Sum Capacity of Equal-Power users in Overloaded ... - CiteSeerX

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N) are decreased as compared to other signature sets (e.g. PN), since they suffer from interference caused by the excess users only; note that no interference occurs when K ≤ N. One possibility is to assign random sequences to the excess users, resulting in the Pseudonoise/OCDMA (PN/O) system of [17]. This PN/O system is of particular importance, since its performance corresponds to the average performance over all overloaded systems that contain a subset of N orthogonal sequences1. In order to decrease the interference levels of the excess users as compared to PN/O, one can actually assign other 1 Note that, as mentioned in [9], the performance of PN corresponds to the average performance over all possible spreading sequence sets, including the overloaded signal sets considered here.

> Paper 103-0602.R2
2N. In section II, we will show that all these overloaded systems (TSS, ES, T/O, O/O (including QOS) and m-O) belong to a much broader class of signature sets, consisting of a hierarchy of orthogonal subsets (HOS). While several articles deal with the sum capacity of PN-spread systems (e.g. [9,10]), there remains some unclarity about the information-theoretical potential of the overloaded systems discussed above. The sum capacity of O/O was obtained in [24], where it was shown to be slightly inferior to that of WBE sequences and significantly superior to that of PN. In this article, we extend this analysis to the sum capacity of all signature sets belonging to the HOS family. We also evaluate the sum capacity of PN/O, which will give us the average sum capacity over all signature sets consisting of N orthogonal set1 users and (K-N) excess users. We will focus both on realvalued and complex-valued signature sequences. In section II, we discuss the various overloaded signature sets and we introduce the HOS family. In section III, we briefly recall the concept of sum capacity under the EAE constraint. We evaluate in section IV the sum capacity resulting from PN/O sets and signature sets belonging to the HOS family, assuming that the EAE constraint holds. Finally, conclusions are drawn in section V. II.

SIGNATURE SETS FOR OVERLOADED CHANNELS

A. Random spreading (PN) We can distinguish between random spreading with binary (quaternary) sequences and with unconstrained sequences. In the first case, every user is assigned in every symbol interval completely at random one of the sequences from N

N

 1+ j −1+ j 1− j −1− j   1 −1  , , Α= , ,  ( Α' =   ). N 2. N  2. N 2. N  N  2. N In the unconstrained case, each signature is drawn uniformly from the surface of the unit N-sphere in every symbol interval. Accordingly, the sequences sk assigned to user k admit the representation g (1) sk = k gk

where gk = [gk,1 … gk,N]T (real signatures) or gk = [gk,1(1)+jgk,1(2) … gk,N(1)+jgk,N(2)]T (complex signatures) are built up from the independent zero-mean Gaussian random variables gk,j, gk,j(1) and gk,j(2) (j = 1, …, N) that have identical variance. B. HOS Family

Consider a CDMA system with spreading factor N, that has to accommodate a total number of K users. The set of users is partitioned into L subsets. The qth subset contains Mq ≤ N users, so that K = M1 + M2 + M3 + … + ML. The subsets are indexed such that their cardinality forms a non-increasing sequence: Mq ≤ Mq-1 for q = 2, ..., L. We define a broad family of signature sequence sets based on a hierarchy of orthogonal subsets (HOS). A signature set belongs to the HOS family, when user j of subset q is assigned a sequence sj(q) ∈ CN (C is the set of complex numbers) from the orthonormal subset S(q) = {s1(q),…, s (Mq ) }, under the restriction that the sequences from q

S(q) (q = 2, …, L) are linear combinations of the sequences from S(q-1). In this way, the overall signature set is composed of L layers of orthonormal signature subsets, that are organized in a hierarchical structure. Various channel overloading schemes that have been described in the literature turn out to be subfamilies of the HOS family: • TSS [13], and ES [14,15] can schematically be described by means of a ‘tree-structure’. The first subset S(1) is a complete orthonormal basis: S(1) = {s1(1), …, sN(1)} (M1 = N). For q ≥ 2, every sk(q) is a linear combination of the sequences from a subset Bk(q) ⊆ S(q-1), such that Bj(q) ∩ Bk(q) = {} if j ≠ k. By construction, each subset S(q) contains orthogonal sequences, and N ≥ M2≥ M3≥ … ≥ ML. Further, no two sequences in successive subsets are allowed to be equal or opposite, so that Mq+1 ≤ Mq/2 (q = 1, …, L-1); this implies that the total number of users has to be smaller than 2N. The advantage of the TSS scheme is the low complexity of the associated optimal multi-user detector. • The family TSS can be extended by relaxing the requirement (Bj(q) ∩ Bk(q) = {} if j≠k) to (Bj(q) = Bk(q) or Bj(q) ∩ Bk(q) = {} if j ≠ k). The hybrid TDMA/OCDMA scheme [16] belongs to this 'extended TSS' family. We obtain a special case of extended TSS, by selecting M1 = … = ML-1 = N and 0 < ML ≤ N, so that every subset S(q) (q = 1, ..., L-1) is a complete basis of the N-dimensional space CN. This particular type of channel overloading is denoted m-O, and has been introduced in [22]. Finally, quasi-orthogonal sequences (QOS) [20,21] minimize over all possible binary m-O sequence sets the maximum crosscorrelation among the users. C. PN/O In the case of PN/O [17], the first N users are assigned orthonormal sequences, whereas additional users are assigned random sequences. Note that the performance of PN/O represents the average performance over all signature sets that contain a complete orthogonal subset. III. SUM CAPACITY OF MULTIPLE ACCESS CHANNELS Consider a discrete-time symbol-synchronous CDMA channel model with spreading factor N and K users, where the receiver observes the complex received vector r(i) during the ith symbol interval (i = 1, …, n):

> Paper 103-0602.R2
Paper 103-0602.R2
1. Since the sum capacity of PN/O represents the average sum capacity over all signature sets with a complete orthogonal subset, this implies that m-O is a subset of PN/O that is superior to the average signature set of PN/O. The superiority of m-O over PN/O is mostly apparent at high loads (say, K/N >2). For PN/O, the maximum of ∆ over the region (K/N, SNR) ∈ Γ amounts to 4.6%. • It is clear from figure 2 that, although ∆ decreases with increasing load, PN-CDMA is significantly inferior to both m-O and PN/O with respect to capacity loss over the considered range of SNR and load. For PN-CDMA, the highest value of the relative capacity loss over (K/N,SNR) ∈ Γ is reached for K/N = 1 and amounts to 21.4%. V. CONCLUSION In this paper, we have introduced a family of signature sequences with spreading factor N, consisting of a hierarchy of orthogonal subsets (HOS). This family covers most of the channel overloading signature sets proposed in the literature that contain an orthogonal subset of dimension N, including the already implemented QOS of cdma2000. We have evaluated the sum capacity, of all signature sets belonging to the HOS family, under the EAE constraint, and found that the sum capacity is dependent only on the number of orthogonal subsets and on the number of sequences in every subset. Over HOS, the sum capacity is maximized by the m-O sequence set, which shows only a slight capacity loss (not exceeding 2.4%) as compared to the WBE sequences. Based on the low capacity loss of PN/O (lower than 4.6%), we can even conclude that many of the overloading schemes, where the

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Relative performance loss (%)

C. Performance Comparison We compare the performance of m-O, (asymptotic) PN/O and (asymptotic) PN based on the ‘relative capacity loss’ ∆, which quantifies the loss in sum capacity of the considered signature set as compared to optimal WBE sequences with the same value of γ and SNR, and the same load K/N: upper (13) ∆ (S) = 1 − C sum C sum

first N users are assigned orthogonal sequences, have a sum capacity which is only a few percent smaller than that of WBE sequences. This has to be contrasted to random spreading (PN), where substantial capacity losses are inevitable. Finally, it was noted that restricting the signature chips to a binary or quaternary alphabet has no impact on the sum capacity of PN/O and HOS. PN/O

SNR = 2dB

4 6dB 3

10dB 15dB

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6dB

20dB

2dB 1

20dB 15dB

10dB

m-O

0 1

1.5

2

load K/N

2.5

3

3.5

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Fig. 1: Comparison of the relative performance loss (in %) for m-O and (asymptotic) PN/O for SNR= 2, 6, 10, 15 and 20dB. 25

relative performance loss (%)

is clear from (12) that the specific relation between the set of N orthogonal sequences and the PN sequences has no influence at all on the sum capacity. So, restricting the chips of the sequences to a finite alphabet (binary/quaternary) does not incur any loss in the achievable sum capacity of PN/O.

4

20 PN 15 10dB 6dB

20dB

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SNR = 2dB 15dB 5 PN/O

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1.5

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2.5

load K/N

3

3.5

4

Fig. 2: Relative performance loss (in %) for SNR = 2, 6, 10, 15 and 20dB for (asymptotic) PN as compared to (asymptotic) PN/O. APPENDIX In order to determine the sum capacity of PN/O and HOS, we consider the eigenvalues of the crosscorrelation matrix R of a signature set SK that contains a complete orthogonal subset SO: SK = SO ∪ SM, with SM = {s1e,…,sMe} the set of (unit norm) signatures of the M excess users. The crosscorrelation matrix R is given by B I (I) R = S HK .S K =  NH P  B where P = (SM)H.SM and B = (SO)H.SM. We denote the rank of P by κ. Since rank(SKH.SK) ≤ rank(SK), the rank of R can not be higher than N. At the other hand, the first N columns of R are linearly independent, so that the rank of R is at least N. As a consequence, rank(R) = N, implying that λ0R = 0 is an

> Paper 103-0602.R2
Paper 103-0602.R2< [18] F. Vanhaverbeke, M. Moeneclaey and H. Sari, “DS/CDMA with Two Sets of Orthogonal Sequences and Iterative Detection,” IEEE Commun. Letters, vol. 4, pp. 289-291, Sept. 2000. [19] H. Sari, F. Vanhaverbeke and M. Moeneclaey, “Multiple Access Using Two Sets of Orthogonal Signal Waveforms,” IEEE Commun. Letters, vol. 4, pp. 4-6, Jan. 2000. [20] K. Yang, Y.-K. Kim and P. V. Kumar, “Quasi-orthogonal Sequences for Code-Division Multiple-Access Systems,” IEEE Trans. Inform. Theory, vol. 46, pp. 982-993, May 2000. [21] H. D. Schotten and H. Hadinejad-Mahram, “Analysis of a CDMA downlink with non-orthogonal spreading sequences for fading channels,” VTC 2000 Spring, pp. 1782-1786, Tokyo, May 2000. [22] F. Vanhaverbeke, M. Moeneclaey and H. Sari, “Increasing CDMA Capacity Using Multiple Sets of Orthogonal Spreading Sequences and Successive Interference Cancellation,” ICC 2002, New York, paper D10-1 April 2002. [23] TIA/EIA 3GPP2 C.S0002-B, “Physical Layer Standard for cdma2000 Spread Spectrum Systems, Release B," Jan. 16, 2001. [24] F. Vanhaverbeke and M. Moeneclaey, “Sum Capacity of the OCDMA/OCDMA Signature Sequence Set,” IEEE Commun. Letters, vol. 6, pp. 340-342, August 2002. [25] S. Verdu, “Capacity Region of Gaussian CDMA Channels : The Symbol-Synchronous Case,” in Proc. 24th Annu. Allerton Conf. Communication, Control, and Computing, Oct. 1986, pp. 1025-1039. [26] .M. Cover and J.A. Thomas, Elements of Information theory, New York: Wiley, 1991. [27] A. Soshnikov, “A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance matrices,” Journal of Statistical Physics, vol. 108, pp. 1033-1056, September 2002. [28] Z. D. Bai, “Methodologies in spectral analysis of large-dimensional random matrices, a review,” Statistica Sinica, vol. 9, pp. 611-677, July 1999. F. Vanhaverbeke graduated as an electrical engineer at the University of Gent, Gent, Belgium, in 1996. From 1996 to 1998, he performed research on spatial light modulators at the Electronics and Information Systems (ELIS) department of Ghent University. In 1998, he joined the Department of Telecommunication and Information Processing (TELIN), Ghent University. His research interests include spread-spectrum, mobile communication, multiuser detection, information theory and coding. More in particular, the focus of his current research is on overloaded channels for spread spectrum communication. He is the author and co-author of about 50 papers in international journals and conference proceedings. M. Moeneclaey (M'93 - SM'99 - F'02) received the diploma of electrical engineering and the Ph.D. degree in electrical engineering from the University of Gent, Gent, Belgium, in 1978 and 1983, respectively. In the period from 1978 to 1999, he held at Ghent University various positions for the Belgian National Fund for Scientific Research (NFWO), from Research Assistant to Research Director. He is presently a Professor in the Department of Telecommunications and Information Processing (TELIN), Ghent University. His research interests include statistical communication theory, carrier and symbol synchronization, bandwidth- efficient modulation and coding, spread-spectrum, satellite and mobile communication. He is the author of more than 250 scientific papers in international journals and conference proceedings. Together with Prof. H. Meyr (RWTH Aachen) and Dr. S. Fechtel (Siemens AG), he co-authors the book Digital Communication Receivers - Synchronization, Channel Estimation, and Signal Processing (New York: J. Wiley, 1998). From 1992-1994, he served as Editor for Synchronization, for the IEEE Transactions on Communications. He was co-guest editor for the Dec. 2001 IEEE JSAC special issue on Signal Synchronization in Digital Transmission Systems. From 1993 to 2002, he has been an executive Committee Member of the IEEE Communications and Vehicular Technology Society Joint Chapter, Benelux Section. He has been active in various international conferences as Technical Program Committee member and Session chairman.

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