PERFORMANCE ANALYSIS OF BPSK SYSTEM WITH HARD DECISION (63, 36) BCH CODE

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1 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. PERFORMANCE ANALYSIS OF BPSK SYSTEM WITH HARD DECISION (6, 6 BCH CODE MAHMOUD A. SMADI Departent of Electrcal Engneerng The Hashete Unversty Zarqa, 5, JORDAN Eal: sad@hu.edu.jo ABSTRACT The purpose of ths paper s to study the Bose, Chaudhur, and Hocquenghe (BCH code, wth an a to sulate the encodng and decodng processes. The gan of the proposed code n nvestgated through applyng t to bnary phase sft keyng (BPSK odulaton schee n syetrc addtve whte Gaussan nose (AWGN channel. The bt error probablty (BEP of coded (6, 6 BCH syste was evaluated and copared wth the perforance of un-coded syste. Keywords: BCH Code, Bnary PSK, Bt error probablty, AWGN.. INTRODUCTION Channel codng for error detecton and correcton helps the councaton syste desgners tgate the effects of a nosy transsson channel. Error control codng theory has been the subject of ntense study snce the 90s and now beng wdely used n councaton systes. As llustrated by Shannon n hs paper publshed n 98 [], for each physcal channel there s a paraetrc quantty called the channel capacty C that s a functon of the channel nput output characterstcs. Shannon showed that there exst error control codes such that arbtrary sall error probablty of the data to be transtted over the channel can be acheved as long as the data transsson rate s less than. A generc block dagra of dgtal councaton syste nvolvng coded wavefors s shown n Fg. []. The bnary nforaton sequence at the encoder nput has a rate of R bts/sec. Manly there are two types of channel encodng technques. The frst s the block codng, by whch a blocks of k nforaton bts are encoded nto correspondng n bts blocks. Each n burst s called a code word wth a total nuber of k possble code words. The code rate, defned as the rato Rc = k / n s a easure of the aount of the redundancy ntroduced by the specfc block codng technque. The second type of encodng s the lnear convoluton encodng. A convoluton encoder converts the entre nforaton sequence strea, regardless of ts length, nto a sngle code word []. The encoder output sequence s a set of lnear cobnatons of the nput sequence that can be perfored usng a fnte-state shft regster approach. The code rate n ths case R c s defned as the recprocal of the nuber of the shft regster output bts for each data bt. In both cases, the bt rate at the encoder output s R / R c. Another desgned paraeter assocated wth the codng schee to be used s the error correctng capablty of ths schee. That s, how any errors that ay be ntroduced by the channel can ths code guarantee to correct. Hence, a good code s the one that nsure a certan error correctng 7

2 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. Bnary nforaton sequence Coded Encoder Modulator Channel Sequense Decoder Deodulator Fg.: Generc dgtal councaton syste wth channel codng []. capablty at a nu output encoder rate R / R c. R c or axu The bnary dgts fro the encoder are fed nto a odulator, whch aps the nto one of the known dgtal odulaton wavefors, say BPSK or BFSK. The channel over whch the wavefors are transtted wll corrupt the wavefors n general by a ultplcatve fadng nose besdes the tradtonal theral AWGN. The resultng receved nosy sgnal s deodulated to ts bnary rege and decoded back to the orgnal bnary nforaton sequence. The decodng decson schee ay be one of two possble decodng schees hard or softdecson schee. In the hard decson decodng, the deodulator quantzed the ncong sgnal nto two levels, denoted as 0 and. The nforaton sequence bts are then recovered by the decoder that wll have a certan error correctng capablty. On the other hand, f the unquantzed (analog deodulator output s fed to the decoder we call ths decodng schee softdecson decodng. In the hard decson case, the typcal BPSK or BFSK result n a syetrc transsson error probablty n whch the probablty that s transtted and 0 s detected P ( 0/ s equal to the probablty that 0 s transtted and s detected P ( /0 = p. Ths channel s called a bnary syetrc channel (BSC. So any papers deal wth the perforance analyss of coded dgtal councaton systes. The authors n [] and [5] proposed several decodng technques for the BCH codes and evaluate ther perforance. The perforance of dgtal rado councaton systes wth a BCH codng schee under a crowave oven nterference envronent s nvestgated n [6]. The authors found that perforance proveent could be obtaned by cobnng BCH codes wth bt nterleavng. The proble of effcent axu-lkelhood soft decson decodng of bnary BCH codes s consdered by the authors n [7]. On the other hand, the BER perforance of severely punched codes and the equvalent systeatc codes s obtaned assung axu lkelhood decodng for (6,57 Hang code n [8]. In contrast, the authors n [9] proposed an proved Hang code ethod whch s shown to be hghly scalable wthout such overhead. Furtherore, the paper [0] analyzes the perforance of concatenated codng systes operatng over the BSC by exanng the loss of capacty resultng fro each of the processng steps. Fnally, two schees for dfferental encodng of block coded M-ary PSK sgnals are presented and copared n []. In ths paper, the perforance of BPSK coded syste wll be sulated. We wll base our analyss on lnear BCH block codng schee wth a hard decson decodng over AWGN bnary syetrc channel. The rest of ths paper s organzed as follows. A bref descrpton of lnear block codes and algebrac feld concepts wll be gven n the next secton. The BCH codes wll be treated n deep n secton. Nuercal Results are gven n secton. Fnally, bref conclusons are provded n secton 5.. LINEAR BLOCK CODES A block code C s constructed by breakng up the essage data strea nto blocks of length k has the for{ 0,,..., k }, and appng these blocks nto code words nc. The resultng code conssts of a set of M code words{ C0, C,..., CM }. Each code word has a fxed length denoted by n and has a 7

3 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. for c0, c,..., cn. The eleents of the code word are selected fro an alphabet feld of q eleents. In the bnary code case, the feld conssts of two eleents, 0 and. On the other hand, when the eleents of the code word are selected fro a feld that has q alphabet eleents, the code s nonbnary code. As a specal case when q s a power of (.e. q = where s a postve nteger, each eleent n the feld can be represented as a set of dstnct bts. As ndcated above, codes are constructed fro felds wth a fnte nuber of q eleents called Galos feld and denoted by GF (q. In general, fnte feld GF (q can be constructed f q s a pre or a power of pre nuber. When q s a pre, the GF( q consst of the eleents { 0,,,... q } wth addton and ultplcaton operatons are defned as a odulo-q. If q s a power of pre (.e. q = p where s any postve nteger, t s possble to extend the feld GF q = p. Ths s called p to the feld GF the extenson feld of GF ( p and n ths case ultplcaton and addton operatons are based on odulo- p arthetc. To construct the eleents of the extenson GF ( q = fro the bnary GF( wth eleents 0 and, a new sybol s defned wth ultplcaton operaton propertes as: 0. =.0 = 0,. =. = and j j j. =. = +. The eleents of the GF ( q = that satsfy the above propertes j are{ 0,,,,...,,..}. As the feld should has eleents and be closed under ultplcaton q should satsfes the condton =. Hence; the eleents of the extenson GF ( q = are q { 0,,,,..., } whch s a coutatve group under an addton and ultplcaton (excludng the zero eleent operatons. s called a prtve eleent snce t can generate all other feld eleents and t s a root of a prtve polynoal px. As entoned before, each eleent n the feld can be represented as a set of -tuple bts. To ake the pcture clear, Table I shows the three representaton for the eleents of GF( wth a p x = + x + x []. prtve polynoal Besdes the code rate paraeter R c defned early, an portant paraeter of the code word s ts nu dstance denoted by d n. As the code weght defned as the nuber of nonzero eleents n the code, the nu dstance of a block code s the nu dstance between all dstnct pars of code words whch s the sae as the nu weght of the code. The nu dstance s a easure of the separaton between code words and thus a code wth nu dstance d n can detect any error pattern of weght less than or equal to d n []. The lnearty property of a code s farly a sple concept. Suppose that C and C are two code words n an ( nk, block code and let and are any two of the feld eleents over where the code s defned, then the code s called a lnear code f and only f C + C j s also a code word n C.. The generator atrx and party check atrx let = ( x, x,..., x X k be the k nforaton bte at the encoder nput and C = c, c,..., c s the encoder output n vector. The encodng operaton perfored n lnear bnary block encoder can be represented n atrx for as C = X G ( where G s called the generator atrx of the code, defned as g g g. g n. g g g gn G = =..... g g g. g k k k kn j ( and hence; any code word s a lnear cobnaton of the rows { g } of G,.e., 7

4 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. Table I: Representatons of the eleents of GF( wth p( x = + x + x []. Power representaton Polynoal representaton -Tuple representaton 0 0 ( ( ( C x g x g x g ( = k k Snce the lnear ( nk, code wth k dstnct code words s a subset of denson k, the rows of G ust be a set of lnearly ndependent rows, and hence, G s not unque. Any generator atrx of ( nk, lnear block code can be reduced by row operaton whch wll keep the lnearly ndependence property of G to a syetrc for gven as p p. p, nk 0. 0 p p. p, 0. 0 nk G = P I k = pk pk. pk, nk 0 0. ( where I s a k k dentty atrx and P s a k ( n k atrx that deternes the code redundant bts. In ths case the last k bts of each code word are dentcal to the k nforaton bts. (0 0 0 ( ( (0 0 + ( ( 0 + ( ( ( ( ( + + ( 0 + ( Assocated wth any lnear ( nk, block code there s a lnear ( nn, k dual code wth nk code words whch s the null space of the ( nk, code. The generator atrx assocated wth the dual code, conssts of ( n k lnearly ndependent rows and denoted by H. Snce G and H are n the null space of each other, any code word generated by G s orthogonal to every row n H. That s T T C H = 0 OR GH = 0 (5 Now f the block code s n syetrc for, t follows fro the last equaton that H = I P nk (6 and snce for lnear block code the nu dstance s equal to the nu weght of the code, another concluson one can draw fro (5 s that the nu dstance for a lnear block code s the nu nuber of coluns n H that ay add up to the zero vector.. Cyclc codes BCH code s a subset of a general lnear block codes called cyclc codes. Cyclc codes are a class

5 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. of lnear codes whch satsfy the followng cyclc shft property: f C s a code word of a cyclc code, then any cyclcally shfts of C s also a code word. In dscussng cyclc code and later a BCH code, t's ore convenent to deal wth polynoals representaton rather than atrces representaton of the code. So, to develop the algebrac propertes of a cyclc code, we represent the coponents of a code word C = ( c0, c,..., cn as the coeffcents of a polynoal called a code polynoal as follows n c( x = c0 + cx + cx cn x (7 t can be shown that the code polynoal resultng fro cyclcally shftng the code word C -th tes denoted by ( c ( x s the reander resultng n fro dvdng the polynoal x cx by x + []. We can generate a bnary ( nk, cyclc code by usng a generator polynoal g ( x of degree n k has the for n k g ( x = g0 + g x gn kx (8 where g s s ether 0 or n the bnary code case. A nuber of portant propertes of the generator polynoal can be suarzed []: a The coeffcents g 0 and g n k have to be. b Any code word polynoal c( x s ultple of g ( x (.e. c( x = ( x g ( x where k... x = 0 + x + x + + k x s the essage polynoal. n c g ( x s a factor of x +. The last property says that any factor of n x + wth degree n k, generates an n ( nk, cyclc code. For large n, x + ay have any factors of degree n k. Soe of theses polynoals generate good codes and others generate bad codes []. To be consstent wth the atrx representaton of a general block code as dscussed n the pervous subsecton, the generator cyclc code can be derved fro the generator polynoal gven n (8 as atrx for ( nk, g0 g g.. gnk g0 g g.. g nk G = 0 0 g0 g g.. gn k g0 g g. g nk (9 wth g 0 = gn k=. A syetrc cyclc code can be obtaned slarly by row operatons.. BCH CODES BCH codes are a large class of cyclc codes that nclude both bnary and non-bnary codes. Bnary ( nk, wth any postve nteger BCH codes can be constructed wth the followng paraeters n = n k t (0 dn t + = δ where t s the error correctng capablty and δ s called the code desgn dstance. That s a BCH code wth specfed paraeters gven n (0, guarantees to correct t or less nuber of errors n the receved n block bts. The generator polynoal g ( x of the t -error correctng BCH code s the lowest degree polynoal over GF (, whch has the t consecutve,,..., as ts roots (.e. g ( = 0, =,,..,t. Let φ ( x be the nal polynoal (the nu degree polynoal that has and ts conjugates as a roots correspondng to, the generator polynoal ust be the least coon ultple (LCM of φ ( x, =,,...,t. That s [] g ( x = LCM { φ( x, φ( x, φ( x,..., φ t ( x } ( All the feld eleents of the for j = ( l, l and s odd, are called conjugate of and all of the over the defned feld have the sae nal polynoal (.e. φj( x = φ( x. Hence, every even power of n ( has the sae nal polynoal as the precedng odd power of. Ths reduces the nuber of ters n ( to t ters, so g ( x = LCM { φ( x, φ( x,..., φ t ( x } ( the BCH codes defned above are called prtve, narrow-sense BCH codes. Snce any code word polynoal n cx = c0 + cx cn x s a ultple of the generator polynoal g ( x (.e cx = x gx, t cx has,,..., as a roots, then ( n c( = c0 + c cn = 0, =,,..,t ( or n atrx for 76

6 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. ( c0, c,..., cn = 0 (. ( n by cobnng ( wth (5, the party check atrx of the BCH codes n 'sfor can be wrtten as n. n ( (. ( n H = ( (. ( ( t t t n ( ( ( whch can be wrtten n 0, for by represent 'sn ts tuples for. Exaple: Consder a prtve, narrow-sense BCH code wth n = =5 andt =. It follows fro ( that ths code s generated by g ( x = LCM { φ( x, φ( x } (6 where 8 φ ( x = ( x + ( x + ( x + ( x + = + x + x (7 6 9 φ ( x = ( x + ( x + ( x + ( x + = + x + x + x + x snce there s no coon factor between φ( x and φ ( x g ( x = φ( x φ( x = + x + x + x + x (8 thus the resultng code s a prtve (5, 7 BCH code wthd n t + = 5. The party check atrx for ths code s. H = 6 9 (9. The general defnton of bnary BCH codes s as follow. Let β be an eleent of GF( and b any nonnegatve nteger. Then a bnary BCH code wth desgn dstance δ has a generator polynoal g ( x wth the followng consecutve powers of b b+ b+ δ β as roots β, β,..., β let ψ ( x and n be the nal polynoal and b the order of β +, respectvely. Then g ( x = LCM { ψ( x, ψ( x,..., ψ b + ( x } (0 wth a code length n = LCM { n, n,..., n b + } ( The BCH code defned above s called a nonprtve, wde-sense bnary BCH code wth a desgn dstanceδ. As a specal cases when b = the code becoes narrow-sense and f β s a prtve eleent, the code s prtve code.. Decodng of bnary BCH codes Decodng process of the BCH codes s the ost challengng task. Manly, we have two decodng algorths for BCH codes, naely: Peterson- Gorenten-Zerler algorth and Berlekap- Massey algorth. Assue that the receved code word ( r0, r,..., r n s dffers fro the sent code word ( c0, c,..., cn n x, x,..., x postons, then the error code word wll have a nonzero eleents at these postons and the error polynoal can be wrtten as ex = x + x x ( Both of these algorths need the coputaton of the syndroes of the receved code polynoal rx. Defne the syndroe S j to be ( j S j = r = e(, j =,,...,t ( or S = X + X X S = X + X X..... ( t t t S t = X + X X l where X l = s the error locatons. Defnng what s called error locator polynoal as Λ ( x =Λ x +Λ x Λ x + = ( xx ( xx...( xx (5 that has zeros at x = X l. It can be shown that ( and (5 can be coupled together n atrx for and wrtten as S S. St Λt St+ S S. S t t S + Λ = t (6 St St+. S t Λ St A Peterson s algorth s based on solvng (6 for Λ 's. If A s found to be sngular that eans we have less than t errors n the receved code word. In ths case we have to reconstruct a new syndroe atrx by deletng the two rght ost coluns and the two botto rows fro A and 77

7 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. solve a gan for Λ 's excludng Λt and so on. After Λ 's are found the error correct polynoal defned n (5 s constructed. Fnally, the roots of Λ( x are to be found usng Chen s search algorth and the error locatons set to be the recprocal of these roots... Berlekap s decodng algorth Berlekap s algorth s uch ore dffcult to understand than Peterson s approach, but results n ore effcent pleentaton. Berlekap s algorth for bnary BCH codes decodng s a recursve algorth that s suarzed n the followng steps [] a Defne the syndroe polynoal t S ( x = Sx + Sx S t x b Set the ntal condtons 0 0 k = 0, Λ x =, andt x = ( k c Let be the coeffcent of x k + n the ( k product Λ ( x + S ( x. d Copute ( k + ( k ( k ( k Λ x = Λ ( x + xt. ( x. e Copute ( k ( x ( k xt f = 0 or ( k deg ( x Λ > k ( k + T ( x = ( k xλ ( x ( k f 0 and ( k ( k deg ( x Λ k f Set k = k +. f k < t then go to step k Λ x = Λ x If the g Deterne the roots of roots are dstnct, then correct the correspondng locatons n the receved code word and STOP. h Declare a decodng falure and STOP.. RESULTS The a of ths paper s to sulate the perforance of BPSK n an AWGN envronent wth hard decson detecton usng (6,6 bnary BCH code wth error-correctng capabltyt = 5. The key thng here; as I beleve; s to buld up the alpha table for the code whch contans the - tuple representaton of the eleents n GF 6 6 ( = 6{ 0,,,,..., }. The power of for each entry n the table s evaluated usng ndex. functon. Note that the ndces are {,0,,,...,6} wth refers to the 0 eleent n the feld. Ths s done so that addng two eleents n the feld s perfored by addng the Fg. Average BEP for BPSK systes. 78

8 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. two -tuple of the eleents usng the alpha table. The ndex for the resultng tuple wll be the resultng eleent. Whle the ultplcaton operaton s perfored by addng the two ndces of the two eleents. The average BEP for the sulated syste s shown n Fg.. For coparson purpose, the BEP for un-coded BPSK s also shown usng the analytcal forula Pb = erfc( γ b, where γ b = Eb / N 0 s the average SNR/bt. As we see, for low SNR, the un-coded syste perforance s better than the coded one. That s because at low SNR let us say db, the BEP for the encoded case s about 0.0. For coded syste, the SNR wll be reduced by k / n = 0.57so that t wll be about 0. db and hence the BEP s around 0., whch eans that out of 6 code word bts 6 bts wll be n error after the hard decson detecton. So, the decodng algorth wll fal n detectng these errors, whch leads to worse perforance. 5. CONCLUSIONS The BEP for coded BPSK syste n syetrc AWGN channel based on a hard decson decodng was sulated. The codng schee that used s bnary BCH code wth error-correctng capablty. The syste s perforance proveent usng channel codng at reasonable SNR s consderable n ost cases. Snce ths proveent s due to the redundancy that s nserted by the codng technque, the prce to be pad for ths proveent s the hgher transsson data rate and hence hgher transsson bandwdth s requred. Generally, analytcal evaluaton of the coded syste perforance s very tough, so sulaton should be carred to do that. 6. ACKNOWLEDGEMENT Ths work was supported by the Deanshp of Scentfc Research and Graduate Studes, The Hashete Unversty, Zarqa, JORDAN. REFERENCES [] C. E. Shannon: "A atheatcal theory of councaton," Bell Syste Techncal Journal, vol. 7, October 98, pp. 79. [] J. Proaks, Dgtal councatons, NY: McGraw Hll, 00. [] S. Ln and D.J. Costello, Jr. Error Control Codng: Fundaentals and Applcatons, Englewood Clffs, NJ: Prentce Hall, 98. [] Joner, L.L. and Koo, J.J., "Decodng bnary BCH codes," IEEE Southeastcon Proceedngs, Mar 995, pp [5] Salah A. Aly, Andreas Klappenecker and Pradeep Kran Sarvepall," On Quantu and Classcal BCH Codes," IEEE Transactons on Inforaton Theory, vol. 5, March 007, pp [6] Kaneoto, H.; Myaoto, S.; Mornaga, N., "Perforance of dgtal rado councaton syste wth BCH codng under crowave oven nterference," IEE Electroncs Letters, vol., Jul 998, pp [7] Vardy, A. and Beapos;ery, Y., "Maxulkelhood soft decson decodng of BCH codes," IEEE Transactons on Inforaton Theory, vol. 0, Mar 99, pp [8] Mchelson, A.M. and Freean, D.F., "Bterror rate perforance of the (6,57 Hang code and a severely punctured convoluton code wth axu lkelhood decodng," IEEE Mltary Councatons Conference, -5 Oct 99. [9] Kuar, U.K. and Uashankar, B.S. "Iproved Hang Code for Error Detecton and Correcton," IEEE Internatonal Syposu on Wreless Pervasve Coputng, 5-7 Feb 007. [0] MacMullan, S.J. and Collns, O.M., "The capacty of bnary channels that use lnear codes and decoders," IEEE Transactons on Inforaton Theory, vol., pp. 97, Jan 998. [] Sayegh, S. and Heat, F. "Dfferentally encoded M-PSK block codes," IEEE Transactons on Councatons, vol., pp. 6 8, Sep

9 Journal of Theoretcal and Appled Inforaton Technology JATIT. All rghts reserved. BIOGRAPHY: Mahoud A. Sad was born n An-Naeh, Jordan, n 975. He receved hs B.Sc. and M.Sc. degrees n Electrcal and Electroncs Engneerng fro the Jordan Unversty of Scence and Technology, Irbd, Jordan, n 998 and 000, respectvely, and the Ph.D. degree fro the Unversty of Texas at Arlngton, Texas, n 005. Currently, he s wth the Departent of Electrcal Engneerng, Hashete Unversty, Zarqa, Jordan. Hs current research nterests nclude wreless councatons wth carrer synchronzaton, spread-spectru councatons (CDMA, and sulaton of councaton systes. 80