Optimal CDMA Signatures: A Finite-Step Approach

Joel A. Tropp Inderjit. S. Dhillon Robert W. Heath Jr. Inst. for Comp. Engr. and Sci. (ICES) Dept. of Comp. Sci. Dept. of Elect. and Comp. Engr. The University of Texas at Austin The University of Texas at Austin The University of Texas at Austin 1 University Station C0200 1 University Station C0500 1 University Station C0803 Austin, TX 78712 Austin, TX 78712 Austin, TX 78712-1084 USA [email protected] [email protected] [email protected]

Abstract— A description of optimal sequences for direct-spread II. BACKGROUND code division multiple access is a byproduct of recent character- izations of the sum capacity. This papers restates the sequence A. Synchronous DS-CDMA design problem as an inverse singular value problem and shows Consider the uplink of an S-CDMA system with N users that it can be solved with finite-step algorithms from matrix and a processing gain of d. Assume that N>d, since the anal- analysis. Relevant algorithms are reviewed and a new one-sided ysis of the other case is straightforward. Define a d×N matrix construction is proposed that obtains the sequences directly def instead of computing the Gram matrix of the optimal signatures. whose columns are the signatures: S = s1 s2 ... sN . Let S ∗ to denote the (conjugate) transpose of S. Note that ∗ (S S)nn =1for each n =1,...,N. Assume that user n has I. INTRODUCTION an average power wn and collect them in the diagonal matrix def The problem of sequence sequences to maximize the sum W =diag(w1,w2,...,wN ). It is often more convenient to capacity of symbol-synchronous direct-spread code division absorb the power constraints into the signatures, so we also def 1/2 multiple access system (henceforth S-CDMA) has received define the weighted signature matrix X = SW . Denote the significant attention in the information theory community over n-th column of X as xn. For each n, one has the relationship the last decade, e.g. [1]–[7]. Despite this work, the algorithms ∗ 2 (X X )nn = xn = wn. (1) proposed to find optimal signatures have not exploited the 2 observation that the signature design problem can be character- Finally, let Σ denote the covariance matrix of the noise. ized as an inverse singular value problem [8]. Thus researchers Viswanath and Anantharam have proven in [5] that, for have been unable to exploit the wealth of algorithms developed real signatures, the sum capacity of the S-CDMA channel per in the matrix theory community during the past two decades degree of freedom is given by the expression (in the complex [9], [10]. case, the sum capacity differs by a constant factor) In this paper we present the signature design problem from 1 −1 ∗ a new perspective: as the solution to an inverse singular value Csum = max log det(Id + Σ SWS ). (2) d S problem. In short, optimal sequences are intimately connected 2 to matrices that satisfy certain conditions on their column The basic sequence design problem is to produce a signature norms and singular spectrum; these constraints are determined matrix S that solves the optimization problem (2). Various by the user power requirements and the noise covariance. This specializations have been considered with equal user powers, matrix analysis point-of-view allows us to leverage existing white noise, unequal user powers, and colored noise. We will algorithms for solving inverse singular value problems to show how each of these cases results in an inverse singular develop numerically stable, finite algorithms for solving the value problem. sequence design problem. Unlike iterative algorithms, e.g. [4], [7], [11], convergence is not an issue for finite-step algorithms B. A Sum Capacity Bound since they are guaranteed to solve the stated problem. In [1], Rupf and Massey produced an upper bound on the The specific contributions of this paper are two-fold. First sum capacity under white noise with variance σ2: we present a summary of relevant finite-step algorithms from 1 Tr W matrix analysis that will be useful for researchers working Csum ≤ log 1+ 2 σ2 d (3) in sequence design. We focus on the algorithms by Bendel- Mickey [9], Chan-Li [10] and Davies-Higham [12]. Second, where Tr (·) indicates the trace operator. They also established we present a new one-sided, finite algorithm that produces a necessary and sufficient condition on the signatures for the signature sequences directly instead of computing their equality to be attained in the bound (3): Gram matrix. This method has very modest time and space ∗ ∗ Tr W requirements in comparison with the other techniques we dis- XX = SWS = Id. d (4) cuss. Throughout we focus on the case of complex signatures exclusively because it subsumes the real case without any A matrix X that satisfies (4) is known as a tight frame [13] additional difficulty of argument. or a general Welch-Bound-Equality sequence (gWBE) [2]. A condition equivalent to (4) is that The key result of [2] is a complete characterization of Tr W the sum capacity of the S-CDMA channel under white X ∗X = P d (5) noise. Viswanath and Anantharam demonstrate that oversized users—those whose power constraints are too large relative P where the matrix represents an orthogonal projector from to the others for the majorization condition to hold—must CN d onto a subspace of dimension . Recall that an orthogonal receive their own orthogonal channels to maximize the sum P2 = P projector is an idempotent, Hermitian matrix. That is, capacity of the system, and they provide a simple method of P = P∗ and . An orthogonal projector is also characterized as determining which users are oversized. The other users share a Hermitian matrix whose nonzero eigenvalues are identically the remaining dimensions equitably. equal to one. In light of equation (1), the problem of con- Suppose that there are mrandom matrix of specified spectrum. The orthogonal matrix of the form command is gallery(’randcorr’, ...). cs It should be clear that a similar algorithm can be applied Q A = −sc to any Hermitian matrix to produce another Hermitian matrix with the same spectrum but whose diagonal entries where c2 + s2 =1[16]. The corresponding plane rotation are identically equal to Tr A/N . ∗ in the jk-plane is the N-dimensional identity matrix with its The columns of S must form an orthogonal basis for the def ∗ jj, jk, kj and kk entries replaced by the entries of the two- column space of A = S S according to (6). Therefore, one dimensional rotation. Let jsymmetric matrix with diagonal entries w and 1, we can construct an orthogonal transformation Q so that eigenvalues λ. Q∗(diag λ)Q has diagonal entries w. 1) Initialize A =diagλ, and put n =1. def Consider N-dimensional vectors for which w
λ. Let A = 2) Find the least j>nso that aj−1,j−1 ≤ wn ≤ ajj. diag λ. The majorization condition implies that λ1 ≤ w1 ≤ 3) Use a symmetric permutation to set an+1,n+1 equal to wN ≤ λN , so it is always possible to select a least integer ajj while shifting diagonal entries n +1,...,j− 1 one j>1 so that λj−1 ≤ w1 ≤ λj. Let P1 be a permutation place down the diagonal. matrix for which 4) Define a rotation Q in the (n, n +1)-plane with ∗ P AP1 =diag(λ1,λj,λ2,...,λj−1,λj+1,...,λN ). an+1,n+1 − wn wn − ann 1 c = ,s= . an+1,n+1 − ann an+1,n+1 − ann Observe that λ1 ≤ w1 ≤ λj and λ1 ≤ λ1 + λj − w1 ≤ ∗ λj. Thus we may use equation (9), replacing λ2 with λj, 5) Replace A by Q AQ. 6) Use a symmetric permutation to re-sort the diagonal powers, i.e. a unit-norm tight frame. Alternately, the Davies- entries of A in ascending order. Higham algorithm can be used to produce the factors directly. 7) Increment n, and repeat Steps 2–7 while nnso that xj−12 ≤ wn ≤xj2. j X n iteration for interference avoidance to the global optimum,” IEEE Trans. 3) Move the -th column of to the ( +1)-st column, IT, vol. 49, no. 4, pp. 873–885, April 2003. shifting the displaced columns to the right. [7] J. A. Tropp, R. W. Heath Jr., and T. Strohmer, “Optimal CDMA signature 4) Define a rotation Q in the (n, n +1)-plane with sequences, inverse eigenvalue problems and alternating projection,” in Proceedings of the 2003 IEEE International Symposium on Information x 2 − w w −x 2 Theory, Yokohama, July 2003, p. 407. n+1 2 n n n 2 [8] M. T. Chu and G. H. Golub, “Structured inverse eigenvalue problems,” c = 2 2 ,s= 2 2 . xn+1 −xn xn+1 −xn Acta Numerica, 2001. 2 2 2 2 [9] R. B. Bendel and M. R. Mickey, “Population correlation matrices for 5) Replace X by XQ. sampling experiments,” Commun. Statist. Simul. Comp., vol. B7, no. 2, n ,...,N pp. 163–182, 1978. 6) Sort columns ( +1) in order of increasing norm. [10] N. N. Chan and K.-H. Li, “Diagonal elements and eigenvalues of a real 7) Increment n, and repeat Steps 2–7 while n