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Quasi-Orthogonal Sequences for Code-Division Multiple-Access Systems Kyeongcheol Yang, Member, IEEE, Young-Ky Kim, and P

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Quasi-Orthogonal Sequences for Code-Division Multiple-Access Systems Kyeongcheol Yang, Member, IEEE, Young-Ky Kim, and P 982 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000 Quasi-Orthogonal Sequences for Code-Division Multiple-Access Systems Kyeongcheol Yang, Member, IEEE, Young-Ky Kim, and P. Vijay Kumar, Member, IEEE Abstract—In this paper, the notion of quasi-orthogonal se- correlation between two binary sequences and of quence (QOS) as a means of increasing the number of channels the same length is given by in synchronous code-division multiple-access (CDMA) systems that employ Walsh sequences for spreading information signals and separating channels is introduced. It is shown that a QOS sequence may be regarded as a class of bent (almost bent) functions possessing, in addition, a certain window property. Such sequences while increasing system capacity, minimize interference where is computed modulo for all . It is easily to the existing set of Walsh sequences. The window property gives shown that where denotes the the system the ability to handle variable data rates. A general procedure of constructing QOS's from well-known families of Hamming distance of two vectors and . Two sequences are binary sequences with good correlation, including the Kasami and said to be orthogonal if their correlation is zero. Gold sequence families, as well as from the binary Kerdock code Let be a family of binary is provided. Examples of QOS's are presented for small lengths. sequences of period . The family is said to be orthogonal if Some examples of quaternary QOS's drawn from Family are any two sequences are mutually orthogonal, that is, also included. for any and . For example, the Walsh sequence family of Index Terms—Bent functions, code-division multiple-access sys- length is orthogonal. Where there is no chance of confusion, tems, Gold sequences, Kasami sequences, Kerdock codes, quasi-or- we will abbreviate and write instead of . thogonal sequences, Walsh sequences. Consider a synchronous system without multipath time dis- persion, where a sequence family is employed to I. INTRODUCTION both spread the signal bandwidth as well as distinguish between ODE-division multiple-access (CDMA) systems use different users. We consider the case when binary phase-shift C pseudo-noise binary sequences as signature sequences keying (BPSK) is used to modulate the signal. Let to distinguish between the signals of different users. Several be the binary information signal of the th user at spread-spectrum communication systems also use them as the th information bit time. Then each bit is spread into spreading codes that help achieve a low probability of intercept chips by the signature sequence of the th-user channel by spreading the signal energy over a large bandwidth. De- during the th information bit time as follows: sirable characteristics of pseudo-noise binary sequences used for such applications include long-period, low out-of-phase autocorrelation values, low crosscorrelation values, large linear Ignoring noise added to the signal in the channel, the received span, symbol balance, low nontrivial partial-period correlation signal during the th information bit time is given by values, large family size, and ease of implementation [7], [10], [20], [22]. Let be a sequence of length over . We will sometimes identify the sequence with the binary vector . The The receiver at the output of the th-user channel computes Manuscript received November 30, 1998; revised October 22, 1999. This work was supported in part by the Korea Research Foundation for the pro- gram year of 1998 and in part by the National Science Foundation under Grant CCR-9612864. The material in this paper was presented in part at the Con- ference on Difference Sets, Sequences and Their Correlation Properties, Bad (1) Windsheim, Germany, Aug. 2–14, 1998. K. Yang is with the Department of Electronic and Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Kyungbuk 790-784, Korea (e-mail: [email protected]). If is orthogonal, then and can be Y. Kim is with Telecommunications R&D Center, Samsung Electronics Co., easily determined from the sign of . When is not or- Pundang P.O. 32, Sungnam, Kyungki-Do 463-050, Korea (e-mail: ykim@ metro.telecom.samsung.co.kr). thogonal, it is necessary to minimize the second term in (1), P. V. Kumar is with Communication Sciences Institute, Electrical Engi- i.e., the interference from other channels. Since is typi- neering–Systems, University of Southern California, Los Angeles, CA 90089- cally assumed to take on values or with equal likelihood, it 2565 USA (e-mail: [email protected]). Communicated by S. W. Golomb, Associate Editor for Sequences. is necessary to minimize the absolute value of the correlations Publisher Item Identifier S 0018-9448(00)02889-3. between two distinct sequences in . 0018–9448/00$10.00 © 2000 IEEE YANG et al.: QUASI-ORTHOGONAL SEQUENCES FOR CDMA SYSTEMS 983 CDMA systems such as the IS-95 system employ Walsh II. QUASI-ORTHOGONAL SEQUENCES sequences of length in the forward link both as spreading Let be an integer. For a positive integer , let sequences and also to separate the different user channels be the sequence of length ,given [24]. Since the IS-95 system is synchronous in the forward by link, it is the inner product between the vectors associated with different user sequences rather than periodic correlation, that is a measure of interference from other channels. Walsh sequences are perfect in the sense that there is no interference (2) between any pair of sequences, as Walsh sequences form an . orthogonal family. However, the orthogonality limits the size . of an orthogonal family—there are only Walsh sequences of length . For this reason, it is impossible to increase the The sequences are said to be the ca- number of channels without either increasing the sequence nonical sequences of length . We will abbreviate and write length or else losing orthogonality between a pair of user se- instead of when there is no chance of confusion. quences. Any binary sequence of length has a Boolean expres- There are many situations where it is not appropriate to in- sion of the form crease the length of the sequence. A loss in orthogonality is in- evitable in such situations and gives rise to interference from (3) other channels. In [1], Bottomley proposed a set of signature sequences drawn from a Kerdock code for a synchronous direct sequence CDMA system where orthogonal spreading is used. where However, in this scheme, signature and spreading sequences are is a sequence obtained via bitwise multiplica- different and there is no requirement on correlation between sig- tion, and the addition of sequences is carried out bitwise modulo nature sequences over a subblock. [14]. For example, the sequence of length In this paper, the notion of quasi-orthogonal sequence may be expressed in the form . With (3) and (QOS) as a means of increasing the number of channels in (2) in mind, we will interchangeably write synchronous code-division multiple-access (CDMA) systems in place of . that employ Walsh sequences for spreading information signals For any integer , , we have the binary expan- and separating channels is introduced. It is shown that a QOS sion , where . sequence may be regarded as a class of bent (almost bent) Using this expansion, we define to be a linear combina- functions possessing, in addition, a certain window property. tion of 's, given by Such sequences while increasing system capacity, minimize interference to the existing set of Walsh sequences. The (4) window property gives the system the ability to handle variable data rates. A general procedure of constructing QOS's from It is not difficult to show that and are mutually or- well-known families of binary sequences with good correlation, thogonal for any . The family , defined by including the Kasami and Gold sequence families, as well as from the binary Kerdock code is provided. Examples of QOS's (5) are presented for small lengths. Complex or quaternary sequences, specifically sequences is a complete set of orthogonal sequences of length drawn from Family [2], [23], in place of binary sequences, known as the family of Walsh sequences of length . were first considered in [5] in order to expand the set of The Walsh family can be interpreted as a subcode of the binary Walsh sequences for CDMA systems. In the final part first-order Reed–Muller code , where is given by of this paper, the correlation properties in “windows” of the sequences in Family are studied. Computer searches were conducted (under a restriction on the subspaces associated with the windows) and used to provide examples of sets of QOS's for all lengths of the form , . where is the all-one sequence and The paper is organized as follows. In Section II, we give are the canonical sequences of length ,given some preliminaries and introduce the concept of quasi-orthog- in (2) (see [14]). In many applications, it is important to find onal sequences. The properties of a QOS sequence are studied the correlation between a binary sequence of length and the here. In Section III, a general procedure for the construction of code sequences in . This correlation is closely related to the QOS's is provided from well-known families including Kasami determination of a parameter of , known as the covering sequences, Gold sequences, and binary Kerdock codes. Quater- radius given by nary quasi-orthogonal sequences are discussed in Section IV, and some examples drawn from the Family are provided. Fi- (6) nally, concluding remarks and some open problems are given in Section V. where runs through all the binary vectors of length in . 984 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000 The covering radius of the first-order Reed–Muller TABLE I code has been intensively studied [4], [9], [15].
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