Computer Organization I. Lecture 8: Boolean Algebra and Circuit Optimization

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Computer Organization I. Lecture 8: Boolean Algebra and Circuit Optimization"

Transcription

1 Computer Organization I Lecture 8: Boolean Algebra and Circuit Optimization

2 Overview The simplification from SOM to SOP and their circuit implementation Basics of Logic Circuit Optimization: Cost Criteria Boolean Function Optimization (1) Algebraic Manipulation (2) Karnaugh map

3 Objectives Understand how to compare implementation cost of two circuits according to cost criteria To know how to use algebraic manipulation to simplify the logic function To understand the basic concepts of Karnaugh map

4 A Simplification Example from SOM to SOP A Simplification Example: F (A,B,C) = Ʃm (1,4,5,6,7) Writing the minterm expression: F = A B C + A B C + A B C + ABC + ABC Simplifying: F = A + B C Simplified F contains 3 literals compared to 15 in minterm F

5 AND/OR Two-level Diagram of SOM and SOP The two implementations for F are shown below it is quite apparent which is simpler! F in SOM include 15 literals and 6 logic gates; while F in SOP include 3 literals and 2 logic gates A B C A B C A B C A B C A B C F A B C We assume the input variables are directly available in their complemented and uncomplemented forms F

6 Some Conclusions on SOM/SOP and SOM/POM So far we have known: Standard Forms (Sum-of-minterms, Product-of-Maxterms), or alternative standard form (SOP, POS) differ in complexity Boolean algebra can be used to manipulate equations into simpler forms. Simpler equations lead to simpler two-level implementations Questions left to us: Is there only one minimum cost circuit? apply the cost criteria to compare complexity of circuits How can we attain a simplest expression? apply Boolean algebraic manipulations or graphical technique (i.e. Karnaugh maps, K-maps) to simply (or optimize) expression

7 Cost Criteria of Logic Circuit Implementation Literal Cost (L) Gate Input Cost (G) Gate Input Cost with NOTs (GN)

8 Cost Criteria of Logic Circuit Implementation - Literal Cost Literal a variable or its complement Literal cost the number of literal appearances in a Boolean expression corresponding to the logic circuit diagram Examples: F = BD + AB C + ACD L = 8 F = BD + ABC + ABD + ABC L = 11 F = (A + B)(A + D)(B + C + D)(B + C + D) L = 10 Which solution is best?

9 Cost Criteria of Logic Circuit Implementation - Gate Input Cost Gate Input Costs - the number of inputs to the gates in the implementation corresponding exactly to the given equation or equations. It can be found from the equation(s) by finding the sum of: all literal appearances the number of terms excluding single literal terms, and optionally, the number of distinct complemented single literals Gate input cost denoted by G if inverters not counted or GN if inverters counted. Example: F = BD + ABC + ACD G = 11, GN = 14 F = BD + ABC + ABD + ABC G =?, GN =? F = (A + B)(A + D)(B + C + D)(B + C + D) G =?, GN =?

10 Cost Criteria of Logic Circuit Implementation - An Example Example 1: F = A + B C + B C GN = G + 2 = 9 L = 5 G = L + 2 = 7 B C A F L (literal count) counts the AND inputs and the single literal OR input. G (gate input count) adds the remaining OR gate inputs GN(gate input count with NOTs) adds the inverter inputs Chapter 2 -Part 2 10

11 Cost Criteria of Logic Circuit Implementation - Another Example Example 2: F = A B C + A B C L = 6 G = 8 GN = 11 F = (A + C)(B + C)(A + B) L = 6 G = 9 GN = 12 Same function and same literal cost But first circuit has better gate input count and better gate input count with NOTs Select it! A B C A B C F F

12 Boolean Function Optimization - What means an optimal function An optimal function can minimize the gate input (or literal) cost of a (a set of) Boolean equation(s) and thus reducing circuit cost. An optimal function has a minimum gate input cost is reasonable since it measures directly number of transistors and wires used in implementing a circuit. The simplest expression is not necessarily unique, sometimes 2 or more expressions satisfy cost criterion applied, in this case either solution is satisfactory from the cost standpoint

13 Boolean Function Optimization - Approach 1: Algebraic Manipulation Four common rules are frequently applied for simplifying logic functions (1) complement rule: A + A = 1 e.g. A (BC + BC) + ABC + ABC = A (2) absorption rule: A + AB = A e.g. AB + ABC (D + E) = AB (3) a common rule: A + AB = A + B e.g. AB + AC + BC = AB + C (4) an expansion rule: A 1 = A and A + A = 1 e.g. AB + ABC + BC = AB + AC

14 Why use K-map? Boolean Function Optimization - Approach 2: Karnaugh Maps (K-map) The algebraic manipulation depends on your familiarity with all the laws, rules of Boolean algebra, the K-map, however, Provides a systematic means for: Finding optimum or near optimum SOP and POS forms, and two-level AND/OR and OR/AND circuit implementations for functions with small numbers of variables (usually number of variables no more than 4) Visualizing concepts related to manipulating Boolean expressions, and Demonstrating concepts used by computer-aided design programs to simplify large circuits

15 Boolean Function Optimization - Approach 2: Karnaugh Maps (K-map) What is K-map? K-map is an array of squares, in which Each square represents a minterm; number of squares equal to number of minterms; given Num. of Variables n, total num. of squares is 2 n The array of squares is a graphical representation of a Boolean function and can be considered as an reorganization of truth table y x m 0 m 1 m 2 m 3 y z x m 0 m 3 m 2 m 4 m 1 m 5 m 7 m 6

16 Boolean Function Optimization - Approach 2: Karnaugh Maps (K-map) Main Feature of K-map? In K-map, each square is physically adjacent to the squares that are immediately next to it on any of its four side Any two squares are physically adjacent means they are logically adjacent (i.e. the two squares differ in one or only one literal which appears uncomplemented in one and complemented in the other) Logically, squares in the top row are adjacent to the corresponding row in the bottom, similarly for squares in the outer left and outer right y z y x m 0 m 1 m 2 m 3 x m 0 m 3 m 2 m 4 m 1 m 5 m 7 m 6

17 Karnaugh Maps (K-map) - An Example: Two Variable K-map A 2-variable Karnaugh Map: Note that minterm m 0 and minterm m 1 are adjacent and differ in the value of the variable y x y Similarly, minterm m 0 and x y minterm m 2 differ in the x variable. Also, m 1 and m 3 differ in the x variable as well. Finally, m 2 and m 3 differ in the value of the variable y y = 0 y = 1 x = 0 m 0 = m 1 = x y x = 1 m 2 = m 3 = x y

18 Summary Simplification from SOM to SOP Cost Criteria for Comparing Complexity of Circuit Implementation Basics of K-map, including why use K-map, what is K- map and its adjacent attribute.

19 Thank you Q & A

Points Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 5: Logic Simplication & Karnaugh Map

Points Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 5: Logic Simplication & Karnaugh Map Points Addressed in this Lecture Lecture 5: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London (Floyd 4.5-4.) (Tocci 4.-4.5) Standard form of Boolean Expressions

More information

Karnaugh Map. Alternative way to Boolean Function Simplification. Karnaugh Map. Description of Kmap & Terminology

Karnaugh Map. Alternative way to Boolean Function Simplification. Karnaugh Map. Description of Kmap & Terminology Alternative way to Boolean Function Simplification Karnaugh Map CIT 595 Spring 2010 Simplification of Boolean functions leads to simpler (and usually faster) digital circuits Simplifying Boolean functions

More information

BOOLEAN ALGEBRA & LOGIC GATES

BOOLEAN ALGEBRA & LOGIC GATES BOOLEAN ALGEBRA & LOGIC GATES Logic gates are electronic circuits that can be used to implement the most elementary logic expressions, also known as Boolean expressions. The logic gate is the most basic

More information

Karnaugh Maps. Circuit-wise, this leads to a minimal two-level implementation

Karnaugh Maps. Circuit-wise, this leads to a minimal two-level implementation Karnaugh Maps 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 also be directly translated

More information

Simplifying Logic Circuits with Karnaugh Maps

Simplifying Logic Circuits with Karnaugh Maps Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified

More information

Karnaugh Maps & Combinational Logic Design. ECE 152A Winter 2012

Karnaugh Maps & Combinational Logic Design. ECE 152A Winter 2012 Karnaugh Maps & Combinational Logic Design ECE 52A Winter 22 Reading Assignment Brown and Vranesic 4 Optimized Implementation of Logic Functions 4. Karnaugh Map 4.2 Strategy for Minimization 4.2. Terminology

More information

4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION

4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION 4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION 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

Combinational Logic Circuits

Combinational Logic Circuits Chapter 2 Combinational Logic Circuits J.J. Shann Chapter Overview 2-1 Binary Logic and Gates 2-2 Boolean Algebra 2-3 Standard Forms 2-4 Two-Level Circuit Optimization 2-5 Map Manipulation 補 充 資 料 :Quine-McCluskey

More information

CSE140: Midterm 1 Solution and Rubric

CSE140: Midterm 1 Solution and Rubric CSE140: Midterm 1 Solution and Rubric April 23, 2014 1 Short Answers 1.1 True or (6pts) 1. A maxterm must include all input variables (1pt) True 2. A canonical product of sums is a product of minterms

More information

CSEE 3827: Fundamentals of Computer Systems. Standard Forms and Simplification with Karnaugh Maps

CSEE 3827: Fundamentals of Computer Systems. Standard Forms and Simplification with Karnaugh Maps CSEE 3827: Fundamentals of Computer Systems Standard Forms and Simplification with Karnaugh Maps Agenda (M&K 2.3-2.5) Standard Forms Product-of-Sums (PoS) Sum-of-Products (SoP) converting between Min-terms

More information

The equation for the 3-input XOR gate is derived as follows

The equation for the 3-input XOR gate is derived as follows The equation for the 3-input XOR gate is derived as follows The last four product terms in the above derivation are the four 1-minterms in the 3-input XOR truth table. For 3 or more inputs, the XOR gate

More information

Boolean Algebra Part 1

Boolean Algebra Part 1 Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems

More information

Chapter 2: Boolean Algebra and Logic Gates. Boolean Algebra

Chapter 2: Boolean Algebra and Logic Gates. Boolean Algebra The Universit Of Alabama in Huntsville Computer Science Chapter 2: Boolean Algebra and Logic Gates The Universit Of Alabama in Huntsville Computer Science Boolean Algebra The algebraic sstem usuall used

More information

United States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1

United States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1 United States Naval Academy Electrical and Computer Engineering Department EC262 Exam 29 September 2. Do a page check now. You should have pages (cover & questions). 2. Read all problems in their entirety.

More information

l What have discussed up until now & why: l C Programming language l More low-level then Java. l Better idea about what s really going on.

l What have discussed up until now & why: l C Programming language l More low-level then Java. l Better idea about what s really going on. CS211 Computer Architecture l Topics Digital Logic l Transistors (Design & Types) l Logic Gates l Combinational Circuits l K-Maps Class Checkpoint l What have discussed up until now & why: l C Programming

More information

Chapter 2 Combinational Logic Circuits

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits

More information

1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1.

1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1. File: chap04, Chapter 04 1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1. 2. True or False? A gate is a device that accepts a single input signal and produces one

More information

Gates, Circuits, and Boolean Algebra

Gates, Circuits, and Boolean Algebra Gates, Circuits, and Boolean Algebra Computers and Electricity A gate is a device that performs a basic operation on electrical signals Gates are combined into circuits to perform more complicated tasks

More information

DESIGN OF GATE NETWORKS

DESIGN OF GATE NETWORKS DESIGN OF GATE NETWORKS DESIGN OF TWO-LEVEL NETWORKS: and-or and or-and NETWORKS MINIMAL TWO-LEVEL NETWORKS KARNAUGH MAPS MINIMIZATION PROCEDURE AND TOOLS LIMITATIONS OF TWO-LEVEL NETWORKS DESIGN OF TWO-LEVEL

More information

CDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012

CDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Multi-Level Gate Circuits NAND and NOR Gates Design of Two-Level Circuits Using NAND and NOR Gates

More information

Digital Logic Design. Basics Combinational Circuits Sequential Circuits. Pu-Jen Cheng

Digital Logic Design. Basics Combinational Circuits Sequential Circuits. Pu-Jen Cheng Digital Logic Design Basics Combinational Circuits Sequential Circuits Pu-Jen Cheng Adapted from the slides prepared by S. Dandamudi for the book, Fundamentals of Computer Organization and Design. Introduction

More information

CH3 Boolean Algebra (cont d)

CH3 Boolean Algebra (cont d) CH3 Boolean Algebra (cont d) Lecturer: 吳 安 宇 Date:2005/10/7 ACCESS IC LAB v Today, you ll know: Introduction 1. Guidelines for multiplying out/factoring expressions 2. Exclusive-OR and Equivalence operations

More information

Programmable Logic Devices (PLDs)

Programmable Logic Devices (PLDs) Programmable Logic Devices (PLDs) Lesson Objectives: In this lesson you will be introduced to some types of Programmable Logic Devices (PLDs): PROM, PAL, PLA, CPLDs, FPGAs, etc. How to implement digital

More information

Boolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris Hui-Ru Jiang Spring 2010

Boolean Algebra (cont d) UNIT 3 BOOLEAN ALGEBRA (CONT D) Guidelines for Multiplying Out and Factoring. Objectives. Iris Hui-Ru Jiang Spring 2010 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic simplification Proving validity of an

More information

Karnaugh Maps (K-map) Alternate representation of a truth table

Karnaugh Maps (K-map) Alternate representation of a truth table Karnaugh Maps (K-map) lternate representation of a truth table Red decimal = minterm value Note that is the MS for this minterm numbering djacent squares have distance = 1 Valuable tool for logic minimization

More information

CSE140: Components and Design Techniques for Digital Systems

CSE140: Components and Design Techniques for Digital Systems CSE4: Components and Design Techniques for Digital Systems Tajana Simunic Rosing What we covered thus far: Number representations Logic gates Boolean algebra Introduction to CMOS HW#2 due, HW#3 assigned

More information

Chapter 4. Gates and Circuits. Chapter Goals. Chapter Goals. Computers and Electricity. Computers and Electricity. Gates

Chapter 4. Gates and Circuits. Chapter Goals. Chapter Goals. Computers and Electricity. Computers and Electricity. Gates Chapter Goals Chapter 4 Gates and Circuits Identify the basic gates and describe the behavior of each Describe how gates are implemented using transistors Combine basic gates into circuits Describe the

More information

CHAPTER 3 Boolean Algebra and Digital Logic

CHAPTER 3 Boolean Algebra and Digital Logic CHAPTER 3 Boolean Algebra and Digital Logic 3.1 Introduction 121 3.2 Boolean Algebra 122 3.2.1 Boolean Expressions 123 3.2.2 Boolean Identities 124 3.2.3 Simplification of Boolean Expressions 126 3.2.4

More information

Unit 3 Boolean Algebra (Continued)

Unit 3 Boolean Algebra (Continued) Unit 3 Boolean Algebra (Continued) 1. Exclusive-OR Operation 2. Consensus Theorem Department of Communication Engineering, NCTU 1 3.1 Multiplying Out and Factoring Expressions Department of Communication

More information

Two-level logic using NAND gates

Two-level logic using NAND gates CSE140: Components and Design Techniques for Digital Systems Two and Multilevel logic implementation Tajana Simunic Rosing 1 Two-level logic using NND gates Replace minterm ND gates with NND gates Place

More information

Logic Reference Guide

Logic Reference Guide Logic eference Guide Advanced Micro evices INTOUCTION Throughout this data book and design guide we have assumed that you have a good working knowledge of logic. Unfortunately, there always comes a time

More information

2 1 Implementation using NAND gates: We can write the XOR logical expression A B + A B using double negation as

2 1 Implementation using NAND gates: We can write the XOR logical expression A B + A B using double negation as Chapter 2 Digital Logic asics 2 Implementation using NND gates: We can write the XOR logical expression + using double negation as + = + = From this logical expression, we can derive the following NND

More information

Logic Design 2013/9/5. Introduction. Logic circuits operate on digital signals

Logic Design 2013/9/5. Introduction. Logic circuits operate on digital signals Introduction Logic Design Chapter 2: Introduction to Logic Circuits Logic circuits operate on digital signals Unlike continuous analog signals that have an infinite number of possible values, digital signals

More information

Gates & Boolean Algebra. Boolean Operators. Combinational Logic. Introduction

Gates & Boolean Algebra. Boolean Operators. Combinational Logic. Introduction Introduction Gates & Boolean lgebra Boolean algebra: named after mathematician George Boole (85 864). 2-valued algebra. digital circuit can have one of 2 values. Signal between and volt =, between 4 and

More information

Introduction. The Quine-McCluskey Method Handout 5 January 21, 2016. CSEE E6861y Prof. Steven Nowick

Introduction. The Quine-McCluskey Method Handout 5 January 21, 2016. CSEE E6861y Prof. Steven Nowick CSEE E6861y Prof. Steven Nowick The Quine-McCluskey Method Handout 5 January 21, 2016 Introduction The Quine-McCluskey method is an exact algorithm which finds a minimum-cost sum-of-products implementation

More information

Lecture 5: Gate Logic Logic Optimization

Lecture 5: Gate Logic Logic Optimization Lecture 5: Gate Logic Logic Optimization MAH, AEN EE271 Lecture 5 1 Overview Reading McCluskey, Logic Design Principles- or any text in boolean algebra Introduction We could design at the level of irsim

More information

Understanding Logic Design

Understanding Logic Design Understanding Logic Design ppendix of your Textbook does not have the needed background information. This document supplements it. When you write add DD R0, R1, R2, you imagine something like this: R1

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

Module-3 SEQUENTIAL LOGIC CIRCUITS

Module-3 SEQUENTIAL LOGIC CIRCUITS Module-3 SEQUENTIAL LOGIC CIRCUITS Till now we studied the logic circuits whose outputs at any instant of time depend only on the input signals present at that time are known as combinational circuits.

More information

Lecture 8: Synchronous Digital Systems

Lecture 8: Synchronous Digital Systems Lecture 8: Synchronous Digital Systems The distinguishing feature of a synchronous digital system is that the circuit only changes in response to a system clock. For example, consider the edge triggered

More information

Digital circuits make up all computers and computer systems. The operation of digital circuits is based on

Digital circuits make up all computers and computer systems. The operation of digital circuits is based on Digital Logic Circuits Digital circuits make up all computers and computer systems. The operation of digital circuits is based on Boolean algebra, the mathematics of binary numbers. Boolean algebra is

More information

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

More information

Lecture 12: More on Registers, Multiplexers, Decoders, Comparators and Wot- Nots

Lecture 12: More on Registers, Multiplexers, Decoders, Comparators and Wot- Nots Lecture 12: More on Registers, Multiplexers, Decoders, Comparators and Wot- Nots Registers As you probably know (if you don t then you should consider changing your course), data processing is usually

More information

Binary Adders: Half Adders and Full Adders

Binary Adders: Half Adders and Full Adders Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

ENGIN 112 Intro to Electrical and Computer Engineering

ENGIN 112 Intro to Electrical and Computer Engineering ENGIN 112 Intro to Electrical and omputer Engineering Lecture 11 NND and XOR Implementations Overview Developing NND circuits from K-maps Two-level implementations onvert from ND/OR to NND (again!) Multi-level

More information

2.0 Chapter Overview. 2.1 Boolean Algebra

2.0 Chapter Overview. 2.1 Boolean Algebra Thi d t t d ith F M k 4 0 2 Boolean Algebra Chapter Two Logic circuits are the basis for modern digital computer systems. To appreciate how computer systems operate you will need to understand digital

More information

FORDHAM UNIVERSITY CISC 3593. Dept. of Computer and Info. Science Spring, 2011. The Binary Adder

FORDHAM UNIVERSITY CISC 3593. Dept. of Computer and Info. Science Spring, 2011. The Binary Adder FORDHAM UNIVERITY CIC 3593 Fordham College Lincoln Center Computer Organization Dept. of Computer and Info. cience pring, 2011 1 Introduction The Binar Adder The binar adder circuit is an important building

More information

Online EFFECTIVE AS OF JANUARY 2013

Online EFFECTIVE AS OF JANUARY 2013 2013 A and C Session Start Dates (A-B Quarter Sequence*) 2013 B and D Session Start Dates (B-A Quarter Sequence*) Quarter 5 2012 1205A&C Begins November 5, 2012 1205A Ends December 9, 2012 Session Break

More information

Karnaugh Maps. Example A B C X 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1. each 1 here gives a minterm e.g.

Karnaugh Maps. Example A B C X 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1. each 1 here gives a minterm e.g. Karnaugh Maps Yet another way of deriving the simplest Boolean expressions from behaviour. Easier than using algebra (which can be hard if you don't know where you're going). Example A B C X 0 0 0 0 0

More information

3.Basic Gate Combinations

3.Basic Gate Combinations 3.Basic Gate Combinations 3.1 TTL NAND Gate In logic circuits transistors play the role of switches. For those in the TTL gate the conducting state (on) occurs when the baseemmiter signal is high, and

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

More information

ENEE 244 (01**). Spring 2006. Homework 4. Due back in class on Friday, April 7.

ENEE 244 (01**). Spring 2006. Homework 4. Due back in class on Friday, April 7. ENEE 244 (**). Spring 26 Homework 4 Due back in class on Friday, April 7.. Implement the following Boolean expression with exclusive-or and AND gates only: F = AB'CD' + A'BCD' + AB'C'D + A'BC'D. F = AB

More information

Points Addressed in this Lecture

Points Addressed in this Lecture Points Addressed in this Lecture Lecture 3: Basic Logic Gates & Boolean Expressions Professor Peter Cheung Department of EEE, Imperial College London (Floyd 3.1-3.5, 4.1) (Tocci 3.1-3.9) What are the basic

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

Take-Home Exercise. z y x. Erik Jonsson School of Engineering and Computer Science. The University of Texas at Dallas

Take-Home Exercise. z y x. Erik Jonsson School of Engineering and Computer Science. The University of Texas at Dallas Take-Home Exercise Assume you want the counter below to count mod-6 backward. That is, it would count 0-5-4-3-2-1-0, etc. Assume it is reset on startup, and design the wiring to make the counter count

More information

Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 313 Lecture #10 2.2: The Inverse of a Matrix Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

More information

Digital Logic: Boolean Algebra and Gates

Digital Logic: Boolean Algebra and Gates Digital Logic: Boolean Algebra and Gates Textbook Chapter 3 CMPE2 Summer 28 Basic Logic Gates CMPE2 Summer 28 Slides by ADB 2 Truth Table The most basic representation of a logic function Lists the output

More information

Course Requirements & Evaluation Methods

Course Requirements & Evaluation Methods Course Title: Logic Circuits Course Prefix: ELEG Course No.: 3063 Sections: 01 & 02 Department of Electrical and Computer Engineering College of Engineering Instructor Name: Justin Foreman Office Location:

More information

Introduction. Logic. Most Difficult Reading Topics. Basic Logic Gates Truth Tables Logical Functions. COMP370 Introduction to Computer Architecture

Introduction. Logic. Most Difficult Reading Topics. Basic Logic Gates Truth Tables Logical Functions. COMP370 Introduction to Computer Architecture Introduction LOGIC GATES COMP370 Introduction to Computer Architecture Basic Logic Gates Truth Tables Logical Functions Truth Tables Logical Expression Graphical l Form Most Difficult Reading Topics Logic

More information

NAND and NOR Implementation

NAND and NOR Implementation University of Wisconsin - Madison EE/omp ci 352 Digital ystems Fundamentals harles R. Kime ection 2 Fall 200 hapter 2 ombinational Logic ircuits Part 7 harles Kime & Thomas Kaminski NND and NOR Implementation

More information

The Inverse of a Matrix

The Inverse of a Matrix The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

More information

Let s put together a Manual Processor

Let s put together a Manual Processor Lecture 14 Let s put together a Manual Processor Hardware Lecture 14 Slide 1 The processor Inside every computer there is at least one processor which can take an instruction, some operands and produce

More information

Basic Logic Gates Richard E. Haskell

Basic Logic Gates Richard E. Haskell BASIC LOGIC GATES 1 E Basic Logic Gates Richard E. Haskell All digital systems are made from a few basic digital circuits that we call logic gates. These circuits perform the basic logic functions that

More information

DIGITAL SYSTEM DESIGN LAB

DIGITAL SYSTEM DESIGN LAB EXPERIMENT NO: 7 STUDY OF FLIP FLOPS USING GATES AND IC S AIM: To verify various flip-flops like D, T, and JK. APPARATUS REQUIRED: Power supply, Digital Trainer kit, Connecting wires, Patch Chords, IC

More information

ENGI 241 Experiment 5 Basic Logic Gates

ENGI 241 Experiment 5 Basic Logic Gates ENGI 24 Experiment 5 Basic Logic Gates OBJECTIVE This experiment will examine the operation of the AND, NAND, OR, and NOR logic gates and compare the expected outputs to the truth tables for these devices.

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Digital Circuits. Electrical & Computer Engineering Department (ECED) Course Notes ECED2200. ECED2200 Digital Circuits Notes 2012 Dalhousie University

Digital Circuits. Electrical & Computer Engineering Department (ECED) Course Notes ECED2200. ECED2200 Digital Circuits Notes 2012 Dalhousie University 1 Digital Circuits Electrical & Computer Engineering Department (ECED) Course Notes ECED2200 2 Table of Contents Digital Circuits... 7 Logic Gates... 8 AND Gate... 8 OR Gate... 9 NOT Gate... 10 NOR Gate...

More information

NEW adder cells are useful for designing larger circuits despite increase in transistor count by four per cell.

NEW adder cells are useful for designing larger circuits despite increase in transistor count by four per cell. CHAPTER 4 THE ADDER The adder is one of the most critical components of a processor, as it is used in the Arithmetic Logic Unit (ALU), in the floating-point unit and for address generation in case of cache

More information

Sum-of-Products and Product-of-Sums expressions

Sum-of-Products and Product-of-Sums expressions Sum-of-Products and Product-of-Sums expressions This worksheet and all related files are licensed under the reative ommons ttribution License, version.. To view a copy of this license, visit http://creativecommons.org/licenses/by/./,

More information

The Inverse of a Square Matrix

The Inverse of a Square Matrix These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

Lecture 6. Inverse of Matrix

Lecture 6. Inverse of Matrix Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

More information

Introduction to Matrices for Engineers

Introduction to Matrices for Engineers Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

More information

L2: Combinational Logic Design (Construction and Boolean Algebra)

L2: Combinational Logic Design (Construction and Boolean Algebra) L2: Combinational Logic Design (Construction and oolean lgebra) cknowledgements: Materials in this lecture are courtesy of the following sources and are used with permission. Prof. Randy Katz (Unified

More information

Digital Electronics Part I Combinational and Sequential Logic. Dr. I. J. Wassell

Digital Electronics Part I Combinational and Sequential Logic. Dr. I. J. Wassell Digital Electronics Part I Combinational and Sequential Logic Dr. I. J. Wassell Introduction Aims To familiarise students with Combinational logic circuits Sequential logic circuits How digital logic gates

More information

Ex. Convert the Boolean function F = x + y z into a sum of minterms by using a truth table.

Ex. Convert the Boolean function F = x + y z into a sum of minterms by using a truth table. Section 3.5 - Minterms, Maxterms, Canonical Fm & Standard Fm Page 1 of 5 3.5 Canonical Fms In general, the unique algebraic expression f any Boolean function can be obtained from its truth table by using

More information

Digital Integrated Circuits EECS 312

Digital Integrated Circuits EECS 312 14 12 10 8 6 Fujitsu VP2000 IBM 3090S Pulsar 4 IBM 3090 IBM RY6 CDC Cyber 205 IBM 4381 IBM RY4 2 IBM 3081 Apache Fujitsu M380 IBM 370 Merced IBM 360 IBM 3033 Vacuum Pentium II(DSIP) 0 1950 1960 1970 1980

More information

Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1. Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

More information

Numerical and Algebraic Fractions

Numerical and Algebraic Fractions Numerical and Algebraic Fractions Aquinas Maths Department Preparation for AS Maths This unit covers numerical and algebraic fractions. In A level, solutions often involve fractions and one of the Core

More information

Lecture 10: A Design Example - Traffic Lights

Lecture 10: A Design Example - Traffic Lights Lecture 10: A Design Example - Traffic Lights In this lecture we will work through a design example from problem statement to digital circuits. The Problem The traffic department is trying out a new system

More information

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

Introduction to Digital Logic with Laboratory Exercises

Introduction to Digital Logic with Laboratory Exercises Introduction to Digital Logic with Laboratory Exercises Introduction to Digital Logic with Laboratory Exercises James Feher Copyright 29 James Feher Editor-In-Chief: James Feher Associate Editor: Marisa

More information

ELEC 2210 - EXPERIMENT 1 Basic Digital Logic Circuits

ELEC 2210 - EXPERIMENT 1 Basic Digital Logic Circuits Objectives ELEC - EXPERIMENT Basic Digital Logic Circuits The experiments in this laboratory exercise will provide an introduction to digital electronic circuits. You will learn how to use the IDL-00 Bit

More information

Chapter 6 Digital Arithmetic: Operations & Circuits

Chapter 6 Digital Arithmetic: Operations & Circuits Chapter 6 Digital Arithmetic: Operations & Circuits Chapter 6 Objectives Selected areas covered in this chapter: Binary addition, subtraction, multiplication, division. Differences between binary addition

More information

Elementary Logic Gates

Elementary Logic Gates Elementary Logic Gates Name Symbol Inverter (NOT Gate) ND Gate OR Gate Truth Table Logic Equation = = = = = + C. E. Stroud Combinational Logic Design (/6) Other Elementary Logic Gates NND Gate NOR Gate

More information

Lecture 2 Matrix Operations

Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

More information

Computer Engineering 290. Digital Design: I. Lecture Notes Summer 2002

Computer Engineering 290. Digital Design: I. Lecture Notes Summer 2002 Computer Engineering 290 Digital Design: I Lecture Notes Summer 2002 W.D. Little Dept. of Electrical and Computer Engineering University of Victoria 1 Preface These lecture notes complement the material

More information

Logic in Computer Science: Logic Gates

Logic in Computer Science: Logic Gates Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers

More information

COMPUTER SCIENCE. Paper 1 (THEORY)

COMPUTER SCIENCE. Paper 1 (THEORY) COMPUTER SCIENCE Paper 1 (THEORY) (Three hours) Maximum Marks: 70 (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time) -----------------------------------------------------------------------------------------------------------------------

More information

2015 Junior Certificate Higher Level Official Sample Paper 1

2015 Junior Certificate Higher Level Official Sample Paper 1 2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list

More information

4. MATRICES Matrices

4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

More information

5.3 The Cross Product in R 3

5.3 The Cross Product in R 3 53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

More information

Chapter 7 Memory and Programmable Logic

Chapter 7 Memory and Programmable Logic NCNU_2013_DD_7_1 Chapter 7 Memory and Programmable Logic 71I 7.1 Introduction ti 7.2 Random Access Memory 7.3 Memory Decoding 7.5 Read Only Memory 7.6 Programmable Logic Array 77P 7.7 Programmable Array

More information

Mixed Logic A B A B. 1. Ignore all bubbles on logic gates and inverters. This means

Mixed Logic A B A B. 1. Ignore all bubbles on logic gates and inverters. This means Mixed Logic Introduction Mixed logic is a gate-level design methodology used in industry. It allows a digital logic circuit designer the functional description of the circuit from its physical implementation.

More information

Combinational Logic Design Process

Combinational Logic Design Process Combinational Logic Design Process Create truth table from specification Generate K-maps & obtain logic equations Draw logic diagram (sharing common gates) Simulate circuit for design verification Debug

More information

Digital Fundamentals

Digital Fundamentals Digital Fundamentals Tenth Edition Floyd hapter 5 2009 Pearson Education, Upper 2008 Pearson Saddle River, Education NJ 07458. ll Rights Reserved Summary ombinational Logic ircuits In Sum-of-Products (SOP)

More information