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82-449: The reflected binary code ( RBC ), also known as reflected binary ( RB ) or Gray code after Frank Gray , is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representation of the decimal value "1" in binary would normally be " 001 " and "2" would be " 010 ". In Gray code, these values are represented as " 001 " and " 011 ". That way, incrementing

164-401: A 0 bit leaves the order of the code words unchanged, prepending a 1 bit reverses the order of the code words. If the bits at position i {\displaystyle i} of codewords are inverted, the order of neighbouring blocks of 2 i {\displaystyle 2^{i}} codewords is reversed. For example, if bit 0 is inverted in a 3 bit codeword sequence,

246-422: A digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause

328-552: A soft-decision algorithm to demodulate digital data from an analog signal corrupted by noise. Many FEC decoders can also generate a bit-error rate (BER) signal which can be used as feedback to fine-tune the analog receiving electronics. FEC information is added to mass storage (magnetic, optical and solid state/flash based) devices to enable recovery of corrupted data, and is used as ECC computer memory on systems that require special provisions for reliability. The maximum proportion of errors or missing bits that can be corrected

410-527: A Gray code into a binary number can be described as a prefix sum of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two. To construct the binary-reflected Gray code iteratively, at step 0 start with the c o d e 0 = 0 {\displaystyle \mathrm {code} _{0}={\mathtt {0}}} , and at step i > 0 {\displaystyle i>0} find

492-440: A binary value into the corresponding Gray code. Each bit is inverted if the next higher bit of the input value is set to one. This can be performed in parallel by a bit-shift and exclusive-or operation if they are available: the n th Gray code is obtained by computing n ⊕ ⌊ n 2 ⌋ {\displaystyle n\oplus \left\lfloor {\tfrac {n}{2}}\right\rfloor } . Prepending

574-400: A binary-reflected Gray code (BRGC). However, mathematicians have discovered other kinds of Gray codes. Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a Hamming distance of 1 from the next word). It is possible to construct binary Gray codes with n bits with a length of less than 2 , if the length is even. One possibility

656-468: A constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise . Digital logic designers use Gray codes extensively for passing multi-bit count information between synchronous logic that operates at different clock frequencies. The logic is considered operating in different "clock domains". It is fundamental to the design of large chips that operate with many different clocking frequencies. If

738-414: A dual-port FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains. The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each domain. Each bit of the pointers is sampled non-deterministically for this clock domain transfer. So for each bit, either

820-515: A fixed ECC method as long as the ECC can handle the error rate, then switch to ARQ when the error rate gets too high; adaptive modulation and coding uses a variety of ECC rates, adding more error-correction bits per packet when there are higher error rates in the channel, or taking them out when they are not needed. The two main categories of ECC codes are block codes and convolutional codes . There are many types of block codes; Reed–Solomon coding

902-913: A hard decision is made whether it corresponds to a one or a zero bit. In contrast, convolutional codes are typically decoded using soft-decision algorithms like the Viterbi, MAP or BCJR algorithms, which process (discretized) analog signals, and which allow for much higher error-correction performance than hard-decision decoding. Nearly all classical block codes apply the algebraic properties of finite fields . Hence classical block codes are often referred to as algebraic codes. In contrast to classical block codes that often specify an error-detecting or error-correcting ability, many modern block codes such as LDPC codes lack such guarantees. Instead, modern codes are evaluated in terms of their bit error rates. Most forward error correction codes correct only bit-flips, but not bit-insertions or bit-deletions. In this setting,

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984-404: A large code-rate close to 1 implies a weak code. The redundant bits that protect the information have to be transferred using the same communication resources that they are trying to protect. This causes a fundamental tradeoff between reliability and data rate. In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR (signal-to-noise-ratio) decreasing

1066-443: A more uniform distribution of errors. Therefore, interleaving is widely used for burst error-correction . The analysis of modern iterated codes, like turbo codes and LDPC codes , typically assumes an independent distribution of errors. Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word. For turbo codes, an interleaver is an integral component and its proper design

1148-402: A pattern of 4 on, 4 off; the i -th least significant bit a pattern of 2 on 2 off. The most significant digit is an exception to this: for an n -bit Gray code, the most significant digit follows the pattern 2 on, 2 off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2 places. The four-bit version of this is shown below: For decimal 15

1230-457: A signal is close to a codeword by only looking at a small number of positions of the signal. Not all testing codes are locally decoding and testing of codes Not all locally decodable codes (LDCs) are locally testable codes (LTCs) neither locally correctable codes (LCCs), q-query LCCs are bounded exponentially while LDCs can have subexponential lengths. Interleaving is frequently used in digital communication and storage systems to improve

1312-466: A single bit-change can cause a big leap and lead to new properties. Gray codes are also used in labelling the axes of Karnaugh maps since 1953 as well as in Händler circle graphs since 1958, both graphical methods for logic circuit minimization . In modern digital communications , 1D- and 2D-Gray codes play an important role in error prevention before applying an error correction . For example, in

1394-540: A system has to cycle sequentially through all possible combinations of on-off states of some set of controls, and the changes of the controls require non-trivial expense (e.g. time, wear, human work), a Gray code minimizes the number of setting changes to just one change for each combination of states. An example would be testing a piping system for all combinations of settings of its manually operated valves. A balanced Gray code can be constructed, that flips every bit equally often. Since bit-flips are evenly distributed, this

1476-471: A time, faster algorithms exist. On newer processors, the number of ALU instructions in the decoding step can be reduced by taking advantage of the CLMUL instruction set . If MASK is the constant binary string of ones ended with a single zero digit, then carryless multiplication of MASK with the grey encoding of x will always give either x or its bitwise negation. In practice, "Gray code" almost always refers to

1558-424: A time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in

1640-418: A time, so there is never any ambiguity of position, resulting in codes assigning to each of a contiguous set of integers , or to each member of a circular list, a word of symbols such that no two code words are identical and each two adjacent code words differ by exactly one symbol. These codes are also known as unit-distance , single-distance , single-step , monostrophic or syncopic codes , in reference to

1722-403: A two-way mechanically scanned TV system in 1930. With Pierre Mertz, Gray wrote the classic paper on the mathematics of raster scan systems in 1934. He later participated in the early days of the digital revolution, with Raymond W. Sears, William M. Goodall, John Robinson Pierce , and others at Bell Labs, by providing the binary code used by Sears in his PCM tube , a beam deflection tube of

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1804-567: A value from 1 to 2 requires only one bit to change, instead of two. Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice. Many devices indicate position by closing and opening switches. If that device uses natural binary codes , positions 3 and 4 are next to each other but all three bits of

1886-405: Is G 1  = ( 0,1 ). This can be thought of as built recursively as above from a zero-bit Gray code G 0  = (  Λ  ) consisting of a single entry of zero length. This iterative process of generating G n +1 from G n makes the following properties of the standard reflecting code clear: These characteristics suggest a simple and fast method of translating

1968-409: Is a relatively inefficient ECC. Better ECC codes typically examine the last several tens or even the last several hundreds of previously received bits to determine how to decode the current small handful of bits (typically in groups of 2 to 8 bits). ECC could be said to work by "averaging noise"; since each data bit affects many transmitted symbols, the corruption of some symbols by noise usually allows

2050-404: Is also used for the evolution of CDMA2000 1x specifically for Internet access, 1xEV-DO (TIA IS-856). Like 1x, EV-DO was developed by Qualcomm , and is sold by Verizon Wireless , Sprint , and other carriers (Verizon's marketing name for 1xEV-DO is Broadband Access , Sprint's consumer and business marketing names for 1xEV-DO are Power Vision and Mobile Broadband , respectively). Sometimes it

2132-637: Is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit . Predating LDPC codes in terms of practical application, they now provide similar performance. One of the earliest commercial applications of turbo coding was the CDMA2000 1x (TIA IS-2000) digital cellular technology developed by Qualcomm and sold by Verizon Wireless , Sprint , and other carriers. It

2214-430: Is completely lost and the missing letters can be recovered with minimal guesswork. Use of interleaving techniques increases total delay. This is because the entire interleaved block must be received before the packets can be decoded. Also interleavers hide the structure of errors; without an interleaver, more advanced decoding algorithms can take advantage of the error structure and achieve more reliable communication than

2296-501: Is crucial for good performance. The iterative decoding algorithm works best when there are not short cycles in the factor graph that represents the decoder; the interleaver is chosen to avoid short cycles. Interleaver designs include: In multi- carrier communication systems, interleaving across carriers may be employed to provide frequency diversity , e.g., to mitigate frequency-selective fading or narrowband interference. Transmission without interleaving : Here, each group of

2378-413: Is determined by the design of the ECC, so different forward error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available bandwidth, which reduces the effective bit-rate while improving the received effective signal-to-noise ratio . The noisy-channel coding theorem of Claude Shannon can be used to compute

2460-434: Is impossible to make all bits change at exactly the same time. If, at the moment the position is sampled, some bits have changed and others have not, the sampled position will be incorrect. In the case of absolute encoders, the indicated position may be far away from the actual position and, in the case of incremental encoders, this can corrupt position tracking. In contrast, the Gray code used by position encoders ensures that

2542-709: Is noteworthy for its widespread use in compact discs , DVDs , and hard disk drives . Other examples of classical block codes include Golay , BCH , Multidimensional parity , and Hamming codes . Hamming ECC is commonly used to correct NAND flash memory errors. This provides single-bit error correction and 2-bit error detection. Hamming codes are only suitable for more reliable single-level cell (SLC) NAND. Denser multi-level cell (MLC) NAND may use multi-bit correcting ECC such as BCH or Reed–Solomon. NOR Flash typically does not use any error correction. Classical block codes are usually decoded using hard-decision algorithms, which means that for every input and output signal

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2624-425: Is only necessary to decode single bits of the message, or to check whether a given signal is a codeword, and do so without looking at the entire signal. This can make sense in a streaming setting, where codewords are too large to be classically decoded fast enough and where only a few bits of the message are of interest for now. Also such codes have become an important tool in computational complexity theory , e.g., for

2706-418: Is optimal in the following way: balanced Gray codes minimize the maximal count of bit-flips for each digit. George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already. A typical use of Gray code counters is building a FIFO (first-in, first-out) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such

2788-486: Is seen as one dimension. When the French engineer Émile Baudot changed from using a 6-unit (6-bit) code to 5-unit code for his printing telegraph system, in 1875 or 1876, he ordered the alphabetic characters on his print wheel using a reflected binary code, and assigned the codes using only three of the bits to vowels. With vowels and consonants sorted in their alphabetical order, and other symbols appropriately placed,

2870-484: Is sometimes misattributed to 19th century electrical device inventor Elisha Gray . Reflected binary codes were applied to mathematical puzzles before they became known to engineers. The binary-reflected Gray code represents the underlying scheme of the classical Chinese rings puzzle , a sequential mechanical puzzle mechanism described by the French Louis Gros in 1872. It can serve as a solution guide for

2952-644: Is to start with a balanced Gray code and remove pairs of values at either the beginning and the end, or in the middle. OEIS sequence A290772 gives the number of possible Gray sequences of length 2 n that include zero and use the minimum number of bits. 0 → 000 1 → 001 2 → 002 10 → 012 11 → 011 12 → 010 20 → 020 21 → 021 22 → 022 100 → 122 101 → 121 102 → 120 110 → 110 111 → 111 112 → 112 120 → 102 121 → 101 122 → 100 200 → 200 201 → 201 202 → 202 210 → 212 211 → 211 212 → 210 220 → 220 221 → 221 Frank Gray (researcher) Frank Gray (13 September 1887 – 23 May 1969)

3034-476: The Hamming distance is the appropriate way to measure the bit error rate . A few forward error correction codes are designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes. The Levenshtein distance is a more appropriate way to measure the bit error rate when using such codes. The fundamental principle of ECC is to add redundant bits in order to help the decoder to find out

3116-409: The Hamming distance of 1 between adjacent codes. In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code , or BRGC . Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. Frank Gray introduced

3198-602: The Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883. Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes. Martin Gardner wrote a popular account of the Gray code in his August 1972 Mathematical Games column in Scientific American . The code also forms a Hamiltonian cycle on a hypercube , where each bit

3280-546: The 5-bit character code has been recognized as a reflected binary code. This code became known as Baudot code and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT-1) in 1932. About the same time, the German-Austrian Otto Schäffler  [ de ] demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for

3362-459: The Shannon limit. However, capacity achieving ECCs are usually extremely complex to implement. The most popular ECCs have a trade-off between performance and computational complexity. Usually, their parameters give a range of possible code rates, which can be optimized depending on the scenario. Usually, this optimization is done in order to achieve a low decoding error probability while minimizing

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3444-461: The binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous. To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment

3526-402: The binary representation differ: The problem with natural binary codes is that physical switches are not ideal: it is very unlikely that physical switches will change states exactly in synchrony. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce ,

3608-426: The bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any ECC (i.e., a code-rate equal to 1) uses the full channel for information transfer purposes, at the cost of leaving the bits without any additional protection. One interesting question is the following: how efficient in terms of information transfer can an ECC be that has a negligible decoding error rate? This question

3690-447: The bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise . Despite the fact that Stibitz described this code before Gray,

3772-723: The bit position of the least significant 1 in the binary representation of i {\displaystyle i} and flip the bit at that position in the previous code c o d e i − 1 {\displaystyle \mathrm {code} _{i-1}} to get the next code c o d e i {\displaystyle \mathrm {code} _{i}} . The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, .... See find first set for efficient algorithms to compute these values. The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at

3854-443: The case of satellites orbiting distant planets, retransmission due to errors would create a delay of several hours. FEC is also widely used in modems and in cellular networks . FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial analog-to-digital conversion in the receiver. The Viterbi decoder implements

3936-406: The code rolls over to decimal 0 with only one switch change. This is called the cyclic or adjacency property of the code. In modern digital communications , Gray codes play an important role in error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that

4018-462: The codes for any two consecutive positions will differ by only one bit and, consequently, only one bit can change at a time. In this case, the maximum position error will be small, indicating a position adjacent to the actual position. Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms . They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally

4100-809: The constituent SPC codes in parallel. LDPC codes were first introduced by Robert G. Gallager in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of Reed–Solomon codes, they were mostly ignored until the 1990s. LDPC codes are now used in many recent high-speed communication standards, such as DVB-S2 (Digital Video Broadcasting – Satellite – Second Generation), WiMAX ( IEEE 802.16e standard for microwave communications), High-Speed Wireless LAN ( IEEE 802.11n ), 10GBase-T Ethernet (802.3an) and G.hn/G.9960 (ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within 3GPP MBMS (see fountain codes ). Turbo coding

4182-407: The construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, it is possible that differences in the arrival times of

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4264-462: The convolutional decoder. Single pass decoding with this family of error correction codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended. Concatenated codes have been standard practice in satellite and deep space communications since Voyager 2 first used the technique in its 1986 encounter with Uranus . The Galileo craft used iterative concatenated codes to compensate for

4346-416: The current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code, add one to it with a standard binary adder, and then convert the result back to Gray code. Other methods of counting in Gray code are discussed in a report by Robert W. Doran , including taking the output from the first latches of the master-slave flip flops in a binary ripple counter. As

4428-444: The design of probabilistically checkable proofs . Locally decodable codes are error-correcting codes for which single bits of the message can be probabilistically recovered by only looking at a small (say constant) number of positions of a codeword, even after the codeword has been corrupted at some constant fraction of positions. Locally testable codes are error-correcting codes for which it can be checked probabilistically whether

4510-472: The execution of program code typically causes an instruction memory access pattern of locally consecutive addresses, bus encodings using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the CPU power consumption in some low-power designs. The binary-reflected Gray code list for n bits can be generated recursively from

4592-450: The form of a Gray code. Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals. Regardless of the mechanism or precision of a moving encoder, position measurement error can occur at specific positions (at code boundaries) because the code may be changing at the exact moment it is read (sampled). A binary output code could cause significant position measurement errors because it

4674-501: The impact to the data rate. Another criterion for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication. Classical (algebraic) block codes and convolutional codes are frequently combined in concatenated coding schemes in which a short constraint-length Viterbi-decoded convolutional code does most of the work and a block code (usually Reed–Solomon) with larger symbol size and block length "mops up" any errors made by

4756-417: The list for n  − 1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary 0 , prefixing the entries in the reflected list with a binary  1 , and then concatenating the original list with the reversed list. For example, generating the n  = 3 list from the n  = 2 list: The one-bit Gray code

4838-419: The maximum achievable communication bandwidth for a given maximum acceptable error probability. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, and hence gives no insight of how to build a capacity achieving code. After years of research, some advanced FEC systems like polar code come very close to

4920-589: The most significant bit), and b i {\displaystyle b_{i}} is the i {\displaystyle i} th binary-coded bit ( b 0 {\displaystyle b_{0}} being the most-significant bit), the reverse translation can be given recursively: b 0 = g 0 {\displaystyle b_{0}=g_{0}} , and b i = g i ⊕ b i − 1 {\displaystyle b_{i}=g_{i}\oplus b_{i-1}} . Alternatively, decoding

5002-519: The old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used. Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at

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5084-415: The order of codewords intact if b i + 1 = 0 {\displaystyle b_{i+1}={\mathtt {0}}} , and reverses the order of blocks of 2 i + 1 {\displaystyle 2^{i+1}} codewords if b i + 1 = 1 {\displaystyle b_{i+1}={\mathtt {1}}} . Now, this is exactly the same operation as

5166-504: The order of two neighbouring codewords is reversed If bit 1 is inverted, blocks of 2 codewords change order: If bit 2 is inverted, blocks of 4 codewords reverse order: Thus, performing an exclusive or on a bit b i {\displaystyle b_{i}} at position i {\displaystyle i} with the bit b i + 1 {\displaystyle b_{i+1}} at position i + 1 {\displaystyle i+1} leaves

5248-435: The original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user data. Most telecommunication systems use a fixed channel code designed to tolerate the expected worst-case bit error rate , and then fail to work at all if the bit error rate is ever worse. However, some systems adapt to the given channel error conditions: some instances of hybrid automatic repeat-request use

5330-510: The output are systematic , while those that do not are non-systematic . A simplistic example of ECC is to transmit each data bit 3 times, which is known as a (3,1) repetition code . Through a noisy channel, a receiver might see 8 versions of the output, see table below. This allows an error in any one of the three samples to be corrected by "majority vote", or "democratic voting". The correcting ability of this ECC is: Though simple to implement and widely used, this triple modular redundancy

5412-408: The performance of forward error correcting codes. Many communication channels are not memoryless: errors typically occur in bursts rather than independently. If the number of errors within a code word exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating

5494-460: The possibility that, when multiple bits change in the binary representation of a position, a misread will result from some of the bits changing before others. For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks). Each track has a stationary metal spring contact that provides electrical contact to the conductive code pattern. Together, these contacts produce output signals in

5576-645: The receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a reverse channel to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth. The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code . FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in multicast . Long-latency connections also benefit; in

5658-459: The reflect-and-prefix method to generate the Gray code. A similar method can be used to perform the reverse translation, but the computation of each bit depends on the computed value of the next higher bit so it cannot be performed in parallel. Assuming g i {\displaystyle g_{i}} is the i {\displaystyle i} th Gray-coded bit ( g 0 {\displaystyle g_{0}} being

5740-563: The reflected binary code was later named after Gray by others who used it. Two different 1953 patent applications use "Gray code" as an alternative name for the "reflected binary code"; one of those also lists "minimum error code" and "cyclic permutation code" among the names. A 1954 patent application refers to "the Bell Telephone Gray code". Other names include "cyclic binary code", "cyclic progression code", "cyclic permuting binary" or "cyclic permuted binary" (CPB). The Gray code

5822-584: The same letter represents a 4-bit one-bit error-correcting codeword. The codeword cccc is altered in one bit and can be corrected, but the codeword dddd is altered in three bits, so either it cannot be decoded at all or it might be decoded incorrectly . With interleaving : In each of the codewords "aaaa", "eeee", "ffff", and "gggg", only one bit is altered, so one-bit error-correcting code will decode everything correctly. Transmission without interleaving : The term "AnExample" ends up mostly unintelligible and difficult to correct. With interleaving : No word

5904-420: The same purpose, in 1874. Frank Gray , who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using vacuum tube -based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953, and the name of Gray stuck to the codes. The " PCM tube " apparatus that Gray patented

5986-458: The term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process". In the standard encoding of the Gray Code the least significant bit follows a repetitive pattern of 2 on, 2 off ( … 11001100 … ); the next digit

6068-411: The theoretical maximum given by the Shannon channel capacity under the hypothesis of an infinite length frame. ECC is accomplished by adding redundancy to the transmitted information using an algorithm. A redundant bit may be a complicated function of many original information bits. The original information may or may not appear literally in the encoded output; codes that include the unmodified input in

6150-430: The transition might look like 011 — 001 — 101 — 100 . When the switches appear to be in position 001 , the observer cannot tell if that is the "real" position 1, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic , then the sequential system may store a false value. This problem can be solved by changing only one switch at

6232-413: The true message that was encoded by the transmitter. The code-rate of a given ECC system is defined as the ratio between the number of information bits and the total number of bits (i.e., information plus redundancy bits) in a given communication package. The code-rate is hence a real number. A low code-rate close to zero implies a strong code that uses many redundant bits to achieve a good performance, while

6314-658: The type that Sears and Pierce collaborated on, which was used in Goodall's "Television by pulse code modulation". Gray graduated from Purdue University in 1911 with a degree in Physics. With Herbert E. Ives as co-inventor, Gray filed for two US patents in 1927: "Electro-optical system" (US 2,037,471, issued 14 April 1936) and "Electro-optical transmission" (US 1,759,504, issued 20 May 1930), and one in just his own name: "Television system" (US 2,113,254, issued 5 April 1938). He patented many other similar-sounding inventions over

6396-463: The very high error rate conditions caused by having a failed antenna. Low-density parity-check (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the channel capacity (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length. Practical implementations rely heavily on decoding

6478-464: The years that followed. His 1953 patent "Pulse Code Communication" with the Gray code was filed in 1947. This article about a United States engineer, inventor or industrial designer is a stub . You can help Misplaced Pages by expanding it . Forward error correction The central idea is that the sender encodes the message in a redundant way, most often by using an error correction code or error correcting code ( ECC ). The redundancy allows

6560-574: Was a physicist and researcher at Bell Labs who made numerous innovations in television, both mechanical and electronic, and is remembered for the Gray code . The Gray code, or reflected binary code (RBC), appearing in Gray's 1953 patent, is a binary numeral system often used in electronics , but with many applications in mathematics . Gray conducted pioneering research on the development of television ; he proposed an early form of " flying spot scanner " for early TV systems in 1927, and helped develop

6642-443: Was answered by Claude Shannon with his second theorem, which says that the channel capacity is the maximum bit rate achievable by any ECC whose error rate tends to zero: His proof relies on Gaussian random coding, which is not suitable to real-world applications. The upper bound given by Shannon's work inspired a long journey in designing ECCs that can come close to the ultimate performance boundary. Various codes today can attain almost

6724-464: Was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, who credited Gray for the idea of the reflected binary code. Gray was most interested in using the codes to minimize errors in converting analog signals to digital; his codes are still used today for this purpose. Gray codes are used in linear and rotary position encoders ( absolute encoders and quadrature encoders ) in preference to weighted binary encoding. This avoids

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