Parity bit error detection and correction techniques
In the previous section, we noted that bit-level error detection and correction — detecting and correcting the corruption of bits in a data-link-layer frame sent from one node to another physically-connected neighboring node — are two services often provided by the data link layer. We saw in Chapter 3 that error detection and correction services are also often offered at the transport layer as well.
In this section, we'll examine a few of the simplest techniques that can be used to detect and, in some cases, correct such bit errors. A full treatment of the theory and implementation of this topic is itself the topic of many textbooks e.
Our goal here is to develop an intuitive feel for the capabilities that error detection and correction techniques provide, and to see how a few simple techniques work and are used in practice in the data link layer. At the sending node, data, Dto be "protected" against parity bit error detection and correction techniques errors is augmented with error detection and correction bits, EDC.
Typically, the data to be protected includes not only the datagram passed down from the network layer for transmission across the link, but also link-level addressing information, sequence numbers, and other fields in the data link frame header.
Both D parity bit error detection and correction techniques EDC are sent to the receiving node in a link-level frame. At the receiving node, a sequence of bits, D ' and EDC ' are received. Error detection and correction scenario. The receiver's challenge is to determine whether or not D ' is the same as the original Dgiven that it has only received D ' and EDC '. The exact wording of the receiver's decision in Figure 5.
Error detection and correction techniques allow the receiver to sometimes, but not alwaysdetect that bit errors have occurred. That is, even with the use of error detection bits there will still be a possibility that undetected bit errors will occur, i. As a consequence, the receiver might deliver a corrupted datagram to the network layer, or be unaware that the contents of some other field in the frame's header have been corrupted. We thus want to choose an error detection scheme so that the probability of such occurrences is small.
Generally, more sophisticated error detection and correction techniques i. Let's now examine three techniques for detecting errors in the transmitted data -- parity checks to illustrate the basic parity bit error detection and correction techniques behind error detection and correctionchecksumming methods which are more typically employed in the transport layer and cyclic redundancy checks which are typically employed in the data link layer.
Perhaps the simplest form of error detection is the use of a single parity bit. Suppose that the information to be sent, D in Figure 5. For odd parity schemes, the parity bit value is chosen such that there are an odd number of 1's. Receiver operation is also simple with a single parity bit. If an odd number of 1-valued bits are found with an even parity scheme, the receiver knows that at least one bit error has occurred.
More precisely, it knows that some odd number of bit errors have occurred. But what happens if an even number of bit errors occur?
You should convince yourself that this would result in an undetected error. If the probability of bit errors is small and errors can be assumed to occur independently from one bit to the next, the probability of multiple bit errors in a packet would be extremely small. In this case, a single parity bit might suffice. However, measurements have shown that rather than occurring independently, errors are often clustered together in "bursts.
Clearly, a more robust error detection scheme is needed and, fortunately, is used in practice! But before examining error detection schemes that are used in practice, let's parity bit error detection and correction techniques a simple generalization of one-bit parity that will provide us with insight into error correction techniques.
Here, the d bits in D are divided into i rows and j columns. A parity value is computed for each row and for each column. Suppose now that a single bit error occurs in the original d bits of information. With this two-dimensional parity scheme, the parity of both the column and the row containing the flipped bit will be in error.
The receiver can thus not only detect the fact that a single bit error has occurred, but can use the column and row indices of the column and row with parity errors to parity bit error detection and correction techniques identify the bit that was corrupted and correct that error!
Although our discussion has focussed on the original d bits of information, a single error in the parity bits themselves is also detectable and correctable. Two dimensional parity can also detect but not correct! Other properties of the two-dimensional parity scheme are explored in the problems at the end of the chapter.
The ability of the receiver to both detect and correct errors is known as forward error correction FEC. These techniques are commonly used in audio storage and playback devices such as audio CD's. FEC techniques are valuable because they can decrease the number of sender retransmissions required. Perhaps more importantly, they allow for immediate correction of errors at the receiver. This avoids having to wait the round-trip propagation delay needed for the sender to receive a NAK packet and for the retransmitted packet to propagate back to the receiver -- a potentially important advantage for real-time network applications [ Rubenstein ].
Recent work examining the use of FEC in error control protocols include [ BiersackNonnenmacherByersShacham ]. In checksumming techniques, the d bits of data in Figure 5. One parity bit error detection and correction techniques checksumming method is to simply sum these k -bit integers and use the resulting sum as the error detection bits.
The so-called Internet checksum [ RFC ] is based on this approach -- bytes of data are treated as bit integers and their ones-complement sum forms the Internet checksum.
A receiver calculates the checksum it calculates over the received data and checks whether it matches the checksum carried in the received packet. In other protocols, e. McAuley [ McAuley ] describe improved weighted checksum codes that are suitable for high-speed software implementation and Feldmeier [ Feldmeier ] presents fast software implementation techniques for not only weighted checksum codes, but CRC see below and other codes as well.
An error detection technique used widely in today's computer networks is based on cyclic redundancy check CRC codes. CRC codes are also known as polynomial codessince it is possible parity bit error detection and correction techniques view the bit string to be sent as a polynomial whose coefficients are the 0 and 1 values in the bit string, with operations on the bit string interpreted as polynomial arithmetic.
CRC codes operate as follows. Consider the d-bit piece of data, Dthat the sending node wants to send to the receiving node. We will require that the most significant leftmost bit of G be a 1. The key idea behind CRC codes is shown in Figure 5. The process of error checking with CRC's is thus simple: If the remainder is non-zero, the receiver knows that an error has occurred; otherwise the data is accepted as being correct.
All CRC calculations are done in modulo 2 arithmetic without carries in addition or borrows in subtraction. This means that addition and subtraction are identical, and both are equivalent to the bitwise exclusive-or XOR of the operands. Multiplication and division are the same as in base 2 arithmetic, except that any required addition or subtraction is done without carries or borrows. As in regular binary arithmetic, multiplication by 2 k left shifts a bit pattern by k places.
Let us now turn to the crucial question of how the sender computes R. Recall that we want to find R such that there is an n such that.
If we exclusive-or parity bit error detection and correction techniques. In other words, we can calculate R as. An example CRC calculation. The nine bits transmitted in this case are International standards have been defined for 8-,and bit generators, G. The theory behind CRC codes and even more powerful codes is beyond the scope of this text. The text [ Schwartz ] provides an excellent introduction parity bit error detection and correction techniques this topic.
Mitzenmacher, A Rege, "A digital fountain approach to reliable distribution of bulk data," Proc.
In information theory and coding theory with applications in computer science and telecommunicationerror detection and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noiseand parity bit error detection and correction techniques errors may be introduced during transmission from the source parity bit error detection and correction techniques a receiver.
Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many parity bit error detection and correction techniques. The modern development of error-correcting codes in is due to Richard W. The general idea for achieving error detection and correction is to add some redundancy i.
Error-detection and correction schemes can be either systematic or non-systematic: In a systematic scheme, the transmitter sends the original data, and attaches a fixed number of check bits or parity datawhich are derived from the data bits by some deterministic algorithm. If only error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission.
In a system that uses a parity bit error detection and correction techniques code, the original message is transformed into an encoded message that has at least as many bits as the original message. Good error control performance requires the scheme to be selected based on the characteristics of the communication channel.
Common channel models include memory-less models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in bursts. Some codes can also be suitable for a mixture of random errors and burst errors. If the channel capacity cannot be determined, or is highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as automatic repeat request ARQand is most notably used in the Internet.
An alternate approach for error control is hybrid automatic repeat request HARQwhich is a combination of ARQ and error-correction coding.
ARQ and FEC may be combined, such that minor errors are corrected without retransmission, and major errors are corrected via a request for retransmission: Error detection is most commonly realized using a suitable hash function or checksum algorithm.
A hash function adds a fixed-length tag to parity bit error detection and correction techniques message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided. There exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors e.
A random-error-correcting code based on minimum distance coding can provide a strict guarantee on the number of detectable errors, but it may not protect against a preimage attack.
A repetition code, described in the section below, is a special case of error-correcting code: A repetition code is a coding scheme that repeats the bits across a channel to achieve error-free communication.
Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to parity bit error detection and correction techniques the bit pattern "", the four-bit block can be repeated three times, thus producing " ".
However, if this twelve-bit pattern was received as " " — where the first block is unlike the other two — it can be determined that an error has occurred. A repetition code is very inefficient, and can be susceptible to problems if the error occurs in exactly the same place for each group e.
The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of numbers stations. A parity bit is a bit that is added to a group of source bits to ensure that the number of set bits i. It is a very simple scheme that can be used to detect single or any other odd number i. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous.
Extensions and variations on the parity bit mechanism are horizontal redundancy checksvertical redundancy checksand "double," "dual," or "diagonal" parity used in RAID-DP. A checksum of a message is a modular arithmetic sum of message code words of a fixed word length e. The sum may be negated by means of a ones'-complement operation prior to transmission to detect errors resulting in all-zero messages. Checksum schemes include parity bitscheck digitsand longitudinal redundancy checks.
Some checksum schemes, such as the Damm algorithmthe Luhn algorithmand the Verhoeff algorithmare specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers.
A cyclic redundancy check CRC is a non-secure hash function designed to detect accidental changes to digital data parity bit error detection and correction techniques computer networks; as a result, it is not suitable for detecting maliciously introduced errors. It is characterized by specification of what is called a generator polynomialwhich is used as the divisor in a polynomial long division over a finite fieldtaking the input data as the dividendsuch that the remainder becomes the result.
A cyclic code has favorable properties that make it well suited for detecting burst errors. CRCs are particularly easy to implement in hardware, and are therefore commonly used in digital networks and storage devices such as hard disk drives. The output of a cryptographic hash functionalso known as a message digestcan provide strong assurances about data integritywhether changes of the data are accidental e.
Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is infeasible to find some input data other than the one given that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a keyed hash or message authentication code MAC can be used for additional security.
Without knowing the key, it is not possible for the attacker easily or conveniently calculate the correct keyed hash value for a modified message.
Any error-correcting code can be used for error detection. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired. The parity bit is an example of a single-error-detecting code. An acknowledgment is a message sent by the receiver to indicate that it has correctly received a data frame.
Usually, when the transmitter does not receive the acknowledgment before the timeout occurs i. ARQ is appropriate if the communication channel has varying or unknown capacitysuch as is the case on the Internet.
However, ARQ requires the availability of a back channelresults in possibly increased latency due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of network congestion can put a strain on the server and overall network capacity.
An parity bit error detection and correction techniques code ECC or forward error correction FEC code is a parity bit error detection and correction techniques of adding redundant data, parity bit error detection and correction techniques parity datato a message, such that it can be recovered by a receiver even when a number of errors up to parity bit error detection and correction techniques capability of the code being used were introduced, either during the process of transmission, or on storage.
Since the receiver does not have to ask the sender for retransmission of the data, a backchannel is not required in forward error correction, and it is therefore suitable for simplex communication such as broadcasting.
Error-correcting codes are frequently used in lower-layer communication, as well as for reliable storage in media such as CDsDVDshard disksand RAM. Error-correcting codes are usually distinguished between convolutional codes and block codes:. Shannon's theorem is an important theorem in forward error correction, and describes the maximum information rate at which reliable communication is possible over a channel that has a certain error probability or signal-to-noise ratio SNR.
This strict upper limit is expressed in terms of the channel capacity. More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a discrete memoryless channel can be made arbitrarily small, provided that the code rate is smaller than the channel capacity. The actual maximum code rate allowed depends on the error-correcting code used, and may be lower.
This is because Shannon's proof was only of existential nature, and did not show how to construct codes which are both optimal and have efficient encoding and decoding algorithms. There are two basic approaches: The latter approach is particularly attractive on an erasure channel when using a rateless erasure code. By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be any good.
Applications where the transmitter immediately parity bit error detection and correction techniques the information as soon as it is sent such as most television cameras cannot use ARQ ; they must use FEC because when an error occurs, the original data is no longer available.
Applications that require extremely low parity bit error detection and correction techniques rates such as digital money transfers must use ARQ. Reliability and inspection engineering also make use of the theory of error-correcting codes. Development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution parity bit error detection and correction techniques signal power over interplanetary distances, and the limited power availability aboard parity bit error detection and correction techniques probes.
Whereas early missions sent their data uncoded, starting from digital error correction was implemented in the form of sub-optimally decoded convolutional codes and Reed—Muller codes. The Voyager 1 and Voyager 2 missions, which started inwere designed to deliver color imaging amongst scientific information of Jupiter and Saturn. The Voyager 2 craft additionally supported an implementation of a Reed—Solomon code: Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as Turbo codes or LDPC codes.
The different kinds of deep space and orbital missions that are conducted suggest that trying to find a "one size fits all" error correction system will be an ongoing problem for some time to come.
For missions close to Earth the nature of the noise in the communication channel is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise gets larger.
The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television including new channels and High Definition TV and IP data. Transponder availability and bandwidth constraints have limited this growth, because transponder capacity is determined by the selected modulation scheme and Forward error correction FEC rate.
Error detection and correction codes are often used to improve the reliability of data storage media. The "Optimal Rectangular Code" used in group coded recording tapes not only detects but also corrects single-bit errors.
Reed Solomon codes are used in compact discs to correct errors caused by scratches. Modern hard drives use CRC codes to detect and Reed—Solomon codes to correct minor errors in sector reads, and to recover data from sectors that have "gone bad" and store that data in the spare sectors.
Filesystems such as ZFS or Btrfsas well as some RAID implementations, support data scrubbing and resilvering, which allows bad blocks to be detected and hopefully recovered before they are used.
The recovered data may be re-written to exactly the parity bit error detection and correction techniques physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware. DRAM memory may provide increased protection against soft errors by relying on error correcting codes.
Such error-correcting memoryknown as ECC or EDAC-protected memory, is particularly desirable for high fault-tolerant applications, such as servers, as well as deep-space applications due to increased radiation. Error-correcting memory controllers traditionally use Hamming codesalthough some use triple modular redundancy.
Interleaving allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a single event upset SEU does not exceed the error threshold e. In addition to hardware providing features required for ECC memory to operate, operating systems usually contain parity bit error detection and correction techniques reporting facilities that are used to provide notifications when soft errors are transparently recovered.
An increasing rate of soft errors might indicate that a DIMM module needs replacing, and such feedback information would not be easily available without the related reporting capabilities.
One example is the Linux kernel 's EDAC subsystem previously known as bluesmokewhich collects the data from error-checking-enabled components inside a computer system; beside collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the PCI bus. Parity bit error detection and correction techniques few systems also support memory scrubbing.
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