Error Correction Codes

Error Correction Codes

Contents

Recall the following terms: i. Block Codes ii. Linear Block Codes iii. Hamming Distance & Hamming Weight iv. Minimum Distance & Error Detection v. Correction Capabilities 1

Relate the minimum Hamming Distance and Error detecting and Correcting Capabilities 2

Recall the Generator Matrix and Parity – Check Matrix 2

Describe Encoding and Decoding of Linear Block Codes 3

Recall Hamming Bound and Hamming Code 4

Recall the Cyclic Codes and its Types 5

Describe the generation of Non-Systematic Cyclic Code 5

Describe the generation of Systematic Cyclic Code 6

Describe the Generator and Parity Check Matrix of Cyclic Code 7

Define the Convolution Codes 8

Describe the Generator of Convolution Codes 9

Describe the Graphical Representation of Convolution Codes Encoder 10

Recall the Viterbi Decoding Algorithm 11

Describe Burst Error Correction Codes 13

Describe Turbo codes 14

Describe Alphanumeric Codes and its types 14

Describe ASCII and EBCDIC Code 15

Describe UNICODE and Baudot Code 16

Recall the following terms: i. Block Codes ii. Linear Block Codes iii. Hamming Distance & Hamming Weight iv. Minimum Distance & Error Detection v. Correction Capabilities

i. Block codes: Block codes are a type of error correction code that operate on blocks of symbols, rather than on individual symbols. They are used to detect and correct errors in transmitted messages or signals by adding redundancy to the transmitted data.

ii. Linear block codes: Linear block codes are a type of block code that can be represented as a linear combination of a set of codewords. They are characterized by the property that the sum of any two codewords is also a valid codeword. Linear block codes are widely used in communication systems due to their simple structure and good error correction capabilities.

iii. a) Hamming distance: The Hamming distance between two codewords is the number of bit positions in which the two codewords differ. It is a measure of the similarity between two codewords and is an important concept in error correction coding.
b) Hamming weight: The Hamming weight of a codeword is the number of 1’s in the codeword. It is a measure of the complexity of the codeword and is often used as a measure of the error correction capabilities of a code.

iv. a) Minimum distance: The minimum distance of a code is the smallest Hamming distance between any two codewords in the code. It is an important measure of the error correction capabilities of the code, as it determines the number of errors that can be corrected by the code.
b) Error detection: Error detection is the process of detecting errors in transmitted messages or signals. It is an important aspect of communication systems, as it allows the receiver to identify and correct errors in the received data.

v. Correction capabilities: The correction capabilities of a code refer to the number of errors that the code can correct. A code with high correction capabilities is able to correct a large number of errors, while a code with low correction capabilities is only able to correct a small number of errors. The correction capabilities of a code are determined by its minimum distance and its error detection capabilities.

Relate the minimum Hamming Distance and Error detecting and Correcting Capabilities

The minimum Hamming distance of a code is an important measure of the error correction capabilities of the code. It determines the number of errors that the code can correct and the number of errors that it can detect.

For a given code, the minimum Hamming distance d is related to the error detection and correction capabilities of the code according to the following:

· The code can detect any number of errors up to d-1.

· The code can correct t or fewer errors, where t is equal to floor((d-1)/2).

For example, consider a code with a minimum Hamming distance of 3. This code can detect any number of errors up to 2 and can correct 1 or fewer errors.

On the other hand, consider a code with a minimum Hamming distance of 4. This code can detect any number of errors up to 3 and can correct 2 or fewer errors.

In general, a code with a larger minimum Hamming distance has higher error correction capabilities, as it is able to correct a larger number of errors. However, the minimum Hamming distance of a code also determines the overhead associated with the code, as more bits are required to represent the codewords with a larger minimum Hamming distance. As a result, there is a trade-off between the error correction capabilities of a code and the overhead associated with the code.

Recall the Generator Matrix and Parity – Check Matrix

The generator matrix and parity-check matrix are two important concepts in the design and analysis of error correction codes. They are used to represent the structure of a code and to facilitate the encoding and decoding of messages or signals.

A generator matrix is a matrix that defines the set of codewords in a code. It is used to encode messages or signals by multiplying the message by the generator matrix to produce the codeword.

A parity-check matrix is a matrix that is used to check the parity of a received codeword. It is used to decode messages or signals by multiplying the received codeword by the parity-check matrix and checking the resulting syndrome.

Both the generator matrix and the parity-check matrix are used to represent the structure of a code and to facilitate the encoding and decoding of messages or signals. They are important tools in the design and analysis of error correction codes and are widely used in communication systems.

Describe Encoding and Decoding of Linear Block Codes

Encoding and decoding of linear block codes are the processes of converting a message or signal into a codeword for transmission over a communication channel and recovering the original message or signal from the received codeword, respectively.

Encoding of linear block codes is typically performed using a generator matrix G. The generator matrix is a matrix that defines the set of codewords in the code. To encode a message or signal, the message is first represented as a vector m, and the codeword c is obtained by multiplying the message vector by the generator matrix:

c = mG

where c is the codeword, m is the message vector, and G is the generator matrix.

Decoding of linear block codes is typically performed using a parity-check matrix H. The parity-check matrix is a matrix that is used to check the parity of a received codeword. To decode a received codeword r, the syndrome s is obtained by multiplying the received codeword by the parity-check matrix:

s = rH

where s is the syndrome and H is the parity-check matrix.

If the syndrome is non-zero, it indicates that errors have occurred in the transmission of the codeword. The decoder can then use the syndrome to identify and correct the errors in the received codeword. If the syndrome is zero, it indicates that the codeword has been received without errors and the original message or signal can be recovered.

Encoding and decoding of linear block codes are important processes in communication systems, as they allow messages or signals to be transmitted over a communication channel with error correction. Linear block codes are widely used in communication systems due to their simple structure and good error correction capabilities.

Recall Hamming Bound and Hamming Code

The Hamming bound is a fundamental result in error correction coding that relates the minimum distance d of a code to its error correction and error detection capabilities. It states that the minimum number of codewords in a code with minimum distance d is given by:

2(n-k) ≤ 2(d-1)

where n is the length of the codewords, k is the number of information bits in the codewords, and d is the minimum distance of the code.

The Hamming bound is an important result in error correction coding, as it provides a limit on the minimum number of codewords in a code with a given minimum distance. It shows that the minimum number of codewords in a code increases with the minimum distance of the code, and that the minimum distance of a code determines the error correction and error detection capabilities of the code.

Hamming code is a specific type of error correction code that was developed by Richard Hamming. It is a linear block code that is based on the idea of adding redundant bits to a message or signal to detect and correct errors. Hamming codes are characterized by the property that the minimum distance of the code is 3, which allows them to correct a single error and detect any two errors. They are widely used in communication systems due to their simple structure and good error correction capabilities.

Recall the Cyclic Codes and its Types

Cyclic codes are a special class of linear block codes that are characterized by the property that the codewords are periodic with a period equal to the length of the codewords. This means that the codewords can be represented as cyclic shifts of a single polynomial.

There are two main types of cyclic codes:

1. Cyclic redundancy check (CRC) codes: CRC codes are a type of cyclic code that are used to detect errors in transmitted messages or signals. They work by adding redundant bits to the transmitted data and using a polynomial division to check the parity of the received data. If the parity check fails, it indicates that an error has occurred in the transmission of the data.

2. BCH codes: BCH codes are a type of cyclic code that are used for error correction. They are characterized by the property that they can correct a certain number of errors for a given code length. BCH codes are widely used in communication systems due to their good error correction capabilities.

Cyclic codes are widely used in communication systems due to their simple structure and good error correction capabilities. They are particularly useful for detecting errors in transmitted messages or signals, as they can detect a large number of errors with a relatively small overhead.

Describe the generation of Non-Systematic Cyclic Code

Non-systematic cyclic codes are a type of cyclic code in which the codewords are not the same as the original message or signal. They are generated by multiplying the message or signal by a generator polynomial g(x) to produce the codeword:

c(x) = m(x) * g(x)

where c(x) is the codeword, m(x) is the message or signal, and g(x) is the generator polynomial.

The generator polynomial is chosen such that it has a degree that is equal to or greater than the number of errors that the code is capable of correcting. For example, a generator polynomial of degree 3 can be used to generate a cyclic code that is capable of correcting up to 3 errors.

To decode a received codeword r(x), it is divided by the generator polynomial using polynomial division:

r(x) = q(x) * g(x) + r'(x)

where q(x) is the quotient, g(x) is the generator polynomial, and r'(x) is the remainder. If the remainder r'(x) is non-zero, it indicates that errors have occurred in the transmission of the codeword. The decoder can then use the remainder to identify and correct the errors in the received codeword. If the remainder is zero, it indicates that the codeword has been received without errors and the original message or signal can be recovered.

Non-systematic cyclic codes are widely used in communication systems due to their simple structure and good error correction capabilities. They are particularly useful for detecting errors in transmitted messages or signals, as they can detect a large number of errors with a relatively small overhead.

Describe the generation of Systematic Cyclic Code

Systematic cyclic codes are a type of cyclic code in which the codewords are the same as the original message or signal, except for the addition of redundant bits. They are generated by adding redundant bits to the message or signal to produce the codeword.

The redundant bits are calculated using a generator polynomial g(x) such that the codeword c(x) satisfies the following equation:

c(x) = m(x) * xk + r(x)

where c(x) is the codeword, m(x) is the message or signal, k is the number of redundant bits, and r(x) is the remainder obtained by dividing m(x) by g(x).

To decode a received codeword r(x), it is divided by the generator polynomial using polynomial division:

r(x) = q(x) * g(x) + r'(x)

where q(x) is the quotient, g(x) is the generator polynomial, and r'(x) is the remainder. If the remainder r'(x) is non-zero, it indicates that errors have occurred in the transmission of the codeword. The decoder can then use the remainder to identify and correct the errors in the received codeword. If the remainder is zero, it indicates that the codeword has been received without errors and the original message or signal can be recovered.

Systematic cyclic codes are widely used in communication systems due to their simple structure and good error correction capabilities. They are particularly useful for detecting errors in transmitted messages or signals, as they can detect a large number of errors with a relatively small overhead.

Describe the Generator and Parity Check Matrix of Cyclic Code

The generator matrix and parity-check matrix are two important concepts in the design and analysis of cyclic codes. They are used to represent the structure of a code and to facilitate the encoding and decoding of messages or signals.

A generator matrix for a cyclic code is a matrix that defines the set of codewords in the code. It is used to encode messages or signals by multiplying the message by the generator matrix to produce the codeword.

The generator matrix for a systematic cyclic code can be written as:

G = [Ik | P]

where Ik is the identity matrix of size k, P is the parity matrix, and k is the number of information bits in the codeword.

A parity-check matrix for a cyclic code is a matrix that is used to check the parity of a received codeword. It is used to decode messages or signals by multiplying the received codeword by the parity-check matrix and checking the resulting syndrome.

The parity-check matrix for a cyclic code can be written as:

H = [PT | I{n-k}]

where PT is the transpose of the parity matrix, I{n-k} is the identity matrix of size n-k, and n is the length of the codewords.

Both the generator matrix and the parity-check matrix are used to represent the structure of a cyclic code and to facilitate the encoding and decoding of messages or signals. They are important tools in the design and analysis of cyclic codes and are widely used in communication systems.

Define the Convolution Codes

Convolutional codes are a type of error correction code that are characterized by the property that the codewords are generated by the convolution of the message or signal with a convolutional encoder. They are widely used in communication systems due to their good error correction capabilities and their ability to handle variable-length messages or signals.

In a convolutional code, the codeword is generated by the convolution of the message or signal with a convolutional encoder, which is a finite state machine that generates the codeword bits based on the current state of the machine and the input message or signal bits. The codeword bits are generated according to a set of rules or transitions defined by the convolutional encoder.

Convolutional codes are characterized by their constraint length K and their rate R, which are defined as follows:

· Constraint length K: The constraint length of a convolutional code is the maximum number of input bits that the encoder takes into account when generating a single output bit.

· Rate R: The rate of a convolutional code is the ratio of the number of output bits to the number of input bits. It is a measure of the efficiency of the code, as it determines the amount of redundancy added to the message or signal.

Convolutional codes are widely used in communication systems due to their good error correction capabilities and their ability to handle variable-length messages or signals. They are particularly useful for correcting errors in long messages or signals, as they can correct a large number of errors with a relatively small overhead.

Describe the Generator of Convolution Codes

The generator of a convolutional code is a matrix that defines the set of codewords in the code. It is used to encode messages or signals by multiplying the message by the generator to produce the codeword.

The generator of a convolutional code can be represented as a polynomial G(D) in the form:

G(D) = g0 + g1D + g2D2 + … + gkDk

where gi are the generator coefficients and D is the delay operator. The generator polynomial is of degree k, which is the constraint length of the code.

The generator polynomial defines the set of codewords in the convolutional code. To encode a message or signal, the message is first represented as a vector m, and the codeword c is obtained by multiplying the message vector by the generator polynomial:

c = mG(D)

where c is the codeword, m is the message vector, and G(D) is the generator polynomial.

The generator of a convolutional code is an important concept in the design and analysis of convolutional codes. It is used to represent the structure of the code and to facilitate the encoding and decoding of messages or signals.

Describe the Graphical Representation of Convolution Codes Encoder

Convolutional codes can be represented graphically using a trellis diagram, which provides a visual representation of the encoding process. The trellis diagram consists of nodes and branches that represent the states and transitions of the convolutional code.

Here’s a graphical representation of a convolutional code encoder using a trellis diagram:

0Gp989Ljmxzmj7+Lucgaaaoluvkpvkt07Qaaqmfpdvrf2Zlvvntrn8Jvfk8Ledu9Ptgwiaadhlctcaaaaabwwngwaaaiccem4Aaaaackq4Aqaaacgo4Qyaaacgoiqbaaaagiisbgaaaaaksrgbaaaakcjhbgaaakcghbsaaacaghjuaaaaaapkuaeaaaaokoegaaaaokcegwaaaiccem4Aaaaackq4Aqaaacgo4Qyaaacgoiqbaaaagiisbgaaaaaksrgbaaaakcjhbgaaakcghbsaaacaghjuaaaaaapkuaeaaaaokoegaaaaokcegwaaaiccem4Aaaaackq4Aqaaacgo4Qyaaacgop4Ogj3Q+Q4A3Qgaaaaasuvork5Cyii=

In this example, the trellis diagram represents a convolutional code with a constraint length of 2, meaning it takes into account the current input bit and the previous input bit to determine the encoded output. The nodes represent the possible states of the encoder, while the branches represent the possible transitions between states based on the input bit.

The branches are labeled with two numbers separated by a slash (/). The first number represents the input bit that leads to the transition, and the second number represents the encoded output bit associated with that transition.

Trellis Path for the state diagram:

Wgpjiqyhd6Z1Gaaaabjru5Erkjggg==

There are different types of convolutional codes based on their trellis structure. The most common types are:

  1. Rate 1/n Convolutional Codes: In this type, each input bit is encoded into n output bits. The trellis diagram for rate 1/2 convolutional code is shown above.
  2. Rate k/n Convolutional Codes: This type encodes k input bits into n output bits. It has a higher coding rate compared to rate 1/n codes. The trellis diagram for rate 2/3 convolutional code is an example of a rate k/n code.

Convolutional codes are widely used in various communication systems for error detection and correction. The trellis diagram provides a graphical representation that helps in understanding the encoding process and in designing decoding algorithms such as the Viterbi decoding algorithm.

Recall the Viterbi Decoding Algorithm

The Viterbi decoding algorithm is an algorithm used for decoding convolutional codes, which are commonly used in error correction systems. The algorithm aims to find the most likely sequence of transmitted bits given a received sequence of symbols that may contain errors.

Here’s a step-by-step explanation of the Viterbi decoding algorithm:

  1. Initialization:
    • Initialize the trellis, which represents the possible states and transitions of the convolutional code.
    • Assign a metric value of zero to the initial state and infinite metric values to all other states in the trellis.
    • Start with an empty path.
  2. Branch Metric Calculation:
    • Calculate the branch metric for each transition in the trellis. The branch metric represents the difference between the received symbol and the expected symbol based on the current state and transition.
  3. Path Metric Calculation:
    • For each state at each time step, calculate the path metric, which is the cumulative metric value obtained by adding the branch metric to the metric of the previous state in the previous time step.
    • Update the path metric for each state.
  4. Survivor Selection:
    • For each state at each time step, select the survivor path, which is the path with the lowest path metric.
    • Update the path metric and path history for each survivor path.
  5. Repeat Steps 2-4 for each received symbol:
    • Calculate branch metrics.
    • Calculate path metrics.
    • Select survivor paths.
  6. Traceback:
    • At the end of the received sequence, determine the most likely sequence of transmitted bits by performing a traceback operation.
    • Start at the final state and follow the path with the lowest path metric back through the trellis.
    • The sequence of bits obtained from the traceback operation represents the decoded output.

The Viterbi decoding algorithm utilizes dynamic programming principles to efficiently search through the trellis and find the most likely transmitted sequence. By considering the accumulated metrics and selecting survivor paths, it takes advantage of the structure of convolutional codes to correct errors and retrieve the original information.

It’s worth noting that the Viterbi decoding algorithm requires knowledge of the trellis structure and the specific characteristics of the convolutional code being used. The algorithm can be implemented in hardware or software to perform efficient decoding in various communication and error correction systems.

Describe Burst Error Correction Codes

Burst error correction codes are a type of error correction code that are specifically designed to correct errors that occur in bursts or runs of consecutive bits. They are widely used in communication systems where errors tend to occur in bursts, such as in satellite communication or in digital storage systems.

Burst error correction codes work by adding redundant bits to the message or signal to create a codeword that can withstand a certain number of errors. The redundant bits are calculated such that they can be used to identify and correct errors in the received codeword.

There are various methods for designing burst error correction codes, including block codes, convolutional codes, and Reed-Solomon codes.

Block codes are a type of error correction code that divide the message or signal into fixed-length blocks and add redundant bits to each block to create the codeword. They are particularly effective at correcting errors that occur in blocks of consecutive bits.

Convolutional codes are a type of error correction code that are generated by the convolution of the message or signal with a convolutional encoder. They are particularly effective at correcting errors that occur in bursts of consecutive bits.

Reed-Solomon codes are a type of error correction code that are based on polynomial division and are particularly effective at correcting errors that occur in bursts of consecutive bits.

Burst error correction codes are widely used in communication systems due to their ability to correct errors that occur in bursts of consecutive bits. They are an important tool for ensuring the reliability and integrity of transmitted messages or signals.

Describe Turbo codes

Turbo codes are a type of error correction code that are based on the principle of iterative decoding. They are widely used in communication systems due to their good error correction capabilities and their ability to operate at very low signal-to-noise ratios.

Turbo codes are constructed by combining two or more convolutional codes in a parallel structure, with interleavers between the convolutional codes. The codeword is produced by encoding the message or signal with the convolutional codes and interleavers.

To decode a received codeword, the Turbo decoder uses an iterative process to refine its estimate of the transmitted message or signal. It starts with an initial estimate of the message or signal and then uses the convolutional codes and interleavers to generate a soft-decision estimate of the message or signal. The soft-decision estimate is then used to update the initial estimate, and the process is repeated until the estimate of the message or signal converges.

Turbo codes are widely used in communication systems due to their good error correction capabilities and their ability to operate at very low signal-to-noise ratios. They are particularly useful for correcting errors in long messages or signals, as they can correct a large number of errors with a relatively small overhead.

Describe Alphanumeric Codes and its types

Alphanumeric codes are codes that represent alphabetic and numeric characters using a combination of bits. They are widely used in communication systems to represent text and other data in a compact and efficient manner.

There are several types of alphanumeric codes, including ASCII, Unicode, and EBCDIC.

ASCII (American Standard Code for Information Interchange) is a widely used alphanumeric code that represents alphabetic and numeric characters using 7 bits. It includes 128 different characters, including upper and lower case letters, digits, and punctuation marks. ASCII is the most common alphanumeric code used in communication systems.

Unicode is a standardized alphanumeric code that represents alphabetic and numeric characters from a wide range of scripts and languages using 16 or 32 bits. It includes over 128,000 different characters, making it suitable for representing a wide variety of languages and scripts.

EBCDIC (Extended Binary Coded Decimal Interchange Code) is an alphanumeric code that represents alphabetic and numeric characters using 8 bits. It is used primarily on IBM mainframe computers and is not as widely used as ASCII or Unicode.

Alphanumeric codes are an important tool for representing text and other data in communication systems. They allow for the efficient representation of characters and data and are widely used in a variety of applications.

Describe ASCII and EBCDIC Code

ASCII and EBCDIC are both character encoding schemes used to represent text characters in digital form. Here’s a brief description of each:

  1. ASCII (American Standard Code for Information Interchange): ASCII is a character encoding scheme that uses 7 bits to represent each character. This allows for a total of 128 possible characters, including uppercase and lowercase letters, digits, punctuation marks, and control characters such as newline and tab. ASCII was developed in the 1960s and is widely used in modern computing systems. ASCII codes are used to represent text in many file formats, including plain text files, HTML documents, and programming code.
  2. EBCDIC (Extended Binary Coded Decimal Interchange Code): EBCDIC is a character encoding scheme developed by IBM in the 1960s. It uses 8 bits to represent each character, allowing for a total of 256 possible characters. EBCDIC includes the same basic characters as ASCII, but also includes additional characters such as currency symbols and international characters. EBCDIC was widely used in IBM mainframe systems, but is less common in modern computing systems.

The key difference between ASCII and EBCDIC is the number of bits used to represent each character. ASCII uses 7 bits, while EBCDIC uses 8 bits. This means that ASCII can represent a smaller set of characters than EBCDIC, but is more widely used in modern computing systems.

Describe UNICODE and Baudot Code

Unicode is a standardised alphanumeric code that represents alphabetic and numeric characters from a wide range of scripts and languages using 16 or 32 bits. It includes over 128,000 different characters, making it suitable for representing a wide variety of languages and scripts. Unicode is widely used in communication systems and software applications to represent text and other data in a way that is independent of the specific language or script being used.

Baudot code is an alphanumeric code that represents alphabetic and numeric characters using 5 bits. It was developed in the 1870s and was originally used to transmit telegraph messages. Baudot code is not as widely used as ASCII, Unicode, or other modern alphanumeric codes.

Unicode and Baudot code are both alphanumeric codes that are used to represent alphabetic and numeric characters in communication systems. However, they differ in the number of bits used to represent each character and in the set of characters that they include. Unicode uses 16 or 32 bits to represent over 128,000 different characters, while Baudot code uses 5 bits to represent a smaller set of characters. Unicode is widely used in modern communication systems and software applications, while Baudot code is no longer in common use.