- THE THREE VARIABLE KARNAUGH MAP.
- THE FOUR VARIABLE KARNAUGH MAP.
- Karnaugh Map Simplification of SOP Expressions.
- KARNAUGH MAP PRODUCT OF SUM (POS) SIMPLIFICATION 5-VARIABLE K-MAPS.
- DON'T CARE CONDITIONS.
The document provides an introduction to Karnaugh maps (K-maps), which are a graphical method for simplifying Boolean logic expressions. It outlines the basics of K-maps, including their advantages, different types (2, 3, 4 and 5 variable maps), and how to group cells and derive simplified logic expressions. The document also discusses sum-of-products (SOP) form, product-of-sums (POS) form, don't care conditions, prime implicants, and includes examples of mapping truth tables to K-maps and simplifying expressions.
The Karnaugh map is a graphical method for simplifying Boolean algebra expressions. It arranges the terms of a Boolean function in a grid according to their binary values, making it easier to identify redundant terms. Groups of adjacent 1s in the map correspond to product terms that can be combined. Common map sizes include 2x2 for 2 variables, 2x4 for 3 variables, and 4x4 for 4 variables. The map can be used to find both Sum of Products and Product of Sum expressions.
This document discusses Karnaugh maps (K-maps) for simplifying logic functions with up to 5 variables. It provides examples of 4-variable and 5-variable K-maps, showing how to group minterms into rectangles to find sum of products (SoP) and product of sums (PoS) expressions. Don't care conditions are also explained, where certain minterm values can be treated as 0 or 1 to allow for more simplification. Practice problems are included for the reader to simplify functions using K-maps.
This presentation summarizes Karnaugh maps, which are a graphical technique for simplifying Boolean expressions. Karnaugh maps arrange the terms of a truth table in a two-dimensional grid, making common factors between terms visible. They can be used for functions with up to five variables. Examples show how to identify groupings of terms and simplify expressions using Karnaugh maps for two, three, and four variables. The presentation concludes with an example of a five variable Karnaugh map.
This presentation contains information about don't care conditions alongwith its examples, Karnaugh-Map i.e. K-map simplification using don't care conditions and seven-segment display with don't care conditions.
K-MAP(KARNAUGH MAP)BY THE SILENT PROGRAMMERkunal kishore
In return for this presentation please subscribe THE SILENT PROGRAMMER,,,,,,,,
where you can get different tricks to fool your pc and increase its perfomance
link:->>
https://www.youtube.com/channel/UCFxFchFfxRRxfWNCtUL7x8A
1) The document provides instructions for Assignment 51 which is due on Monday January 30th and includes checking Inventory #5 with a partner.
2) Lesson 51 discusses that 9.315 x 1012 represents a very large number, specifically 9,315,000,000,000, and examples of writing numbers in scientific and standard notation.
3) Students are given examples to write numbers in scientific and standard notation, compare numbers, and write a number in words.
This document contains a 13 question test on database and networking concepts for a Class 12 Computer Science exam. The questions cover topics like SQL commands, database concepts, network devices, topologies and security. They involve tasks like identifying errors, writing SQL queries, differentiating between command types, expanding acronyms, defining terms, drawing network diagrams and making recommendations for a medical center network setup.
The document provides an introduction to Karnaugh maps (K-maps), which are a graphical method for simplifying Boolean logic expressions. It outlines the basics of K-maps, including their advantages, different types (2, 3, 4 and 5 variable maps), and how to group cells and derive simplified logic expressions. The document also discusses sum-of-products (SOP) form, product-of-sums (POS) form, don't care conditions, prime implicants, and includes examples of mapping truth tables to K-maps and simplifying expressions.
The Karnaugh map is a graphical method for simplifying Boolean algebra expressions. It arranges the terms of a Boolean function in a grid according to their binary values, making it easier to identify redundant terms. Groups of adjacent 1s in the map correspond to product terms that can be combined. Common map sizes include 2x2 for 2 variables, 2x4 for 3 variables, and 4x4 for 4 variables. The map can be used to find both Sum of Products and Product of Sum expressions.
This document discusses Karnaugh maps (K-maps) for simplifying logic functions with up to 5 variables. It provides examples of 4-variable and 5-variable K-maps, showing how to group minterms into rectangles to find sum of products (SoP) and product of sums (PoS) expressions. Don't care conditions are also explained, where certain minterm values can be treated as 0 or 1 to allow for more simplification. Practice problems are included for the reader to simplify functions using K-maps.
This presentation summarizes Karnaugh maps, which are a graphical technique for simplifying Boolean expressions. Karnaugh maps arrange the terms of a truth table in a two-dimensional grid, making common factors between terms visible. They can be used for functions with up to five variables. Examples show how to identify groupings of terms and simplify expressions using Karnaugh maps for two, three, and four variables. The presentation concludes with an example of a five variable Karnaugh map.
This presentation contains information about don't care conditions alongwith its examples, Karnaugh-Map i.e. K-map simplification using don't care conditions and seven-segment display with don't care conditions.
K-MAP(KARNAUGH MAP)BY THE SILENT PROGRAMMERkunal kishore
In return for this presentation please subscribe THE SILENT PROGRAMMER,,,,,,,,
where you can get different tricks to fool your pc and increase its perfomance
link:->>
https://www.youtube.com/channel/UCFxFchFfxRRxfWNCtUL7x8A
1) The document provides instructions for Assignment 51 which is due on Monday January 30th and includes checking Inventory #5 with a partner.
2) Lesson 51 discusses that 9.315 x 1012 represents a very large number, specifically 9,315,000,000,000, and examples of writing numbers in scientific and standard notation.
3) Students are given examples to write numbers in scientific and standard notation, compare numbers, and write a number in words.
This document contains a 13 question test on database and networking concepts for a Class 12 Computer Science exam. The questions cover topics like SQL commands, database concepts, network devices, topologies and security. They involve tasks like identifying errors, writing SQL queries, differentiating between command types, expanding acronyms, defining terms, drawing network diagrams and making recommendations for a medical center network setup.
The document discusses finding the sum of two functions f(x) and g(x). It provides two examples:
1) f(x)=x^2 and g(x)=2. The sum function h(x)=x^2+2 is graphed by adding the y-values.
2) f(x)=x^2 and g(x)=3x. The sum function p(x)=x^2+3x is graphed by adding the y-values. The domain is all real numbers and the range is all real numbers greater than -9/4, as determined by finding the midpoint of the function.
This document discusses Karnaugh maps, which are a graphical technique for simplifying Boolean expressions into minimal sum-of-products form. It covers how to:
1) Construct Karnaugh maps for 2, 3, and 4 variable expressions from truth tables or sums of minterms.
2) Group adjacent squares containing common literals to find the simplest product terms.
3) Read the minimal sum-of-products form from the groupings on the map.
Examples are provided to demonstrate how to simplify expressions and construct Karnaugh maps.
This document illustrates the difference between autocorrelation and convolution. Autocorrelation is defined by an equation that involves integrating the product of a function and a shifted version of itself. Convolution is defined by an equation that involves integrating the product of two functions - one shifted over the other. Both autocorrelation and convolution have Fourier transforms that are related to the Fourier transform of the original functions in different ways, as described by theorems. The document provides equations and applets to demonstrate calculating autocorrelation, convolution, and their Fourier transforms.
This document contains lecture notes from a Math 1a class on October 17, 2007. It discusses increasing and decreasing functions, and concavity and the second derivative. It provides definitions of increasing, decreasing, concave up, and concave down functions on an interval based on the sign of the function. It also lists several facts, such as if a function is increasing on an interval then it is greater than or equal to 0 on that interval.
This document discusses Karnaugh maps (K-maps), a method for simplifying Boolean algebra expressions. It begins by stating that K-maps allow for minimized results with less calculation compared to Boolean algebra alone. The document then covers the basics of K-maps, including how to represent different numbers of variables and the rules for grouping ones and zeros. It provides examples of using K-maps to minimize functions with 2 to 5 variables. Finally, it discusses extensions of K-maps, such as incorporating don't cares, using maxterms instead of minterms, and when the Quine-McCluskey method is preferable to K-maps for problems with many variables.
Given a function f, the derivative f' can be used to get important information about f. For instance, f is increasing when f'>0. The second derivative gives useful concavity information.
Este documento presenta un proyecto de plan de lectura para el curso 2013/2014 desarrollado por un grupo de estudiantes de la Facultad de Educación de la Universidad de Castilla La-Mancha. El plan de lectura incluye objetivos generales, secuencialización de objetivos y contenidos por niveles de primaria, criterios de evaluación, metodología, competencias básicas y actividades sugeridas. El objetivo principal es lograr que los estudiantes sean lectores competentes y desarrollen el hábito de la lectura de forma
Programme EXPOGEP // Programa EXPOGEP (Brasília, 02/02/2014)COLUFRAS
Programme provisoire d’un séminaire international axé sur la formation et la rémunération des professionnels de la santé responsables des soins primaires, comme condition de la pérennité des systèmes publics de santé. // Programa provisório de um seminário internacional focado na formação e na remuneração dos profissionais da atenção primária em saúde (aps), como condição de sustentabilidade dos sistemas públicos de saúde.
O documento descreve o relatório e o parecer prévio de uma sessão do Tribunal de Contas do Estado de Pernambuco que julgou as contas da Prefeitura Municipal de Santa Cruz do Capibaribe do exercício de 2008 como regulares com ressalvas, dando quitação ao prefeito. O Ministério Público de Contas recorreu da decisão pedindo a reforma para julgamento de irregularidade das contas e emissão de parecer prévio recomendando rejeição.
Photography developed through the contributions of scientists and artists around the world. Early photographic processes created individual plates or prints, but later processes used paper or albumen to create multiple copies. Key developments included Daguerre's daguerreotype process in 1839, which used iodine and mercury to create positive prints on silver plates, as well as Talbot's calotype process in 1841, which used paper negatives. Over time, emulsions containing light-sensitive silver salts and applied to surfaces like film or glass became the standard.
Getting sh*t done: How design is changing the way Coolblue works - JeffreyCoolblue
When you order something at Coolblue a lot of things happen behind the scenes. Thank goodness there's Vanessa, our back-office software that helps colleagues get sh*t done. Started nearly 15 years ago, she's due for retirement. Let us talk you through how we're redesigning such a complex application and what we've learned along the way.
With the price of crude remaining around $30 a barrel, now more than ever it is vital that operators fully assess their artificial lift selection criteria to increase production efficiency, reduce equipment & energy costs, and mitigate failure. The Artificial Lift Optimization Congress North America 2016, coming to Houston on March 22-23, 2016, will deliver production optimization case studies to maximize production rates at minimal costs.
In the lead-up to Houston, we interviewed Zach Awny, Production Engineer at ConocoPhillips, who discussed reducing lifting cost per barrel and the increased reliance of "smart" wells now and going forward.
How to attract clients for your own businessSameer Nawab
The document provides tips on how to attract and keep customers for a business. It defines marketing and customers, and outlines nine tips for attracting customers, including understanding what makes your business unique, understanding your target customer, focusing your marketing efforts on your customer, using social media and referrals, maintaining contact with existing customers through newsletters and calls, following up with customers properly, offering guarantees to reduce risk, and branding your content.
This document discusses different types of logic gates and their functions. It begins by defining boolean variables and truth tables. It then explains the operations of OR, AND, NOT, NOR, NAND, exclusive-OR and exclusive-NOR gates. Various integrated circuit logic families are also covered such as diode logic, resistor-transistor logic, diode-transistor logic, transistor-transistor logic and CMOS logic. The document concludes by comparing the performance characteristics of different logic gate technologies.
FYBSC IT Digital Electronics Unit II Chapter II Minterm, Maxterm and Karnaugh...Arti Parab Academics
Minterm, Maxterm and Karnaugh Maps:
Introduction, minterms and sum of minterm form, maxterm and Product
of maxterm form, Reduction technique using Karnaugh maps –
2/3/4/5/6 variable K-maps, Grouping of variables in K-maps, K-maps
for product of sum form, minimize Boolean expression using K-map
and obtain K-map from Boolean expression, Quine Mc Cluskey
Method.
The document discusses Karnaugh map methods for minimizing Boolean functions. It introduces Karnaugh maps as a tool for representing Boolean functions with up to six variables. The key points covered are:
1. Karnaugh maps arrange the variables in a grid so that logically adjacent cells correspond to inputs that differ in only one variable.
2. Cells are marked with 1s or 0s based on the function's truth table. Adjacent 1s can be combined to eliminate variables and simplify the function.
3. Examples show how to use Karnaugh maps to minimize Boolean functions expressed as sums of products (SOP) or products of sums (POS).
4. "Don't care"
This document discusses Karnaugh maps (K-maps), which are a tool for simplifying Boolean expressions into their minimum sum-of-products (SOP) or product-of-sums (POS) forms. It explains how to map standard and non-standard SOP expressions onto K-maps by placing 1s in the cells corresponding to the terms. Groups of adjacent 1s can then be identified to find the minimum SOP expression. The same process applies to POS expressions, except 0s are placed on the map instead of 1s. Examples are provided to demonstrate how to group cells and determine the minimum expressions from K-maps.
The document discusses finding the sum of two functions f(x) and g(x). It provides two examples:
1) f(x)=x^2 and g(x)=2. The sum function h(x)=x^2+2 is graphed by adding the y-values.
2) f(x)=x^2 and g(x)=3x. The sum function p(x)=x^2+3x is graphed by adding the y-values. The domain is all real numbers and the range is all real numbers greater than -9/4, as determined by finding the midpoint of the function.
This document discusses Karnaugh maps, which are a graphical technique for simplifying Boolean expressions into minimal sum-of-products form. It covers how to:
1) Construct Karnaugh maps for 2, 3, and 4 variable expressions from truth tables or sums of minterms.
2) Group adjacent squares containing common literals to find the simplest product terms.
3) Read the minimal sum-of-products form from the groupings on the map.
Examples are provided to demonstrate how to simplify expressions and construct Karnaugh maps.
This document illustrates the difference between autocorrelation and convolution. Autocorrelation is defined by an equation that involves integrating the product of a function and a shifted version of itself. Convolution is defined by an equation that involves integrating the product of two functions - one shifted over the other. Both autocorrelation and convolution have Fourier transforms that are related to the Fourier transform of the original functions in different ways, as described by theorems. The document provides equations and applets to demonstrate calculating autocorrelation, convolution, and their Fourier transforms.
This document contains lecture notes from a Math 1a class on October 17, 2007. It discusses increasing and decreasing functions, and concavity and the second derivative. It provides definitions of increasing, decreasing, concave up, and concave down functions on an interval based on the sign of the function. It also lists several facts, such as if a function is increasing on an interval then it is greater than or equal to 0 on that interval.
This document discusses Karnaugh maps (K-maps), a method for simplifying Boolean algebra expressions. It begins by stating that K-maps allow for minimized results with less calculation compared to Boolean algebra alone. The document then covers the basics of K-maps, including how to represent different numbers of variables and the rules for grouping ones and zeros. It provides examples of using K-maps to minimize functions with 2 to 5 variables. Finally, it discusses extensions of K-maps, such as incorporating don't cares, using maxterms instead of minterms, and when the Quine-McCluskey method is preferable to K-maps for problems with many variables.
Given a function f, the derivative f' can be used to get important information about f. For instance, f is increasing when f'>0. The second derivative gives useful concavity information.
Este documento presenta un proyecto de plan de lectura para el curso 2013/2014 desarrollado por un grupo de estudiantes de la Facultad de Educación de la Universidad de Castilla La-Mancha. El plan de lectura incluye objetivos generales, secuencialización de objetivos y contenidos por niveles de primaria, criterios de evaluación, metodología, competencias básicas y actividades sugeridas. El objetivo principal es lograr que los estudiantes sean lectores competentes y desarrollen el hábito de la lectura de forma
Programme EXPOGEP // Programa EXPOGEP (Brasília, 02/02/2014)COLUFRAS
Programme provisoire d’un séminaire international axé sur la formation et la rémunération des professionnels de la santé responsables des soins primaires, comme condition de la pérennité des systèmes publics de santé. // Programa provisório de um seminário internacional focado na formação e na remuneração dos profissionais da atenção primária em saúde (aps), como condição de sustentabilidade dos sistemas públicos de saúde.
O documento descreve o relatório e o parecer prévio de uma sessão do Tribunal de Contas do Estado de Pernambuco que julgou as contas da Prefeitura Municipal de Santa Cruz do Capibaribe do exercício de 2008 como regulares com ressalvas, dando quitação ao prefeito. O Ministério Público de Contas recorreu da decisão pedindo a reforma para julgamento de irregularidade das contas e emissão de parecer prévio recomendando rejeição.
Photography developed through the contributions of scientists and artists around the world. Early photographic processes created individual plates or prints, but later processes used paper or albumen to create multiple copies. Key developments included Daguerre's daguerreotype process in 1839, which used iodine and mercury to create positive prints on silver plates, as well as Talbot's calotype process in 1841, which used paper negatives. Over time, emulsions containing light-sensitive silver salts and applied to surfaces like film or glass became the standard.
Getting sh*t done: How design is changing the way Coolblue works - JeffreyCoolblue
When you order something at Coolblue a lot of things happen behind the scenes. Thank goodness there's Vanessa, our back-office software that helps colleagues get sh*t done. Started nearly 15 years ago, she's due for retirement. Let us talk you through how we're redesigning such a complex application and what we've learned along the way.
With the price of crude remaining around $30 a barrel, now more than ever it is vital that operators fully assess their artificial lift selection criteria to increase production efficiency, reduce equipment & energy costs, and mitigate failure. The Artificial Lift Optimization Congress North America 2016, coming to Houston on March 22-23, 2016, will deliver production optimization case studies to maximize production rates at minimal costs.
In the lead-up to Houston, we interviewed Zach Awny, Production Engineer at ConocoPhillips, who discussed reducing lifting cost per barrel and the increased reliance of "smart" wells now and going forward.
How to attract clients for your own businessSameer Nawab
The document provides tips on how to attract and keep customers for a business. It defines marketing and customers, and outlines nine tips for attracting customers, including understanding what makes your business unique, understanding your target customer, focusing your marketing efforts on your customer, using social media and referrals, maintaining contact with existing customers through newsletters and calls, following up with customers properly, offering guarantees to reduce risk, and branding your content.
This document discusses different types of logic gates and their functions. It begins by defining boolean variables and truth tables. It then explains the operations of OR, AND, NOT, NOR, NAND, exclusive-OR and exclusive-NOR gates. Various integrated circuit logic families are also covered such as diode logic, resistor-transistor logic, diode-transistor logic, transistor-transistor logic and CMOS logic. The document concludes by comparing the performance characteristics of different logic gate technologies.
FYBSC IT Digital Electronics Unit II Chapter II Minterm, Maxterm and Karnaugh...Arti Parab Academics
Minterm, Maxterm and Karnaugh Maps:
Introduction, minterms and sum of minterm form, maxterm and Product
of maxterm form, Reduction technique using Karnaugh maps –
2/3/4/5/6 variable K-maps, Grouping of variables in K-maps, K-maps
for product of sum form, minimize Boolean expression using K-map
and obtain K-map from Boolean expression, Quine Mc Cluskey
Method.
The document discusses Karnaugh map methods for minimizing Boolean functions. It introduces Karnaugh maps as a tool for representing Boolean functions with up to six variables. The key points covered are:
1. Karnaugh maps arrange the variables in a grid so that logically adjacent cells correspond to inputs that differ in only one variable.
2. Cells are marked with 1s or 0s based on the function's truth table. Adjacent 1s can be combined to eliminate variables and simplify the function.
3. Examples show how to use Karnaugh maps to minimize Boolean functions expressed as sums of products (SOP) or products of sums (POS).
4. "Don't care"
This document discusses Karnaugh maps (K-maps), which are a tool for simplifying Boolean expressions into their minimum sum-of-products (SOP) or product-of-sums (POS) forms. It explains how to map standard and non-standard SOP expressions onto K-maps by placing 1s in the cells corresponding to the terms. Groups of adjacent 1s can then be identified to find the minimum SOP expression. The same process applies to POS expressions, except 0s are placed on the map instead of 1s. Examples are provided to demonstrate how to group cells and determine the minimum expressions from K-maps.
After successfully completing this module students should be able to:
Understand the Need to simplify (minimize) expressions
List Different Methods for Minimization
Karnaugh Maps
Algebraic method
Use Karnaugh Map method to minimize the Boolean expression
The document discusses Karnaugh maps, a method for minimizing Boolean expressions. It begins by introducing Karnaugh maps and their inventor. It then covers drawing K-maps, the minimization steps including forming groups to reduce literals, and rules for simplification. The example minimizes an expression into groups using octets, quads, pairs and single cells to arrive at the final minimized expression.
The document discusses the minimization of Boolean expressions using Karnaugh maps (K-maps). It explains that K-maps provide an alternative graphical method for simplifying logic circuits by arranging the 1s and 0s from a truth table. The key steps involve drawing the K-map based on the number of variables, entering the function values, forming the largest possible groups of adjacent 1s, and selecting the minimum number of groups needed to cover all 1s while avoiding redundancy. Examples demonstrate how octet, quad, pair and single cell reductions can minimize expressions by reducing the number of literals. Rules for valid groupings using K-maps in simplifying expressions are also outlined.
The document describes how to use a Karnaugh map to simplify sum of products (SOP) expressions:
1) Construct a K-map according to the number of variables.
2) Place 1s in cells for min-terms in the SOP expression and 0s elsewhere.
3) Encircle isolated 1s and octets, quads, and pairs formed by rolling and overlapping.
4) Eliminate redundant groups and write the simplified expression by ORing products for each circled group.
Two examples are given to simplify expressions using a two-variable and four-variable K-map.
This document discusses various methods for minimizing switching functions, including:
1. The Karnaugh map method, which represents truth tables graphically to find logically adjacent terms that can be combined.
2. Prime implicants and essential prime implicants, which are product terms that cover minterms. The essential ones must be included in the minimal expression.
3. Don't care conditions, which allow further simplification by treating unspecified minterms as don't cares.
4. The Quine-McCluskey tabulation method, which systematically generates prime implicants and finds the essential ones and minimal cover.
Okay, here are the steps to solve this problem:
* Florante's location is given as point (2, 1)
* Laura's x-coordinate is given as 8
* The slope of the line containing their locations is given as 1/3
* We can use the slope formula to find Laura's y-coordinate:
Slope (m) = (Change in y) / (Change in x)
* Change in x from Florante to Laura is 8 - 2 = 6
* Change in y is what we want to find and is represented by y - 1
* We are given: m = Rise/Run = 1/3
* Plugging into the slope formula:
1/
The document discusses Karnaugh maps (K-maps), which are a tool for representing and simplifying Boolean functions with up to six variables. K-maps arrange the variables in a grid according to their binary values. Adjacent cells that differ in only one variable can be combined to simplify the function by eliminating that variable. The document provides examples of using K-maps to minimize Boolean functions in sum of products and product of sums form. It also discusses techniques like combining cells into the largest groups possible and handling don't-care conditions to further simplify expressions.
Digital electronics k map comparators and their functionkumarankit06875
This document provides an overview of a digital electronics presentation covering K-maps, comparators, and their applications. The agenda includes an introduction to K-maps and how they are used to simplify Boolean expressions. It also covers comparators, their operation and function. Examples are given of using K-maps to minimize logic expressions and identify prime implicants. The applications of K-maps in digital circuit design optimization are discussed. Comparators and their use in examples is briefly outlined.
This document discusses various methods for minimizing switching functions, including:
1. The Karnaugh map method, which graphically represents truth tables to find logically adjacent terms that can be combined.
2. Prime implicants, which are product terms obtained by maximally combining adjacent squares on a K-map. Essential prime implicants must be included in the minimal expression.
3. Don't care conditions, which allow for unspecified minterms that provide further simplification opportunities on the K-map.
4. The Quine-McCluskey tabular method, which systematically generates prime implicants and uses a cover table to find the essential prime implicants and minimal cover.
Program Derivation of Operations in Finite Fields of Prime OrderCharles Southerland
This document summarizes Charles Southerland's presentation on program derivation of operations in finite prime fields Fp. It begins with an introduction and thanks section. Then, it outlines the topics to be covered: finite fields, program derivation, and deriving a program to find the multiplicative inverse in Fp. It provides background on finite fields, Dijkstra's guarded command language, the weakest precondition predicate transformer, and the process of program derivation. It also discusses multiplicative inverses in finite fields, the greatest common divisor algorithm, exploring Bezout's identity, and deriving a program to find the multiplicative inverse using a loop invariant based on Bezout's identity and the gcd algorithm.
The document discusses Karnaugh maps, which are a method for simplifying Boolean algebraic expressions with 2-4 variables. It provides rules for constructing and using Karnaugh maps, including that groups cannot include zeros, must be rectangular, and can wrap around the map. Steps are outlined to solve expressions using Karnaugh maps by placing 1s or 0s on the map based on whether solving for sum of products or product of sums form, grouping variables in powers of 2, and determining the product or sum terms from the groups. Examples are given of using Karnaugh maps to minimize 3 and 4 variable expressions in both sum of products and product of sums form.
The document discusses AND/OR graphs and the AO* algorithm for searching AND/OR trees. Some problems can be represented as having subgoals that can be achieved simultaneously or independently (AND) or as OR options. The AO* algorithm extends A* search to AND/OR trees. It examines multiple nodes simultaneously, selecting the most promising path and expanding nodes to generate successors. It computes heuristic values (h) for nodes and propagates new information up the graph as the search progresses until a solution is found or all paths are determined to be unsolvable. An example demonstrates how AO* searches an AND/OR graph and labels nodes as it proceeds.
This document outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
167
c h a p t e r
4
Optimized Implementation of Logic
Functions
Chapter Objectives
In this chapter you will learn about:
• Synthesis of logic functions
• Analysis of logic circuits
• Techniques for deriving minimum-cost implementations of logic functions
• Graphical representation of logic functions in the form of Karnaugh maps
• Cubical representation of logic functions
• Use of CAD tools and VHDL to implement logic functions
168 C H A P T E R 4 • Optimized Implementation of Logic Functions
In Chapter 2 we showed that algebraic manipulation can be used to find the lowest-cost implementations of
logic functions. The purpose of that chapter was to introduce the basic concepts in the synthesis process.
The reader is probably convinced that it is easy to derive a straightforward realization of a logic function in
a canonical form, but it is not at all obvious how to choose and apply the theorems and properties of section
2.5 to find a minimum-cost circuit. Indeed, the algebraic manipulation is rather tedious and quite impractical
for functions of many variables.
If CAD tools are used to design logic circuits, the task of minimizing the cost of implementation does
not fall to the designer; the tools perform the necessary optimizations automatically. Even so, it is essential to
know something about this process. Most CAD tools have many features and options that are under control
of the user. To know when and how to apply these options, the user must have an understanding of what the
tools do.
In this chapter we will introduce some of the optimization techniques implemented in CAD tools and
show how these techniques can be automated. As a first step we will discuss a graphical approach, known as
the Karnaugh map, which provides a neat way to manually derive minimum-cost implementations of simple
logic functions. Although it is not suitable for implementation in CAD tools, it illustrates a number of key
concepts. We will show how both two-level and multilevel circuits can be designed. Then we will describe a
cubical representation for logic functions, which is suitable for use in CAD tools. We will also continue our
discussion of the VHDL language.
4.1 Karnaugh Map
In section 2.6 we saw that the key to finding a minimum-cost expression for a given logic
function is to reduce the number of product (or sum) terms needed in the expression, by
applying the combining property 14a (or 14b) as judiciously as possible. The Karnaugh map
approach provides a systematic way of performing this optimization. To understand how it
works, it is useful to review the algebraic approach from Chapter 2. Consider the function
f in Figure 4.1. The canonical sum-of-products expression for f consists of minterms m0,
m2, m4, m5, and m6, so that
f = x1x2x3 + x1x2x3 + x1x2x3 + x1x2x3 + x1x2x3
The combining property 14a allows us to replace two minterms that differ in the value of
only one variable with a single product term that does not include that variable a.
Similar to Logic Design - Chapter 4: Karnaugh Maps (19)
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
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Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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2. Contents
THE THREE VARIABLE KARNAUGH MAP
THE FOUR VARIABLE KARNAUGH MAP
Karnaugh Map Simplification of SOP Expressions
KARNAUGH MAP PRODUCT OF SUM (POS)
SIMPLIFICATION
5-VARIABLE K-MAPS
DON'T CARE CONDITIONS
2
5. Karnaugh Map Simplification of SOP
Expressions
Grouping
the 1s
1. A group must contain either 1, 2, 4, 8,
or 16 cells.
2. Each cell in a group must be adjacent to
one or more cells in that same group.
3. Always include the largest possible
number of 1s in a group.
4. Each 1 on the map must be included in at
least one group.
Karnaugh Maps
6. Determining the Minimum SOP
Expression from the Map
1. Group the cells that have 1s.
2. Determine the minimum product terms for each group.
For a 3-variable map:
?
For a 5-variable map
A 1-cell group yields a 3-variable product term
A 2-cell group yields a 2-variable product term
A 4-cell group yields a 1-variable term
An 8-cell group yields a value of 1 for the expression
For a 4-variable map
(1)
(2)
(3)
(4)
?
3. When all the minimum product terms are derived from
the Karnaugh map, they are summed to form the minimum
SOP expression.
Karnaugh Maps
11. KARNAUGH MAP PRODUCT OF
SUM (POS) SIMPLIFICATION
If we mark the empty squares by 0's and
combine them into valid adjacent squares,
we obtain a simplified expression of the
complement of the function, i.e., of F'.
The complement of F' gives us back the
function F.
Because of Demorgan's theorem, the
function so obtained is automatically in the
product of sums form.
11
12. EXAMPLE
Simplify the following Boolean function in (a) sum
of products and (b) product of sums.
F(w,x,y,z) = Σ (0,1,2,3,10,11,14)
12
13. EXAMPLE
Use a Karnaugh map to
minimize the following
POS expression.
(x+y+z)(w+x+y'+z)
(w'+x+y+z')
(w+x'+y+z)
(w'+x'+y+z)
(w+x+y+z)(w'+x+y+z)
(w+x+y'+z)
(w'+x+y+z')
(w+x'+y+z)
(w'+x'+y+z)
=Π(0,8,2,9,4,12)
13
14. 5-VARIABLE K-MAPS (1/2)
K-maps of more than 4 variables are more
difficult to use because the geometry (hypercube
configurations) for combining adjacent squares
becomes more involved.
For 5 variables, e.g. v, w, x, y, z, we need 25 =
32 squares.
Each square has 5 neighbours.
ELC224C
Karnaugh Maps
15. 5-VARIABLE K-MAPS (2/2)
Organized as two 4-variable K-maps. One for v'
and the other for v.
v'
wx
v
y
yz
00
01
11
m0
m1
m3
0 m4
1
m5
m7
00
m12
w
wx
10
00
01
11
10
m2
00 m16
m17
m19
m18
m6
0 m20
1
m21
m23
m22
m28
m29
m31
m30
m24
m25
m27
m26
x
m13
m15
m14
w
11
m8
m9
m11
y
yz
11
m10
10
x
10
z
z
Corresponding squares of each map are adjacent.
Can visualise this as one 4-variable K-map being on
TOP of the other 4-variable K-map.
ELC224C
Karnaugh Maps
16. DON'T CARE CONDITIONS
Sometimes a situation arises in which some input
variable combinations are not allowed.
Since these unallowed states will never occur in
an application involving the BCD code, they can
be treated as "don't care" terms
for each "don't care" term, an X is placed in the
cell.
When grouping the 1's, Xs can be treated as 1's
to make a larger grouping or as 0s if they cannot
be used to advantage.
Be careful do not make a group entirely of
x's.
16