Poeengs of the 5th WSEAS Intentonl Confeene on Sgnl Poessng Istnbul Tukey y 7-9 6 (pp73-78 Convolutonl Dt Tnssson Syste Usng Rel-Vlue Self-Othogonl Fnte-Length Sequenes Jong LE n Yoshho TANADA Gute Shool of Sene n Engneeng Yguh Unvesty -6- Tokw Ube Yguh 755-86 JAPAN Abstt: A el-vlue self-othogonl fnte-length sequene hs n pulsve utooelton funton wth zeo selobes eept both shft ens Ths ppe pesents onvolutonl t tnssson syste usng the sequene A synhonzng sequene n t tn e onvolve wth nothe sequene to be hnge to Gussn sgnl The sgnl s quntze to n ntege tn n tnstte though hnnel n eeve sgnl s oelte wth the espetve sequenes Ths syste suppesses hnnel nose n the stoton on the pltue ltton n the quntzton n bngs n pulsve synhonzng pulse to esue ultpth ey-wos: self-othogonl fnte-length el-vlue sequene onvoluton oelton pltue ltton quntzton stoton ultpth t tnssson syste Intouton Fo the net geneton oble ountons the othogonl fequeny vson ultpleng (OFD t tnssson syste[] s nonte s one of the leng systes Howeve the OFD t tnssson syste hs the efets s pek stoton oheeny slp n nte e ntefeene whh ust be pove by the oplte ethos[][3] In oe to vo suh oplte ethos n keep hgh te tnssson we popose onvolutonl t tnssson syste whh kes the best use of the popetes of fnte-length pseuonose sequene The sequene s el-vlue self-othogonl fnte-length sequene n ts utooelton funton hs no selobes eept left n ght shft ens[4] In ths syste t tn s onvolve wth sequene to be hnge to Gussn sgnl The sgnl s quntze to n ntege tn n tnstte though hnnel The eeve sgnl s poesse by oelto Nuel epeents show tht the stoton bse on pltue ltton n quntzton s suppesse though the oelton poessng Rel-Vlue Self-Othogonl Fnte-Length Sequene An el pseuonose sequene whose utooelton funton hs no selobes eept t both shft ens s lle self-othogonl fnte-length sequene[4] sne ts shfte sequenes e othogonl n lte shft nge In ths seton the self-othogonl fnte-length sequene s eplne fo the pplton lte on An peo utooelton funton of the self-othogonl fnte-length sequene l of length ebe l n onl s epesse s ρ l l l l ε ( ε ( elsewhee whee l fo < n > n s shft n enotes ople onjugte ε s shft-en ople vlue gven by j ε ε e ϕ ( whee ϕ s phse n j The sequene s solve by the of n pulse tn weghte by the sequene vlue t evey te hp T ( t δ ( t T (3 l l
Poeengs of the 5th WSEAS Intentonl Confeene on Sgnl Poessng Istnbul Tukey y 7-9 6 (pp73-78 n ts Foue tnsfo ( A f l whee (t l (4 δ s D s elt funton of te t n jπft e n f s fequeny Fo Eq(4 n l ( t s elte to the enegy spetu of l l utooelton funton ρ of the sequene l s A ( l f ρ l l ( (5 By substtutng Eq( to Eq(5 n ftozng polynol wth espet to we obtn soluton fo the sequene n spetl on[4] gven by A whee l ε l e l α o β α l β + α 4 ε ε jϕ l ϕ j l e e (+ π j (6 (7 (8 (9 ( Fo the el-vlue sequene the phse of the shft-en oelton vlue s ϕ o π n the tes wth onjugte vlues n Eq(6 e obne Thus we hve fou foule ong to postve o negtve shft-en vlue of ε n even o o length Fo postve ε n o we hve the spetu soluton of the sequene l s A l ε os l + ( π + l l by eplng wth + 3 ± ± L ( ± n Eq(6 whee l l ( Fo negtve ε n o we hve the spetu soluton of the sequene s A l ε ( 3 os π l l ( l + l l + l (3 by eplng L wth 3 ± L ± n Eq(6 whee l 3 ( l l l (4 n the petes wth the k e efne s slly s those wthout the k We n synthesze the sequene of length l n shft-en negtve vlue ε fo the sequene λ of o length n shft-en postve vlue ε n the sequene of o length n shft-en negtve λ ε ε vlue Fo Eqs( n (3 we obtn the followng spetu of the sequene s l A ( f A A ( f l λ λ whee ε ε α α ε α ε ε Eq(8 s obtne fo the elton of Eq(7 Eq(5 s ewtten by the stn fo s (5 (6 (7 (8
Poeengs of the 5th WSEAS Intentonl Confeene on Sgnl Poessng Istnbul Tukey y 7-9 6 (pp73-78 A whee l ( 3 os ε n nπ l λ λ l + ( l l n l n + λ ; n l n λ ; n l (9 ( ( Slly we n synthesze the sequene of o length n shft-en negtve vlue fo the sequenes of the se even lengths n shft-en postve n negtve vlues wth the se gntue In othe wos the sequene of o length n shft-en negtve vlue s obtne fo the onvoluton between p of sequenes wth the shote se length n the shft-en postve n negtve vlues of the se gntue 3 Convolutonl Dt Tnssson by Rel-Vlue Self-Othogonl Fnte-Length Sequenes The pseuonose popetes of the el-vlue self-othogonl fnte-length sequene e effetvely utlze to onvolutonl t tnssson syste Fg shows sgnl llotent of the onvolutonl t tnssson syste usng the el-vlue self-othogonl fnte-length sequenes In Fg( synhonzng sequene w λ s llotte t the fst hlf of synhonzng ntevl ( T whee w s the weght fo the blne wth t powe n tn of t k + k L n s llotte t the font of the t ntevl N ( T Convolvng the sgnls of Fg( wth sequene λ kes the sgnls of Fg(b whee the sgnl n ts vege powe P n the t ntevl e gven by n k k λ k ( N ( P N ( + n (3 N ( + n the synhonzng sgnl s n ts vege powe P s n the synhonzng ntevl e gven by s P s w w w λ + ( l + ( λ (4 (5 Fg Sgnl llotent of onvolutonl t tnssson syste
Poeengs of the 5th WSEAS Intentonl Confeene on Sgnl Poessng Istnbul Tukey y 7-9 6 (pp73-78 Fo the lge nube n of t the heght of the sgnl pesents Gussn stbuton ue to the entl lt theoe The heght of the synhonzng sgnl s pesents ppotely Gussn stbuton fo the lost sequene If we l just these sgnl powe to σ P Ps then the stbuton of the heght of the sgnls n s ppotely epesente by s σ q( e (6 π σ In the se of n ( N ( + N >> >> n ε << we obtn the followngs fo Eqs(6 (8 (3 n (5 σ (7 (8 w (9 The sgnls n s of Fg(b e quntze to n ntege tn to be tnstte The vlue of the sgnl s lte by levels n n onvete to the ntege ˆ between A n A s follows: ˆ ; < < (3 A ; ˆ (3 A ; whee A n s oeffent so tht the powe of the ppote sgnl ˆ ght nely equl the powe of the pltue-lte sgnl n the quntzton nput The heght stbuton of the pltue-lte el-vlue sgnl s gven by q ( q( ; < < (3 ~ Qδ ( ; q ( ~ (33 Qδ ( + ; ~ whee δ ( s D s elt funton of n Q q( ef ( σ (34 The powe of the pltue-lte sgnl s lulte s P q ( σ ef ( ( σ σ ef ( σ e π σ σ + Q σ e π σ (35 n P 78 σ ; σ P 96 σ ; σ (36 P 998 ; 3 σ σ The eeve sgnl ˆ s poesse by oelto wth the efeene sequene λ to poue the sgnl ~ shown n Fg( whh s nlogous to the sgnl n Fg( The sgnl ~ s poesse by nothe oelto wth the efeene sequene to gve the synhonzng pulse s shown n λ Fg( We n etet the t k n Fg( by the synhonzng pulse n Fg( 4 Nuel Epeents We ene the popose t tnssson syste by nuel epeents whee 65 N 4 n ( N ( + 93 n w Fg shows vlue pttens of p of sequenes 65 λ n 65 λ n ths epeent whee ε / 65 λ 8574 [5] 65 λ 9585 65 λ 499 n ε / 65 λ 5438969 [5] λ 9963 λ 49 67 65 65 Fg Pttens of p of sequenes λ 65 65 λ
Poeengs of the 5th WSEAS Intentonl Confeene on Sgnl Poessng Istnbul Tukey y 7-9 6 (pp73-78 Fg3 shows the sgnls oesponng to those of Fg In Fg3( the synhonzng sequene 65 λ s ultple by n 93 t wth ± vlues est n the t ntevl 56 T whee T s otte n the fgue In Fg3(b the onvolve synhonzng sgnl s ppes the lge n gntue thn the onvolve t sgnl Fg3( shows the ntege sgnl ˆ wth the lt level 3σ 4 n the oesponng u ntege A 3 when quntze wth equl vson[5] Fg3( shows the oelte sgnl ~ wth the efeene sequene λ tht ontns the 65 sll stotons bse on the pltue ltton the quntzton n the sll ntefeenes on the shft-en oelton Fg3(e shows the oelte sgnl between ~ n the nothe efeene sequene whee the synhonzng pulse s obtne 65 λ wth pek n the low selobes Fg3 Sgnls n t tnssson syste ( Synhonzng sequene n t (b Convolve sgnl ( Quntze sgnl ˆ ( Coelte sgnl ~ wth 65 λ (e Coelte sgnl between ~ n 65 λ Fg4 shows the heght stbutons of the t sgnl n the synhonzng sgnl s In Fg4( the 47 sets of 93 no t e loe n the vlues of the sgnl e esue t the nge 8 whee the sgnl epesents suh sttstl popety s the sgnl of the vey long t length Fg4(b pesents the se stbuton s the vlue stbuton of 65 9 l Fg4 Heght stbutons of sgnls n s Fg5 shows the heght stbutons of the oelte t sgnl ~ fo sets of 93 no t when A 3 The evton fo the en vlue of t eeses s the lt level neses n the en vlues fo σ 8 6 σ 6 n 3σ 4 gve the vlues
Poeengs of the 5th WSEAS Intentonl Confeene on Sgnl Poessng Istnbul Tukey y 7-9 6 (pp73-78 ne 78 96 n 998 espetvely fo Eq(36 If we eese the te of the loe t we hve the slle stoton thn the stoton fo the fully loe t Fo the esults n Fg6 the seleton of 3σ n A fo to 4 s ppopte fo the ptl syste n the pesene of tve nose 5 Conluson The onvolutonl t tnssson syste usng the el-vlue self-othogonl fnte-length sequenes s popose In the tnstte the synhonzng sequene n two-vlue t e onvolve wth the othe sequene n quntze to be tnstte In the eeve the fst oelton eonstuts the t n the seon oelton gves the pulsve synhonzng pulse Ths syste suppesses the stoton on the pltue ltton n the quntzton s well s hnnel nose n bngs n pulsve synhonzng pulse to esue the ultpth Fg5 Heght stbutons of oelte t sgnl ~ wth 65 λ ( σ 8 6 (b σ 6 ( 3σ 4 ~ Fg6 shows the stoton D of the t sgnl fo sets of 93 no t σ σ 3 σ n A below 5 whee the stoton s gven by σ D log (37 µ n σ s the stn evton of the stoton n µ s the bsolute en vlue of the t Refeenes: [] RNee n RPs OFD fo Weless ulte Countons Ateh House [] SShephee JOss n SBton Aystot Lts n Pek Envelope Powe Reuton by Reunnt Cong n Othogonl Fequeny -Dvson ultple oulton IEEE Tn Coun Vol4 No pp5- Jnuy 998 [3] TShl n DCCo Robust Fequeny n Tng Synhonzton fo OFD IEEE Tn Coun Vol45 No pp63-6 Deebe 997 [4] YTn Othogonl Fnte-Length Sequene Sets Wth Ipulsve Autooelton Funton WSEAS Tn on Systes Vol3 Issue 6 pp4-46 August 4 [5] JLe n YTn Quntztons of Rel-Vlue Self-Othogonl Fnte-Length Sequene n The Effets on Coelton Pefone Po of the n Intentonl Wokshop on Sequene Desgn n Its Appltons n Countons pp-67 Otobe 5 Fg6 Output stoton