# CSE 220: Systems Fundamentals I Unit 7: Logic Gates; Digital Logic Design: Boolean Equations and Algebra

Save this PDF as:

Size: px
Start display at page:

Download "CSE 220: Systems Fundamentals I Unit 7: Logic Gates; Digital Logic Design: Boolean Equations and Algebra"

## Transcription

1 CSE 220: Systems Fundamentals I Unit 7: Logic Gates; Digital Logic Design: Boolean Equations and Algebra Logic Gates Logic gatesare simple digital circuits that take one or more binary inputs and produce a binary output The relationship between the inputs and outputs can be described by a truth tableor a Boolean equation Each type of logic gate has its own corresponding symbol Inputs to a circuit are generally drawn at the left or top of a circuit diagramand the outputs on the right or bottom Usually we ll use letters A, B, C, for inputs and the letter Y for the output We will also see examples where subscripts are used to label inputs and outputs (e.g.,,,,,, ) 1 2 The NOT Gate Buffers Takes one input and produces one output The little circle is called a bubbleand indicates that negation is taking place The overline indicates negation Say it like this: Yequals NOT A Negation can also be denoted. I will use both notations in lecture notes, examples, quizzes and exams. A NOT gate is also called an inverter because it inverts its input In these truth tables, 0 indicates falseand 1 indicates true A buffertakes one input and reproduces that input on its output This seems pointless, but buffers can be used for useful tasks like changing voltage or amplifying a signal Could we implement a buffer using NOT gates? 3 4

2 The AND Gate The OR Gate The AND gate performs a logical AND operation over bits It produces a 1 only when both inputs are 1; otherwise, it produces a 0 Also denoted with a dot: Yequals AAND B (not Atimes B ) Note the order in which the values for A and Bare listed. This is the reverse order of what you typically see in a discrete mathematics class (e.g., CSE 215) If Aand Bare taken together to form a 2-bit number, we are counting in binary from 0 to 3 The OR gate performs a logical OR operation over bits It produces a 1 when either input or both inputs are 1; otherwise, it produces a 0 Sometimes called inclusive ORbecause the gate produces a 1 when both inputs are 1 Also denoted with a plus sign: + Yequals AOR B (not Aplus B ) 5 6 The XOR Gate The NAND Gate The XOR gate performs a logical exclusive-or operator over bits It produces a 1 when only one of the inputs is 1; otherwise, it produces a 0 In general, an N-input XOR gate produces a 1 if an odd number of its inputs are 1 The NAND gate produces a 0 only when both inputs are 1; otherwise, it produces a 1 In other words, it behaves like an AND gate with a NOT gate connected to the AND gate s output wire 7 8

3 The NOR Gate The XNOR Gate The NOR gate outputs a 1 if neither input is 1 Stated another way, a NOR gate produces a 1 only when both inputs are 0 It behaves like an OR gate with a NOT gate connected to the OR gate s output wire NAND and NOR gates are preferred in circuit manufacturing because each requires only 2 transistors, whereas AND and OR gates require 3 transistors each Fewer transistors smaller, cheaper circuit; more gates can fit on a silicon die The XNOR gate produces a 1 when both inputs are the same It behaves like an XOR gate with a NOT gate connected to the XOR gate s output wire 9 10 Multiple-Input Logic Gates Multiple-Input Logic Gates Multiple-input logic gates exist for most of the gates we have studied so far (NOT being an obvious exception) An N-input AND gate produces a 1 if all the inputs are 1 An N-input OR gate produces a 1 if at least one input is 1 What would be the behavior of a 3-input NOR gate? Answer: produce a 1 only when all three inputs are 0s 11 12

4 Digital Circuit Design Digital Circuit Design A logic circuit is composed of: Inputs and outputs Functional specification (what does it do) Timing specification (how long does it take to do that) A circuit as a black box of inputs and outputs. Digital circuits can be combinational or sequential A combinational circuit s output depends only on its current inputs In contrast, sequential circuits have feedback the outputs are connected to the inputs Sequential circuits can be used to create memory Nodes(wires) Inputs: A, B, C Outputs: Y, Z Internal: n1 Circuit elements E1, E2, E3 Each is a circuit Specification vs. Implementation As is the case with algorithms, there could be different ways to implement a circuit s specification Below is a specification for a circuit ( CL indicates that it is implemented using combinational logic) Rules of Combinational Composition These rules tell us how we can build a large combinational circuit from smaller combinational circuits 1. Every element is combinational 2. Every node is either an input or connects to exactly one output 3. The circuit contains no cyclic paths It could be implemented in a variety of ways. Two possibilities are shown on the right

5 Rules of Combinational Composition Boolean Equations Which of these are combinational circuits and which ones aren t? A Boolean equation gives a formula that gives a functional specification of outputs in terms of inputs Example: =,, =,, Some Definitions Complement: variable with a bar over it,, Remember: can also be written. Literal: variable or its complement,,,,, Implicant: a product of literals,, Minterm: a product that includes all input variables For a 3-input circuit (inputs: A, B, C),, and would be minterms, but and would not be Maxterm: a sum that includes all input variables For a 3-input circuit, + + and + + would be maxterms, but + and + would not be Sum-of-Products (SOP) Form Each row in a truth table corresponds with a minterm Each mintermis TRUE for that row (and only that row) The rows are labeled 0 through 2 1where Nis the number of inputs The row number can be determined by taking the Ninputs as binary digits of an N-bit number 19 20

6 Sum-of-Products (SOP) Form Any equation can be written in sum-of-products canonical formas a sum (OR) of products (AND) Form the function by ORingonly those mintermsfor which the output is TRUE Sum-of-Products (SOP) Form SOP form can also be written in sigma notation For example, = + could also be written, = Σ, or, = Σ(1,3)or, = Σ(1,3) Note that you MUST list the input variables in the same order as given above: start with 0 and increment the count from there For the function given above, its SOP form is = Product-of-Sums (POS) Form Product-of-Sums (POS) Form Any equation can be written in product-of-sums canonical formas a product (AND) of sums (OR) Each row in a truth table corresponds with a maxterm Each maxtermis FALSE for that row (and only that row) Form the function by ANDingonly those maxtermsfor which the output is FALSE For the function given above, its POS form is = + + POS form can also be written in pi notation For example, = + + could also be written, = Π #,# or, = Π(0,2)or, = Π#(0,2) 23 24

7 SOP vs. POS The sum-of-products form of a Boolean equation and its corresponding product-of-sums form are dualsof each other: they are equivalent ways of expressing the same function Generally we use SOP form when a truth table has only a few rows where the output is TRUE. Why? Because it leads to a shorter equations. Likewise, we use POS form when a truth table has only a few rows where the output is FALSE The logic used in SOP and POS forms is called two-level logicbecause it consists of two levels of gates: AND-OR or OR-AND, depending on the form (SOP = AND-OR, POS = OR-AND) SOP vs. POS (cont.) We use the generic word termto refer to the minimal unit of a Boolean expression Depending on the form of the function, a term is the firstlevel gate unit Examples: ()has one term, but + has two terms ( + +)has one term, but has two terms Boolean Equation Example You are going to the cafeteria for lunch You won t eat lunch (%) if it s not open (&)or if they only serve corndogs () Write a truth table for determining if you will eat lunch (%) and both the SOP and POS forms of the Boolean equation Boolean Algebra Boolean equations written in SOP and POS form lead to circuits that have many gates, which increases cost of the circuit and makes it slower than equivalent circuits that have simpler (but logically equivalent) Boolean expressions We can use axioms and theorems of Boolean algebra to simplify complex equations Boolean algebra is similar to regular algebra, but we have Boolean values (1 and 0) instead of numerical values Most of these axioms and theorems exhibit duality, meaning you can exchange AND for OR and also 0 for 1 and the theorem still holds 27 29

8 Boolean Algebra Axioms Axiom Dual = 0if 1 = 1if 0 0 = 1 1 = = = = = = 1 0 = = = 1 Boolean Algebra Theorems Theorem Dual Name 1 = 0 = 0 = = + 0 = Identity + 1 = 1 Null element + = Idempotency Involution = 0 + = 1 Complement Boolean algebra theorems allow us to simplify Boolean expressions, which simplify the corresponding digital circuits All of these theorems involve only one variable. We will see theorems for two or more variables shortly Boolean Algebra Theorems Boolean Algebra Theorems The identity theoremallows us to replace an AND or OR gate with a wire connected to the input The idempotency theorem allows us to replace an AND or ORgate with a wire connected to the input The null element theorem allows us to replace an AND or ORgate with a wire tied to high (1) or low (0) The involution theorem allows us to replace two inverters wired in series with a wire connected to the input 32 33

9 Boolean Algebra Theorems Boolean Algebra Theorems The complement theorem allows us to replace an AND or ORgate with a wire connected to high or low Theorem Dual Name = + = + Commutativity ( = ( () + + ( = + ( + () Associativity + ( = + ( + + ( = + ( Distributivity + = + = Absorption + = + + = Combining + ( + ( = + ( + + ( + ( = + + ( Consensus + = + + = No name = = De Morgan s Laws De Morgan s Laws as Circuits Simplifying Boolean Expressions De Morgan s Laws give us an interesting result that is helpful in implementing circuits: a NAND gate is equivalent to an OR gate with inverted inputs, and a NOR gate is equivalent to an AND gate with inverted inputs Since it is often advantageous to use exclusively NAND or NOR gates in a circuits, we find that de Morgan s laws will always let us replace AND gates with NOR gates, and OR gates with NAND gates First look for terms that you can group together and factor out elements that evaluate to 1. Example: + = + = (why?) Example: + = 1+ = (why?) Second, if you cannot find any terms to group together and simplify, then multiply elements by identities equivalent to 1 (but only using variables which don tappear in the term) Example of this on the next slide 36 37

10 Simplifying Boolean Expressions Simplifying Boolean Expressions Example: + + = identity = complement = distributivity = identity = (1 + ) distributivity = + null element This is a proof of the consensus theorem When all else fails, the best bet is to expand out all the terms into minterms, use the idempotency theorem to duplicate terms, and then regroup them Example: minimize + + = idempotency = + + ( + ) distributivity (used 2x) = complement (used 2x) = + identity (used 2x) It s not always obvious (i) that an expression is fully simplified, or (ii) if it isn t simplified, how to simplify it Later I will teach you a visual technique called Karnaugh maps that makes this process easier From Logic to Gates Example: From Logic to Gates To make circuit diagrams (schematics) as easy as possible to draw and read, we will adopt a few guidelines: Inputs are on the left (or top) Outputs are on right (or bottom) Gates flow from left to right Use straight wires only Wires always connect at a T junction A dot where wires cross indicates a connection between the wires Wires crossing withouta dot make no connection Let s draw a schematic for =

11 Example: From Logic to Gates Recall that we can simplify = + + to = + This means we can draw a much simpler diagram, resulting in a cheaper, faster circuit (hopefully!) Example: From Logic to Gates But wait, we can do even better! Both inputs to the top AND gate are complemented, and nowhere else do we use. This means we can use a NOR gate (drawn using AND) and eliminate one gate (an inverter) Example: Majority Circuit Example: Majority Circuit The majority function takes several 1-bit inputs and outputs 1 if a majority of the inputs is 1 Consider the 3-bit majority function (,, ) A B C (,,) Write Fin sum-of-products form and draw its circuit diagram

12 Example: Majority Circuit Example: Majority Circuit The majority function takes several 1-bit inputs and outputs 1 if a majority of the inputs is 1 Consider the 3-bit majority function (,, ) A B C (,,) Write Fin product-of-sums form and draw its circuit diagram Multiple-Output Circuits 4-bit Priority Circuit Example: Priority circuit the output is asserted which corresponds to the most significant TRUE input Suppose we have four inputs:,,, and they are given in order of priority,,, (i.e., has highest priority and has lowest) We have four outputs,,, and corresponding to these four inputs, respectively is TRUE whenever asserted is TRUE if is asserted and is not is TRUE if is asserted and neither nor is is TRUE if is asserted and no other is 0 if no input is asserted How would we implement this circuit? First, derive Boolean expressions for,,, 55 56

13 4-bit Priority Circuit 4-bit Priority Circuit: Don t Cares In that truth table, we actually don t care what certain values are in many cases For example, if is asserted, it doesn t matter what the other three values are; they could be any combination of 1s and 0s and we don t care We write the letter X when we want to indicate that we don t care whata particular input or output is We will see in a later unit how to exploit don t cares to simplify circuits Example: p. 99, #2.18(a) Simplify =

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Chapter 5: Sequential Circuits (LATCHES)

Chapter 5: Sequential Circuits (LATCHES) Latches We focuses on sequential circuits, where we add memory to the hardware that we ve already seen Our schedule will be very similar to before: We first show

### 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

### 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

### 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

### 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

### 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

### 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)

### 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

### 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

### 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

### 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/./,

### Boolean Algebra. Boolean Algebra. Boolean Algebra. Boolean Algebra

2 Ver..4 George Boole was an English mathematician of XIX century can operate on logic (or Boolean) variables that can assume just 2 values: /, true/false, on/off, closed/open Usually value is associated

### Switching Algebra and Logic Gates

Chapter 2 Switching Algebra and Logic Gates The word algebra in the title of this chapter should alert you that more mathematics is coming. No doubt, some of you are itching to get on with digital design

### 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

### CHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.

CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:

### 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

### 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.

### 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

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### 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

### 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

### 2011, The McGraw-Hill Companies, Inc. Chapter 4

Chapter 4 4.1 The Binary Concept Binary refers to the idea that many things can be thought of as existing in only one of two states. The binary states are 1 and 0 The 1 and 0 can represent: ON or OFF Open

### 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

### 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

### Gate: A simple electronic circuit (a system) that realizes a logical operation.

Gates Gate: simple electronic circuit (a system) that realizes a logical operation. The direction of information flow is from the input terminals to the output terminal. The number of input and output

### Chapter 19 Operational Amplifiers

Chapter 19 Operational Amplifiers The operational amplifier, or op-amp, is a basic building block of modern electronics. Op-amps date back to the early days of vacuum tubes, but they only became common

### Digital Electronics Detailed Outline

Digital Electronics Detailed Outline Unit 1: Fundamentals of Analog and Digital Electronics (32 Total Days) Lesson 1.1: Foundations and the Board Game Counter (9 days) 1. Safety is an important concept

### 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

### 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

### Binary Adder. sum of 2 binary numbers can be larger than either number need a carry-out to store the overflow

Binary Addition single bit addition Binary Adder sum of 2 binary numbers can be larger than either number need a carry-out to store the overflow Half-Adder 2 inputs (x and y) and 2 outputs (sum and carry)

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### Using Logic to Design Computer Components

CHAPTER 13 Using Logic to Design Computer Components Parallel and sequential operation In this chapter we shall see that the propositional logic studied in the previous chapter can be used to design digital

### 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

### EE360: Digital Design I Course Syllabus

: Course Syllabus Dr. Mohammad H. Awedh Fall 2008 Course Description This course introduces students to the basic concepts of digital systems, including analysis and design. Both combinational and sequential

### 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

### 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

### 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

### FORDHAM UNIVERSITY CISC 3593. Dept. of Computer and Info. Science Spring, 2011. Lab 2. The Full-Adder

FORDHAM UNIVERSITY CISC 3593 Fordham College Lincoln Center Computer Organization Dept. of Computer and Info. Science Spring, 2011 Lab 2 The Full-Adder 1 Introduction In this lab, the student will construct

### 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

### Chapter 4 Register Transfer and Microoperations. Section 4.1 Register Transfer Language

Chapter 4 Register Transfer and Microoperations Section 4.1 Register Transfer Language Digital systems are composed of modules that are constructed from digital components, such as registers, decoders,

### Figure 2.1(a) Bistable element circuit.

3.1 Bistable Element Let us look at the inverter. If you provide the inverter input with a 1, the inverter will output a 0. If you do not provide the inverter with an input (that is neither a 0 nor a 1),

### Lecture Summary Module 2 Combinational Logic Circuits

Lecture Summary Module 2 Combinational Logic Circuits Learning Outcome: an ability to analyze and design combinational logic circuits Learning Objectives: 2-1. identify minterms (product terms) and maxterms

### Having read this workbook you should be able to: recognise the arrangement of NAND gates used to form an S-R flip-flop.

Objectives Having read this workbook you should be able to: recognise the arrangement of NAND gates used to form an S-R flip-flop. describe how such a flip-flop can be SET and RESET. describe the disadvantage

### 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

### #5. Show and the AND function can be constructed from two NAND gates.

Study Questions for Digital Logic, Processors and Caches G22.0201 Fall 2009 ANSWERS Digital Logic Read: Sections 3.1 3.3 in the textbook. Handwritten digital lecture notes on the course web page. Study

### 4 Combinational Components

Chapter 4 Combinational Components Page of 8 4 Combinational Components In constructing large digital circuits, instead of starting with the basic gates as building blocks, we often start with larger building

### CS311 Lecture: Sequential Circuits

CS311 Lecture: Sequential Circuits Last revised 8/15/2007 Objectives: 1. To introduce asynchronous and synchronous flip-flops (latches and pulsetriggered, plus asynchronous preset/clear) 2. To introduce

### 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

### 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.

### 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

### 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

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Combinational Logic Design

Chapter 4 Combinational Logic Design The foundations for the design of digital logic circuits were established in the preceding chapters. The elements of Boolean algebra (two-element switching algebra

### 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

### 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.

### 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

### 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...

### A Course Material on DIGITAL PRINCIPLES AND SYSTEM DESIGN

A Course Material on By MS.G.MANJULA ASSISTANT PROFESSOR DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM 638 56 QUALITY CERTIFICATE This is to certify

### Chapter 4. Arithmetic for Computers

Chapter 4 Arithmetic for Computers Arithmetic Where we've been: Performance (seconds, cycles, instructions) What's up ahead: Implementing the Architecture operation a b 32 32 ALU 32 result 2 Constructing

### Flip-Flops, Registers, Counters, and a Simple Processor

June 8, 22 5:56 vra235_ch7 Sheet number Page number 349 black chapter 7 Flip-Flops, Registers, Counters, and a Simple Processor 7. Ng f3, h7 h6 349 June 8, 22 5:56 vra235_ch7 Sheet number 2 Page number

### 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

### 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

### exclusive-or and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576

exclusive-or and Binary Adder R eouven Elbaz reouven@uwaterloo.ca Office room: DC3576 Outline exclusive OR gate (XOR) Definition Properties Examples of Applications Odd Function Parity Generation and Checking

### 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

### Sistemas Digitais I LESI - 2º ano

Sistemas Digitais I LESI - 2º ano Lesson 6 - Combinational Design Practices Prof. João Miguel Fernandes (miguel@di.uminho.pt) Dept. Informática UNIVERSIDADE DO MINHO ESCOLA DE ENGENHARIA - PLDs (1) - The

### Introduction to Fractions

Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying

### 6. BOOLEAN LOGIC DESIGN

6. OOLEN LOGI DESIGN 89 Topics: oolean algebra onverting between oolean algebra and logic gates and ladder logic Logic examples Objectives: e able to simplify designs with oolean algebra 6. INTRODUTION

### NAND/NOR 1: 1 2: 4 3: 6 4: 8 5: 10 6: 11 7: 14 8: 18 9: 20 10: 24 11: 26 12: 28 13: BCD

Table of Contents Experiment 1: Elementary Boolean Functions...1 Experiment 2: The Properties of Boolean Functions...4 Experiment 3: Theorems and Canonical Forms...6 Experiment 4: Two-Level Functions...8

### 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

### Binary full adder. 2-bit ripple-carry adder. CSE 370 Spring 2006 Introduction to Digital Design Lecture 12: Adders

SE 370 Spring 2006 Introduction to Digital Design Lecture 12: dders Last Lecture Ls and Ls Today dders inary full 1-bit full omputes sum, carry-out arry-in allows cascaded s = xor xor = + + 32 ND2 11 ND2

Adder.T(//29) 5. Lecture 3 Adder ircuits Objectives Understand how to add both signed and unsigned numbers Appreciate how the delay of an adder circuit depends on the data values that are being added together

### Circuits and Boolean Expressions

Circuits and Boolean Expressions Provided by TryEngineering - Lesson Focus Boolean logic is essential to understanding computer architecture. It is also useful in program construction and Artificial Intelligence.

### Part 2: Operational Amplifiers

Part 2: Operational Amplifiers An operational amplifier is a very high gain amplifier. Op amps can be used in many different ways. Two of the most common uses are a) as comparators b) as amplifiers (either

### 7. Latches and Flip-Flops

Chapter 7 Latches and Flip-Flops Page 1 of 18 7. Latches and Flip-Flops Latches and flip-flops are the basic elements for storing information. One latch or flip-flop can store one bit of information. The

### Figure 8-1 Four Possible Results of Adding Two Bits

CHPTER EIGHT Combinational Logic pplications Thus far, our discussion has focused on the theoretical design issues of computer systems. We have not yet addressed any of the actual hardware you might find

### Integer multiplication

Integer multiplication Suppose we have two unsigned integers, A and B, and we wish to compute their product. Let A be the multiplicand and B the multiplier: A n 1... A 1 A 0 multiplicand B n 1... B 1 B