CSE140: Components and Design Techniques for Digital Systems. Introduction. Instructor: Mohsen Imani
|
|
- Mildred Ruth French
- 5 years ago
- Views:
Transcription
1 CSE4: Components and Design Techniques for Digital Systems Introduction Instructor: Mohsen Imani Slides from: Prof.Tajana Simunic Rosing & Dr.Pietro Mercati
2 Welcome to CSE 4! Instructor: Mohsen Imani please put CSE4 in the subject line Class: Tue/Thr :am-:5pm at PCYNH 22 Office Hours: Tue 3-5pm CSE 227 Discussion sessions: Tue/Thr -am at PCYNH 22 Course website: Announcements and online discussion: SIGN UP SOON!!!! TAs and Tutors: Office hours listed on the class website/piazza
3 Grading: Grading Class participation using iclicker: 5% 4 Homeworks: 5% each Midterm: 3% Final: 45% Homeworks are out on Tuesdays and are due on the following Tuesday before am Tue discussion session will go over the HW solution Submission via gradescope: Only a subset of problems will be graded Late submissions will not be accepted! Homework will be out right after this class
4 Academic Honesty Some Class Policies Studying together in groups is encouraged Turned-in work must be completely your own Both giver and receiver are equally culpable Cheating on HW/ exams: F in the course. Any instance of cheating will be referred to Academic Integrity Office
5 Textbooks and Recommended Readings Recommended textbook: digital design by F. Vahid Other recommended textbooks: Digital Design by F. Vahid Digital Design & Computer Arch. by David & Sarah Harris Contemporary Logic Design by R. Katz & G. Borriello Lecture slides are derived from the slides designed for all three books
6 CSE4 & other CSE classes A pplication S oftw are Layers of abstraction program s O perating S ystem s device drivers CSE 3 CSE 4 CSE 4 focus of this course A rchitecture M icroarchitecture Logic D igital C ircuits instructions registers datapaths controllers adders m em ories A N D gates N O T gates Abstraction: A way to simplify by hiding details from other layers A nalog C ircuits D evices P h ys ic s am plifiers filte rs transistors diodes electrons
7 Why Study Digital Design? Look under the hood of your processors You become a better programmer when you understand hardware your code runs on Nvidia Tegra 2 die photo
8 The Scope of CSE4 We start with Boolean algebra Y = A and B End up with the design of a simple CPU What s next? CSE4 more complex CPU architectures Nvidia Tegra 2 die photo
9 Lets get started! Number representations Analog vs. Digital Digital representations: Binary, Hexadecimal, Octal Switches, MOS transistors, Logic gates What is a switch How a transistor operates Logic gates Building larger functions from logic gates Universal gates Boolean algebra Properties How Boolean algebra can be used to design logic circuits
10 What Does Digital Mean? Analog signal Infinite possible values Ex: voltage on a wire created by microphone Digital signal Finite possible values Ex: button pressed on a keypad Sound waves microphone value move the membrane, which moves the magnet, which creates current in the nearby wire analog signal Possible values:.,., 2.9,... infinite possibilities Which is analog? A) Wind speed B) Radio Signal C) Clicker response D) A) & B) E) All of the above value digital signal Possible values:,, 2, 3, or 4. That s it. time time
11 Encoding Numbers Base & 2 Each position represents a quantity; symbol in position means how many of that quantity Base ten (decimal) Ten symbols:,, 2,..., 8, and 9 More than 9 -- next position So each position power of Nothing special about base -- used because we have fingers Base two (binary) Two symbols: and More than -- next position So each position power of 2 Decimal to binary: - Divide by two - Report the reminder - Concatenate the reminders NUMBER REMINDER 523 =
12
13 Encoding Numbers Base & 2 Each position represents a quantity; symbol in position means how many of that quantity Base ten (decimal) Ten symbols:,, 2,..., 8, and 9 More than 9 -- next position So each position power of Nothing special about base -- used because we have fingers Base two (binary) Two symbols: and More than -- next position So each position power of 2 Binary to decimal: - Each position is a power of 2 - Sum up all the powers of 2 where you have a = 523
14 Bases Sixteen & Eight hex binary 8 A F A F hex 8 9 A B C D E F binary Base sixteen Used as compact way to write binary numbers Basic digits: -9, A-F Known as hexadecimal, or just hex Base eight Basic digits: -7 Known as octal octal binary
15 Bases Sixteen & Eight hex binary hex binary Hexadecimal to binary: Expand each hexadecimal digit into the corresponding binary string A B C D E F 8 A F = Binary to hexadecimal: - Pad the binary string with zeros on the left until you have a number of digits multiple of 4 - Group digits 4-by-4 starting from the right and convert into the corresponding hexadecimal symbol 2 D
16 Question AB is a hex value. What s the equivalent value in Octal? A. 534 B. 526 C. AB D. 342 E. None
17 Question 38 is a decimal value. What s the equivalent value in hex? A. 6 B. 2A C. 4 D. E. None
18 Combinational circuit building blocks: Switches & CMOS transistors
19 CMOS circuit CMOS Switches Consists of N and PMOS transistors Both N and PMOS are similar to basic switches A positive voltage here......attracts electrons here, turning the channel between source and drain into a conductor. nmos gate source gate oxide drain IC package conducts d o e s n o t conduct pmos gate (a) IC Silicon -- not quite a conductor or insulator: Semiconductor does not conduct conducts
20 Transistor Circuit Design nmos: Turns on when gate is connected to When turned on, nmos passes zeros well, but not ones, so connect source to GND nmos forms a pull-down network pmos: Turns on when gate is connected to When turned on, pmos passes ones well, but not zeros, so connect source to V DD pmos forms a pull-up network Note: Vahid s textbook shows some circuits with pmos connected to GND and nmos to Vdd: this is NOT normally done in practice! inputs V DD pmos pull-up network nmos pull-down network output
21 CMOS Switches The following is true for CMOS switches: A. nmos turns on when gate is connected to logic B. pmos is an open switch when gate is connect to logic C. All of the above D. None of the above A positive voltage here......attracts electrons here, turning the channel between source and drain into a conductor. nmos g a t e gate source oxide drain IC package pmos gate (a) IC
22 Logic gates: CMOS NOT Gate NOT V DD A Y = A Y A P Y N A Y GND A P N Y
23 CMOS Two Input NAND Gate A B NAND Y = AB Y A B Y V DD P2 P Y A B N N2 GND A B P P2 N N2 Y
24 Common Logic Gates a b AND a b a b OR a+b a NOT a a BUF a (a b) (a+b) ab + a b ab + a b A B Y N2 N P2 P V DD A Y GND N P a b NAND a b NOR a b XNOR a b XOR
25 B = {, } Variables represent or only Operators return or only Basic operators Boolean algebra Intersection: is logical AND: a AND b returns only when a= & b= Union: + is logical OR: a OR b returns if either a= or b= (or both) Complement: is logical NOT: NOT a returns the opposite of a AND OR NOT BUFFER a b a b AND Derived operators: a+b a b OR NAND NOR XOR XNOR a b NAND (a b) (a+b) a b NOR a a a b NOT XOR a a a b BUF XNOR
26 Boolean Algebra X and Y are Boolean variables with X=, Y= What is X+X+Y? A. B. C. 2 D. None of the above
27 BREAK!
28 Universal Gate: NAND Any logic function can be implemented using just NAND gates. Boolean algebra s basic operators are AND, OR and NOT
29 F= A + B C Gate level Implementation
30 F= (A+B).(C+D) Gate level Implementation What s the minimum number of NAND gates to implement this function? A. 4 B. 8 C. 6 D.
31 Universal Gate: NOR Any logic function can be implemented using just NOR gates. Boolean algebra needs AND, OR and NOT
32 Boolean Axioms & Theorems
33 Boolean theorems of multiple variables
34 Boolean Duality Derived by replacing by +, + by, by, and by & leaving variables unchanged Generalized duality: X + Y +... X Y... f (X,X 2,...,X n,,,+, ) f(x,x 2,...,X n,,,,+) Any theorem that can be proven is also proven for its dual! Note: this is NOT demorgan s Law
35 Covering Theorem Explained Covering Theorem: A*(A+B) = A+A*B = A Venn Diagrams
36 Combining Theorem Explained Combining Theorem: AB+AB = (A+B)*(A+B ) = A
37 Proving theorems with Boolean Algebra Using the axioms of Boolean algebra: e.g., prove the consensus theorem: X Y + X Y = X distributivity X Y + X Y = X (Y + Y ) complementarity X (Y + Y ) = X () identity X () = X e.g., prove the covering theorem: X + X Y = X identity X + X Y = X + X Y distributivity X + X Y = X ( + Y) identity X ( + Y) = X () identity X () = X
38 Consensus Theorem of 3 Variables Consensus Theorem: AB+B C+AC = AB+B C
39 Proof of Consensus Theorem with Boolean Algebra Consensus Theorem: (X Y) + (Y Z) + (X Z) = X Y + X Z (X Y) + (Y Z) + (X Z) identity (X Y) + () (Y Z) + (X Z) complementarity (X Y) + (X + X) (Y Z) + (X Z) distributivity (X Y) + (X Y Z) + (X Y Z) + (X Z) commutativity (X Y) + (X Y Z) + (X Y Z) + (X Z) factoring (X Y) ( + Z) + (X Z) ( + Y) null (X Y) () + (X Z) () identity (X Y) + (X Z)
40 Applying Boolean Algebra Theorems Which of the following is CB+BA+C A equal to? A. AB+AC B. BC+AC C. AB+BC D. None of the above
41 DeMorgan s Theorem (Bubble Pushing) Y = AB = A + B A B A B Y Y Y = A + B = A B A B A B Y Y
42 Example of Transforming Circuits with Bubble Pushing 42
43 F = X Y + Z Implement using only NORs
44 Proving Theorems with Perfect Induction Using perfect induction = complete the truth table: e.g., de Morgan s: (X + Y) = X Y NOR is equivalent to AND with inputs complemented X Y X Y (X + Y) X Y (X Y) = X + Y NAND is equivalent to OR with inputs complemented X Y X Y (X Y) X + Y
45 Specifying Logic Problems with Truth Tables Half Adder: Truth table -> Boolean Eq. -> Logic Circuit a Truth Table a b carry sum b Sum Carry 45
46 Going from a Truth Table to a Boolean Equation Truth Table a b carry sum Boolean Equation: Sum (a, b) = a b + ab = = a XOR b Carry (a, b) = ab 46
47 How do we get a Boolean Equation from the Truth Table? Truth Table a b carry sum How? Boolean Equation Sum (a, b) = a b + ab Carry (a, b) = ab 47
48 Need Some Definitions Complement: variable with a bar over it or a A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Implicate: sum of literals (A+B+C), (A+C), (B+C) Minterm: AND that includes all input variables ABC, A BC, AB C Maxterm: OR that includes all input variables (A+B+C), (A +B+C), (A +B +C) 48
49 Minterms of Two Input Functions A B Minterm Maxterm A B A B AB AB A+B A+B A +B A +B
50 CSE4: Components and Design Techniques for Digital Systems Canonical representation 5
51 Canonical Form -- Sum of Minterms Truth tables are too big for numerous inputs Use standard form of equation instead Known as canonical form Regular algebra: group terms of polynomial by power ax 2 + bx + c (3x 2 + 4x + 2x > 5x 2 + 4x + 4) Boolean algebra: create a sum of minterms (or a product of maxterms) 5
52 Canonical form? Is F(a,b)=ab+a in canonical form? A) True B) False 52
53 Sum-of-products Canonical Form Also known as disjunctive normal form Minterm expansion: F = F = A B C + A BC + AB C + ABC + ABC A B C F F F = A B C + A BC + AB C 53
54 Sum-of-products canonical form (cont d) Product minterm ANDed product of literals input combination for which output is each variable appears exactly once, true or inverted (but not both) You can write a canonical form in multiple ways A B C minterms A B C m A B C m A BC m2 A BC m3 AB C m4 AB C m5 ABC m6 ABC m7 short-hand notation for minterms of 3 variables F F in canonical form: F(A, B, C) = m(,3,5,6,7) = m + m3 + m5 + m6 + m7 = A B C + A BC + AB C + ABC + ABC canonical form minimal form F(A, B, C) = A B C + A BC + AB C + ABC + ABC = (A B + A B + AB + AB)C + ABC = ((A + A)(B + B))C + ABC = C + ABC = ABC + C = AB + C 54
55 Sum of Products Canonical Form Minterm A B Carry Sum A B A B AB AB I. sum(a,b) = II. carry(a,b)= 55
56 Sum of Products Canonical Form A B Y Does the following SOP canonical expression correctly express the above truth table: A.Yes B.No Y(A,B)= Σm(2,3) 56
57 Product-of-sums canonical form Also known as conjunctive normal form Also known as maxterm expansion Implements zeros of a function A B C F F F = F = (A + B + C) (A + B + C) (A + B + C) F = (A + B + C ) (A + B + C ) (A + B + C ) (A + B + C) (A + B + C ) 57
58 Product-of-sums canonical form (cont d) Sum term (or maxterm) ORed sum of literals input combination for which output is false each variable appears exactly once, true or inverted (but not both) A B C maxterms A+B+C M A+B+C M A+B +C M2 A+B +C M3 A +B+C M4 A +B+C M5 A +B +C M6 A +B +C M7 F F in canonical form: F(A, B, C) = M(,2,4) = M M2 M4 = (A + B + C) (A + B + C) (A + B + C) canonical form minimal form 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) = (A + C) (B + C) short-hand notation for maxterms of 3 variables 58
59 What we reviewed thus far: Summary Number representations Switches, logic gates Universal gates Boolean algebra Using Boolean algebra to simplify Boolean equations There is an easier way! What is next: Combinational logic: Minimization (the easier way) Designs of common combinational circuits Adders, comparators, subtractors, multipliers, etc.
60 Claude Shannon: Historical Note In 936, Shannon began his graduate studies in electrical engineering at MIT While studying the complicated ad hoc circuits of this analyzer, Shannon designed switching circuits based on Boole's concepts. In 937, he wrote his master's degree thesis, A Symbolic Analysis of Relay and Switching Circuits, [9] A paper from this thesis was published in 938. [] In this work, Shannon proved that his switching circuits could be used to simplify the arrangement of the electromechanical relays that were used then in telephone call routing switches. Next, he expanded this concept, proving that these circuits could solve all problems that Boolean algebra could solve. In the last chapter, he presents diagrams of several circuits, including a 4-bit full adder. [9] Using this property of electrical switches to implement logic is the fundamental concept that underlies all electronic digital computers
CSE140: Components and Design Techniques for Digital Systems. Introduction. Prof. Tajana Simunic Rosing
CSE4: Components and Design Techniques for Digital Systems Introduction Prof. Tajana Simunic Rosing Welcome to CSE 4! Instructor: Tajana Simunic Rosing Email: tajana.teach.ucsd@gmail.com; please put CSE4
More informationCombinational Logic. Review of Combinational Logic 1
Combinational Logic! Switches -> Boolean algebra! Representation of Boolean functions! Logic circuit elements - logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational
More informationSwitches: basic element of physical implementations
Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by re-writing and by perfect induction Winter 200 CSE370 - II - Boolean Algebra Switches: basic element
More informationCPE100: Digital Logic Design I
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu CPE100: Digital Logic Design I Midterm01 Review http://www.ee.unlv.edu/~b1morris/cpe100/ 2 Logistics Thursday Oct. 5 th In normal lecture (13:00-14:15)
More informationLecture A: Logic Design and Gates
Lecture A: Logic Design and Gates Syllabus My office hours 9.15-10.35am T,Th or gchoi@ece.tamu.edu 333G WERC Text: Brown and Vranesic Fundamentals of Digital Logic,» Buy it.. Or borrow it» Other book:
More information2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS
CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS What will we learn? 2 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0 Terminology 3 Digital
More informationEECS Variable Logic Functions
EECS150 Section 1 Introduction to Combinational Logic Fall 2001 2-Variable Logic Functions There are 16 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs X
More informationEEE130 Digital Electronics I Lecture #4
EEE130 Digital Electronics I Lecture #4 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi Topics to be discussed 4-1 Boolean Operations and Expressions 4-2 Laws and Rules of Boolean
More informationChapter 2: Switching Algebra and Logic Circuits
Chapter 2: Switching Algebra and Logic Circuits Formal Foundation of Digital Design In 1854 George Boole published An investigation into the Laws of Thoughts Algebraic system with two values 0 and 1 Used
More informationAdministrative Notes. Chapter 2 <9>
Administrative Notes Note: New homework instructions starting with HW03 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2
More informationEECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive
EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150 - Lec19-cl1 Page 1 Boolean Algebra I (Representations of Combinational
More informationIntro To Digital Logic
Intro To Digital Logic 1 Announcements... Project 2.2 out But delayed till after the midterm Midterm in a week Covers up to last lecture + next week's homework & lab Nick goes "H-Bomb of Justice" About
More informationFloating Point Representation and Digital Logic. Lecture 11 CS301
Floating Point Representation and Digital Logic Lecture 11 CS301 Administrative Daily Review of today s lecture w Due tomorrow (10/4) at 8am Lab #3 due Friday (9/7) 1:29pm HW #5 assigned w Due Monday 10/8
More informationChapter 2 Combinational logic
Chapter 2 Combinational logic Chapter 2 is very easy. I presume you already took discrete mathemtics. The major part of chapter 2 is boolean algebra. II - Combinational Logic Copyright 24, Gaetano Borriello
More informationContents. Chapter 2 Digital Circuits Page 1 of 30
Chapter 2 Digital Circuits Page 1 of 30 Contents Contents... 1 2 Digital Circuits... 2 2.1 Binary Numbers... 2 2.2 Binary Switch... 4 2.3 Basic Logic Operators and Logic Expressions... 5 2.4 Truth Tables...
More informationEECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits)
EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4-bool1 Page 1, 9/5 9am Outline Review of
More informationCS 226: Digital Logic Design
CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions
More informationLecture 2 Review on Digital Logic (Part 1)
Lecture 2 Review on Digital Logic (Part 1) Xuan Silvia Zhang Washington University in St. Louis http://classes.engineering.wustl.edu/ese461/ Grading Engagement 5% Review Quiz 10% Homework 10% Labs 40%
More informationOutline. EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined
EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) January 30, 2003 John Wawrzynek Outline Review of three representations for combinational logic:
More informationCombinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions.
Combinational logic Possible logic functions of two variables Logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,... Minimal set xioms and theorems of oolean algebra Proofs by re-writing
More informationBoolean Algebra. The Building Blocks of Digital Logic Design. Section. Section Overview. Binary Operations and Their Representation.
Section 3 Boolean Algebra The Building Blocks of Digital Logic Design Section Overview Binary Operations (AND, OR, NOT), Basic laws, Proof by Perfect Induction, De Morgan s Theorem, Canonical and Standard
More informationNumber System conversions
Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number
More informationSchool of Computer Science and Electrical Engineering 28/05/01. Digital Circuits. Lecture 14. ENG1030 Electrical Physics and Electronics
Digital Circuits 1 Why are we studying digital So that one day you can design something which is better than the... circuits? 2 Why are we studying digital or something better than the... circuits? 3 Why
More informationELEC Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2)
ELEC 2200-002 Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2) Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn,
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev
E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean
More informationCombinational Logic Design Principles
Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems
More informationfor Digital Systems Simplification of logic functions Tajana Simunic Rosing Sources: TSR, Katz, Boriello & Vahid
SE140: omponents and Design Techniques for Digital Systems Simplification of logic functions Tajana Simunic Rosing 1 What we covered thus far: Number representations Where we are now inary, Octal, Hex,
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates
Digital Circuits & Systems Lecture Transparencies (Boolean lgebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean lgebra & Logic Gates Major topics Boolean algebra NND & NOR gates Boolean algebra
More informationL2: Combinational Logic Design (Construction and Boolean Algebra)
L2: Combinational Logic Design (Construction and Boolean Algebra) Acknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. Borriello, Contemporary Logic Design (second edition), Pearson
More informationCh 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1
Ch 2. Combinational Logic II - Combinational Logic Contemporary Logic Design 1 Combinational logic Define The kind of digital system whose output behavior depends only on the current inputs memoryless:
More informationCS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman
CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions
More informationLecture 22 Chapters 3 Logic Circuits Part 1
Lecture 22 Chapters 3 Logic Circuits Part 1 LC-3 Data Path Revisited How are the components Seen here implemented? 5-2 Computing Layers Problems Algorithms Language Instruction Set Architecture Microarchitecture
More informationDigital Circuit And Logic Design I. Lecture 3
Digital Circuit And Logic Design I Lecture 3 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinational-circuit analysis 4. Combinational-circuit synthesis Panupong
More informationPossible logic functions of two variables
ombinational logic asic logic oolean algebra, proofs by re-writing, proofs by perfect induction logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,..., minimal set Logic realization
More informationReview. EECS Components and Design Techniques for Digital Systems. Lec 06 Minimizing Boolean Logic 9/ Review: Canonical Forms
Review EECS 150 - Components and Design Techniques for Digital Systems Lec 06 Minimizing Boolean Logic 9/16-04 David Culler Electrical Engineering and Computer Sciences University of California, Berkeley
More informationSAMPLE ANSWERS MARKER COPY
Page 1 of 12 School of Computer Science 60-265-01 Computer Architecture and Digital Design Fall 2012 Midterm Examination # 1 Tuesday, October 23, 2012 SAMPLE ANSWERS MARKER COPY Duration of examination:
More informationCSE140: Components and Design Techniques for Digital Systems. Logic minimization algorithm summary. Instructor: Mohsen Imani UC San Diego
CSE4: Components and Design Techniques for Digital Systems Logic minimization algorithm summary Instructor: Mohsen Imani UC San Diego Slides from: Prof.Tajana Simunic Rosing & Dr.Pietro Mercati Definition
More informationIntroduction to Computer Engineering ECE 203
Introduction to Computer Engineering ECE 203 Northwestern University Department of Electrical Engineering and Computer Science Teacher: Robert Dick Office: L477 Tech Email: dickrp@ece.northwestern.edu
More informationBOOLEAN ALGEBRA. Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra
BOOLEAN ALGEBRA Introduction 1854: Logical algebra was published by George Boole known today as Boolean Algebra It s a convenient way and systematic way of expressing and analyzing the operation of logic
More informationCombinational Logic Fundamentals
Topic 3: Combinational Logic Fundamentals In this note we will study combinational logic, which is the part of digital logic that uses Boolean algebra. All the concepts presented in combinational logic
More informationSlides for Lecture 10
Slides for Lecture 10 ENEL 353: Digital Circuits Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 30 September, 2013 ENEL 353
More informationChapter 2 Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates The most common postulates used to formulate various algebraic structures are: 1. Closure. N={1,2,3,4 }, for any a,b N we obtain a unique c N by the operation
More informationChapter 2: Boolean Algebra and Logic Gates
Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a
More informationCHAPTER1: Digital Logic Circuits Combination Circuits
CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.
More informationCs302 Quiz for MID TERM Exam Solved
Question # 1 of 10 ( Start time: 01:30:33 PM ) Total Marks: 1 Caveman used a number system that has distinct shapes: 4 5 6 7 Question # 2 of 10 ( Start time: 01:31:25 PM ) Total Marks: 1 TTL based devices
More informationECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering
ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of
More informationUNIVERSITI TENAGA NASIONAL. College of Information Technology
UNIVERSITI TENAGA NASIONAL College of Information Technology BACHELOR OF COMPUTER SCIENCE (HONS.) FINAL EXAMINATION SEMESTER 2 2012/2013 DIGITAL SYSTEMS DESIGN (CSNB163) January 2013 Time allowed: 3 hours
More informationChapter 1 :: From Zero to One
Chapter 1 :: From Zero to One Digital Design and Computer Architecture David Money Harris and Sarah L. Harris Copyright 2007 Elsevier 1- Chapter 1 :: Topics Background The Game Plan The Art of Managing
More informationChapter 2: Princess Sumaya Univ. Computer Engineering Dept.
hapter 2: Princess Sumaya Univ. omputer Engineering Dept. Basic Definitions Binary Operators AND z = x y = x y z=1 if x=1 AND y=1 OR z = x + y z=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 Boolean Algebra
More informationChap 2. Combinational Logic Circuits
Overview 2 Chap 2. Combinational Logic Circuits Spring 24 Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization Two-Level Optimization
More informationCombinational Logic (mostly review!)
ombinational Logic (mostly review!)! Logic functions, truth tables, and switches " NOT, N, OR, NN, NOR, OR,... " Minimal set! xioms and theorems of oolean algebra " Proofs by re-writing " Proofs by perfect
More information4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra
4 Switching Algebra 4.1 Axioms; Signals and Switching Algebra To design a digital circuit that will perform a required function, it is necessary to manipulate and combine the various input signals in certain
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter
More informationMC9211 Computer Organization
MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (29-2/ODD) (29 - / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra
More informationWhy digital? Overview. Number Systems. Binary to Decimal conversion
Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not
More informationEC-121 Digital Logic Design
EC-121 Digital Logic Design Lecture 2 [Updated on 02-04-18] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What
More informationCprE 281: Digital Logic
CprE 281: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Design Examples CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev
More informationChapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms
Chapter 2 (Lect 2) Canonical and Standard Forms Sum of Minterms Product of Maxterms Standard Form Sum of products Product of sums Other Logic Operators Logic Gates Basic and Multiple Inputs Positive and
More informationE40M. Binary Numbers. M. Horowitz, J. Plummer, R. Howe 1
E40M Binary Numbers M. Horowitz, J. Plummer, R. Howe 1 Reading Chapter 5 in the reader A&L 5.6 M. Horowitz, J. Plummer, R. Howe 2 Useless Box Lab Project #2 Adding a computer to the Useless Box alows us
More informationEx: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC.
Boolean Expression Forms: Sum-of-products (SOP) Write an AND term for each input combination that produces a 1 output. Write the input variable if its value is 1; write its complement otherwise. OR the
More informationcontrol in out in out Figure 1. Binary switch: (a) opened or off; (b) closed or on.
Chapter 2 Digital Circuits Page 1 of 18 2. Digital Circuits Our world is an analog world. Measurements that we make of the physical objects around us are never in discrete units but rather in a continuous
More informationSlide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary
Slide Set 3 for ENEL 353 Fall 2016 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2016 SN s ENEL 353 Fall 2016 Slide Set 3 slide
More informationGates and Logic: From Transistors to Logic Gates and Logic Circuits
Gates and Logic: From Transistors to Logic Gates and Logic Circuits Prof. Hakim Weatherspoon CS 3410 Computer Science Cornell University The slides are the product of many rounds of teaching CS 3410 by
More informationGates and Logic: From switches to Transistors, Logic Gates and Logic Circuits
Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University See: P&H ppendix C.2 and C.3 (lso, see C.0 and
More informationCPE100: Digital Logic Design I
Chapter 2 Professor Brendan Morris, SEB 326, brendan.morris@unlv.edu http://www.ee.unlv.edu/~bmorris/cpe/ CPE: Digital Logic Design I Section 4: Dr. Morris Combinational Logic Design Chapter 2 Chapter
More informationWeek-I. Combinational Logic & Circuits
Week-I Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and
More informationCprE 281: Digital Logic
CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Examples of Solved Problems CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander
More informationDigital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit
Digital Techniques 1. Binary System The digital computer is the best example of a digital system. A main characteristic of digital system is its ability to manipulate discrete elements of information.
More informationUC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits
2 Wawrzynek, Garcia 2004 c UCB UC Berkeley College of Engineering, EECS Department CS61C: Representations of Combinational Logic Circuits 1 Introduction Original document by J. Wawrzynek (2003-11-15) Revised
More information211: Computer Architecture Summer 2016
211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic - Storage: Recap - Review: cache hit rate - Project3 - Digital Logic: - truth table => SOP - simplification: Boolean
More informationCMPE12 - Notes chapter 1. Digital Logic. (Textbook Chapter 3)
CMPE12 - Notes chapter 1 Digital Logic (Textbook Chapter 3) Transistor: Building Block of Computers Microprocessors contain TONS of transistors Intel Montecito (2005): 1.72 billion Intel Pentium 4 (2000):
More informationComputer Organization: Boolean Logic
Computer Organization: Boolean Logic Representing and Manipulating Data Last Unit How to represent data as a sequence of bits How to interpret bit representations Use of levels of abstraction in representing
More informationECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2
ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 Instructor: Andrew B. Kahng (lecture) Email: abk@ucsd.edu Telephone: 858-822-4884 office, 858-353-0550 cell Office: 3802
More informationKP/Worksheets: Propositional Logic, Boolean Algebra and Computer Hardware Page 1 of 8
KP/Worksheets: Propositional Logic, Boolean Algebra and Computer Hardware Page 1 of 8 Q1. What is a Proposition? Q2. What are Simple and Compound Propositions? Q3. What is a Connective? Q4. What are Sentential
More informationCSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication
CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm
More informationBoolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.
The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or
More informationTotal Time = 90 Minutes, Total Marks = 50. Total /50 /10 /18
University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Midterm Examination Instructor: M. Sachdev October 23rd, 2007 Total Time = 90 Minutes, Total
More informationChapter-2 BOOLEAN ALGEBRA
Chapter-2 BOOLEAN ALGEBRA Introduction: An algebra that deals with binary number system is called Boolean Algebra. It is very power in designing logic circuits used by the processor of computer system.
More informationLearning Objectives 10/7/2010. CE 411 Digital System Design. Fundamental of Logic Design. Review the basic concepts of logic circuits. Dr.
/7/ CE 4 Digital ystem Design Dr. Arshad Aziz Fundamental of ogic Design earning Objectives Review the basic concepts of logic circuits Variables and functions Boolean algebra Minterms and materms ogic
More informationLab 1 starts this week: go to your session
Lecture 3: Boolean Algebra Logistics Class email sign up Homework 1 due on Wednesday Lab 1 starts this week: go to your session Last lecture --- Numbers Binary numbers Base conversion Number systems for
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Chapter 2 - Part 1 2 Chapter 2 - Part 1 3 Chapter 2 - Part 1 4 Chapter 2 - Part
More informationFundamentals of Computer Systems
Fundamentals of Computer Systems Boolean Logic Stephen A. Edwards Columbia University Summer 2015 Boolean Logic George Boole 1815 1864 Boole s Intuition Behind Boolean Logic Variables X,,... represent
More informationCPE100: Digital Logic Design I
Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris Combinational Logic Design Chapter
More informationLogic Design. Chapter 2: Introduction to Logic Circuits
Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits perform operation on digital signal Digital signal: signal values are restricted to a few discrete values Binary logic
More informationGates and Logic: From Transistors to Logic Gates and Logic Circuits
Gates and Logic: From Transistors to Logic Gates and Logic Circuits Prof. Hakim Weatherspoon CS 3410 Computer Science Cornell University The slides are the product of many rounds of teaching CS 3410 by
More informationUnit 8A Computer Organization. Boolean Logic and Gates
Unit 8A Computer Organization Boolean Logic and Gates Announcements Bring ear buds or headphones to lab! 15110 Principles of Computing, Carnegie Mellon University - CORTINA 2 Representing and Manipulating
More informationFundamentals of Computer Systems
Fundamentals of Computer Systems Boolean Logic Stephen A. Edwards Columbia University Fall 2011 Boolean Logic George Boole 1815 1864 Boole s Intuition Behind Boolean Logic Variables x, y,... represent
More informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines Basic
More informationCS61c: Representations of Combinational Logic Circuits
CS61c: Representations of Combinational Logic Circuits J. Wawrzynek March 5, 2003 1 Introduction Recall that synchronous systems are composed of two basic types of circuits, combination logic circuits,
More informationMidterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes
Page 1 of 10 School of Computer Science 60-265-01 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name
More informationCSE140: Components and Design Techniques for Digital Systems. Decoders, adders, comparators, multipliers and other ALU elements. Tajana Simunic Rosing
CSE4: Components and Design Techniques for Digital Systems Decoders, adders, comparators, multipliers and other ALU elements Tajana Simunic Rosing Mux, Demux Encoder, Decoder 2 Transmission Gate: Mux/Tristate
More informationCMSC 313 Lecture 17. Focus Groups. Announcement: in-class lab Thu 10/30 Homework 3 Questions Circuits for Addition Midterm Exam returned
Focus Groups CMSC 33 Lecture 7 Need good sample of all types of CS students Mon /7 & Thu /2, 2:3p-2:p & 6:p-7:3p Announcement: in-class lab Thu /3 Homework 3 Questions Circuits for Addition Midterm Exam
More informationIntroduction to Computer Engineering. CS/ECE 252, Spring 2017 Rahul Nayar Computer Sciences Department University of Wisconsin Madison
Introduction to Computer Engineering CS/ECE 252, Spring 2017 Rahul Nayar Computer Sciences Department University of Wisconsin Madison Chapter 3 Digital Logic Structures Slides based on set prepared by
More informationEvery time has a value associated with it, not just some times. A variable can take on any value within a range
Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationNumber System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary
Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION
More informationChapter 2. Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Basic Definitions A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common
More informationCSE 140, Lecture 2 Combinational Logic CK Cheng CSE Dept. UC San Diego
CSE 140, Lecture 2 Combinational Logic CK Cheng CSE Dept. UC San Diego 1 Combinational Logic Outlines 1. Introduction 1. Scope 2. Review of Boolean lgebra 3. Review: Laws/Theorems and Digital Logic 2.
More informationStandard Expression Forms
ThisLecture will cover the following points: Canonical and Standard Forms MinTerms and MaxTerms Digital Logic Families 24 March 2010 Standard Expression Forms Two standard (canonical) expression forms
More information