Vajira Thambawita, profile picture
Uploaded byVajira Thambawita
PDF, PPTX390 views

Lec 03 - Combinational Logic Design

The document provides a comprehensive overview of combinational logic design in digital electronics, highlighting the distinction between combinational and sequential circuits. It discusses Boolean equations, their simplification using Boolean algebra, and canonical forms such as sum-of-products and product-of-sums. Additionally, it addresses important concepts like minterms, maxterms, and De Morgan's Theorems in relation to digital logic design.

Education◦
In this document
Powered by AI
See More

Introduces Combinational Logic Design focusing on digital circuits that process discrete inputs and outputs.

Differentiates between combinational circuits (memoryless, outputs depend on current inputs) and sequential circuits (outputs depend on past and current inputs).

Discusses Boolean equations for digital logic representation, including Sum-of-Products and Product-of-Sums forms.

Explains the simplification of Boolean equations via Boolean algebra axioms and their application in circuit design.

Discusses single and several variable theorems, highlighting De Morgan's Theorem and its implications in circuit design.

Describes Boolean functions, their truth tables, gate implementations, and finding complements of functions.

Explains minterms and maxterms, how to form them with binary variables, and combinations for n variables.

Describes the conversion between canonical forms, expressing functions as product-of-sums and examples.

Introduces standard forms such as sum of products and product of sums in Boolean functions.

Discusses the basic digital gates, their functionalities, and extensions to multiple inputs.

Explains the concepts of positive and negative logic systems concerning logic levels representation.

Embed presentation

Download as PDF, PPTX
Combinational Logic Design D.R.V.L.B Thambawita August 27, 2017 D.R.V.L.B Thambawita Combinational Logic Design
Table of contents D.R.V.L.B Thambawita Combinational Logic Design
INTRODUCTION In digital electronics, a circuit is a network that processes discrete valued variables. one or more discrete-valued input terminals one or more discrete-valued output terminals a functional specification describing the relationship between inputs and outputs a timing specification describing the delay between inputs changing and outputs responding D.R.V.L.B Thambawita Combinational Logic Design
combinational or sequential combinational circuit circuits outputs depend only on the current values of the inputs; in other words, it combines the current input values to compute the output.A combinational circuit is memoryless. sequential circuit circuits outputs depend on both current and previous values of the inputs; in other words, it depends on the input sequence. A sequential circuit has memory. D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN EQUATIONS Boolean equations deal with variables that are either TRUE or FALSE, so they are perfect for describing digital logic. D.R.V.L.B Thambawita Combinational Logic Design
Sum-of-Products Form Do you know? Sum-of-Products Form This is called the sum-of-products canonical form of a function because it is the sum (OR) of products (ANDs forming minterms). D.R.V.L.B Thambawita Combinational Logic Design
Product-of-Sums Form Do you know? An alternative way of expressing Boolean functions is the product-of sums canonical form. Each row of a truth table corresponds to a maxterm that is FALSE for that row. D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN ALGEBRA Just as you use algebra to simplify mathematical equations, you can use Boolean algebra to simplify Boolean equations. Boolean algebra is based on a set of axioms that we assume are correct. Axioms are unprovable in the sense that a definition cannot be proved. From these axioms, we prove all the theorems of Boolean algebra. Do you know? These theorems have great practical significance, because they teach us how to simplify logic to produce smaller and less costly circuits. D.R.V.L.B Thambawita Combinational Logic Design
Axioms of Boolean algebra Do you know? Axioms and theorems of Boolean algebra obey the principle of duality. If the symbols 0 and 1 and the operators • (AND) and + (OR) are interchanged, the statement will still be correct. D.R.V.L.B Thambawita Combinational Logic Design
Boolean theorems of one variable What are the benefits? D.R.V.L.B Thambawita Combinational Logic Design
Theorems of Several Variables D.R.V.L.B Thambawita Combinational Logic Design
Basic Theorems D.R.V.L.B Thambawita Combinational Logic Design
Basic Theorems D.R.V.L.B Thambawita Combinational Logic Design
De Morgans Theorem This is a particularly powerful tool in digital design. According to De Morgans theorem, a NAND gate is equivalent to an OR gate with inverted inputs. A NOR gate is equivalent to an AND gate with inverted inputs. D.R.V.L.B Thambawita Combinational Logic Design
De Morgans Theorem you can imagine that pushing a bubble through the gate causes it to come out at the other side and flips the body of the gate from AND to OR or vice versa Pushing bubbles backward (from the output) or forward (from the inputs) changes the body of the gate from AND to OR or vice versa. Pushing a bubble from the output back to the inputs puts bubbles on all gate inputs. Pushing bubbles on all gate inputs forward toward the output puts a bubble on the output. D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Boolean algebra is an algebra that deals with binary variables and logic operations. A Boolean function can be represented in a truth table. Example F1 = x + y z The function F1 is equal to 1 if x is equal to 1 or if both y’ and z are equal to 1. F1 is equal to 0 otherwise F2 = x y z + x yz + xy D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Truth Tables for F1 and F2 D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Gate implementation of F1 = x + y z D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Gate implementation of F2 = x y z + x yz + xy D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS F2 = x y z + x yz + xy = x y(y + y) + xy = x y + xy D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. 4 xy + x z + yz = xy + x z + yz(x + x ) = xy + x z + xyz + x yz = xy(1 + z) + x z(1 + y) = xy + x z. 5 (x + y)(x + z)(y + z) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. 4 xy + x z + yz = xy + x z + yz(x + x ) = xy + x z + xyz + x yz = xy(1 + z) + x z(1 + y) = xy + x z. 5 (x + y)(x + z)(y + z) = (x + y)(x + z) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS 1 x(x + y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. 4 xy + x z + yz = xy + x z + yz(x + x ) = xy + x z + xyz + x yz = xy(1 + z) + x z(1 + y) = xy + x z. 5 (x + y)(x + z)(y + z) = (x + y)(x + z) by duality from function 4 D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Complement of a Function (A + B + C) = (A + x) let B + C = x = A x by DeMorgan = A (B + C) substitute B + C = x = A (B C ) by DeMorgan = ABC by associative D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Complement of a Function (A + B + C + D + ... + F) = A B C D ...F (ABCD...F) = A + B + C + D + ... + F D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Find the complement of the functionsF1 = x yz + x y z and F2 = x(y z + yz) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Find the complement of the functionsF1 = x yz + x y z and F2 = x(y z + yz) D.R.V.L.B Thambawita Combinational Logic Design
BOOLEAN FUNCTIONS Do you know? D.R.V.L.B Thambawita Combinational Logic Design
More about Minterms and Maxterms Minterms Consider two binary variables x and y combined with an AND operation D.R.V.L.B Thambawita Combinational Logic Design
More about Minterms and Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy D.R.V.L.B Thambawita Combinational Logic Design
More about Minterms and Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy Each of these four AND terms is called a minterm, or a standard product. D.R.V.L.B Thambawita Combinational Logic Design
More about Minterms and Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy Each of these four AND terms is called a minterm, or a standard product. n variables can be combined to form 2n minterms D.R.V.L.B Thambawita Combinational Logic Design
More about Minterms and Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy Each of these four AND terms is called a minterm, or a standard product. n variables can be combined to form 2n minterms D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Maxterms N variables forming an OR term, with each variable being primed or unprimed, provide 2n possible combinations, called maxterms, or standard sums. D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Maxterms N variables forming an OR term, with each variable being primed or unprimed, provide 2n possible combinations, called maxterms, or standard sums. The eight maxterms for three variables. D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Maxterms N variables forming an OR term, with each variable being primed or unprimed, provide 2n possible combinations, called maxterms, or standard sums. The eight maxterms for three variables. D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Minterms and Maxterms for Three Binary Variables D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Example: Functions of Three Variables D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Fuction f1 and f2 for above truth table D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Fuction f1 and f2 for above truth table f1 = x y z + xy z + xyz = m1 + m4 + m7 D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Fuction f1 and f2 for above truth table f1 = x y z + xy z + xyz = m1 + m4 + m7 f2 = x yz + xy z + xyz + xyz = m3 + m5 + m6 + m7 D.R.V.L.B Thambawita Combinational Logic Design
Minterms and Maxterms Fuction f1 and f2 for above truth table f1 = x y z + xy z + xyz = m1 + m4 + m7 f2 = x yz + xy z + xyz + xyz = m3 + m5 + m6 + m7 D.R.V.L.B Thambawita Combinational Logic Design
PRODUCT-OF-SUMS Following table shows the truth table for a Boolean function, Y, and its complement, ¯Y . Using De Morgans Theorem, derive the product-of-sums canonical form of Y from the sum-of-products form of ¯Y . D.R.V.L.B Thambawita Combinational Logic Design
PRODUCT-OF-SUMS Solution: ¯Y = ¯A ¯B + ¯AB (1) ¯¯y = y = ¯A ¯B + ¯AB = ( ¯A ¯B)( ¯AB) = (A + B)(A + ¯B) (2) D.R.V.L.B Thambawita Combinational Logic Design
PRODUCT-OF-SUMS To express a Boolean function as a product of maxterms, it must first be brought into a form of OR terms. This may be done by using the distributive law, x + yz = (x + y)(x + z). Example: Express the Boolean function F = xy + xz as a product of maxterms D.R.V.L.B Thambawita Combinational Logic Design
PRODUCT-OF-SUMS To express a Boolean function as a product of maxterms, it must first be brought into a form of OR terms. This may be done by using the distributive law, x + yz = (x + y)(x + z). Example: Express the Boolean function F = xy + xz as a product of maxterms F = xy + x y = (xy + x )(xy + z) = (x + x )(y + x )(x + z)(y + z) = (x + y)(x + z)(y + z) D.R.V.L.B Thambawita Combinational Logic Design Z
Product of Maxterms The function has three variables: x, y, and z. Each OR term is missing one variable; therefore D.R.V.L.B Thambawita Combinational Logic Design F = (x + y + z)(x + y + z)(x + y + z)(x + y + z) = M0M2M4M5
Conversion between Canonical Forms The complement of a function expressed as the sum of minterms equals the sum of minterms missing from the original function Now, if we take the complement of F’ by DeMorgan’s theorem, we obtain F in a different form: D.R.V.L.B Thambawita Combinational Logic Design
Conversion between Canonical Forms D.R.V.L.B Thambawita Combinational Logic Design
Standard Forms Another way to express Boolean functions is in standard form The sum of products is a Boolean expression containing AND terms, called product terms, with one or more literals each A product of sums is a Boolean expression containing OR terms, called sum terms. D.R.V.L.B Thambawita Combinational Logic Design
Standard Forms D.R.V.L.B Thambawita Combinational Logic Design
Standard Forms: Three and twolevel implementation D.R.V.L.B Thambawita Combinational Logic Design
OTHER LOGIC OPERATIONS There are 22n functions for n binary variables. Thus, for two variables, n=2, and the number of possible Boolean functions is 16. Truth Tables for the 16 Functions of Two Binary Variables D.R.V.L.B Thambawita Combinational Logic Design
OTHER LOGIC OPERATIONS Boolean Expressions for the 16 Functions of Two Variables D.R.V.L.B Thambawita Combinational Logic Design
OTHER LOGIC OPERATIONS Boolean Expressions for the 16 Functions of Two Variables D.R.V.L.B Thambawita Combinational Logic Design
DIGITAL LOGIC GATES D.R.V.L.B Thambawita Combinational Logic Design
DIGITAL LOGIC GATES D.R.V.L.B Thambawita Combinational Logic Design
DIGITAL LOGIC GATES D.R.V.L.B Thambawita Combinational Logic Design
DIGITAL LOGIC GATES D.R.V.L.B Thambawita Combinational Logic Design
Extension to Multiple Inputs The gates shown can be extended to have more than two inputs. A gate can be extended to have multiple inputs if the binary operation it represents is commutative and associative. Ex: OR function x + y = y + x(commutative) (x + y) + z = x + (y + z) = x + y + z(associative) D.R.V.L.B Thambawita Combinational Logic Design
Extension to Multiple Inputs NAND and NOR functions are commutative NAND and NOR operators are not associative D.R.V.L.B Thambawita Combinational Logic Design Z
Extension to Multiple Inputs Threeinput exclusiveOR function Figure: Threeinput exclusive-OR gate Do you know? F is equal to 1 if only one input is equal to 1 or if all three inputs are equal to 1. (when the total number of 1’s in the input variables is odd) D.R.V.L.B Thambawita Combinational Logic Design
Positive and Negative Logic Do you know? Choosing the highlevel H to represent logic 1 defines a positive logic system. Choosing the lowlevel L to represent logic 1 defines a negative logic system. Figure: Signal assignment and logic polarity D.R.V.L.B Thambawita Combinational Logic Design
Positive and Negative Logic D.R.V.L.B Thambawita Combinational Logic Design
Positive and Negative Logic D.R.V.L.B Thambawita Combinational Logic Design
Positive and Negative Logic D.R.V.L.B Thambawita Combinational Logic Design

More Related Content

PPT
Chapter 2 Boolean Algebra (part 2)
byFrankie Jones
 
PPT
Krishna
byKrishna Godugunuru
 
PPT
Logic design and switching theory
byjomerson remorosa
 
PDF
Ascii
byGajendra Saini
 
PPTX
Polyadic predicates
byReymark Lubon
 
PPT
Boolean algebra
byRania H
 
PPTX
Clock-8086 bus cycle
byRani Rahul
 
PPTX
Back tracking and branch and bound class 20
byKumar
 
Chapter 2 Boolean Algebra (part 2)
byFrankie Jones
 
Krishna
byKrishna Godugunuru
 
Logic design and switching theory
byjomerson remorosa
 
Ascii
byGajendra Saini
 
Polyadic predicates
byReymark Lubon
 
Boolean algebra
byRania H
 
Clock-8086 bus cycle
byRani Rahul
 
Back tracking and branch and bound class 20
byKumar
 

What's hot

PDF
Reconfigurable computing
bySudhanshu Janwadkar
 
PDF
AVR Micro controller Interfacing
byRaghav Shetty
 
PDF
Algorithms Lecture 7: Graph Algorithms
byMohamed Loey
 
PPT
Type Checking(Compiler Design) #ShareThisIfYouLike
byUnited International University
 
PPTX
Type checking compiler construction Chapter #6
byDaniyal Mughal
 
PPTX
DLD Lecture No 18 Analysis and Design of Combinational Circuit.pptx
bySaveraAyub2
 
PDF
Processes and Threads
byEmery Berger
 
PDF
DESIGN OF COMBINATIONAL LOGIC
byPUNE VIDYARTHI GRIHA'S COLLEGE OF ENGINEERING, NASHIK
 
PPTX
MS SQL DATABASE PROJECT ON RAILWAYS.pptx
byMohammedMustafa294978
 
PPTX
Cpld and fpga mod vi
byAgi George
 
PDF
Python algorithm
byProf. Dr. K. Adisesha
 
PPTX
OPTIMAL BINARY SEARCH
byCool Guy
 
PPTX
Multiplexer & Demultiplexer
byJatin Sharma
 
PPT
Digital Logic circuit
bykavitha muneeshwaran
 
PPTX
Multiplexer and DeMultiplexer
byEstiak Khan
 
PDF
Compiler Design Lecture Notes
byFellowBuddy.com
 
PPTX
Serial communication in LPC2148
bysravannunna24
 
PPT
Data representation and Arithmetic Algorithms
byMumthas Shaikh
 
PPT
Time complexity
byKatang Isip
 
PDF
Data+Structures+and+Algorithm+Analysis+in+C+-+Mark+Allen+Weiss.pdf
byRameshBabuKellamapal
 
Reconfigurable computing
bySudhanshu Janwadkar
 
AVR Micro controller Interfacing
byRaghav Shetty
 
Algorithms Lecture 7: Graph Algorithms
byMohamed Loey
 
Type Checking(Compiler Design) #ShareThisIfYouLike
byUnited International University
 
Type checking compiler construction Chapter #6
byDaniyal Mughal
 
DLD Lecture No 18 Analysis and Design of Combinational Circuit.pptx
bySaveraAyub2
 
Processes and Threads
byEmery Berger
 
DESIGN OF COMBINATIONAL LOGIC
byPUNE VIDYARTHI GRIHA'S COLLEGE OF ENGINEERING, NASHIK
 
MS SQL DATABASE PROJECT ON RAILWAYS.pptx
byMohammedMustafa294978
 
Cpld and fpga mod vi
byAgi George
 
Python algorithm
byProf. Dr. K. Adisesha
 
OPTIMAL BINARY SEARCH
byCool Guy
 
Multiplexer & Demultiplexer
byJatin Sharma
 
Digital Logic circuit
bykavitha muneeshwaran
 
Multiplexer and DeMultiplexer
byEstiak Khan
 
Compiler Design Lecture Notes
byFellowBuddy.com
 
Serial communication in LPC2148
bysravannunna24
 
Data representation and Arithmetic Algorithms
byMumthas Shaikh
 
Time complexity
byKatang Isip
 
Data+Structures+and+Algorithm+Analysis+in+C+-+Mark+Allen+Weiss.pdf
byRameshBabuKellamapal
 

Similar to Lec 03 - Combinational Logic Design

PPTX
digital logic circuits, digital component
byRai University
 
PPTX
B.sc cs-ii-u-1.5 digital logic circuits, digital component
byRai University
 
PPT
2dig circ
byAremarati Ramakanth
 
PPTX
Bca 2nd sem-u-1.5 digital logic circuits, digital component
byRai University
 
PPTX
Digital logic and computer design with explanation
bykathanshgamer
 
PPTX
B sc cs i bo-de u-ii logic gates
byRai University
 
PDF
boolean-algebra.pdf
byRameshK531901
 
PDF
DM2020 boolean algebra
byRobert Geofroy
 
PPT
Basic Digital Logic(Notes-DE) for engineering .ppt
byroshh1
 
PPT
Boolean algebra And Logic Gates
byKumar
 
PPT
2dig circ
bySaravanan Muthukrishnan
 
PPT
Boolean Algebra and Logic gates ( Chapter 2)
bydiganta das
 
PPT
Boolean
byJeane Paguio
 
PDF
boolean_algebra.pdf for discrete mathematics
bysomnathmule3
 
PDF
Boolean algebra and Logic gates
byProf. Dr. K. Adisesha
 
PDF
boolean algebra for logic circuits and switching
byDayNonzGetino
 
PDF
Chapter 4 logic design
byBurhanuddin Mohammad
 
PPT
boolean_algebra.ppt of all of us and the family will not even have to worry
byBifaHirpo1
 
DOCX
M3 PPT 22ESC143.docx
byJyothi122118
 
DOCX
M3 PPT 22ESC143.docx
byJyothi122118
 
digital logic circuits, digital component
byRai University
 
B.sc cs-ii-u-1.5 digital logic circuits, digital component
byRai University
 
2dig circ
byAremarati Ramakanth
 
Bca 2nd sem-u-1.5 digital logic circuits, digital component
byRai University
 
Digital logic and computer design with explanation
bykathanshgamer
 
B sc cs i bo-de u-ii logic gates
byRai University
 
boolean-algebra.pdf
byRameshK531901
 
DM2020 boolean algebra
byRobert Geofroy
 
Basic Digital Logic(Notes-DE) for engineering .ppt
byroshh1
 
Boolean algebra And Logic Gates
byKumar
 
2dig circ
bySaravanan Muthukrishnan
 
Boolean Algebra and Logic gates ( Chapter 2)
bydiganta das
 
Boolean
byJeane Paguio
 
boolean_algebra.pdf for discrete mathematics
bysomnathmule3
 
Boolean algebra and Logic gates
byProf. Dr. K. Adisesha
 
boolean algebra for logic circuits and switching
byDayNonzGetino
 
Chapter 4 logic design
byBurhanuddin Mohammad
 
boolean_algebra.ppt of all of us and the family will not even have to worry
byBifaHirpo1
 
M3 PPT 22ESC143.docx
byJyothi122118
 
M3 PPT 22ESC143.docx
byJyothi122118
 

More from Vajira Thambawita

PDF
Lecture 4 principles of parallel algorithm design updated
byVajira Thambawita
 
PDF
Lecture 3 parallel programming platforms
byVajira Thambawita
 
PDF
Lecture 2 more about parallel computing
byVajira Thambawita
 
PDF
Lecture 1 introduction to parallel and distributed computing
byVajira Thambawita
 
PDF
Lecture 12 localization and navigation
byVajira Thambawita
 
PDF
Lecture 11 neural network principles
byVajira Thambawita
 
PDF
Lecture 10 mobile robot design
byVajira Thambawita
 
PDF
Lecture 09 control
byVajira Thambawita
 
PDF
Lecture 08 robots and controllers
byVajira Thambawita
 
PDF
Lecture 07 more about pic
byVajira Thambawita
 
PDF
Lecture 06 pic programming in c
byVajira Thambawita
 
PDF
Lecture 05 pic io port programming
byVajira Thambawita
 
PDF
Lecture 04 branch call and time delay
byVajira Thambawita
 
PDF
Lecture 03 basics of pic
byVajira Thambawita
 
PDF
Lecture 02 mechatronics systems
byVajira Thambawita
 
PDF
Lecture 1 - Introduction to embedded system and Robotics
byVajira Thambawita
 
PDF
Lec 09 - Registers and Counters
byVajira Thambawita
 
PDF
Lec 08 - DESIGN PROCEDURE
byVajira Thambawita
 
PDF
Lec 07 - ANALYSIS OF CLOCKED SEQUENTIAL CIRCUITS
byVajira Thambawita
 
PDF
Lec 06 - Synchronous Sequential Logic
byVajira Thambawita
 
Lecture 4 principles of parallel algorithm design updated
byVajira Thambawita
 
Lecture 3 parallel programming platforms
byVajira Thambawita
 
Lecture 2 more about parallel computing
byVajira Thambawita
 
Lecture 1 introduction to parallel and distributed computing
byVajira Thambawita
 
Lecture 12 localization and navigation
byVajira Thambawita
 
Lecture 11 neural network principles
byVajira Thambawita
 
Lecture 10 mobile robot design
byVajira Thambawita
 
Lecture 09 control
byVajira Thambawita
 
Lecture 08 robots and controllers
byVajira Thambawita
 
Lecture 07 more about pic
byVajira Thambawita
 
Lecture 06 pic programming in c
byVajira Thambawita
 
Lecture 05 pic io port programming
byVajira Thambawita
 
Lecture 04 branch call and time delay
byVajira Thambawita
 
Lecture 03 basics of pic
byVajira Thambawita
 
Lecture 02 mechatronics systems
byVajira Thambawita
 
Lecture 1 - Introduction to embedded system and Robotics
byVajira Thambawita
 
Lec 09 - Registers and Counters
byVajira Thambawita
 
Lec 08 - DESIGN PROCEDURE
byVajira Thambawita
 
Lec 07 - ANALYSIS OF CLOCKED SEQUENTIAL CIRCUITS
byVajira Thambawita
 
Lec 06 - Synchronous Sequential Logic
byVajira Thambawita
 

Recently uploaded

PDF
Types of eggs and Egg membranes (Developmental Zoology)
bySudhandraKarthi
 
PDF
The voice of the rain poem class 11th.pdf
byReeta Baral
 
PPTX
Isolation, Identification and Analysis of phytoconstituents.
byD. Y. Patil College of Pharmacy, Kolhapur.
 
PPTX
Unit 3: Gender Issues and Provisions F.Y.BEd Course
byRejoshaRajendran
 
PPTX
Steve Hale - Developing Elite Young Goalkeepers 2024.pptx
bypjmfd
 
PDF
IMANI Africa files RTI request seeking full disclosure on 2026 SIM registrati...
bynservice241
 
PPTX
B. Pharmacy Unit-3 (Fats and Oils Reaction and Analytical Constant)
bySandeshSul
 
PDF
Madhumathi Endowment Quiz Finals by Sreejesh P S and Shyam Krishnan P - Keral...
bySreejesh P S
 
PPTX
cis.jackson.alaysia.ResearchPresentation 1.Microsoft Teams vs Zoom vs Webex v...
byMerritt College
 
PPTX
Partial Correlation - Values of r₁₂.₃, r₂₃.₁ & r₁₃.₂ r₁₂, r₁₃ and r₂₃
bySundar B N
 
PPTX
REVISED DEFENSE MECHANISM / ADJUSTMENT MECHANISM-FINAL
byShimla
 
PDF
Environmental Studies Module 2 BBA,BCCA Sem 1 NEP.pdf
byJayanti Pande
 
PPTX
Multiple Correlation - Introduction, Meaning, Examples, Options, Formulas and...
bySundar B N
 
PDF
Advanced Process Flow and Micromachining Techniques for MEMS Fabrication: A D...
byGS Virdi
 
PDF
Environmental Studies Module 1 BBA,BCCA Sem 1 NEP.pdf
byJayanti Pande
 
PPTX
How to Track a Link Using Odoo 18 SMS Marketing
byCeline George
 
PPTX
PARENTAL ROUTES OF DRUGS ADMINISTRATION .pptx
byAneetaSharma15
 
PPTX
How to Create a Spreadsheet in Odoo 18
byCeline George
 
PDF
Multiple Myeloma , definition, etiology, PP, CM , DE and management
byNisha Borsa
 
PPTX
Multiple Linear Regression - Least Square Method X₁ on X₂ and X₃
bySundar B N
 
Types of eggs and Egg membranes (Developmental Zoology)
bySudhandraKarthi
 
The voice of the rain poem class 11th.pdf
byReeta Baral
 
Isolation, Identification and Analysis of phytoconstituents.
byD. Y. Patil College of Pharmacy, Kolhapur.
 
Unit 3: Gender Issues and Provisions F.Y.BEd Course
byRejoshaRajendran
 
Steve Hale - Developing Elite Young Goalkeepers 2024.pptx
bypjmfd
 
IMANI Africa files RTI request seeking full disclosure on 2026 SIM registrati...
bynservice241
 
B. Pharmacy Unit-3 (Fats and Oils Reaction and Analytical Constant)
bySandeshSul
 
Madhumathi Endowment Quiz Finals by Sreejesh P S and Shyam Krishnan P - Keral...
bySreejesh P S
 
cis.jackson.alaysia.ResearchPresentation 1.Microsoft Teams vs Zoom vs Webex v...
byMerritt College
 
Partial Correlation - Values of r₁₂.₃, r₂₃.₁ & r₁₃.₂ r₁₂, r₁₃ and r₂₃
bySundar B N
 
REVISED DEFENSE MECHANISM / ADJUSTMENT MECHANISM-FINAL
byShimla
 
Environmental Studies Module 2 BBA,BCCA Sem 1 NEP.pdf
byJayanti Pande
 
Multiple Correlation - Introduction, Meaning, Examples, Options, Formulas and...
bySundar B N
 
Advanced Process Flow and Micromachining Techniques for MEMS Fabrication: A D...
byGS Virdi
 
Environmental Studies Module 1 BBA,BCCA Sem 1 NEP.pdf
byJayanti Pande
 
How to Track a Link Using Odoo 18 SMS Marketing
byCeline George
 
PARENTAL ROUTES OF DRUGS ADMINISTRATION .pptx
byAneetaSharma15
 
How to Create a Spreadsheet in Odoo 18
byCeline George
 
Multiple Myeloma , definition, etiology, PP, CM , DE and management
byNisha Borsa
 
Multiple Linear Regression - Least Square Method X₁ on X₂ and X₃
bySundar B N
 

Lec 03 - Combinational Logic Design

  • 1.
    Combinational Logic Design D.R.V.L.BThambawita August 27, 2017 D.R.V.L.B Thambawita Combinational Logic Design
  • 2.
    Table of contents D.R.V.L.BThambawita Combinational Logic Design
  • 3.
    INTRODUCTION In digital electronics,a circuit is a network that processes discrete valued variables. one or more discrete-valued input terminals one or more discrete-valued output terminals a functional specification describing the relationship between inputs and outputs a timing specification describing the delay between inputs changing and outputs responding D.R.V.L.B Thambawita Combinational Logic Design
  • 4.
    combinational or sequential combinationalcircuit circuits outputs depend only on the current values of the inputs; in other words, it combines the current input values to compute the output.A combinational circuit is memoryless. sequential circuit circuits outputs depend on both current and previous values of the inputs; in other words, it depends on the input sequence. A sequential circuit has memory. D.R.V.L.B Thambawita Combinational Logic Design
  • 5.
    BOOLEAN EQUATIONS Boolean equationsdeal with variables that are either TRUE or FALSE, so they are perfect for describing digital logic. D.R.V.L.B Thambawita Combinational Logic Design
  • 6.
    Sum-of-Products Form Do youknow? Sum-of-Products Form This is called the sum-of-products canonical form of a function because it is the sum (OR) of products (ANDs forming minterms). D.R.V.L.B Thambawita Combinational Logic Design
  • 7.
    Product-of-Sums Form Do youknow? An alternative way of expressing Boolean functions is the product-of sums canonical form. Each row of a truth table corresponds to a maxterm that is FALSE for that row. D.R.V.L.B Thambawita Combinational Logic Design
  • 8.
    BOOLEAN ALGEBRA Just asyou use algebra to simplify mathematical equations, you can use Boolean algebra to simplify Boolean equations. Boolean algebra is based on a set of axioms that we assume are correct. Axioms are unprovable in the sense that a definition cannot be proved. From these axioms, we prove all the theorems of Boolean algebra. Do you know? These theorems have great practical significance, because they teach us how to simplify logic to produce smaller and less costly circuits. D.R.V.L.B Thambawita Combinational Logic Design
  • 9.
    Axioms of Booleanalgebra Do you know? Axioms and theorems of Boolean algebra obey the principle of duality. If the symbols 0 and 1 and the operators • (AND) and + (OR) are interchanged, the statement will still be correct. D.R.V.L.B Thambawita Combinational Logic Design
  • 10.
    Boolean theorems ofone variable What are the benefits? D.R.V.L.B Thambawita Combinational Logic Design
  • 11.
    Theorems of SeveralVariables D.R.V.L.B Thambawita Combinational Logic Design
  • 12.
    Basic Theorems D.R.V.L.B ThambawitaCombinational Logic Design
  • 13.
    Basic Theorems D.R.V.L.B ThambawitaCombinational Logic Design
  • 14.
    De Morgans Theorem Thisis a particularly powerful tool in digital design. According to De Morgans theorem, a NAND gate is equivalent to an OR gate with inverted inputs. A NOR gate is equivalent to an AND gate with inverted inputs. D.R.V.L.B Thambawita Combinational Logic Design
  • 15.
    De Morgans Theorem youcan imagine that pushing a bubble through the gate causes it to come out at the other side and flips the body of the gate from AND to OR or vice versa Pushing bubbles backward (from the output) or forward (from the inputs) changes the body of the gate from AND to OR or vice versa. Pushing a bubble from the output back to the inputs puts bubbles on all gate inputs. Pushing bubbles on all gate inputs forward toward the output puts a bubble on the output. D.R.V.L.B Thambawita Combinational Logic Design
  • 16.
    BOOLEAN FUNCTIONS Boolean algebrais an algebra that deals with binary variables and logic operations. A Boolean function can be represented in a truth table. Example F1 = x + y z The function F1 is equal to 1 if x is equal to 1 or if both y’ and z are equal to 1. F1 is equal to 0 otherwise F2 = x y z + x yz + xy D.R.V.L.B Thambawita Combinational Logic Design
  • 17.
    BOOLEAN FUNCTIONS Truth Tablesfor F1 and F2 D.R.V.L.B Thambawita Combinational Logic Design
  • 18.
    BOOLEAN FUNCTIONS Gate implementationof F1 = x + y z D.R.V.L.B Thambawita Combinational Logic Design
  • 19.
    BOOLEAN FUNCTIONS Gate implementationof F2 = x y z + x yz + xy D.R.V.L.B Thambawita Combinational Logic Design
  • 20.
    BOOLEAN FUNCTIONS F2 =x y z + x yz + xy = x y(y + y) + xy = x y + xy D.R.V.L.B Thambawita Combinational Logic Design
  • 21.
    BOOLEAN FUNCTIONS 1 x(x+ y) D.R.V.L.B Thambawita Combinational Logic Design
  • 22.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy D.R.V.L.B Thambawita Combinational Logic Design
  • 23.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy D.R.V.L.B Thambawita Combinational Logic Design
  • 24.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. D.R.V.L.B Thambawita Combinational Logic Design
  • 25.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y D.R.V.L.B Thambawita Combinational Logic Design
  • 26.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) D.R.V.L.B Thambawita Combinational Logic Design
  • 27.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) D.R.V.L.B Thambawita Combinational Logic Design
  • 28.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. D.R.V.L.B Thambawita Combinational Logic Design
  • 29.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) D.R.V.L.B Thambawita Combinational Logic Design
  • 30.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy D.R.V.L.B Thambawita Combinational Logic Design
  • 31.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) D.R.V.L.B Thambawita Combinational Logic Design
  • 32.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. D.R.V.L.B Thambawita Combinational Logic Design
  • 33.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. 4 xy + x z + yz = xy + x z + yz(x + x ) = xy + x z + xyz + x yz = xy(1 + z) + x z(1 + y) = xy + x z. 5 (x + y)(x + z)(y + z) D.R.V.L.B Thambawita Combinational Logic Design
  • 34.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. 4 xy + x z + yz = xy + x z + yz(x + x ) = xy + x z + xyz + x yz = xy(1 + z) + x z(1 + y) = xy + x z. 5 (x + y)(x + z)(y + z) = (x + y)(x + z) D.R.V.L.B Thambawita Combinational Logic Design
  • 35.
    BOOLEAN FUNCTIONS 1 x(x+ y) = xx + xy = 0 + xy = xy. 2 x + x y = (x + x )(x + y) = 1(x + y) = x + y. 3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x. 4 xy + x z + yz = xy + x z + yz(x + x ) = xy + x z + xyz + x yz = xy(1 + z) + x z(1 + y) = xy + x z. 5 (x + y)(x + z)(y + z) = (x + y)(x + z) by duality from function 4 D.R.V.L.B Thambawita Combinational Logic Design
  • 36.
    BOOLEAN FUNCTIONS Complement ofa Function (A + B + C) = (A + x) let B + C = x = A x by DeMorgan = A (B + C) substitute B + C = x = A (B C ) by DeMorgan = ABC by associative D.R.V.L.B Thambawita Combinational Logic Design
  • 37.
    BOOLEAN FUNCTIONS Complement ofa Function (A + B + C + D + ... + F) = A B C D ...F (ABCD...F) = A + B + C + D + ... + F D.R.V.L.B Thambawita Combinational Logic Design
  • 38.
    BOOLEAN FUNCTIONS Find thecomplement of the functionsF1 = x yz + x y z and F2 = x(y z + yz) D.R.V.L.B Thambawita Combinational Logic Design
  • 39.
    BOOLEAN FUNCTIONS Find thecomplement of the functionsF1 = x yz + x y z and F2 = x(y z + yz) D.R.V.L.B Thambawita Combinational Logic Design
  • 40.
    BOOLEAN FUNCTIONS Do youknow? D.R.V.L.B Thambawita Combinational Logic Design
  • 41.
    More about Mintermsand Maxterms Minterms Consider two binary variables x and y combined with an AND operation D.R.V.L.B Thambawita Combinational Logic Design
  • 42.
    More about Mintermsand Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy D.R.V.L.B Thambawita Combinational Logic Design
  • 43.
    More about Mintermsand Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy Each of these four AND terms is called a minterm, or a standard product. D.R.V.L.B Thambawita Combinational Logic Design
  • 44.
    More about Mintermsand Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy Each of these four AND terms is called a minterm, or a standard product. n variables can be combined to form 2n minterms D.R.V.L.B Thambawita Combinational Logic Design
  • 45.
    More about Mintermsand Maxterms Minterms Consider two binary variables x and y combined with an AND operation There are four possible combinations: x’y’, x’y, xy’, and xy Each of these four AND terms is called a minterm, or a standard product. n variables can be combined to form 2n minterms D.R.V.L.B Thambawita Combinational Logic Design
  • 46.
    Minterms and Maxterms Maxterms Nvariables forming an OR term, with each variable being primed or unprimed, provide 2n possible combinations, called maxterms, or standard sums. D.R.V.L.B Thambawita Combinational Logic Design
  • 47.
    Minterms and Maxterms Maxterms Nvariables forming an OR term, with each variable being primed or unprimed, provide 2n possible combinations, called maxterms, or standard sums. The eight maxterms for three variables. D.R.V.L.B Thambawita Combinational Logic Design
  • 48.
    Minterms and Maxterms Maxterms Nvariables forming an OR term, with each variable being primed or unprimed, provide 2n possible combinations, called maxterms, or standard sums. The eight maxterms for three variables. D.R.V.L.B Thambawita Combinational Logic Design
  • 49.
    Minterms and Maxterms Mintermsand Maxterms for Three Binary Variables D.R.V.L.B Thambawita Combinational Logic Design
  • 50.
    Minterms and Maxterms Example:Functions of Three Variables D.R.V.L.B Thambawita Combinational Logic Design
  • 51.
    Minterms and Maxterms Fuctionf1 and f2 for above truth table D.R.V.L.B Thambawita Combinational Logic Design
  • 52.
    Minterms and Maxterms Fuctionf1 and f2 for above truth table f1 = x y z + xy z + xyz = m1 + m4 + m7 D.R.V.L.B Thambawita Combinational Logic Design
  • 53.
    Minterms and Maxterms Fuctionf1 and f2 for above truth table f1 = x y z + xy z + xyz = m1 + m4 + m7 f2 = x yz + xy z + xyz + xyz = m3 + m5 + m6 + m7 D.R.V.L.B Thambawita Combinational Logic Design
  • 54.
    Minterms and Maxterms Fuctionf1 and f2 for above truth table f1 = x y z + xy z + xyz = m1 + m4 + m7 f2 = x yz + xy z + xyz + xyz = m3 + m5 + m6 + m7 D.R.V.L.B Thambawita Combinational Logic Design
  • 55.
    PRODUCT-OF-SUMS Following table showsthe truth table for a Boolean function, Y, and its complement, ¯Y . Using De Morgans Theorem, derive the product-of-sums canonical form of Y from the sum-of-products form of ¯Y . D.R.V.L.B Thambawita Combinational Logic Design
  • 56.
    PRODUCT-OF-SUMS Solution: ¯Y = ¯A¯B + ¯AB (1) ¯¯y = y = ¯A ¯B + ¯AB = ( ¯A ¯B)( ¯AB) = (A + B)(A + ¯B) (2) D.R.V.L.B Thambawita Combinational Logic Design
  • 57.
    PRODUCT-OF-SUMS To express aBoolean function as a product of maxterms, it must first be brought into a form of OR terms. This may be done by using the distributive law, x + yz = (x + y)(x + z). Example: Express the Boolean function F = xy + xz as a product of maxterms D.R.V.L.B Thambawita Combinational Logic Design
  • 58.
    PRODUCT-OF-SUMS To express aBoolean function as a product of maxterms, it must first be brought into a form of OR terms. This may be done by using the distributive law, x + yz = (x + y)(x + z). Example: Express the Boolean function F = xy + xz as a product of maxterms F = xy + x y = (xy + x )(xy + z) = (x + x )(y + x )(x + z)(y + z) = (x + y)(x + z)(y + z) D.R.V.L.B Thambawita Combinational Logic Design Z
  • 59.
    Product of Maxterms Thefunction has three variables: x, y, and z. Each OR term is missing one variable; therefore D.R.V.L.B Thambawita Combinational Logic Design F = (x + y + z)(x + y + z)(x + y + z)(x + y + z) = M0M2M4M5
  • 60.
    Conversion between CanonicalForms The complement of a function expressed as the sum of minterms equals the sum of minterms missing from the original function Now, if we take the complement of F’ by DeMorgan’s theorem, we obtain F in a different form: D.R.V.L.B Thambawita Combinational Logic Design
  • 61.
    Conversion between CanonicalForms D.R.V.L.B Thambawita Combinational Logic Design
  • 62.
    Standard Forms Another wayto express Boolean functions is in standard form The sum of products is a Boolean expression containing AND terms, called product terms, with one or more literals each A product of sums is a Boolean expression containing OR terms, called sum terms. D.R.V.L.B Thambawita Combinational Logic Design
  • 63.
    Standard Forms D.R.V.L.B ThambawitaCombinational Logic Design
  • 64.
    Standard Forms: Threeand twolevel implementation D.R.V.L.B Thambawita Combinational Logic Design
  • 65.
    OTHER LOGIC OPERATIONS Thereare 22n functions for n binary variables. Thus, for two variables, n=2, and the number of possible Boolean functions is 16. Truth Tables for the 16 Functions of Two Binary Variables D.R.V.L.B Thambawita Combinational Logic Design
  • 66.
    OTHER LOGIC OPERATIONS BooleanExpressions for the 16 Functions of Two Variables D.R.V.L.B Thambawita Combinational Logic Design
  • 67.
    OTHER LOGIC OPERATIONS BooleanExpressions for the 16 Functions of Two Variables D.R.V.L.B Thambawita Combinational Logic Design
  • 68.
    DIGITAL LOGIC GATES D.R.V.L.BThambawita Combinational Logic Design
  • 69.
    DIGITAL LOGIC GATES D.R.V.L.BThambawita Combinational Logic Design
  • 70.
    DIGITAL LOGIC GATES D.R.V.L.BThambawita Combinational Logic Design
  • 71.
    DIGITAL LOGIC GATES D.R.V.L.BThambawita Combinational Logic Design
  • 72.
    Extension to MultipleInputs The gates shown can be extended to have more than two inputs. A gate can be extended to have multiple inputs if the binary operation it represents is commutative and associative. Ex: OR function x + y = y + x(commutative) (x + y) + z = x + (y + z) = x + y + z(associative) D.R.V.L.B Thambawita Combinational Logic Design
  • 73.
    Extension to MultipleInputs NAND and NOR functions are commutative NAND and NOR operators are not associative D.R.V.L.B Thambawita Combinational Logic Design Z
  • 74.
    Extension to MultipleInputs Threeinput exclusiveOR function Figure: Threeinput exclusive-OR gate Do you know? F is equal to 1 if only one input is equal to 1 or if all three inputs are equal to 1. (when the total number of 1’s in the input variables is odd) D.R.V.L.B Thambawita Combinational Logic Design
  • 75.
    Positive and NegativeLogic Do you know? Choosing the highlevel H to represent logic 1 defines a positive logic system. Choosing the lowlevel L to represent logic 1 defines a negative logic system. Figure: Signal assignment and logic polarity D.R.V.L.B Thambawita Combinational Logic Design
  • 76.
    Positive and NegativeLogic D.R.V.L.B Thambawita Combinational Logic Design
  • 77.
    Positive and NegativeLogic D.R.V.L.B Thambawita Combinational Logic Design
  • 78.
    Positive and NegativeLogic D.R.V.L.B Thambawita Combinational Logic Design