean algebra. For small examples it is easy. For larger expressions with 3 and more variables it because much harder. In this lecture we will study a method called Karnaugh maps you can use to quickly find the minimum standard form of a Boolean function of 4 or fewer variables. Later we will study an algorithmic method that works for functions of any number of variables. The karnaugh, or k-map, method is fast and best carried out by a human. The algorithm we will study later is tedious for humans but is easy to program using any high-level programming language.

2 COM111 Introduction to Computer Engineering (Fall ) NOTES 5 -- page 2 of 5 K-Maps A K-map shows the value of a function for every combination of input values just like a truth table, but a K-map spatially arranges the values so it is easy to find common terms that can be factored out. The image below shows the form of a 2,3 and 4 variable K-map. The numbers inside of the boxes above refer to the corresponding row in the truth table for a function of the same number of variables. (You may want to use the truth table to the right to compare K-map entries and truth table entries.) Notice how the columns are numbered on the 3 variable K-map and how both the rows and columns are numbered on the 4 variable K-map ( ). This numbering guarantees that adjacent terms differ by only one term. For example, in the 3 variable K-map above the square with a 2 in it represents the minterm A'BC' and the square with a 6 in it represents the minterm ABC'. If both of these terms appear as minterms in an expression we could factor out the A: A'BC' + ABC' = (A' + A)BC' = BC' Also notice that entries at the ends of the K-map differ by only one element with entries on the other side of the K-map. For example, Row A B C Minterm F A' B' C' A' B' C A' B C' A' B C A B' C' A B' C A B C' A B C 1 A'B'C' & AB'C' We start with a general example of how to use a K-map to simplify a function and then describe a more precise procedure. Example 1. Use a K-map to simplify the following Boolean function: F(A,B,C) = m(2,4,5,6,7) Since this is a function of 3 variables we first draw the outline for a 3-variable K-map. Since the function is given in terms of minterms we write the minterm number inside the box that represents that minterm. Next we put a 1 in each square where the function has the value 1. Finally, we circle groups of 1's so that all 1's are circled. We circle only groups that are powers of 2 and try to create circles as large as possible.

3 COM111 Introduction to Computer Engineering (Fall ) NOTES 5 -- page 3 of 5 F(A,B,C) = A + BC' The example above gives the general idea. What follows is a more formal description of how to proceed. The first step to understanding the formal procedure is understanding some terms. Implicant - a single minterm or group of minterms that can be combined together on the K-map. For example the implicants in the example on the right are A'B'C', A'BC', A'BC, ABC, A'C', A'B, and BC. Prime Implicant - Implicant that can not be combined with another one to remove a literal. For example: A'C', A'B, BC. Each product term in the minimum sum of products expression is a prime implicant. Essential Prime Implicant - A prime imlpicant that includes a minterm not covered by any other prime implicant. For example: A'C' and BC. Note, that with the example above if you're not careful you could end up with an expression with too many prime implicants. For example, if you choose A'B first you would then have to choose A'C' and BC to cover every implicant in the on-set. This observation is the motivation for the formal K-map procedure that follows. To derive the minimized expression from a K-map: 1. Draw the K-map and put a 1 in each square that corresponds to a minterm of the function. You can draw the K-map from a truth table, Boolean expression, etc. 2. Find the prime implicants. To do this find the largest adjacent groups of 1's. Groups must be "square" and the number of

4 COM111 Introduction to Computer Engineering (Fall ) NOTES 5 -- page 4 of 5 1's in a group must be a power of Find the essential prime implicants. An essential prime implicant is a prime implicant that includes a 1 not covered by any other prime implicants. 4. Write down the minimized expression. First write down all essential prime implicants. If there are any 1's not covered by prime implicants, carefully select prime implicants to cover the remaining 1's. Note, you may have to try several selections to find the minimal form of the expression. Example 2. Use a K-map to simplify the following Boolean function: M(0,1,2,4,9,11,15) m(3,5,6,7,8,10,12,13,14) BC'D + AD' + A'CD + BCD' Don't Cares Sometimes the output of a function for a specific input value is "don't care". For example, the output of a BCD increment by one circuit given the input 1010 would be XXXX, or don't care because the input (1010) 2 = (10) 10 would never be seen. Don't cares act like joker in a deck of playing cards--we can make them whatever we want. When you are simplifying a function using a K-map these don't care values help you to form larger groups of 1's which give you smaller prime implicants. Example 3. Use a K-map to simplify the following Boolean function: m(0,4,5,8,10,15) + d(2,7,9,13)

5 COM111 Introduction to Computer Engineering (Fall ) NOTES 5 -- page 5 of 5 C'A'B +DB + D'B'

Points Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 4: Logic Simplication & Karnaugh Map

Points Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 4: Logic Simplication & Karnaugh Map Points Addressed in this Lecture Lecture 4: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London Standard form of Boolean Expressions Sum-of-Products (SOP),

More information

Gate Level Minimization Map Method

Gate Level Minimization Map Method Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically

More information

A B AB CD Objectives:

A B AB CD Objectives: Objectives:. Four variables maps. 2. Simplification using prime implicants. 3. "on t care" conditions. 4. Summary.. Four variables Karnaugh maps Minterms A A m m m3 m2 A B C m4 C A B C m2 m8 C C m5 C m3

More information

Specifying logic functions

Specifying logic functions CSE4: Components and Design Techniques for Digital Systems Specifying logic functions Instructor: Mohsen Imani Slides from: Prof.Tajana Simunic and Dr.Pietro Mercati We have seen various concepts: Last

More information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard

More information

Chapter 2 Combinational

Chapter 2 Combinational Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization HOANG Trang 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean Equations Binary Logic

More information

Experiment 4 Boolean Functions Implementation

Experiment 4 Boolean Functions Implementation Experiment 4 Boolean Functions Implementation Introduction: Generally you will find that the basic logic functions AND, OR, NAND, NOR, and NOT are not sufficient to implement complex digital logic functions.

More information

Combinational Logic Circuits

Combinational Logic Circuits Chapter 3 Combinational Logic Circuits 12 Hours 24 Marks 3.1 Standard representation for logical functions Boolean expressions / logic expressions / logical functions are expressed in terms of logical

More information

Combinational Circuits Digital Logic (Materials taken primarily from:

Combinational Circuits Digital Logic (Materials taken primarily from: Combinational Circuits Digital Logic (Materials taken primarily from: http://www.facstaff.bucknell.edu/mastascu/elessonshtml/eeindex.html http://www.cs.princeton.edu/~cos126 ) Digital Systems What is a

More information

Chapter 3. Gate-Level Minimization. Outlines

Chapter 3. Gate-Level Minimization. Outlines Chapter 3 Gate-Level Minimization Introduction The Map Method Four-Variable Map Five-Variable Map Outlines Product of Sums Simplification Don t-care Conditions NAND and NOR Implementation Other Two-Level

More information

DKT 122/3 DIGITAL SYSTEM 1

DKT 122/3 DIGITAL SYSTEM 1 Company LOGO DKT 122/3 DIGITAL SYSTEM 1 BOOLEAN ALGEBRA (PART 2) Boolean Algebra Contents Boolean Operations & Expression Laws & Rules of Boolean algebra DeMorgan s Theorems Boolean analysis of logic circuits

More information

Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples

Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples Lecture B: Logic Minimization Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples Incompletely specified functions

More information

University of Technology

University of Technology University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year Lecture 5 & 6 Minimization with Karnaugh Maps Karnaugh maps lternate way of representing oolean function ll rows

More information

Ch. 5 : Boolean Algebra &

Ch. 5 : Boolean Algebra & Ch. 5 : Boolean Algebra & Reduction elektronik@fisika.ui.ac.id Objectives Should able to: Write Boolean equations for combinational logic applications. Utilize Boolean algebra laws and rules for simplifying

More information

Gate Level Minimization

Gate Level Minimization Gate Level Minimization By Dr. M. Hebaishy Digital Logic Design Ch- Simplifying Boolean Equations Example : Y = AB + AB Example 2: = B (A + A) T8 = B () T5 = B T Y = A(AB + ABC) = A (AB ( + C ) ) T8 =

More information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian ذو الحجة 1438 ه Winter

More information

Gate-Level Minimization

Gate-Level Minimization MEC520 디지털공학 Gate-Level Minimization Jee-Hwan Ryu School of Mechanical Engineering Gate-Level Minimization-The Map Method Truth table is unique Many different algebraic expression Boolean expressions may

More information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in View Show

More information

Chapter 2. Boolean Expressions:

Chapter 2. Boolean Expressions: Chapter 2 Boolean Expressions: A Boolean expression or a function is an expression which consists of binary variables joined by the Boolean connectives AND and OR along with NOT operation. Any Boolean

More information

Gate-Level Minimization

Gate-Level Minimization Gate-Level Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2011 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method

More information

Combinational Logic Circuits Part III -Theoretical Foundations

Combinational Logic Circuits Part III -Theoretical Foundations Combinational Logic Circuits Part III -Theoretical Foundations Overview Simplifying Boolean Functions Algebraic Manipulation Karnaugh Map Manipulation (simplifying functions of 2, 3, 4 variables) Systematic

More information

CSCI 220: Computer Architecture I Instructor: Pranava K. Jha. Simplification of Boolean Functions using a Karnaugh Map

CSCI 220: Computer Architecture I Instructor: Pranava K. Jha. Simplification of Boolean Functions using a Karnaugh Map CSCI 22: Computer Architecture I Instructor: Pranava K. Jha Simplification of Boolean Functions using a Karnaugh Map Q.. Plot the following Boolean function on a Karnaugh map: f(a, b, c, d) = m(, 2, 4,

More information

CMPE223/CMSE222 Digital Logic

CMPE223/CMSE222 Digital Logic CMPE223/CMSE222 Digital Logic Optimized Implementation of Logic Functions: Strategy for Minimization, Minimum Product-of-Sums Forms, Incompletely Specified Functions Terminology For a given term, each

More information

4 KARNAUGH MAP MINIMIZATION

4 KARNAUGH MAP MINIMIZATION 4 KARNAUGH MAP MINIMIZATION A Karnaugh map provides a systematic method for simplifying Boolean expressions and, if properly used, will produce the simplest SOP or POS expression possible, known as the

More information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian محرم 1439 ه Winter

More information

Gate-Level Minimization

Gate-Level Minimization Gate-Level Minimization ( 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 The Map Method

More information

Literal Cost F = BD + A B C + A C D F = BD + A B C + A BD + AB C F = (A + B)(A + D)(B + C + D )( B + C + D) L = 10

Literal Cost F = BD + A B C + A C D F = BD + A B C + A BD + AB C F = (A + B)(A + D)(B + C + D )( B + C + D) L = 10 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm

More information

CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey

CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey CHAPTER-2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, K-Map and Quine-McCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input

More information

Graduate Institute of Electronics Engineering, NTU. CH5 Karnaugh Maps. Lecturer: 吳安宇教授 Date:2006/10/20 ACCESS IC LAB

Graduate Institute of Electronics Engineering, NTU. CH5 Karnaugh Maps. Lecturer: 吳安宇教授 Date:2006/10/20 ACCESS IC LAB CH5 Karnaugh Maps Lecturer: 吳安宇教授 Date:2006/0/20 CCESS IC L Problems in lgebraic Simplification The procedures are difficult to apply in a systematic way. It is difficult to tell when you have arrived

More information

ENGIN 112 Intro to Electrical and Computer Engineering

ENGIN 112 Intro to Electrical and Computer Engineering ENGIN 2 Intro to Electrical and Computer Engineering Lecture 8 Minimization with Karnaugh Maps Overview K-maps: an alternate approach to representing oolean functions K-map representation can be used to

More information

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University Logic Design First Stage Lecture No.6 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Outlines Boolean Operations Laws of Boolean Algebra Rules of

More information

ECE380 Digital Logic

ECE380 Digital Logic ECE38 Digital Logic Optimized Implementation of Logic Functions: Strategy for Minimization, Minimum Product-of-Sums Forms, Incompletely Specified Functions Dr. D. J. Jackson Lecture 8- Terminology For

More information

CS470: Computer Architecture. AMD Quad Core

CS470: Computer Architecture. AMD Quad Core CS470: Computer Architecture Yashwant K. Malaiya, Professor malaiya@cs.colostate.edu AMD Quad Core 1 Architecture Layers Building blocks Gates, flip-flops Functional bocks: Combinational, Sequential Instruction

More information

A graphical method of simplifying logic

A graphical method of simplifying logic 4-5 Karnaugh Map Method A graphical method of simplifying logic equations or truth tables. Also called a K map. Theoretically can be used for any number of input variables, but practically limited to 5

More information

Date Performed: Marks Obtained: /10. Group Members (ID):. Experiment # 04. Boolean Expression Simplification and Implementation

Date Performed: Marks Obtained: /10. Group Members (ID):. Experiment # 04. Boolean Expression Simplification and Implementation Name: Instructor: Engr. Date Performed: Marks Obtained: /10 Group Members (ID):. Checked By: Date: Experiment # 04 Boolean Expression Simplification and Implementation OBJECTIVES: To understand the utilization

More information

Switching Circuits & Logic Design

Switching Circuits & Logic Design Switching Circuits & Logic Design Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 23 5 Karnaugh Maps K-map Walks and Gray Codes http://asicdigitaldesign.wordpress.com/28/9/26/k-maps-walks-and-gray-codes/

More information

Digital Logic Lecture 7 Gate Level Minimization

Digital Logic Lecture 7 Gate Level Minimization Digital Logic Lecture 7 Gate Level Minimization By Ghada Al-Mashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. K-map principles. Simplification using K-maps. Don t-care

More information

Experiment 3: Logic Simplification

Experiment 3: Logic Simplification Module: Logic Design Name:... University no:.. Group no:. Lab Partner Name: Mr. Mohamed El-Saied Experiment : Logic Simplification Objective: How to implement and verify the operation of the logical functions

More information

IT 201 Digital System Design Module II Notes

IT 201 Digital System Design Module II Notes IT 201 Digital System Design Module II Notes BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.

More information

UNIT II. Circuit minimization

UNIT II. Circuit minimization UNIT II Circuit minimization The complexity of the digital logic gates that implement a Boolean function is directly related to the complexity of the algebraic expression from which the function is implemented.

More information

Module -7. Karnaugh Maps

Module -7. Karnaugh Maps 1 Module -7 Karnaugh Maps 1. Introduction 2. Canonical and Standard forms 2.1 Minterms 2.2 Maxterms 2.3 Canonical Sum of Product or Sum-of-Minterms (SOM) 2.4 Canonical product of sum or Product-of-Maxterms(POM)

More information

1. Mark the correct statement(s)

1. Mark the correct statement(s) 1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another

More information

Chapter 6. Logic Design Optimization Chapter 6

Chapter 6. Logic Design Optimization Chapter 6 Chapter 6 Logic Design Optimization Chapter 6 Optimization The second part of our design process. Optimization criteria: Performance Size Power Two-level Optimization Manipulating a function until it is

More information

2.1 Binary Logic and Gates

2.1 Binary Logic and Gates 1 EED2003 Digital Design Presentation 2: Boolean Algebra Asst. Prof.Dr. Ahmet ÖZKURT Asst. Prof.Dr Hakkı T. YALAZAN Based on the Lecture Notes by Jaeyoung Choi choi@comp.ssu.ac.kr Fall 2000 2.1 Binary

More information

LSN 4 Boolean Algebra & Logic Simplification. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology

LSN 4 Boolean Algebra & Logic Simplification. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology LSN 4 Boolean Algebra & Logic Simplification Department of Engineering Technology LSN 4 Key Terms Variable: a symbol used to represent a logic quantity Compliment: the inverse of a variable Literal: a

More information

To write Boolean functions in their standard Min and Max terms format. To simplify Boolean expressions using Karnaugh Map.

To write Boolean functions in their standard Min and Max terms format. To simplify Boolean expressions using Karnaugh Map. 3.1 Objectives To write Boolean functions in their standard Min and Max terms format. To simplify Boolean expressions using. 3.2 Sum of Products & Product of Sums Any Boolean expression can be simplified

More information

Digital Logic Design. Outline

Digital Logic Design. Outline Digital Logic Design Gate-Level Minimization CSE32 Fall 2 Outline The Map Method 2,3,4 variable maps 5 and 6 variable maps (very briefly) Product of sums simplification Don t Care conditions NAND and NOR

More information

Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions

Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions 1. Convert the following SOP expression to an equivalent POS expression. 2. Determine the values of A, B, C, and D that make

More information

CS8803: Advanced Digital Design for Embedded Hardware

CS8803: Advanced Digital Design for Embedded Hardware CS883: Advanced Digital Design for Embedded Hardware Lecture 2: Boolean Algebra, Gate Network, and Combinational Blocks Instructor: Sung Kyu Lim (limsk@ece.gatech.edu) Website: http://users.ece.gatech.edu/limsk/course/cs883

More information

Simplification of Boolean Functions

Simplification of Boolean Functions Simplification of Boolean Functions Contents: Why simplification? The Map Method Two, Three, Four and Five variable Maps. Simplification of two, three, four and five variable Boolean function by Map method.

More information

CprE 281: Digital Logic

CprE 281: Digital Logic CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Minimization CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev Administrative

More information

2.6 BOOLEAN FUNCTIONS

2.6 BOOLEAN FUNCTIONS 2.6 BOOLEAN FUNCTIONS Binary variables have two values, either 0 or 1. A Boolean function is an expression formed with binary variables, the two binary operators AND and OR, one unary operator NOT, parentheses

More information

BOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits.

BOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits. COURSE / CODE DIGITAL SYSTEMS FUNDAMENTAL (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) BOOLEAN ALGEBRA Boolean Logic Boolean logic is a complete system for logical operations. It is used in countless

More information

Review: Standard forms of expressions

Review: Standard forms of expressions Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and NOT. These operations can be combined to form complex expressions, which can

More information

Slide Set 5. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary

Slide Set 5. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary Slide Set 5 for ENEL 353 Fall 207 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 207 SN s ENEL 353 Fall 207 Slide Set 5 slide

More information

ENGIN 112. Intro to Electrical and Computer Engineering

ENGIN 112. Intro to Electrical and Computer Engineering ENIN 2 Intro to Electrical and Computer Engineering Lecture 6 More Boolean Algebra ENIN2 L6: More Boolean Algebra September 5, 23 A B Overview Epressing Boolean functions Relationships between algebraic

More information

Outcomes. Unit 9. Logic Function Synthesis KARNAUGH MAPS. Implementing Combinational Functions with Karnaugh Maps

Outcomes. Unit 9. Logic Function Synthesis KARNAUGH MAPS. Implementing Combinational Functions with Karnaugh Maps .. Outcomes Unit I can use Karnaugh maps to synthesize combinational functions with several outputs I can determine the appropriate size and contents of a memory to implement any logic function (i.e. truth

More information

ece5745-pla-notes.txt

ece5745-pla-notes.txt ece5745-pla-notes.txt ========================================================================== Follow up on PAL/PROM/PLA Activity ==========================================================================

More information

5. Minimizing Circuits

5. Minimizing Circuits 5. MINIMIZING CIRCUITS 46 5. Minimizing Circuits 5.. Minimizing Circuits. A circuit is minimized if it is a sum-of-products that uses the least number of products of literals and each product contains

More information

Combinational Logic & Circuits

Combinational Logic & Circuits Week-I Combinational Logic & Circuits Spring' 232 - Logic Design Page Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other

More information

Karnaugh Map (K-Map) Karnaugh Map. Karnaugh Map Examples. Ch. 2.4 Ch. 2.5 Simplification using K-map

Karnaugh Map (K-Map) Karnaugh Map. Karnaugh Map Examples. Ch. 2.4 Ch. 2.5 Simplification using K-map Karnaugh Map (K-Map) Ch. 2.4 Ch. 2.5 Simplification using K-map A graphical map method to simplify Boolean function up to 6 variables A diagram made up of squares Each square represents one minterm (or

More information

Combinational Logic Circuits

Combinational Logic Circuits Chapter 2 Combinational Logic Circuits J.J. Shann (Slightly trimmed by C.P. Chung) Chapter Overview 2-1 Binary Logic and Gates 2-2 Boolean Algebra 2-3 Standard Forms 2-4 Two-Level Circuit Optimization

More information

Summary. Boolean Addition

Summary. Boolean Addition Summary Boolean Addition In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. A single variable can only have a value of or 0. The complement represents the inverse

More information

EEE130 Digital Electronics I Lecture #4_1

EEE130 Digital Electronics I Lecture #4_1 EEE130 Digital Electronics I Lecture #4_1 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi 4-6 Standard Forms of Boolean Expressions There are two standard forms: Sum-of-products form

More information

Unit-IV Boolean Algebra

Unit-IV Boolean Algebra Unit-IV Boolean Algebra Boolean Algebra Chapter: 08 Truth table: Truth table is a table, which represents all the possible values of logical variables/statements along with all the possible results of

More information

2. BOOLEAN ALGEBRA 2.1 INTRODUCTION

2. BOOLEAN ALGEBRA 2.1 INTRODUCTION 2. BOOLEAN ALGEBRA 2.1 INTRODUCTION In the previous chapter, we introduced binary numbers and binary arithmetic. As you saw in binary arithmetic and in the handling of floating-point numbers, there is

More information

3.4 QUINE MCCLUSKEY METHOD 73. f(a, B, C, D, E)¼AC ĒþB CD þ BCDþĀBD.

3.4 QUINE MCCLUSKEY METHOD 73. f(a, B, C, D, E)¼AC ĒþB CD þ BCDþĀBD. 3.4 QUINE MCCLUSKEY METHOD 73 FIGURE 3.22 f(a, B, C, D, E)¼B CD þ BCD. FIGURE 3.23 f(a, B, C, D, E)¼AC ĒþB CD þ BCDþĀBD. A¼1map are, 1, and 1, respectively, whereas the corresponding entries in the A¼0

More information

Mid-Term Exam Solutions

Mid-Term Exam Solutions CS/EE 26 Digital Computers: Organization and Logical Design Mid-Term Eam Solutions Jon Turner 3/3/3. (6 points) List all the minterms for the epression (B + A)C + AC + BC. Epanding the epression gives

More information

Chapter 3 Simplification of Boolean functions

Chapter 3 Simplification of Boolean functions 3.1 Introduction Chapter 3 Simplification of Boolean functions In this chapter, we are going to discuss several methods for simplifying the Boolean function. What is the need for simplifying the Boolean

More information

Boolean Function Simplification

Boolean Function Simplification Universit of Wisconsin - Madison ECE/Comp Sci 352 Digital Sstems Fundamentals Charles R. Kime Section Fall 200 Chapter 2 Combinational Logic Circuits Part 5 Charles Kime & Thomas Kaminski Boolean Function

More information

Boolean Analysis of Logic Circuits

Boolean Analysis of Logic Circuits Course: B.Sc. Applied Physical Science (Computer Science) Year & Sem.: IInd Year, Sem - IIIrd Subject: Computer Science Paper No.: IX Paper Title: Computer System Architecture Lecture No.: 7 Lecture Title:

More information

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University

Bawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University Logic Design First Stage Lecture No.5 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Boolean Operations Laws of Boolean Algebra Rules of Boolean Algebra

More information

Gate-Level Minimization. section instructor: Ufuk Çelikcan

Gate-Level Minimization. section instructor: Ufuk Çelikcan Gate-Level Minimization section instructor: Ufuk Çelikcan Compleity of Digital Circuits Directly related to the compleity of the algebraic epression we use to build the circuit. Truth table may lead to

More information

TWO-LEVEL COMBINATIONAL LOGIC

TWO-LEVEL COMBINATIONAL LOGIC TWO-LEVEL COMBINATIONAL LOGIC OVERVIEW Canonical forms To-level simplification Boolean cubes Karnaugh maps Quine-McClusky (Tabulation) Method Don't care terms Canonical and Standard Forms Minterms and

More information

SWITCHING THEORY AND LOGIC CIRCUITS

SWITCHING THEORY AND LOGIC CIRCUITS SWITCHING THEORY AND LOGIC CIRCUITS COURSE OBJECTIVES. To understand the concepts and techniques associated with the number systems and codes 2. To understand the simplification methods (Boolean algebra

More information

2008 The McGraw-Hill Companies, Inc. All rights reserved.

2008 The McGraw-Hill Companies, Inc. All rights reserved. 28 The McGraw-Hill Companies, Inc. All rights reserved. 28 The McGraw-Hill Companies, Inc. All rights reserved. All or Nothing Gate Boolean Expression: A B = Y Truth Table (ee next slide) or AB = Y 28

More information

Synthesis of combinational logic

Synthesis of combinational logic Page 1 of 14 Synthesis of combinational logic indicates problems that have been selected for discussion in section, time permitting. Problem 1. A certain function F has the following truth table: A B C

More information

(Refer Slide Time 6:48)

(Refer Slide Time 6:48) Digital Circuits and Systems Prof. S. Srinivasan Department of Electrical Engineering Indian Institute of Technology Madras Lecture - 8 Karnaugh Map Minimization using Maxterms We have been taking about

More information

Digital Techniques. Lecture 1. 1 st Class

Digital Techniques. Lecture 1. 1 st Class Digital Techniques Lecture 1 1 st Class Digital Techniques Digital Computer and Digital System: Digital computer is a part of digital system, it based on binary system. A block diagram of digital computer

More information

Presented By :- Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak

Presented By :- Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak Presented By :- Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak Content - Introduction -2 Feature -3 Feature of BJT -4 TTL -5 MOS -6 CMOS -7 K- Map - Introduction Logic IC ASIC: Application Specific

More information

B.Tech II Year I Semester (R13) Regular Examinations December 2014 DIGITAL LOGIC DESIGN

B.Tech II Year I Semester (R13) Regular Examinations December 2014 DIGITAL LOGIC DESIGN B.Tech II Year I Semester () Regular Examinations December 2014 (Common to IT and CSE) (a) If 1010 2 + 10 2 = X 10, then X is ----- Write the first 9 decimal digits in base 3. (c) What is meant by don

More information

Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic

Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic Department of Electrical Engineering McGill University ECSE 221 Introduction to Computer Engineering Assignment 2 Combinational Logic Question 1: Due October 19 th, 2009 A convenient shorthand for specifying

More information

Programmable Logic Devices

Programmable Logic Devices Programmable Logic Devices Programmable Logic Devices Fig. (1) General structure of PLDs Programmable Logic Device (PLD): is an integrated circuit with internal logic gates and/or connections that can

More information

Chapter 2 Part 5 Combinational Logic Circuits

Chapter 2 Part 5 Combinational Logic Circuits Universit of Wisconsin - Madison ECE/Comp Sci 352 Digital Sstems Fundamentals Kewal K. Saluja and Yu Hen Hu Spring 2002 Chapter 2 Part 5 Combinational Logic Circuits Originals b: Charles R. Kime and Tom

More information

SEE1223: Digital Electronics

SEE1223: Digital Electronics SEE223: Digital Electronics 3 Combinational Logic Design Zulkifil Md Yusof Dept. of Microelectronics and Computer Engineering The aculty of Electrical Engineering Universiti Teknologi Malaysia Karnaugh

More information

Gate-Level Minimization. BME208 Logic Circuits Yalçın İŞLER

Gate-Level Minimization. BME208 Logic Circuits Yalçın İŞLER Gate-Level Minimization BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com Complexity of Digital Circuits Directly related to the complexity of the algebraic expression we use to

More information

Presentation 4: Programmable Combinational Devices

Presentation 4: Programmable Combinational Devices Presentation 4: Programmable Combinational Devices Asst. Prof Dr. Ahmet ÖZKURT DEUEEE Based on the Presentation by Prof. Kim, Young Ho Dept. of Information Computer Engineering E-mail : yhkim@hyowon.cs.pusan.ac.kr

More information

Gate-Level Minimization

Gate-Level Minimization Gate-Level Minimization Mano & Ciletti Chapter 3 By Suleyman TOSUN Ankara University Outline Intro to Gate-Level Minimization The Map Method 2-3-4-5 variable map methods Product-of-Sums Method Don t care

More information

Homework. Update on website issue Reading: Chapter 7 Homework: All exercises at end of Chapter 7 Due 9/26

Homework. Update on website issue Reading: Chapter 7 Homework: All exercises at end of Chapter 7 Due 9/26 Homework Update on website issue Reading: hapter 7 Homework: All exercises at end of hapter 7 Due 9/26 opyright c 22 28 UMaine omputer Science Department / 2 OS 4: Foundations of omputer Science Karnaugh

More information

Digital Logic Design (CEN-120) (3+1)

Digital Logic Design (CEN-120) (3+1) Digital Logic Design (CEN-120) (3+1) ASSISTANT PROFESSOR Engr. Syed Rizwan Ali, MS(CAAD)UK, PDG(CS)UK, PGD(PM)IR, BS(CE)PK HEC Certified Master Trainer (MT-FPDP) PEC Certified Professional Engineer (COM/2531)

More information

Get Free notes at Module-I One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)

More information

數位系統 Digital Systems 朝陽科技大學資工系. Speaker: Fuw-Yi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷

數位系統 Digital Systems 朝陽科技大學資工系. Speaker: Fuw-Yi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷 數位系統 Digital Systems Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝陽科技大學資工系 Speaker: Fuw-Yi Yang 楊伏夷 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象,

More information

ADAPTIVE MAP FOR SIMPLIFYING BOOLEAN EXPRESSIONS

ADAPTIVE MAP FOR SIMPLIFYING BOOLEAN EXPRESSIONS ABSTRACT ADAPTIVE MAP FOR SIMPLIFYING BOOLEAN EXPRESSIONS Dr. Mohammed H. AL-Jammas Department of Computer and Information Engineering, College of Electronics Engineering, University of Mosul, Mosul -

More information

Digital Circuits ECS 371

Digital Circuits ECS 371 Digital Circuits ECS 37 Dr. Prapun Suksompong prapun@siit.tu.ac.th Lecture 7 Office Hours: KD 36-7 Monday 9:-:3, :3-3:3 Tuesday :3-:3 Announcement HW2 posted on the course web site Chapter 4: Write down

More information

QUESTION BANK FOR TEST

QUESTION BANK FOR TEST CSCI 2121 Computer Organization and Assembly Language PRACTICE QUESTION BANK FOR TEST 1 Note: This represents a sample set. Please study all the topics from the lecture notes. Question 1. Multiple Choice

More information

EE292: Fundamentals of ECE

EE292: Fundamentals of ECE EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 22 121115 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Binary Number Representation Binary Arithmetic Combinatorial Logic

More information

Department of Electrical and Computer Engineering University of Wisconsin - Madison. ECE/CS 352 Digital System Fundamentals.

Department of Electrical and Computer Engineering University of Wisconsin - Madison. ECE/CS 352 Digital System Fundamentals. Department of Electrical and Computer Engineering University of Wisconsin - Madison ECE/C 352 Digital ystem Fundamentals Quiz #2 Thursday, March 7, 22, 7:15--8:3PM 1. (15 points) (a) (5 points) NAND, NOR

More information

S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017

S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017 S1 Teknik Telekomunikasi Fakultas Teknik Elektro FEH2H3 2016/2017 Karnaugh Map Karnaugh maps Last time we saw applications of Boolean logic to circuit design. The basic Boolean operations are AND, OR and

More information

Programmable Logic Devices. Programmable Read Only Memory (PROM) Example

Programmable Logic Devices. Programmable Read Only Memory (PROM) Example Programmable Logic Devices Programmable Logic Devices (PLDs) are the integrated circuits. They contain an array of AND gates & another array of OR gates. There are three kinds of PLDs based on the type

More information

Synthesis 1. 1 Figures in this chapter taken from S. H. Gerez, Algorithms for VLSI Design Automation, Wiley, Typeset by FoilTEX 1

Synthesis 1. 1 Figures in this chapter taken from S. H. Gerez, Algorithms for VLSI Design Automation, Wiley, Typeset by FoilTEX 1 Synthesis 1 1 Figures in this chapter taken from S. H. Gerez, Algorithms for VLSI Design Automation, Wiley, 1998. Typeset by FoilTEX 1 Introduction Logic synthesis is automatic generation of circuitry

More information