3. Error Correcting Codes

Transcription

1 3. Error Correctig Codes Refereces V. Bhargava, Forward Error Correctio Schemes for Digital Commuicatios, IEEE Commuicatios Magazie, Vol 21 No , Jauary 1983 Mischa Schwartz, Iformatio Trasmissio Modulatio ad Noise, 4 th ed, Chater 7, Mc Graw Hill 199 Shu Li, Daiel J Costello, Error Cotrol Codig: Fudametals ad Alicatios, Pretice Hall, 1983 Irvig S Reed, Xuemi Che, Error-Cotrol Codig for Data Networks, Kluwer Academic, 1999 Robert Morelos-Zaragoza, The Art of Error Correctig Codig, 2 d ed., Wiley 26 Itroductio I statistical commuicatio theory, usig biary PSK trasmissio, over a biary symmetric chael, BSC, the robability of a bit error evet over a additive white Gaussia oise chael, (AWGN)is give by 1 Eb Pe erfc (3.1) 2 N where E b is the symbol eergy ad is the sectral oise eergy, ad the erfc(x) is the comlemetary error fuctio erfc(x) = 1 erf(x) where erf ( x) 2 x e t 2 2 dt erfc( x) x e 2 t dt (3.2) This is also give i terms of 2 Eb 1 x Pe Q where Q( x) erfc( ) (3.3) N 2 2 The classical Digital Commuicatio System is show i Fig 3.1. I this sectio we are lookig at the chael ecodig ad decodig i the resece of oise that ca be additive or burst oise. The chael ca be ay tye from cable to satellite, as well as electromagetic storage ivolvig CD s ad other data storage media.

2 Basic jargo associated with error correctig codes (i) a block code ca be of various tyes. But all tyes have a block of k data bits associated to a codeword of bits. This is called a (, k) code. The code rate, which idicates the efficiecy of the code, is k/ ad is always a fractio. (ii) A biary code s caability is defied i terms of the XOR outcome betwee ay two of its codewords. A (,k) code has 2 k codewords sread over a vector sace of 2 -tules. The outcome is called the distace betwee two codewords. The miimum distace of a code is the miimum distace betwee ay two codewords, deoted by t. (iii) The error correctig caability i bits, give by d, is obtaied as d 2t + 1 So a code with miimum distace 3, ca correct 1 bit i error. (iv) There are also codes that have memory over the blocks, called covolutioal codes. These tye of codes are defied o 3 arameters, (, k, r) where r deotes the umber of codewords beig geerated that are iflueced by ay block of data beig used. Clearly there must be a mechaism of how to obtai codes, based o some techique that gives the best, or good, miimum distace. Further for large k ad there must be techiques of geeratig automatically the codewords, ad ot have a memory through which to comare ad associate the curret data to its aroriate codeword. There must also be techiques of automatically checkig the received -tule to be able to decide o whether it is correct, ad if it is ot correct, to automatically correct the error. Liear Algebra theory based o grous, fields, Galois Fields of the tye GF(2) ad GF(2 m ), is used extesively as a basis for geeratig efficietly good error correctig codes. The above is based also o the remise that the received -tule does ot have error bits that exceed the caabilities of the code, as this may result i udetectable errors. These are to be avoided as much as ossible, (i fact i some alicatios at all costs), ad therefore alicatio tye, sigal ower, ad exected chael oise, have to be

3 aalysed to decide o the ecessary error code caability, before decidig o which code to use. Further there are alicatios where two tyes of codes are used to icrease the robustess of the codig. Block Codes A Liear Block Codes I a liear block code the sum of two codewords, based o XOR, results i aother codeword. This meas that the codewords form a closed subsace over the field of GF(2). Iitially we will cosider the ecessary algebra to uderstad the GF(2). (a) Defiitio of a grou: A set of elemets G with a oeratio * is called a grou if: (i) the biary oeratio U* is associative; (ii) G cotais a elemet e, the idetity elemet, such that A*e = e*a = a (iii) for ay elemet a, there exists a iverse elemet a such that a*a = a *a = e A grou is commutative if for ay two elemets a, b A*b = b*a The biary set,1 is a grou uder the oeratio of XOR. The itegers to (N-1) are a grou uder modulo(n) additio. The itegers {1,2,,} are also a grou uder modulo (N) multilicatio IF is rime Table of iteger modulo-5 uder multilicatio (b) Defiitio of a field I this case there are two oerators ad i geeral oe ca talk about additio, subtractio, multilicatio ad divisio, where oeratio o two elemets of a set {F} still results i aother elemet of set {F} (i) F is commutative uder additio, (+), ad the idetity elemet is. (ii) F is commutative uder multilicatio (.). The idetity elemt is the uit elemet, 1 (iii) Multilicatio is distributive over additio a.(b+c) = a.b + a.c A field with a fiite umber of elemets is called a fiite field. Fiite fields are also called Galois Fields. Galois Fields with a umber of elemets that is rime, i articular GF(2) lay a imortat role i codig theory.

4 Modulo-2 Additio Modulo-2 Multilicatio GF(2) Sets of itegers {,1,2,,-1} of elemets, where is a rime costitute a Galois Field GF(). Note that the rocess of additio ivolves addig the iverse elemet uder additio, while divisio ivolves multilyig the iverse elemet uder multilicatio. A extesio of GF(, a rime, to a GF( m ) is also ossible. We will be makig extesive use of Galois fields of the tye GF(2 m ). A liear block code is a (,k) code where every codeword is the modulo-2 sum of ay other two codewords. A liear block code is characterized by a geerator matrix. The geerator matrix is made u of a arity check ad a Idetity matrix. Ex. For a (7,4) code, the geerator matrix cosists of g g G g g a arity check [4 x 3] matrix followed by a Idetity [4 x 4] matrix to make u a [4 x 7] matrix. I this format the liear block code is also called a systematic code sice the structure of the code searates out the derived arity checks ad the origial data block ito searate subblocks. Examle: Liear Block Code (7,4) has a geerator matrix g g G g g ad results i the 2 4 = 16 codewords listed below Data Codeword

5 The Hammig Weight of every codeword is at least three. The miimum distace of the code is the miimum differece betwee ay two codewords, but sice ay codeword is the sum of ay other two codewords, the miimum Hammig distace is 3. The arity-check matrix of a (,k) liear block code is derived from the [k x ] geerator matrix, as a [(-k) x ] matrix give by H = I -k.p T geerator matrix. where I is the idetity matrix ad P the arity-check art of the For the (7,4) code above, the arity-check matrix H is a [3 x 7] matrix, give by H The imortace of the arity-check matrix is due to the fact that G. H T = ; also imlies that v. H T = ; where v is ay valid codeword Sydrome ad Error Detectio I geeral, whe a received word is assed through the H T matrix it gives rise to a [-k] bit atter called the sydrome. A o-zero sydrome idicates a error has bee detected. Oe ca write the received vector as r = e + v where r is the received -tule that ca be cosidered to be made u of a codeword v ad a error vector e. However there is still the ossibility of a udetectable error if the error vector e alters oe codeword to aother. There are (2 k 1) received -tules that ca be udetectable errors. The rest, i.e. 2 2 k are detectable. The robability of a udetectable error, deoted by P u (E) is give by

6 P i i u Ai (1 ) (3.4) i1 where A is the Hammig weight of each codeword ad e is the robability of bit error, (trasitio error robability) o a BSC. Ad (1-) c is the robability of a correctly received bit For the (7,4) code above the weight distributio is A = 1, A 3 = A 4 = 7, A 1 = A 2 = A 5 = A 6 =. Therefore (1 ) 7 (1 ) P u ad for a = 1-2, the P u (E) = 7 x 1-6, cosiderably lower tha. Sydrome circuit From H T of the (7,4) code above the sydrome bits are s = r + r 3 + r 4 + r 6 ; s 1 = r 1 + r 3 + r 5 + r 6 ; s 2 = r 2 + r 4 + r 5 + r 6 ; ad they ca be obtaied from a XOR logic circuit give below. The ext imortat issue is whether give a sydrome, ad the cosequet idicatio of a error, there is the ossibility of automatically correctig the error. For a (7,4) code there are = 128 detectable error atters, that are detected by (2 3-1) = 7 o-zero sydrome atters. Clearly there is a maig of 16 to 1. Usig maximum likelihood, the error vector with the miimum Hammig weight from the set of 16, should be chose. Deotig this vector by e, the resultat codeword would be v = r + e The 16 atters that are a set for each sydrome atter, together with the sixtee codewords whe the sydrome is zero, form the 2 7 = 128 ossible 7-bit tules for the (7,4) code. For a liear block code these give rise to a stadard array. The atter with the miimum Hammig weight i each of the eight cosets, is used as a coset leader.

7 Stadard Array for the (7,4)code The coset leaders are i the first colum of each coset ad are the 7-bit atter with miimum weight i that coset. Fially, the relatioshi of the sydrome to the coset leaders is give by Sydrome Coset Leader Note that the order of the sydrome atters corresod to the row order of the H T. For a k greater tha 8, the size of the table becomes cosiderable, ad the search for the corresodig error vector to add to the received 7-bit atter to obtai the codeword becomes log. There is a relatioshi betwee a geerator matrix ad its arity check matrix i terms of dual codes. If H is used for code C, the the same H is the G for the dual code C d. This ca be used to calculate the P u (E) whe it is ot easy to get the codewords weight distributio A i of the code because k is big. Defiig the two weight distributios i terms of olyomials as C A(z) = A + A 1 z A z ad for its dual (3.5) C d B(z) = B + B 1 z B z where z is give by (3.6) z = /(1-) ad A(z) ca be obtaied i terms of B(z) from ( k ) 1 z A( z) 2 (1 z) B( ) (3.7) 1 z ad usig (3.4) give by P i i u Ai (1 ), by rearragig to obtai i1 i Pu (1 ) Ai ( ) (3.8) i1 1 ad sice A = 1, the usig the two equatios (3.7) ad (3.8), a alterative to (3.4) ca be obtaied as ( k ) P 2 B(1 2 ) (1 ) (3.9) u

8 where i B( 1 2 ) B i (1 2 ) (3.1) i Either 3.4 or (3.9) ca be used to obtai P u (E) deedig o which of k or (-k) is suitable. Examle: For the (7.4) code havig G ad H above, the C d is a (7,3) code whose G is give by G ad the eight codewords are give by The weight distributio gives B(z) = 1 + 7z 4 Hece from the dual code P u 2 [1 7(1 2 ) ] (1 ) which is equal to the revious aswer. (*Check it out by workig out the olyomial i for both cases). I geeral, for large (.k) it is ot easy to work out P u (E). It ca be show that a uer boud exists give by ( k ) P 2 [1 (1 ) ] ad sice [1 (1-) ] is 1 the P u (E) 2 -(-k) u Hammig Codes These are liear codes based o the followig roerties. For ay m 3, a Hammig code ca be built with = 2 m 1; k = 2 m m 1; -k = m. Note that the (7,4) code above is a Hammig code. Oe articular roerty, because of the ature of the umber of sydrome atters available, is that a Hammig code is a erfect code. The class of t-error correctig codes that has i its stadard array all atters of t or less weight as the coset leaders, is called a erfect code. Perfect codes are rare. Besides the Hammig Codes, the other otrivial erfect code is the (23,12) Golay code. The Hammig code therefore caot detect two or more errors. However this ca be doe usig a shorteed Hammig code obtaied by removig from the H matrix all the colums whose weight is eve. If l colums are removed, the code ow becomes Code legth = 2 m - l 1 = l Iformatio bits k = 2 m m - l 1 Parity check bits -k = m Miimum distace d 3 Usig this shorteed cyclic code decodig is doe as follows: 1. If the sydrome is zero a correct codeword is received 2. If the sydrome is odd, the a sigle error occurred which ca be corrected 3. If the sydrome is eve there is a ucorrectable error atter