Why digital? Overview. Number Systems. Binary to Decimal conversion
|
|
- Marianna McCarthy
- 5 years ago
- Views:
Transcription
1 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 affect much. Digital takes only two voltage levels either Low or High. It is because of switching devices we use like Diode, Transistors and relays. Number Systems inary rithmetic inary Codes Logic Gates oolean lgebra Karnaugh Map Combinational Circuit Design / 59 / 59 Number Systems ecause of two levels, binary numbers are used in digital systems. High means 5 V. Low means 0 V. In positive logic, High In negative logic Each binary digit is called a bit. Low 0 High 0 Low inary to Decimal conversion The right most bit of the binary number is the LS (least significant bit) whose weight is 0 =. The left most bit is the MS (most significant bit) whose weight depends on the size of binary number. Example: 00 = = = 0 In a fractional number, the left most bit is the MS whose weight is = = = / 59 4 / 59 Decimal to inary conversion systematic method of converting decimal to binary is repeated division by. Example : Convert 4 to binary Remainder 4 0 LS MS * - Stop when the whole number quotient is zero. Decimal fractions to inary Decimal fractions can be converted to binary by repeated multiplication by. Example : Convert 0.65 to binary. Carry 0.65 =.5 MS 0.5 = =.0 LS * - Continue to the desired number of decimal places or stop when the fractional part is all zeros = = / 59 6 / 59 Hexadecimal Numbers. It has sixteen characters.. It is very easy to convert between binary and hexadecimal. 3. When we program in machine language such as writing an instruction in microprocessor, it is easy to write in hexadecimal system Decimal inary Hexadecimal Decimal inary Hexadecimal C D E F 7 / 59 inary to Hexadecimal conversion reak the binary number in 4-bit groups, starting at the right most bit and replace each 4-bit group with the equivalent hexadecimal symbol. Example : convert 0000 and 0 to hexadecimal. 00 }{{} 00 }{{} = }{{} }{{} F 0 = F 6 * - two zeros have been added to complete a 4-bit group 8 / 59
2 Hexadecimal to inary conversion Replace each hexadecimal symbol with a 4-bit binary number. Example: Convert 0 6 and C 6 }{{} }{{} = 0000 }{{} C }{{} C 6 = Octal Numbers. It has eight digits.. Like hexadecimal number system, it provides a convenient way to express binary numbers. 3. It is very easy to convert between octal and binary. Decimal inary Octal However, it is used less frequently than hexadecimal. 9 / 59 0 / 59 inary to Octal conversion. reak the binary number in 3-bit groups, starting at the right most bit and replace each 3-bit group with the equivalent octal digit.. If there are not three bits in the left most group, add enough zeros to complete a 3-bit group. Example: Convert 0000 and 00 }{{} 0 }{{} = }{{} }{{} 0 6 Octal to inary conversion Replace each octal digit with a 3-bit binary number. Example: Convert 5 8 and 50 8 }{{} }{{} = 0 }{{} 5 }{{} = = 6 8 *- one zero has been added to complete a 3-bit group. / 59 / 59 inary ddition = = + 0 = + = 0 and carry to the next column Carrying a is equivalent to adding to that column. Example : dd 4 and 0 in binary 4 0 = = = 4 0 inary Subtraction 0-0 = = and borrow from the next column - 0 = - = 0 orrowing a from a column is equivalent to subtracting from that column. Example : Subtract 5 from 8 binary 8 0 = = = / 59 4 / 59 s and s complements Finding s complement s and s complements are mainly used to represent negative numbers. Finding s complement: s complement of a binary number is found by changing all s to 0s and all 0s to s. Example : inary number 0 0 s complement The s complement of a binary number is found by adding to the LS of the s component. Example : Find the s complement of inary number 0 0 s complement dd s complement 5 / 59 6 / 59
3 Subtraction Using s complement Example : Subtract 5 from 8. Find s complement of binary number corresponding to 5 and add with binary number of 8. s complement of 00 is Example : Subtract 8 from Discard carry and the answer is 00. Since there is no carry, the answer = -( s complement of 0) = / 59 inary Coded Decimal (CD) CD means that each decimal digit, 0 through 9, is represented by a binary code of four bits. It is very easy to convert from CD to decimal and vice versa. Only 0 combinations are used. The remaining 6 combinations are invalid. Decimal Digit CD pplications: Seven segment displays use CD codes. Example: Digital clocks, digital meters. 8 / 59 CD - Conversion Example: Convert 5 and 00 to CD (a) Example: Convert the CD codes 0000 and 0000 to decimal. 00 }{{} 00 }{{} 9 5 (c) (b) 0 }{{} 000 }{{} 7 (d) Logic Gates - NOT Gate It is also called as an inverter. F = F 0 0 = High, 0= Low 9 / 59 0 / 59 Logic Gates - ND Gate Logic Gates - OR Gate F = F = + For a input ND Gate, output F is High iff and are High. For a input OR Gate, output F is High if either or is High or both and are high = High, 0= Low F / 59 = High, 0= Low F / 59 Logic Gates - NND Gate The NND Gate operation is opposite that of ND Gate. Logic Gates - NOR Gate The NOR Gate operation is opposite that of OR Gate. F = F = + For a input NND Gate, output F is Low iff and are High. For a input NOR Gate, output F is High iff and are Low. F F = High, 0= Low 3 / 59 = High, 0= Low 4 / 59
4 Logic Gates - Exclusive-OR Gate Logic Gates - Exclusive-NOR Gate Exclusive-NOR is opposite that of Exclusive-OR. F = = + F = = + The output of an exclusive-or gate is High only when the two inputs are at opposite logic levels. The output of an exclusive-nor gate is High only when the two inputs are at the same level. F F = High, 0= Low 5 / 59 = High, 0= Low 6 / 59 NND - Universal Gate. Invert (NOT) NOR - Universal Gate. Invert (NOT) F = F =. ND Gate. ND Gate 3. OR Gate F = F = F = = + 3. OR Gate F = + F = + = F = + 7 / 59 8 / 59 Logic Circuits oolean lgebra ny system/subsystem/arithmetic circuit can be expressed as functional blocks containing logic gates. There are two types of logic circuits.. Combinational logic : Outputs depend only on the present inputs. Example: Logic gates, Code converters. Sequential logic : Outputs not only depend on the present inputs but also on past inputs. Example: Counter George oole, an English mathematician, developed this algebra in 854. That is why it is named after him. oolean algebra provides a mathematics of digital systems. oolean algebra is a systematic way of analyzing and designing the logic circuits. ll digital systems are sequential logic circuits. In EE0, we design combinational logic circuits. 9 / / 59 Laws of oolean lgebra Rules of oolean lgebra. Commutative laws + = + =. ssociative laws + ( + C) = ( + ) + C (C) = ()C + 0 = + = 0 = 0 = + = + = = = 0 = + = + = + ( + )( + C) = + C 3. Distributive Law ( + C) = + C,, or C can represent a single variable or a combination of variables. 3 / 59 3 / 59
5 DeMorgan s Theorems Example. The complement of a product of variables is equal to the sum of the complements of the variables. = +. The complement of a sum of variables is equal to the product of the complements of the variables. + = Use DeMorgan s theorems to simplify the following expressions.. ( + C). ( + )(C + D) Solution:. ( + C) = + + C = + C. ( + )(C + D) = ( + )+(C + D) = +C D = +C D 33 / / 59 Simplification using oolean lgebra Check yourself. oolean algebra is used to simplify a particular expression to its simplest form.. simplified boolean expression uses the fewest gates possible to implement the given expression. Example: ( + )C + C C + C + C C + C( + ) C + C ( + )C Simplify. + C + C C + C ( + )( + C) + C + + C + C + C + + C + C + C + C + C 35 / / 59 Standard form of oolean expressions Implementation of SOP Example: + CD ny oolean expressions can be converted into either of the following two forms.. Sum of Products (SOP) Form Examples C + C + C + C. Product of Sums (POS) Form Examples ( + + C)( + + C) ( + ) C D CD + CD 37 / / 59 Implementation of POS Karnaugh Map Example : ( + )( + C + D) C D ( + ) ( + C + D) ( + )( + C + D) Karnaugh map provides a systematic way of minimizing the boolean expressions. If properly used, it will produce the simplest form possible.. It is an array of cells in which each cell represents a binary value of the input variables.. The number of cells in a Karnaugh map is equal to the total number of possible input variable combinations. 3. For variables, no of cells is = 4. For 3, 3 = 8 and so on. 4. The cells are arranged in such a way that there is a single variable change (djacency) between adjacent cells. 39 / / 59
6 Karnaugh Map for 3-variables C C C C C C C C C Notice here, the cells are arranged so that there is a single variable change between adjacent cells. It is called as djacency. For example, 00 cell is adjacent to 0, 000, and cell is not adjacent 00,, 00, and 0. Mapping of SOP in Karnaugh Map Example: Example : C + C + C + C C / 59 4 / 59 Simplification - Karnaugh Map fter mapping Sum of Products (SOP) in Karnaugh Map,. Group the s. For grouping use the following procedure.. group must contain either,, 4, 8 or 6 cells. ( powers of two). lways include the largest possible number of s in a group..3 Each cell in a group must be adjacent to one or more cells in that same group..4 Each on the map must be included in at least one group. The s already in a group can be included in another group.. Determine the minimum product term for each group.. For -variable map.. -cell group gives a -variable term.. -cell group gives a -variable term..3 4-cell group gives a value.. For 3-variable map.. -cell group gives a 3-variable term.. -cell group gives a -variable term..3 4-cell group gives a -variable term..4 8-cell group gives a value 3. Sum all the terms to form the minimum SOP expression. 43 / 59 Example Let us simplify C + C + C + C C Minimum SOP expression 0 44 / 59 Check yourself SOP expression from Truth Table Simplify C + C + C + C C Minimum SOP expression 0 + C Consider the truth table for three input variables logic circuit. The output F is true for the following combinations. C, C, C and C. These terms are called minterms. C F F = C + C + C + C It is called as Sum of Products (SOP) expression. F = m(,, 6, 7) m indicates minterm. 45 / / 59 POS expression from Truth Table Consider the same truth table. The output F is false for the following combinations. C, C, C and C. C F F = C + C + C + C y DeMorgan s theorems, F = (++C)(++C)(++C)(++C) It is called as Product of Sums (POS) expression. F = M(0, 3, 4, 5) Check yourself SimplifyF = m(, 3, 4, 6) to minimum SOP. C Minimum SOP expression 0 F = C + C M indicates maxterm. 47 / / 59
7 Check yourself SimplifyF = M(0,, 5, 7) to minimum POS. C F = C + C Minimum POS expression If simplified, F = ( + C)( + C) F = + C + C + C C F = C + C same minimum SOP expression... (see previous slide) Don t Care Conditions In practice, in some applications the function is not specified for certain combinations of the variables. In most applications, we simply don t care what value is assigned to them. These combinations are called don t care conditions. These conditions will provide further simplification of the oolean expression. Example: Map the oolean function F = (, 3, 4, 6) that has the don t care conditions. d = (0, ) C X X 49 / / 59 Combinational Circuit Design Design procedure involves the following steps.. From the specifications given, identify the number of inputs and outputs.. Construct the truth table that defines the required relationship between inputs and outputs 3. Map the minterms into Karnaugh map and obtain the simplified oolean expression for each output. 4. Draw the logic diagram and verify it manually. Design - Example Let us design a circuit with three inputs and one output for the following condition. The output is when the binary value of the inputs is an even number. Solution: Let us name the inputs as,, and C and the output F. Truth Table C F / 59 5 / 59 Design - Example (contd...) K- map C F = C Logic diagram Decoder decoder is a digital circuit that detects the presence of a specified combination of bits on its inputs and indicates the presence of that code by a specified output level. It has N input lines and N output lines pplications: Decoders are used in computer to select input/output devices. CD-to-7-Segment Decoder C F 53 / / 59 -to-4-decoder Let us design. It has inputs and 4 outputs. Depending on the input configuration, decoder will produce high at specified output. Input Output D 0 D D D D 0 = D = D = D 3 = -to-4-decoder - Logic Circuit D 0 = D = D = D 3 = 55 / / 59
8 Encoder Decimal-to-CD Encoder It converts familiar symbols like decimal to coded form like binary or CD. It is just opposite to Decoder. It has N input lines and N output lines. pplication: Keyboard encoder : The ten decimal digits on the keyboard must be encoded for processing by the logic circuitry. When one of the keys pressed, the decimal digit is encoded to the corresponding CD code It has ten inputs (one for each decimal digit) and four outputs corresponding to the CD code. 0 3 Decimal CD Digit / / 59 Decimal-to-CD Encoder - Logic Circuit 3 0 LS MS 59 / 59
Part 5: Digital Circuits
Characteristics of any number system are: Part 5: Digital Circuits 5.: Number Systems & Code Conversions. ase or radix is equal to the number of possible symbols in the system 2. The largest value of digit
More informationProve that if not fat and not triangle necessarily means not green then green must be fat or triangle (or both).
hapter : oolean lgebra.) Definition of oolean lgebra The oolean algebra is named after George ool who developed this algebra (854) in order to analyze logical problems. n example to such problem is: Prove
More informationCOMP2611: Computer Organization. Introduction to Digital Logic
1 OMP2611: omputer Organization ombinational Logic OMP2611 Fall 2015 asics of Logic ircuits 2 its are the basis for binary number representation in digital computers ombining bits into patterns following
More informationLecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University
Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Aby K George, ECE Department, Wayne State University Contents The Map method Two variable
More informationTheorem/Law/Axioms Over (.) Over (+)
material prepared by: MUKESH OHR Follow me on F : http://www.facebook.com/mukesh.sirji4u OOLEN LGER oolean lgebra is a set of rules, laws and theorems by which logical operations can be mathematically
More informationAppendix A: Digital Logic. Principles of Computer Architecture. Principles of Computer Architecture by M. Murdocca and V. Heuring
- Principles of Computer rchitecture Miles Murdocca and Vincent Heuring 999 M. Murdocca and V. Heuring -2 Chapter Contents. Introduction.2 Combinational Logic.3 Truth Tables.4 Logic Gates.5 Properties
More informationCHAPTER 7. Exercises 17/ / /2 2 0
CHAPTER 7 Exercises E7. (a) For the whole part, we have: Quotient Remainders 23/2 /2 5 5/2 2 2/2 0 /2 0 Reading the remainders in reverse order, we obtain: 23 0 = 0 2 For the fractional part we have 2
More informationChapter 7 Logic Circuits
Chapter 7 Logic Circuits Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary
More informationLogic. Basic Logic Functions. Switches in series (AND) Truth Tables. Switches in Parallel (OR) Alternative view for OR
TOPIS: Logic Logic Expressions Logic Gates Simplifying Logic Expressions Sequential Logic (Logic with a Memory) George oole (85-864), English mathematician, oolean logic used in digital computers since
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 informationKarnaugh Map & Boolean Expression Simplification
Karnaugh Map & Boolean Expression Simplification Mapping a Standard POS Expression For a Standard POS expression, a 0 is placed in the cell corresponding to the product term (maxterm) present in the expression.
More 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 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 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 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 informationLecture 1. Notes. Notes. Notes. Introduction. Introduction digital logic February Bern University of Applied Sciences
Output voltage Input voltage 3.3V Digital operation (Switch) Lecture digital logic February 26 ern University of pplied Sciences Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev.
More informationDigital Logic (2) Boolean Algebra
Digital Logic (2) Boolean Algebra Boolean algebra is the mathematics of digital systems. It was developed in 1850 s by George Boole. We will use Boolean algebra to minimize logic expressions. Karnaugh
More 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 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 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 informationUNIT 4 MINTERM AND MAXTERM EXPANSIONS
UNIT 4 MINTERM AND MAXTERM EXPANSIONS Spring 2 Minterm and Maxterm Expansions 2 Contents Conversion of English sentences to Boolean equations Combinational logic design using a truth table Minterm and
More informationCMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates
CMSC 33 Lecture 6 Postulates & Theorems of oolean lgebra Semiconductors CMOS Logic Gates UMC, CMSC33, Richard Chang Last Time Overview of second half of this course Logic gates & symbols
More informationECE/Comp Sci 352 Digital System Fundamentals Quiz # 1 Solutions
Last (Family) Name: KIME First (Given) Name: Student I: epartment of Electrical and omputer Engineering University of Wisconsin - Madison EE/omp Sci 352 igital System Fundamentals Quiz # Solutions October
More informationSimplifying 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 informationBoolean Algebra and Logic Simplification
S302 Digital Logic Design Boolean Algebra and Logic Simplification Boolean Analysis of Logic ircuits, evaluating of Boolean expressions, representing the operation of Logic circuits and Boolean expressions
More informationBinary addition (1-bit) P Q Y = P + Q Comments Carry = Carry = Carry = Carry = 1 P Q
Digital Arithmetic In Chapter 2, we have discussed number systems such as binary, hexadecimal, decimal, and octal. We have also discussed sign representation techniques, for example, sign-bit representation
More informationLecture 3. Title goes here 1. level Networks. Boolean Algebra and Multi-level. level. level. level. level
Lecture 3 Dr Richard Reilly Dept. of Electronic & Electrical Engineering Room 53, Engineering uilding oolean lgebra and Multi- oolean algebra George oole, little formal education yet was a brilliant scholar.
More informationChapter 2. Review of Digital Systems Design
x 2-4 = 42.625. Chapter 2 Review of Digital Systems Design Numbering Systems Decimal number may be expressed as powers of 10. For example, consider a six digit decimal number 987654, which can be represented
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 informationPart 1: Digital Logic and Gates. Analog vs. Digital waveforms. The digital advantage. In real life...
Part 1: Digital Logic and Gates Analog vs Digital waveforms An analog signal assumes a continuous range of values: v(t) ANALOG A digital signal assumes discrete (isolated, separate) values Usually there
More informationIntroduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra
oolean lgebra Introduction 1854: Logical algebra was published by George oole known today as oolean lgebra It s a convenient way and systematic way of expressing and analyzing the operation of logic circuits.
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 informationof Digital Electronics
26 Digital Electronics 729 Digital Electronics 26.1 Analog and Digital Signals 26.3 Binary Number System 26.5 Decimal to Binary Conversion 26.7 Octal Number System 26.9 Binary-Coded Decimal Code (BCD Code)
More informationThis form sometimes used in logic circuit, example:
Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (k-map). 1. Deriving of logical expression from
More informationLogic Design Combinational Circuits. Digital Computer Design
Logic Design Combinational Circuits Digital Computer Design Topics Combinational Logic Karnaugh Maps Combinational uilding locks Timing 2 Logic Circuit logic circuit is composed of: Inputs Outputs Functional
More informationReview for Test 1 : Ch1 5
Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8
More 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 informationGoals for Lecture. Binary Logic and Gates (MK 2.1) Binary Variables. Notation Examples. Logical Operations
Introduction to Electrical Engineering, II LETURE NOTES #2 Instructor: Email: Telephone: Office: ndrew. Kahng (lecture) abk@ucsd.edu 858-822-4884 office 3802 P&M lass Website: http://vlsicad.ucsd.edu/courses/ece20b/wi04/
More informationDigital- or Logic Circuits. Outline Logic Circuits. Logic Voltage Levels. Binary Representation
Outline Logic ircuits Introduction Logic Systems TTL MOS Logic Gates NOT, OR, N NOR, NN, XOR Implementation oolean lgebra ombinatorial ircuits Multipleer emultipleer rithmetic ircuits Simplifying Logic
More informationDigital Logic. CS211 Computer Architecture. l Topics. l Transistors (Design & Types) l Logic Gates. l Combinational Circuits.
CS211 Computer Architecture Digital Logic l Topics l Transistors (Design & Types) l Logic Gates l Combinational Circuits l K-Maps Figures & Tables borrowed from:! http://www.allaboutcircuits.com/vol_4/index.html!
More informationCOMBINATIONAL LOGIC FUNCTIONS
COMBINATIONAL LOGIC FUNCTIONS Digital logic circuits can be classified as either combinational or sequential circuits. A combinational circuit is one where the output at any time depends only on the present
More informationII. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):
ENGI 386 Digital Logic II. COMBINATIONAL LOGIC DESIGN Combinational Logic output of digital system is only dependent on current inputs (i.e., no memory) (a) Boolean Algebra - developed by George Boole
More informationLearning Objectives. Boolean Algebra. In this chapter you will learn about:
Ref. Page Slide /78 Learning Objectives In this chapter you will learn about: oolean algebra Fundamental concepts and basic laws of oolean algebra oolean function and minimization Logic gates Logic circuits
More informationLogic Gates and Boolean Algebra
Logic Gates and oolean lgebra The ridge etween Symbolic Logic nd Electronic Digital Computing Compiled y: Muzammil hmad Khan mukhan@ssuet.edu.pk asic Logic Functions and or nand nor xor xnor not 2 Logic
More informationCMSC 313 Lecture 15 Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo
CMSC 33 Lecture 5 Good-bye ssembly Language Programming Overview of second half on Digital Logic DigSim Demo UMC, CMSC33, Richard Chang Good-bye ssembly Language What a pain! Understand
More informationL2: Combinational Logic Design (Construction and Boolean Algebra)
L2: Combinational Logic Design (Construction and oolean lgebra) cknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. orriello, Contemporary Logic Design (second edition), Pearson Education,
More informationEE40 Lec 15. Logic Synthesis and Sequential Logic Circuits
EE40 Lec 15 Logic Synthesis and Sequential Logic Circuits Prof. Nathan Cheung 10/20/2009 Reading: Hambley Chapters 7.4-7.6 Karnaugh Maps: Read following before reading textbook http://www.facstaff.bucknell.edu/mastascu/elessonshtml/logic/logic3.html
More informationCombinational Logic Circuits Part II -Theoretical Foundations
Combinational Logic Circuits Part II -Theoretical Foundations Overview Boolean Algebra Basic Logic Operations Basic Identities Basic Principles, Properties, and Theorems Boolean Function and Representations
More informationSAU1A FUNDAMENTALS OF DIGITAL COMPUTERS
SAU1A FUNDAMENTALS OF DIGITAL COMPUTERS Unit : I - V Unit : I Overview Fundamentals of Computers Characteristics of Computers Computer Language Operating Systems Generation of Computers 2 Definition of
More informationCHAPTER 7. Solutions for Exercises
CHAPTER 7 Solutions for Exercises E7.1 (a) For the whole part we have: Quotient Remainders 23/2 11 1 11/2 5 1 5/2 2 1 2/2 1 0 1/2 0 1 Reading the remainders in reverse order we obtain: 23 10 = 10111 2
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 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 informationL2: 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 people and used with permission. - Randy H. Katz (University of
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 informationCOSC3330 Computer Architecture Lecture 2. Combinational Logic
COSC333 Computer rchitecture Lecture 2. Combinational Logic Instructor: Weidong Shi (Larry), PhD Computer Science Department University of Houston Today Combinational Logic oolean lgebra Mux, DeMux, Decoder
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 informationElectronics. Overview. Introducction to Synthetic Biology
Electronics Introducction to Synthetic iology E Navarro Montagud P Fernandez de Cordoba JF Urchueguía Overview Introduction oolean algebras Logical gates Representation of boolean functions Karnaugh maps
More informationSave from: cs. Logic design 1 st Class أستاذ المادة: د. عماد
Save from: www.uotiq.org/dep cs Logic design 1 st Class أستاذ المادة: د. عماد استاذة المادة: م.م ميساء Contents Lectured One: Number system operation 1- Decimal numbers. 2- Binary numbers. 3- Octal numbers.
More informationChapter 5. Digital systems. 5.1 Boolean algebra Negation, conjunction and disjunction
Chapter 5 igital systems digital system is any machine that processes information encoded in the form of digits. Modern digital systems use binary digits, encoded as voltage levels. Two voltage levels,
More informationCS/COE0447: Computer Organization and Assembly Language
CS/COE0447: Computer Organization and Assembly Language Logic Design Introduction (Brief?) Appendix B: The Basics of Logic Design Dept. of Computer Science Logic design? Digital hardware is implemented
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 informationShow that the dual of the exclusive-or is equal to its compliment. 7
Darshan Institute of ngineering and Technology, Rajkot, Subject: Digital lectronics (2300) GTU Question ank Unit Group Questions Do as directed : I. Given that (6)0 = (00)x, find the value of x. II. dd
More informationZ = F(X) Combinational circuit. A combinational circuit can be specified either by a truth table. Truth Table
Lesson Objectives In this lesson, you will learn about What are combinational circuits Design procedure of combinational circuits Examples of combinational circuit design Combinational Circuits Logic circuit
More informationWeek No. 06: Numbering Systems
Week No. 06: Numbering Systems Numbering System: A numbering system defined as A set of values used to represent quantity. OR A number system is a term used for a set of different symbols or digits, which
More informationSignals and Systems Digital Logic System
Signals and Systems Digital Logic System Prof. Wonhee Kim Chapter 2 Design Process for Combinational Systems Step 1: Represent each of the inputs and outputs in binary Step 1.5: If necessary, break the
More information12/31/2010. Overview. 05-Boolean Algebra Part 3 Text: Unit 3, 7. DeMorgan s Law. Example. Example. DeMorgan s Law
Overview 05-oolean lgebra Part 3 Text: Unit 3, 7 EEGR/ISS 201 Digital Operations and omputations Winter 2011 DeMorgan s Laws lgebraic Simplifications Exclusive-OR and Equivalence Functionally omplete NND-NOR
More informationLecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps
EE210: Switching Systems Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps Prof. YingLi Tian Feb. 21/26, 2019 Department of Electrical Engineering The City College of New York
More 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 informationChapter 4: Combinational Logic Solutions to Problems: [1, 5, 9, 12, 19, 23, 30, 33]
Chapter 4: Combinational Logic Solutions to Problems: [, 5, 9, 2, 9, 23, 3, 33] Problem: 4- Consider the combinational circuit shown in Fig. P4-. (a) Derive the Boolean expressions for T through T 4. Evaluate
More informationPhiladelphia University Student Name: Student Number:
Philadelphia University Student Name: Student Number: Faculty of Engineering Serial Number: Final Exam, Second Semester: 2015/2016 Dept. of Computer Engineering Course Title: Logic Circuits Date: 08/06/2016
More informationTextbook: Digital Design, 3 rd. Edition M. Morris Mano
: 25/5/ P-/70 Tetbook: Digital Design, 3 rd. Edition M. Morris Mano Prentice-Hall, Inc. : INSTRUCTOR : CHING-LUNG SU E-mail: kevinsu@yuntech.edu.tw Chapter 3 25/5/ P-2/70 Chapter 3 Gate-Level Minimization
More informationDigital Design 2. Logic Gates and Boolean Algebra
Digital Design 2. Logic Gates and oolean lgebra József Sütő ssistant Lecturer References: [1] D.M. Harris, S.L. Harris, Digital Design and Computer rchitecture, 2nd ed., Elsevier, 213. [2] T.L. Floyd,
More informationNumbers & Arithmetic. Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University. See: P&H Chapter , 3.2, C.5 C.
Numbers & Arithmetic Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University See: P&H Chapter 2.4-2.6, 3.2, C.5 C.6 Example: Big Picture Computer System Organization and Programming
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 informationUnit 2 Session - 6 Combinational Logic Circuits
Objectives Unit 2 Session - 6 Combinational Logic Circuits Draw 3- variable and 4- variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the Product-of-Sums
More informationMinimization techniques
Pune Vidyarthi Griha s COLLEGE OF ENGINEERING, NSIK - 4 Minimization techniques By Prof. nand N. Gharu ssistant Professor Computer Department Combinational Logic Circuits Introduction Standard representation
More informationBoolean Algebra and Digital Logic 2009, University of Colombo School of Computing
IT 204 Section 3.0 Boolean Algebra and Digital Logic Boolean Algebra 2 Logic Equations to Truth Tables X = A. B + A. B + AB A B X 0 0 0 0 3 Sum of Products The OR operation performed on the products of
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 informationDigital Electronics Part 1: Binary Logic
Digital Electronics Part 1: Binary Logic Electronic devices in your everyday life What makes these products examples of electronic devices? What are some things they have in common? 2 How do electronics
More informationENGR 303 Introduction to Logic Design Lecture 3. Dr. Chuck Brown Engineering and Computer Information Science Folsom Lake College
Introduction to Logic Design Lecture 3 Dr. Chuck rown Engineering and Computer Information Science Folsom Lake College Outline for Todays Lecture Logic Circuits SOP / POS oolean Theorems DeMorgan s Theorem
More informationCHAPTER 3 LOGIC GATES & BOOLEAN ALGEBRA
CHPTER 3 LOGIC GTES & OOLEN LGER C H P T E R O U T C O M E S Upon completion of this chapter, student should be able to: 1. Describe the basic logic gates operation 2. Construct the truth table for basic
More informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two- and three-variable Four-variable Determination of minimum expressions using essential prime implicants Five-variable
More informationLecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions
EE210: Switching Systems Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions Prof. YingLi Tian Feb. 15, 2018 Department of Electrical Engineering The City College of New York The
More informationDigital Circuits. 1. Inputs & Outputs are quantized at two levels. 2. Binary arithmetic, only digits are 0 & 1. Position indicates power of 2.
Digital Circuits 1. Inputs & Outputs are quantized at two levels. 2. inary arithmetic, only digits are 0 & 1. Position indicates power of 2. 11001 = 2 4 + 2 3 + 0 + 0 +2 0 16 + 8 + 0 + 0 + 1 = 25 Digital
More informationCSC9R6 Computer Design. Practical Digital Logic
CSC9R6 Computer Design Practical Digital Logic 1 References (for this part of CSC9R6) Hamacher et al: Computer Organization App A. In library Floyd: Digital Fundamentals Ch 1, 3-6, 8-10 web page: www.prenhall.com/floyd/
More information2. Associative Law: A binary operator * on a set S is said to be associated whenever (A*B)*C = A*(B*C) for all A,B,C S.
BOOLEAN ALGEBRA 2.1 Introduction Binary logic deals with variables that have two discrete values: 1 for TRUE and 0 for FALSE. A simple switching circuit containing active elements such as a diode and transistor
More informationFundamentals of Boolean Algebra
UNIT-II 1 Fundamentals of Boolean Algebra Basic Postulates Postulate 1 (Definition): A Boolean algebra is a closed algebraic system containing a set K of two or more elements and the two operators and
More informationDepartment of Electrical & Electronics EE-333 DIGITAL SYSTEMS
Department of Electrical & Electronics EE-333 DIGITAL SYSTEMS 1) Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X -Y and (b) Y - X using 2's complements. a) X = 1010100
More informationLecture 2. Notes. Notes. Notes. Boolean algebra and optimizing logic functions. BTF Electronics Fundamentals August 2014
Lecture 2 Electronics ndreas Electronics oolean algebra and optimizing logic functions TF322 - Electronics Fundamentals ugust 24 Exercise ndreas ern University of pplied Sciences Rev. 946f32 2. of oolean
More informationENG2410 Digital Design Combinational Logic Circuits
ENG240 Digital Design Combinational Logic Circuits Fall 207 S. Areibi School of Engineering University of Guelph Binary variables Binary Logic Can be 0 or (T or F, low or high) Variables named with single
More 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 informationCMSC 313 Lecture 16 Announcement: no office hours today. Good-bye Assembly Language Programming Overview of second half on Digital Logic DigSim Demo
CMSC 33 Lecture 6 nnouncement: no office hours today. Good-bye ssembly Language Programming Overview of second half on Digital Logic DigSim Demo UMC, CMSC33, Richard Chang Good-bye ssembly
More informationCOSC 243. Introduction to Logic And Combinatorial Logic. Lecture 4 - Introduction to Logic and Combinatorial Logic. COSC 243 (Computer Architecture)
COSC 243 Introduction to Logic And Combinatorial Logic 1 Overview This Lecture Introduction to Digital Logic Gates Boolean algebra Combinatorial Logic Source: Chapter 11 (10 th edition) Source: J.R. Gregg,
More informationFundamentals of Digital Design
Fundamentals of Digital Design Digital Radiation Measurement and Spectroscopy NE/RHP 537 1 Binary Number System The binary numeral system, or base-2 number system, is a numeral system that represents numeric
More informationBinary addition example worked out
Binary addition example worked out Some terms are given here Exercise: what are these numbers equivalent to in decimal? The initial carry in is implicitly 0 1 1 1 0 (Carries) 1 0 1 1 (Augend) + 1 1 1 0
More informationCombinational Logic. By : Ali Mustafa
Combinational Logic By : Ali Mustafa Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer How to Analyze any combinational circuit like this? Analysis Procedure To obtain the output
More informationDIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA
DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA 1 Learning Objectives Understand the basic operations and laws of Boolean algebra. Relate these operations and laws to circuits composed of AND gates, OR gates, INVERTERS
More informationMODULAR CIRCUITS CHAPTER 7
CHAPTER 7 MODULAR CIRCUITS A modular circuit is a digital circuit that performs a specific function or has certain usage. The modular circuits to be introduced in this chapter are decoders, encoders, multiplexers,
More 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 information10/14/2009. Reading: Hambley Chapters
EE40 Lec 14 Digital Signal and Boolean Algebra Prof. Nathan Cheung 10/14/2009 Reading: Hambley Chapters 7.1-7.4 7.4 Slide 1 Analog Signals Analog: signal amplitude is continuous with time. Amplitude Modulated
More information