Multiparameter Golay 2-complementary sequences and transforms

Transcription

1 Mulipaamee Golay -plemeay sequeces ad asfoms V.G. Labues, V.P. Chasovsih, E. Osheime Ual Sae Foes Egieeig Uivesiy, Sibisy a, 37, Eaeibug, Russia, 6000 Capica LLC, Pompao Beach, Floida, USA Absac. I his wo, we develop a ew uified appoach o he so-called geealized Golay- Rudi-Shapio (GRS -plemeay sequeces. I based o a ew geealized ieaio geeaig cosucio. Keywods: geealized plemeay sequeces, mulipaamee Fouie-Golay-Rudi- Shapio asfoms, OFDM elemuicaio sysems.. Ioducio Biay ± -valued Golay-Rudi-Shapio sequeces (-GRS associaed wih he cyclic goup Z wee ioduced idepedely by Golay [,,3] i , Shapio [4,5] ad Rudi [6] i 95. M.J.E. Golay [] ioduced he geeal cocep of plemeay pais" of fiie sequeces all of whose eies ae ±. This was moivaed by a highly o-ivial applicaio o ifaed specomey. The he gave a explici cosucio fo biay Golay plemeay pais of legh m ad lae oed ha he cosucio implies he exisece of a leas m m!/ biay Golay sequeces of his legh. β γ They ae ow o exis fo all leghs N 0 6, whee βγ,, ae ieges ad βγ,, 0 (Tuy, [7], bu do o exis fo ay legh N havig a pime faco cogue o he modulo 4 (Eliahou e al., [8]. I 95, H. S. Shapio [4,5] ioduced wha became ow, afe 963, as he Rudi-Shapio polyomial pais. Shapio's wo was eiely i pue mahemaics. Budisi [9,0,] usig he wo of Sivaswamy [] gave a moe geeal ecusive cosucio fo Golay plemeay pais ad showed ha he se of all biay Golay plemeay pais of legh m obaiable fom i coicides wih hose give explicily by Golay. Fo a suvey of esuls o biay ad obiay Golay plemeay pais, see Byes ad Fa, Dael, [4], especively. I 999, Davis ad Jedwab [5] gave a explici descipio of a lage class of Golay plemeay sequeces i ems of ceai coses of he fis ode Reed-Mulle codes. Discee classical Fouie-Golay-Rudi-Shapio Tasfoms (FGRST i bases of diffee Golay- Rudi-Shapio sequeces ca be used i may sigal pocessig applicaios: muliesoluio by discee ohogoal wavele deposiio, digial audiio, digial video boadcasig, muicaio sysems (Ohogoal Fequecy Divisio Muliplexig - OFDM, Muli-Code-Divisio Muliple Access - MCDA, ada, ad cypogaphic sysems. IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08

2 V.G. Labues e al. Fo buildig he classical FGRSTs i bases of classical Golay-Rudi-Shapio sequeces he followig acos ae used: he Abelia goup Z, -poi Fouie asfom F, ad 3 he plex field C, i.e., hese asfoms ae associaed wih he iple ( Z, F, C. I his wo, we develop a ew appoach o he so-called geealized plex-, GF( p -, ad Cliffod-valued plemeay sequeces. The appoach is based o a ew ieaio geeaig cosucio. This Z, F, C, bu wih cosucio has a ich algebaic sucue. I is associaed o wih he iple ( ( Z, { F, F,..., F( }, A lg, whee {,,..., ( } F F F a se of abiay uiay iθ ( -asfoms of ype F (, (whee e : e A lg, e isead of F, A lg is a algebas (fo example, Cliffod algebas o fiie igs Z, o fiie Galois fields N GF ( q isead of he plex field C. The es of he pape is ogaized as follows: i Secio, he objec of he sudy (Golay-Rudi- Shapio biay sequeces is descibed. I Secio 3, he ieaio ule fo desig of he Golya maix is ioduced. I Secio 4, he poposed mehod based o ew geealized ieaio cosucio is explaied.. The objec of he sudy. Ieaio cosucio of Golay maices We begi by descibig he oigial Golay -plemeay ± -valued sequeces. 0 Defiiio. Le : ( c, c,..., cn ad ( s0, s,..., sn, whee c, s B { ± }. The 0 sequeces, pai ove { } 0 ae called he -plemeay ( 0 ±, if COR ( τ COR ( τ Nδτ, i ± -valued o Goley plemeay o 0 COM ( z COM ( z N, whee 0 0 COR ( τ, COR ( τ ae he peiodic coelaio fucios of, ad { } { } COM 0 ( z 0, COM ( z Z Z ae hei Z asfoms. Ay sequece, which is a membe of a Golay plemeay pai, is called he Golay sequece ad is Z asfom COM ( z Z is called he Golay-Shapio-Rudi polyomial (GSRP. { } We use wo symbols Z ad [0, ] [0, ] Z i z fo umeaio of Golay sequeces ad discee ime, especively. Fo iege [0, ] ad [0, ] we shall use biay codes (,,...,, (,,...,, whee i, i { 0, }, i,,...,. Le (,,..., ad (,,..., be biay codes, he defie i i (,,..., i, (,,..., i i i be ieges whose biay codes ae (,,..., ad (,,...,, whee, ae less sigifica bis (LSB ad, ae mos sigifica bis (MSB of (,,..., ad,,...,, especively. Obviously, IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 04

3 V.G. Labues e al. Z, Z,, Z Z Z,, Z Z,, Z Z Z,, Z Z, ( ( 3 ( ( (, Z Z Z, (, Z Z. Z, Z, (, Z Z Z, (, Z Z, 3, Z Z Z,, Z Z, Z Z Z,, Z Z, (, whee { } 0,... Z Z Z Z [ ] [ ] Le (,0 (, ad { 0,,,..., } Z., ( be a se of pais of plemeay sequeces of legh The he followig maix of deph has size. ( is called he Golay maix, whee he symbol of he veical cocaeaio of ( maices [ ] (,0( ae a pai of plemeay sequeces ad [ ] (, ( [ ] (,0(. Fo example, [ ] (, ( is IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 05

4 V.G. Labues e al. 3. Mehods [ ] [] [] [ ] [ ] The maix G is cosuced by a ieaio cosucio G G... G G. The iiial maix G is fomed by saig wih he Fouie-Walsh ( maix [] [] 0 G F [] ad by epeaed applicaio of he ieaio cosucio o pais of [ ] ows i he maix. Le us o suppose ha we have he Golay maix G. We eed o cosuc he ex Golay maix [ G ] [] usig oly G ad F G. The maix G have sucue simila (: [ ] [ ] Fo cosucig G [ ] fom [ G ] we ae each plemeay pai fom ( i he fom of ad cosuc shifed vesa of hei poes whee 0, ad s T is he shif opeao o s s posiios i ime domai: T f( : f( s. Now we cosuc he geeal buildig blocs fo he Golay ( [ ] -maix G : whee (0 ( ( (0 ( T T diag F (0 {, T T }. T T IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 06

5 V.G. Labues e al. Usig hese buildig blocs of ( -maix [ ] G we cosuc he Golay ( -maix [ ] G accodig o he followig ieaio ule [6]: [ ] [ ] (0 (,0 I T (,0 F [ ] Z [ ] ( (, (,,0 I T [ ] ( [ ] [ ] (, ( (,0 [ ] ] F T I (,0 [ ] (, [ ] T I (, [ ] [ ] [ ] (,0 ( (, ( (,0,0 ( [ ] [ ] [ ] (,0 ( (, ( (, 0, ( [ ] [ ], (, ( (,0 ( (,,0 ( (, (,0 ( [ ] (,, ( whee [ ] [ ] [ ] ( ( (, (,0,0 (,0 (, ( ( (, [ ] [ ] [ ] (,0, (,0 (, ( ( (, [ ] [ ] [ ] (,,0 (, (,0 ( ( (. [ ] [ ] [ ] (,, (, (,0 ae plemeay sequeces of wice legh, belogig o G. Hece, [ ] o [ ] [ ] ( β β [ ] ( (,, T (, β β 0 ( ( ( ( 0 ( (. [ ] [ ] (,0 (, IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 07

6 V.G. Labues e al. Sice (,, he believig ( ( β, we obai ( ( [ ] [ ] (,, (, 0 (, ( Hece, [ ] (,. 0 ( ( (,. [ ] [ ] (,, (, I is fially ecue elaio bewee plemeay sequeces fom G Example. [] (0( [] [] G F [] (, ( ( G ( [] (0,0 [] [] (0, ( [] [] (, (,0 ( [] (, ( (, [ ] ( diag { } diag{ ( }, ad G [] ( ( ( ( 0 ( ( ( 3 ( a whee 0, 3 0, G 3 (0,0,0( 3 (0,0, ( 3 (0,,0 ( 3 (0,, ( 3 (,0,0( 3 (,0, ( 3 (,,0 ( 3 (,, ( 3 [] ( 3 3 (,, 3 (,, 3 (, 3 (, ( ( 3 ( 3 3 { } ( ( 0 ( ( ( 3 ( 3( 3 4 { } ( 3 ( 3 3 ( 3 dig dig, whee 0, 4 0. [ ] Fom (5 we obai wo expessios fo (, [ ] ( i i i i i ( : [ ] i i ( i i i ii ( 3 i i i, [ ]. (4 (5 (6 (7 IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 08

7 V.G. Labues e al. whee 0, 0. New sequeces i (5 ae ohogoal ad plemeay sequeces. 4. Geealizaio Ou geealizaio uses he followig ieaio cosucio F F( 3 F( 4 [] [] [ F ] [ F F ] 3 [ F F F 3 ] G G, G,,... F F( [ ] [ ] G [ F F F ] G [ F F F F ]...,,...,,,...,,, based o a sequece of uiay asfoms: e iϕ F( e, e lg,,,...,. e A e Fo beviy le U : { F, F,..., F } ad U { F F F F } whee : (,,...,, : (,,...,, [ ] [ ] [ ] [ ] [ G ] [,,..., ] G G F, F,..., F(. We eed o cosuc he ex Golay maix [ ] [ ] [ G ] [ ( ] G U [ ] [ ] G [ F, F,..., F(, ] F usig oly G [ U ] ad ( [ ] [ ] [ ] maix G [ ] G [ U ] G [ F F F ] he same sucue as i ( :,,...,,,. Le us assume ha we have he Golay maix U F F F (depedig o pevious asfoms,,..., ( F. We ae goig o use fo Golay [ ] Fo cosucig G [ ] fom [ ] [ ] (,0 ( [ ] (, ( he followig ieaio ule (8 [ ] G we ae each plemeay pai i he fom of [ ] fom (8. The Golay ( -maix [ ] G is cosuced accodig o [ ] [ ] (0 (,0 I T (,0 F_ ( [ ] Z [ ] ( (,,0 [ ] I T (, [ ] [ ] (, ( (,0 [ ] ] (,0 F_ ( T I [ ] (, [ ] T (, I [ ] [ ] [ ] (,0 ( (, ( (,0,0 ( [ ] [ ] [ ] (,0 ( (, ( co m ( [ ],0, ( [ ], (, ( (,0 ( (,,0 ( (, ( (,0 ( [ ] (,, ( whee [ ] ( [ ] ( [ ] (, (,0,0 (,0 (, ( ( (, [ ] [ ] [ ] (,0, (,0 (, ( ( (, [ ] [ ] [ ] (,,0 (, (,0 ( ( (. [ ] [ ] [ ] (,, (, (,0 IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 09

8 V.G. Labues e al. [ ] ae plemeay sequeces of wice legh, belogig o [ ] G. Hece, (9 o [ ] [ ] ( ( ( (,, 0 ( 0 [ ] [ ] T (,0 T (, β 0 ( β [ ] T (, β β β ( ( [ ] β β (,. β β β 0 Sice, (, he believig β, we obai ( ( [ ] [ ] (,, T (, 0 0 ( [ ] (, [ ] (, 0 (, Theefoe, (. [ ] [ ] (,, (, (,,. (0 [ ] The Golay( -maix G [ ] is mulipaamee maix, depedig o paamees (,,...,,. I is easy o poof, ha sequeces (0 ae plemeay ad uiay sequeces. IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 00

9 V.G. Labues e al. Example. Le us cosuc [] [ ] [] G, G [ ] ad G 3 [ ] G 3 : ( a [] diag{ } diag { }, G [] (0,0(, [] (0, (, (, [] (, (, (,0 (, [] (, (, [] ( ( ( ( ( (, ( ( ( a a [] [] [] (0 a F ( ( a ( [] ( ( G [] [] a ( a ( a a ( a { } { } { } { } a [] G diag ( diag diag diag, [] ( a 3 a3 ( a 3 G 3 (,, 3 (,, ( 3,, 3 (, 3 (, ( a a 3 { 3 } G { } 3 3 diag diag. The esulig maix sill has he ohogoal ows ad evey pai is plemeay i he Golay- Rudi-Shapio sese. Fom (0 we see ha [ ] [ ] a { } { } a a [ ] ( G (,,...,,,.., (,,..., diag 3... a a a diag..., G [ ] ai [ ] i (,,.., i (,,.., i i i (,,...,. ( jθ jθ jθ e e, e e,..., e e C ae plex umbes, he G is he plex-valued ( C - If valued Fouie-Golay-Rudi-Shapio asfom (FGRST. [ ] If,,..., GF( p, e G (,,..., is he umbe heoeical Galois-Golay-Rudi- Shapio asfom (GGRS-NTT, if,,..., Clif, whee Clif is he Cliffod algeba, he [ ] G (,,..., is he Cliffod-Golay-Rudi-Shapio asfom, if,,..., Ham, whee Ham is he quaeio Hamilo algeba, he [ G ] (,,..., is he Hamilo-Golay-Rudi- Shapio asfom ad so o. 5. Coclusio I his pape, we have show a ew uified appoach o he so-called geealized plex-, GF( p -,o Cliffod-valued plemeay sequeces. The appoach is based o a ew ieaio geeaig cosucio. This cosucio has a ich algebaic sucue. I is associaed o wih he iple Z, F, C, bu wih Z,{ F (, F (,..., F ( }, A lg, whee { F, F,..., F ( } a se of ( ( iθ abiay uiay ( -asfoms of ype F(, (whee e : e A lg, e isead of IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 0

10 V.G. Labues e al. F, A lg is a algebas (fo example, Cliffod algebas o fiie igs Z, o fiie Galois N fields GF ( q isead of he plex field C. 6. Acowledgmes This wo was suppoed by gas he RFBR ad by Ual Sae Foes Egieeig s Cee of Excellece i Quaum ad Classical Ifomaio Techologies fo Remoe Sesig Sysems. 7. Refeeces [] Golay, M.J.E. Mulispli specomey / M.J.E. Golay // J. Opical Sociey Am., 949. P [] Golay, M.J.E. Complemeay seies / M.J.E. Golay // IRE Tas. Ifomaio Theoy. 96. Vol. IT-7. P Golay, M.J.E. Sieves fo low auocoelaio biay sequeces / M.J.E. Golay // IEEE Tas. Ifom. Theoy Vol. IT-3. P [4] Shapio, H.S. Exemal poblems fo polyomials ad powe seies / H.S. Shapio // ScM. Thesis, Massachuses Isiue of Techology, 95. [5] Shapio, H.S. A powe seies wih small paial sums / H.S. Shapio // Noices of he AMS Vol. 6(3. P [6] Rudi, W. Some heoems o Fouie coefficies / W. Rudi // Poc. Ame. Mah. Soc Vol. 0. P [7] Tuy, R.J. Hadamad maices, Baume-Hall uis, fou-symbol sequeces, pulse pessio, ad suface wave ecodigs / R.J. Tuy // J. Combi. Theoy (A Vol. 6. P [8] Eliahou, S. A ew esicio o he leghs of Golay plemeay sequeces / S. Eliahou, M. Kevaie, B. Saffai // J. Combi. Theoy (A Vol. 55. P [9] Budisi, S.Z. New plemeay pais of sequeces / S.Z. Budisi // Eleco. Le Vol. 6. P [0] Budisi, S.Z. Efficie pulse pesso fo Golay plemeay sequeces / S.Z. Budisi // Eleco. Le. 99. Vol. 7. P [] Budisi, S.Z. Desigig ada sigals usig plemeay sequeces / S.Z. Budisi, B.M. Popovic, L.M. Idji // Poc. IEE Cof. RADAR, 987. [] Sivaswamy, R. Muliphase plemeay codes / R. Sivaswamy // IEEE Tas. Ifom. Theoy Vol. IT-4. P Byes, J.S. Quadaue mio files, low ces faco aays, fucios achievig opimal uceaiy piciple bouds, ad plee ohoomal sequeces a uified appoach / J.S. Byes // Applied ad Compuaioal Hamoic Aalysis P [4] Fa, P. Sequece Desig fo Commuicaios Applicaios (Commuicaios Sysems, Techologies ad Applicaios / P. Fa, M. Daell // Tauo. U.K: Res. Sudies, 996. [5] Davis, J.A. Pea-o-Mea Powe Cool i OFDM, Golay Complemeay Sequeces, ad Reed-Mulle Codes / J.A. Davis, J. Jedwab // IEEE Tas. Ifom. Theoy Vol. IT- 45(7. P [6] Rudblad-Osheime, E. Uified appoach o Fouie-Cliffod-Pomeheus sequeces, asfoms ad file bas / E. Rudblad-Osheime, I. Niii, V. Labues // Compuaioal Nomuaive Algeba ad Applicaios. Eds. by Byes J, Osheime G. Dodec, Boso, Lodo: Kluwe Academic Publishes, 003. P IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-08 0