Performance Comparison of Two Inner Coding Structures in Concatenated Codes for Frequency-Hopping Spread Spectrum Multiple-Access Communications

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1 Iteratoal Joural o Recet ad Iovato Treds Computg ad Commucato IN: Volume: 3 Issue: erformace Comparso of Two Ier Codg tructures Cocateated Codes for Frequecy-Hoppg pread pectrum Multple-Access Commucatos e Hoo Lee Departmet of lectroc ad IT Meda geerg eoul Natoal Uversty of cece ad Techology eoul Korea y.lee@sut.ac.r Abstract I ths paper we compare the performace of two er codg structures for cocateated codes slow frequecyhoppg spread spectrum multple-access commucato systems. It s assumed that two outer code symbols are trasmtted durg a hop. The frst structure cossts of oe er codeword per oe outer code symbol whle the secod structure cossts of oe er codeword per two outer code symbols. We aalyze the overall bloc error probablty asymptotc rego ad show that the performace of the secod scheme s superor to the frst oe. Keywords- Cocateated code; er code; slow hoppg; frequecy hoppg; spread spectrum; multple access ***** I. INTRODUCTION We cosder a frequecy-hopped (FH) spread-spectrum multple-access (MA) commucato system utlzg cocateated codes. The basc features of the FH commucato system are dscussed [1 ad the ey characterstcs of the cocateated codes are dscussed [. I FH MA systems may users are desred to share a gve badwdth wth a gve error probablty ad errors occur prmarly due to multple access terferece (MAI). To mtgate the MAI a cocateated codg system s employed ths paper. We assume throughout ths paper that two outer code symbols are trasmtted durg a hop. Cocateated codes form a class of error-correctg codes that are derved by combg a er code ad a outer code. They were coceved as a soluto to the problem of fdg a code that has both expoetally decreasg error probablty wth creasg bloc legth ad polyomal-tme decodg complexty. Cocateated codes became wdely used space commucatos the 1970s ad recetly adopted Dgtal Televso Terrestral Broadcastg (DTTB) [3. The most atural choce for outer codes s Reed-olomo (R) codes cocateated codes. Because the R codes beg maxmum-dstace-separable codes mae hghly effcet use of redudacy ad well suted to burst error correcto [4. We wll use R codes as outer codes throughout ths wor. The er code we cosder ths paper s error detectg or correctg bary bloc code. The er code corrects e c errors ad detects e d errors provded e c + e d < d m where d m s the mmum dstace of the er code. Whe a error s detected every symbol of the er code s erased. There are however errors that are ot detected or corrected by the er code whch results errors at the output of the er decoder. The purpose of the outer code s to correct the errors ad erasures of the er code. ce we assume that two outer code symbols are trasmtted durg a hop we ca cosder two d of er codg structures the cocateated code. The frst structure cossts of oe er codeword per oe outer code symbol; we wll call ths (N K)(N K) scheme. Ad the secod structure cossts of oe er codeword per two outer code symbols; we wll call ths (N K) scheme. I ths paper we aalyse the overall bloc error probablty of the cocateated codes wth (N K)(N K) ad (N K) schemes asymptotc rego ad the performace of two er codg schemes. Ths paper s orgazed as follows. The system ad the chael model are troduced ecto II. The (N K)(N K) ad (N K) schemes are preseted ecto III ad ecto IV respectvely. The asymptotc aalyss ad umercal results are provded ecto V. Fally a cocluso s made ecto VI. Fgure 1. IJRITCC February II. TM AND CHANNL MODL A FH MA commucato system wth I smultaeous users We cosder a FH/ pacet rado etwor whch I users wsh to commucate smultaeously over a commo chael as show Fgure 1. ach source geerates messages whch are depedet of other users. There are I separate ecoders oe for each source. The j th ecoder receves oly the message from the j th source ad produces a codeword (x j1 x j...x j ) x j X where X s a commo put alphabet. At the j th recever the j th frequecy dehopper whch has owledge of the j th hoppg patter dehops the receved sgal. Ad the dehopped sgal s demodulated to produce the output vector (y j1 y j...y j ) y j 741

2 Iteratoal Joural o Recet ad Iovato Treds Computg ad Commucato IN: Volume: 3 Issue: where s a commo output alphabet. Decodg s doe where depedetly at each of the I recever ad thus there s o cooperato betwee users o ether the trasmttg ad the! recevg sde. The dvdual chaels ca be characterzed te t! e! t e! by (y x j ) j {1 I 1} whch s detcal for all users [1[5. We assume that all FH trasmtters adjust ther tmgs of III. (N K)(N K) INNR CODING CHM frequecy chages (sychroous frequecy hoppg) ad trasmt two outer code symbols durg a hop (slow frequecy TABL I. TH KIND OF VNT JOINT ROBABILITI AND NUMBR OF hoppg). Thus the mult-user terferece level durg a hop OCCURRNC IN TH (N K)(N K) CHM wll rema costat throughout the hop. We assume that the Kds of vets Jot robablty No. of Occurreces hoppg patter s essetally radom whch maes the d 1 d d1 d terferece durg a hop depedet of that of the other hop 1 tervals. Whe I users trasmt ther pacets smultaeously t ud 1 ud ud1 ud s probable that +1 {0 1 I-1} users occupy a c 1 c c1 c 3 partcular frequecy slot smultaeously. If a frequecy slot s d occuped by +1 users the slot ca be modeled by a bary 1 ud d1ud 4 symmetrc chael (BC) wth a chael crossover ud 1 d ud1d 5 probablty p gve by 1 p 1 O the other had the probablty of the chael I beg chose.e. the probablty of +1 users occupyg the same frequecy slot hi () s gve by I 1 I q q ( ) I 1. hi Fgure. M-ary erasures ad errors chael model At the recever sde the demodulated data s frst decoded wth the er decoder ad the the symbols from the er decoder are decoded wth the outer decoder. Whe a er code s used for detectg errors a super chael created by the er code ca be modeled by M-ary erasures ad errors chael as show Fgure. The purpose of the outer code s to correct the errors ad erasures of the er code. From the mmum dstace property [4 the ( ) Reed-olomo code wth bouded dstace decodg ca correct up to e = erasures or up to t = ( )/ errors. More geerally t ca correct ay combato of l erasures ad m errors provded that m + l does ot exceed. Thus the probablty of overall bloc (a outer codeword) error for the memoryless chael s gve by ( 1 ) e t t e d ud d ud te te. IJRITCC February d1 c c1 d ud1 c c1 ud d 1 c 6 c 1 d 7 ud 1c 8 c 1ud 9 I the (N K)(N K) scheme cosdered ths secto the two er decoder operate depedetly. The the two outer symbols whch have passed through the super chael have jot probabltes ad the umber of occurreces are gve Table I. I Table I represets that the evet of a outer d ud c where d ud c represet detected error symbol s udetected error ad corrected error respectvely. 1 j deotes that the frst outer symbol s ad the secod oe s j j d ud c. evets occurrg. Hece the codto express the umber of 9 / should be 1 satsfed sce there are / trasmssos to sed a outer codeword. The dstrbuto of / outer symbol pars passed though the super chael created by the er code s gve by multomal. The the overall bloc error probablty s gve by Where! cc 1 te! 1 3 d d ud ud d1ud d1c ud1c e t I (5) the probabltes { } j j d ud c are derved 1 Appedx A. IV. (N K) INNR CODING CHM If we employ oe er codeword to lmt two outer symbols the super chael model created by the er code s M-ary erasures ad errors chael whch s the same as Fgure 74

3 Iteratoal Joural o Recet ad Iovato Treds Computg ad Commucato IN: Volume: 3 Issue: Whe errors are detected or corrected by the er code the artcularly as the chael traffc (umber of users er codeword s decoded to correspodg two outer erasures smultaeously accessg a gve frequecy slot) gets larger the or corrected symbols respectvely. But f errors are udetected performace dffereces betwee (N K) ad (N K)(N K) by the er code t causes two cases. Oe s that the er scheme becomes more sgfcat. But we ca also ote that codeword s decoded to two udetected outer symbols. The there s o performace dffereces whe N = 0 ad N = 30. other s that t s decoded to a udetected (or corrected) ad a Ths meas that the shorter the redudacy of the er code s corrected (or udetected) outer symbol respectvely. We gore the better the performace of (N K) scheme s tha (N K)(N the latter case sce the probablty of the latter case occurrg K) scheme. s much lower tha that of the former. Thus we ca show that the overall bloc error probablty s gve by! t! e! t e! d ud d ud 4te e t t e 1. I (7) d ud ad c ca be derved from total probablty theorem [6. V. NUMRICAL RULT AND DICUION It s very dffcult to compare equato (5) wth equato (7) a fte rego sce we must fd all combato of to calculate the overall bloc error probablty equato (5). It s a very tme cosumg job computer computato. Hece we compare these two er codg schemes asymptotc rego. From Wea Law of Large Numbers [6 t ca be show that (see Appedx B) where 0 1 lm 1 r 1 r 1 r 1 Fgure 3. Maxmum achevable outer code rate by varyg I. (q = 5 = K = 048 K = 11 N = 15) 3 d1 c ud1 c d1 d d1 ud ud1 ud ad where 0 1 lm 1 r 1 r 1 r 1 d ud The we compare 1 wth 1 whch are the maxmum achevable outer code rates for error free commucato asymptotc rego. We have plotted 1 ad 1 as a fucto of smultaeously actve users I for N = Fgure respectvely. We assume outer bloc legth s 048 whch s large eough to smulate asymptotc rego. Ier message legth K s therefore 11. We ca see several facts from Fgure 3 Fgure 4 ad Fgure 5. Frst we ca see that the (N K) scheme gves better performace tha the (N K)(N K) scheme whe N =15. Fgure 4. Maxmum achevable outer code rate by varyg I. (q = 5 = K = 048 K = 11 N = 1) IJRITCC February

4 Iteratoal Joural o Recet ad Iovato Treds Computg ad Commucato IN: Volume: 3 Issue: where we assume that two receved er codewords are depedet. Ths assumpto ca be proved as follows. Based o the memoryless chael we have put vector X ( X1X ) ( x1x1 xn ) ad output vector 1 ( y y y ). From the defto 1 N N 1 ( ) ( X ) ( y x ) whch mples the chael crossover probablty p s tme varat ad depedet from bt to bt. The (15) s wrtte by N N N ( X) ( y x ) ( y x ) ( X ) ( X ). 1 1 Therefore each output symbols depeds oly o correspodg each put symbols dcatg that we ca assume two put symbols X1 X are depedet. Fgure 5. Maxmum achevable outer code rate by varyg I. (q = 5 = K = 048 K = 11 N = 30) ANDIX B I ths appedx we frst provde the proof of (8). Let radom varables X / ad Z /be defed as VI. CONCLUION I ths paper we cosdered two ds of er codg structures of cocateated codes for slow FH/ systems where partcularly two outer code symbols are trasmtted durg a hop. The frst structure cossts of oe er codeword per oe outer code symbol ad secod structure cossts of oe er codeword per two outer code symbols. We vestgated the overall bloc error probablty of two schemes asymptotc rego ad showed that the performace of the secod scheme s superor to the frst oe. ANDIX A I ths appedx we derve the probabltes { } j j d ud c (5). Let us defe 1 ad The 1 ( s selected) 1 j th ( j outer symbols s ) j 1 for { d ud c} j {1}. ad f the two receved symbols 0 are both corrected f oe receved symbol s 1 corrected ad the other s erased f the two receved symbols are X both erased or oe s corrected ad the other s error f oe receved symbol s erased 3 ad the other s error f the two receved symbols 4 are both error Z / / 1 X [ / / / Var[ / I 1 ( s selected) 1j j 1 1 I 1 1 ( s selected) 1 1 I 1 ( s selected) j 1 I 1 ( s selected) j 1 j respectvely. I (19) [ / ad Var [ / are the mea ad the varace of /. The / s the total umber of erasures ad twce the umber of errors the receved outer codeword. Therefore the overall bloc error probablty s gve by / / (1 r) /. From the wea law of large umber [6 IJRITCC February

5 Iteratoal Joural o Recet ad Iovato Treds Computg ad Commucato IN: Volume: 3 Issue: / [ X d 1c 4 ud1c d1d 6 d1ud 4 ud1ud lm X 0 / Combg (3) (5) ad (6) yelds For ay ths mples (1 r) X 1 X [ r 1 /. (1 r) X 0 X [ r 1 / lm (1 r) Therefore we get X [ 1 r 1. X [ 0 r 1 Here we assume X 1 / are the depedet ad detcally dstrbuted radom varable. The from cetral lmt theorem [6 1 u / lm ( Z / z) e du. / Also t ca be also show that We ow have z X [ lm 0.5 f r 1. 0 r 1 3 d1c ud1c d1d d1ud ud1ud d1c ud1c d1d d1ud ud1ud lm 0.5 r r 1 d 3 1c ud1c d1d d1ud ud1ud Followg aga the above procedure the proof of (10) s straghtforward. ACKNOWLDGMNT Ths study was supported by the Research rogram fuded by the eoul Natoal Uversty of cece ad Techology ( ). RFRNC [1. Geraots ad M. B. ursley rror probabltes for slow-frequecyhopped spread-spectrum multple-access commucatos over fadg chaels I Tras. Commuos vol. COM-31 pp May 198. [ G. D. Forey Cocateated Codes MIT ress Cambrdge MA [3 Z. ag M. Ha C. a A. Me ad L. ag A ovel scheme of codg ad modulato for dgtal televso terrestral broadcastg roc. of I IMRC pp [4 T. Blahut Theory ad ractce of rror Cotrol Codes Addso Wesley ub. Co [5 H. ato Two-user commucato chaels I Tras. o Iform. Theory vol. IT-3 pp May [6 A. apouls robablty Radom Varables ad tochastc rocesses McGrawHll Ic Kogausha Ltd IJRITCC February

K-Map 1. In contrast, Karnaugh map (K-map) method provides a straightforward procedure for simplifying Boolean functions.

K-Map 1. In contrast, Karnaugh map (K-map) method provides a straightforward procedure for simplifying Boolean functions. K-Map Lesso Objectves: Eve though Boolea expressos ca be smplfed by algebrac mapulato, such a approach lacks clear regular rules for each succeedg step ad t s dffcult to determe whether the smplest expresso