# Error Detection And Correction Codes In Digital Electronics Tutorial Pdf

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- Error Detection & Correction Codes
- Error Correction and Detection Codes
- Error Detection and Correction Code

## Error Detection & Correction Codes

The chapter gives an overview of the various types of errors encountered in a communication system. It discusses the various error detection and error correction codes. The role of polynomials in error detection and error correction is discussed in detail with the architecture for practical implementation of the codes in a communication channel. Coding Theory. Different types of errors are encountered during data transmission because of physical defects in the communication medium as well as environmental interference.

Environmental interference and physical defects in the communication medium can cause random bit errors during data transmission. Error coding is a method of detecting and correcting these errors to ensure that there are no errors in the information when it is sent from source to destination.

Data transmission errors occur in terrestrial mobile communication due to multipath fading, diffractions or scattering in cellular wireless communications, low signal-to-noise ratio, and limited transmitted power and energy resources in satellite communication [ 1 ]. Error coding uses mathematical formulae to encode data bits at the source into longer bit words for transmission.

The code word consists of extra bits, which provide redundancy, and at the destination, it will decode the data to find out whether the communication channel introduced any error and some schemes can even correct the errors so that there is no need to resend the data.

There are two ways to deal with errors. One way is to introduce redundant information along with the data to be transmitted, which will enable the receiver to deduce the information that has been transmitted. The second way is to include only enough redundancy to allow the receiver to detect that error has occurred, but not which error and the receiver makes a request for retransmission. Consider a frame having m data bits message to be sent and r redundant bits used for checking.

The number of bits in which the codewords are different is called Hamming distance. Suppose the code words are Hamming distance d- apart, it will require d single-bit errors to connect one code word to another.

The properties of error detection and error correction depend on the Hamming distance. Thus the error is detected at the receiver. When the data travels from the sender to receiver, different types of errors are encountered in the communication channel [ 2 ].

When the data travel through a conductor, there are different influences such as sound waves, electrical signals, noise such as electricity from motors, power switches, impulse noise, because of which data can be corrupted or destroyed. Old conductors are unable to handle these types of interference and heavy data traffic, hence the data transmission suffers.

Burst errors are large clumps of bits and occur when there are a number of inter-connected bit errors which occur at many places. These types of errors may occur because of some wrong placement in the data chain. It may contain several hundred or thousand-bit errors.

Data sent on a communication channel consists of thousands of data bits, sent in a particular order or sequence. However, there is a probability that the bits may be rearranged by accident in the transmission process. These types of errors are known as random bit errors. Cross talk occurs when the transmission cable through which the data is transmitted, is surrounded by other transmission lines. The data and code words, which are traveling in the neighboring line crosses over and gets superimposed on the transmission cable.

Echo is similar to cross talk; how ever, it occurs in a single transmission line, through which multiple computer ports are sending data at the same time. The data from one port will echo into another, thus resulting in data corruption Figure 1. Error detection uses additional bits in the message to be transmitted. This adds redundancy and facilitates detection and correction of errors. Popular techniques of error detection are, Simple parity check. This technique is most common and cheap mechanism for detection.

The data unit is appended with a redundant bit known as the parity bit. At the receiver end, the parity is computed of the block of data received and compared with the received parity bit.

Two-dimensional parity check improves the performance. Here, the data bits are organized in the form of a table, computed for each row as well each column and are sent along with the data. The parity is computed for the received data and compared with the received data bits. Two-dimension parity checking is mainly used to detect burst errors. It detects a burst error of more than n bits with a high probability.

However, this mechanism will not be able to detect the errors if two bits in one data unit are damaged. Example if is changed to and is changed to the error will not be detected.

This scheme divides the data bits to be sent into k segments. Each segment consists of m bits. Checksum is obtained by complementing the sum, and the data segments are transmitted together. The sum generated is complemented. The receiver accepts the data if the result of complementing is zero.

The checksum mechanism detects all errors consisting of odd number of bits. It also detects most errors having even number of bits. Cyclic redundancy check is the most powerful and easy to implement error detection mechanism. Checksum uses addition, whereas CRC is based on binary division. In CRC, the data unit is appended at the end by a sequence of redundant bits, called cyclic redundancy check bits. The resulting data unit is exactly divisible by a second, predetermined binary number.

At the receiver end, the incoming data unit is divided by the same predetermined binary number. If the remainder is zero, the data unit is assumed to be error-free and is accepted. A remainder indicates that the data unit has encountered an error in transit and therefore is rejected at the receiver.

The generalized technique to generate the CRC bits is explained below:. Consider there is a k bit message to be transmitted. The transmitter generates an r -bit sequence called as FCS frame check sequence.

The r -bit FCS is generated by dividing the k bit message, appended by r zeros, by a predetermined number. The r -bit FCS is generated as the remainder of binary division. If the remainder is zero, it means there was no error, and the frame is accepted by the receiver. Operations at both the sender and receiver end are shown in Figure 2. CRC is widely used in data communications, data storage, and data compression as a powerful method for detecting errors in the data. It is also used in testing of integrated circuits and the detection of logical faults.

A cyclic redundancy code is a non-secure hash function designed to detect accidental changes to raw computer data. CRCs are popular because they are simple to implement in binary hardware, are easy to analyze mathematically, and are particularly good at detecting common errors caused by noise in transmission channels. CRC guarantees Cyclic redundancy codes are a subset of cyclic [ 4 , 5 ] codes that are also a subset of linear block codes. They use a binary alphabet, 0 and 1. The CRC method treats the data frame as a large Binary number.

This number is then divided at the generator end by a fixed binary number the generator polynomial and the resulting CRC value, known as the FCS Frame Check Sequence , is appended to the end of the data frame and transmitted. The receiver divides the message including the calculated CRC , by the same polynomial used during transmission and compares its CRC value with the generated CRC value. If it does not match, the system requests for re-transmission of the data frame.

CRC codes are often used for error detection over frames or vectors of a certain length. The frame can be expressed as a polynomial in x, where the exponent of x is the place marker of the coefficient. The vector length L is represented by the degree L -1 polynomial. CRC coding is a generalization of the parity check bit. Parity bits are used for short vectors to detect one-bit error. However, if there are errors in two-bit positions, it will not detect the error. Let the data to be transmitted consist of a length k binary vector, and represent it by the degree k -1 polynomial.

Then, to add redundant bits, so the total length of the code word is n , we should add n-k bits. These redundant bits, which are the CRC bits, can be represented by the degree n-k- 1 polynomial. All coefficients of the polynomial are binary, and modulo-2 arithmetic is used [ 4 ]. To see how the receiver side can use this code word to detect errors, we first need to derive some properties of it. Let z x denote the quotient in the division hence, following is the data polynomial.

In modulo-2 arithmetic, addition and subtraction are alike, and the codeword polynomial can be written as. This gives rise to the following theorem [ 5 ]. A polynomial c x with deg. If c x is transmitted over a channel and there occur errors, they can be represented by an addition of the polynomial e x , and the received polynomial is. Thus g x is a factor of each transmitted codeword which can be used by the receiver to detect the error. The error is detected if g x is not a factor.

This quantity is known as Syndrome. The syndrome plays an important role in coding theory. CRC is a very effective and popular error detection technique.

The error detection capabilities of CRC depend on the chosen generator polynomial. CRC has capacity to detect all single-bit errors.

## Error Correction and Detection Codes

Hamming code is a set of error-correction codes that can be used to detect and correct the errors that can occur when the data is moved or stored from the sender to the receiver. It is technique developed by R. Hamming for error correction. Redundant bits are extra binary bits that are generated and added to the information-carrying bits of data transfer to ensure that no bits were lost during the data transfer. The number of redundant bits can be calculated using the following formula:. Parity bits are used for error detection.

## Error Detection and Correction Code

Transmitted data can be corrupted during communication. It is likely to be affected by external noise or other physical failures. In such a situation, the input data can't be the same as the output data. This mismatch is known as "Error. Most of the data transfer in digital systems will be in the form of 'Bit transfer.

During transmission, digital signals suffer from noise that can introduce errors in the binary bits travelling from sender to receiver. That means a 0 bit may change to 1 or a 1 bit may change to 0. To avoid this, we use error-detecting codes which are additional data added to a given digital message to help us detect if any error has occurred during transmission of the message.

In computing , telecommunication , information theory , and coding theory , an error correction code , sometimes error correcting code , ECC is used for controlling errors in data over unreliable or noisy communication channels. The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. The American mathematician Richard Hamming pioneered this field in the s and invented the first error-correcting code in the Hamming 7,4 code. ECC contrasts with error detection in that errors that are encountered can be corrected, not simply detected.

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