Optimal CDMA Signatures: A Finite-Step Approach

Transcription

1 Optimal CDMA Signatures: A Finite-Step Approach Joel A. Tropp Inst. for Comp. Engr. an Sci. (ICES) 1 University Station C000 Austin, TX 7871 jtropp@ices.utexas.eu Inerjit. S. Dhillon Dept. of Comp. Sci. 1 University Station C0500 Austin, TX 7871 inerjit@cs.utexas.eu Robert W. Heath Jr. Dept. of Elect. an Comp. Engr. 1 University Station C0803 Austin, TX USA rheath@ece.utexas.eu Abstract A escription of optimal sequences for irect-sprea coe ivision multiple access is a byprouct of recent characterizations of the sum capacity. This papers restates the sequence esign problem as an inverse singular value problem an shows that it can be solve with finite-step algorithms from matrix analysis. Relevant algorithms are reviewe an a new one-sie construction is propose that obtains the sequences irectly instea of computing the Gram matrix of the optimal signatures. I. INTRODUCTION The problem of sequence sequences to maximize the sum capacity of symbol-synchronous irect-sprea coe ivision multiple access system (henceforth S-CDMA) has receive significant attention in the information theory community over the last ecae, e.g. [1] [7]. Despite this work, the algorithms propose to fin optimal signatures have not exploite the observation that the signature esign problem can be characterize as an inverse singular value problem [8]. Thus researchers have been unable to exploit the wealth of algorithms evelope in the matrix theory community uring the past two ecaes [9], [10]. In this paper we present the signature esign problem from a new perspective: as the solution to an inverse singular value problem. In short, optimal sequences are intimately connecte to matrices that satisfy certain conitions on their column norms an singular spectrum; these constraints are etermine by the user power requirements an the noise covariance. This matrix analysis point-of-view allows us to leverage existing algorithms for solving inverse singular value problems to evelop numerically stable, finite algorithms for solving the sequence esign problem. Unlike iterative algorithms, e.g. [4], [7], [11], convergence is not an issue for finite-step algorithms since they are guarantee to solve the state problem. The specific contributions of this paper are two-fol. First we present a summary of relevant finite-step algorithms from matrix analysis that will be useful for researchers working in sequence esign. We focus on the algorithms by Benel- Mickey [9], Chan-Li [10] an Davies-Higham [1]. Secon, we present a new one-sie, finite algorithm that prouces the signature sequences irectly instea of computing their Gram matrix. This metho has very moest time an space requirements in comparison with the other techniques we iscuss. Throughout we focus on the case of complex signatures exclusively because it subsumes the real case without any aitional ifficulty of argument. II. BACKGROUND A. Synchronous DS-CDMA Consier the uplink of an S-CDMA system with N users an a processing gain of. Assume that N>, since the analysis of the other case is straightforwar. Define a N matrix whose columns are the signatures: S ef = s 1 s... s N. Let S to enote the (conjugate) transpose of S. Note that (S S) nn =1for each n =1,...,N. Assume that user n has an average power w n an collect them in the iagonal matrix W ef =iag(w 1,w,...,w N ). It is often more convenient to absorb the power constraints into the signatures, so we also efine the weighte signature matrix X ef = SW 1/. Denote the n-th column of X as x n. For each n, one has the relationship (X X ) nn = x n = w n. (1) Finally, let Σ enote the covariance matrix of the noise. Viswanath an Anantharam have proven in [5] that, for real signatures, the sum capacity of the S-CDMA channel per egree of freeom is given by the expression (in the complex case, the sum capacity iffers by a constant factor) C sum = 1 max log et(i + Σ 1 SWS ). () S The basic sequence esign problem is to prouce a signature matrix S that solves the optimization problem (). Various specializations have been consiere with equal user powers, white noise, unequal user powers, an colore noise. We will show how each of these cases results in an inverse singular value problem. B. A Sum Capacity Boun In [1], Rupf an Massey prouce an upper boun on the sum capacity uner white noise with variance σ : ( C sum 1 log 1+ Tr W ) σ (3) where Tr ( ) inicates the trace operator. They also establishe a necessary an sufficient conition on the signatures for equality to be attaine in the boun (3): XX = SWS = Tr W I. (4) A matrix X that satisfies (4) is known as a tight frame [13] or a general Welch-Boun-Equality sequence (gwbe) []. A

2 conition equivalent to (4) is that X X = Tr W P (5) where the matrix P represents an orthogonal projector from C N onto a subspace of imension. Recall that an orthogonal projector is an iempotent, Hermitian matrix. That is, P = P an P = P. An orthogonal projector is also characterize as a Hermitian matrix whose nonzero eigenvalues are ientically equal to one. In light of equation (1), the problem of constructing optimal signature sequences in the present setting is closely relate to the problem of constructing an orthogonal projector with a specifie iagonal. C. White Noise, Equal Powers Consier the case where the power constraints are equal, viz. W = w I N for some positive number w. Then conition (4) for equality to hol in (3) becomes w 1 XX = SS = N I. (6) A matrix S which satisfies (6) is known as a unit-norm tight frame (UNTF) [13] or a Welch-Boun-Equality sequence (WBE) [1]. In fact, there always exist signature matrices that satisfy conition (6), an so the upper boun on the sum capacity can always be attaine when the users power constraints are equal [1]. The equation (6) can also be interprete as a restriction on the singular values of the signature matrix. Uner the assumptions of white noise an equal power constraints, a matrix S yiels optimal signatures if an only if 1) each column of S has unit-norm an ) the nonzero singular values of S are ientically equal to N/. Therefore, this sequence esign problem falls into the category of structure inverse singular value problems. Note that conition 1) must hol irrespective of the type of noise. D. Majorization The boun (3) cannot be met for an arbitrary set of power constraints. The explanation requires some backgroun in majorization [14]. Essentially, majorization efines a partial orering that allows two vectors of equal norm to be compare. A vector w majorizes λ (written w λ ) if all the partial sums of the entries of w are greater than or equal to the partial sums of λ. It turns out that majorization efines the precise relationship between the iagonal entries of a Hermitian matrix an its spectrum (see [14], [15] for etails). Theorem 1 (Schur-Horn [15]): The iagonal entries of a Hermitian matrix majorize its eigenvalues. Conversely, if w λ, there exists a Hermitian matrix with iagonal elements liste by w an eigenvalues liste by λ. E. White Noise, Unequal Powers The Schur-Horn Theorem forbis the construction of an orthogonal projector with arbitrary iagonal entries. For this reason, (5) cannot always hol, an the upper boun (3) cannot always be attaine. The key result of [] is a complete characterization of the sum capacity of the S-CDMA channel uner white noise. Viswanath an Anantharam emonstrate that oversize users those whose power constraints are too large relative to the others for the majorization conition to hol must receive their own orthogonal channels to maximize the sum capacity of the system, an they provie a simple metho of etermining which users are oversize. The other users share the remaining imensions equitably. Suppose that there are m < oversize users, whose signatures form the columns of S 0. Let the columns of S 1 list the signatures of the (N m) remaining users, an let the iagonal matrix W 1 list their power constraints. The conitions for achieving sum capacity follow. 1) The m oversize users receive orthogonal signatures: S 0 S 0 = I m. ) The remaining (N m) signatures are also orthogonal to the oversize users signatures: S 0 S 1 = 0. 3) The remaining users signatures satisfy S 1 W 1 S1 = Tr W 1 m I m. Repeat the foregoing arguments to see that the sequence esign problem still amounts to constructing a matrix with given column norms an singular spectrum. It is therefore an inverse singular value problem. F. Colore Noise, Unequal Powers When the noise is colore, the situation is somewhat more complicate. Nevertheless, optimal sequence esign still boils own to constructing a matrix with given column norms an singular spectrum. Viswanath an Anantharam show that the following proceure will solve the problem [5]. 1) Compute an eigenvalue ecomposition of the noise covariance matrix, Σ = QDQ, where D =iagσ for some non-negative vector σ. ) Use Algorithm A of [5] to etermine µ, the Schurminimal element of the set of possible eigenvalues of SWS + Σ. 3) Form the vector λ ef = µ σ. 4) Compute an auxiliary signature matrix T with unit-norm columns so that TWT =iagλ. 5) The optimal signature matrix is S ef = QT. The computation in step (4) is equivalent to proucing a N matrix X ef = TW 1/. The columns of X must have square norms liste by the iagonal of W. The vector λ must list the nonzero square singular values of X. This is another inverse singular value problem. III. CONSTRUCTING CORRELATION MATRICES A positive semi-efinite Hermitian matrix with a unit iagonal is known as a correlation matrix [1]. The Gram matrix A ef = S S of an optimal signature matrix S is always a correlation matrix. Every correlation matrix with the appropriate spectrum can be factore to prouce optimal signature matrices [1]. Therefore, we begin with a technique for constructing correlation matrices with a preassigne spectrum.

3 A. The Benel-Mickey Algorithm In 1978, Benel an Mickey presente an algorithm that uses a finite sequence of rotations to convert an arbitrary N N Hermitian matrix with trace N into a unit-iagonal matrix that has the same spectrum [9]. We follow the superb exposition of Davies an Higham [1]. Brief iscussions appear on page 76 of Horn an Johnson [15] an in Problems an 8.4. of Golub an van Loan [16]. Suppose that A M N is a Hermitian matrix with Tr A = N. (Let M N enote the set of complex N N matrices an M,N to enote the set of complex N matrices.) If A oes not have a unit iagonal, one can locate two iagonal elements so that a jj < 1 <a kk ; otherwise, the trace conition woul be violate. It is then possible to construct a real plane rotation Q in the jk-plane so that (Q AQ) jj =1. The transformation A Q AQ preserves the conjugate symmetry an the spectrum of A but reuces the number of non-unit iagonal entries by at least one. Thus, at most (N 1) rotations are require before the resulting matrix has a unit iagonal. The appropriate form of the rotation is easy to iscover, but the following erivation is essential to ensure numerical stability. Recall that a two-imensional plane rotation is an orthogonal matrix of the form c s Q = s c where c + s =1[16]. The corresponing plane rotation in the jk-plane is the N-imensional ientity matrix with its jj, jk, kj an kk entries replace by the entries of the twoimensional rotation. Let j<kbe inices so that a jj < 1 <a kk or a kk < 1 <a jj. The esire plane rotation yiels the matrix equation c s ajj a jk c s 1 ãjk s c a = jk a kk s c ã jk ã kk where c + s =1. The equality of the upper-left entries can be state as c a jj sc Re a jk + s a kk =1. This equation is quaratic in t = s/c: whence (a kk 1) t t Re a jk +(a jj 1) = 0 t = Re a jk ± (Re a jk ) (a jj 1)(a kk 1). (7) a kk 1 Notice that the choice of j an k guarantees a positive iscriminant. As is stanar in numerical analysis, the ± sign in (7) must be taken to avoi cancelations. If necessary, one can extract the other root using the fact that the prouct of the roots equals (a jj 1)/(a kk 1). Finally, 1 1+t an s = ct. (8) Floating-point arithmetic is inexact, so the rotation may not yiel a jj =1. A better implementation sets a jj =1explicitly. Davies an Higham prove that the Benel-Mickey algorithm is backwar stable, so long as it is implemente the way we have escribe [1]. We restate the algorithm. Algorithm 1 (Benel-Mickey): Given Hermitian A M N with Tr A = N, this algorithm yiels a correlation matrix whose eigenvalues are ientical with those of A. 1) While some iagonal entry a jj 1, repeat Steps 4. ) Fin an inex k (without loss of generality j<k) for which a jj < 1 <a kk or a kk < 1 <a jj. 3) Determine a plane rotation Q in the jk-plane using equations (7) an (8). 4) Replace A by Q AQ. Set a jj =1. Since the loop executes no more than (N 1) times, the total cost of the algorithm is no more 1N real floating-point operations, to highest orer, if conjugate symmetry is exploite. The plane rotations never nee to be generate explicitly, an all the intermeiate matrices are Hermitian. Therefore, the algorithm must store only N(N +1)/ complex floatingpoint numbers. MATLAB 6 contains a version of Algorithm 1 that starts with a ranom matrix of specifie spectrum. The comman is gallery( rancorr,...). It shoul be clear that a similar algorithm can be applie to any Hermitian matrix A to prouce another Hermitian matrix with the same spectrum but whose iagonal entries are ientically equal to Tr A/N. The columns of S must form an orthogonal basis for the column space of A ef = S S accoring to (6). Therefore, one can use a rank-revealing QR factorization to extract a signature sequence S from the output A of Algorithm 1 [16]. B. The Davies-Higham Algorithm The methos of the last section can be moifie to compute the signature sequence irectly without recourse to an aitional QR factorization. Any correlation matrix A M N can be expresse as the prouct S S where S M r,n has columns of unit norm an imension r rank A. With this factorization, the two-sie transformation A Q AQ is equivalent to a one-sie transformation S SQ. In consequence, the machinery of the Benel-Mickey algorithm requires little ajustment to prouce these factors. We refer to the one-sie version as the Davies-Higham algorithm in view of [1]. We have observe that it can also be use to fin the factors of an N-imensional correlation matrix with rank r<n, in which case S may take imensions N for any r. Algorithm (Davies-Higham): Given S M,N for which Tr S S = N, this proceure yiels a N matrix with the same singular values as S but with unit-norm columns. 1) Calculate an store the column norms of S. ) While some column has norm s j 1, repeat Steps ) Fin inices j<kfor which s j < 1 < s k or s k < 1 < s j.

4 4) Form the quantities a jj = s j, a jk = s k, s j an a kk = s k. 5) Determine a rotation Q in the jk-plane using equations (7) an (8). 6) Replace S by SQ. 7) Upate the two column norms that have change. Step (1) requires 4N real floating-point operations, an the remaining steps require 1N real floating-point operations to highest orer. The algorithm requires the storage of N complex floating-point numbers an N real numbers for the current column norms. Davies an Higham remark that the algorithm is numerically stable [1]. IV. CHAN-LI ANDITS VARIANTS Now we iscuss a simple technique for constructing a limite selection of Hermitian matrices with prescribe iagonal an spectrum. This is the core problem for the unequal power case with white or colore noise because every optimal weighte signature sequence has a Gram matrix A ef = X X with fixe iagonal an spectrum (an conversely). The algorithm begins with a iagonal matrix of eigenvalues an applies a sequence of rotations to impose the power constraints. A similar technique can be use to buil optimal weighte signature sequences X irectly. A. The Chan-Li Algorithm Chan an Li present a beautiful, constructive proof of converse part of the Schur-Horn Theorem [10]. Suppose that w an λ are N-imensional, real vectors for which w λ. Using inuction on the imension, we show how to construct a Hermitian matrix with iagonal w an spectrum λ. In the sequel, assume without loss of generality that the entries of w an λ have been sorte in ascening orer. Therefore, w (k) = w k an λ (k) = λ k for each k. Suppose that N =. The majorization relation implies λ 1 w 1 w λ. Let A ef =iagλ. We can explicitly construct a plane rotation Q so that the iagonal of Q AQ equals w: [ ] 1 Q ef λ w = 1 w1 λ 1 λ λ 1. (9) w 1 λ 1 λ w 1 Since Q is orthogonal, Q AQ retains spectrum λ but gains iagonal entries w. Suppose that, whenever w λ for vectors of length N 1, we can construct an orthogonal transformation Q so that Q (iag λ)q has iagonal entries w. Consier N-imensional vectors for which w λ. Let A ef = iag λ. The majorization conition implies that λ 1 w 1 w N λ N, so it is always possible to select a least integer j > 1 so that λ j 1 w 1 λ j. Let P 1 be a permutation matrix for which P 1 AP 1 =iag(λ 1,λ j,λ,...,λ j 1,λ j+1,...,λ N ). Observe that λ 1 w 1 λ j an λ 1 λ 1 + λ j w 1 λ j. Thus we may use equation (9), replacing λ with λ j, to construct a plane rotation Q that sets the first entry of Q (iag (λ 1,λ j ))Q to w 1. If we efine the rotation ef Q 0 P = 0 I N then P P1 w1 v AP 1 P = v A N 1 where v is an appropriate vector an A N 1 = iag (λ 1 + λ j w 1,λ,...,λ j 1,λ j+1,...,λ N ). To apply the inuction hypothesis, it remains to check that the vector (w,w 3,...,w N ) majorizes the iagonal of A N 1. We accomplish this in three steps. First, recall that λ k w 1 for k =,...,j 1. Therefore, w k (m 1) w 1 λ k for each m =,...,j 1. The sum on the right-han sie obviously excees the sum of the smallest (m 1) entries of iag A N 1, so the first (j ) majorization inequalities are in force. Secon, use the fact that w λ to calculate that w k = w k w 1 λ k w 1 k=1 k=1 j 1 =(λ 1 + λ j w 1 )+ λ k + λ k k=j+1 for m = j,...,n. Once again, observe that the sum on the right-han sie excees the sum of the smallest (m 1) entries of iag A N 1, so the remaining majorization inequalities are in force. Finally, rearranging the relation N k=1 w k = N k=1 λ k yiels N w k =TrA N 1. In consequence, the inuction furnishes a rotation Q N 1 which sets the iagonal entries of A N 1 equal to the numbers (w,...,w N ). Define ef P 3 = Q N 1 Conjugating A by the orthogonal matrix P = P 1 P P 3 transforms the iagonal entries of A to w while retaining the spectrum λ. The proof yiels the following algorithm. Algorithm 3 (Chan-Li): Let w an λ be vectors with ascening entries an such that w λ. The following proceure computes a real, symmetric matrix with iagonal entries w an eigenvalues λ. 1) Initialize A =iagλ, an put n =1. ) Fin the least j>nso that a j 1,j 1 w n a jj. 3) Use a symmetric permutation to set a n+1,n+1 equal to a jj while shifting iagonal entries n +1,...,j 1 one place own the iagonal. 4) Define a rotation Q in the (n, n +1)-plane with a n+1,n+1 w n a n+1,n+1 a nn, s = 5) Replace A by Q AQ. w n a nn a n+1,n+1 a nn.

5 6) Use a symmetric permutation to re-sort the iagonal entries of A in ascening orer. 7) Increment n, an repeat Steps 7 while n<n. This algorithm requires about 6N real floating-point operations. It requires the storage of about N(N +1)/ real floating-point numbers, incluing the vector w. It is conceptually simpler to perform the permutations escribe in the algorithm, but it can be implemente without them. We have observe that the algorithm given by Viswanath an Anantharam [] for constructing gwbes is similar to the Chan-Li algorithm. Nevertheless, we feel that the simplicity of Chan an Li s presentation merits repetition. B. One-Sie Chan-Li The Chan-Li algorithm only prouces a Gram matrix, which must be factore to obtain the weighte signature matrix. Referring back to the ieas of Davies an Higham, we propose a one-sie version of the Chan-Li algorithm. The benefits are several. It requires less storage an less computation than the Chan-Li algorithm. At the same time, it constructs the factors explicitly. This algorithm has not been publishe before. Algorithm 4 (One-Sie Chan-Li): Suppose that w an λ are non-negative vectors of length N with ascening entries. Assume, moreover, that the first (N ) components of λ are zero an that w λ. The following algorithm prouces a N matrix X whose column norms are liste by w an whose square singular values are liste by λ. 1) Initialize n =1, an set λn +1 X = λn ) Fin the least j>nso that x j 1 w n x j. 3) Move the j-th column of X to the (n +1)-st column, shifting the isplace columns to the right. 4) Define a rotation Q in the (n, n +1)-plane with x n+1 w n w n x n x n+1 x n, s = x n+1 x n. 5) Replace X by XQ. 6) Sort columns (n+1),...,n in orer of increasing norm. 7) Increment n, an repeat Steps 7 while n<n. The algorithm requires 3N real floating-point operations an storage of N( +) real floating-point numbersm incluing the esire column norms an the current column norms. The proceure can be implemente without permutations. V. DISCUSION We have iscusse a group of four algorithms that can be use to prouce sum-capacity-optimal S-CDMA sequences in a wie variety of circumstances. The first algorithm, Benel- Mickey, constructs a Hermitian matrix with a constant iagonal an a prescribe spectrum. This matrix can be factore to yiel an optimal signature sequence for the case of equal user powers, i.e. a unit-norm tight frame. Alternately, the Davies- Higham algorithm can be use to prouce the factors irectly. The thir algorithm, Chan-Li, constructs a Hermitian matrix with an arbitrary iagonal an prescribe spectrum, subject to the majorization conition. The resulting matrix can be factore to obtain an optimal signature sequence for the case of unequal receive powers, i.e. a tight frame. We have also introuce a new variant, the one-sie Chan-Li algorithm, that can calculate the factors irectly. The Benel-Mickey an Davies-Higham algorithms can potentially calculate every correlation matrix an its factors. If they are initialize with ranom matrices, one may interpret the output as a ranom correlation matrix. The factors can be interprete as ranom unit-norm signature sequences. On the other han, the output of the Chan-Li an one-sie Chan-Li algorithms is not encyclopeic. They can construct only a few matrices for each pair (w, λ). These matrices are also likely to have many zero entries, which is unesirable for some applications. In aition, these algorithms only buil real matrices, whereas complex matrices are often of more interest. REFERENCES [1] M. Rupf an J. L. Massey, Optimum sequence multisets for synchronous coe-ivision multiple-access channels, IEEE Trans. Inform. Theory, vol. 40, no. 4, pp , July [] P. Viswanath an V. Anantharam, Optimal sequences an sum capacity of synchronous CDMA systems, IEEE Trans. Inform. Th., vol. 45, no. 6, pp , Sept [3] S. Ulukus an R. D. 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