The document discusses different number systems and digital logic concepts. It describes the decimal, binary, octal and hexadecimal number systems. It also covers number system conversions, signed and complement representations, coding systems like BCD and Gray code, and universal gates like NAND and NOR gates. All digital circuits are ultimately based on the binary number system and these fundamental concepts.
Digital design uses computer skills and creativity to design visuals for electronic technology. It includes fields like web design, digital imaging, and 3D modeling. Digital design creates graphics and designs for the web, TV, print, and portable devices using computers, graphics tablets, and other electronic tools. It is an evolving industry that explores new technologies. Digital design has many applications including web design, 3D modeling for movies, architectural planning, and product design. 3D modeling involves creating mathematical representations of objects, placing them in virtual scenes, and rendering them into images. Popular 3D modeling programs include 3Ds Max, Maya, SketchUp, Rhino, CATIA, and SolidWorks.
This document discusses parallel prefix adders. It provides background on parallel prefix operations and defines binary addition as a parallel prefix problem. The key steps of carry lookahead adders are described, including precomputing propagate and generate values, calculating carries through a carry generation block, and combining carries and propagates to generate the sum. Several parallel prefix adder architectures are introduced, including the Sklansky conditional adder, Kogge-Stone adder, and Ladner-Fischer adder, which aim to optimize parameters like depth, node count, and fan-out.
Image Interpolation Techniques with Optical and Digital Zoom Conceptsmmjalbiaty
Digital image concepts and interpolation techniques for optical and digital zoom are discussed. There are three main types of interpolation used for resizing images: nearest neighbor, bilinear, and bicubic. Nearest neighbor is the simplest but produces the lowest quality, while bicubic is the most complex but highest quality. Optical zoom uses lens magnification before sensing, whereas digital zoom interpolates after sensing, resulting in lower quality than optical zoom. Interpolation methods assign pixel values to new locations during resizing based on weighting patterns around the original pixel values.
The document discusses the career prospects for graduating engineers in India's VLSI industry. It notes the growing demand for skilled VLSI engineers and the talent shortage India faces. It highlights the knowledge and skills gap in fresh graduates, including lack of VLSI design skills, problem solving, and soft skills. The document proposes that a holistic VLSI training program encompassing all key skills could help alleviate the talent crunch by making freshers job-ready for India's booming semiconductor industry.
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
This document discusses various algorithms for polygon scan conversion and filling, including:
- The scan line polygon fill algorithm which determines pixel color by calculating polygon edge intersections with scan lines and using an odd-even rule.
- Methods for handling special cases like horizontal edges and vertex intersections.
- Using a sorted edge table and active edge list to incrementally calculate edge intersections across scan lines.
- Flood fill and depth/z-buffer algorithms for hidden surface removal when rendering overlapping polygons.
Digital design uses computer skills and creativity to design visuals for electronic technology. It includes fields like web design, digital imaging, and 3D modeling. Digital design creates graphics and designs for the web, TV, print, and portable devices using computers, graphics tablets, and other electronic tools. It is an evolving industry that explores new technologies. Digital design has many applications including web design, 3D modeling for movies, architectural planning, and product design. 3D modeling involves creating mathematical representations of objects, placing them in virtual scenes, and rendering them into images. Popular 3D modeling programs include 3Ds Max, Maya, SketchUp, Rhino, CATIA, and SolidWorks.
This document discusses parallel prefix adders. It provides background on parallel prefix operations and defines binary addition as a parallel prefix problem. The key steps of carry lookahead adders are described, including precomputing propagate and generate values, calculating carries through a carry generation block, and combining carries and propagates to generate the sum. Several parallel prefix adder architectures are introduced, including the Sklansky conditional adder, Kogge-Stone adder, and Ladner-Fischer adder, which aim to optimize parameters like depth, node count, and fan-out.
Image Interpolation Techniques with Optical and Digital Zoom Conceptsmmjalbiaty
Digital image concepts and interpolation techniques for optical and digital zoom are discussed. There are three main types of interpolation used for resizing images: nearest neighbor, bilinear, and bicubic. Nearest neighbor is the simplest but produces the lowest quality, while bicubic is the most complex but highest quality. Optical zoom uses lens magnification before sensing, whereas digital zoom interpolates after sensing, resulting in lower quality than optical zoom. Interpolation methods assign pixel values to new locations during resizing based on weighting patterns around the original pixel values.
The document discusses the career prospects for graduating engineers in India's VLSI industry. It notes the growing demand for skilled VLSI engineers and the talent shortage India faces. It highlights the knowledge and skills gap in fresh graduates, including lack of VLSI design skills, problem solving, and soft skills. The document proposes that a holistic VLSI training program encompassing all key skills could help alleviate the talent crunch by making freshers job-ready for India's booming semiconductor industry.
The document discusses finite fields and related algebraic concepts. It begins by defining groups, rings, and fields. It then focuses on finite fields, particularly GF(p) fields consisting of integers modulo a prime p. It discusses finding multiplicative inverses in such fields using the extended Euclidean algorithm. As an example, it finds the inverse of 550 modulo 1759.
This document discusses various algorithms for polygon scan conversion and filling, including:
- The scan line polygon fill algorithm which determines pixel color by calculating polygon edge intersections with scan lines and using an odd-even rule.
- Methods for handling special cases like horizontal edges and vertex intersections.
- Using a sorted edge table and active edge list to incrementally calculate edge intersections across scan lines.
- Flood fill and depth/z-buffer algorithms for hidden surface removal when rendering overlapping polygons.
This document describes Bresenham's circle algorithm for efficiently scan converting a circle. It begins by explaining the symmetry of a circle and prior inefficient polynomial and trigonometric methods. It then presents Bresenham's algorithm which takes advantage of the circle's symmetry and uses a decision variable to determine whether to move in the x or y direction to plot each pixel, ensuring points are always closest to the true circle. The algorithm is presented with variables initialized and steps to iterate through the first octant to plot all pixels.
This document discusses numerical data representation and digital and analog systems. It notes that there are two types of numerical representation: analog and digital. It also describes two types of systems: analog systems, which use analog representations, and digital systems, which use digital representations. The document outlines several advantages of digital systems, such as being easier to design, having greater accuracy and precision, and being less affected by noise. It also notes a limitation of digital systems is that the real world is analog. To take advantage of digital techniques, analog inputs must be converted to digital forms, operations performed digitally, and digital outputs converted back to analog forms.
This document provides an outline for a course on digital logic design. It includes the course title and credit hours, topics that will be covered such as Boolean algebra, logic gates, combinational and sequential circuits, programmable logic devices, and memory. It also lists recommended textbooks and provides the grading breakdown. Examples of analogue and digital quantities, signals, and number systems are given. Common logic gates such as AND, OR, NOT, NAND and NOR are described along with their truth tables and applications. Combinational circuits, functional devices, sequential circuits and memory are also introduced.
Cubic curves are commonly used in graphics because curves of lower order commonly have too little flexibility, while curves of higher order are usually considered unnecessarily complex and make it easy to introduce undesired wiggles.
This document discusses various 2D transformations in computer graphics including translation, rotation, and scaling. Translation moves an object by adding offsets to the x and y coordinates. Rotation uses trigonometric functions and rotation matrices to reposition objects around a central point. Scaling enlarges or shrinks objects by multiplying their coordinates by scaling factors. Homogeneous coordinates generalize these transformations into matrix operations.
This document contains information about 3D display methods in computer graphics presented by a group of 5 students. It discusses parallel projection, perspective projection, depth cueing, visible line identification, and surface rendering techniques. The goal is to generate realistic 3D images and correctly display depth relationships between objects.
Comprehensive coverage of fundamentals of computer graphics.
3D Transformations
Reflections
3D Display methods
3D Object Representation
Polygon surfaces
Quadratic Surfaces
Digital logic design is the basis of electronic systems like computers and cell phones. It uses binary numbers (zeros and ones) to represent information and process input/output operations. Digital logic employs logic gates that perform functions like AND, OR, and NOT to translate binary input signals into specific outputs. Career opportunities include developing device infrastructures using components like information storage, signal transmission, and information processing. Engineers work to improve performance, decrease energy usage, and debug issues.
This document provides information about a digital logic design course taught by Dr. Javaid Khurshid including the instructor and lab instructor contact details, lecture and lab schedule, grading policy, textbooks, and syllabus. The syllabus covers topics such as number systems, logic gates, Boolean algebra, combinational and sequential logic, memory, and microprocessors.
This document discusses various two-dimensional geometric transformations including translations, rotations, scaling, reflections, shears, and composite transformations. Translations move objects without deformation using a translation vector. Rotations rotate objects around a fixed point or pivot point. Scaling transformations enlarge or shrink objects using scaling factors. Reflections produce a mirror image of an object across an axis. Shearing slants an object along an axis. Composite transformations combine multiple basic transformations using matrix multiplication.
1) The document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how to represent and convert between quantities in these number systems.
2) Techniques for converting between the different number systems are presented, including multiplying/dividing by powers of the base and grouping bits.
3) Examples are provided for converting between decimal, binary, octal, and hexadecimal numbers as well as fractions.
Digital images are represented by a matrix of numeric values where each value corresponds to the intensity of a pixel at a specific location. Images can be binary, representing black and white, or they can have multiple intensity levels represented by integers to capture shades of gray. Standard image file formats specify the spatial resolution in pixels and color encoding using a certain number of bits per pixel. When stored, an image is saved as a two-dimensional array of values, each representing intensity data for a pixel. Bitmap images use a one-dimensional matrix for monochrome and greater bit depth for more colors. Popular graphics software programs allow for image editing, painting and drawing.
This document discusses different types of number complements that allow subtraction to be performed using addition. It explains that subtraction using borrowing is inefficient for computers, so instead subtraction is done by taking the complement of one of the numbers and then adding. For binary numbers, the 2's complement is commonly used, where the 2's complement of a number is found by flipping all the bits and adding 1. The document provides examples of calculating 1's, 2's, and other bases' complements.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It explains that number systems have a radix or base, which determines the set of symbols used and their positional values. The key representations for binary numbers discussed are sign-magnitude, one's complement, and two's complement, which provide different methods for representing positive and negative numbers. The document provides examples of addition, subtraction, multiplication, and division operations in binary.
The document describes the Scale-invariant feature transform (SIFT) algorithm. It outlines the key steps: 1) constructing scale space by generating blurred images at different scales, 2) calculating difference of Gaussian images to find keypoints, 3) assigning orientations to keypoints, and 4) generating 128-element feature vectors for each keypoint to uniquely describe local image features in a way that is invariant to scale, rotation, and illumination changes. The SIFT algorithm allows for reliable object recognition and image stitching.
The document discusses VLSI (Very Large Scale Integration) systems and the VLSI design flow. VLSI systems integrate millions of electronic components into a small chip area. The objectives of VLSI design are high circuit speed, low power consumption, and minimizing design area. The VLSI design flow involves idea conception, specifying requirements, designing architecture, register transfer level coding in VHDL, RTL verification through simulation, synthesis into logic gates, sending to a foundry for fabrication, and producing the final integrated circuit chip.
The document describes different algorithms for filling polygon and area shapes, including scanline fill, boundary fill, and flood fill algorithms. The scanline fill algorithm works by determining intersections of boundaries with scanlines and filling color between intersections. Boundary fill works by starting from an interior point and recursively "painting" neighboring points until the boundary is reached. Flood fill replaces a specified interior color. Both can be 4-connected or 8-connected. The document also discusses problems that can occur and more efficient span-based approaches.
Chapter 2. Digital Image Fundamentals.pdfDngThanh44
This document discusses digital image fundamentals including image sensing and acquisition, sampling and quantization, relationships between pixels, and basic mathematical tools used in digital image processing. Specifically, it covers how images are captured using single sensing elements, line sensors and sensor arrays. It also describes how continuous images are converted to digital images through sampling and quantization of coordinate and intensity values. Key concepts covered include spatial and intensity resolution, image interpolation methods, and definitions of pixel neighbors, regions, boundaries and connectivity in digital images.
This presentation was made for student batch 2017-2018 of MBSTU. Here we will get
IEEE 32 bit floating representation .
IEEE 754 floating point representation
32 bit floating point Addition
Digital Electronics- Number systems & codes VandanaPagar1
This document covers number systems including decimal, binary, hexadecimal and their representations. It discusses how to convert between different number bases including binary to decimal and hexadecimal to decimal. Binary operations like addition, subtraction and codes like binary coded decimal are explained. Non-weighted codes such as gray code are also introduced. Reference books on digital electronics and number systems are provided.
This document describes Bresenham's circle algorithm for efficiently scan converting a circle. It begins by explaining the symmetry of a circle and prior inefficient polynomial and trigonometric methods. It then presents Bresenham's algorithm which takes advantage of the circle's symmetry and uses a decision variable to determine whether to move in the x or y direction to plot each pixel, ensuring points are always closest to the true circle. The algorithm is presented with variables initialized and steps to iterate through the first octant to plot all pixels.
This document discusses numerical data representation and digital and analog systems. It notes that there are two types of numerical representation: analog and digital. It also describes two types of systems: analog systems, which use analog representations, and digital systems, which use digital representations. The document outlines several advantages of digital systems, such as being easier to design, having greater accuracy and precision, and being less affected by noise. It also notes a limitation of digital systems is that the real world is analog. To take advantage of digital techniques, analog inputs must be converted to digital forms, operations performed digitally, and digital outputs converted back to analog forms.
This document provides an outline for a course on digital logic design. It includes the course title and credit hours, topics that will be covered such as Boolean algebra, logic gates, combinational and sequential circuits, programmable logic devices, and memory. It also lists recommended textbooks and provides the grading breakdown. Examples of analogue and digital quantities, signals, and number systems are given. Common logic gates such as AND, OR, NOT, NAND and NOR are described along with their truth tables and applications. Combinational circuits, functional devices, sequential circuits and memory are also introduced.
Cubic curves are commonly used in graphics because curves of lower order commonly have too little flexibility, while curves of higher order are usually considered unnecessarily complex and make it easy to introduce undesired wiggles.
This document discusses various 2D transformations in computer graphics including translation, rotation, and scaling. Translation moves an object by adding offsets to the x and y coordinates. Rotation uses trigonometric functions and rotation matrices to reposition objects around a central point. Scaling enlarges or shrinks objects by multiplying their coordinates by scaling factors. Homogeneous coordinates generalize these transformations into matrix operations.
This document contains information about 3D display methods in computer graphics presented by a group of 5 students. It discusses parallel projection, perspective projection, depth cueing, visible line identification, and surface rendering techniques. The goal is to generate realistic 3D images and correctly display depth relationships between objects.
Comprehensive coverage of fundamentals of computer graphics.
3D Transformations
Reflections
3D Display methods
3D Object Representation
Polygon surfaces
Quadratic Surfaces
Digital logic design is the basis of electronic systems like computers and cell phones. It uses binary numbers (zeros and ones) to represent information and process input/output operations. Digital logic employs logic gates that perform functions like AND, OR, and NOT to translate binary input signals into specific outputs. Career opportunities include developing device infrastructures using components like information storage, signal transmission, and information processing. Engineers work to improve performance, decrease energy usage, and debug issues.
This document provides information about a digital logic design course taught by Dr. Javaid Khurshid including the instructor and lab instructor contact details, lecture and lab schedule, grading policy, textbooks, and syllabus. The syllabus covers topics such as number systems, logic gates, Boolean algebra, combinational and sequential logic, memory, and microprocessors.
This document discusses various two-dimensional geometric transformations including translations, rotations, scaling, reflections, shears, and composite transformations. Translations move objects without deformation using a translation vector. Rotations rotate objects around a fixed point or pivot point. Scaling transformations enlarge or shrink objects using scaling factors. Reflections produce a mirror image of an object across an axis. Shearing slants an object along an axis. Composite transformations combine multiple basic transformations using matrix multiplication.
1) The document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how to represent and convert between quantities in these number systems.
2) Techniques for converting between the different number systems are presented, including multiplying/dividing by powers of the base and grouping bits.
3) Examples are provided for converting between decimal, binary, octal, and hexadecimal numbers as well as fractions.
Digital images are represented by a matrix of numeric values where each value corresponds to the intensity of a pixel at a specific location. Images can be binary, representing black and white, or they can have multiple intensity levels represented by integers to capture shades of gray. Standard image file formats specify the spatial resolution in pixels and color encoding using a certain number of bits per pixel. When stored, an image is saved as a two-dimensional array of values, each representing intensity data for a pixel. Bitmap images use a one-dimensional matrix for monochrome and greater bit depth for more colors. Popular graphics software programs allow for image editing, painting and drawing.
This document discusses different types of number complements that allow subtraction to be performed using addition. It explains that subtraction using borrowing is inefficient for computers, so instead subtraction is done by taking the complement of one of the numbers and then adding. For binary numbers, the 2's complement is commonly used, where the 2's complement of a number is found by flipping all the bits and adding 1. The document provides examples of calculating 1's, 2's, and other bases' complements.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It explains that number systems have a radix or base, which determines the set of symbols used and their positional values. The key representations for binary numbers discussed are sign-magnitude, one's complement, and two's complement, which provide different methods for representing positive and negative numbers. The document provides examples of addition, subtraction, multiplication, and division operations in binary.
The document describes the Scale-invariant feature transform (SIFT) algorithm. It outlines the key steps: 1) constructing scale space by generating blurred images at different scales, 2) calculating difference of Gaussian images to find keypoints, 3) assigning orientations to keypoints, and 4) generating 128-element feature vectors for each keypoint to uniquely describe local image features in a way that is invariant to scale, rotation, and illumination changes. The SIFT algorithm allows for reliable object recognition and image stitching.
The document discusses VLSI (Very Large Scale Integration) systems and the VLSI design flow. VLSI systems integrate millions of electronic components into a small chip area. The objectives of VLSI design are high circuit speed, low power consumption, and minimizing design area. The VLSI design flow involves idea conception, specifying requirements, designing architecture, register transfer level coding in VHDL, RTL verification through simulation, synthesis into logic gates, sending to a foundry for fabrication, and producing the final integrated circuit chip.
The document describes different algorithms for filling polygon and area shapes, including scanline fill, boundary fill, and flood fill algorithms. The scanline fill algorithm works by determining intersections of boundaries with scanlines and filling color between intersections. Boundary fill works by starting from an interior point and recursively "painting" neighboring points until the boundary is reached. Flood fill replaces a specified interior color. Both can be 4-connected or 8-connected. The document also discusses problems that can occur and more efficient span-based approaches.
Chapter 2. Digital Image Fundamentals.pdfDngThanh44
This document discusses digital image fundamentals including image sensing and acquisition, sampling and quantization, relationships between pixels, and basic mathematical tools used in digital image processing. Specifically, it covers how images are captured using single sensing elements, line sensors and sensor arrays. It also describes how continuous images are converted to digital images through sampling and quantization of coordinate and intensity values. Key concepts covered include spatial and intensity resolution, image interpolation methods, and definitions of pixel neighbors, regions, boundaries and connectivity in digital images.
This presentation was made for student batch 2017-2018 of MBSTU. Here we will get
IEEE 32 bit floating representation .
IEEE 754 floating point representation
32 bit floating point Addition
Digital Electronics- Number systems & codes VandanaPagar1
This document covers number systems including decimal, binary, hexadecimal and their representations. It discusses how to convert between different number bases including binary to decimal and hexadecimal to decimal. Binary operations like addition, subtraction and codes like binary coded decimal are explained. Non-weighted codes such as gray code are also introduced. Reference books on digital electronics and number systems are provided.
Introduction
Number Systems
Types of Number systems
Inter conversion of number systems
Binary addition ,subtraction, multiplication and
division
Complements of binary number(1’s and 2’s
complement)
Grey code, ASCII, Ex
3,BCD
FYBSC IT Digital Electronics Unit I Chapter I Number System and Binary Arithm...Arti Parab Academics
Number System:
Analog System, digital system, numbering system, binary number
system, octal number system, hexadecimal number system, conversion
from one number system to another, floating point numbers, weighted
codes binary coded decimal, non-weighted codes Excess – 3 code, Gray
code, Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code,
Hollerith Code, Morse Code, Teletypewriter (TTY), Error detection
and correction, Universal Product Code, Code conversion.
The document discusses number systems. It defines a number system as a system for writing and representing numbers using digits or symbols in a consistent manner. It allows for arithmetic operations and provides a unique representation for every number. The four most common number systems are decimal, binary, octal, and hexadecimal. Binary uses only two digits, 0 and 1, and is used to represent electrical signals in computers. Decimal uses base 10 with digits 0-9 in place values. [END SUMMARY]
This document discusses analog and digital electronics. It begins by explaining that analog electronics deals with continuous signals while digital electronics deals with discrete signals. It then discusses how digital techniques grew tremendously after 1938 when Claude Shannon systemized George Boole's theoretical work. Finally, it covers various number systems such as binary, decimal, octal, and hexadecimal and how to convert between them.
This document discusses number systems and conversions between number systems. It begins by introducing analog and digital electronics, and analog and digital signals. It then discusses different number systems including binary, decimal, octal and hexadecimal. The main methods covered are:
1) Converting a decimal number to binary, octal or hexadecimal using repeated division and noting the remainders.
2) Converting a binary, octal or hexadecimal number to decimal by multiplying each digit by its place value weight.
3) Conversions can also be done between binary and octal by grouping bits into groups of three.
This document discusses analog and digital electronics. It begins by explaining that analog electronics deals with continuous signals while digital electronics deals with discrete signals. It then discusses how digital techniques grew tremendously after 1938 when Claude Shannon systemized George Boole's theoretical work. Finally, it covers various number systems such as binary, decimal, octal, and hexadecimal and how to convert between them.
The document discusses various number systems used in digital computers including binary, decimal, octal, and hexadecimal. It provides details on:
1) How numbers are represented positionally in these systems, with different radixes (bases) and the meaning of each digit based on its position.
2) Methods for converting between the different number systems, such as dividing the number by the new base and writing the remainders in reverse order.
3) The steps to convert a decimal number to its binary, octal or hexadecimal equivalent and vice versa by calculating the place values of each digit.
This document provides an introduction to different digital number systems used in computer systems, including binary, decimal, octal, and hexadecimal. It discusses how each system uses different bases and symbols to represent numeric values. Conversion techniques between these number systems are also covered, along with signed and unsigned number representations, overflow detection, and other related topics. Key points covered include how each place value in a number represents different powers of the base, and how binary addition works with signed and unsigned numbers.
This document provides an introduction to digital number systems used in computer science. It discusses binary, decimal, octal, and hexadecimal number systems. For each system, it explains the base, digits used, and how to convert between the number systems and decimal. It also covers signed binary number representations, binary arithmetic, detecting overflow, and binary coded decimal. References are provided at the end for additional reading on number systems and computer data representation.
This document discusses different number systems used in computing including decimal, binary, hexadecimal, and binary-coded decimal (BCD). It explains how each system uses a positional notation with a radix or base to represent values. Binary uses a base of 2 with digits of 0 and 1. Hexadecimal uses a base of 16 with digits 0-9 and A-F. Conversion between decimal, binary, and hexadecimal is described. Signed and unsigned numbers are also discussed, with two's complement being the most common way of representing signed binary numbers. Fixed precision and the concept of overflow are introduced when numbers are represented with a limited number of bits in computers.
This document discusses different number systems used in computing including decimal, binary, hexadecimal, and binary-coded decimal (BCD). It explains how each system uses a positional notation with a radix or base. Binary uses a base of 2 with digits of 0 and 1. Hexadecimal uses a base of 16 with digits 0-9 and A-F. Conversion between decimal, binary, and hexadecimal is described. Signed numbers can be represented using two's complement where the most significant bit indicates the sign. Fixed precision numbers have a limited range determined by the number of bits used.
This document discusses different number systems used in computing including decimal, binary, hexadecimal, and binary-coded decimal (BCD). It explains how each system uses a positional notation with a base or radix to represent numbers. Decimal uses base 10 with digits 0-9, binary uses base 2 with digits 0-1, hexadecimal uses base 16 with digits 0-9 and A-F, and BCD encodes each decimal digit as a 4-bit binary number. The document describes how to convert between these number systems through division and remainder operations. Hexadecimal is commonly used to efficiently represent large binary numbers with fewer digits.
This document discusses different number systems used in digital computers and their conversions. It begins with an introduction to digital number systems and then describes the decimal, binary, octal and hexadecimal number systems. It explains how to represent integers and real numbers in binary. The document also covers number conversions between these systems using different methods like repeated division. Finally, it discusses various ways of representing integers in binary like sign-magnitude, one's complement and two's complement representations.
This document presents an overview of different number systems including decimal, binary, octal, and hexadecimal. It defines each system, their bases, and the digits used. Conversion methods between the systems are described, such as repeated division to convert decimal to binary and multiplying place values to convert in the opposite direction. The relationships between the different bases are shown, including that hexadecimal represents groups of 4 binary digits. Examples are provided for conversions between the various number systems.
This document contains a presentation on digital logic design. It discusses topics like number systems, number base conversion, binary arithmetic operations, weighted and non-weighted binary codes, and binary coded decimal arithmetic. The presentation was created by faculty at the Institute of Aeronautical Engineering for computer science and information technology students as part of a course on digital logic design.
The document discusses data representation in computer systems. It covers different number systems like binary, decimal, hexadecimal and their conversions. It explains how computers use the positional number system to represent numbers. It also discusses signed and unsigned integers, binary arithmetic operations, and character representation using ASCII code.
Sv data types and sv interface usage in uvmHARINATH REDDY
SystemVerilog provides several data types for modeling hardware including basic types like reg, wire, integer, real, time and logic. It also introduces user-defined types like enum, struct, union, typedef and class. Enum allows defining a set of named values. Struct packs different data types together. Union shares the same storage for different types. Typedef defines custom type names. Class defines user-defined objects. Operators allow performing arithmetic, relational, equality and logical operations on data types. Assignment, increment/decrement operators are also supported.
HBM stands for high bandwidth memory and is a type of memory interface used in 3D-stacked DRAM (dynamic random access memory) in GPUs, as well as the server, machine-learning DSP , high-performance computing and networking and client space.
The document describes the roles of sequencers and drivers in UVM. It explains that sequencers generate stimulus data and pass it to drivers for execution. Drivers drive data items to the DUT following the interface protocol. Sequencers and drivers communicate through TLM ports, with the driver fetching items from the sequencer and signaling when items are done. Sequences contain multiple data items that together form a scenario or pattern, while arbitration ensures only one sequence accesses the driver at a time.
This document describes the design and implementation of a real-time flow measurement system using a Hall probe sensor and PC-based SCADA. The objectives are to measure flow rate using a Hall probe sensor with a rotameter and develop, test, and demonstrate the system. It introduces the components of the system including the Hall probe sensor, LCD display, Arduino microcontroller, and other hardware. Diagrams of the Hall probe sensor and system block diagram are presented. The document also provides information on how the Hall probe sensor works and its output signal.
This document discusses SystemVerilog assertions (SVA). It introduces SVA and explains that assertions are used to document design functionality, check design intent is met, and determine if verification tested the design. Assertions can be specified by the design or verification engineer. The document outlines the key building blocks of SVA like sequences, properties, and assertions. It provides examples of different types of assertions and how they are used. Key concepts discussed include implication, timing windows, edge detection, and repetition operators.
This document summarizes the design and implementation of an optimized differential Gaussian frequency-shift keying (GFSK) demodulator. It includes blocks for a low-noise amplifier, mixer, voltage-controlled oscillator, complex filter, limiter, received signal strength indicator, and digital demodulator. The demodulator aims to improve bit-error-rate performance over conventional designs in additive white Gaussian noise and flat fading channels. It also addresses the phase wrapping problem that occurs during phase differential detection in the demodulator implementation.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
2. Numbering System
The number system is used for representing the information.
The number system has different bases and the most common of them are
the decimal, binary, octal, and hexadecimal.
The base or radix of the number system is the total number of the digit
used in the number system.
If the number system representing the digit from 0 – 9 then the base of the
system is the 10.
4. Decimal Number System
The number system is having digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
The base of a system, more properly called the RADIX, is the number of
different values that can be expressed using a single digit.
When writing a number, the digits used give its value, but the number is
scaled by its RADIX POINT.
For example, 456.210 is ten times bigger than 45.6210 although the digits are
the same.
5. Binary Number System
Binary has only two values 0 and 1. If larger values than 1 are needed, extra
columns are added to the left.
Each column value is now twice the value of the column to its right. For
example the decimal value three is written 11 in binary (1 two + 1 one).
The digital electronic equipment's are works on the binary number system
and hence the decimal number system is converted into binary system.
6. Octal Number System
Octal has eight values 0 to 7. If larger values than 7 are needed, extra
columns are added to the left.
The octal system has the base of eight as it uses eight digits 0, 1, 2, 3, 4, 5,
6, 7.
The next digit in the octal number is represented by 10, 11, 12, which are
equivalent to decimal digits 8, 9, 10 respectively.
The main advantage of using octal number system is that it can be
converted directly to binary in a very easy manner.
7. Hexadecimal Number System
• The hexadecimal number system has a base of 16, and hence it consists of
the following sixteen number of digits.
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
• This Hexadecimal system is used in computer registers to store the
addresses of the data. If we have to give a large number of binary strings.
• For suppose 1011110110001011111010110001101, it is very much difficult
and create a lot of confusion. So computer uses Hexadecimal numbers in
representation of such strings.
8. Number System Conversions
Any radix to Decimal number system D= 𝑖=−𝑛
𝑝−1
𝑑𝑖. 𝑟 𝑖
.
Where p is No. of digits to the left of the radix point.
n is No. of digits to the right of the radix point.
d is value of the number.
r is radix of the number system.
The number based conversions are essential in digital electronics. Why
because, in all digital system, we have the input in decimal format.
While computation system need binary conversion and result will be
Hexadecimal format by inverse conversion.
9. Hexadecimal to binary conversion
To convert a hexadecimal number to a binary number, convert each
hexadecimal digit to its four digit equivalent.
For example, consider the hexadecimal number 9AF which is converted into
a binary digit. The conversions are explained below.
10. Binary to Hexadecimal conversion
To convert the given binary number into its equivalent hexadecimal number
rewrite the binary number of the sets of four digits.
Then place the hexadecimal digit in front of each four digit set of a binary
number as explained by the following number.
11. Hexadecimal to Decimal conversion
The base of the hexadecimal number system is 16, therefore the weights
corresponding to various positions of the digits will be as shown below.
For instance, consider the conversion of hexadecimal number E8F6.27 into
its equivalent binary number.
Therefore E8f6.27 written in decimal as 59638.1523437.
12. Decimal to Hexadecimal conversion
The conversion of the given decimal number into hexadecimal number
requires the application of hex-dabble method.
Consider the conversion of the decimal number 3749 into its hexadecimal
equivalent number.
The third reminder 13 is equivalent to D in a hexadecimal number system.
Thus the equivalent hexadecimal number D97.
14. Number System Conversions
Binary to Decimal:
(1010.01)2
1×23 + 0x22 + 1×21+ 0x20 + 0x2 -1 + 1×2 -2 = 8+0+2+0+0+0.25 =
10.25
(1010.01)2 = (10.25)10
Decimal to Octal
(10.25)10
(10)10 = (12)8 ; And Fractional part:0.25 x 8 = 2.00
(10.25)10 = (12.2)8
15. Number System Conversions
Octal to Decimal
(12.2)8
1 x 81 + 2 x 80 +2 x 8-1 = 8+2+0.25 = 10.25
(12.2)8 = (10.25)10
Hexadecimal to octal Number
First the individual digits are converted into its binary bits. After that
the subsequent bits are grouped into 3 bits.
(ABCD)16
A (1010) , B (1011), C (1100), D(1101)
001 010 101 111 001 101 (pairing 3 binary bits)
So, (ABCD)16 =(125715)8
16. Signed Magnitude Representation
MSB of a bit string is used as the sign bit and the lower bits contain the
magnitude.
Ex: (1111)2= (15)10 unsigned number representation.
(01111)2= +(15)10
(11111)2= −(15)10 signed number representation.
Range of the n bit signed magnitude integer is given as
- (2 𝑛−1
-1) to + (2 𝑛−1
-1) to
17. Complement of Numbers
There are two types of complements for each base-r system.
1. r's complement , 2. (r -1)'s complement.
Ex: The 9’s complement of 546700 is 999999-546700=453299
For binary numbers, r = 2 and r –1 = 1, so the 1's complement of N is
(2^n -1) –N.
Ex: The 1’s complement of 1011000is 0100111.
18. Complement of Numbers
Radix Complement
The r's complement of an n-digit number N in base r is defined as 𝑟 𝑛–N for
N ≠ 0 and as 0 for N = 0.
Comparing with the (r -1) 's complement, we note that the r’s complement
is obtained by adding 1 to the (r-1) 's complement, since
𝑟 𝑛–N = [(𝑟 𝑛-1)–N] + 1.
Ex: The 10's complement of 012398 is 987602
Ex: The 2's complement of 1101100 is 0010100
19. Compliment of Numbers
The subtraction of two n-digit unsigned numbers M –N in base r can be
done as follows:
20. Compliment of Numbers
Using 10's complement, subtract 72532 –3250.
Using 10's complement, subtract 3250 –72532
Here no end carry, Therefore, the answer is –(10's complement of 30718) is
- 69282.
21. CODES
In the coding, when numbers or letters are represented by a specific group
of symbols, that group of symbols is called as code or Binary code.
If the code has positional weights, then it is said to be weighted code.
Otherwise, it is an unweighted code.
Codes are required to conveniently input data into digital system and
interpret results.
23. CODES
Weighted codes: In weighted codes, each digit is assigned a specific weight
according to its position. For example, in 8421BCD code, 1001 the weights
of 1, 0, 0, 1 (from left to right) are 8, 4, 2 and 1 respectively.
The codes 8421BCD, 2421BCD, 5211BCD are all weighted codes.
Non-weighted codes: The non-weighted codes are not positionally
weighted. In other words, each digit position within the number is not
assigned a fixed value ( or weight ).
Excess-3 and gray code are non-weighted codes.
24. CODES
Reflective codes: A code is reflective when the code is self complementing.
In other words, when the code for 9 is the complement the code for 0, 8 for
1, 7 for 2, 6 for 3 and 5 for 4.
2421BCD, 5421BCD and Excess-3 code are reflective codes.
Sequential codes: In sequential codes, each succeeding code is one binary
number greater than its preceding code. This property helps in
manipulation of data.
8421 BCD and Excess-3 are sequential codes.
25. CODES
Alphanumeric codes: Codes used to represent numbers, alphabetic
characters, symbols and various instructions necessary for conveying
intelligible information.
ASCII, EBCDIC, UNICODE are the most-commonly used alphanumeric codes.
Error detecting and correcting codes: Codes which allow error detection
and correction are called error detecting and correcting codes. Hamming
code is the mostly commonly used error detecting and correcting code.
26. Binary Coded Decimal(BCD) Code
In this code each decimal digit is represented by a 4-bit binary number.
BCD is a way to express each of the decimal digits with a binary code.
In the BCD, with four bits we can represent sixteen numbers (0000 to 1111).
But in BCD code only first ten of these are used (0000 to 1001).
The remaining six code combinations i.e. 1010 to 1111 are invalid in BCD.
Ex: (874)10= (1000 0111 0100) 𝑏𝑐𝑑
27. Excess-3 Code
• It is non-weighted code used to express decimal numbers. The Excess-3
code words are derived from the 8421 BCD code.
28. Gray Code
It is the non-weighted code and it is not arithmetic codes. That means there
are no specific weights assigned to the bit position.
It has a very special feature that, only one bit will change each time.
the gray code is called as a unit distance code.
The gray code is a cyclic code.
Gray code cannot be used for arithmetic operation.
For Low power applications Gray code will be useful.
29. Codes Conversion
There are many methods or techniques which can be used to convert code
from one format to another. We'll demonstrate here the following.
1.Binary to BCD Conversion
2.BCD to Binary Conversion
3.BCD to Excess-3
4.Excess-3 to BCD
30. Binary to BCD conversion
Step 1 -- Convert the binary number to decimal.
Step 2 -- Convert decimal number to BCD.
• Ex: binary number is (11101)2
• Binary Number −> (11101)2 = Decimal Number −> (29)10
(29)10 =(00101001)BCD
31. BCD to Binary conversion
Step 1 -- Convert the BCD number to decimal.
Step 2 -- Convert decimal to binary.
Ex: convert (00101001)BCD to Binary.
(00101001)BCD => 00102 10012 => 210 910 =>
Decimal Number −> (29)10
Decimal Number −> 2910 = Binary Number −> (11101)2
32. BCD to Excess-3 conversion
Step 1 -- Convert BCD to decimal.
Step 2 -- Add (3)10 to this decimal number.
Step 3 -- Convert into binary to get excess-3 code.
Ex: convert (1001)BCD to Excess-3.
Step 1 − Convert to decimal
(1001)BCD = 910
Step 2 − Add 3 to decimal
(9)10 + (3)10 = (12)10
Step 3 − Convert to Excess-3
(12)10 = (1100)2
33. Excess-3 to BCD conversion
• Step 1 -- Subtract (0011)2 from each 4 bit of excess-3 digit to obtain the
corresponding BCD code.
Ex: convert (10011010)XS-3 to BCD.
Ex-3 -> 1001 1010
subtract (0011)2 -> 0011 0011
result BCD -> 0110 0111
(10011010)XS-3 = (01100111)BCD
34. Binary to Gray Conversion
The first bit(MSB) of the gray code is the same as the first bit of the binary
number
The second bit of the gray code equals the exclusive OR of the first and
second bits of the binary number from MSB
The third bit of the gray code equals the exclusive OR of the second and
third bits of the binary number and so on.
Ex: (01001)2= (01101) gray
35. Gray to Binary Conversion
The M.S.B of the binary number will be equal to the M.S.B of the given gray
code.
Now if the second gray bit is 0 the second binary bit will be same as the
previous or the first bit. If the gray bit is 1 the second binary bit will alter. If
it was 1 it will be 0 and if it was 0 it will be 1.
This step is continued for all the bits to do Gray code to binary conversion.
Ex: (01101) gray = (01001)2
36. UNIVERSEL GATES
A Gate which can be use to create any Logic Gate is called Universal Gate.
NAND and NOR Gates are called Universal Gates because all the other Gates can be created by
using these Gates.
NAND and NOR Gates can implement any logical Boolean expression.
In practice, this is advantageous since NAND and NOR gates are economical and easier to
fabricate and are the basic gates used in all IC digital logic families.
37. NAND Gate
NAND function is compliment of the AND function.
NAND consist of an AND graphic symbol followed by a small circle.
Its name is an abbreviation of NOT AND .
NAND output logical expression is given as z = 𝑥. 𝑦
x y z
0 0 1
0 1 1
1 0 1
1 1 0
x
y
z
Truth table for NAND Gate
Fig:Logic symbol for NAND
Gate
38. Inverter implementation by NAND Gate
All NAND input pins connect to the input signal A gives an output 𝐴.
One NAND input pin is connected to the input signal A while all other input
pins are connected to logic 1. The output will be 𝐴.
A 𝐴 𝐴
A
1
A ~A
0 1
1 0
Truth table for NOT
Gate
Fig: NOT Gate implementation by NAND Gate
39. AND Gate implementation by NAND Gate
The AND is replaced by a NAND gate with its output complemented by a
NAND gate inverter .
A 𝑨. 𝑩
Truth table for AND
Gate
Fig:AND Gate implementation by NAND Gate
A
B
𝑨. 𝑩
A B A.B
0 0 0
0 1 0
1 0 0
1 1 1
40. OR Gate implementation by NAND Gate
The OR gate is replaced by a NAND gate with all its inputs complemented by
NAND gate inverters .
Truth table for OR
Gate
Fig:OR Gate implementation by NAND Gate
A
B
A B A+B
0 0 0
0 1 1
1 0 1
1 1 1
𝑨
𝑩
𝑨. 𝑩
41. NOR Gate implementation by NAND Gate
A NOR gate is simply an OR gate with an inverted output:
Truth table for NOR Gate
Fig:NOR Gate implementation by NAND Gate
A B
Q=𝑥 + 𝑦
0 0 1
0 1 0
1 0 0
1 1 0
42. XOR Gate implementation by NAND Gate
The output of an XOR gate is true only when one of its inputs is true.
If both of an XOR gate's inputs are false, or if both of its inputs are true,
then the output of the XOR gate is false.
Logical symbol is given as
Truth table for XOR
Gate
Fig:XOR Gate implementation by NAND Gate
A B Q=
0 0 0
0 1 1
1 0 1
1 1 0
A ⊕ B
43. XNOR Gate implementation by NAND Gate
The output of an XNOR gate is true when all of its inputs are true or when
all of its inputs are false.
If some of its inputs are true and others are false, then the output of the
XNOR gate is false.
Logical symbol is given like
Truth table for XNOR
Gate
Fig: XNOR Gate implementation by
NAND Gate
A B Q=
0 0 1
0 1 0
1 0 0
1 1 1
A
B
𝐴 ⊕ 𝐵
𝐴 ⊕ 𝐵
44. NOR Gate
NOR function is compliment of the AND function.
NOR consist of an OR graphic symbol followed by a small circle.
Its name is an abbreviation of NOT OR .
NOR output logical expression is given as z = 𝑥 + 𝑦
x y z
0 0 1
0 1 0
1 0 0
1 1 0
x
y
z
Truth table for NOR Gate
Fig: Logic symbol for NOR
Gate
45. Inverter implementation by NOR Gate
All NOR input pins connect to the input signal A gives an output 𝐴.
One NOR input pin is connected to the input signal A while all other input
pins are connected to logic 0. The output will be 𝐴.
Fig: NOT Gate implementation by NOR Gate
46. AND Gate implementation by NOR Gate
An AND gate gives a 1 output when both inputs are 1;
a NOR gate gives a 1 output only when both inputs are 0.
Therefore, an AND gate is made by inverting the inputs to a NOR gate
Truth table for AND
Gate
Fig: AND Gate implementation by NOR Gate
A B
Q=A.B
0 0 0
0 1 0
1 0 0
1 1 1
47. OR Gate implementation by NOR Gate
The OR gate is simply a NOR gate followed by a NOT gate.
Truth table for OR
Gate
Fig: OR Gate implementation by NOR Gate
A B Q=
A+B
0 0 0
0 1 1
1 0 1
1 1 1
48. NAND Gate implementation by NOR Gate
A NAND gate is made using an AND gate in series with a NOT gate
A B
Q=𝐴. 𝐵
0 0 1
0 1 1
1 0 1
1 1 0Truth table for NAND Gate
Fig: NAND Gate implementation by NOR Gate
49. XOR Gate implementation by NOR Gate
An XOR gate is made by connecting the output of 3 NOR gates.
This expresses the logical formula (A AND B) NOR (A NOR B).
This construction require a propagation delay three times that of a single
NOR gate and uses five gates.
Logical symbol is given as
Truth table for XOR
Gate
Fig:XOR Gate implementation by NOR Gate
A B Q=
0 0 0
0 1 1
1 0 1
1 1 0
A ⊕ B
50. XNOR Gate implementation by NOR Gate
An XNOR gate can be constructed from four NOR gates implementing the
expression (A NOR N) NOR (B NOR N) where N = A NOR B.
This construction entails a propagation delay three times that of a single
NOR gate and uses four gates.
Logical symbol is given as
Truth table for XNOR
Gate
Fig:XNOR Gate implementation by NOR Gate
A B Q=
0 0 1
0 1 0
1 0 0
1 1 1
𝐴 ⊕ 𝐵
51. Canonical and Standard forms
In Boolean algebra, Boolean function can be expressed as Canonical
Disjunctive Normal Form known as minterm .
And some are expressed as Canonical Conjunctive Normal Form known as
maxterm .
minterm for each combination of the variables that produces a 1 in the
function and then taking the OR of all those terms.
maxterm for each combination of the variables that produces a 0 in the
function and then taking the AND of all those terms
Boolean functions expressed as a sum of minterms(SOP) or product of
maxterms(POS) are said to be in canonical form.
52. Truth table Notation for Minterms and Maxterm
• Example: Assume 3 Literals x,y,z .
53. Sum of minterm
With ‘n’ variable, maximum possible minterms are 2^n.
Ex: Express the Boolean function F = A + B’C as a sum of minterms.
• First term A = A(B + B’) = AB + AB’
A = AB(C + C’) + AB'(C + C’) = ABC + ABC’+ AB’C + AB’C’
• second term B’C = B’C(A + A’) = AB’C + A’B’C
F = A + B’C = ABC + ABC’ + AB’C + AB’C’ + A’B’
here AB’C appears twice, from Boolean theorems
F = A’B’C + AB’C + AB’C + ABC’ + ABC= m1 + m4 + m5 + m6 + m7
SOP is represented as ∑1, 4, 5, 6, 7) .
54. Product of maxterm
Ex: Express the Boolean function F = xy + x’z as a product of maxterms
sol: F = xy + x’z
= (xy + x’)(xy + z)
= (x + x’)(y + x’)(x + z)(y + z)
= (x’ + y)(x + z)(y + z)
x’ + y = x’ + y + zz’
= (x’+ y + z)(x’ + y + z’)
x + z = x + z + yy’
= (x + y + z)(x + y’ + z)
y + z = y + z + xx’
= (x + y + z)(x’ + y + z)
F = (x + y + z)(x + y’ + z)(x’ + y + z)(x’ + y + z’)
= M0*M2*M4*M5
POS is represented as ∏(0, 2, 4, 5)
With ‘n’ variable, maximum possible maxterms are 2^n.
55. Conversion between canonical Forms
Replace ∑ with ∏ (or vice versa) and replace those j’s that appeared in the
original form with those that do not.
Example:
f1(a,b,c) = a’b’c + a’bc’ + ab’c’ + abc’
= m1 + m2 + m4 + m6
= ∑(1,2,4,6)
= ∏(0,3,5,7)
= (a+b+c)•(a+b’+c’)•(a’+b+c’)•(a’+b’+c’)
56. Standard Forms
Standard forms are like canonical forms, except that not all variables need
appear in the individual product (SOP) or sum (POS) terms.
Example:
f1(a,b,c) = a’b’c + bc’ + ac’
is a standard sum-of-products form
f1(a,b,c) = (a+b+c)•(b’+c’)•(a’+c’)
is a standard product-of-sums form.
57. Conversion of SOP from standard to canonical form
Expand non-canonical terms by inserting equivalent of 1 in each missing
variable x:
(x + x’) = 1
Remove duplicate minterms
f1(a,b,c) = a’b’c + bc’ + ac’
= a’b’c + (a+a’)bc’ + a(b+b’)c’
= a’b’c + abc’ + a’bc’ + abc’ + ab’c’
= a’b’c + abc’ + a’bc + ab’c’
58. Conversion of POS from standard to canonical form
Expand noncanonical terms by adding 0 in terms of missing variables (e.g.,
xx’ = 0) and using the distributive law
Remove duplicate maxterms
f1(a,b,c) = (a+b+c)•(b’+c’)•(a’+c’)
= (a+b+c)•(aa’+b’+c’)•(a’+bb’+c’)
= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•
(a’+b+c’)•(a’+b’+c’)
= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•(a’+b+c’)