Chapter 4. Coding systems. 4.1 Binary codes Gray (reflected binary) code

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1 Chapter 4 Codin systems Codin systems define how information is mapped to numbers. Different codin systems try to store/transmit information more efficiently [Sec. 4.], or protect it from damae while it is stored/transmitted on unreliable media [Sec. 4.]. 4. Binary codes A binary code defines the mappin between the members of a set of symbols and their correspondin binary numbers. 4.. Gray (reflected binary) code Gray code is a non-positional, non-weihted binary encodin of numbers. It is a minimum-chane code, in which adjacent values differ by only one bit. Gray code is also called reflected binary code. A -bit Gray code consists of the diits and, in that order. Given an existin n-bit code (n ) the n +-bit code is constructed by repeatin the existin n codes in reverse order (hence reflected ), and then prependin s in front of the n oriinal codes and s in front of the n repeated and reflected codes [Fi. 4.]. To convert a n-bit binary number b to a Gray code representation, each bit i is formed by addin the correspondin bit b i and the bit immediately to its left b i+ usin modulo- addition. (There is an implicit to the left of the most sinificant bit in b.) To convert a n-bit Gray code representation to a binary number b, each bit b i (startin with the N binary Gray bit reflect -bit reflect Fiure 4.: Reflected binary code binary = implicit + Gray = + binary = Fiure 4.: Gray code conversion most sinificant) is formed by addin the previous result bit b i+ to the correspondin bit i usin modulo- addition. (There is an implicit to the left of the most sinificant bit in b.) Binary to Gray code conversion is equivalent to takin the bitwise exclusive or of the input with itself shifted riht one bit [Al. 4.]. Each bit in the output from Gray code to binary conversion is the result of a cumulative exclusive or between the correspondin input bit and all input bits to the left of it [Al. 4.]. -bit 5

2 Alorithm 4. Convertin from binary to Gray code input: a binary number b output: the Gray code representation of b function BINARYTOGRAY(b) return b (b rshift ). is exclusive or end function Alorithm 4. Convertin from Gray code to binary input: a Gray code representation output: the correspondin binary value b function GRAYTOBINARY() b repeat b b rshift.xrshift y is x shifted riht by y bits until = return b end function Minimum-chane codes have many practical applications; for example, in telemetry, error correction, electro-mechanical sensor desin, enetic alorithms, simplification of combinational diital circuits [Sec. 5..], and decouplin of clocks in sequential diital circuits [Sec. 5.6]. 4.. ASCII The American Standard Code for Information Interchane (ASCII), is a fixed-width binary code that maps the latin alphabet, arabic numerals, and some common punctuation symbols to a binary representation. Each code is seven bits wide. The most sinificant bit is normally zero, althouh it can be used for parity [Sec. 4..] or to indicate the use of an ASCII-compatible extended codin system such as UTF 8 [Sec. 4..5]. The followin table shows the hexadecimal code for each of the 8 ASCII symbols. most sinificant diit least sinificant diit A B C D E F NUL SOH STX ETX EOT ENQ ACK BEL BS HT NL VT NP CR SO SI DLE DC DC DC DC4 NAK SYN ETB CAN EM SUB ESC FS GS RS US SP! " # $ % & ( ) * +, -. / : ; < = >? A B C D E F G H I J K L M N O 5 P Q R S T U V W X Y Z [ \ ] ^ 6 a b c d e f h i j k l m n o 7 p q r s t u v w x y z { } ~ DEL The first codes, and the final code, are non-printin control codes that communicate with the device receivin the data [App. A]. 6

3 4.. Unicode and UTF Unicode is a mappin of more than 8, characters (diits, puctuation, and symbols spannin 5 lanuaes and writin systems) to -bit code points (numbers) in the rane 6 FFFF 6. Most of the characters and symbols used in modern lanuaes have code points in the rane 6 FFFF 6 (the basic multilinual plane, BMP ) and fit into a 6- bit inteer. Code points in the rane 6 7E 6 correspond directly to the (non-control) ASCII characters. Code points in the rane 6 FFFF 6 (the supplemental planes) contain symbols for ancient lanuaes, mathematics, music notation, etc. Several encodins exist for Unicode, called Unicode Transformation Formats (UTF). The most common Unicode encodins are UTF, UTF 6, and UTF 8. UTF is a fixed-width encodin of Unicode in which each -bit code point is stored directly within a -bit inteer UTF 6 UTF 6 is a variable-width encodin of Unicode in which each -bit code point is represented as one or two 6-bit words. Code points in the basic multilinual plane can be represented directly as a sinle 6-bit word in the rane 6 FFFF 6. A code point in one of the supplemental planes ( 6 to FFFF 6 ) is encoded as two words. 6 is first subtracted from the code leavin a -bit number in the rane 6 FFFFF 6. The most sinificant ten bits of this result are added to D8 6 to create the hih surroate in the rane D8 6 DBFF 6. The least sinificant ten bits of the result are added to DC 6 to create the low surroate in the rane DC 6 DFFF 6. The code point is then stored or transmitted as the hih surroate followed by the low surroate. U+A (ASCII * ) UTF 6: A 6 U+65 (star symbol F ) UTF 6: 65 6 U+F (shootin star symbol ) F 6 6 = F 6 = D8 6 + = D8C 6 (hih surroate) DC 6 + = DF 6 (low surroate) UTF 6: D8C DF 6 Unicode does not assin any characters to code points in the rane D8 6 DFFF 6. There is no ambiuity between surroates and code points in the basic multilinual plane UTF 8 UTF 8 is an efficient, variable-width encodin of Unicode in which each -bit code point is represented usin between one and four 8-bit bytes. The first 8 code points ( 6 7F 6 ) are represented directly as a sinle 8-bit byte whose most sinificant bit is zero (makin UTF 8 a compatible superset of ASCII). Code points from 8 6 onwards are represented as a multibyte sequence in which each byte has its most sinificant bit set. The fist byte in an N-byte sequence has the most sinificant N bits set to, followed by a bit. The remainin bits encode the most sinificant 7

4 8 N bits of the code point. The next N bytes in the sequence have the most sinificant two bits set to, followed by the next six bits of the code point. number of bits code points byte byte byte byte F 6 xxxxxxx (5+6) 8 6 7FF 6 xxxxx xxxxxx 6 (4+6+6) 8 6 FFFF 6 xxxx xxxxxx xxxxxx (+6+6+6) 6 FFFF 6 xxx xxxxxx xxxxxx xxxxxx U+A (ASCII * ) UTF 8: A 6 U+65 (star symbol F ) 65 6 = UTF 8: E U+F (shootin star symbol ) F 6 = UTF 8: F8 9F 8C A 6 4. Source codin Source codin deals with encodin information efficiently, so that it requires less storae space or can be exchaned more rapidly. 4.. Huffman code Huffman codes are created by analysin a known messae text and constructin an optimal encodin specifically for that text. An alphabet is a set of symbols that can appear in a messae. Encodin beins by considerin the frequency of each alphabet symbol in the known messae. For the strin oolin, the alpbabet is {olin} with frequencies ranin from to. Dividin the frequency of a iven symbol by the total lenth of the strin ives the probability of that symbol appearin in the messae. f symbol P (s) l i n /8 o /8 /8 The code is constructed as a binary tree, from the bottom (leaves) up (towards the root), by repeatedly coalescin subtrees. Each subtree is labelled with the probability that a symbol within it will appear in the messae. An initial list of subtrees is created from the symbols of the alphabet, each of which becomes a leaf node labelled with its probability. The followin steps are then repeated find the two subtrees p and q havin the lowest probabilities; create a new subtree r with children p and q and probability P (r) =P (p)+p (q); remove p and q from, and insert r into, the list of subtrees until the list of subtrees contains only one element. The sinle subtree remainin in the list is the root of the Huffman encodin tree for the oriinal messae. 8

5 . An initial list of subtrees is formed from the alphabet symbols, each labelled with its probability. l i n o. The smallest two subtrees (nodes for symbols l and i ) are coalesced into a new subtree. The two symbol nodes are removed from, and the new subtree is inserted into, the list. n o l i. The smallest two subtrees (nodes for symbols n and o ) are coalesced into a new subtree. The two symbol nodes are removed from, and the new subtree is inserted into, the list. l i n o 4. The smallest two subtrees (subtree for il and node for symbol ) are coalesced into a new subtree. n 5. The smallest two subtrees (for no and li ) are coalesced into a new subtree. Only one subtree remains, and the coalescin process is complete. The root of the final subtree will always have a probability of (in this example, 8/8). Labellin left and riht children with and, respectively, yields the Huffman encodin tree for the oriinal n o messae. Each symbol now has a binary code correspondin to the strin of or bits that label the path from the root of the tree to the symbol s leaf node. Less probable symbols were coalesced first, and have the lonest codes. More probable symbols were coalesced later, and have the shortest codes. The encodin is unambiuous (no code is a prefix of any other code). It is specific to the messae, so sender and receiver must both aree on the encodin tree bein used. The oriinal messae can now be encoded as a sequence of bits, or a sequence of bits decoded into the oriinal messae. o o l i n o 8 l l i i 5 5 symbol encodin n o l i Stored as 8-bit bytes, the 8-character example messae occupies 64 bits. Stored usin its Huffman encodin, the 8-character example messae occupies 8 bits, a compression ratio of 64 8 = Channel codin Channel codes are desined to protect information from damae when stored on, or transmitted over, unreliable media. Parity bits [Sec. 4..] and CRC codes [Sec. 4..4] are used to detect errors. Hammin codes [Sec. 4..5] are used to correct small errors. 9

6 4.. Parity bits A binary number with an even number of s has even parity. A binary number with an odd number of s has odd parity. A parity bit is transmitted (or stored) alon with a number to uarantee an expected parity. In a sequence of numbers that are expected to have even parity, any number with odd parity must be damaed. (Any number with even parity is either undamaed, or has suffered an even number of bit inversions.) number parity even odd odd even odd even parity expected data p error (five s) error (three s) odd parity expected data p error (four s) error (four s) These lateral parity bits reliably detect only sinle-bit errors. 4.. Lonitudinal parity A parity word is added to the end of a fixed-lenth block of data. Each bit is a lonitudinal parity bit, calculated for the bits in a specific position with the block of data. If each bit position is expected to have odd parity, any position with even parity must be damaed. even lonitudinal parity parity word odd lonitudinal parity parity word Only sinle-bit errors can be detected. 4.. Block parity Block parity combines lateral parity bits and lonitudinal parity words for each block of data. Knowin both the damaed word and the damaed bit position allows the error to be corrected.

7 even lateral parity data p odd parity word odd lateral parity data p even parity word Block parity can reliably correct only a sinle-bit error Cyclic Reduncancy Check A cyclic redundancy check (CRC) adds a redundant checksum to each block of data, calculated usin a cyclic divison alorithm. CRCs are particularly ood for detectin burst errors. Burst errors are contiuous sequences of many incorrect bits, frequently encountered in communication channels (because of noise) and physical storae media (because of material defects). CRCs operate on one block of bits at a time, e.., a sinle byte (8 bits) or an Ethernet packet (up to 5 bytes =,4 bits). The number of check bits is usually smaller than the block size (Ethernet packets have check bits). Usin N check bits, burst errors of up to N bits in lenth (and some patterns loner than N bits) can be reliably detected. To calculate N check bits the data block bits are used as a dividend, and divided numerically by a N +-bit divisor. (An 8-bit CRC therefore requires a 9-bit divisor.) The result is a quotient and a N-bit remainder. The quotient is discarded and the N remainder bits are used as the CRC check bits. Unsined binary lon division is used to calculate the remainder [Sec..4]. N s are first appended to the riht of the dividend (the oriinal data block), to hold the remainder. The dividend and N +-bit divisor are left-alined, and the followin steps repeated the divisor shifted riht, if necessary, until its most sinificant bit is alined with the most sinificant bit of the dividend; and the divisor is subtracted from the dividend usin modulo- arithmetic (borrows between adjacent bits are inored), includin any result to the riht of the dividend in the N remainder bits until the dividend becomes zero. The N bits to the riht of the dividend are the remainder, and can be used as the CRC check bits for the oriinal data. (Modulo- subtraction is equivalent to performin a bitwise exclusive-or between the dividend and the divisor [Sec. 5..], makin it very efficient. Note that the N remainder bits, formed by appendin N zeros to the riht of the dividend, are included in the subtraction operations but are not considered when checkin if the dividend has reached zero.) Calculatin a -bit CRC for a 6-bit data block usin a -bit divisor :

8 dividend (oriinal data) bits ######## remainder (CRC) bits modulo- subtraction ( XOR operation) transmitted data (includin CRC) = = new dividend = alin divisor with MS dividend bit = alin divisor with MS dividend bit = dividend is zero, remainder (CRC) = The interity of the data can be verified by performin the same division, usin the received CRC bits as the initial remainder before performin the division. If the data was undamaed, the remainder bits (on completion of the division operation) will be zero. Burst errors of N bits or less in the data will enerate a non-zero remainder. = = = = dividend, remainder = ) no error A -bit burst error, forcin the third and fourth bits to zero, enerates a non-zero remainder: two-bit burst error ## = = = = = non-zero remainder ) error! Transmission and reception of the 6-bit data block usin a -bit CRC with divisor proceeds as follows:

9 CRC eneration 4..5 Hammin code! network transmission (or store and retrieve) CRC verification no error Hammin code is a form of parity that allows correction of sinle-bit errors and detection of double-bit errors, while bein more efficient than block parity. To enerate a Hammin code for a N-bit block of data: number the code bit positions in binary, startin from any bit position whose number is a power of two (,, 4, 8, etc.) is a parity bit p i all other bit positions are data bits d j stop numberin code bits when all N data bits have received a numbered position consider the binary number of each parity bit p n (a power of two, n ) all code bit positions whose binary number has bit n set to are to be included in the parity calculation for p n all other code bit positions are excluded from the calculation of p n Each data bit will be included in the calculation of a unique combination of two or more parity bits. bit number: content: p p d p 4 d d d 4 p 8 d 5 d 6 d 7 d 8 d 9 d d p 6 d d d 4 d 5 d 6 p p p 4 p 8 p 6 parity coverae (The pattern continues in the obvious way, for as many bit positions as are required.) Even or odd parity can be used. Usin even parity, the Hammin code for the 8-bit block is constructed as follows:

10 bit number: content: p p p 4 p 8 p p p 4 p 8 parity coverae Final code: When data is received, if all parity bits are correct then there is no error. Otherwise the sum of the bit positions of the incorrect parity bits identify the erroneous bit. If the bit in position 9 were to be damaed, parity bits p and p 8 would be incorrect, indicatin a problem with bit position +8=9. If the bit in position 4 were to be damaed, parity bit p 4 would be incorrect, indicatin a problem with bit position 4 (the parity bit p 4 itself). 4