Symmetric Capacity of Optimal Sequences in Quasi-scalable CDMA Systems Frederik Vanhaverbeke and Marc Moeneclaey TELIN/DIGCOM Ghent University, Sint-Pietersnieuwstraat 41 B-9000 Gent, Belgium Abstract-We evaluate the symmetric capacity of the OCDMA/OCDMA (O/O) signature sets that maximize the sum capacity in quasi-scalable CDMA systems. It is found that the symmetric capacity is equal to the sum capacity for randomly scrambled O/O. I. INTRODUCTION As long as the number of users K in a synchronous direct-sequence spread-spectrum Code-Division Multiple Access (CDMA) system remains lower than the spreading factor N of the system, interference between the users can be avoided by assigning orthogonal signatures to the users [1,2]. A nontrivial question is how to choose the signatures if K exceeds N (i.e. an oversaturated system). In [3], it was found that signature sets that achieve the Welch lower bound on Total Squared Correlation [4] maximize the (information- theoretical) sum capacity of oversaturated CDMA channels under the Equal Average input Energy (EAE) constraint. In addition to this, it turned out in [5] that these so-called Welch Bound Equality (WBE) sequences maximize the symmetric capacity as well. Unfortunately, WBE sequence sets are plagued by the drawback that they are valid for one particular choice (K,N) only. Hence, they are to be recalculated and reassigned each time the number of users K changes. This implies that the application of WBE sequences results in a completely unscalable system, making these sequences unattractive for practical applications [6,7]. As pointed out in [8], it is impossible to combine strict scalability 1 with the requirement that all users are orthogonal each time K drops below N. However, it was shown in [8] that the latter requirement can be realized easily by relaxing the requirement of strict scalability to quasi-scalability 2 . Under the constraint of quasi- scalability and the EAE constraint, it was found in [8] that OCDMA/OCDMA (O/O) systems [9-11] maximize the sum capacity. The sum capacity of O/O was even 1 In a strictly scalable system with K users and signature set SK = {s1, …, sK}, the signature set of the users turns into SK+1 = SK ∪ {sK+1} if an additional user (K+1) enters the system, or in SK-1 = SK\{sm} if user m (1 ≤ m ≤ K) leaves the system. 2 In a quasi-scalable system with K users and signature set SK = {s1, …, sK}, the signature set of the users turns into SK+1 = SK ∪ {sK+1} if an additional user (K+1) enters the system, or in SK-1 = SK\{sK} if user m (1 ≤ m ≤ K) leaves the system. shown to be only slightly lower than that of WBE systems [12], indicating that quasi-scalability results in a minor loss of sum capacity as compared to systems without scalabity requirement. In this paper, we are interested in the evaluation of the symmetric capacity of the quasi-scalable O/O system. In section II, we briefly recall some essential background information about the information-theoretical capacity in CDMA channels. This is used to evaluate the symmetric capacity of O/O in section III. Finally, section IV summarizes the main contributions of the paper. II. INFORMATION THEORETICAL CONCEPTS A. Capacity region of a CDMA channel Consider a symbol-synchronous CDMA system with spreading factor N, accommodating K users who transmit over a non-dispersive channel. The receiver observes the sum of the transmitted signals embedded in additive white Gaussian noise w(t). The output after chip matched filtering in symbol interval i (i = 1, …, n) of the multiple- access channel under consideration is the real-valued vector r i , given by ) ( ). ( ) ( ) ( 1 i i b i with i k k k i K k k i s X w X r ≡ + = ∑ = (1) In this expression, • s k (i) is the unit-norm signature sequence of user k in symbol interval i. We denote the NxK signature matrix corresponding to the ith symbol interval as [ ] ) ( ... ) ( ) ( 1 i i i K s s S = . • w i is a Gaussian noise vector with zero mean and E[w i .w j T ] = σ 2 .δ i-j .I N , where I N is the identity matrix of order N. • The data of each user k are encoded into a codeword (b k (1), …, b k (n)) of length n and symbol rate 1/T. The only constraint on the symbols b k (i) is the requirement that the codeword of every user k is power-constrained: E[b k 2 (i)] ≤ A k 2 . Throughout the paper, we focus on the EAE constraint: A k 2 = A 2 (k = 1, …, K). The average number of information bits per symbol b k (i.e. the information rate of user k) is denoted as ρ k .