A Central Limit Theorem for the SINR at the LMMSE Estimator Output for Large Dimensional Signals

Transcription

1 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT A Central Limit Theorem for the SINR at the LMMSE Estimator Output for Large Dimensional Signals arxiv: v [cs.it] 0 Oct 2008 Abla ammoun ), Malika harouf 2), Walid Hachem 3) and Jamal Najim 3) Abstract This paper is devoted to the performance study of the Linear Minimum Mean Squared Error estimator for multidimensional signals in the large dimension regime. Such an estimator is frequently encountered in wireless communications and in array processing, and the Signal to Interference and Noise Ratio SINR) at its output is a popular performance index. The SINR can be modeled as a random quadratic form which can be studied with the help of large random matrix theory, if one assumes that the dimension of the received and transmitted signals go to infinity at the same pace. This paper considers the asymptotic behavior of the SINR for a wide class of multidimensional signal models that includes general multiantenna as well as spread spectrum transmission models. The expression of the deterministic approximation of the SINR in the large dimension regime is recalled and the SINR fluctuations around this deterministic approximation are studied. These fluctuations are shown to converge in distribution to the Gaussian law in the large dimension regime, and their variance is shown to decrease as the inverse of the signal dimension. Index Terms Antenna Arrays, CDMA, Central Limit Theorem, LMMSE, Martingales, MC-CDMA, MIMO, Random Matrix Theory. ) Telecom ParisTech ENST), Paris, France. abla.kammoun@enst.fr 2) Telecom ParisTech ENST) and Casablanca University, Morocco. malika.kharouf@enst.fr 3) CNRS / ENST, Paris, France. walid.hachem, jamal.najim@enst.fr October 0, 2008

2 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 2 I. INTRODUCTION Large Random Matrix Theory LRMT) is a powerful mathematical tool used to study the performance of multi-user and multi-access communication systems such as Multiple Input Multiple Output MIMO) digital wireless systems, antenna arrays for source detection and localization, spread spectrum communication systems as Code Division Multiple Access CDMA) and Multi-Carrier CDMA MC-CDMA) systems. In most of these communication systems, the N dimensional received random vector r C N is described by the model r = Σs + n ) where s = [s 0,s,...,s ] T is the unknown random vector of transmitted symbols with size + satisfying Ess = I +, the noise n is an independent Additive White Gaussian Noise AWGN) with covariance matrix Enn = ρi N whose variance ρ > 0 is known, and matrix Σ represents the known channel in the wide sense whose structure depends on the particular system under study. One typical problem addressed by LRMT concerns the estimation performance by the receiver of a given transmitted symbol, say s 0. In this paper we focus on one of the most popular estimators, namely the linear Wiener estimator, also called LMMSE for Linear Minimum Mean Squared Error estimator: the LMMSE estimate ŝ 0 = g r of signal s 0 is the one for which the N vector g minimizes E ŝ 0 s 0 2. If we partition the channel matrix as Σ = [y Y] where y is the first column of Σ and where matrix Y has dimensions N, then it is well known that vector g is given by g = ΣΣ + ρi N ) y. Usually, the performance of this estimator is evaluated in terms of the Signal to Interference plus Noise Ratio SINR) at its output. Writing the received vector r as r = s 0 y + r in where s 0 y is the relevant term and r in represents the so-called interference plus noise term, the SINR is given by β = g y 2 /E g r in 2. Plugging the expression of g given above into this expression, one can prove that the SINR β is given by the well-known expression: β = y YY + ρi N ) y. 2) In general, this expression does not provide a clear insight on the impact of the channel model parameters such as the load factor N, the power distribution of the transmission data streams, or the correlation structure of the channel paths in the context of multi-antenna transmissions) on the performance of the LMMSE estimator. An alternative approach, justified by the fluctuating nature of the channel paths in the context of MIMO communications and by the pseudo-random nature of the spreading sequences in spread spectrum October 0, 2008

3 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 3 applications consists to model matrix Σ as a random matrix in this case, β becomes a random SINR). The simplest random matrix model for Σ, corresponding to the most canonical MIMO or CDMA transmission channels, corresponds to independent and identically distributed i.i.d.) entries with mean zero and variance N. In that case, LRMT shows that when and the load factor N converges to a limiting load factor α > 0, the SINR β converges almost surely a.s.) to an explicit deterministic quantity βα, ρ) which simply depends on the limiting load factor α and on the noise variance ρ. As a result, the impact of these two parameters on the LMMSE performance can be easily evaluated [], [2]. The LMMSE SINR large dimensional behavior for more sophisticated random matrix models has also been thoroughly studied cf. [], [3] [9]) and it has been proved that there exists a deterministic sequence β ), generally defined as the solution of an implicit equation, such that β β 0 almost surely as and N remains bounded away from zero and from infinity. Beyond the convergence β β 0, a natural question arises concerning the accuracy of β for finite values of. A first answer to this question consists in evaluating the Mean Squared Error MSE) of the SINR E β β 2 for large. A further problem is the computation of outage probability, that is the probability for β β to be below a certain level. Both problems can be addressed by establishing a Central Limit Theorem CLT) for β β. In this paper, we establish such a CLT Theorem 3 below) for a large class of random matrices Σ. We prove that there exists a sequence Θ 2 = O) such that Θ β β ) converges in distribution to the standard normal law N0,) in the asymptotic regime. One can therefore infer that the MSE asymptotically behaves like Θ2 and that the outage probability can be simply approximated by a Gaussian tail function. The class of random matrices Σ we consider in this paper is described by the following statistical model: Assume that Σ = Σ nk ) N, n=,k=0 = σnk W nk ) N, n=,k=0 where the complex random variables W nk are i.i.d. with EW nk = 0, EW 2 nk = 0 and E W nk 2 = and where σ 2 nk ; n N; 0 k ) is an array of real numbers. Due to the fact that E Σ nk 2 = σ2 nk, the array σnk 2 ) is referred to as a variance profile. An important particular case is when σ2 nk is separable, that is, writes: σ 2 nk = d n d k, 4) where d,...,d N ) and d 0,..., d ) are two vectors of real positive numbers. 3) October 0, 2008

4 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 4 Applicative contexts. Among the applicative contexts where the channel is described appropriately by model 3) or by its particular case 4), let us mention: Multiple antenna transmissions with + distant sources sending their signals toward an array of N antennas. The corresponding transmission model is r = Ξs + n where Ξ = HP /2, matrix H is a N +) random matrix with complex Gaussian elements representing the radio channel, P = diagp 0,...,p ) is the deterministic) matrix of the powers given to the different sources, and n is the usual AWGN satisfying Enn = ρi N. Write H = [h 0 h ], and assume that the columns h k are independent, which is realistic when the sources are distant one from another. Let C k be the covariance matrix C k = Eh k h k and let C k = U k Λ k U k be a spectral decomposition of C k where Λ k = diagλ nk ; n N) is the matrix of eigenvalues. Assume now that the eigenvector matrices U 0,...,U are all equal to some matrix U, for instance), a case considered in e.g. [0] note that sometimes they are all identified with the Fourier N N matrix []). Let Σ = U Ξ. Then matrix Σ is described by the statistical model 3) where the W nk are standard Gaussian i.i.d., and σ 2 nk = λ nkp k. If we partition Ξ as Ξ = [x X] similarly to the partition Σ = [y Y] above, then the SINR β at the output of the LMMSE estimator for the first element of vector s in the transmission model r = Ξs + n is β = x XX + ρi N ) x = y YY + ρi N ) y due to the fact that U is a unitary matrix. Therefore, the problem of LMMSE SINR convergence for this MIMO model is a particular case of the general problem of convergence of the right-hand member of 2) for model 3). It is also worth to say a few words about the particular case 4) in this context. If we assume that Λ 0 = = Λ and these matrices are equal to Λ = diagλ,...,λ N ), then the model for H is the well-known ronecker model with correlations at reception [2]. In this case, Σ = U Ξ = U HP /2 = Λ /2 WP /2 5) where W is a random matrix with iid standard Gaussian elements. This model coincides with the separable variance profile model 4) with d n = λ n and d k = p k. CDMA transmissions on flat fading channels. Here N is the spreading factor, + is the number of users, and Σ = VP /2 6) October 0, 2008

5 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 5 where V is the N + ) signature matrix assumed here to have random i.i.d. elements with mean zero and variance N, and where P = diagp 0,...,p ) is the users powers matrix. In this case, the variance profile is separable with d n = and d k = N p k. Note that elements of V are not Gaussian in general. Cellular MC-CDMA transmissions on frequency selective channels. In the uplink direction, the matrix Σ is written as: Σ = [H 0 v 0 H + v + ], 7) where H k = diagh k exp2ıπn )/N); n N) is the radio channel matrix of user k ı = ) in the discrete Fourier domain here N is the number of frequency bins) and V = [v0,,v ] is the N +) signature matrix with i.i.d. elements as in the CDMA case above. Modeling this time the channel transfer functions as deterministic functions, we have σ 2 nk = N h kexp2ıπn )/N)) 2. In the downlink direction, we have Σ = HVP /2 8) where H = diaghexp2ıπn )/N); n N) is the radio channel matrix in the discrete Fourier domain, the N + ) signature matrix V is as above, and P = diagp 0,...,p ) is the matrix of the powers given to the different users. Model 8) coincides with the separable variance profile model 4) with d n = N hexp2ıπn )/N)) 2 and d k = p k. About the literature. The asymptotic approximation β first order result) is connected with the asymptotic eigenvalue distribution of Gram matrices YY where elements of Y are described by the model 3), and can be found in the mathematical LRMT literature in the work of Girko [3] see also [4] and [5]). Applications in the field of wireless communications can be found in e.g. [6] in the separable case and in [8] in the general variance profile case. Concerning the CLT for β β second order result), only some particular cases of the general model 3) have been considered in the literature among which the i.i.d. case σnk 2 = ) is studied in [6] and based on a result of [7] pertaining to the asymptotic behavior of the eigenvectors of YY ). The more general CDMA model 6) has been considered in [8], using a result of [9]. The model used in this paper includes the models of [6] and [8] as particular cases. October 0, 2008

6 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 6 ) Fluctuations of other performance indexes such as Shannon s mutual information E log det ΣΣ ρ + I N have also been studied at length. Let us cite [20] where the CLT is established in the separable case and [2] for a CLT in the general variance profile case. Similar results concerning the mutual information are found in [22] and in [23]. Limiting expressions vs -dependent expressions. As one may check in Theorems 2 and 3 below, we deliberately chose to provide deterministic expressions β and Θ 2 which remain bounded but do not necessarily converge as. For instance, Theorem 2 only states that β β 0 almost surely. No conditions which would guarantee the convergence of β are added. This approach has two advantages: ) such expressions for β and Θ 2 exist for very general variance profiles σnk 2 ) while limiting expressions may not, and 2) they provide a natural discretization which can easily be implemented. The statements about these deterministic approximations are valid within the following asymptotic regime: Note that N to 9)., lim inf N > 0 and lim sup N <. 9) is not required to converge. In the remainder of the paper, the notation will refer We note that in the particular case where N α > 0 and the variance profile is obtained by a regular ) sampling of a continuous function f i.e. σnk 2 = f n N, k +, it is possible to prove that β and Θ 2 converge towards limits that can be characterized by integral equations. Principle of the approach. The approach used here is simple and powerful. It is based on the approximation of β by the sum of a martingale difference sequence and on the use of the CLT for martingales [24]. We note that apart from the LRMT context, such a technique has been used recently in [25] to establish a CLT on general quadratic forms of the type z Az where A is a deterministic matrix and z is a random vector with i.i.d. elements. Paper organization. In Section II, first-order results, whose presentation and understanding is compulsory to state the CLT, are recalled. The CLT, which is the main contribution of this paper, is provided in Section III. In Section October 0, 2008

7 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 7 IV, simulations and numerical illustrations are provided. The proof of the main theorem Theorem 3) in given in Section V while the Appendix gathers proofs of intermediate results. Notations. Given a complex N N matrix X = [x ij ] N i,j=, denote by X its spectral norm, and by X its maximum row sum norm, i.e., X = max N i N j= x ij. Denote by the Euclidean norm of a vector and by its max or l ) norm. II. FIRST ORDER RESULTS: THE SINR DETERMINISTIC APPROXIMATION In the sequel, we shall often show explicitly the dependence on in the notations. Consider the quadratic form 2): β = y YY + ρi N ) y, where the sequence of matrices Σ) = [y) Y)] is given by Let us state the main assumptions: ) Σ) = Σ nk )) N, n=,k=0 = σnk ) N, W nk n=,k=0 A: The complex random variables W nk ; n, k 0) are i.i.d. with EW 0 = 0, EW 2 0 = 0, E W 0 2 = and E W 0 8 <. A2: There exists a real number σ max < such that sup max σ n N nk ) σ max. 0 k Let a m ; m M) be complex numbers, then diaga m ; m M) refers to the M M diagonal matrix whose diagonal elements are the a m s. If A = a ij ) is a square matrix, then diaga) refers to the matrix diaga ii ). Consider the following diagonal matrices based on the variance profile along the columns and the rows of Σ: A3: The variance profile satisfies D k ) = diagσk 2 ),,σ2 Nk )), 0 k 0) D n ) = diagσn 2 ),,σ2 n )), n N. lim inf min 0 k trd k) > 0. Since E W 0 2 =, one has E W 0 4. The following is needed:. October 0, 2008

8 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 8 A4: At least one of the following conditions is satisfied: E W 0 4 > or lim inf 2tr D 0 ) ) D k ) > 0. Remark : If needed, one can attenuate the assumption on the eighth moment in A. For instance, one can adapt without difficulty the proofs in this paper to the case where E W 0 4+ǫ < for ε > 0. We assumed E W 0 8 < because at some places we rely on results of [2] which are stated with the assumption on the eighth moment. Assumption A3 is technical. It has already appeared in [26]. Assumption A4 is necessary to get a non-vanishing variance Θ 2 k= in Theorem 3. The following definitions will be of help in the sequel. A complex function tz) belongs to class S if tz) is analytical in the upper half plane C + = {z C ; imz) > 0}, if tz) C + for all z C + and if imz) tz) is bounded over the upper half plane C +. Denote by Q z) and Q z) the resolvents of Y)Y) and Y) Y) respectively, that is the N N and matrices defined by: Q z) = Y)Y) zi N ) and Q z) = Y) Y) zi ). A. The SINR Deterministic approximation It is known [3], [26] that there exists a deterministic diagonal N N matrix function Tz) that approximates the resolvent Qz) in the following sense: Given a test matrix S with bounded spectral norm, the quantity tr SQz) Tz)) converges a.s. to zero as. It is also known that the approximation β of the SINR β is simply related to Tz) cf. Theorem 2). As we shall see, matrix Tz) also plays a fundamental role in the second order result Theorem 3). In the following theorem, we recall the definition and some of the main properties of Tz). Theorem : The following hold true: ) [26, Theorem 2.4] Let σnk 2 ); n N; k ) be a sequence of arrays of real numbers and consider the matrices D k ) and D n ) defined in 0). The system of N + functional equations t n, z) = t k, z) = ), z + tr D n ) T z)) n N z + trd k)t z)) ), k ) October 0, 2008

9 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 9 where T z) = diagt, z),...,t N, z)), T z) = diag t, z),..., t, z)) admits a unique solution T, T) among the diagonal matrices for which the t n, s and the t k, s belong to class S. Moreover, functions t n, z) and t k, z) admit an analytical continuation over C R + which is real and positive for z,0). 2) [26, Theorem 2.5] Assume that Assumptions A and A2 hold true. Consider the sequence of random matrices Y)Y) where Y has dimensions N and whose entries are given by Y nk = σnk W nk. For every sequence S of N N diagonal matrices and every sequence S of diagonal matrices with the following limits hold true almost surely: lim lim ) supmax S, S <, tr S Q z) T z)) = 0, z C R +, tr S Q z) T ) z) = 0, z C R +. The following lemma which reproduces [27, Lemma 2.7] will be used throughout the paper. It characterizes the asymptotic behavior of an important class of quadratic forms: Lemma : Let x = [X,...,X N ] T be a N vector where the X n are centered i.i.d. complex random variables with unit variance. Let A be a deterministic N N complex matrix. Then, for any p 2, there exists a constant C p depending on p only such that E N x Ax N tra) p C p E X N p 4 traa ) ) p/2 + E X 2p tr AA ) p/2)). 2) Noticing that traa ) N A 2 and that tr AA ) p/2) N A p, we obtain the simpler inequality E N x Ax N tra) p C p N p/2 A p E X 4) p/2 + E X 2p) 3) which is useful in case one has bounds on A. Using Theorem and Lemma, we are in position to characterize the asymptotic behavior of the quadratic form β given by 2). We begin by rewriting β as β = w 0 D/2 0 YY + ρi N ) D /2 0 w 0 = w 0 D/2 0 Q ρ)d /2 0 w 0 4) October 0, 2008

10 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 0 where the N vector w 0 is given by w 0 = [W 0,...,W N0 ] T and the diagonal matrix D 0 is given by 0). Recall that w 0 and Q are independent and that D 0 σmax 2 by A2. Furthermore, one can easily notice that Q ρ) = YY + ρi) /ρ. Denote by E Q the conditional expectation with respect to Q, i.e. E Q = E Q). From Inequality 3), there exists a constant C > 0 for which EE Q β trd 4 0Q ρ) C ) N 2 2 E D 0 Q 4 E W 0 4 ) 2 + E W 8) 0 C ) N 2 ) σ 2 4 max E W0 4 2 ) 2 + E W 8) 0 ρ ) = O 2. By the Borel-Cantelli Lemma, we therefore have β trd 0Q ρ)) 0 a.s. Using this result, simply apply Theorem 2) with S = D 0 recall that D 0 σmax 2 ) to obtain: Theorem 2: Let β = trd 0)T ρ)) where T is given by Theorem ). Assume A and A2. Then β β 0 a.s. B. The deterministic approximation in the separable case In the separable case σ nk ) = d n ) d k ), matrices D k ) and D n ) are written as D k ) = d k )D) and D n ) = d n ) D) where D) and D) are the diagonal matrices D) = diagd ),...,d N )), D) = diag d ),..., d )). 5) and one can check that the system of N + equations leading to T and T simplifies into a system of two equations, and Theorem takes the following form: Proposition : [26, Sec. 3.2] ) Assume σnk 2 ) = d n) d k ). Given ρ > 0, the system of two equations δ ρ) = D tr ρi N + δ ) ) ρ)d) ) ) 6) δ ρ) = D tr ρi + δ ρ) D) October 0, 2008

11 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT where D and D are given by 5) admits a unique solution δ ρ), δ ρ)). Moreover, in this case matrices T ρ) and T ρ) provided by Theorem ) coincide with T ρ) = ρ I + δρ)d) and T ρ) = ρ I + δρ) D). 7) 2) Assume that A and A2 hold true. Let matrices S and S be as in Theorem 2). Then, almost surely tr S Q ρ) T ρ))) 0 and S Q tr ρ) T )) ρ) 0 as. With these equations we can adapt the result of Theorem 2 to the separable case. Notice that D 0 = d 0 D and that δρ) given by the system 6) coincides with trdt), hence Proposition 2: Assume that σ 2 nk ) = d n) d k ), and that A and A2 hold true. Then where δ ρ) is given by Proposition ). β d0 δ ρ) 0 a.s. Let us provide a more explicit expression of δ which will be used in Section IV to illustrate the SINR behavior for the MIMO Model 5) and for MC-CDMA downlink Model 8). By combining the two equations in System 6), it turns out that δ = δ ρ) is the unique solution of the implicit equation δ = N n=0 d n ρ + d n k= p k +p kδ. 8) Recall that in the case of the MIMO model 5), d n = λ n and d k = p k, while in the case of the MC- CDMA downlink model 8), d n = N hexp2ıπn )/N) 2 and d k = p k again. Here d 0 = p 0 is the power of the user of interest user 0), and therefore β / d 0 is the normalized SINR of this user. Notice that δ ρ) is almost the same for all users, hence the normalized SINRs for all users are close to each other for large. Their common deterministic approximation is given by 8) which is the discrete analogue of the integral equation 6) in [6]. This example will be continued in Section III. III. SECOND ORDER RESULTS: THE CENTRAL LIMIT THEOREM The following theorem is the main result of this paper. Its proof is postponed to Section V. October 0, 2008

12 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 2 Theorem 3: ) Assume that A2, A3 and A4 hold true. Let A and be the matrices [ A = trd ] ld m T ρ) 2 + trd lt ρ) ) 2 and 9) l,m= = diag + ) 2 trd lt ρ) ; l ), where T is defined in Theorem ). Let g be the vector [ g = trd 0D T ρ) 2,, ] T trd 0D T ρ) 2. Then the sequence of real numbers is well defined and furthermore Θ 2 = gt I A) g + E W 0 4 ) trd2 0T ρ) 2 20) 0 < lim inf Θ2 lim sup Θ 2 <. 2) Assume in addition A. Then the sequence β = y YY + ρi) y satisfies ) β β Θ N0,) in distribution where β = trd 0T is defined in the statement of Theorem 2. Remark 2: Comparison with other performance indexes) It is interesting to compare the Mean Squared Error MSE) related to the SINR β : MSEβ ) = Eβ β ) 2, with the MSE related to Shannon s mutual information per transmit dimension I = log detρσσ +I) studied in [2], [22] for instance): ) ) MSEβ ) O while M SEI) O 2. Remark 3: On the achievability of the minimum of the variance) Recall that the variance writes Θ 2 = gt I A) g + E W 0 4 ) trd2 0T 2. As E W 0 2 =, one clearly has E W with equality if and only if W 0 = with probability one. Moreover, we shall prove in the sequel Section V-B) that lim inf D 0)T 2 > 0. Therefore E W 0 4 ) trd2 0 T2 is nonnegative, and is zero if and only if W 0 = with probability one. As a consequence, Θ 2 is minimum with respect to the distribution of the W nk if and only if these random variables have their values on the unit circle. In the context of CDMA and MC-CDMA, this is the case when the signature matrix elements are elements of a PS constellation. In multi-antenna systems, the October 0, 2008

13 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 3 W nk s are frequently considered as Gaussian which induces a penalty on the SINR asymptotic MSE with respect to the unit norm case. In the separable case, Θ 2 = d 2 0 Ω2 where Ω2 is given by the following corollary. Corollary : Assume that A2 is satisfied and that σnk 2 = d n d k. Assume moreover that ) min lim inf trd)),lim inf tr D)) > 0 2) where D and D are given by 5). Let γ = trd2 T 2 and γ = tr D 2 T2. Then the sequence ρ Ω 2 2 γ γ = γ ρ 2 γ γ + E W 0 4 ) ) satisfies 0 < lim inf Ω 2 lim sup Ω 2 <. If, in addition, A holds true, then: ) δ Ω β d0 N0,) in distribution. Remark 4: Condition 2) is the counterpart of Assumption A3 in the case of a separable variance profile and suffices to establish 0 < lim inf ρ 2 γ γ) lim sup ρ 2 γ γ) < see for instance [20]), hence the fact that 0 < lim inf Ω 2 lim sup Ω 2 <. The remainder of the proof of Corollary is postponed to Appendix B. Remark 5: As a direct application of Corollary to be used in Section IV below), let us provide the expressions of γ and γ for the MIMO Model 5) or MC-CDMA downlink Model 8). From 5) 7), we get γ = γ = N n=0 d n ρ + ρd n δ p k ρ + ρp k δ k= ) 2 = ) 2 N n=0 d n ρ + d n k= p k +p kδ where we recall that d n = λ n for Model 5), d n = N hexp2ıπn )/N) 2 for Model 8), and δ is the solution of 8). ) 2 22) IV. SIMULATIONS A. The general non necessarily separable) case In this section, the accuracy of the Gaussian approximation is verified by simulation. In order to validate the results of Theorems 2 and 3 for practical values of, we consider the example of a MC- CDMA transmission in the uplink direction. We recall that is the number of interfering users in this October 0, 2008

14 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 4 context. In the simulation, the discrete time channel impulse response of user k is represented by the vector with L = 5 coefficients g k = [g k,0,...,g k,l ] T. In the simulations, these vectors are generated pseudo-randomly according to the complex multivariate Gaussian law CN0,/LI L ). Setting the number of frequency bins to N, the channel matrix H k for user k in the frequency domain see Eq. 7)) is H k = diagh k exp2ıπn )/N); n N) where h k z) = L l=0 g k,lz l, the norm g k is the Euclidean norm of g k and P k is the power received from user k. Concerning the distribution of the user powers P k, we assume that these are arranged into five power classes with powers P,2P,4P,8P and 6P with relative frequencies given by Table IV-A. The user of interest User 0) is assumed to belong Pk g k TABLE I POWER CLASSES AND RELATIVE FREQUENCIES Class Power P 2P 4P 8P 6P Relative frequency /8 /4 /4 /8 /4 to Class. Finally, we assume that the number of interfering users is set to = N/2. In Figure, the Signal over Noise Ratio SNR) P/ρ for the user of interest is fixed to 0 db. The evolution of Eβ β ) 2 /Θ 2 for this user where Eβ β ) 2 is measured numerically) is shown with respect to. We note that this quantity is close to one for values of as small as = 8. In Figure 2, is set to = 64, and the SINR normalized MSE Eβ β ) 2 /Θ 2 is plotted with respect to the input SNR P/ρ. This figure also confirms the fact that the MSE asymptotic approximation is highly accurate. Figure 3 shows the histogram of β β )/Θ for N = 6 and N = 64. This figure gives an idea of the similarity between the distribution of β β )/Θ and N0,). More precisely, Figure 4 quantifies this similarity through a Quantile-Quantile plot. B. The separable case In order to test the results of Proposition 2 and Corollary, we consider the following multiple antenna MIMO) model with exponentially decaying correlation at reception: Σ = Ψ /2 WP /2 October 0, 2008

15 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 5 E ) 2 β β /Θ Fig.. SINR normalized MSE vs 0.8 Eβ β ) 2 /Θ SNR Fig. 2. SINR normalized MSE vs SNR October 0, 2008

16 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT hi stogram of β β) f or N=6 Θ 250 hi stogram of β β) f or N=64 Θ Fig. 3. Histogram of β β ) for N = 6 and N = QQplot for N=6 4 QQplot for N= Normal Quantiles 0 Normal Quantiles Empirical Quantiles Empirical Quantiles Fig. 4. Q-Q plot for β β ), N = 6 and N = 64; dash doted line is the 45 degree line. October 0, 2008

17 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 7 where Ψ = [a m n ] N m,n=0 with 0 < a < is the covariance matrix that accounts for the correlations at the receiver side, P = diag p 0,,p ) is the matrix of the powers given to the different sources and W is a N + ) matrix with Gaussian standard iid elements. Let P u denote the vector containing the powers of the interfering sources. We set P u up to a permutation of its elements) to: [4P 5P] if = 2 P u = [P P 2P 4P] if = 4 [P P 2P 2P 2P 4P 4P 4P 8P 6P 6P 6P ] if = 2. For = 2 p with 3 p 7, we assume that the powers of the interfering sources are arranged into 5 classes as in Table IV-A. We set the SNR P/ρ to 0 db and a to 0.. We investigate in this section the accuracy of the Gaussian approximation in terms of the outage probability. In Fig.5, we compare the empirical % outage SINR with the one predicted by the Central Limit Theorem. We note that the Gaussian approximation tends to under estimate the % outage SINR. We also note that it has a good accuracy for small values of α and for enough large values of N N 64). Observe that all these simulations confirm a fact announced in Remark 2 above: compared with functionals of the channel singular values such as Shannon s mutual information, larger signal dimensions are needed to attain the asymptotic regime for quadratic forms such as the SINR see for instance outage probability approximations for mutual information in [22] and in [23]). This observation holds for first order as well as for second order results. V. PROOF OF THEOREM 3 This section is devoted to the proof of Theorem 3. We begin with mathematical preliminaries. A. Preliminaries The following lemma gathers useful matrix results, whose proofs can be found in [28]: Lemma 2: Assume X = [x ij ] N i,j= and Y are complex N N matrices. Then ) For every i,j N, x ij X. In particular, diagx) X. 2) XY X Y. 3) For ρ > 0, the resolvent XX + ρi) satisfies XX + ρi) ρ. 4) If Y is Hermitian nonnegative, then trxy) X try). October 0, 2008

18 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 8 8 N=6 Empirical Theoretical 8 N=32 Empirical Theoretical SINR in db 0 SINR in db α α 0 N=64 9 N=28 Empirical Theoretical 8 Empirical Theoretical SINR in db 4 2 SINR in db α α Fig. 5. Theoretical and empirical % outage SINR Let X = UΛV be a spectral decomposition of X where Λ = diagλ,...,λ n ) is the matrix of singular values of X. For a real p, the Schatten l p -norm of X is defined as X p = λ p i )/p. The following bound over the Schatten l p -norm of a triangular matrix will be of help for a proof, see [25], [29, page 278]): Lemma 3: Let X = [x ij ] N i,j= be a N N complex matrix and let X = [x ij i>j ] N i,j= be the strictly lower triangular matrix extracted from X. Then for every p, there exists a constant C p depending on p only such that X p C p X p. October 0, 2008

19 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 9 The following lemma lists some properties of the resolvent Q and the deterministic approximation matrix T. Its proof is postponed to Appendix A. Lemma 4: The following facts hold true: ) Assume A2. Consider matrices T ρ) = diagt ρ),...,t N ρ)) defined by Theorem ). Then for every n N, ρ + σ 2 max t n ρ) ρ. 23) 2) Assume in addition A and A3. Let Q ρ) = YY + ρi) and let matrices S be as in the statement of Theorem 2). Then B. Proof of Theorem 3 ) sup E tr S Q T ) 2 <. 24) We introduce the following notations. Assume that X is a real matrix, by X 0 we mean X ij 0 for every element X ij. For a vector x, x 0 is defined similarly. In the remainder of the paper, C = Cρ,σmax,lim 2 inf N,sup N ) < denotes a positive constant whose value may change from line to line. The following lemma, which directly follows from [2, Lemma 5.2 and Proposition 5.5], states some important properties of the matrices A defined in the statement of Theorem 3. true: Lemma 5: Assume A2 and A3. Consider matrices A defined by 9). Then the following facts hold ) Matrix I A is invertible, and I A ) 0. 2) Element k,k) of the inverse satisfies [ I A ) ] for every k. k,k 3) The maximum row sum norm of the inverse satisfies lim sup I A ) <. Due to Lemma 5 ), Θ 2 is well defined. Let us prove that lim sup Θ 2 right-hand side of 20) satisfies <. The first term of the gt I A ) g g I A ) g g I A ) g g 2 I A ) 25) October 0, 2008

20 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 20 due to. Recall that T ρ by Lemma 4 ). Therefore, any element of g satisfies trd 0D k T 2 N D 0 D k T 2 N σmax 4 ρ 2 26) by A2, hence sup g C. From Lemma 5 3) and 25), we then obtain lim sup gt I A ) g C. 27) We can prove similarly that the second term in the right-hand side of 20) satisfies sup E W 0 4 ) trd2 0 T ρ)2 ) C. Hence lim sup Θ 2 <. Let us prove that lim inf Θ 2 > 0. We have gt I A ) g a) b) c) d) gt diag I A ) ) g + N + N + N C σ 2 max ρ σ 2 max ρ σ 2 max ρ 2tr D 0 ) 2 k= ) 2 trd 0D k T 2 ) ) 2 ) 2 2tr D 0 D k T 2 k= ) 2 ) 2 ρ + σ 2 max ) 4 2tr D 0 D k ) 2 D k, where a) follows from the fact that I A ) 0 Lemma 5 ), and the straightforward inequalities 0 and g 0), b) follows from Lemma 5 2) and + N k= k= σ 2 max ρ )2, c) follows from the elementary inequality n x 2 i n x i ) 2, and d) is due to Lemma 4 ). Similar derivations yield: E W 0 4 ) trd2 0 T E W 0 4 ) 2 ρ + σmax) 2 2 trd 0 CE W 0 4 ) by A3. Therefore, if A4 holds true, then lim inf Θ 2 > 0 and Theorem 3 ) is proved. C. Proof of Theorem 3 2) Recall that the SINR β is given by Equation 4). The random variable be decomposed as Θ β β ) = Θ β β ) can therefore ) w0 D/2 0 QD /2 0 w 0 trd 0 Q) + trd 0 Q T))) Θ Θ = U, + U 2,. 28) October 0, 2008

21 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 2 Thanks to Lemma 4 2) and to the fact that lim inf Θ 2 > 0, we have EU2,2 < C which implies that U,2 0 in probability as. Hence, in order to conclude that β β Θ ) N0,) in distribution, it is sufficient by Slutsky s theorem to prove that U, N0,) in distribution. The remainder of the section is devoted to this point. Remark 6: Decomposition 28) and the convergence to zero in probability) of U 2, yield the following interpretation: The fluctuations of β β ) are mainly due to the fluctuations of vector w 0. Indeed the contribution of the fluctuations of trd 0Q, due to the random nature of Y, is negligible. Denote by E n the conditional expectation E n [ ] = E[ W n,0,w n+,0,...,w N,0,Y]. Put E N+ [ ] = E[ Y] and note that E N+ w0 D/2 0 QD /2 0 w 0 ) = trd 0 Q. With these notations at hand, we have: U, = E n E n+ ) w 0 D/2 0 QD /2 0 w 0 = Z n,. 29) Θ Θ n= Consider the increasing sequence of σ fields n= F N, = σw N,0,Y),, F, = σw,0,,w N,0,Y). Then the random variable Z n, is integrable and measurable with respect to F n, ; moreover it readily satisfies E n+ Z n, = 0. In particular, the sequence Z N,,...,Z, ) is a martingale difference sequence with respect to F N,,, F, ). The following CLT for martingales is the key tool to study the asymptotic behavior of U, : Theorem 4: Let X N,,X N,,...,X, be a martingale difference sequence with respect to the increasing filtration G N,,..., G,. Assume that there exists a sequence of real positive numbers s 2 such that s 2 n= E [ X 2 n, G n+, ] in probability. Assume further that the Lyapunov condition holds: α > 0, s 2+α) n= E X n, 2+α 0, In fact, one may prove that the fluctuation of trd0q T) are of order, i.e. trd0q T) asymptotically behaves as a Gaussian random variable. Such a speed of fluctuations already appears in [2], when studying the fluctuations of the mutual information. October 0, 2008

22 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 22 Then s N n= X n, converges in distribution to N0,) as. Remark 7: This theorem is proved in [24], gathering Theorem 35.2 which is expressed under the weaker Lindeberg condition) together with the arguments of Section 27 where it is proved that Lyapunov s condition implies Lindeberg s condition). In order to prove that U, = Θ n= Z n, N0,) in distribution, 30) we shall apply Theorem 4 to the sum Θ N n= Z n, and the filtration F n, ). The proof is carried out into four steps: Step : We first establish Lyapunov s condition. Due to the fact that lim inf Θ 2 need to show that α > 0, n= Step 2: We prove that V = N n= E n+zn, 2 satisfies E W0 4 2 ) V Step 3: We first show that tr D 2 0diagQ)) 2) + trd 0QD 0 Q) > 0, we only E Z n, 2+α 0. 3) ) 0 in probability. 32) trd2 0diagQ)) 2 trd2 0T 2 0 in probability. 33) In order to study the asymptotic behavior of trd 0QD 0 Q), we introduce the random variables U l = trd 0QD l Q) for 0 l the one of interest being U 0 ). We then prove that the U l s satisfy the following system of equations: where U l = c lk = c lk U k + trd 0D l T 2 + ǫ l, 0 l, 34) k= trd ld k T ρ) 2 + trd kt ρ) ) 2, 0 l, k 35) and the perturbations ǫ l satisfy E ǫ l C 2 where we recall that C is independent of l. October 0, 2008

23 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 23 Step 4: We prove that U 0 = trd 0QD 0 Q satisfies U 0 = trd2 0T 2 + gt I A) g + ǫ 36) with E ǫ C 2. This equation combined with 32) and 33) yields n E n+z 2 n, Θ2 0 in probability. As lim inf Θ 2 > 0, this implies Θ n E n+zn, 2 in probability, which proves 30) and thus ends the proof of Theorem 3. have Write B = [b ij ] N i,j= = D/2 0 QD /2 0 and recall from 29) that Z n, = E n E n+ )w0 Bw 0. We Hence n E n w0 Bw 0 = b ll + l= Z n, = Wn0 2 ) b nn + Wn0 l,l 2=n l=n+ W l 0 W l 20b ll 2. W l0 b nl + W n0 N l=n+ W l0 b ln ). 37) Step : Validation of the Lyapunov condition: The following inequality will be of help to check Lyapunov s condition. Lemma 6 Burkholder s inequality): Let X k be a complex martingale difference sequence with respect to the increasing sequence of σ fields F k. Then for p 2, there exists a constant C p for which p E X k C p E E [ X k 2 ] ) p/2 F k + E X k p. k k k Recall Assumption A. Eq. 37) yields: ) Z n, 4 W n ρσmax W n0 W l0 b nl l=n+ 23 Wn0 2 ) W n0 W l0 b nl 38) ρσ 2 max l=n+ where we use the fact that b nn ρσ 2 max ) cf. Lemma 2 )) and the convexity of x x 4. Due to Assumption A, we have: E W n0 2 + ) E W n0 8 + ) <. 39) October 0, 2008

24 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 24 Considering the second term at the right-hand side of 38), we write 4 E W n0 W l0 b nl = E W n0 4 4 E W l0 b nl, l=n+ l=n+ a) N ) 2 C E E W l0 2 ) b nl 2 + b) C E l=n+ N l=n+ b nl 2 ) 2 + l=n+ l=n+ E b nl 2, E W l0 4 )E b nl 4 ), where a) follows from Lemma 6 Burkholder s inequality), the filtration being F N,,..., F n+, and b) follows from the bound b nl 4 b nl 2 max b nl 2 b nl 2 σ 2 max ρ ) 2 cf. Lemma 2 )). Now, notice that l=n+ b nl 2 < b nl 2 = l= [ ] D /2 0 QD 0 QD /2 0 nn D/2 0 QD 0 QD /2 0 σ4 max ρ 2. This yields E W n0 N l=n+ W l 0b nl 4 C. Gathering this result with 39), getting back to 38), taking the expectation and summing up finally yields: E Z n, 4 n= which establishes Lyapunov s condition 3) with α = 2. Step 2: Proof of 32): Eq. 37) yields: E n+ Zn, 2 = E W0 4 ) b 2 nn + E n+ Wn0 +2b nn E W 0 W 0 2) N l=n+ Note that the second term of the right-hand side writes: C 0 l=n+ W l0 b nl + W n0 W l0 b nl + 2b nn E W0 W 0 2) N l=n+ N l=n+ W l0 b ln W l0 b ln ) ) 2. E n+ W n0 l=n+ W l0 b nl + W n0 Therefore, V = N n= E n+z 2 n, writes: V = E W0 4 ) b 2 nn + 2 n= N l=n+ n= l,l 2=n+ W l0 b ln) 2 = 2 W l0w l 20 b nl b l2n l,l 2=n+ + 2 R E W 0 W 0 2) N W l0w l 20 b nl b l2n. N b nn n= l=n+ W l0 b nl ), October 0, 2008

25 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 25 where R denotes the real part of a complex number. We introduce the following notations: R = r ij ) N i,j= = b ij i>j ) N i,j= and Γ = N b nn n= l=n+ W l0 b nl. Note in particular that R is the strictly lower triangular matrix extracted from D /2 0 QD /2 0. We can now rewrite V as: V = E W0 4 ) tr D 2 0diagQ)) 2) + 2 w 0RR w 0 + 2R Γ EW 0 W 0 2). 40) We now prove that the third term of the right-hand side vanishes, and find an asymptotic equivalent for the second one. Using Lemma 2, we have: E N+ Γ 2 = 2 n,m= N b nn b mm b nl b ml l>n l>m = 2tr diagb)r RdiagB)) l= = ) 2tr D /2 0 diagq)d /2 0 R RD /2 0 diagq)d /2 0 2 D 0 2 Q 2 trr R) D 0 2 Q 4 In particular, E Γ 2 0 and σmax 4 ρ 4 2 D 0 2 Q 2 trb 2 ) R EW 0 W 0 2) Γ ) Consider now the second term of the right-hand side of Eq. 40). We prove that: 0. 2 D 0 4 Q 2 trq 2 ) 0 in probability. 4) w 0RR w 0 trrr ) 0 in probability. 42) By Lemma Ineq. 2)), we have E w 0RR w 0 ) 2 trrr ) C 2E W 0 4 )trrr RR ). Notice that trrr RR ) = R 4 4 where R 4 is the Schatten l 4 -norm of R. Using Lemma 3, we have: Therefore, R 4 4 C D /2 0 QD / NC D /2 0 QD /2 0 4 N Cσ8 max ρ 4. E w 0RR w 0 ) 2 trrr ) C N 2 0 October 0, 2008

26 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 26 which implies 42). Now, due to the fact that B = B, we have 2 trrr = 2 = n= l=n+ n,l= Gathering 40 43), we obtain 32). Step 2 is proved. b nl 2 b nl 2 b nn 2 n= = trd 0QD 0 Q trd2 0 diagq))2 43) Step 3: Proof of 33) and 34): We begin with some identities. Write Qz) = [q ij z)] N i,j= and Qz) = [ q ij z)] i,j=. Denote by y k the column number k of Y and by ξ n the row number n of Y. Denote by Y k the matrix that remains after deleting column k from Y and by Y n the matrix that remains after deleting row n from Y. Finally, write Q k z) = Y k Y k zi) and Q n z) = Y n Y n zi). The following formulas can be established easily see for instance [28, and 0.7.4]): q nn ρ) = ρ + ξ n Qn ρ)ξ n ), q kk ρ) = ρ + y k Q k ρ)y k ), 44) Q = Q k Q ky k y k Q k + y k Q ky k 45) Lemma 7: The following hold true: ) Rank one perturbation inequality) The resolvent Q k ρ) satisfies traq Q k ) A /ρ for any N N matrix A. 2) Let Assumptions A A3 hold. Then, max Eq nn ρ) t n ρ)) 2 C n N. 46) The same conclusion holds true if q nn and t n are replaced with q kk and t k respectively. We are now in position to prove 33). First, notice that: E q 2 nn t 2 n = E q nn t n q nn + t n ) Eq nn t n ) 2 Eq nn + t n ) 2 2 ρ Eqnn t n ) 2. 47) October 0, 2008

27 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 27 Now, E trd 2 0 diagq)2 T 2 ) n= 2σ4 maxn ρ σ0,n 4 E qnn 2 t2 n σ4 maxn max n N Eq nn t n ) 2 max E qnn 2 n N t2 n where the last inequality follows from 47) together with Lemma 7 2). Convergence 33) is established. We now establish the system of equations 34). Our starting point is the identity Q = T + TT Q )Q = T + ρ T diagtr D T,...,tr DN T)Q TYY Q. Using this identity, we develop U l = trd 0QD l Q as U l = trd 0QD l T + ρ 2trD 0QD l Tdiagtr D T,...,tr DN T)Q trd 0QD l TYY Q = X + X 2 X 3. 48) Lemma 4 2) with S = D 0 D l T yields: X = trd 0D l T 2 + ǫ 49) where E ǫ Eǫ 2 C/. Consider now the term X 3 = k= trd 0QD l Ty k yk Q. Using 44) and 45), we have Hence X 3 = ρ = ρ yk Q = y k Qy ) k + yk Qy yk Q k = ρ q kk yk Q k. k q kk yk Q kd 0 QD l Ty k k= t k yk Q kd 0 QD l Ty k + ρ k= By Cauchy-Schwartz inequality, 0, q kk t k )yk Q kd 0 QD l Ty k k= = X 3 + ǫ 2. 50) E ǫ 2 ρ k= E q kk t k ) 2 Ey k Q kd 0 QD l Ty k ) 2. We have Ey k Q kd 0 QD l Ty k ) 2 σ 8 maxρ 6 E y k 4 C. Using in addition Lemma 7 2), we obtain E ǫ 2 C. October 0, 2008

28 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 28 Consider X 3. From 44) and 45), we have Q = Q k ρ q kk Q k y k y k Q k. Hence, we can develop X 3 as X 3 = ρ k= = X 4 + X 5. t k y k Q kd 0 Q k D l Ty k ρ2 t k q kk yk Q kd 0 Q k y k yk Q kd l Ty k k= 5) Consider X 4. Notice that y k and Q k are independent. Therefore, by Lemma, we obtain y k Q kd 0 Q k D l Ty k = trd kq k D 0 Q k D l T + ǫ 3 = trd kqd 0 QD l T + ǫ 3 + ǫ 4 where Eǫ 2 3 < C by Ineq. 3). Applying twice Lemma 7 ) to ǫ 4 = trd kq k D 0 Q k D l T trd k QD 0 QD l T) yields ǫ 4 < C. Note in addition that t k D k = diagtr D T,...,tr DN T). Thus, we obtain X 4 = ρ 2tr t k D k )QD 0 QD l T + ǫ 5 k= where ǫ 5 = ǫ 3 + ǫ 4, which yields E ǫ 5 C 2. = X 2 + ǫ 5, 52) We now turn to X 5. First introduce the following random variable: ) ) ǫ 6 = t k q kk yk Q kd 0 Q k y k yk Q kd l Ty k t k q kk trd kq k D 0 Q k trd kq k D l T Then ǫ 6 ρ 2y k Q kd 0 Q k y k yk Q kd l Ty k trd kq k D l T + ρ 2 y k Q kd 0 Q k y k trd kq k D 0 Q k trd kq k D l T and one can prove that E ǫ 6 < C 2 with help of Lemma, together with Cauchy-Schwarz inequality. In addition, we can prove with the help of Lemma 7 that: ) ) ) ) t k q kk trd kq k D 0 Q k trd kq k D l T = t 2 k trd kqd 0 Q trd kqd l T + ǫ 7 ) ) = t 2 k trd kqd 0 Q trd kd l T 2 + ǫ 7 + ǫ 8 where ǫ 7 and ǫ 8 are random variables satisfying E ǫ 7 < C 2 by Lemma 7, and max k,l E ǫ 8 max k,l E ǫ8 2 C 2 by Lemma 4 2). Using the fact that ρ 2 t 2 k = + trd kt) 2, we end up with X 5 = ρ2 k= t 2 k ) ) trd kqd 0 Q trd kd l T 2 + ǫ 9 = c lk U k + ǫ 9 53) k= October 0, 2008

29 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 29 where c lk is given by 35), and where E ǫ 9 < C 2. Plugging Eq. 49) 53) into 48), we end up with U l = k= c lku k + trd 0D l T 2 + ǫ with E ǫ < C 2. Step 3 is established. Step 4 : Proof of 36): We rely on results of Section V-B, in particular on Lemma 5. Define the following + ) vectors: [ ] u = [U k ] k=0, d = trd 0D k T 2, ǫ = [ǫ k ] k=0, k=0 where the U k s and ǫ k s are defined in 34). Recall the definition of the c lk s for 0 l and k, define c l 0 = 0 for 0 l and consider the + ) + ) matrix C = [c lk ] l,k=0. With these notations, System 34) writes I + C)u = d + ǫ. 54) Let α = trd2 0 T2 and β = + trd 0T) 2. We have in particular d = α, C = 0 gt g 0 A T recall that A, and g are defined in the statement of Theorem 3). Consider a square matrix X which first column is equal to [,0,...,0] T, and partition X as X =. Recall that the inverse of X exists if and only if X xt 0 0 X [X ] 0 of X is given by [ X ] 0 = [ x T 0X ] exists, and in this case the first row see for instance [28]). We now apply these results to the system 54). Due to 54), U 0 can be expressed as U 0 = [I C) ] 0 d + ǫ). By Lemma 5 ), I A T ) exists hence I C) exists, [I + C) ] [ ] = 0 gt I A T ) and U 0 = α + gt I A T) g + ǫ0 + gt I A T) ǫ with ǫ = [ǫ,...,ǫ ] T. Gathering the estimates of Section V-B together with the fact that Eǫ C 2, we get 36). Step 4 is established, so is Theorem 3., October 0, 2008

30 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 30 A. Proof of Lemma 4 APPENDIX Let us establish 23). The lower bound immediately follows from the representation t n = ρ + k= σ 2 nk + P N l= σ2 lk tl a) ρ + σ 2 max where a) follows from A2 and t l ρ) 0. The upper bound requires an extra argument: As proved in [26, Theorem 2.4], the t n s are Stieltjes transforms of probability measures supported by R +, i.e. there exists a probability measure µ n over R + such that t n z) = µ ndt) t z. Thus and 23) is proved. t n ρ) = 0 µ n dt) t + ρ ρ, We now briefly justify 24). We have E trsq T) 2 = E trsq EQ) 2 + trseq T) 2. In [2, Lemma 6.3] it is stated that sup E trsq EQ) 2 <. Furthermore, in the proof of [2, Theorem 3.3] it is shown that sup EQ T <, hence trseq T) SEQ T) EQ T S < by Lemma 2 2). The result follows. B. Proof of Corollary Recall that in the separable case, D k = d k D and D n = d n D. Let d be the vector d = [ dk ] k=. In the separable case, Eq. 20) is written Θ 2 d 2 0 = d g T I A) g + γe W ), 55) 0 where γ is defined in statement of the corollary. Here, vector g and matrix A are given by [ g = γ d 0 d and A = trd ] ld m T 2 + trd lt ) 2 = γ d dt. By the matrix inversion lemma [28], we have Noticing that d g T I A) g = 2 0 d T d = γ2 l,m= d T γ d d T) d = γ2 d T k= + γ d 2 k + trd kt ) 2 = ρ2 γ d T d d dt d 2 k t 2 k = ρ2 γ, k= ) d. October 0, 2008

31 A CENTRAL LIMIT THEOREM FOR THE SINR AT THE LMMSE ESTIMATOR OUTPUT 3 we obtain d g T I A) g = γ ρ2 γ γ 2 ρ 0 2 γ γ. Plugging this equation into 55), we obtain 22). C. Proof of Lemma 7 The proof of Part can be found in [2, Proof of Lemma 6.3] see also [4, Lemma 2.6]). Let us prove Part 2. We have from Equations ) and 44) q nn ρ) t n ρ) = ρ + tr D n T) + ξn Qn ξ n) ξ Q n n ξ n tr D n T ρ ξ Q n n ξ n tr D n T. Hence, Eq nn t n ) 2 2 ξ ρ E n Qn ξ n ) 2 tr D n Q + 2 ρ 2 E tr D n Q T) ) 2 C by Lemma and Lemma 4 2), which proves 46). REFERENCES [] D.N.C Tse and S. Hanly, Linear multi-user receiver: Effective interference, effective bandwidth and user capacity, IEEE Trans. on Information Theory, vol. 45, no. 2, pp , Mar [2] S. Verdú and Sh. Shamai, Spectral efficiency of CDMA with random spreading, IEEE Trans. on Information Theory, vol. 45, no. 2, pp , Mar [3] J. Evans and D.N.C Tse, Large system performance of linear multiuser receivers in multipath fading channels, IEEE Trans. on Information Theory, vol. 46, no. 6, pp , Sept [4] E. Biglieri, G. Caire, and G. Taricco, CDMA system design through asymptotic analysis, IEEE Trans. on Communications, vol. 48, no., pp , Nov [5] W. Phoel and M.L. Honig, Performance of coded DS-CDMA with pilot-assisted channel estimation and linear interference suppression, IEEE Trans. on Communications, vol. 50, no. 5, pp , May [6] J.-M. Chaufray, W. Hachem, and Ph. Loubaton, Asymptotic Analysis of Optimum and Sub-Optimum CDMA Downlink MMSE Receivers, IEEE Trans. on Information Theory, vol. 50, no., pp , Nov [7] L. Li, A.M. Tulino, and S. Verdú, Design of Reduced-Rank MMSE Multiuser Detectors Using Random Matrix Methods, IEEE Trans. on Information Theory, vol. 50, no. 6, pp , June [8] A.M. Tulino, L. Li, and S. Verdú, Spectral Efficiency of Multicarrier CDMA, IEEE Trans. on Information Theory, vol. 5, no. 2, pp , Feb [9] M.J.M. Peacock, I.B. Collings, and M.L. Honig, Asymptotic spectral efficiency of multiuser multisignature CDMA in frequency-selective channels, IEEE Trans. on Information Theory, vol. 52, no. 3, pp. 3 29, Mar [0] J.H. otecha and A.M. Sayeed, Transmit signal design for optimal estimation of correlated MIMO channels, IEEE Trans. on Signal Processing, vol. 52, no. 2, pp , Feb October 0, 2008