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82-422: In information theory , a low-density parity-check ( LDPC ) code is a linear error correcting code , a method of transmitting a message over a noisy transmission channel. An LDPC code is constructed using a sparse Tanner graph (subclass of the bipartite graph ). LDPC codes are capacity-approaching codes , which means that practical constructions exist that allow the noise threshold to be set very close to

164-433: A source of information. A memoryless source is one in which each message is an independent identically distributed random variable , whereas the properties of ergodicity and stationarity impose less restrictive constraints. All such sources are stochastic . These terms are well studied in their own right outside information theory. Information rate is the average entropy per symbol. For memoryless sources, this

246-641: A Reed-Solomon outer code. The DVB-S2, the DVB-T2 and the DVB-C2 standards all use a BCH code outer code to mop up residual errors after LDPC decoding. 5G NR uses polar code for the control channels and LDPC for the data channels. Although LDPC code has had its success in commercial hard disk drives, to fully exploit its error correction capability in SSDs demands unconventional fine-grained flash memory sensing, leading to an increased memory read latency. LDPC-in-SSD

328-467: A factor of four, the subcarrier spacing is reduced by the same factor. This introduces OFDM symbols that are four times longer: in 802.11ac, an OFDM symbol takes 3.2 microseconds to transmit. In 802.11ax, it takes 12.8 microseconds (both without guard intervals ). The 802.11ax amendment brings several key improvements over 802.11ac . 802.11ax addresses frequency bands between 1 GHz and 6 GHz. Therefore, unlike 802.11ac, 802.11ax also operates in

410-452: A higher spectral efficiency . 802.11ax Wi-Fi has a main feature called OFDMA , similar to how cell technology works with Wi-Fi . This brings better spectrum use, improved power control to avoid interference, and enhancements like 1024‑ QAM , MIMO and MU-MIMO for faster speeds. There are also reliability improvements such as lower power consumption and security protocols like Target Wake Time and WPA3 . The 802.11ax standard

492-448: A measurement in bytes per symbol, and a logarithm of base 10 will produce a measurement in decimal digits (or hartleys ) per symbol. Intuitively, the entropy H X of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known. The entropy of a source that emits a sequence of N symbols that are independent and identically distributed (iid)

574-405: A patent-free alternative of similar performance. Since then, advances in low-density parity-check codes have seen them surpass turbo codes in terms of error floor and performance in the higher code rate range, leaving turbo codes better suited for the lower code rates only. In 2003, an irregular repeat accumulate (IRA) style LDPC code beat six turbo codes to become the error-correcting code in

656-442: A random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon (Sh) as unit: The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: ( X , Y ) . This implies that if X and Y are independent , then their joint entropy is the sum of their individual entropies. For example, if ( X , Y ) represents

738-594: A signal; noise, periods of silence, and other forms of signal corruption often degrade quality. 802.11ax Wi-Fi 6 , or IEEE 802.11ax , is an IEEE standard from the Wi-Fi Alliance , for wireless networks ( WLANs ). It operates in the 2.4 GHz and 5 GHz bands, with an extended version, Wi-Fi 6E , that adds the 6 GHz band. It is an upgrade from Wi-Fi 5 ( 802.11ac ), with improvements for better performance in crowded places. Wi-Fi 6 covers frequencies in license-exempt bands between 1 and 7.125 GHz, including

820-400: A single random variable. Another useful concept is mutual information defined on two random variables, which describes the measure of information in common between those variables, which can be used to describe their correlation. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with

902-429: A sparse parity-check matrix . This sparse matrix is often randomly generated, subject to the sparsity constraints— LDPC code construction is discussed later . These codes were first designed by Robert Gallager in 1960. Below is a graph fragment of an example LDPC code using Forney's factor graph notation . In this graph, n variable nodes in the top of the graph are connected to ( n − k ) constraint nodes in

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984-447: A statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source. This division of coding theory into compression and transmission is justified by the information transmission theorems, or source–channel separation theorems that justify the use of bits as the universal currency for information in many contexts. However, these theorems only hold in

1066-486: A theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation W = K log m (recalling the Boltzmann constant ), where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley 's 1928 paper, Transmission of Information , uses

1148-511: A unit or scale or measure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers. Much of the mathematics behind information theory with events of different probabilities were developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs . Connections between information-theoretic entropy and thermodynamic entropy, including

1230-400: A zero in that position would satisfy the constraint. This procedure is then iterated. The new value for the fourth bit can now be used in conjunction with the first constraint to recover the first bit as seen below. This means that the first bit must be a one to satisfy the leftmost constraint. Thus, the message can be decoded iteratively. For other channel models, the messages passed between

1312-433: Is N ⋅ H bits (per message of N symbols). If the source data symbols are identically distributed but not independent, the entropy of a message of length N will be less than N ⋅ H . If one transmits 1000 bits (0s and 1s), and the value of each of these bits is known to the receiver (has a specific value with certainty) ahead of transmission, it is clear that no information is transmitted. If, however, each bit

1394-408: Is symmetric : Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) between the posterior probability distribution of X given the value of Y and the prior distribution on X : In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y . This is often recalculated as

1476-404: Is a way of comparing two distributions: a "true" probability distribution ⁠ p ( X ) {\displaystyle p(X)} ⁠ , and an arbitrary probability distribution ⁠ q ( X ) {\displaystyle q(X)} ⁠ . If we compress data in a manner that assumes ⁠ q ( X ) {\displaystyle q(X)} ⁠

1558-558: Is added to row 3. Step 3: Row 2 and 3 are swapped. Step 4: Row 1 is added to row 3. From this, the generator matrix G can be obtained as [ I k | P ] {\displaystyle {\begin{bmatrix}I_{k}|P\end{bmatrix}}} (noting that in the special case of this being a binary code P = − P {\displaystyle P=-P} ), or specifically: Finally, by multiplying all eight possible 3-bit strings by G , all eight valid codewords are obtained. For example,

1640-517: Is an effective approach to deploy LDPC in SSD with a very small latency increase, which turns LDPC in SSD into a reality. Since then, LDPC has been widely adopted in commercial SSDs in both customer-grades and enterprise-grades by major storage venders. Many TLC (and later) SSDs are using LDPC codes. A fast hard-decode (binary erasure) is first attempted, which can fall back into the slower but more powerful soft decoding. LDPC codes functionally are defined by

1722-466: Is defined as: It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose. The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding . Communications over a channel is the primary motivation of information theory. However, channels often fail to produce exact reconstruction of

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1804-467: Is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of X relative to Y is given by: where SI ( S pecific mutual Information) is the pointwise mutual information . A basic property of the mutual information is that That is, knowing Y , we can save an average of I ( X ; Y ) bits in encoding X compared to not knowing Y . Mutual information

1886-466: Is independently equally likely to be 0 or 1, 1000 shannons of information (more often called bits) have been transmitted. Between these two extremes, information can be quantified as follows. If X {\displaystyle \mathbb {X} } is the set of all messages { x 1 , ..., x n } that X could be, and p ( x ) is the probability of some x ∈ X {\displaystyle x\in \mathbb {X} } , then

1968-519: Is iterated until a valid codeword is achieved or decoding is exhausted. This type of decoding is often referred to as sum-product decoding. The decoding of the SPC codes is often referred to as the "check node" processing, and the cross-checking of the variables is often referred to as the "variable-node" processing. In a practical LDPC decoder implementation, sets of SPC codes are decoded in parallel to increase throughput. In contrast, belief propagation on

2050-435: Is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is: that is, the conditional entropy of a symbol given all the previous symbols generated. For the more general case of a process that is not necessarily stationary, the average rate is: that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result. The information rate

2132-454: Is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric). Another interpretation of the KL divergence is the "unnecessary surprise" introduced by a prior from the truth: suppose a number X is about to be drawn randomly from a discrete set with probability distribution ⁠ p ( x ) {\displaystyle p(x)} ⁠ . If Alice knows

2214-766: Is not symmetric. The I ( X n → Y n ) {\displaystyle I(X^{n}\to Y^{n})} measures the information bits that are transmitted causally from X n {\displaystyle X^{n}} to Y n {\displaystyle Y^{n}} . The Directed information has many applications in problems where causality plays an important role such as capacity of channel with feedback, capacity of discrete memoryless networks with feedback, gambling with causal side information, compression with causal side information, real-time control communication settings, and in statistical physics. Other important information theoretic quantities include

2296-429: Is the conditional mutual information I ( X 1 , X 2 , . . . , X i ; Y i | Y 1 , Y 2 , . . . , Y i − 1 ) {\displaystyle I(X_{1},X_{2},...,X_{i};Y_{i}|Y_{1},Y_{2},...,Y_{i-1})} . In contrast to mutual information, directed information

2378-474: Is the average conditional entropy over Y : Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use. A basic property of this form of conditional entropy is that: Mutual information measures the amount of information that can be obtained about one random variable by observing another. It

2460-458: Is the distribution underlying some data, when, in reality, ⁠ p ( X ) {\displaystyle p(X)} ⁠ is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus defined Although it is sometimes used as a 'distance metric', KL divergence is not a true metric since it

2542-581: Is used. A common unit of information is the bit or shannon , based on the binary logarithm . Other units include the nat , which is based on the natural logarithm , and the decimal digit , which is based on the common logarithm . In what follows, an expression of the form p log p is considered by convention to be equal to zero whenever p = 0 . This is justified because lim p → 0 + p log ⁡ p = 0 {\displaystyle \lim _{p\rightarrow 0+}p\log p=0} for any logarithmic base. Based on

Low-density parity-check code - Misplaced Pages Continue

2624-597: The Gilbert–Varshamov bound for linear codes over general fields. Impractical to implement when first developed by Gallager in 1963, LDPC codes were forgotten until his work was rediscovered in 1996. Turbo codes , another class of capacity-approaching codes discovered in 1993, became the coding scheme of choice in the late 1990s, used for applications such as the Deep Space Network and satellite communications . LDPC codes then received renewed interest as

2706-630: The Rényi entropy and the Tsallis entropy (generalizations of the concept of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information . Also, pragmatic information has been proposed as a measure of how much information has been used in making a decision. Coding theory is one of the most important and direct applications of information theory. It can be subdivided into source coding theory and channel coding theory. Using

2788-612: The Wi-Fi 802.11 standard as an optional part of 802.11n and 802.11ac , in the High Throughput (HT) PHY specification. LDPC is a mandatory part of 802.11ax (Wi-Fi 6). Some OFDM systems add an additional outer error correction that fixes the occasional errors (the "error floor") that get past the LDPC correction inner code even at low bit error rates . For example: The Reed-Solomon code with LDPC Coded Modulation (RS-LCM) uses

2870-416: The binary erasure channel is particularly simple where it consists of iterative constraint satisfaction. For example, consider that the valid codeword, 101011, from the example above, is transmitted across a binary erasure channel and received with the first and fourth bit erased to yield ?01?11. Since the transmitted message must have satisfied the code constraints, the message can be represented by writing

2952-470: The binary symmetric channel is an NP-complete problem, shown by reduction from 3-dimensional matching . So assuming P != NP , which is widely believed, then performing optimal decoding for an arbitrary code of any useful size is not practical. However, sub-optimal techniques based on iterative belief propagation decoding give excellent results and can be practically implemented. The sub-optimal decoding techniques view each parity check that makes up

3034-548: The forward error correction (FEC) system for the ITU-T G.hn standard. G.hn chose LDPC codes over turbo codes because of their lower decoding complexity (especially when operating at data rates close to 1.0 Gbit/s) and because the proposed turbo codes exhibited a significant error floor at the desired range of operation. LDPC codes are also used for 10GBASE-T Ethernet, which sends data at 10 gigabits per second over twisted-pair cables. As of 2009, LDPC codes are also part of

3116-605: The log is in base 2. In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him. Directed information , I ( X n → Y n ) {\displaystyle I(X^{n}\to Y^{n})} , is an information theory measure that quantifies the information flow from the random process X n = { X 1 , X 2 , … , X n } {\displaystyle X^{n}=\{X_{1},X_{2},\dots ,X_{n}\}} to

3198-473: The probability mass function of each source symbol to be communicated, the Shannon entropy H , in units of bits (per symbol), is given by where p i is the probability of occurrence of the i -th possible value of the source symbol. This equation gives the entropy in the units of "bits" (per symbol) because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called

3280-406: The shannon in his honor. Entropy is also commonly computed using the natural logarithm (base e , where e is Euler's number), which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas. Other bases are also possible, but less commonly used. For example, a logarithm of base 2 = 256 will produce

3362-532: The unit ban . The landmark event establishing the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948. Historian James Gleick rated the paper as the most important development of 1948, noting that

Low-density parity-check code - Misplaced Pages Continue

3444-580: The DVB-S2 rate 2/3 code the encoded block size is 64800 symbols (N=64800) with 43200 data bits (K=43200) and 21600 parity bits (M=21600). Each constituent code (check node) encodes 16 data bits except for the first parity bit which encodes 8 data bits. The first 4680 data bits are repeated 13 times (used in 13 parity codes), while the remaining data bits are used in 3 parity codes (irregular LDPC code). For comparison, classic turbo codes typically use two constituent codes configured in parallel, each of which encodes

3526-438: The LDPC as an independent single parity check (SPC) code. Each SPC code is decoded separately using soft-in-soft-out (SISO) techniques such as SOVA , BCJR , MAP , and other derivates thereof. The soft decision information from each SISO decoding is cross-checked and updated with other redundant SPC decodings of the same information bit. Each SPC code is then decoded again using the updated soft decision information. This process

3608-618: The LDPC concept in his doctoral dissertation at the Massachusetts Institute of Technology in 1960. However, LDPC codes require computationally expensive iterative decoding, so they went unused for decades. In 1993 the newly invented turbo codes demonstrated that codes with iterative decoding could far outperform other codes used at that time, but turbo codes were patented and required a fee for use. This raised renewed interest in LDPC codes, which were shown to have similar performance, but were much older and patent-free. Now that

3690-575: The ability of the Informed Dynamic Scheduling (IDS) algorithm to overcome trapping sets of near codewords. Information theory A key measure in information theory is entropy . Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process . For example, identifying the outcome of a fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying

3772-416: The analysis of music , art creation , imaging system design, study of outer space , the dimensionality of space , and epistemology . Information theory studies the transmission, processing, extraction, and utilization of information . Abstractly, information can be thought of as the resolution of uncertainty. In the case of communication of information over a noisy channel, this abstract concept

3854-480: The assertion: With it came the ideas of: Information theory is based on probability theory and statistics, where quantified information is usually described in terms of bits. Information theory often concerns itself with measures of information of the distributions associated with random variables. One of the most important measures is called entropy , which forms the building block of many other measures. Entropy allows quantification of measure of information in

3936-547: The bottom of the graph. This is a popular way of graphically representing an ( n ,  k ) LDPC code. The bits of a valid message, when placed on the T's at the top of the graph, satisfy the graphical constraints. Specifically, all lines connecting to a variable node (box with an '=' sign) have the same value, and all values connecting to a factor node (box with a '+' sign) must sum, modulo two, to zero (in other words, they must sum to an even number; or there must be an even number of odd values). Ignoring any lines going out of

4018-484: The channel capacity. These codes can be roughly subdivided into data compression (source coding) and error-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both codes and ciphers ). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis , such as

4100-411: The codeword for the bit-string '101' is obtained by: where ⊙ {\displaystyle \odot } is symbol of mod 2 multiplication. As a check, the row space of G is orthogonal to H such that G ⊙ H T = 0 {\displaystyle G\odot H^{T}=0} The bit-string '101' is found in as the first 3 bits of the codeword '101011'. During

4182-408: The commonly used 2.4 GHz and 5 GHz, as well as the broader 6 GHz band . This standard aims to boost data speed ( throughput -per-area ) in crowded places like offices and malls. Though the nominal data rate is only 37% better than 802.11ac, the total network speed increases by 300%, making it more efficient and reducing latency by 75%. The quadrupling of overall throughput is made possible by

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4264-412: The design of well performing low rate codes is easier for turbo codes. As a practical matter, the hardware that forms the accumulators is reused during the encoding process. That is, once a first set of parity bits are generated and the parity bits stored, the same accumulator hardware is used to generate a next set of parity bits. As with other codes, the maximum likelihood decoding of an LDPC code on

4346-536: The divergence from the product of the marginal distributions to the actual joint distribution: Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ test : mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution. The Kullback–Leibler divergence (or information divergence , information gain , or relative entropy )

4428-478: The encoding of a frame, the input data bits (D) are repeated and distributed to a set of constituent encoders. The constituent encoders are typically accumulators and each accumulator is used to generate a parity symbol. A single copy of the original data (S 0,K-1 ) is transmitted with the parity bits (P) to make up the code symbols. The S bits from each constituent encoder are discarded. The parity bit may be used within another constituent code. In an example using

4510-498: The entire input block (K) of data bits. These constituent encoders are recursive convolutional codes (RSC) of moderate depth (8 or 16 states) that are separated by a code interleaver which interleaves one copy of the frame. The LDPC code, in contrast, uses many low depth constituent codes (accumulators) in parallel, each of which encode only a small portion of the input frame. The many constituent codes can be viewed as many low depth (2 state) " convolutional codes " that are connected via

4592-490: The entropy, H , of X is defined: (Here, I ( x ) is the self-information , which is the entropy contribution of an individual message, and E X {\displaystyle \mathbb {E} _{X}} is the expected value .) A property of entropy is that it is maximized when all the messages in the message space are equiprobable p ( x ) = 1/ n ; i.e., most unpredictable, in which case H ( X ) = log n . The special case of information entropy for

4674-401: The first 3 bits of the codeword. While illustrative, this erasure example does not show the use of soft-decision decoding or soft-decision message passing, which is used in virtually all commercial LDPC decoders. In recent years, there has also been a great deal of work spent studying the effects of alternative schedules for variable-node and constraint-node update. The original technique that

4756-443: The fundamental patent for turbo codes has expired (on August 29, 2013), LDPC codes are still used for their technical merits. LDPC codes have been shown to have ideal combinatorial properties. In his dissertation, Gallager showed that LDPC codes achieve the Gilbert–Varshamov bound for linear codes over binary fields with high probability. In 2020 it was shown that Gallager's LDPC codes achieve list decoding capacity and also achieve

4838-416: The given distribution can be reliably compressed. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution. The choice of logarithmic base in the following formulae determines the unit of information entropy that

4920-487: The important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory . In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with

5002-473: The interference between clients is reduced, and the overall throughput is increased, since multiple clients can receive data simultaneously. With 802.11ax, a similar multiplexing is introduced in the frequency domain : OFDMA . With OFDMA, multiple clients are assigned to different Resource Units in the available spectrum. By doing so, an 80 MHz channel can be split into multiple Resource Units, so that multiple clients receive different types of data over

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5084-403: The limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent. Coding theory is concerned with finding explicit methods, called codes , for increasing the efficiency and reducing the error rate of data communication over noisy channels to near

5166-459: The most are the ones that need to be updated first. Highly reliable nodes, whose log-likelihood ratio (LLR) magnitude is large and does not change significantly from one update to the next, do not require updates with the same frequency as other nodes, whose sign and magnitude fluctuate more widely. These scheduling algorithms show greater speed of convergence and lower error floors than those that use flooding. These lower error floors are achieved by

5248-415: The new DVB-S2 standard for digital television . The DVB-S2 selection committee made decoder complexity estimates for the turbo code proposals using a much less efficient serial decoder architecture rather than a parallel decoder architecture. This forced the turbo code proposals to use frame sizes on the order of one half the frame size of the LDPC proposals. In 2008, LDPC beat convolutional turbo codes as

5330-613: The outcome from a roll of a die (which has six equally likely outcomes). Some other important measures in information theory are mutual information , channel capacity , error exponents , and relative entropy . Important sub-fields of information theory include source coding , algorithmic complexity theory , algorithmic information theory and information-theoretic security . Applications of fundamental topics of information theory include source coding/ data compression (e.g. for ZIP files ), and channel coding/ error detection and correction (e.g. for DSL ). Its impact has been crucial to

5412-576: The paper was "even more profound and more fundamental" than the transistor . He came to be known as the "father of information theory". Shannon outlined some of his initial ideas of information theory as early as 1939 in a letter to Vannevar Bush . Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs , all implicitly assuming events of equal probability. Harry Nyquist 's 1924 paper, Certain Factors Affecting Telegraph Speed , contains

5494-651: The picture, the parity-check matrix representing this graph fragment is In this matrix, each row represents one of the three parity-check constraints, while each column represents one of the six bits in the received codeword. In this example, the eight codewords can be obtained by putting the parity-check matrix H into this form [ − P T | I n − k ] {\displaystyle {\begin{bmatrix}-P^{T}|I_{n-k}\end{bmatrix}}} through basic row operations in GF(2) : Step 1: H. Step 2: Row 1

5576-438: The picture, there are eight possible six-bit strings corresponding to valid codewords: (i.e., 000000, 011001, 110010, 101011, 111100, 100101, 001110, 010111). This LDPC code fragment represents a three-bit message encoded as six bits. Redundancy is used, here, to increase the chance of recovering from channel errors. This is a (6, 3) linear code , with n  = 6 and k  = 3. Again ignoring lines going out of

5658-433: The position of a chess piece— X the row and Y the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece. Despite similar notation, joint entropy should not be confused with cross-entropy . The conditional entropy or conditional uncertainty of X given random variable Y (also called the equivocation of X about Y )

5740-450: The random process Y n = { Y 1 , Y 2 , … , Y n } {\displaystyle Y^{n}=\{Y_{1},Y_{2},\dots ,Y_{n}\}} . The term directed information was coined by James Massey and is defined as where I ( X i ; Y i | Y i − 1 ) {\displaystyle I(X^{i};Y_{i}|Y^{i-1})}

5822-444: The received message on the top of the factor graph. In this example, the first bit cannot yet be recovered, because all of the constraints connected to it have more than one unknown bit. In order to proceed with decoding the message, constraints connecting to only one of the erased bits must be identified. In this example, only the second constraint suffices. Examining the second constraint, the fourth bit must have been zero, since only

5904-475: The repeat and distribute operations. The repeat and distribute operations perform the function of the interleaver in the turbo code. The ability to more precisely manage the connections of the various constituent codes and the level of redundancy for each input bit give more flexibility in the design of LDPC codes, which can lead to better performance than turbo codes in some instances. Turbo codes still seem to perform better than LDPCs at low code rates, or at least

5986-407: The same spectrum, simultaneously. To support OFDMA , 802.11ax needs four times as many subcarriers as 802.11ac. Specifically, for 20, 40, 80, and 160 MHz channels, the 802.11ac standard has, respectively, 64, 128, 256 and 512 subcarriers while the 802.11ax standard has 256, 512, 1024, and 2048 subcarriers. Since the available bandwidths have not changed and the number of subcarriers increases by

6068-417: The situation where one transmitting user wishes to communicate to one receiving user. In scenarios with more than one transmitter (the multiple-access channel), more than one receiver (the broadcast channel ) or intermediary "helpers" (the relay channel ), or more general networks , compression followed by transmission may no longer be optimal. Any process that generates successive messages can be considered

6150-747: The success of the Voyager missions to deep space, the invention of the compact disc , the feasibility of mobile phones and the development of the Internet and artificial intelligence . The theory has also found applications in other areas, including statistical inference , cryptography , neurobiology , perception , signal processing , linguistics , the evolution and function of molecular codes ( bioinformatics ), thermal physics , molecular dynamics , black holes , quantum computing , information retrieval , intelligence gathering , plagiarism detection , pattern recognition , anomaly detection ,

6232-492: The theoretical maximum (the Shannon limit ) for a symmetric memoryless channel. The noise threshold defines an upper bound for the channel noise, up to which the probability of lost information can be made as small as desired. Using iterative belief propagation techniques, LDPC codes can be decoded in time linear in their block length. LDPC codes are also known as Gallager codes , in honor of Robert G. Gallager , who developed

6314-460: The true distribution ⁠ p ( x ) {\displaystyle p(x)} ⁠ , while Bob believes (has a prior ) that the distribution is ⁠ q ( x ) {\displaystyle q(x)} ⁠ , then Bob will be more surprised than Alice, on average, upon seeing the value of X . The KL divergence is the (objective) expected value of Bob's (subjective) surprisal minus Alice's surprisal, measured in bits if

6396-459: The variable nodes and check nodes are real numbers , which express probabilities and likelihoods of belief. This result can be validated by multiplying the corrected codeword r by the parity-check matrix H : Because the outcome z (the syndrome ) of this operation is the three × one zero vector, the resulting codeword r is successfully validated. After the decoding is completed, the original message bits '101' can be extracted by looking at

6478-425: The word information as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as H = log S = n log S , where S was the number of possible symbols, and n the number of symbols in a transmission. The unit of information was therefore the decimal digit , which since has sometimes been called the hartley in his honor as

6560-514: Was approved on September 1, 2020, with Draft 8 getting 95% approval. Subsequently, on February 1, 2021, the standard received official endorsement from the IEEE Standards Board. Notes In 802.11ac (802.11's previous amendment), multi-user MIMO was introduced, which is a spatial multiplexing technique. MU-MIMO allows the access point to form beams towards each client , while transmitting information simultaneously. By doing so,

6642-419: Was formalized in 1948 by Claude Shannon in a paper entitled A Mathematical Theory of Communication , in which information is thought of as a set of possible messages, and the goal is to send these messages over a noisy channel, and to have the receiver reconstruct the message with low probability of error, in spite of the channel noise. Shannon's main result, the noisy-channel coding theorem , showed that, in

6724-433: Was used for decoding LDPC codes was known as flooding . This type of update required that, before updating a variable node, all constraint nodes needed to be updated and vice versa. In later work by Vila Casado et al. , alternative update techniques were studied, in which variable nodes are updated with the newest available check-node information. The intuition behind these algorithms is that variable nodes whose values vary

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