Convergence of Linear Interference Cancellation Multiuser Receivers

Transcription

1 Convergence of Linear Interference Cancellation Multiuser Receivers Alex Grant and Christian Schlegel Abstract We consider the convergence in norm of several iterative implementations of linear multiuser receivers, under the assumption of long random spreading sequences. We find that asymptotically, linear parallel interference cancellation diverges for systems loads of greater than about 7%. Using known results from the theory of iterative solutions for linear systems we derive optimal or near optimal relaxation parameters for parallel (first and second order stationary, Chebyshev) and serial cancellation (successive relaxation) methods. An analytic comparison of the asymptotic convergence factor for the various methods is given. Simulations are used to verify results for finite size systems. Keywords Interference cancellation, iterative methods, multi-stage receivers, multi-user receivers, CDMA, random sequences I. Introduction Interference cancellation is a simple multiuser detection technique in which estimates (or statistics) of other users interference are subtracted from the received signal, in order to improve the estimate for a given user. This process can proceed iteratively (sometimes called a multi-stage receiver), by repeating the subtraction process. Such iterations may be linear or non-linear. Linear methods use linear estimates of the interference, whereas non-linear methods have no such restriction. We mainly restrict our attention to linear schemes, previously considered in [], [2], [3], [4], [5], [6], [7], [8], [9], [0], [], [2], [3], [4], [5], [6], [7], [8], [9]. For an introduction to multiuser detection, see [20]. It is known that iterated linear interference cancellation for the detection of linear multiple access channels is equivalent to well known iterative methods for the solution of systems of linear equations [0], [6], [7]. In particular, parallel interference cancellation is equivalent to the method of Jacobi, while serial cancellation corresponds to the Gauss-Seidel method. Our research was motivated by the realization that parallel interference cancellation converges to the decorrelator [4], [5], [8], [9]. Related to these receivers are the approximate decorrelators [], [5], [20], based on the viewpoint of series expansions for the linear filters. Given the mathematical literature devoted to the iterative solution of linear systems (see [2], [22], [23], [24], for textbook treatments), it makes sense to apply those results to the design of multiuser detectors, rather than try to derive known results from engineering applications. In the recent engineering literature there has been a num- A. Grant is with the Institute for Telecommunications Research, University of South Australia C. Schlegel is with the Deptartment of Electrical Engineering University of Utah ber of papers on iterative detection and decoding. While most of these detectors use some form of non-linear cancellation, usually in the form of a hard-decision device placed in the iteration loop (or between decoder stages), their signal flow structure is identical to the well-known iterative linear algorithms described in this paper. The original multistage detector [25], [26] corresponds to the Jacobi iteration while the fine tuned successive interference canceler discussed by Divsalar et. al. [27], [28], [29] is in fact a minor modification of the successive relaxation method discussed in Section III-E. Iterative methods have also been successfully applied to coded multiuser systems [30], [3], [32]. We restrict our scope to detection of uncoded data. We analyze the convergence properties of several linear iterative implementations of both the decorrelator and the minimum mean squared error filter. Convergence is measured with respect to a given norm (e.g. Euclidean). In particular, we consider the method of Jacobi and improvements thereon, corresponding to weighted parallel cancellation (such as the Chebyshev method). We also consider the Gauss-Seidel, and more generally, the successive relaxation methods corresponding to (weighted) serial cancellation. Finally we also present results concerning the conjugate gradient method, which is a parameter free iteration. For the case of random spreading sequences use the theory of large random matrices to give asymptotically (for large systems) optimal weighting parameters that are easily calculated knowing only the number of users and the length of the spreading sequences. We also consider the convergence speeds of the various methods (with respect to an arbitrary given norm). Related work may be found in [33], where the authors use random matrix results to find the spectral efficiency of various iterative detectors. In Section II we introduce our discrete-time, synchronous, linear model and describe the optimal receiver followed by the decorrelator and minimum mean squared error filters. Section III is the main part of the paper, where we consider the various iterative implementations of the linear receivers. In Section IV we use computer simulations to suggest that our results are also valid for small systems. We conclude in Section V with a discussion and comparison of the various implementations. In the Appendix we present those results from random matrix theory required in our proofs. We shall use the following notations. The vector x C n is an column vector with complex entries x i, i =, 2,..., n. Likewise A C m n is a matrix with complex entries A ij, i =,..., m, j =,..., n. The superscripts T, and denote respectively transposition, Hermitian adjoint and matrix inverse. I n is the n n identity matrix. Denote by

2 2 S (R) the spectrum [34, p. 35] of R. The spectral radius of a matrix A is ρ (A) = max{ λ, λ S (A)}. We shall also refer to λ min = min S (A) and λ max = max S (A). The operator E denotes expectation. II. Linear Multiple Access and Linear Detectors We assume a linear multiple access system (e.g. Code- Division Multiple Access) where K users access the channel each using a signature signal of duration N chips. Transmission is symbol-synchronous. Such a system can be described by the linear relation [20] r = AWd + n, where A C N K has unit energy columns which are the discrete signature signals of the K users, W = diag(w,..., w K ) is a diagonal matrix of the users amplitudes, d {, +} K has elements d i, the transmitted binary symbol for user i, and n C N is a sampled circularly symmetric complex noise vector with covariance matrix E [nn ] = σ 2 I N. We shall be particularly interested in the case where the signature sequences A are chosen randomly (in a fashion known to the receiver) for each symbol interval. The optimal detector outputs the hypothesis ˆd which maximizes the conditional probability distribution p(r d). In additive Gaussian noise this is equivalent to minimizing the squared Euclidean distance ˆd = arg min r AWd 2, d {,+} K in turn equivalent to minimizing the quadratic ˆd = arg min d T W T RWd 2d T W Re(y), () d {,+} K where R = A A is the K K normalized correlation matrix between the signature signals of the users and y = A r is the output of a bank of matched filters. Brute force evaluation of () requires the computation of 2 K terms. In certain special cases however optimal detection is possible with polynomial complexity [35], [36], [37], [38]. Of theoretical and practical interest are the class of linear detectors which form decisions based on the output of various linear filters ˆd = sgn Hr. We are particularly interested in the decorrelator [39] and the linear minimum mean squared error (LMMSE) [40], [4], [42], [43] filters. We shall consider the synchronous case. The asynchronous extension is not difficult [20]. The decorrelator may be derived by relaxing the constraints on the minimization (). Instead of a combinatorial minimization, we find ˆd = arg min d R K r AWd 2, which has the closed form solution ˆd = A r, where A is the Moore-Penrose generalized inverse of A [34, pp ]. If A is non-singular, A = A. If A has linearly independent columns, A = (A A) A. Note that A exists for any matrix A and can always be expressed in terms of the singular value decomposition of A. Minimizing the expected Euclidean distance from the filter output to the transmitted data, the LMMSE filter is given by [20] H = arg min E [ Hr d 2] M C K N = ( WRW + σ 2 I ) WA The calculation of either the LMMSE filter or the decorrelator requires a matrix inversion of complexity O ( K 3), which is a large computational burden if the signature sequences are changing from symbol interval to symbol interval. For the remainder of the paper we shall assume that the entries of A are chosen randomly according to the following rule, modeling the use of spreading sequences with period much longer than the data symbol duration. Definition : For each symbol interval, the real or complex spreading sequences A shall be selected randomly as follows. Each element A ij, i =,..., N, j =,..., K shall be chosen independently from a given distribution, with mean E[A ij ] = 0, variance E[A 2 ij ] = /N and finite higher moments. Such sequence selection is quite general, and includes the ubiquitous uniform selection of chips from {+, }. Our convergence analyses will draw upon results from random matrix theory, summarized in Appendix V. III. Iterative Solution Methods For a given n n real or complex matrix M and n-vector b, solving a system of linear equations Mx = b, (2) for the unknown vector x requires in general O ( n 3) operations (unless M has some special property that makes it easily invertible). It is possible however to design an iteration that converges to the solution. Letting M = S T (a splitting of M), we see that Sx = Tx + b has the same solution as (2). Furthermore, for any initial guess x 0, is is easy to show that the iteration Sx k+ = Tx k + b, k = 0,, 2,... converges in any given norm to the true solution if and only if the spectral radius of the iteration matrix B = S T satisfies ρ (B) <. See [2], [23] for details. For a useful iteration we require that S be easily invertible (resulting in low complexity iteration steps) and ρ ( S T ) be as small as possible (increasing the rate of convergence). For linear multiuser detection, we are interested in the iterative solution of Md = b where for the decorrelator, M = R, b = y and for the LMMSE M = WRW + σ 2 I, b = Wy. Let D, L and U be the diagonal, strictly lower triangular and strictly upper triangular parts of M respectively. We shall consider linear parallel cancellation, based on S = D (Jacobi, stationary methods and Chebyshev) and linear serial cancellation based on S = D + ωl for a real parameter ω [0, 2] (Gauss-Seidel, successive relaxation). We shall consider convergence criteria for large systems which henceforth means β = K/N is held constant and K.

3 3 A. General Convergence Issues We consider convergence in normed spaces [34, Chapter 5]. While this may not be directly related to bit error rate, it is still a useful criteria for the following reasons. Firstly it is tractable and gives insight into the iterative process. Secondly, the Euclidean norm is of interest for Gaussian channels and systems using linear filters as pre-processors for minimum distance decoding. We have already stated that for fast convergence we should minimize ρ (B). In practice it depends on the norm of interest and the number of steps that will be used. Let e k = ˆx x k, where ˆx is a solution to (2). It is easy to show that the error vector at step k is e k = B k e 0. Thus for any relevant vector norm, and its induced matrix norm [34, Definition 5.6.] we have e k = B k e 0 B k e 0, according to the properties of induced matrix norms [34, Theorem 5.6.2]. Thus the error at step k at most B k times the error of the initial guess. We therefore make the following definition [23]. Definition 2: Let B = S T be the iteration matrix. For a given matrix norm,, the convergence factor for k steps is given by B k. If it is known a-priori how many steps will be used (or can be afforded) for a particular iteration, it is B k that should be minimized. However it is known that for any matrix norm, lim k B k /k = ρ (B), i.e. the average convergence factor, per step for k steps approaches the spectral radius for large k [34, Corollary 5.6.4]. Note also that for B symmetric, B k /k 2 = ρ (B), which means that the error of the initial guess in Euclidean space is decreased geometrically with rate ρ (B). In light of these results (especially the latter, pertaining to the Euclidean norm) and for tractability we hereafter concentrate on the spectral radius of the iteration matrix, sometimes called the asymptotic convergence factor. The techniques we describe could be applied to the finite k case, or for the use of any other norm. See [44], [45] and [20, Problem 5.2] for optimization of convergence for some finite k receivers. Not only is the spectral radius of the iteration matrix of interest, but clearly reduction of the norm of the initial error is also important. This amounts to choosing an inital guess that is not too far (at least on average) from the solution. Of course this guess must be obtained with low computational burden (otherwise we could simply choose the exact solution as our first guess). This issue was considered in [4], where they found via simulation that initialisation using the all-zero vector performed worse than an initial guess consisting of the matched filter output y. The authors also considered an even better sliding window method for initialisation, which was only possible due to the asynchronous nature of their system. In this paper, we shall use x 0 = y as the initial guess throughout, as we wish to concentrate on the properties of the iteration matrices. We consider in greater detail the issue of optimising x 0 in a forthcoming paper [46]. B. Jacobi The Jacobi iteration has a particularly simple implementation, since the diagonal matrix S = D is easily invertible. The iteration is as follows: x k+ = D (y (M I)x k ). (3) For the decorrelator, the Jacobi iteration corresponds to parallel interference cancellation (see Figure, with τ i = and initial guess d 0 = y). d k+ = y (R I)d k. (4) This is equivalent to implementing the matrix inversion via the series expansion R = ( ) i (R I) i, i=0 convergent if ρ (R) < 2. This convergence criterion has been mentioned by several authors [20]. In the following lemma 2, we give a more meaningful convergence criterion for systems using long pseudo-random sequences. Theorem : For large systems, the Jacobi iteration (4) converges to the decorrelator if and only if K < N( 2 ) 2, that is for system loads of less than about 7%. Furthermore, the asymptotic convergence factor, ρ J is given by ρ J = β + 2 β. (5) Proof: The iteration is convergent if and only if ρ (I R) <. Now λ S (R) implies λ S (I R) [34, Theorem..6]. Hence the iteration converges if and only if max λ S(R) λ <. If we fix β = K/N, and send K, ρ (R) converges in probability to ( β ) 2, according to Lemma 2. This implies convergence to the decorrelator if and only if K < N( 2 ) 2. (Since λ min λ max for β [0, ]). [Figure about here.] For the LMMSE filter, S = W 2 + σ 2 I, T = W(I R)W and the iteration is given by x k+ = ( W 2 + σ 2 I ) W [y (R I)Wxk ], (6) which is just a signal-to-noise ratio weighted version of the parallel cancellation (4). Using similar reasoning as before, we can derive convergence criteria for the Jacobi implementation of the LMMSE filter. Theorem 2: For large systems, the Jacobi iteration (6) converges to the LMMSE detector if K < N ( 2 + γ max ) 2 (7) where γ max = max i=,...,k wi 2/σ2. Moreover, for equal power users (7) is also necessary. Corollary : For under-saturated systems (β ), the iteration (6) always converges for γmax 2. 2 This result has since been independently derived in [47].

4 4 Proof: S T = ( W 2 + σ 2 I ) W(I R)W, W ( W 2 + σ 2 I ) W(I R), where denotes similarity. Note that both I R and W ( W 2 + σ 2 I ) W are Hermitian, and hence their spectral radii are equal to their respective spectral norms. From the sub-multiplicative property, ρ ( S T ) = S T 2 (8) W ( W 2 + σ 2 I ) W 2 I R 2 (9) ( = ρ W 2( W 2 + σ 2 I ) ) ρ (I R) (0) = (λ max ) max i=...k w 2 i w 2 i + σ2. () For equal power systems W(W 2 + σ 2 I) W is just a scaled identity matrix, and (9) is an equality. Equation (7) results from similar limiting arguments as for the decorrelator and the definition of γ max. If γ max 2, we see that the condition reduces to K N which is true for under-saturated systems. Corollary corresponds to the strongest user having a signal-to-noise ratio of 3 db or less. The performance of the LMMSE filter would not be very good is that case. C. Stationary Iterative Methods A simple generalization improving on the the Jacobi iteration (3) is brought about by the introduction of a sequence of parameters. By carefully selecting these parameters, the convergence problems of the Jacobi method are defeated, and convergence speed can be improved. A first order iterative solution of (2) is given by (for some initial vector x 0 ) x k+ = x k τ k (Mx k b), k = 0,,... (2) where τ k is a sequence of constants [23, Equation (5.8 )]. If τ k = τ for all k, the method is called stationary 3. It is of interest to select a τ that minimizes the spectral radius of the iteration matrix. Theorem 3: For large systems, β <, the first order stationary iteration d k+ = + β y ( ) + β R I d k, k = 0,,... converges to the decorrelator for any initial vector d 0, and the constant /( + β) is optimal 4. The asymptotic convergence factor, ρ S is given by ρ S = 2 β + β (3) 3 We have used the identity as the pre-conditioning matrix in this case. See [23, Section 5.2] for the easy extension to pre-conditioning matrices. 4 With respect to the Euclidean norm, this constant is optimal for any number of stages, since the iteration matrix I τr is symmetric. Proof: According to [23, Theorem 5.6], for M symmetric positive definite, the parameter that results in fastest convergence of the first order stationary iteration (2) is τ opt = 2/(λ min +λ max ). The proof is completed by application of Lemma 2. Figure shows the implementation of this receiver, with τ i = /( + β). This first order stationary method has been suggested in [45], although the connection to this well known mathematical technique is not made. Given in [45] however is an outline of how to select τ to minimize the mean square error for a given number of steps. See also [44]. A second order iteration, which depends now on the two most recent estimates, is given by [23, Equation (5.9 )] as x k+ = a k x k + ( a k )x k b k (Mx k b), k = 0,,... (4) For the stationary iteration we set a k = a, b k = b for l =, 2,... and a 0 =, b 0 = τ opt 5. Theorem 4: For large systems, β <, the second order iteration d = y + β (R I) d 0 d k+ = y (R ( + β)i) d k βd k, k =, 2,... converges to the decorrelator for any given initial guess d 0, and the parameters chosen are optimal among second order stationary iterations (4). The asymptotic convergence factor, ρ S2 is given by ρ S2 = β. (5) Proof: R is positive definite, hence we may use [23, Theorem 5.9] to find that a = b = + 2 ( ) 2 λmin/λ max +λ min/λ max 2a λ min + λ max. minimizes the asymptotic convergence factor. Lemma 2 completes the proof. Setting M = WRW + σ 2, either the first order iteration (2), or the second order iteration (4) may also be used to implement the LMMSE filter. In this case the limiting values of the extremal eigenvalues must be found, or bounded, using the techniques described in Appendix V. D. Chebyshev Iteration We now consider the optimal parameter sequence for the first order non-stationary iteration (2). This is known as the Chebyshev method. In this case, the optimal τ k are known for a given number of iteration steps, k = 0,, 2,..., k max. It is shown in [23, p. 8] that these parameters are related to the inverses of the zeros of Chebyshev polynomials. Using Lemma 2, we can easily evaluate 5 This choice of b 0 yields the same first step improvement as the first order method.

5 5 the large system τ k for either the decorrelator, or for the LMMSE. In the case of the decorrelator 6, τ k = + β + 2 ( ) k /2 β cos k max + π. (6) Figure shows the implementation of this receiver. As k max, the convergence factor for the first order Chebyshev method approaches that of the second order stationary method [23]. This first order Chebyshev iteration can however be numerically unstable for certain values of k, as the following lemma shows. Lemma : The first order Chebyshev iterative implementation of the decorrelator is numerically stable for β < For larger values of β, the matrix 2 B k is unstable for k /2 k max + > π cos λ min 4 β. (7) For large k max, an approximate condition is k > λ max k max 8 β. Proof: Let B = I τ k R be the iteration matrix for stage k. First, we note that, according to [34, Theorem..6], ρ (B) = max { τ k λ max, τ k λ min }. It is easily verified for 0 < β <, that τ k is positive and monotonically increasing with k. Now if both τ k λ max and τ k λ min are less than, the matrix is stable. For any other case, a necessary condition for instability is τ k λ max > 2. Hence B k is unstable if cos ( ) k /2 k max + π < λ min 4 β. (8) Note that for β < /(7+2 2) the inequality (8) cannot be solved for k, and B k is always stable. For other values of β, we have the solution (7). The approximation is due to cos (x)/π ( x)/2. Note that Lemma only states that for certain values of k, the iteration matrix is unstable. What is important for the method as a whole is that ρ ( kmax k=0 B k ) <. In practice, the small values may balance the larger values. This phenomenon may be avoided by using the second order Chebyshev method [23]. Calculation of the optimal parameters for the second order method is left to the reader, see [23, Section 5.3.]. See also [44], where it is shown that for the LMMSE filter, choices of the τ k exist, such that only K steps are needed for exact convergence. E. Successive Relaxation We have seen that convergence problems exist for both the decorrelator and LMMSE filters implemented using 6 For the equal power LMMSE, simply add σ 2 to the right hand side of (6). the Jacobi iteration. In this section we shall investigate the successive relaxation (SR) technique, which is always convergent for symmetric positive definite matrices 7 and ω [0, 2]. It is also known that successive relaxation converges faster than Jacobi. In the SR method the complex Hermitian matrix M is split into M = D ωl ( ω)l L and after some algebra, we obtain the (simplified) iteration ( ) ( ) ω ω D L x k+ = ω D + L x k + b. (9) Figure 2 shows an implementation of the successive relaxation technique which amounts to a weighted serial interference cancellation process. [Figure 2 about here.] The design problem for successive relaxation is the choice of the parameter ω. This parameter can be chosen from the interval [0, 2] in order to speed up convergence. Since convergence speed depends upon the spectral radius of S T, it makes sense to choose ω to minimize this quantity. For matrices with a certain special property [48], a closed form expression for the optimal value of ω exists. Unfortunately, the matrices that we must consider do not have this property, and we must settle for a selection of ω that minimizes an upper bound on ρ ( S T ). Such analysis is given for positive definite matrices in [23]. We reproduce the main result here for reference. Theorem 5 (Axelsson Theorem 6.43) Let M C n n be Hermitian and positive definite with diagonal part D and strictly lower triangular part L, such that M = D L L. Further let S = ω D+L be the splitting of M according to the SR iteration and let 0 < ω < 2. Then ρ ( S T ) 2 2 ω ( ω ) 2 2 λ min( M) + γ +, (20) ω where M = D 2 MD 2, L = D 2 LD 2 and [ x ] γ = x Mx 2 Lx 4 sup x C n, x = (2) Furthermore if for all unit length complex n-vectors, x Lx < /2, then ω = 2/( + (2λ min ( M)) /2 ) minimizes the upper bound (20). Unfortunately, the last part of Theorem 5 is too restrictive, as the following calculation suggests. For the decorrelator, M = M = R (since D = I) and L = L. Define the numerical radius [34, p. 32], r(m) = max x = x Mx. Using the properties of the numerical radius [23, p. 36], we may write r(r) r(i) + 2r(L) and hence r(l) (r(r) )/2 = (λ max )/2. Thus for large systems r(r) β/2+ β, which is monotonically increasing and equal to /2 at β = ( 2 + ) 2.7. Hence for β > ( 2 + ) 2, the required property 7 A sufficient condition only. There are other types of matrices that can also result in convergence.

6 6 x Lx < /2 is violated. We therefore seek another approach. Note that for 0 < ω < 2, the second term of (20) is positive, since we know that in this range the spectral radius is less than. The upper bound results from γ, which is a maximization over all unit length complex vectors x, rather than setting x equal to the eigenvector corresponding to the largest eigenvalue of S T (see [23, p. 242] for details). Instead of choosing γ according to (2), we propose to use an (appropriately maximized) lower bound to this value. This means that we no longer get an upper bound for the spectral radius, however we shall see that the resulting ω compares well with simulation, as shown in Figure 4. Let M and L be defined as in the Theorem. Then (since M is positive definite), x x Mx = D 2 (D L L ) D 2 x + 2 x Lx. Hence x 2 Lx 4 (x Mx ) 2. Thus for all unit x C n, we have γ [ x ] x Mx 2 Lx 4 x Mx = 4 x Mx 2. [ 4 (x Mx ) 2 4 Hence γ λ max ( M)/4 /2 and we choose as our relaxation parameter ω as follows 2 ω = arg min ω ( ω [0,2] ω 2) 2 λ min ( M) λmax( M) ω λ min ( M)λ max ( M) = 2 λ min ( M)λ max ( M). (22) For the decorrelator, M = R and we can substitute the limiting values for the extremal eigenvalues from Lemma 2 to find ω DEC = 2 2 β. (23) Substitution of ω into the right hand side of (20) results in an asymptotic convergence factor, ρ SR of (bearing in mind the approximations used) ρ SR = β 4. (24) Hence the asymptotic convergence factor for successive relaxation is always worse than that for the second order stationary method. In the case of the LMMSE filter with unit power users, M = (R + σ 2 I)/( + σ 2 ), and ω for large systems may be found easily from Equation (22), with λ min ( M) = (λ min + σ 2 )/( + σ 2 ) and λ max ( M) = (λ max + σ 2 )/( + σ 2 ). ] F. Conjugate Gradient The conjugate gradient method [22] was considered for multiple-access detection in [4], [7]. This method is parameter free, provides fast convergence and has a finite termination property. The latter property means that in the absence of numerical errors, the method finds the exact solution of a K K system after K iterations. For a system matrix M with condition number κ = λ max /λ min, the error at iteration k with respect to the weighted norm e M = M /2 e 2 obeys (κ ) k e k M 2 ( κ + ) 2k + ( κ ) 2k e 0 M (25) Taking k, we obtain the asymptotic convergence factor ( κ )/( κ + ). Considering the decorrelator and large systems this results in an asymptotic convergence factor ρ CG = β, which is the same as the second order stationary method. This result however is misleading. In practice more accurate approximations than predicted by (25) are obtained. IV. Simulation Results The analysis presented in this paper has been for very large systems, K. We now present simulation results that indicate the validity of the results for more realistically sized systems, (bearing in mind that if the system is small enough, we can always implement direct inversion). Figure 3 shows simulated bit error rates for the decorrelator and for 5 steps of the Jacobi method. The number of users was kept fixed, K = 8, equal powers were assigned to the users, corresponding to a signal to noise ratio of E b /N 0 = 7dB. [Figure 3 about here.] From Figure 3, we see that the performance of the Jacobi method starts to diverge from the decorrelator for β >.5, agreeing well with the value of.7 predicted by theory. The curve labeled Hard is the result of performing the Jacobi iteration, taking however a hard decision at each stage (i.e. a non-linear iteration). It is interesting to note that although the hard canceler works better than Jacobi, it still has similar convergence problems (not shared by the other methods we have discussed). Note that the hard-decision cancellation method offers only slight improvement over the decorrelator, and only for very lightly loaded systems. This result should be compared to the non-linear cancellation literature, such as [25], [28]. In fact, the iteration given in [25] is identical to Figure 2 with a hard-decision device inserted before the weighing coefficients R ij. Figure 4 compares the optimal parameters for successive relaxation, obtained by simulation and by analysis, Equation (23). The simulation was performed by generating 200 sets of unit energy, length N = 32 binary random sequences for K = through K = 32 users, and finding the ω which minimized the spectral radius of the corresponding iteration matrix. The resulting ω were averaged for each K, N pair.

7 7 [Figure 4 about here.] From this figure, we see that the analysis agrees fairly well with the simulation, (especially for small β), bearing in mind that the analysis was not for the expected value of ω, and that it is only an approximation. In any case, as the next figure shows, the choice of ω for successive relaxation may perhaps be better approached from an entirely different perspective. We remark that for low β (the region in which linear receivers work well), we predict ω close to. It may therefore be desirable to use the Gauss-Seidel method, which would avoid some multiplications. Figure 5 shows the simulated bit error rate versus iteration for conjugate gradient (CG), successive relaxation (SOR), Gauss-Seidel (GS), Chebyshev, first (S) and second order (S2) stationary methods. The simulation parameters are K = 8, N = 6, E b /N 0 = 0dB (all users transmit with equal powers). The SOR method uses ω as the relaxation parameter. Jacobi is not shown, as β = /2 is outside its convergence region. The exact decorrelator and LMMSE performances are included for comparison. A maximum of eight iterations is shown. This is because with eight users, direct inversion would be more efficient than performing nine or more iterations. [Figure 5 about here.] In terms of BER, the conjugate gradient method achieves the decorrelator performance first, requiring only three iterations. Note that the convergence of this method is not monotone, and for iterations 3-6 conjugate gradient obtains better performance than the decorrelator. We also see that although the optimized SOR converges slightly faster than Gauss-Seidel, that the latter method out-performs SOR. Once again this is because convergence is from below (this was also observed in [4]). Of the parallel methods, the Chebyshev method performs best, both in terms of BER, and in terms of providing a smooth convergence (once again from below). It is interesting to note that the Chebyshev method actually achieves the best BER of any method (approaching the LMMSE performance), due to its overshoot of the decorrelator performance. Similar phenomena have been observed by other researchers, and have been exploited in [44]. The stationary methods display a ping-pong effect, which has been previously reported by others. It is clear that further research is required in order to optimize convergence, or to minimize the BER of such methods. Figure 6 shows the BER performance of the iterative implementations of the LMMSE filter. The iterative methods are labeled as for Figure 5. Once again, the simulation parameters are K = 8, N = 6, E b /N 0 = 0dB. From the figure, we see that the conjugate gradient method achieves the LMMSE performance after only 5 iterations. Successive relaxation takes an extra iteration and Gauss-Seidel requires 7 iterations. Note however that for iteration and 2, Gauss-Seidel outperforms SOR. Of the parallel methods, Chebyshev again outperforms the stationary methods as expected. We now see that whereas for the decorrelator, the slow convergence of the Chebyshev method was to its advantage in obtaining good performance, it now converges from above, making it less attractive. [Figure 6 about here.] In order to investigate the convergence properties for a more lightly loaded system, Figure 7 shows the simulated bit error rate performance versus iteration for a system with K = 8 and N = 32. The SNR is once again 0 db. Only the results for implementation of the LMMSE are shown, since at this load, the performance of the decorrelator and LMMSE are not that different. [Figure 7 about here.] The stationary methods are still slow to converge. We now see that the conjugate gradient, successive relaxation and Gauss-Seidel all offer similar performance. This is in contrast to remarks made in [4]. We further investigated this phenomenon with larger dimension simulations (32 users), and we observed similar characteristics. In practice, there are other issues in selection of an implementation method. Although conjugate gradient is in general fastest, it is slightly more complex that either parallel or serial cancellation (it requires the computation of an additional three inner products). The parallel and serial cancellation methods also have the advantage that they can be pipelined, to reduce latency. V. Discussion Using the theory of large random matrices, we have presented convergence analyses of various iterative implementations for the decorrelator and LMMSE. Let us now compare these methods, based upon the asymptotic convergence factor, which may be of more interest for coded systems which use these receivers as a first stage (we have already seen in Section IV that uncoded BER and convergence factor are only loosely related). Figure 8, shows the asymptotic convergence factor for the Jacobi (5), first (3) and second (5) order stationary and successive relaxation (24) methods. For the latter three methods, optimal or near optimal choice of parameters has been used. [Figure 8 about here.] Starting with small spreading factors, we see that Jacobi, the first order stationary method and successive relaxation have similar performance, whereas the second order stationary method (and asymptotically, the first order Chebyshev method) outperforms all, even for highly loaded systems. Notable is the cut-off at β 0.7 for the Jacobi method, a problem that does not affect the other methods. Note also how these methods approach for highly loaded β systems. Because the first order stationary method is only a simple modification of the Jacobi method, but avoids the convergence problems, it may be desirable for low spreading factors. Even so, its convergence factor increases quite quickly with β, and if better performance is desired, the second

8 8 order method would be used, resulting in a slight, but perhaps justified increase in complexity. Lying in between the first and second order methods is successive relaxation. Table I summarizes the main results of this paper. Further research is needed into the convergence properties of non-linear cancellation schemes. Results in this direction can be found in [47]. Extensions of our results to spacetime processing may be found in [49]. [Table about here.] Appendix We are interested in the asymptotic behavior of the eigenvalues of R = A A, where A is chosen according to Definition. The following lemma shows that the extremal eigenvalues of R converge to specific values. Lemma 2 (Extremal eigenvalues of R) Let λ min = min S (R) and λ max = max S (R) be the smallest and largest eigenvalues respectively of the complex positive semi-definite matrix R = A A, where A is chosen randomly according to Definition. Then, holding β = K/N < constant, and sending K, ( ) 2 λ min β λ max ( β + ) 2, almost surely. The proof for this lemma can be found in [50] (smallest eigenvalue) and [5] for the largest eigenvalue. In fact the distribution of the eigenvalues tends to a deterministic limit [52], [53], [50], [54], [55], a phenomenon exploited by several authors to prove asymptotic results concerning the capacity and spectral efficiency of the randomsequence CDMA channel, both for optimal joint decoding [56] and using the decorrelator and LMMSE [57], [58]. A stronger result than Lemma 2 is in fact true, namely that in the limit, with probability, there are no eigenvalues outside the support of the limiting spectral distribution [59]. In order to derive results for the LMMSE filter, we shall have occasion to find the extremal eigenvalues of WRW + σ 2 I. Using [34, Theorem..6] we need only find the extreme eigenvalues of WRW (or alternatively W 2 R, by the obvious similarity transform). In the equal power case (where without loss of generality we can assume unit power), the limiting values of the eigenvalues are found easily, and under the same limiting conditions as Lemma 2 are given by min S ( R + σ 2 I ) ( ) 2 β + σ 2 max S ( R + σ 2 I ) ( β + ) 2 + σ 2. In the non-equal power case, there is a little more work to do. For large R, chosen according to Definition, we can find the support of the limiting spectral distribution of WRW using the method described in [60], [59]. Suppose that for large systems, the spectral distribution of W converges to a fixed distribution H(t), and let z(m) = m + β tdh(t) + tm. (26) (according to Equation (.2) in [59]). The support of the limiting spectral distribution of WRW is given by the range of values of (26) where z(m) is increasing. 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10 0 Alex Grant Alex Grant received the B.E. and Ph.D. degrees from the University of South Australia in 993 and 996 respectively. In 997, he was a post-doctoral research fellow at the Laboratory for Signal and Information Processing, Swiss Federal Institute of Technology (ETH), Zurich. Since March 998 he has been with the Institute for Telecommunication Research, where he is now a Senior Lecturer. Alex s research interests include information theory and its applications to multiple user and other communications problems. Alex is Leader of the Coding and Information Theory Research Group at ITR and serves as Chariman for the Australian Chapter of the IEEE Information Theory Society. Christian Schlegel received the Dipl. El. Ing. ETH degree from the Federal Institute of Technology, Zurich, in 984, and the M.S. and Ph.D. degrees in electrical engineering from the University of Notre Dame, Notre Dame, IN, in 986 and 988, respectively. In 988 he joined the Communications Group at the research center of Asea Brown Boveri, Ltd., in Baden, Switzerland, where he was involved in mobile communications research. He spent the 99/92 academic year as Visiting Assistant Professor at the University of Hawaii at Manoa, Hawaii, before joining the Digital Communications Group at the University of South Australia in Adelaide, Australia, where he headed the mobile research centre (MCRC). From he was with the University of Texas at San Antonio, and since 996 he has been with the University of Utah, Salt Lake City. His interests are in the area of digital communications, coded modulation, mobile radio systems, multiple access communications, space-time coding for high-capacity wireless systems, as well as in the implementation of complex communications systems and the design of analog VLSI decoders for digital systems. He is the author of the research monograph Trellis Coding published by IEEE Press in 997, and is currently working on his new book entitled Coordinated Multiple User Communications, coauthored with Dr. Alex Grant. Dr. Schlegel received an NSF 997 Career Award in support of his research in multiuser communications. Dr. Schlegel is a senior member of the IEEE Information Theory and Communication Societies. In the past he has organized conference sessions and served as consultant on digital communications and error control coding projects on numerous occasions. He is an associate editor for coding theory and techniques for the IEEE Transactions on Communications, and the technical co-chair of the IEEE Information Theory Workshop 200 to be held in Cairns, Australia. He has also served on several technical progam committees, most recently for the IEEE VTC 99 conference in Houston, TX, the IEEE VTC 2000 conference held in Tokyo, Japan, and Globecom 200 to be held in San Antonio, TX.