The document discusses different number systems and digital coding techniques. It describes the decimal, binary, octal and hexadecimal number systems. Conversion methods between these systems are provided, including complement representations. Common codes like binary coded decimal, excess-3, and gray codes are defined along with their properties. NAND and NOR gates are identified as universal gates that can be used to implement any logical function. Methods for constructing common logic gates using only NAND gates are presented.
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.
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.
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.
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 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.
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 contains lecture slides on digital logic design. It covers topics like number systems, binary arithmetic operations, weighted and non-weighted binary codes, binary coded decimal, excess-3 codes, representation of signed numbers, and error detecting codes. The document includes examples and explanations of converting between different number bases, performing binary operations, and using various coding schemes. It is intended as teaching material for a course on digital logic design.
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.
CDS Fundamentals of digital communication system UNIT 1 AND 2.pdfshubhangisonawane6
The document discusses various number systems including decimal, binary, hexadecimal and their conversions. It explains binary addition and subtraction using 2's complement. Binary coded decimal and gray codes are also covered. The last part discusses ASCII codes for alphanumeric representation. Key points discussed are:
- Decimal, binary and hexadecimal number systems and inter-conversions between them.
- Binary addition and subtraction using 2's complement.
- Binary coded decimal and gray codes for number representation.
- ASCII codes for alphanumeric representation in computers.
The document discusses different numeral systems used in computing including binary, decimal, octal and hexadecimal. It explains how each system uses a different base and symbol set. Binary uses base-2 with symbols 0-1. Decimal is base-10 with 0-9. Octal is base-8 with 0-7. Hexadecimal is base-16 with 0-9 and A-F. The document also provides examples and methods for converting between these different numeral systems that are commonly used for representing numbers, instructions and other data in computers.
The document discusses different number systems including binary, octal, hexadecimal, and decimal. It provides examples and steps for converting between these number systems. Specifically, it explains that a number system defines a set of values to represent quantities using digits. The main types covered are the binary, octal, hexadecimal, and decimal systems. Conversion between these systems involves dividing the number by the base and recording the remainders to get the digits in the target system.
The document discusses different number systems used in digital technologies, including decimal, binary, octal, and hexadecimal systems. It provides details on how each system works, such as having 10 symbols in decimal, 2 symbols in binary, 8 symbols in octal, and 16 symbols in hexadecimal. The document also covers error detection codes like parity and checksums that are used to detect errors in digital data transmission and storage.
This document discusses number systems and data representation in computers. It covers topics like binary, decimal, hexadecimal, and ASCII number systems. Some key points covered include:
- Computers use the binary number system and positional notation to represent data precisely.
- Different number systems have different bases (like binary base-2, decimal base-10, hexadecimal base-16).
- Methods for converting between number systems like binary to decimal and hexadecimal to decimal.
- Signed and unsigned integers, ones' complement, twos' complement representation of negative numbers.
- ASCII encoding of characters and how to convert between character and numeric representations.
Digital computers represent data by means of an easily identified symbol called a digit. The data may
contain digits, alphabets or special character, which are converted to bits, understandable by the computer.
In Digital Computer, data and instructions are stored in computer memory using binary code (or
machine code) represented by Binary digIT’s 1 and 0 called BIT’s.
The number system uses well-defined symbols called digits.
Number systems are classified into two types:
o Non-positional number system
o Positional number system
This document provides an overview of satellite communication principles and the evolution of communication satellites. It discusses how Arthur C. Clarke first conceived of the idea of communication satellites in geostationary orbits in 1945. It then summarizes the key milestones in the development of communication satellites, including early satellites launched by the US and USSR in the late 1950s, the establishment of international cooperation organizations like INTELSAT and Comsat in the 1960s, and the growth of satellite capabilities over time. The document also provides details about Bangladesh's first communication satellite, Bangabandhu Satellite-1, and describes different types of communication satellites.
This document introduces alternating current (AC), which regularly changes direction unlike direct current (DC). It defines key terms used to describe AC quantities like amplitude, time period, frequency, instantaneous value, and angular frequency. It also provides an example problem calculating the maximum value, frequency, time period, and instantaneous value of a given sinusoidal current. Finally, it discusses average value and instantaneous and average power of AC circuits.
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.
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 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.
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 contains lecture slides on digital logic design. It covers topics like number systems, binary arithmetic operations, weighted and non-weighted binary codes, binary coded decimal, excess-3 codes, representation of signed numbers, and error detecting codes. The document includes examples and explanations of converting between different number bases, performing binary operations, and using various coding schemes. It is intended as teaching material for a course on digital logic design.
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.
CDS Fundamentals of digital communication system UNIT 1 AND 2.pdfshubhangisonawane6
The document discusses various number systems including decimal, binary, hexadecimal and their conversions. It explains binary addition and subtraction using 2's complement. Binary coded decimal and gray codes are also covered. The last part discusses ASCII codes for alphanumeric representation. Key points discussed are:
- Decimal, binary and hexadecimal number systems and inter-conversions between them.
- Binary addition and subtraction using 2's complement.
- Binary coded decimal and gray codes for number representation.
- ASCII codes for alphanumeric representation in computers.
The document discusses different numeral systems used in computing including binary, decimal, octal and hexadecimal. It explains how each system uses a different base and symbol set. Binary uses base-2 with symbols 0-1. Decimal is base-10 with 0-9. Octal is base-8 with 0-7. Hexadecimal is base-16 with 0-9 and A-F. The document also provides examples and methods for converting between these different numeral systems that are commonly used for representing numbers, instructions and other data in computers.
The document discusses different number systems including binary, octal, hexadecimal, and decimal. It provides examples and steps for converting between these number systems. Specifically, it explains that a number system defines a set of values to represent quantities using digits. The main types covered are the binary, octal, hexadecimal, and decimal systems. Conversion between these systems involves dividing the number by the base and recording the remainders to get the digits in the target system.
The document discusses different number systems used in digital technologies, including decimal, binary, octal, and hexadecimal systems. It provides details on how each system works, such as having 10 symbols in decimal, 2 symbols in binary, 8 symbols in octal, and 16 symbols in hexadecimal. The document also covers error detection codes like parity and checksums that are used to detect errors in digital data transmission and storage.
This document discusses number systems and data representation in computers. It covers topics like binary, decimal, hexadecimal, and ASCII number systems. Some key points covered include:
- Computers use the binary number system and positional notation to represent data precisely.
- Different number systems have different bases (like binary base-2, decimal base-10, hexadecimal base-16).
- Methods for converting between number systems like binary to decimal and hexadecimal to decimal.
- Signed and unsigned integers, ones' complement, twos' complement representation of negative numbers.
- ASCII encoding of characters and how to convert between character and numeric representations.
Digital computers represent data by means of an easily identified symbol called a digit. The data may
contain digits, alphabets or special character, which are converted to bits, understandable by the computer.
In Digital Computer, data and instructions are stored in computer memory using binary code (or
machine code) represented by Binary digIT’s 1 and 0 called BIT’s.
The number system uses well-defined symbols called digits.
Number systems are classified into two types:
o Non-positional number system
o Positional number system
This document provides an overview of satellite communication principles and the evolution of communication satellites. It discusses how Arthur C. Clarke first conceived of the idea of communication satellites in geostationary orbits in 1945. It then summarizes the key milestones in the development of communication satellites, including early satellites launched by the US and USSR in the late 1950s, the establishment of international cooperation organizations like INTELSAT and Comsat in the 1960s, and the growth of satellite capabilities over time. The document also provides details about Bangladesh's first communication satellite, Bangabandhu Satellite-1, and describes different types of communication satellites.
This document introduces alternating current (AC), which regularly changes direction unlike direct current (DC). It defines key terms used to describe AC quantities like amplitude, time period, frequency, instantaneous value, and angular frequency. It also provides an example problem calculating the maximum value, frequency, time period, and instantaneous value of a given sinusoidal current. Finally, it discusses average value and instantaneous and average power of AC circuits.
1. The document provides a syllabus for RMS and average values, steady state analysis of RLC circuits with sinusoidal excitation, self and mutual inductances, and resonance in series and parallel circuits.
2. Key concepts covered include RMS and average values, form factors, steady state analysis using phasors, self and mutual inductances, dot convention, bandwidth and Q factor.
3. Example calculations are provided for average value, RMS value, form factor, and peak factor of different waveforms.
1. The document discusses AC circuits and components including inductors, capacitors, resistors, and transformers.
2. Key concepts covered include inductive and capacitive reactance, impedance, phase relationships between voltage and current, and calculations of effective voltage and current.
3. Transformers can be used to step up or down voltages in an AC circuit by changing the ratio of turns in the primary and secondary coils. An ideal transformer does not lose energy.
1) Binary codes represent numbers, letters, and other data using groups of bits or symbols. Weighted binary codes follow a positional weighting principle where each bit position represents a specific weight.
2) Non-weighted codes like excess-3 code and Gray code do not assign positional weights. Gray code is used in shaft position encoders to prevent multiple bit changes that can cause problems.
3) BCD (binary coded decimal) represents each decimal digit with a 4-bit binary number, allowing representation of numbers from 0-9. BCD addition can result in numbers outside the valid 0-9 range, requiring carries between digits.
The document discusses number base conversions between binary, decimal, octal, and hexadecimal number systems. It provides examples and steps for converting between these different number systems. Conversions include changing the integral and fractional parts of decimal numbers, grouping bits into the correct number of bits for the target base, and multiplying/dividing by the place value of each position.
The document discusses iron loss in the armature of a DC machine. It explains that hysteresis loss occurs in the armature core due to magnetic field reversal as it passes under poles of different polarity. It also explains that eddy current loss is the power loss from eddy currents induced in the armature core by the magnetic field as the armature rotates. The maximum efficiency condition for a DC machine is also mentioned.
This document discusses lap and wave winding methods for electrical generators. Lap winding involves connecting the ends of coils to the same segment of a commutator. Wave winding connects the starting end of one coil to the end of the next coil of the same polarity. An example problem calculates the induced EMF and armature current of a short-shunt compound generator delivering 30A at 220V. The document provides guidance to refer to notes for better understanding generator equations and examples involving speed and load current calculations. Students are assigned math problems related to the content covered.
The document discusses the construction and working principles of a simple DC generator. It explains that a rotating coil inside a magnetic field will generate an alternating current in the coil. To make the output current unidirectional, slip rings are replaced with a split-ring commutator. The commutator has two segments that are insulated from each other and connected to the coil ends. As the coil rotates, the commutator reverses the direction of the current flow, rectifying it to produce a unidirectional current in the external circuit. An actual DC generator consists of additional key components like magnetic poles, field coils, an armature core and windings, commutator, and brushes to generate a steady direct
There are two types of generators: DC generators and AC generators. Both work on the principle of Faraday's laws of electromagnetic induction to convert mechanical energy into electrical energy. DC generators produce direct current, while AC generators produce alternating current. The generator consists of a coil of wire placed in a changing magnetic field. According to Faraday's laws, any change in the magnetic flux through the coil will induce an electromotive force (emf) in the coil.
This document summarizes the performance analysis of non-line-of-sight (NLOS) ultraviolet (UV) communication using serial relay. It discusses how using relays can help overcome challenges of NLOS communication such as high path loss, low signal-to-noise ratio, and limited range. It presents the system model, analyzes path loss, received power, SNR, outage probability, and bit error rate with different numbers of relays. Simulation results show that using relays can enhance signal strength and reduce attenuation effects compared to NLOS without relays. The analysis demonstrates relays improve feasibility and performance of NLOS UV communication.
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
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
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.
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
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
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.
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.
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. 𝑟𝑖
.
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
- (2𝑛−1
-1) to + (2𝑛−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 𝑟𝑛–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
𝑟𝑛–N = [(𝑟𝑛-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 0
Truth 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’)