A New Class of Ternary Zero Correlation Zone Sequence Sets Based on Mutually Orthogonal Complementary Sets
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1 IOSR Joural of Electroics ad Commuicatio Egieerig (IOSR-JECE) e-issn: ,p- ISSN: Volume 0, Issue 3, Ver. I (May - Ju.205), PP A New Class of Terary Zero Correlatio Zoe Sequece Sets Based o Mutually Orthogoal Complemetary Sets B. Fassi, H. Mimou, R. Messaoudi, M. Addad Telecommuicatios ad Digital Sigal Processig Laboratory, Djillali Liabes Uiversity of Sidi Bel Abbes, Algeria Abstract: This paper proposes a ew class of terary zero correlatio zoe (ZCZ) sequece sets based o mutually orthogoal complemetary sets (MOCS), i which the periodic cross-correlatio fuctio ad the outof-phase of the periodic auto-correlatio fuctio of the proposed sequece set are zero i a specified zoe of phase shift. It is show that the proposed zero correlatio zoe sequece set ca achieve the upper boud o the terary ZCZ codes. Keywords - Sequece desig, mutually orthogoal complemetary sets, theoretical upper boud, zero correlatio zoe (ZCZ) sequeces. I. Itroductio Differet types of sequece set used i commuicatios systems have bee studied i order to reduce the Multiple Access Iterferece (MAI) [-6]. Zero correlatio zoe (ZCZ) sequeces, which have vigorously bee studied ow, are defied to be a sequece set with ZCZ which meas the duratio with zero auto-correlatio fuctio ad zero cross-correlatio fuctio at out-of-phase state [-2]. Whe ZCZ sequeces are used as spreadig sequeces i DS-CDMA, it ca effectively elimiate MAI if all multiple access delays are iside ZCZ [-6]. There are various itesive studies o the costructig of ZCZ sequeces icludig biary, terary ad polyphase sequeces [-2]. Geerally, sets of ZCZ sequeces are characterized by the period of sequeces N, the family size, amely the umber of sequeces M, ad the legth of the zero-correlatio zoe Z CZ [7]. A ZCZ (N, M, Z CZ ) sequece set that satisfies the theoretical boud defied by the ratio M Z cz + /N = is called a optimal zero-correlatio zoe sequece set [7, 0]. I this paper, we propose a ew costructio method to obtai terary ZCZ sequece sets based o biary mutually orthogoal complemetary set (MOCS) ad from paddig zero betwee sequeces of MOCS. Our proposed terary zero-correlatio zoe sequece set is almost optimal ZCZ sequece set. The paper is orgaized as follows. Sectio 2 itroduces the otatios required for the subsequet sectios, the proposed scheme for sequece costructio is explaied i sectio 3. Examples of ew ZCZ sequece sets are preseted i Sectio 4. The properties of the proposed sequece sets are show i Sectio 5. Fially, we draw the cocludig remarks. II. Notatios 2. Defiitio : Suppose X j =(x j,0, x j,,. x j,n ) ad, X v =(x v,0, x v,,. x v,n ) are two sequeces of period N. Sequece pair (X j, X v ) is called a biary sequece pair if x j,i, x v,i, +, i = 0,,2, N [2]. The Periodic Correlatio Fuctio (PCF) betwee X j ad X v at a shift τ is defied by [8]: N τ 0, θ X j,x v τ = i=0 x j,i x v, i+τ mod (N) ad θ X v,x j τ = θ X j,x v τ () 2.2 Defiitio 2: A set of M sequeces X 0, X, X 2,, X M is deoted by X j j=0 M A set of sequeces X j j=0 is called zero correlatio zoe sequece set, deoted by Z(N,M,Z CZ ) if the periodic correlatio fuctios satisfy [8] : j, 0 < τ Z cz, θ X j,x j τ =0 (2) j, j v, τ Z cz, θ X j,x v τ =0 (3) 2.3 Defiitio 3: Let each elemet of a H H matrix F () ( 0 ) be a sequece of legth l = 2 l m = 2m+, where lm=2m ad m 0 [5]. M. DOI: / Page
2 A New Class of Terary Zero Correlatio Zoe Sequece Sets Based o Mutually Orthogoal F () = F F 2 F H F 2 F 22 F k F H F 2k F 2H F H2 F Hk F HH (H H) (4) If F satisfies the followig formulas, the F is a MOCS [3]: H k= φ Fj,k (τ) = 0, for j, τ 0 (5) F j,k H k= φ Fj,k (τ) = 0, for j s, τ (6) F s,k Where φ Fj,k F j,k (τ) ad φ Fj,k F s,k (τ) are the aperiodic autocorrelatio ad cross-correlatio fuctios, respectively [0]. III. Proposed Sequece Costructio I this sectio, a ew method for costructig sets of terary ZCZ sequeces is proposed. The costructio is accomplished through three steps. 3. Step : The matrix MOCS of dimesio H H = (2 +, 2 + ). Sice each elemet i F () is a sequece of legth l = 2 l m = 2 m+, each raw has a legth of L = 2 2+m+. The matrix MOCS is ( m+ ) with 0 ad m 0 [5]. () () () () F F 2 F () () () = F 2 F F k F f H () () 22 F 2k F 2H = f j () () () () F H F H2 F Hk F HH f H (7) A set of M = 2H = 2 +2 sequeces d j, each of legth S = 2L + 2 = 2 2+m , is costructed as follows: For j H d j +0 = f j 0 f j 0 =[ F j () (). 0 F j () (). 0] (8) d j + = f j 0 f j 0 =[ F j () (). 0 F j () (). 0] (9) Where [f j 0 f j 0] is called the cocateatio operatio betwee two rows f j ad f j of the matrix F () ad two zeros. 3.2 Step 2: For the first iteratio, p = 0, we ca geerate, based o iterleavig techique i [0], a series of sets T j of M sequeces. A pair of sequeces T j +0 ad T j + of legth N = 2 p+ S = 2 p+ 2 2+m = 2 p+ 2 L+2 is costructed by iterleavig elemets (±Fjk et 0 ) of a sequece pair dj+0 ad dj+ as follows (by usig equatios (8) ad (9)): For j H, T j +0 = [d j +0,, d j +,, d j +0,2, d j +,2,, d j +0,S, d j +,S ] (0) T j +0 = [ F j F j 0 0 F j F j 0 0] ad, T j + = [d j +0,, d j +,, d j +0,2, d j +,2,, d j +0,S, d j +,S ] () T j + = [ F j F j 0 0 F j F j 0 0 ] The member size of the sequece set T j is M = 2H = Step 3: For p > 0, we ca recursively costruct a ew series of set, T j, by iterleavig of actual T j (i equatios (0) ad ()). Both sequeces T j +0 ad T j + are of legth N = 2 p+ S. DOI: / Page
3 A New Class of Terary Zero Correlatio Zoe Sequece Sets Based o Mutually Orthogoal The T j is geerated as follows: For j H, T j +0 = [T j +0,, T j +,, T j +0,2, T j +,2,, T j +0,2S, T j +,2S ] (2) T j +0 = F j F j F j F j 0000F j F j F j ad F j 0000 T j + = [T j +0,, T j +,, T j +0,2, T j +,2,, T j +0,2S, T j +,2S ] (3) T j + = [ F j F j F j F j 0000F j F j F j F j 0000] IV. Example Of Costructio 4. Step : Let F () a matrix MOCS H L) = (4 8, =, m = 0, l = 2 l m = 2 m + = 2. F = F F 2 F 3 F 4 F 2 F 22 F 32 F 42 F 3 F 23 F 33 F 43 F 4 F 24 F 34 F 44 = f f j f 4 F = Where (-) ad (+) are (-) ad (+), respectively. A set of M = 8 sequeces d j, each of legth S = 8, is costructed as follows (usig equatios (8) ad (9)): For j 4, d +0 = f 0, f 0 = F F 2 F 3 F 4 0F F 2 F 3 F 4 0 = [ ] It follows for the other sequeces: d 2+0 = [ ] d 3+0 = d 4+0 = ad, d + = f 0, f 0 = F F 2 F 3 F 4 0F F 2 F 3 F 4 0 = [ ] d 2+ = [ ] d 3+ = [ ] d 4+ = Step 2: For the first iteratio, p = 0. A pair of sequeces T j +0 ad T j + of legth N = 2 p+ S = 36 is costructed as follows (usig equatios (0) ad ()): For j 4, T +0 = [ ] T 2+0 = T 3+0 = T 4+0 = [ ] T + = [ ] T 2+ = [ ] T 3+ = [ ] T 4+ = [ ] DOI: / Page
4 Periodic Auto-Correlatio Fuctio A New Class of Terary Zero Correlatio Zoe Sequece Sets Based o Mutually Orthogoal 4.3 Step 3: For the ext iteratio p =, the T j is geerated as follows (usig equatios (2) ad (3)): T +0 = [ ] T 2+0 = [ ] T 3+0 = [ ] T 4+0 = [ ] T + = [ ] T 2+ = [ ] T 3+ = [ ] T 4+ = [ ] The legth of both sequeces T j +0 ad T j + is equal to N = 2 p+ S = 72. The member size of the sequece set T j is M = 8. Fig. shows the auto-correlatio fuctio give i Equatio of T +0, ad Fig. 2 shows the crosscorrelatio fuctio give i Equatio of T +0 with T The Periodic Auto-Correlatio Fuctio Delay Time Figure. The autocorrelatio fuctio of T +0 with ZCZ(72,8,6) DOI: / Page
5 Periodic Cross-Correlatio Fuctio A New Class of Terary Zero Correlatio Zoe Sequece Sets Based o Mutually Orthogoal 35 The Periodic Cross-Correlatio Fuctio Delay Time Figure 2. The cross-correlatio fuctio betwee T +0 ad T 2+0 with ZCZ(72,8,6) The periodic correlatio fuctio cofirm that T j is a ZCZ (72,8,6) sequece set. V. The Properties Of The Proposed Sequece The proposed terary ZCZ sequece set ca satisfy the ideal autocorrelatio ad cross-correlatio properties i the zero-correlatio zoe. The geerated sequece set satisfies the followig properties: j, τ 0, τ Z CZ θ T j,t j τ =0 (4) ad, j v, τ, τ Z CZ θ T j,t v τ =0. (5) The T j is a terary ZCZ sequece set havig parameters: ) For + m = 0, ZCZ N, M, Z CZ = ZCZ(2 p m +, 2 +2, 2 +m +p+ ) = ZCZ 2 p+ 6,4,2p+. 2) For + m =, ZCZ N, M, Z CZ = ZCZ(2 p m +, 2 +2, 2 +m+p+ 2) = ZCZ 2 p ,2+2,2p ) For + m >, ZCZ N, M, Z CZ = ZCZ(2 p m+, 2 +2, 2 +m+p + 4 p). VI. Coclusio I this paper, we have proposed a ew method for costructig sets of terary ZCZ sequeces based o biary mutually orthogoal complemetary set (MOCS) ad from paddig zero betwee sequeces of MOCS. The periodic auto-correlatio fuctio side lobes ad the periodic cross-correlatio fuctio of the proposed sequece set is zero for the phase shifts withi the zero correlatio zoe. The proposed terary ZCZ sequece set is almost optimal ZCZ sequece set. DOI: / Page
6 A New Class of Terary Zero Correlatio Zoe Sequece Sets Based o Mutually Orthogoal Refereces []. B. Fassi, A. Djebbari, ad A. Taleb-Ahmed, Terary Zero Correlatio Zoe Sequece Sets for Asychroous DS-CDMA, Joural Commuicatios ad Network, 6(4), 204, [2]. S. Reghui, Z. Xiaoqu, Li. Lizhi, Research o Costructio Method of ZCZ Sequece Pairs Set, Joural of Covergece Iformatio Techology, 6(), Jauary 20, [3]. P. Z.Fa, Spreadig Sequece Desig ad Theoretical Limits for quasi-sychroous CDMA Systems, EURASIP Joural o wireless Commuicatios ad Networkig, 2004, 9-3. [4]. H. Doela, T. O Farrell, Large Families of Terary Sequeces with Aperiodic Zero Correlatio Zoes for a MC-DS-CDMA System, Proc. Of 3 th. IEEE Itl. SPIMRC, 5, 2002, [5]. P. Z. Fa, N. Suehiro, N. Kuroyaagi ad X. M. Deg, Class of biary sequeces with zero correlatio zoe, IEE Electroic Letters, 35(0), 999, [6]. N. Suk-Hoo, O ZCZ Sequeces ad its Applicatio to MC-DS-CDMA, Master of Sciece, YONSEI Uiversity, [7]. B. Fassi, A. Djebbari, A Taleb-Ahmed, ad I. Dayoub, A New Class of Biary Zero Correlatio Zoe Sequece Sets, IOSR- JECE, 5(3),203, 5-9 [8]. T. Hayashi, A class of zero-correlatio zoe sequece set usig a perfect sequece, IEEE Sigal ProcessigLetters, 6(4),2009, [9]. T. Hayashi, Y. Wataabe, ad T. Maeda, A Novel Class of Biary Zero-Correlatio Zoe Sequece Sets by usig a Cyclic Differece Set, i ISITA: Iformatio Theory ad its Applicatios, Melboure, Australia, 204, [0]. T. Maeda, S. Kaemoto, T. Hayashi, A Novel Class of Biary Zero-Correlatio Zoe Sequece Sets, N IEEE, TENCON 200. []. T. Hayashi, A class of terary sequece sets havig a zero correlatio zoe for eve ad odd correlatio fuctios, IEEE ISIT Iformatio Theory, Yokohama, Japa, 2003, 434. [2]. H. Torii, M. Nakamura, ad N. Suehiro, A New Class of Polyphase Sequece Sets with Optimal Zero Correlatio Zoes, IEICE Tras. Fudametals, E88(7), 2005, [3]. C.C. Tseg, ad C.L. Liu, Complemetary sets of sequeces, IEEE Trasactios o Iformatio Theory, 8(5), 972, DOI: / Page
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