This document provides an overview of error detection and correction techniques used in digital communications. It discusses different types of errors that can occur like single-bit and burst errors. It explains that extra redundant bits must be sent along with data to detect or correct errors. Various block coding techniques are described like adding parity bits in simple parity codes or generating codewords from datawords in Hamming codes to guarantee detection of certain number of errors. The concepts of Hamming distance, minimum Hamming distance, and linear block codes are covered. Cyclic codes that have the property of rotation are also discussed along with the cyclic redundancy check technique.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur, such as single-bit and burst errors. It describes how redundancy is used to detect and correct errors using block coding techniques. Specific examples are provided to illustrate how block codes are constructed and used to detect and correct errors. Key concepts discussed include linear block codes, Hamming distance, minimum Hamming distance, and how these relate to the error detection and correction capabilities of different coding schemes.
This document provides an overview of error detection and correction techniques in digital communications. It discusses different types of errors that can occur like single-bit and burst errors. It explains how redundancy is used to detect and correct errors. Block coding techniques are described that divide messages into blocks and add redundant bits to create codewords. Linear block codes are introduced where the XOR of two codewords results in another valid codeword. Simple parity-check codes are discussed as a basic error detecting code using a single redundant bit. Examples are provided to illustrate concepts like minimum Hamming distance, error detection and correction capabilities of different codes.
Data Communication And Networking - ERROR DETECTION AND CORRECTIONAvijeet Negel
This document discusses error detection and correction in digital communications. It begins by explaining the different types of errors that can occur like single-bit and burst errors. It then introduces the concept of adding redundant bits to detect or correct errors. The document focuses on block coding, where the message is divided into blocks and redundant bits are added to each block to form a codeword. Error detection codes are able to detect errors but not correct them, while error correction codes can correct errors by encoding extra redundant bits. Examples of block codes for error detection and correction are provided to illustrate these concepts.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur like single-bit and burst errors. It explains that redundancy is needed to detect or correct errors. Various coding techniques are described, including block coding, linear block codes, cyclic codes, and checksums. Specific codes like parity-check codes, Hamming codes, and cyclic redundancy checks are explained in detail with examples. The document emphasizes that error detection codes can only detect certain error types, while error correction codes can correct errors.
This document discusses error detection and correction techniques for digital communications. It covers Hamming codes, cyclic redundancy checks (CRCs), checksums, and other coding schemes. The key aspects covered are the Hamming distance metric, linear block codes, minimum distance requirements for error detection and correction capabilities, and hardware implementations of encoders and decoders. Examples are provided to illustrate error detection and correction using various coding schemes.
This document discusses error detection and correction in digital communications. It begins by introducing different types of errors that can occur during data transmission and explains the need for redundancy to detect and correct errors. It then covers various block coding techniques used for error detection and correction, including linear block codes, Hamming codes, and cyclic codes such as cyclic redundancy checks (CRCs). Key concepts discussed include single-bit and burst errors, minimum Hamming distance, encoding and decoding processes, and the properties and advantages of different coding schemes.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur, such as single-bit and burst errors. It describes how redundancy is used to detect and correct errors using block coding techniques. Specific examples are provided to illustrate how block codes are constructed and used to detect and correct errors. Key concepts discussed include linear block codes, Hamming distance, minimum Hamming distance, and how these relate to the error detection and correction capabilities of different coding schemes.
This document discusses error detection and correction techniques using Hamming codes and cyclic codes. It provides examples of calculating Hamming distance between words and finding the minimum Hamming distance of coding schemes. Linear block codes are introduced where the XOR of two codewords results in another valid codeword. Simple parity check codes that can detect single bit errors are examined. Hamming codes that can detect up to two errors and correct single errors are analyzed. Cyclic codes are discussed as a special type of linear block code where cyclically shifting a codeword results in another valid codeword. Polynomial representations and cyclic redundancy checks are presented.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur, such as single-bit and burst errors. It describes how redundancy is used to detect and correct errors using block coding techniques. Specific examples are provided to illustrate how block codes are constructed and used to detect and correct errors. Key concepts discussed include linear block codes, Hamming distance, minimum Hamming distance, and how these relate to the error detection and correction capabilities of different coding schemes.
This document provides an overview of error detection and correction techniques in digital communications. It discusses different types of errors that can occur like single-bit and burst errors. It explains how redundancy is used to detect and correct errors. Block coding techniques are described that divide messages into blocks and add redundant bits to create codewords. Linear block codes are introduced where the XOR of two codewords results in another valid codeword. Simple parity-check codes are discussed as a basic error detecting code using a single redundant bit. Examples are provided to illustrate concepts like minimum Hamming distance, error detection and correction capabilities of different codes.
Data Communication And Networking - ERROR DETECTION AND CORRECTIONAvijeet Negel
This document discusses error detection and correction in digital communications. It begins by explaining the different types of errors that can occur like single-bit and burst errors. It then introduces the concept of adding redundant bits to detect or correct errors. The document focuses on block coding, where the message is divided into blocks and redundant bits are added to each block to form a codeword. Error detection codes are able to detect errors but not correct them, while error correction codes can correct errors by encoding extra redundant bits. Examples of block codes for error detection and correction are provided to illustrate these concepts.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur like single-bit and burst errors. It explains that redundancy is needed to detect or correct errors. Various coding techniques are described, including block coding, linear block codes, cyclic codes, and checksums. Specific codes like parity-check codes, Hamming codes, and cyclic redundancy checks are explained in detail with examples. The document emphasizes that error detection codes can only detect certain error types, while error correction codes can correct errors.
This document discusses error detection and correction techniques for digital communications. It covers Hamming codes, cyclic redundancy checks (CRCs), checksums, and other coding schemes. The key aspects covered are the Hamming distance metric, linear block codes, minimum distance requirements for error detection and correction capabilities, and hardware implementations of encoders and decoders. Examples are provided to illustrate error detection and correction using various coding schemes.
This document discusses error detection and correction in digital communications. It begins by introducing different types of errors that can occur during data transmission and explains the need for redundancy to detect and correct errors. It then covers various block coding techniques used for error detection and correction, including linear block codes, Hamming codes, and cyclic codes such as cyclic redundancy checks (CRCs). Key concepts discussed include single-bit and burst errors, minimum Hamming distance, encoding and decoding processes, and the properties and advantages of different coding schemes.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur, such as single-bit and burst errors. It describes how redundancy is used to detect and correct errors using block coding techniques. Specific examples are provided to illustrate how block codes are constructed and used to detect and correct errors. Key concepts discussed include linear block codes, Hamming distance, minimum Hamming distance, and how these relate to the error detection and correction capabilities of different coding schemes.
This document discusses error detection and correction techniques using Hamming codes and cyclic codes. It provides examples of calculating Hamming distance between words and finding the minimum Hamming distance of coding schemes. Linear block codes are introduced where the XOR of two codewords results in another valid codeword. Simple parity check codes that can detect single bit errors are examined. Hamming codes that can detect up to two errors and correct single errors are analyzed. Cyclic codes are discussed as a special type of linear block code where cyclically shifting a codeword results in another valid codeword. Polynomial representations and cyclic redundancy checks are presented.
Chapter 10: Error Correction and DetectionJeoffnaRuth
This document discusses error detection and correction techniques. It defines single-bit and burst errors and explains how redundancy is used to detect or correct errors by adding extra bits. It describes the differences between error detection and correction. Various error correction methods are presented, including forward error correction, retransmission, and the use of modular arithmetic and cyclic redundancy checks. Hardware implementations of cyclic redundancy checks are also summarized.
This document discusses error detection and correction in digital communication. It covers the types of errors like single-bit and burst errors. It then explains various error detection techniques like parity checks, longitudinal redundancy checks, and cyclic redundancy checks which work by adding redundant bits. Finally, it discusses Hamming codes, which can not only detect errors but also correct single-bit errors through the strategic placement of redundant bits.
This document discusses error detection and correction techniques used in data transmission. It covers various types of errors that can occur during transmission and different coding schemes used for error detection and correction, including block coding, linear block coding, cyclic codes, and cyclic redundancy checks (CRCs). Specific examples are provided to illustrate how Hamming codes, parity checks, and CRCs can detect and correct single-bit and burst errors. Key concepts covered include redundancy, minimum Hamming distance, encoding/decoding processes, and the use of polynomials to represent binary words in CRC calculations.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur, such as single-bit and burst errors. It describes how redundancy is used to detect and correct errors using block coding techniques. Specific examples are provided to illustrate how block codes are constructed and used to detect and correct errors. Key concepts discussed include linear block codes, Hamming distance, minimum Hamming distance, and how these relate to the error detection and correction capabilities of different coding schemes.
T com presentation (error correcting code)Akshit Jain
This document discusses error correcting codes, which are used to detect and correct errors that occur during data transmission. It covers different types of block codes like Hamming codes and Reed-Muller codes. Hamming codes can detect and correct single bit errors by adding redundant bits. Reed-Muller codes use a generator matrix to encode data and can detect and correct single bit errors through majority decoding. The document provides examples of encoding and decoding data using Hamming codes and Reed-Muller codes to demonstrate how they can detect and correct errors.
The Hamming Code allows for the detection and correction of single bit errors by adding parity bits to the data word. The parity bits are placed in bit positions that are powers of two. Each parity bit checks some subset of the data/parity bits based on its position, and is set to 1 if the number of ones in the checked bits is odd, or 0 if it is even. To locate the position of an error, the parity bits are checked and the positions of any incorrect ones are summed to find the data/parity bit position with the error.
This document discusses error detection and correction techniques in digital communications. It covers Hamming distance, block codes, linear block codes, cyclic codes, cyclic redundancy checks (CRCs), and checksums. Key points include:
- The Hamming distance between words is the number of differing bits
- Minimum Hamming distance determines a code's error detection/correction capability
- Linear block codes allow valid codewords to be created by XORing other codewords
- Cyclic codes maintain codeword validity under cyclic shifts
- CRCs use polynomial division to embed a checksum in transmitted data for error detection
- Checksums provide simple error detection by transmitting a sum of message bits
This document discusses error detection and correction techniques in digital communications. It covers Hamming distance, block codes, linear block codes, cyclic codes, cyclic redundancy checks (CRCs), and checksums. Key points include:
- The Hamming distance between words is the number of differing bits
- Minimum Hamming distance determines a code's error detection/correction capabilities
- CRCs use polynomial division to generate check bits and detect errors
- Cyclic codes have the property that cyclic shifts of codewords are also codewords
- CRC and cyclic codes are often implemented using shift registers and polynomial arithmetic
- Characteristics of good cyclic code generators for detecting various error patterns are discussed
This document discusses error detection and correction in digital communications. It begins by explaining the different types of errors that can occur like single-bit and burst errors. It then introduces the concept of adding redundant bits to detect or correct errors. The document focuses on block coding, where the message is divided into blocks and redundant bits are added to each block to form a codeword. Error detection codes are able to detect errors but not correct them, while error correction codes can correct errors by encoding extra redundant bits. Examples of block codes for error detection and correction are provided to illustrate these concepts.
How were the first error correcting codes constructed? A historical introduct...PadmaGadiyar
Hamming's original construction has been recast into symbolic form. This leads to an elementary historical route to the theory of error correcting codes. The talk is going to be historical and pedagogical in nature
This document discusses various techniques for error detection and correction in digital communications. It begins by describing common types of errors like single-bit and burst errors. It then explains error detection methods like parity checks and cyclic redundancy checks (CRCs). CRCs use cyclic codes and polynomial division to detect errors. Block codes like Hamming codes can detect and correct errors by ensuring a minimum Hamming distance between codewords. Checksums are also discussed as a simpler error detection technique than CRCs. The document provides examples to illustrate how these different error control methods work.
Error detection & correction presentationShamim Hossain
This document discusses error detection and correction in data transmission. It describes two types of errors: single-bit errors, where one bit is corrupted, and burst errors, where multiple contiguous bits are corrupted. Redundancy is added through techniques like forward error correction and coding to detect or correct errors. Specifically, it explains Hamming coding, where redundant bits are added to data blocks to create codewords, allowing the receiver to detect errors when a codeword is received incorrectly.
This document provides an overview of error detection and correction techniques used in digital communication systems. It defines different types of errors like single bit errors and burst errors that can occur during signal transmission. It also describes various error detection methods like parity checking, checksum detection, and cyclic redundancy check (CRC). The document explains concepts of forward error correction (FEC), automatic repeat request (ARQ), and CRC checkers. It provides block diagrams of the basic ARQ system and its operations.
This document summarizes error detection and correction techniques. It discusses types of errors like single-bit errors and burst errors. It covers basic concepts of error detection, including adding redundant bits and using techniques like parity checks. Error correction requires knowing the number and positions of errors. Linear block codes and cyclic codes are introduced. Hamming distance and minimum distance are important metrics for error detection and correction capability. Specific codes like parity codes, Hamming codes, and cyclic redundancy checks (CRCs) are described through examples.
The document is a report submitted to a lecturer that discusses error detection and correction of burst errors. It first defines key concepts like Hamming codes and burst errors. It then describes how burst errors occur when multiple adjacent bits are flipped and explains how redundancy can be used to detect and correct burst errors. Specifically, it notes that linear codes can correct all burst errors of length t or less if the errors occur in distinct cosets, allowing them to be identified and corrected based on their syndrome. The report provides examples of burst errors and definitions of burst error parameters like location and pattern.
This document discusses error detection and correction techniques used at the data link layer. It describes different types of errors that can occur like single-bit and burst errors. Error detection methods like parity checks, cyclic redundancy checks (CRC), and checksums are explained. Forward error correction codes like Hamming codes that allow for error correction are also covered. The document provides examples to illustrate how various error detection and correction schemes work.
This document discusses error detection and correction techniques used at the data link layer. It covers parity checks, cyclic redundancy checks (CRC), checksums, and Hamming codes. Parity checks, CRC, and checksums are used for error detection, while Hamming codes can detect and correct errors. The document provides examples of how these techniques work and compares their abilities to detect single-bit and burst errors.
The document discusses linear block codes and their encoding and decoding. It begins by defining linearity and systematic codes. Encoding can be represented by a linear system using generator matrices G, where the codewords c are a linear combination of the message m and G. Decoding uses parity check matrices H, where the syndrome s is computed as the received word r multiplied by H. For Hamming codes, the syndrome corresponds to the location of a single error.
This document provides an overview of error detection and correction techniques in data transmission. It discusses different types of errors that can occur like single-bit and burst errors. It introduces concepts like redundancy, block coding, linear block codes, cyclic codes, and checksums using cyclic redundancy checks (CRCs). Specific error detection and correction codes are presented like parity checks, Hamming codes, and CRC codes. Worked examples are provided to illustrate how these codes detect and sometimes correct errors. Key metrics like minimum Hamming distance that determine error detection and correction capabilities are explained.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur during transmission, including single-bit and burst errors. Error detection and correction require adding redundant bits to the data. Block coding divides data into blocks and adds redundant bits to each block to create codewords. Simple parity-check codes can detect single-bit errors by adding one redundant bit. Hamming codes have higher error detection and correction capabilities and can detect up to two errors and correct single errors. The document provides examples to illustrate linear block codes, minimum Hamming distance, and how Hamming codes work.
This document discusses error detection and correction in digital communications. It begins with an introduction to different types of errors that can occur like single-bit errors and burst errors. It explains that redundancy is needed to detect or correct errors. It then discusses various block coding techniques used for error detection and correction including linear block codes, cyclic codes, and cyclic redundancy checks. Specific error correcting codes like Hamming codes and parity checks are explained through examples. The key aspects of error detection capability, minimum Hamming distance, and generator polynomials in cyclic codes are covered.
Lecture8_Error Detection and Correction 232.pptxMahabubAlam97
This document discusses error detection and correction. It begins by defining different types of errors that can occur like single-bit and multiple-bit errors. It then discusses concepts like redundancy and error detection vs correction. Specific error detection and correction techniques are covered such as block coding, forward error correction vs retransmission, linear block codes including Hamming codes and cyclic redundancy checks (CRC). Worked examples are provided to illustrate key concepts.
Chapter 10: Error Correction and DetectionJeoffnaRuth
This document discusses error detection and correction techniques. It defines single-bit and burst errors and explains how redundancy is used to detect or correct errors by adding extra bits. It describes the differences between error detection and correction. Various error correction methods are presented, including forward error correction, retransmission, and the use of modular arithmetic and cyclic redundancy checks. Hardware implementations of cyclic redundancy checks are also summarized.
This document discusses error detection and correction in digital communication. It covers the types of errors like single-bit and burst errors. It then explains various error detection techniques like parity checks, longitudinal redundancy checks, and cyclic redundancy checks which work by adding redundant bits. Finally, it discusses Hamming codes, which can not only detect errors but also correct single-bit errors through the strategic placement of redundant bits.
This document discusses error detection and correction techniques used in data transmission. It covers various types of errors that can occur during transmission and different coding schemes used for error detection and correction, including block coding, linear block coding, cyclic codes, and cyclic redundancy checks (CRCs). Specific examples are provided to illustrate how Hamming codes, parity checks, and CRCs can detect and correct single-bit and burst errors. Key concepts covered include redundancy, minimum Hamming distance, encoding/decoding processes, and the use of polynomials to represent binary words in CRC calculations.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur, such as single-bit and burst errors. It describes how redundancy is used to detect and correct errors using block coding techniques. Specific examples are provided to illustrate how block codes are constructed and used to detect and correct errors. Key concepts discussed include linear block codes, Hamming distance, minimum Hamming distance, and how these relate to the error detection and correction capabilities of different coding schemes.
T com presentation (error correcting code)Akshit Jain
This document discusses error correcting codes, which are used to detect and correct errors that occur during data transmission. It covers different types of block codes like Hamming codes and Reed-Muller codes. Hamming codes can detect and correct single bit errors by adding redundant bits. Reed-Muller codes use a generator matrix to encode data and can detect and correct single bit errors through majority decoding. The document provides examples of encoding and decoding data using Hamming codes and Reed-Muller codes to demonstrate how they can detect and correct errors.
The Hamming Code allows for the detection and correction of single bit errors by adding parity bits to the data word. The parity bits are placed in bit positions that are powers of two. Each parity bit checks some subset of the data/parity bits based on its position, and is set to 1 if the number of ones in the checked bits is odd, or 0 if it is even. To locate the position of an error, the parity bits are checked and the positions of any incorrect ones are summed to find the data/parity bit position with the error.
This document discusses error detection and correction techniques in digital communications. It covers Hamming distance, block codes, linear block codes, cyclic codes, cyclic redundancy checks (CRCs), and checksums. Key points include:
- The Hamming distance between words is the number of differing bits
- Minimum Hamming distance determines a code's error detection/correction capability
- Linear block codes allow valid codewords to be created by XORing other codewords
- Cyclic codes maintain codeword validity under cyclic shifts
- CRCs use polynomial division to embed a checksum in transmitted data for error detection
- Checksums provide simple error detection by transmitting a sum of message bits
This document discusses error detection and correction techniques in digital communications. It covers Hamming distance, block codes, linear block codes, cyclic codes, cyclic redundancy checks (CRCs), and checksums. Key points include:
- The Hamming distance between words is the number of differing bits
- Minimum Hamming distance determines a code's error detection/correction capabilities
- CRCs use polynomial division to generate check bits and detect errors
- Cyclic codes have the property that cyclic shifts of codewords are also codewords
- CRC and cyclic codes are often implemented using shift registers and polynomial arithmetic
- Characteristics of good cyclic code generators for detecting various error patterns are discussed
This document discusses error detection and correction in digital communications. It begins by explaining the different types of errors that can occur like single-bit and burst errors. It then introduces the concept of adding redundant bits to detect or correct errors. The document focuses on block coding, where the message is divided into blocks and redundant bits are added to each block to form a codeword. Error detection codes are able to detect errors but not correct them, while error correction codes can correct errors by encoding extra redundant bits. Examples of block codes for error detection and correction are provided to illustrate these concepts.
How were the first error correcting codes constructed? A historical introduct...PadmaGadiyar
Hamming's original construction has been recast into symbolic form. This leads to an elementary historical route to the theory of error correcting codes. The talk is going to be historical and pedagogical in nature
This document discusses various techniques for error detection and correction in digital communications. It begins by describing common types of errors like single-bit and burst errors. It then explains error detection methods like parity checks and cyclic redundancy checks (CRCs). CRCs use cyclic codes and polynomial division to detect errors. Block codes like Hamming codes can detect and correct errors by ensuring a minimum Hamming distance between codewords. Checksums are also discussed as a simpler error detection technique than CRCs. The document provides examples to illustrate how these different error control methods work.
Error detection & correction presentationShamim Hossain
This document discusses error detection and correction in data transmission. It describes two types of errors: single-bit errors, where one bit is corrupted, and burst errors, where multiple contiguous bits are corrupted. Redundancy is added through techniques like forward error correction and coding to detect or correct errors. Specifically, it explains Hamming coding, where redundant bits are added to data blocks to create codewords, allowing the receiver to detect errors when a codeword is received incorrectly.
This document provides an overview of error detection and correction techniques used in digital communication systems. It defines different types of errors like single bit errors and burst errors that can occur during signal transmission. It also describes various error detection methods like parity checking, checksum detection, and cyclic redundancy check (CRC). The document explains concepts of forward error correction (FEC), automatic repeat request (ARQ), and CRC checkers. It provides block diagrams of the basic ARQ system and its operations.
This document summarizes error detection and correction techniques. It discusses types of errors like single-bit errors and burst errors. It covers basic concepts of error detection, including adding redundant bits and using techniques like parity checks. Error correction requires knowing the number and positions of errors. Linear block codes and cyclic codes are introduced. Hamming distance and minimum distance are important metrics for error detection and correction capability. Specific codes like parity codes, Hamming codes, and cyclic redundancy checks (CRCs) are described through examples.
The document is a report submitted to a lecturer that discusses error detection and correction of burst errors. It first defines key concepts like Hamming codes and burst errors. It then describes how burst errors occur when multiple adjacent bits are flipped and explains how redundancy can be used to detect and correct burst errors. Specifically, it notes that linear codes can correct all burst errors of length t or less if the errors occur in distinct cosets, allowing them to be identified and corrected based on their syndrome. The report provides examples of burst errors and definitions of burst error parameters like location and pattern.
This document discusses error detection and correction techniques used at the data link layer. It describes different types of errors that can occur like single-bit and burst errors. Error detection methods like parity checks, cyclic redundancy checks (CRC), and checksums are explained. Forward error correction codes like Hamming codes that allow for error correction are also covered. The document provides examples to illustrate how various error detection and correction schemes work.
This document discusses error detection and correction techniques used at the data link layer. It covers parity checks, cyclic redundancy checks (CRC), checksums, and Hamming codes. Parity checks, CRC, and checksums are used for error detection, while Hamming codes can detect and correct errors. The document provides examples of how these techniques work and compares their abilities to detect single-bit and burst errors.
The document discusses linear block codes and their encoding and decoding. It begins by defining linearity and systematic codes. Encoding can be represented by a linear system using generator matrices G, where the codewords c are a linear combination of the message m and G. Decoding uses parity check matrices H, where the syndrome s is computed as the received word r multiplied by H. For Hamming codes, the syndrome corresponds to the location of a single error.
This document provides an overview of error detection and correction techniques in data transmission. It discusses different types of errors that can occur like single-bit and burst errors. It introduces concepts like redundancy, block coding, linear block codes, cyclic codes, and checksums using cyclic redundancy checks (CRCs). Specific error detection and correction codes are presented like parity checks, Hamming codes, and CRC codes. Worked examples are provided to illustrate how these codes detect and sometimes correct errors. Key metrics like minimum Hamming distance that determine error detection and correction capabilities are explained.
This document discusses error detection and correction techniques for digital data transmission. It introduces different types of errors that can occur during transmission, including single-bit and burst errors. Error detection and correction require adding redundant bits to the data. Block coding divides data into blocks and adds redundant bits to each block to create codewords. Simple parity-check codes can detect single-bit errors by adding one redundant bit. Hamming codes have higher error detection and correction capabilities and can detect up to two errors and correct single errors. The document provides examples to illustrate linear block codes, minimum Hamming distance, and how Hamming codes work.
This document discusses error detection and correction in digital communications. It begins with an introduction to different types of errors that can occur like single-bit errors and burst errors. It explains that redundancy is needed to detect or correct errors. It then discusses various block coding techniques used for error detection and correction including linear block codes, cyclic codes, and cyclic redundancy checks. Specific error correcting codes like Hamming codes and parity checks are explained through examples. The key aspects of error detection capability, minimum Hamming distance, and generator polynomials in cyclic codes are covered.
Lecture8_Error Detection and Correction 232.pptxMahabubAlam97
This document discusses error detection and correction. It begins by defining different types of errors that can occur like single-bit and multiple-bit errors. It then discusses concepts like redundancy and error detection vs correction. Specific error detection and correction techniques are covered such as block coding, forward error correction vs retransmission, linear block codes including Hamming codes and cyclic redundancy checks (CRC). Worked examples are provided to illustrate key concepts.
This document discusses error detection and correction techniques using Hamming codes and cyclic redundancy checks (CRCs). It explains key concepts like Hamming distance, minimum distance, linear block codes, and cyclic codes. Examples are provided to illustrate how Hamming codes can detect up to two errors and correct one error using a minimum distance of 3. CRC codes are also examined, showing how they use polynomial division to detect errors. The advantages of cyclic codes for hardware implementation are noted.
This document discusses error detection and correction techniques using Hamming codes and cyclic redundancy checks (CRCs). It explains key concepts like Hamming distance, minimum distance, linear block codes, and cyclic codes. Examples are provided to illustrate how Hamming codes can detect up to two errors and correct one error using a minimum distance of 3. CRC codes are also examined, showing how they use polynomial division to detect errors. The advantages of cyclic codes for hardware implementation are noted.
This document discusses error detection and correction techniques using Hamming codes and cyclic redundancy checks (CRCs). It explains key concepts like Hamming distance, minimum distance, linear block codes, and cyclic codes. Examples are provided to illustrate how Hamming codes can detect up to two errors and correct one error using a minimum distance of 3. CRC codes are also examined, showing how they use polynomial division to detect errors. The advantages of cyclic codes for hardware implementation are noted.
This document discusses error detection and correction techniques for digital communications. It covers Hamming codes, cyclic redundancy checks (CRCs), checksums, and other coding schemes. Key points include:
- Hamming codes can detect up to two errors and correct single errors by ensuring a minimum Hamming distance of 3 between codewords.
- CRCs use cyclic codes to detect errors, with the CRC value serving as a checksum. CRC division and hardware implementations are examined.
- Cyclic codes have the property that cyclically shifting a codeword results in another valid codeword. This allows for efficient encoding and decoding.
- Checksums involve sending the sum of message bits along with the message for the receiver to
This document discusses error detection and correction techniques for digital communications. It covers Hamming codes, cyclic redundancy checks (CRCs), checksums, and other coding schemes. Key points include:
- Hamming codes can detect up to two errors and correct single errors by ensuring a minimum Hamming distance of 3 between codewords.
- CRCs use cyclic codes to detect errors, with the CRC value serving as a checksum. CRC division and hardware implementations are examined.
- Cyclic codes have the property that cyclically shifting a codeword results in another valid codeword. This allows for efficient encoding and decoding.
- Checksums involve sending the sum of message bits along with the message for the receiver to
This document discusses error detection and correction techniques used in digital communications. It covers Hamming codes, cyclic redundancy checks (CRCs), checksums, and linear block codes. Key points include:
- Hamming codes can detect up to two errors and correct single errors by ensuring a minimum Hamming distance of 3 between codewords.
- CRCs use cyclic codes to detect errors by calculating syndromes based on polynomial division of the received codeword by a generator polynomial.
- Checksums involve sending the sum of data words along with the words to detect errors by verifying the received sum is correct.
- Linear block codes allow the XOR of any two valid codewords to produce another valid codeword.
This document discusses error detection and correction techniques for digital data transmission. It covers various types of errors that can occur like single-bit and burst errors. It also describes different coding schemes used for error detection and correction including block coding, linear block codes, cyclic codes, and checksums. Specific codes discussed include parity-bit codes, Hamming codes, and cyclic redundancy checks (CRCs). The techniques are evaluated based on their error detection and correction capabilities as defined by concepts like minimum Hamming distance.
This presentation covers the type of errors during data transmission on a network. It also explains the strategies needed to detect and correct errors during data transmission. Error occurs when user receives unexpected data that was changed during transmission. There are two types of errors: Single bit error and Burst Error.
This document discusses error detection and correction in digital communications. It introduces block coding where messages are divided into blocks of k bits with r redundant bits added to each block, making the block length n=k+r. Linear block codes are discussed where the XOR of two codewords results in a third valid codeword. The minimum Hamming distance dmin relates to the error detection and correction capabilities of a code, with dmin=s+1 needed to detect s errors and dmin=2t+1 to correct t errors. Examples are given of simple parity-check codes and generating codewords from datawords.
Data Link Layer- Error Detection and Control_2.pptxArunVerma37053
The document discusses the data link layer and its objectives. The data link layer transforms the physical layer into a link responsible for node-to-node communication. Specific responsibilities of the data link layer include framing, addressing, flow control, error control, and media access control. It divides data into frames, adds header information, implements flow control mechanisms, and adds reliability through error detection and retransmission of damaged frames. When multiple devices share the same link, data link protocols determine which device has control over the link.
This document discusses error detection and correction in digital communications. It begins by explaining the different types of errors that can occur like single-bit and burst errors. It then introduces the concept of adding redundancy through extra bits to detect or correct errors. Block coding is described as a method of dividing messages into fixed-length blocks and adding redundant bits to each block. Error detection codes are able to detect errors but not correct them, while error correction codes can correct a certain number of errors without knowing their location. Examples are provided to illustrate how error detection and correction work using simple block codes.
This document discusses error detection and correction techniques in digital communications. It covers Hamming distance, block codes, linear block codes, cyclic codes, cyclic redundancy checks (CRCs), and checksums. Key points include:
- The Hamming distance between words is the number of differing bits
- Minimum Hamming distance determines a code's error detection/correction capability
- Linear block codes allow valid codewords to be created by XORing other codewords
- Cyclic codes maintain codeword validity under cyclic shifts
- CRCs use polynomial division to embed a checksum in transmitted data
- Checksums provide error detection by transmitting a calculated value over transmitted data
This document discusses error detection and correction in data transmission. It begins with an introduction to types of errors like single-bit and burst errors. It then discusses key concepts like error detection, correction, and forward error correction versus retransmission. The document focuses on block coding techniques for error detection and correction. It explains linear block codes and provides examples of parity-check codes and Hamming codes. Parity-check codes can detect single and odd number of errors while Hamming codes can detect and correct errors.
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1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
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2. Note
Data can be corrupted
during transmission.
Some applications require that
errors be detected and corrected.
10.2
3. 10-1 INTRODUCTION
Let us first discuss some issues related, directly or
indirectly, to error detection and correction.
Topics discussed in this section:
Types of Errors
Redundancy
Detection Versus Correction
Forward Error Correction Versus Retransmission
Coding
Modular Arithmetic
10.3
13. 10-2 BLOCK CODING
In block coding, we divide our message into blocks,
each of k bits, called datawords. We add r redundant
bits to each block to make the length n = k + r. The
resulting n-bit blocks are called codewords.
Topics discussed in this section:
Error Detection
Error Correction
Hamming Distance
Minimum Hamming Distance
10.13
15. Example 10.1
The 4B/5B block coding discussed in Chapter 4 is a good
example of this type of coding. In this coding scheme,
k = 4 and n = 5. As we saw, we have 2k = 16 datawords
and 2n = 32 codewords. We saw that 16 out of 32
codewords are used for message transfer and the rest are
either used for other purposes or unused.
10.15
17. Example 10.2
Let us assume that k = 2 and n = 3. Table 10.1 shows the
list of datawords and codewords. Later, we will see
how to derive a codeword from a dataword.
Assume the sender encodes the dataword 01 as 011 and
sends it to the receiver. Consider the following cases:
1. The receiver receives 011. It is a valid codeword. The
receiver extracts the dataword 01 from it.
10.17
18. Example 10.2 (continued)
2. The codeword is corrupted during transmission, and
111 is received. This is not a valid codeword and is
discarded.
3. The codeword is corrupted during transmission, and
000 is received. This is a valid codeword. The receiver
incorrectly extracts the dataword 00. Two corrupted
bits have made the error undetectable.
10.18
19. Table 10.1 A code for error detection (Example 10.2)
10.19
20. Note
An error-detecting code can detect
only the types of errors for which it is
designed; other types of errors may
remain undetected.
10.20
22. Example 10.3
Let us add more redundant bits to Example 10.2 to see if
the receiver can correct an error without knowing what
was actually sent. We add 3 redundant bits to the 2-bit
dataword to make 5-bit codewords. Table 10.2 shows the
datawords and codewords. Assume the dataword is 01.
The sender creates the codeword 01011. The codeword is
corrupted during transmission, and 01001 is received.
First, the receiver finds that the received codeword is not
in the table. This means an error has occurred. The
receiver, assuming that there is only 1 bit corrupted, uses
the following strategy to guess the correct dataword.
10.22
23. Example 10.3 (continued)
1. Comparing the received codeword with the first
codeword in the table (01001 versus 00000), the
receiver decides that the first codeword is not the one
that was sent because there are two different bits.
2. By the same reasoning, the original codeword cannot
be the third or fourth one in the table.
3. The original codeword must be the second one in the
table because this is the only one that differs from the
received codeword by 1 bit. The receiver replaces
01001 with 01011 and consults the table to find the
dataword 01.
10.23
24. Table 10.2 A code for error correction (Example 10.3)
10.24
25. Note
The Hamming distance between two
words is the number of differences
between corresponding bits.
10.25
26. Example 10.4
Let us find the Hamming distance between two pairs of
words.
1. The Hamming distance d(000, 011) is 2 because
2. The Hamming distance d(10101, 11110) is 3 because
10.26
27. Note
The minimum Hamming distance is the
smallest Hamming distance between
all possible pairs in a set of words.
10.27
28. Example 10.5
Find the minimum Hamming distance of the coding
scheme in Table 10.1.
Solution
We first find all Hamming distances.
The dmin in this case is 2.
10.28
29. Example 10.6
Find the minimum Hamming distance of the coding
scheme in Table 10.2.
Solution
We first find all the Hamming distances.
The dmin in this case is 3.
10.29
30. Note
To guarantee the detection of up to s
errors in all cases, the minimum
Hamming distance in a block
code must be dmin = s + 1.
10.30
31. Example 10.7
The minimum Hamming distance for our first code
scheme (Table 10.1) is 2. This code guarantees detection
of only a single error. For example, if the third codeword
(101) is sent and one error occurs, the received codeword
does not match any valid codeword. If two errors occur,
however, the received codeword may match a valid
codeword and the errors are not detected.
10.31
32. Example 10.8
Our second block code scheme (Table 10.2) has dmin = 3.
This code can detect up to two errors. Again, we see that
when any of the valid codewords is sent, two errors create
a codeword which is not in the table of valid codewords.
The receiver cannot be fooled.
However, some combinations of three errors change a
valid codeword to another valid codeword. The receiver
accepts the received codeword and the errors are
undetected.
10.32
35. Note
To guarantee correction of up to t errors
in all cases, the minimum Hamming
distance in a block code
must be dmin = 2t + 1.
10.35
36. Example 10.9
A code scheme has a Hamming distance dmin = 4. What is
the error detection and correction capability of this
scheme?
Solution
This code guarantees the detection of up to three errors
(s = 3), but it can correct up to one error. In other words,
if this code is used for error correction, part of its capability
is wasted. Error correction codes need to have an odd
minimum distance (3, 5, 7, . . . ).
10.36
37. 10-3 LINEAR BLOCK CODES
Almost all block codes used today belong to a subset
called linear block codes. A linear block code is a code
in which the exclusive OR (addition modulo-2) of two
valid codewords creates another valid codeword.
Topics discussed in this section:
Minimum Distance for Linear Block Codes
Some Linear Block Codes
10.37
38. Note
In a linear block code, the exclusive OR
(XOR) of any two valid codewords
creates another valid codeword.
10.38
39. Example 10.10
Let us see if the two codes we defined in Table 10.1 and
Table 10.2 belong to the class of linear block codes.
1. The scheme in Table 10.1 is a linear block code
because the result of XORing any codeword with any
other codeword is a valid codeword. For example, the
XORing of the second and third codewords creates the
fourth one.
2. The scheme in Table 10.2 is also a linear block code.
We can create all four codewords by XORing two
other codewords.
10.39
40. Example 10.11
In our first code (Table 10.1), the numbers of 1s in the
nonzero codewords are 2, 2, and 2. So the minimum
Hamming distance is dmin = 2. In our second code (Table
10.2), the numbers of 1s in the nonzero codewords are 3,
3, and 4. So in this code we have dmin = 3.
10.40
41. Note
A simple parity-check code is a
single-bit error-detecting
code in which
n = k + 1 with dmin = 2.
10.41
44. Example 10.12
Let us look at some transmission scenarios. Assume the
sender sends the dataword 1011. The codeword created
from this dataword is 10111, which is sent to the receiver.
We examine five cases:
1. No error occurs; the received codeword is 10111. The
syndrome is 0. The dataword 1011 is created.
2. One single-bit error changes a1 . The received
codeword is 10011. The syndrome is 1. No dataword
is created.
3. One single-bit error changes r0 . The received codeword
is 10110. The syndrome is 1. No dataword is created.
10.44
45. Example 10.12 (continued)
4. An error changes r0 and a second error changes a3 .
The received codeword is 00110. The syndrome is 0.
The dataword 0011 is created at the receiver. Note that
here the dataword is wrongly created due to the
syndrome value.
5. Three bits—a3, a2, and a1—are changed by errors.
The received codeword is 01011. The syndrome is 1.
The dataword is not created. This shows that the simple
parity check, guaranteed to detect one single error, can
also find any odd number of errors.
10.45
54. Example 10.13
Let us trace the path of three datawords from the sender
to the destination:
1. The dataword 0100 becomes the codeword 0100011.
The codeword 0100011 is received. The syndrome is
000, the final dataword is 0100.
2. The dataword 0111 becomes the codeword 0111001.
The syndrome is 011. After flipping b2 (changing the 1
to 0), the final dataword is 0111.
3. The dataword 1101 becomes the codeword 1101000.
The syndrome is 101. After flipping b0, we get 0000,
the wrong dataword. This shows that our code cannot
correct two errors.
10.54
55. Example 10.14
We need a dataword of at least 7 bits. Calculate values of
k and n that satisfy this requirement.
Solution
We need to make k = n − m greater than or equal to 7, or
2m − 1 − m ≥ 7.
1. If we set m = 3, the result is n = 23 − 1 and k = 7 − 3,
or 4, which is not acceptable.
2. If we set m = 4, then n = 24 − 1 = 15 and k = 15 − 4 =
11, which satisfies the condition. So the code is
C(15, 11)
10.55
57. 10-4 CYCLIC CODES
Cyclic codes are special linear block codes with one
extra property. In a cyclic code, if a codeword is
cyclically shifted (rotated), the result is another
codeword.
Topics discussed in this section:
Cyclic Redundancy Check
Hardware Implementation
Polynomials
Cyclic Code Analysis
Advantages of Cyclic Codes
Other Cyclic Codes
10.57
68. Note
The divisor in a cyclic code is normally
called the generator polynomial
or simply the generator.
10.68
69. Note
In a cyclic code,
If s(x) ≠ 0, one or more bits is corrupted.
If s(x) = 0, either
a. No bit is corrupted. or
b. Some bits are corrupted, but the
decoder failed to detect them.
10.69
70. Note
In a cyclic code, those e(x) errors that
are divisible by g(x) are not caught.
10.70
71. Note
If the generator has more than one term
and the coefficient of x0 is 1,
all single errors can be caught.
10.71
72. Example 10.15
Which of the following g(x) values guarantees that a
single-bit error is caught? For each case, what is the
error that cannot be caught?
a. x + 1
b. x3
c. 1
Solution
a. No xi can be divisible by x + 1. Any single-bit error can
be caught.
b. If i is equal to or greater than 3, xi is divisible by g(x).
All single-bit errors in positions 1 to 3 are caught.
c. All values of i make xi divisible by g(x). No single-bit
error can be caught. This g(x) is useless.
10.72
74. Note
If a generator cannot divide xt + 1
(t between 0 and n – 1),
then all isolated double errors
can be detected.
10.74
75. Example 10.16
Find the status of the following generators related to two
isolated, single-bit errors.
a. x + 1 b. x4 + 1 c. x7 + x6 + 1 d. x15 + x14 + 1
Solution
a. This is a very poor choice for a generator. Any two
errors next to each other cannot be detected.
b. This generator cannot detect two errors that are four
positions apart.
c. This is a good choice for this purpose.
d. This polynomial cannot divide xt + 1 if t is less than
32,768. A codeword with two isolated errors up to
32,768 bits apart can be detected by this generator.
10.75
76. Note
A generator that contains a factor of
x + 1 can detect all odd-numbered
errors.
10.76
77. Note
❏ All burst errors with L ≤ r will be
detected.
❏ All burst errors with L = r + 1 will be
detected with probability 1 – (1/2) r–1.
❏ All burst errors with L > r + 1 will be
detected with probability 1 – (1/2) r.
10.77
78. Example 10.17
Find the suitability of the following generators in relation
to burst errors of different lengths.
a. x6 + 1
b. x18 + x7 + x + 1
c. x32 + x23 + x7 + 1
Solution
a. This generator can detect all burst errors with a length
less than or equal to 6 bits; 3 out of 100 burst errors
with length 7 will slip by; 16 out of 1000 burst errors of
length 8 or more will slip by.
10.78
79. Example 10.17 (continued)
b. This generator can detect all burst errors with a length
less than or equal to 18 bits; 8 out of 1 million burst
errors with length 19 will slip by; 4 out of 1 million
burst errors of length 20 or more will slip by.
c. This generator can detect all burst errors with a length
less than or equal to 32 bits; 5 out of 10 billion burst
errors with length 33 will slip by; 3 out of 10 billion
burst errors of length 34 or more will slip by.
10.79
80. Note
A good polynomial generator needs to
have the following characteristics:
1. It should have at least two terms.
2. The coefficient of the term x0 should
be 1.
3. It should not divide xt + 1, for t
between 2 and n − 1.
4. It should have the factor x + 1.
10.80
82. 10-5 CHECKSUM
The last error detection method we discuss here is
called the checksum. The checksum is used in the
Internet by several protocols although not at the data
link layer. However, we briefly discuss it here to
complete our discussion on error checking
Topics discussed in this section:
Idea
One’s Complement
Internet Checksum
10.82
83. Example 10.18
Suppose our data is a list of five 4-bit numbers that we
want to send to a destination. In addition to sending these
numbers, we send the sum of the numbers. For example,
if the set of numbers is (7, 11, 12, 0, 6), we send (7, 11, 12,
0, 6, 36), where 36 is the sum of the original numbers.
The receiver adds the five numbers and compares the
result with the sum. If the two are the same, the receiver
assumes no error, accepts the five numbers, and discards
the sum. Otherwise, there is an error somewhere and the
data are not accepted.
10.83
84. Example 10.19
We can make the job of the receiver easier if we send the
negative (complement) of the sum, called the checksum.
In this case, we send (7, 11, 12, 0, 6, −36). The receiver
can add all the numbers received (including the
checksum). If the result is 0, it assumes no error;
otherwise, there is an error.
10.84
85. Example 10.20
How can we represent the number 21 in one’s
complement arithmetic using only four bits?
Solution
The number 21 in binary is 10101 (it needs five bits). We
can wrap the leftmost bit and add it to the four rightmost
bits. We have (0101 + 1) = 0110 or 6.
10.85
86. Example 10.21
How can we represent the number −6 in one’s
complement arithmetic using only four bits?
Solution
In one’s complement arithmetic, the negative or
complement of a number is found by inverting all bits.
Positive 6 is 0110; negative 6 is 1001. If we consider only
unsigned numbers, this is 9. In other words, the
complement of 6 is 9. Another way to find the
complement of a number in one’s complement arithmetic
is to subtract the number from 2n − 1 (16 − 1 in this case).
10.86
87. Example 10.22
Let us redo Exercise 10.19 using one’s complement
arithmetic. Figure 10.24 shows the process at the sender
and at the receiver. The sender initializes the checksum
to 0 and adds all data items and the checksum (the
checksum is considered as one data item and is shown in
color). The result is 36. However, 36 cannot be expressed
in 4 bits. The extra two bits are wrapped and added with
the sum to create the wrapped sum value 6. In the figure,
we have shown the details in binary. The sum is then
complemented, resulting in the checksum value 9 (15 − 6
= 9). The sender now sends six data items to the receiver
including the checksum 9.
10.87
88. Example 10.22 (continued)
The receiver follows the same procedure as the sender. It
adds all data items (including the checksum); the result
is 45. The sum is wrapped and becomes 15. The wrapped
sum is complemented and becomes 0. Since the value of
the checksum is 0, this means that the data is not
corrupted. The receiver drops the checksum and keeps
the other data items. If the checksum is not zero, the
entire packet is dropped.
10.88
90. Note
Sender site:
1. The message is divided into 16-bit words.
2. The value of the checksum word is set to 0.
3. All words including the checksum are
added using one’s complement addition.
4. The sum is complemented and becomes the
checksum.
5. The checksum is sent with the data.
10.90
91. Note
Receiver site:
1. The message (including checksum) is
divided into 16-bit words.
2. All words are added using one’s
complement addition.
3. The sum is complemented and becomes the
new checksum.
4. If the value of checksum is 0, the message
is accepted; otherwise, it is rejected.
10.91
92. Example 10.23
Let us calculate the checksum for a text of 8 characters
(“Forouzan”). The text needs to be divided into 2-byte
(16-bit) words. We use ASCII (see Appendix A) to change
each byte to a 2-digit hexadecimal number. For example,
F is represented as 0x46 and o is represented as 0x6F.
Figure 10.25 shows how the checksum is calculated at the
sender and receiver sites. In part a of the figure, the value
of partial sum for the first column is 0x36. We keep the
rightmost digit (6) and insert the leftmost digit (3) as the
carry in the second column. The process is repeated for
each column. Note that if there is any corruption, the
checksum recalculated by the receiver is not all 0s. We
leave this an exercise.
10.92