Un-polr Orhogonl Code : Degn, Anly nd Applon.C.S.Chuhn, MIEEE, hn Ahn, MIEEE, Eleron Engneerng Deprmen HBTI. Knpur (UPTU Lukno Knpur, Ind rm.hb123@gml.om, rhnhn@redffml.om.n.sngh, SMIEEE Elerl Engneerng Deprmen IIT Knpur Knpur, Ind ynngh@k..n Abr The un-polr orhogonl ode, h ode lengh n nd ode egh hvng uo-orrelon nd ro-orrelon onrn equl o 1, n be degned h propoed lgorhm. Th propoed heme degn ll he poble group onnng mxmum number of orhogonl ode per Johnon bound. In h e, eh of he degned group ndependen h oher group of un-polr orhogonl ode hvng mlr lengh, nd egh. The un-polr orhogonl ode from ny of he degned group n be ulzed for generon of opl gnure equene for gnmen o uer of Inoheren Opl Code Dvon Mulple Ae (OCDMA yem n pred perum ommunon over opl fber. All oher degned group of unpolr orhogonl my be ulzed for eury performne mprovemen, nreng perl effeny nd BE performne mprovemen of he OCDMA yem Keyord- uo-orrelon; ro-orrelon; Orhogonl bnry equene; I. INTODUCTION The un-polr orhogonl ode re no perfe orhogonl ode beue he ro orrelon for un-polr orhogonl ode n no be le hn one, hle he perfe orhogonl ode hould hve ro orrelon equl o zero. Hene, he un-polr orhogonl ode re peudo orhogonl ode. Here un-polr ode refer o bnry equene. A un-polr orhogonl odeord of lengh n nd egh nd ll n-1 yllly hfed veron repreen he me odeord. For he ode lengh n, 2 n bnry equene n be produed, ou of hh bnry equene of egh n be eleed. There re n C bnry equene of lengh n nd egh. Thee nlude n yl hfed bnry equene orrepondng o eh odeord. Thu ol number of ode ord ll be n (1/n C (n - 1! /! (n -!, (n - 1(n - 2 (n - ( - 1/.( - 1 2.1 In h propoed lgorhm, ll he un-polr ode of ode lengh n nd ode egh re genered. n Dfferene of Poon (DoP epreenon (DoP. From hh only hoe ode re eleed hh h mxmum uo-orrelon for nonzero hf equl o 1. Therefer ll poble group hvng mxmum number of ode re onrued uh h ll he ode n group hve muul ro-orrelon onrn equl o 1. The mxmum number of orhogonl ode for ode lengh n, ode egh uo-orrelon nd ro-orrelon onrn equl o 1, gven by Johnon bound A [2]. Some heme re propoed n [3-14] for generon of opl orhogonl ode n ngle group onnng un-polr orhogonl ode h uo nd ro-orrelon onrn lyng beeen 1 o -1. In hee heme he ode re genered h ome exr lmon on prmeer n,,, nd C. C number of ode n group. The propoed heme of generng unpolr orhogonl ode h no lmon on prmeer n,,, exep ome onepul lmon lke < (, < < n. Th heme genere ll poble group onnng mxmum number of unpolr orhogonl ode per Johnon bound [2] hle no onvenonl mehod [3-14] for degnng of unpolr orhogonl ode n degn more hn one group of unpolr orhogonl ode. Thee oher degned group of unpolr orhogonl ode my be ulzed for eury performne mprovemen, nreng perl effeny nd BE performne mprovemen of he OCDMA yem. The eon II derbe bou dfferen repreenon of un-polr orhogonl ode hh ne pr of. The eon III derbe he mehod of lulon of uo nd ro-orrelon onrn for unpolr orhogonl ode. In eon IV, he degnng mehod of un-polr orhogonl ode derbed ell ne mehod of lulon of uo nd ro orrelon onrn. Whle eon V derbe bou formon of group of un-polr orhogonl ode h uo nd roorrelon onrn equl o one. The eon VI bou onluon, pplon nd fuure ope of he reerh ork for un-polr orhogonl ode.
II. EPESENTATIONS OF UNIPOL OTHOGONAL CODES A every un-polr orhogonl ode ord lo h n yl hfed veron, n hve n repreenon hh ho he poon of b 1. Th ype of repreenon of un-polr orhogonl odeord my be lled eghed poon repreenon (WP or b 1 poon repreenon. Thee n repreenon n be redued by mkng ompulory poon of b 1 poon zero. Th redue he number of eghed poon repreenon of he orhogonl ode o. Th redued eghed poon repreenon my be lled ompulory eghed poon repreenon or fxed eghed poon repreenon (FWP. Suh repreenon of n orhogonl ode no unque; h repreenon of n orhogonl ode. To mke he repreenon of n orhogonl ode o be unque, Dfferene of Poon of b 1 propoed. An orhogonl odeord repreened n eghed poon repreenon or fxed eghed poon repreenon h fxed dfferene of poon (DoP of b 1 n erl nd yl order. The DoP elemen n be luled by ung modulo n ddon/ubron. There re DoP repreenon of n orhogonl ode ord. Eh of DoP repreenon h DoP elemen. If he loe or hghe DoP elemen fxed fr or l elemen repevely, he orhogonl ode n be repreened n unque mnner lled fxed loe DOP or fxed hghe DOP repreenon. I n be be underood by he follong exmple. Exmple: ode lengh n 13, egh 3 No. of orhogonl ode ord 12.11 / 3.2.1 22 A ode from bove exmple of n 13, 3 Code 111. I n be repreened n eghed poon repreenon (WP (, 2, 6. Th ode h 13 eghed poon repreenon due o yl hfed veron vz. (1,3,7, (2,4,8, (3,5,9, (4,6,1, (5,7,11, (6,8,12, (7,9, or (,7,9, (1,8,1, (2,9,11, (3,1,12, (4,11, or (,4,11, (1,5,12, (2,6, or (,2,6. Thee repreenon of n orhogonl ode n be redued by mkng fxed poon of b 1 zero lke (,2,6, (,4,11, (,7,9. There re 3 fxed eghed poon repreenon. All of he eghed poon repreenon (WP or fxed eghed poon repreenon n be hnged no dfferene of poon of b 1 repreenon n he gven follong mnner, he ode repreenon (,2,6 equvlen o (2-, 6-2, -6mod(13 (2,4,7 n DoP repreenon. Smlrly WP (, 4, 11 DoP (4, 7, 2, WP (, 7, 9 DoP (7, 2, 4, WP (2, 4, 8 DoP (2, 4, 7, nd WP (2, 9, 11 DoP (7, 2, 4, Smlrly for oher WP here re only 3 DoP repreenon vz. DoP (2, 4, 7 or DoP (4, 7, 2 or DoP (7, 2, 4. Thee DoP repreenon of n ode re yl hfed veron of eh oher. Thee DoP n be fxed o loe elemen fr poon DoP (2,4,7 or hghe elemen fxed l poon DoP (2,4,7, here n h e boh repreenon re me bu hee my be dfferen lo n oher e. Hene, unque repreenon of un-polr orhogonl ode 111 DoP (2, 4, 7. The unque repreenon of DoP (, b,, d of egh 4 n be onvered no WP DoP (, b,, d WP (,, +b, +b+. III. CALCULATION OF COELATION CONSTAINTS An un-polr orhogonl ode repreened by n bnry equene for every yle hfng of he ode n WP. The orrelon of n un-polr orhogonl ode h un-hfed bnry equene equl o egh of he ode Suppoe ode h ode lengh n nd egh be (x x 1 x 2... x n-1, x or 1 for n 1 The orrelon of h un-hfed equene gven by n x x hh ll be ly equl o. I lo uoorrelon pek hh pper he deeor for he deeon of bnry d equl o 1 repreened by h odeord. The ode h m un yl lef hfng repreened m (x m x m+1 x m+2... x m-1, x m+ gven under modulo n ddon for m n 1, The orrelon of h m (he yllly hfed veron gven by m x x m < m n 1 The uo-orrelon onrn defned nd gven Mxmum of,,..., m or ( 1 2 n 1 x x < m n 1 For un-polr orhogonl bnry equene, 1
Suppoe ode h ode lengh n nd egh be (y y 1 y 2... y n-1, y or 1 for n 1 The orrelon of h nd rulrly unhfed & hfed bnry equene ( m gven m x y m 1 m n The ro-orrelon onrn defned nd gven Mxmum of (,,..., 1 n 1 or x y m m n 1 For un-polr orhogonl bnry equene 1 [1]. IV. DESIGN OF UNI-POLA OTHOGONAL CODES A n he bove eon for repreenon of unpolr orhogonl ode, found h n un-polr orhogonl bnry equene of lengh n nd egh n be unquely repreened n DoP repreenon by fxng of fr loe elemen or l hghe elemen. Suppoe e re kng he e of fr loe, hen ll poble ode of lengh n nd egh n be genered by rng MATLAB bed progrm. I n be undernd by follong exmple, Code lengh n 13, egh 3, n DoP repreenon, Number of DoP elemen equl o egh.e. 3, Sum of ll DoP elemen equl o ode lengh n.e. 13 Suppoe ode (, b, n (+b n DoP, Fr mehod for degnng of UOC n be derbed follong Code n be genered h follong ondon for 3. (ι >, b >, >, (ιι b, (ιιι <, The ode re genered follong Code #1 (1, b1, 11 Code #2 (1, b2, 1 Code #3 (1, b3, 9 (,, Code #21 (3, b6, 4 Code #22 (4, b4, 5 Number of ode genered (n-1(n-2/.(-11 12.11/3.2.1 22, hene number of ode re verfed. For egh > 3, elemen re nlzed nd vred uh h ll elemen re pove. Fr elemen ly le hn l elemen bu hen le hn or equl o oher elemen, hen fr elemen le hn or equl o eond, eond elemen le hn or equl o hrd nd o on upo l elemen. Seond mehod for degnng of UOC n be derbed follo, n DoP h l hghe elemen, ll ode n be genered for me bove exmple, n13, 3 Number of DoP elemen equl o egh.e. 3, Sum of ll DoP elemen equl o ode lengh n.e. 13 Suppoe ode (, b, n (+b n DoP, Code n be genered h follong ondon for 3. (ι >, b >, >, b, (ιι (ιιι <, The ode re genered follong Code #1 (1, b1, 11 Code #2 (1, b2, 1 Code #3 (1, b3, 9 (,, Code #21 (5, b1, 7 Code #22 (5, b2, 6 For egh > 3, elemen re nlzed nd vred uh h ll elemen re pove. The l elemen ly greer hn fr elemen bu hen greer hn or equl o oher elemen, hen fr elemen le hn or equl o eond, eond elemen le hn or equl o hrd nd o on upo l elemen. All poble ombnon of ode n be genered for ode lengh n nd egh, by rng MATLAB bed progrm for. No nex hllenge o ele only hoe ode hh hve nd equl o 1. Then fr k o fnd he vlue of for hee ll poble ode for ode lengh n nd egh. Aully gven, for n orhogonl ode x x m < n 1 m
In bnry dg repreenon (x x 1 x 2... x n-1, x or 1 for n 1 m (x m x m+1 x m+2... x m-1, x m+ gven under modulo n ddon for < m n 1, The orhogonl ode remn me for eh hf,.e. 1 2 3... n The ode n FWP n be gven F (f, f 1, f 2,, f -1 men h he poon (f, (f 1, (f 2,, (f -1, re 1 ( eghed hle oher poon re. No hfng F by (f 1, (f 2,, (f (-1, un n lef rulrly o ge F1, F2, F(-1. No hfng F by (f 1, (f 2,, (f (-1, un n lef rulrly o ge F1, F2, F(-1. F1 1 (f, f 1, f 2, f -1 F2 2 (f, f 1, f 2, f -1 F(-1 (-1 (f, f 1, f 2, f -1 The orrelon of F h F, F1, F2,..., F(-1 n be luled 1 F F F F F F 1 F f 2 F F1 1 F F 2 2 F1 1 (f, f 1,, f -1 F2 2 (f, f 1,, f -1 F(-1 (-1 (f, f 1,, f -1 The orrelon of F h F1, F2,..., F(-1 n be luled 1 1 ( F F1 F F 1 1 2 ( 2 ( 2 F F f f F F f F F ( 1 Here nd F F F ( 1 ( 1 F repreen he rnpoed mrx of 1, (f. j(f,, j,, 1 or Mx (, 1 F F1 for ohere F F 2 (f (f,, F F ( 1 1 1 Smlrly for o ode nd of me ode lengh n nd egh, he ro-orrelon onrn n be luled n FWP, F (f, f 1, f 2, f -1 F (f, f 1, f 2, f -1 j F F ( 1 F F ( 1 ( 1 1, for (f j(f (f. j(f, ohere, j,, 1 Mx (,,,..., F F F F 1 F F 2 F F ( 1, 1 Suppoe he ode repreened n DoP h fr loe elemen or l hghe elemen, he ode n DoP n be onvered no FWP o fnd nd follo The ode n DoP gven D (d, d 1,... d -1, equvlen n FWP gven F (f, f 1,... f -1, here (f, (f 1 (d, (f 2 (d 1, (f -1 (d -2 One F formed from D, F hfed by (f 1, (f 2,, (f (-1, un n lef rulrly o ge F1, F2, F(-1 repevely. Smlrly F n be genered from D, he ode n DoP. One he F nd F genered hen
nd n be luled for ny pr of ode derbed bove. V. FOMATION OF GOUPS OF UNIPOLA OTHOGONAL CODES In bove eon ll poble un-polr orhogonl ode n be genered. For eh of hee ode he uo-orrelon onrn n be luled. No bg group of hoe un-polr orhogonl ode n be formed n hh eh ode h 1. From h bg group only hoe ode hve o be eleed hh hve 1 h eh oher. Suppoe here re A ode hh hve 1, he orrelon mrx AxA n be formed hh onn ll he vlue of nd of hee A ode h eh oher. Th orrelon mrx n help n eleon of hoe ode hh hve 1 The Johnon bound for ode lengh n, egh, nd 1 gven (n-1/(-1 J A. Th J A gve he mxmum number of orhogonl ode n he group for hh nd 1. The J A ode from A number of ode n be grouped n G A C J A mnner or dfferen group. The J A ode n eh of G group re heked for 1 h eh oher from orrelon mrx AxA. From hee G group H uh group re eleed hh h 1. Thee ndependen eh of H group onn only hoe ode hh hve nd 1. The J A unpolr orhogonl ode of ny of H eleed group n be ued opl gnure equene by pung n opl pule eghed poon n noheren opl dm yem. VI. CONCLUSION The propoed heme for he generon of unpolr orhogonl ode ble o genere no only one e of orhogonl ode bu ll oher poble e of un-polr orhogonl ode h uo nd roorrelon onrn equl o one, for me ode lengh n nd egh. In fuure he ork n be exended for he generon of group of o dmenonl un-polr orhogonl ode of dered or gven vlue of uo nd ro-orrelon onrn. The mn pplon of h heme n he generon of un-polr one-dmenonl opl orhogonl gnure equene o be ulzed n noheren opl CDMA yem. The dfferen genered poble e of orhogonl ode n be ulzed for eury purpoe ell for nreng perl effeny performne nd BE performne mprovemen of opl ode dvon mulple e yem. EFEENCES [1] Fn. K. Chung, Jd A. Sleh, member IEEE, nd Vor K. We, member IEEE Opl Orhogonl Code: Degn, Anly, nd Applon, IEEE Trnon on Informon Theory, Vol. 35, No. 3, My 1989. [2] ej Omrn nd P.Vjy Kumr; Code for Opl CDMA SETA 26, LNCS 486, pp. 34-46, 26. [3] A.A.Shr nd P.A.Dv, Prme equene: Qu-opml equene for hnnel ode dvon mulplexng, Eleron Leer, vol.19, pp. 888-889, Oober 1983. [4] A.S. Holme nd..a. Sym, All-opl dm ung qu-prme ode, Journl of Lghve Tehnology, vol. no.1,pp.279-286, Februry 1992. [5] S.V.Mr, Z.I.Ko, nd E.L.Tlebum, A ne fmly of opl ode equene for ue n pred-perum fberop lol re neork, IEEE Trn. On Communon, vol.41, pp.1217-1221, Augu 1993. [6] S,V.Mr, Ne fmly of lgebrlly degned opl orhogonl ode for ue n dm fber-op neork, Eleron Leer, vol.29, pp.538-539, Mrh 1993 [7] C.Argon nd H.F. Ahmd, Opml opl orhogonl ode degn ung dfferene e nd projeve geomery, Op Communon, vol.118, pp.55-58, Augu 1995. [8] M.Choudhry, P.K.Cherjee, nd J,John, Code equene for fber op dm yem, n Pro. of Nonl Conferene on Communon 1995, IIT Knpur, pp. 35-42, 1995. [9] H. Chung nd P.V. Kumr, Opl orhogonl ode ne bound nd n opml onruon, IEEE Trn. Informon Theory, vol. 36, pp. 866-873, July 199. [1] A.E. Brouer, J.B. Sherer, N.J.A. Slone, nd W.D. Smh, A ne ble of onn egh ode, IEEE Trn. Informon Theory, vol.36, pp. 1334-138, November 199. [11] Q.A. Nguyen, L. Gyorf, nd J.L. Mey, Conruon of bnry onn egh yl ode nd yllly permuble ode, IEEE Trn. Informon Theory, vol.38, pp 94-949, My 1992 [12] S.Bn nd T. Ezon, Conruon for opml onn egh yllly permuble ode nd dfferene fmle, IEEE Trn. Informon Theory, vol. 41, pp. 77-87, Jnury 1995. [13] M. Choudhry, P.K. Cherjee, nd J. John, Opl orhogonl ode ung hdmrd me, n Pro. of Nonl Conferene on Communon 21, IIT Knpur, pp. 29-211, 21. [14] ng, G. C. (1995. Some ne fmle of opl orhogonl ode for ode dvon mulple e fber op neork. IEEE Pro. Commun. 142(6: 363-368.