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A Theoretical Framework for LMS MIMO Communication Systems Performance Analysis Giuseppa Alfano, Antonio De Maio, Senior Member, IEEE, and Antonia Maria Tulino, Senior Member, IEEE
Abstract—A statistical model for Land Mobile Satellite (LMS) channels, where transmitters and receivers are equipped with multiple antennas, is introduced. Several spectral statistics are given, which allow the theoretical performance analysis of the newly proposed channel model from both a communication and an information-theoretic point of view. Specifically, joint and marginal statistics of the squared singular-values of the channel matrix are evaluated, paving the way for the performance analysis under ergodic and nonergodic assumptions on the channel behavior. The capacity-achieving input covariance matrix, and the corresponding ergodic capacity, assuming perfect receive-side information but making different assumptions on the amount of channel knowledge at the transmitter, are derived. We obtain exact results, but for the case when perfect channel knowledge is assumed at both ends of the link, for which we provide an upper bound to the ergodic capacity. In the nonergodic scenario, we compute the outage probability in absence of power-control, and discuss the asymptotic Gaussianity of the mutual information, which strongly depends on the overall number of degrees of freedom available on the channel. Design guidelines for multiantenna LMS channels are gained studying the low signal-to-noise ratio (SNR) behavior of the capacity, still under the assumption of absence of knowledge of the channel matrix (or its statistics) at the transmitter. The results are illustrated through several examples, aimed at assessing the impact on the performance of the diversity order and/or the line-of-sight (LOS) fluctuations. Index Terms—Asymptotic analysis, eigenanalysis, LMS, multiple-input multiple-output (MIMO), noncentral Wishart.
I. INTRODUCTION UE to the growing interest in satellite personal communications and the great advantages of satellite in delivering multicast and broadcast traffic with respect to terrestrial mo-
D
Manuscript received October 28, 2008; revised December 14, 2009. Date of current version October 20, 2010. This work was supported (in part) by VIDES Project, Misura 3.17 of Regione Campania, Italy. The material in this paper was presented (in part) at the IEEE International Waveform Diversity and Design Conference (WDD 2007), Pisa, Italy, June 2007. G. Alfano is with DELEN, Politecnico di Torino, 10129, Torino, Italy (e-mail:
[email protected]). A. De Maio is with the Dipartimento di Ingegneria Biomedica Elettronica e delle Telecomunicazioni, Universitá degli Studi di Napoli “Federico II”, I-80125 Napoli, Italy (e-mail:
[email protected]). A. M. Tulino is with the Dipartimento di Ingegneria Biomedica Elettronica e delle Telecomunicazioni, Universitá degli Studi di Napoli “Federico II”, I-80125 Napoli, Italy, and the Department of Wireless Communications, Bell Laboratories, Alcatel-Lucent, Holmdel, NJ (e-mail:
[email protected];
[email protected]). Corresponding Author: G. Alfano Communicated by H. Bölcskei, Associate Editor for Detection and Estimation. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2010.2070230
bile communication systems, several models for the statistics of the LMS channel have been proposed during the last years. Channel measurements have confirmed the presence in the received signal of a fluctuating line-of-sight (LOS) component whose first order statistical characterization has been studied in [1]. Due to the LOS randomness, such a fading model has been referred to as Shadowed-Rice (SR). In [2], a thorough analysis has been conducted on real data, showing that LOS random fluctuations are conveniently modeled by the Gamma distribution. Relying on these results, the probability density function (pdf) and the moments of the received signal amplitude, averaged over the LOS fluctuations, have been also provided. A further step toward the study of LMS channels has been done in [3], where the sum of independent as well as correlated SR random variables has been characterized, allowing diversity analysis in LMS channels. In [4], the capacity improvement of a multiple antenna LMS link with single satellite over a single-input single-output (SISO) channel is observed through measurements, confirming the convenience of resorting to spatial diversity at both the ends of the link. Moreover, random fluctuations in the (matrix) LOS component have been observed as well. Some improvements to LMS communication systems can be obtained through satellite multiple-input multiple-output (MIMO) [5], [6] (where channels can be separated by space or polarization) which offers improvements to QoS (for instance transmission rate) due to diversity gain using spatial/polarization time coding techniques, and which increases the spectral efficiency exploiting spatial/polarization multiplexing or their suitable combination. The quoted promising performance improvement has thus offered the rationale for the present work, which is aimed at providing a comprehensive statistical characterization of these channels together with their theoretical analysis. Specifically, the contribution of this paper can be summarized as follows. First of all, a statistical MIMO LMS channel model is developed. Hence, the pdf of the Gram channel matrix is provided under the assumption of a matrix-Gamma distribution [7] for the Gram matrix1 of the LOS2. The model accounts for the joint presence of a scattered and a LOS component, assumed statistically independent because due to distinct physical phenomena. However, both the random processes might exhibit a correlation at the more constrained end, namely where there is the smallest number of sensors. This would adequately model a downlink scenario where the receiving mobile terminal is a handheld device, where, due to space limitations, the number of receiving antennas is less than 1Recall
here that the Gram matrix of a matrix
G = C C.
C is defined as the matrix
2This assumption is consistent with those made for the scalar channel modeling and for the case where diversity is exploited at only one end of the link.
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ALFANO et al.: THEORETICAL FRAMEWORK FOR LMS MIMO COMMUNICATION SYSTEMS
the number of transmit antennas. Reasonably, the closeness between the receiving sensors may cause a nonnegligible correlation. It is worth pointing out that the model has been validated in [2] with reference to the SISO case. However, the extension to the MIMO scenario, has not yet been validated on real measurements. This is due to the lack of real data necessary to perform a statistical analysis aimed at ascertain the fitting capabilities of the model for the MIMO scenario. Relying on recent results on spectral statistics of finite dimensional random matrices [8], the density function of the Cholesky factor of the LOS Gram matrix is given, the marginal density distribution of an unordered eigenvalue and the cumulative distribution function (CDF) of the extremal (i.e., maximum and minimum) eigenvalues of the Gram channel matrix are evaluated, following quite standard techniques. Both ergodic and nonergodic channels are considered. In the former case, namely under ergodicity assumption, the capacity-achieving input covariance matrix and the corresponding ergodic capacity, assuming perfect receive-side information but several degrees of knowledge of channel at the transmitter, are derived. Only when the transmitter is provided with perfect information about the channel state, we give an upper bound on the capacity, while in all other cases we obtain exact results. Assuming that the transmitter has no knowledge of the channel matrix (or of regime, we uncover a compact its statistics), for the lowexpansion of capacity, that projects insight on how the correlation and/or the random LOS component impact the tradeoffs between power, bandwidth and rate. Turning the attention to the nonergodic regime, we first evaluate the outage capacity in the case when the transmitter knows the channel matrix but has no power-control capability, exploiting the newly obtained expressions of the maximum eigenvalue distribution. Afterward, we evaluate the second moment of the mutual information, and discuss how its asymptotic Gaussianity is influenced by the degrees of freedom available on the channel. Finally, the paper is complemented with some numerical results. The paper is organized as follows. Section II constitutes the mathematical background, dealing with the problem of evaluating the distribution of a Shadowed-Rice matrix variate, and of its eigenvalues’ statistics. Section III introduces the system model, while the information-theoretic analysis, both in the ergodic as well as in the nonergodic case, is carried out in Section IV. Section V is devoted to some numerical examples and their discussion. Conclusions are given in Section VI, while longest proofs are relegated to the Appendix. Throughout the paper, matrices are denoted by uppercase denotes boldface letters, vectors by lowercase boldface; indicates the conjugate transpose statistical expectation, operator, and , respectively, the determinant and the trace of a square matrix, stands for the euclidean norm, and finally denotes the Vandermonde determinant [9], i.e.,
(1) where
is an .
Hermitian matrix with eigenvalues
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II. MATHEMATICAL BACKGROUND Definition 1: The random matrix is a noncentral ) complex Wishart matrix with degrees of freedom ( and noncentrality matrix , , if the joint distribution of its entries can be written as [10]
(2) is the Bessel hypergeometric function of matrix where argument [10] and , , is the complex multivariate Gamma function [10] Definition 2: The random matrix is a complex ) and matrix Gamma matrix with scalar parameter ( parameter , , if the joint distribution of its entries can be written as [7] (3) Notice that, for integer , it reduces to a central Wishart matrix . [10], with degrees of freedom and parameter matrix random complex lower triangular matrix Let the , with with real positive diagonals, such that distributed according to (3). Then, from (3), it follows that for , the joint distribution of the entries of can be written as (4) This is essentially the Cholesky decomposition which states that the elements of are independent and its squared diagonal elements, , are distributed according to a Gamma distribution with and scale pawith shape parameter with are rameter 1 while its off diagonal elements, standard complex Gaussian. The joint distribution of , in (4), is , obtained recalling that the Cholesky factorizations lower triangular matrix, has the following with an Jacobian [11]: (5) random matrix, Note, finally, that in order to generate a , distributed as in (3), it is enough consider the following with distributed as above. product matrix such that Assume, now, to have a (6) are zero-mean independent comwhere the columns of plex circular Gaussian vectors with common covariance matrix , while is a random matrix, statistically independent on , whose Gram matrix, , is such that . Under these assumptions, the statistical distribution is given by the following. of Proposition 1: Let be given as in (6), then the statistical , can be written as distribution of its Gram matrix,
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shown in (7), at the bottom of the page, where is an hypergeometric function of single matrix argument [10]. . Notice that, if Proof: Observe that, given , , where . Hence, the uncondican be obtained averaging (2) over , tional distribution of i.e.,
with
,
and
where the integration is taken over the space of Hermitian positive definite matrices of size and is performed with the help of [12, eq. 115]. In order to get handy expressions, from now on we set , where is the identity matrix. Property 1: A random matrix, distributed as in (7), is unitarily . invariant whenever Proof: The proof is an immediate generalization of the proof given in [7, Theor. 3.2] for real matrices. Property 2: Let be a random matrix distributed as in (7) and a diagonal matrix. Then, the joint with distribution of its entries equals that of for any values , where is the diagonal matrix whose th of if and 1 otherwise. diagonal element equals Proof: From [13], it follows that the Jacobian of the transis . From this and from (7) formation specialized for a diagonal , it follows immediately the new random matrix admits the following pdf: (8)
the th ordered eigenvalue of . Recall here that is the confluent hypergeometric function of scalar argument [14, Ch. 13]. Proof: See Appendix A. Based on this finding, we can now statistically characterize an unordered eigenvalue for the matrix distributed as in (7). Proposition 2: The marginal density distribution of an un(7), when , is given by (10), as ordered eigenvalue of is the -coshown at the bottom of the page, where matrix whose th entry equals factor of the
denoting the Gauss hypergeometric function of scalar argument [14, Ch. 15]. Proof: See Appendix B. Of further interest are, in general, the extremal eigenvalues, smallest and largest, of , for which we derive herein the CDF. In this concern, we prove the following Proposition 3: Let eigenvalue of (7), when
, then the CDF of the maximum , can be written as
Corollary 1: The joint distribution of the ordered eigenvalues of (7), when , can be expressed as
(11)
(9)
matrix are given by the where the entries of the , with equation at the bottom of the page, when the confluent hypergeometric function of two variables
(7)
(10)
ALFANO et al.: THEORETICAL FRAMEWORK FOR LMS MIMO COMMUNICATION SYSTEMS
[15, 9.261.1]. When
where matrix
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is the cofactor of the whose th entry can be written as
and
. Proof : See Appendix E.
(12)
, distributed as in Proposition 6: Given a random matrix (7), when , the moments of its generalized variance ( ) are given by
with
the Pochhammer symbol [14]. Finally, for , coincides with the CDF of the maximum eigenvalue . of a complex central Wishart matrix Proof : See Appendix C. Proposition 4: Let (7), when eigenvalue of
, then the CDF of the minimum , can be written as
(17) Proof: The result follows from [12, Formula 115]. Corollary 2: Given a random matrix , distributed as in (7), when , the first order logarithmic moment of its generalized variance can be written as
(13) (18) where the entries of the matrix are given by the , and first equation shown at the bottom of the page, when by when , with the Pochhammer symbol [14]. Finally, , is the CDF of the smallest eigenvalue of for [11]. a complex central Wishart matrix Proof : See Appendix D. The characterization of the trace and of the determinant of a SR matrix variate in terms of their moments is finally given by the following propositions, which conclude the Section. Proposition 5: A random matrix , satisfies whenever
with the digamma function [14], the Euler constant, and where the matrix has generic th entry , but for the th column with the entry shown in the second equation at the bottom of the page. Proof : See Appendix F. III. SYSTEM MODEL
, distributed as in (7),
Denoting by and the number of receive and transmit antennas, a generic LMS channel can be described by the following linear vector memoryless channel:
(14)
(19) is the –dimensional input vector such that , is the –dimensional output vector, and is the additive circularly symmetric Gaussian noise such that . All these quantities are complex valued. Finally, in (19), is the ( ) complex channel matrix, whose entries represent the fading coefficients between each transmit and each receive antenna normalized such that where
(15) and
(16) (20)
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Normalized by its energy per dimension, the input covariance is denoted by:
where the normalization ensures that transmitted SNR is
, while the
(21) Since the fading encountered by wireless systems on terrestrial links tends to be either Rayleigh or Ricean, the entries of can be modeled as jointly Gaussian. Hence, the statistical characterization of entails simply determining the mean and the correlation3 between its entries. However, as observed for the single antenna case, the fading which affects the LMS channel can be more adequately described by a SR model [2], namely a Rice fading with random LOS component. It is thus of interest to exploit the matrix variate scenario, based on the results in the previous Section. Specifically, denoting by (22) and defining the
matrix if if
(23)
in the following, we will focus on linear vector memoryless is such that channel as in (19) where the channel matrix , as defined in (23), is distributed according to (7). Notice is tantamount to assuming that the multhat assuming tipath component exhibits a shorter decorrelation distance than the LOS component. In the rest of the paper, we will refer to in (23) as to the Gram channel matrix. IV. PERFECT RECEIVER SIDE INFORMATION This section is devoted to the performance analysis of communication systems on a MIMO SR fading channel with perfect channel state information (CSI) at the receiver and under different types of CSI at transmitter. Several key parameters are 3In
this concern, we adhere to the separable or Kronecker correlation model, (see, e.g., [16] for a comprehensive presentation of widely used correlation models in multiantenna scenario), and assume the link to be correlated only at the more constrained end. It would adequately model a downlink scenario where the receiving mobile terminal is a handheld device. Indeed, due to space limitations, the number of receiving antennas is less than the number of transmit antennas. Moreover, the closeness between the receiving sensors may cause a nonnegligible correlation.
evaluated, either in closed form or by means of a low power expansion, since the communication on LMS channels mainly regime. More specifically, we list the oboccurs in low tained results based on the different assumptions made on CSI availability, either at the transmit and/or the receive side. In the remainder of the section, we first provide bounds to the ergodic capacity when both transmitter and receiver are aware of the channel state, under the guidelines of [17]. Then, the case when only the receiver has access to the CSI is studied, both in the ergodic as well as in the nonergodic framework. The structure of the capacity-acheiving covariance matrix is individuated, relying on the same technique as in [18, Theor. I]. Compact indexes for the performance evaluation in the low-power regime are finally given, thus ending the Section. Before proceeding further, and in order to make the results easily comparable with previously available expressions for the case of nonfluctuating LOS, we provide a definition of the -factor for the SR matrix-variate case. Specifically, the -factor, quantifying the ratio between the LOS and the diffuse energies, can be defined as (24) In the following, we provide performance indexes expressions, exploiting, when needed, their dependence on the generalized -factor defined above. A. Ergodic Capacity Conditioned on the channel matrix and on the input covariance , the mutual information (in bits/s/Hz) is
In this section, we focus on ergodic channels where a codeword spans many realizations of the fading coefficients. The quantity of interest is then the ergodic mutual information, , i.e., the mutual information, averaged over the distribution of the random channel matrix . The evaluation of the capacity will, instead, depends on the type of CSI available at the transmitter. Generalizing the steps given in [17], the following results can be derived: Theorem 1: Consider a LMS link as the one described in (19), whose Gram channel matrix is distributed as in (7) with . When CSI is available at both the transmitter as well as the receive side, the ergodic capacity is upper-bounded by , , and (25), as shown at the bottom of the page, where its eigenvalues ’s are as in Proposition 2, is defined in Corollary 2 and is the Euler digamma function [14, Ch. 6].
(25)
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Fig. 1. C (SNR) for several number of transmit and receive antennas and several values of : Solid indicate analytical upper bound and stars indicate simulation. The transmit antennas are correlated as per (35) with = 0:1 while the receive antennas are uncorrelated.
Proof: When CSI is available at both the transmitter as well as the receive side, the ergodic capacity is given by
where eigenvalues of equality [9]
5,
, in (30), denote the nonzero (31) follows from the well-known in-
(26)
(33)
where the maximum is over all whose trace equals and the expectation is with respect to the distribution of . Let denote the capacity-achieving input covariance matrix4 solution of the optimization problem in (26), then
with denoting an arbitrary Hermitian positive semidefinite matrix, and finally in (32) we have used that . The second expectation in (32) is evaluated using (18), while the first term can be upper-bounded by
(28) (34) (29) (30)
(31)
(32) 4As well known, capacity-achieving input covariance, 8 , admits the folwhere V coincides with the eigenlowing eigendecomposition 8 = , vector matrix of H H while the j th diagonal entry of the diagonal matrix say p is obtained through a waterfill process [19]–[22]
VP V
p
=
0
Plugging (18), (34), and (15) in (32), the bound is obtained. Before proceeding with the analysis, we illustrate the tightness of this bound. Example 1: Depicted in Fig. 1 is for several number of transmit and receive antennas and several values of , both from Theorem 1 and from Montercarlo simulations solving the optimal water-filling policy. Because of reciprocity results for , they apply also to . Note the tight correspondence between the closed-form expression given by analytical upper bound as per Theorem 1 and the exact expression of obtained via simulations, for a wide range of , number of antennas, and . The th entry of the matrix is
P
N (H H )
(35) while the receive antennas are uncorrelated.
(27)
fPg = N
where , the waterfilling normalization coefficient, ensures that Tr
.
5Note that (H H ); j = 1; . . . ; m, correspond to the eigenvalues of the positive definite matrix W = H H , if N > N , and to the eigenvalues of the positive definite matrix W = H H , if f N N .
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Theorem 2: Consider a LMS link as the one described in (19) with , whose Gram channel matrix is distributed as in (7). Suppose that the receiver has perfect knowledge of the channel , while the transmitter has only access to the distribution of the channel matrix, but not to such matrix itself. while is a diagonal matrix, then the capacityIf , is a diagonal maachieving input covariance matrix, trix whose th diagonal element, , is the positive solution to [23, Theorem 4]:
Note that
(39) (40) (41) (42)
(36)
where
is th column of
, and
(37)
if it exists (i.e., if ), otherwise, ; is chosen so that . Proof: In this case, the capacity is . Using Property 2, the optimality of the diagonal structure for capacity-achieving input covariance matrix can be immediately proved following the same technique introduced in [18, Theor. 1]. The expressions of the optimal diagonal elements follow, instead, immediately from [23, Theor. 4].
where in (40) is defined as in (23) and where is a random matrix whose distribution is given in (38). Consequently, the mutual information of a LMS link whose Gram channel matrix is distributed according to (7) admits the same statistical characterization of the mutual information of a LMS link whose Gram channel matrix is distributed according to (38). Theorem 4: Consider a LMS link as the one defined in The), orem 3. If the link has unit-power LOS component (i.e., and , then the capacity-achieving input covariance matrix is the identity matrix. Proof: The above result follows from [18, Proposition 1] 6. using the unitary invariance of (7) for the case of Theorem 5: Consider a LMS link as the one defined in Theorem 1. Suppose that the receiver has perfect knowledge of the channel but the transmitter has no knowledge of the channel matrix (or its statistics). and is arbitrary, then If
Let us now generalize Theorem 2 to an arbitrary positive definite matrix . Theorem 3: Consider a LMS link as the one described in (19) with , whose Gram channel matrix is distributed as in (7). Suppose that the receiver has perfect knowledge of the channel , while the transmitter has only access to the distribution of the channel matrix, but not to such matrix itself. and is arbitrary, then the capacity-achieving input If , where is given by the eigencovariance is vector matrix of while is a diagonal matrix whose th diagonal element is given as in (36) and where in (37) is replaced by its rotated version . Proof: Denote by and the eigenvector and eigenvalue matrix, respectively, of i.e, . Consider the transformation . From [13], we have that its , where Gram channel matrix, Jacobian is , is distributed as in (7). Thus, it follows that the rotated , admits the following Gram channel matrix, pdf:
(43) where is the Exponential Integral [14], and , , and ’s are as in Proposition 2. Proof: In this case, the optimum input distribution is an -dimensional Gaussian vector with independent and identically distributed (i.i.d.) components achieving capacity [24], [25] (44) and the where denotes an arbitrary nonzero eigenvalue of expectation is over its distribution. In order to evaluate the average in (44), we explicitly write the integral over the marginal density of an unordered eigenvalue of (7), namely (45)
(38) The result, finally, follows immediately observing that, given (38), for the rotated Gram channel matrix , the capacity-achieving covariance is diagonal.
where
is given by (10).
W is called unitarily invariant if V W V for any unitary matrix
6Recall here that a Hermitian random matrix the joint distribution of its entries equals that of independent of .
V
W
ALFANO et al.: THEORETICAL FRAMEWORK FOR LMS MIMO COMMUNICATION SYSTEMS
The result in (43) follows from the application of [26, Formula 40]7 to (45), when the confluent hypergeometric function in (10) is expanded as an infinite series. Standard properties of the confluent hypergeometric function , (10) can be imply that, when is an integer number with expanded using a finite sum. This implies that the infinite series in (43) reduces to a finite summation. Specifically: Proposition 7: For integer in the previous Theorem
, with
,
, and
’s as
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the maximal-eigenvalue of . In this case, the mutual information, for a zero-mean Gaussian input vector, is given by (47) The following result quantifies the probability that a certain rate will not be achievable reliably. Theorem 6: Consider a LMS link as the one defined in Theorem 2. Assuming that the input vector is a zero-mean Gaussian vector whose components have equal variance along the orthog, onal directions of the maximal-eigenvalue eigenspace of and zero otherwise, then (48) with
given by (11). Proof: The result in (48) is obtained by applying (11) to (47). (46) while for [27], we get the first equation shown at the bottom of the page. Proof: Plugging (10) in (45) and using the linear Kummer transform (cf. (77)), the result follows immediately. B. Outage Probability Often, however, we may encounter channels that change is held approximately constant during the slowly so that transmission of a codeword. In this case, the average mutual information has no operational significance and a more suitable performance measure is the so-called outage capacity, which regards the mutual information as a random variable whose distribution is induced by the channel matrix . From the distribution of the mutual information, we can quantify the probability that a certain rate will not be achievable reliably. In the case that the transmitter knows but has no ability to do power control (or, equivalently, it knows the maximal eigenbut not the maximal eigenvalue), then value eigenspace of a natural strategy (which turns out to be optimal in low SNR regime) is to beamform, with equal power, in the directions of 7Herein, the Incomplete Gamma function appearing in [26, Formula 40] is further expressed through [14, 5.1.45] as an Exponential Integral.
If the transmitter has neither instantaneous or statistical CSI, in the case of ergodic channels, as already explored in Theorem 5, the ergodic-capacity-achieving input distribution is a Gaussian vector with i.i.d. components [28, Theor. 13],[24], [25]. For nonergodic channel, if the transmitter has neither instantaneous or statistical CSI, there are no results that prove the optimality of isotropic input, however a reasonable choice, is to assume that transmit antennas are fed by independent equalpower streams. Under this assumption, the mutual information, conditioned on the channel realizations, is given by
from which we can prove the following result: Theorem 7: Consider a LMS link as the one defined in Theorem 2. Suppose that the receiver has perfect knowledge of the channel . If the input vector is Gaussian i.i.d., then asymptot, the mutual inically, in the case of integer , as formation converges in distribution to a Gaussian random variable. Furthermore, for finite and , its mean and variance are given, respectively, by (46) and (49), as shown at the bottom of , and with , , the page, with and ’s as in Proposition 2. Herein, denotes the Mejer’s G function [15, Ch. 9]. Proof: See Appendix G.
(49)
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Fig. 2. CDF of I ( ; I) for a LMS channel with receive antennas are uncorrelated.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 11, NOVEMBER 2010
N
= 2 and
N
= 4 and
Example 2: Fig. 2 compares the Gaussian distribution whose mean and variance are given by (46) and (49) respectively, with when , , the empirical distribution of , and respectively. Furthermore, the th entry of the LMS matrix is
alpha
For
=
f3; 6g. The transmit antennas are correlated as per (50), while the
, (49) can be further simplified as
(50) Note that, since and is an integer number, then the matrix models the transmit correlation matrix of the LMS channel. The receive antennas are, instead, uncorrelated. For and and and with the LMS matrix given by (50), we compute the Monte Carlo estimates of the 10% outage capacities as well as the values predicted by Gaussian-distribution approximation. In both cases, the agreement is remarkable. For , and , specifically, we find that at an of 10 dB (respectively, 0 dB) the asymptotic formula yields 3.6830 (respectively, 1.6541) while the Monte Carlo estimate equals 3.6806 (respectively, 1.6436). For , , and in turn, we find that at 10 dB (respectively, 0 dB) the asymptotic formula yields 4.1792 (respectively, 2.0469) while the Monte Carlo estimate is 4.1761 (respectively, 2.0416).
For the more general case of noninteger , the asymptotic normality of the mutual information has not been proved so far although we believe that the result of Theorem 7 still holds. For this case, we have that: Theorem 8: If the transmitter has no knowledge of the channel matrix (or its statistics), then the mean and variance of the conditioned mutual information are given, respectively by (43) and (51), as shown at the bottom of the page, with , and still with , , and ’s as in Proposition 2. Proof: From (75) and (76), the results of Theorem 8 follow expanding the confluent hypergeometric function appearing in (10) as an infinite series.
(51)
ALFANO et al.: THEORETICAL FRAMEWORK FOR LMS MIMO COMMUNICATION SYSTEMS
Fig. 3. CDF of I (
;
I) for a LMS channel with
N
=
f2; 3g and N
= 4 and
Example 3: Fig. 3 compares the Gaussian distribution whose mean and variance given by (43) and (51) respectively, with the with transmit, empirical distribution of and receive antennas, and with a LMS matrix as defined in (50). For and and and with given by (50), we also compute the Monte Carlo estimates of the 10% outage capacities as well as the values predicted by Gaussian-distribution approximation. In both cases, the agreement is remarkable. For , , and , specifically, we find that at an of 10 dB (respectively, 0 dB) the asymptotic formula yields 3.7679 (respectively, 1.7415) while the Monte Carlo estimate equals 3.7382 (respectively, 1.7313). For , , and in turn, we find that at 10 dB (respectively, 0 dB) the asymptotic formula yields 3.7652 (respectively, 1.8135) while the Monte Carlo estimate is 3.7615 (respectively, 1.8100).
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= 3:142.
which, based on the normalization given in (20), corresponds to the received energy per information bit when the input is isotropic8. At low SNR, the first-order behavior of the capacity, as a function of (in dB), takes on the convenient form (cf. [28] and [29]):
(54) with a second-order term. The two quantities, determining (54), are (the minimum energy per information bit required to convey any positive rate reliably) and , the capacity slope therein in bits/s/Hz/(3 dB). We have the following result: Theorem 9: Consider a LMS link as the one defined in Theorem 2. Suppose that the receiver has perfect knowledge of the channel , but the transmitter has no knowledge of the channel matrix (or its statistics). Then
C. The Low-Power Regime Denoting by the channel capacity expressed in bits per (complex) channel use, from (21), it follows that the transmitted energy per information bit, required to operate at capacity, relative to the noise level, is given by [28]
(52)
(55) and we have (56), as shown at the bottom of the next page, with , as in Proposition 2, and and can be evaluated as in Proposition 5. Proof: From [28] and [29], we have that
(53) (57) In the following, we denote: 8In
general, the received energy per information bit, , which in general depends on 8 .
, is
=
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Fig. 4. Marginal density distribution of an unordered eigenvalue for a 2
2 3 (dotted curve) and a 2 2 2 (continuous curve) LMS channel with = 3 and = 0:9.
while
(58) with the dispersion of a ( as [29]:
) (random) matrix
, defined
(59) which achieves its minimum value (equal to 1) when is equal to the identity matrix. As pointed out in [29], the dispersion succinctly quantifies how much the channel and noise depart from those in the canonical model. Using (14) and (16) in (57) and (58), we immediately get the result. From Theorem 9, in follows that is influenced solely by the number of receive antennas, which are those devoted to capture as much signal energy as possible in order to get a reliable communication. We can also notice that for the received minimum energy per bit, we still get the value . The case of integer leads to more compact expressions. In fact, in this case, the LOS matrix is Wishart distributed. Thus using the result derived in [29, Prop. I], we obtain (60)
with defined in (24), which mirrors the result for the correlated Ricean case in [29]. Furthermore notice that, since for , the multivariate SR model (6) itself results in a Wishart channel , this case boils down to the ones already matrix evaluated in [29], leading to the following slope: (61) which immediately offers a key to quantify the impact of the LOS matrix parameters on the overall performance [29, Corollary I]. We now turn our attention to the limiting cases and . As to the former, the expression one obtains particularizing (60) coincides with that given in [29] for the canonical (uncorrelated Rayleigh) case, while . Recall that the low power slope of the MIMO Ricean channel becomes for that of a scalar unfaded channel [29, Formula 24]. In our framework, instead, independence on the antenna number is retained, but the actual slope value does depend on the matrix parameter . Summarizing, up to the first order, the only system parameter affecting the minimum bit energy required to reliably convey a positive rate is the receive antenna number, as expected. Combined effects of the transmit and receive antenna numbers, as well as of the LOS distribution parameters, can be instead observed on the slope value. This behavior perfectly mirrors that of a Ricean channel with unfaded LOS component, but for the limiting case , for which the slope is unaffected by the antenna number and just exibits a dependence on .
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2
=
2
Fig. 5. CDF of for a 2 2 LMS channel with 3, for a 2 3 LMS channel with = 4, and for a 2 correspond to values of = 0:1, dotted to = 0:5, continuous curves to = 0:9.
Fig. 6. Coherent mutual information versus = 0 :9 .
for a 2
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2 4 LMS channel with = 5. Dashed curves
2 2 LMS channel with = 2, 2 2 3 with = 3 and 2 2 4 with = 4. All curves are evaluated for
V. NUMERICAL RESULTS For all examples in this sections, we consider a LMS wireless MIMO channel with uncorrelated transmit and correlated . Fig. 4 contains receive antennas and assume the marginal density distribution of an unordered eigenvalue of , given in (10), for two different values of the ordered pair , , and . Specifically, the blue curve refers to the pair (2, 3) while the red one to the case . The behavior of the marginal law is further investigated around zero in the right part of the figure. Turning the attention to the extremal eigenvalues statistics, in Fig. 5, the CDF of the maximum eigenvalue of a SR Gram channel matrix is plotted. The impact of correlation on the performance is evident from the drawn curves. An appreciable gap between the curves for (dashed) and those obtained setting
(continuous curves) is present. Since the curves are obtained for and , while keeping constant only the different values of number of receive antennas equal to 2, in order to enhance the correlation impact, it is not immediate from the figure to see how spatial diversity impacts performance. However, further numerical results not reported herein confirms that fixing the values of all the remaining parameters, to deploy an increasing number of transmit antennas allows, as expected, to obtain better performance especially in case of negligible correlation. The dependence of the capacity on the channel statistics is depicted in Fig. 6. The curves, obtained setting , , and varying both the number of transmit antennas as well as the LOS fluctuation parameter , correspond to the case when perfect CSI is available at the receiver and neither instantaneous nor statistical CSI are given at the transmitter.
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Fig. 7. Comparison between the spectral efficiency (Start-marked points) as a function of the received energy per bit and its lowfor single-antenna and four-antenna architectures.
expansion at the first order
Fig. 8. Comparison between Rayleigh fading (stars-marked points) and SR fading (full lines) MIMO channels for single-antenna, two-antenna and four-antenna architectures, with as a parameter.
In Figs. 7 and 8, we, again, analyze ascenario wherethe receiver is provided with a perfect information about the channel state and thetransmitterhasnoknowledgeofthechannelmatrix(oritsstatistics).Specifically,Fig.7verifiestheeffectivenessofthelowpower approximation of the capacity. Therein, the actual capacity versus and itslinear approximation, given in (54), are plotted,for both a Rayleigh scalar channel as well as a 4 4 MIMO channel with and . The approximations are very close-by the aclevels. tual curves even for rather ambitious The meaning of as the number of degrees of freedom additionally available on the channel due to the LOS randomness,
with respect to the scattering degrees of freedom inherent the underlying Rayleigh fading process, is explained in Fig. 8. Therein, for the linear approximation of the capacity versus are depicted, both for the Rayleigh as well as for the SR case. Stars-marked points represent the canonical channel behavior,while full lines the SR case. We explicitly notice that, for and , the low power slope of forthe multivariate SR channel is greater than that of the Rayleigh one. For , we consider two scenarios. In the former, , the low power slope of the SR channel is smaller than with that of a Rayleigh channel exploiting the same number of transmit
ALFANO et al.: THEORETICAL FRAMEWORK FOR LMS MIMO COMMUNICATION SYSTEMS
and receive antennas. The latter, with , corresponds in the SR case, to a slope larger than that under the Rayleigh assumpfurther confirm the trend; tion. The curves for specifically, the cases of and are depicted. The last situation shows the performance loss of the SR model due , the to the lack of degrees of freedom. Finally, when low-power approximations of both the Rayleigh as well as the SR faded channel are perfectly coincident. VI. CONCLUSIONS In this paper, the multivariate SR model for MIMO LMS channels has been introduced. Both the pdf and the spectral statistics of the Gram channel matrix are given, allowing the theoretical performance analysis of the newly proposed channel model from both a communication and an information-theoretic point of view. Particular attention has been payed to the analysis of the coherent case, namely the perfect receive-side information scenario, for which several key parameters have been evaluated, mostly in closed form. Numerical examples are provided in order to assess the impact on the performance of the diversity order and/or the LOS fluctuations. The presented setting paves the way for the analysis of intra-sensors correlation impact as well as rank deficiency of the LOS component and multiple polarizations. The presented model, and in particular the eigenanalysis results, offer a theoretical background also for sensor networks capacity and connectivity analysis. Both topics are subjects of ongoing work.
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We finally exploit the representation of hypergeometric functions of matrix arguments9 in [30] to get (9). APPENDIX B MARGINAL DENSITYOF AN UNORDERED EIGENVALUE The proof follows the lines of [8, Theorem I]. We repeat the steps hereinafter for sake of completeness, since randomness in was not accounted for in the referred work. The joint distribution of the unordered eigenvalues of , following a symAs a metry argument [31], can be obtained dividing (9) by consequence, it can be expressed as (62)
be the matrix whose th entry is , Let then namely, the matrix whose determinant coincides with . We now substitute in (62) and then exploit the Laplace determinant expansion, i.e., see the (63), at the bottom of the page, with matrix obtained deleting the first row and the th column from , and the matrix obtained deleting the first row and the th column from . Equation (10) stems from via Cauchy–Binet Lemma, integration of (63) over are evaluated through [10, Formula 28]. and the entries of Finally, we remark that the choice of in (63) has no effect on the result since we started from an unordered distribution.
APPENDIX A JOINT EIGENVALUES
APPENDIX C MAXIMUM EIGENVALUE
The joint distribution of the ordered eigenvalues of a complex is given by [10, Formula 93] random matrix
The CDF of the maximum eigenvalue can be obtained from the joint density of the ordered eigenvalues following the procedure outlined in [32, Theorem I] (64)
where stands for the unitary group of size . The above integral can be evaluated applying the splitting formula [10, Formula 92]
which yields
where
,
. In order to evaluate the integral at the right-hand side of (64), we decompose it as (65) 9We consider the case where has distinct eigenvalues. However, such a requirement can be relaxed resorting to a proper perturbation technique [30]. For sake of brevity, we omit here the analysis.
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where the first term at the right-hand side can be evaluated via , we sub[10, Formula 28]. As to the second term, when stitute [14, Formula 13.2.1] in the integral, namely we express
(66) Exchanging the integration order in the resulting double integral expression, and using the definition of the Incomplete Gamma Function [15, Formula 8.350.2], we get the first equation at the bottom of the next page, which can be evaluated in closed form , it can be via [15, Formula 3.385]. As to the case of handled exploiting the series expression [14, Formula 13.1.2] in the second term at the right-hand side of (65). Integrating term by term, and still using [15, Formula 8.350.2], we get the result in (11). , the expression of in (12) Note that, when does contain an infinite series, which can be nevertheless expressed in a more compact way using algebra like in [33, Appendix III]. We outline in the following the derivation strategy. Let us first consider the generic term in the series given in (12), which we recall below for the reader’s convenience,
relied on being (70) is obtained through [33, eq. (102)]
When is an integer, and , (11) further simplifies by virtue of Kummer transform [14]
Specifically, in this last case the entries of the matrix are given by (71), as shown at the bottom of the page. , then reduces to a complex central Wishart matrix If and the CDF of the maximum eigenvalue can be found in [34]. APPENDIX D MINIMUM EIGENVALUE The CDF of the minimum eigenvalue can be obtained through
(67) Notice that (67) admits also the alternative expressions shown in (68)–(70), at the bottom of the page, where in (68), we used , in (69) we
and, finally,
(72) where uate
and, following [32, Theorem I], we may eval, with, again,
(68) (69) (70)
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(75)
. As a consequence, can be evaluated as in formulas (65)–(70), leading finally to (14).
is given by (17). Expression (17), indeed, alwhere lows for a simple evaluation of the derivative with respect to through the standard formula of the derivative of a product, namely
APPENDIX E MOMENTS OF THE TRACE • We evaluate both (14) and (16) using known results on the moments of the trace of a noncentral Wishart matrix [29]. Specifically, we first exploit
and then perform the external average through
and
[10,
Formula
28]
to
get
. • Expression (16) too can be evaluated via conditional expectation as
Noticing that , and for any square matrix , , we can cope with the first part of the computation. The derivative of the hypergeometric function appearing in (17), instead, can be evaluated by first exploiting the detergiven in [30, eq. (29)]) and then minant form of using the identity [35]: with . APPENDIX G MUTUAL INFORMATION: ASYMPTOTIC NORMALITY
Specifically, the inner average, by virtue of the indepen, can be written as [29, Formula dence between and 47] (73) As to the outer average, we exploit where is an unordered eigenvalue of . The pdf of , in turn, can be obtained applying [8, Theorem I] to (3), leading to . Finally, by virtue of [10, Formula 28], we get . • As to (15), since mula 28] the result immediately follows.
, by [10, For-
boils down to the sum of suitable corFor integer , related central Wishart matrices. Consequently, the asymptotic follows normality of the mutual information, for immediately from the result in [36]. Concerning the expression of the variance for finite and , let us recall here [26, Formula 41] (see (75), as shown at the top of the page). Now (76) is given by (10). where and integer, the confluent hypergeometric funcFor , admits the following expression [14, tion appearing in (13.1.27) and (13.6.9)]:
(77) APPENDIX F LOGARITHMIC MOMENT OF THE GENERALIZED VARIANCE Notice first that we can write
usually referred to as linear Kummer transform. Plugging (10) and (77) in (76), the result of Theorem 7 follows immediately. REFERENCES
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[28] S. Verdú, “Spectral efficiency in the wideband regime,” IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1319–1343, Jun. 2002. [29] A. Tulino, A. Lozano, and S. Verdú, “Multiple antenna capacity in the low-power regime,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 1019–1030, Oct. 2003. [30] A. Y. Orlov, “New solvable matrix integrals,” Int. J. Appl. Phys., vol. 19, pp. 276–293, Jun. 2004. [31] U. Haagerup and S. T. nsen, Random Matrices with Complex Gaussian Entries, to be published. [32] M. Kang and M. S. Alouini, “Largest eigenvalue of complex wishart matrices and performance analysis of MIMO MRC systems,” IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 418–426, Apr. 2003. [33] M. R. McKay, P. J. Smith, H. A. Suraweera, and I. B. Collings, “On the mutual information distribution of ofdm-based spatial multiplexing: Exact variance and outage approximation,” IEEE Trans. Inform. Theory, vol. 54, no. 7, pp. 3260–3278, Jul. 2008. [34] C. G. Khatri, “Distribution of the largest or the smallest characteristic root under null hypothesis concerning complex multivariate normal populations,” Ann. Math. Stat., vol. 35, no. 12, pp. 1807–1810, Dec. 1964. [35] H.-H. Olsen, The derivative of a determinant, 1997, unpublished. [36] W. Hachem, P. Loubaton, and J. Najim, “A CLT for information theoretic statistics of Gram random matrices with a given variance profile,” Ann. Appl. Probabil., 2008, to be published.
Giusi Alfano received the Laurea degree in communication engineering from the University of Naples Federico II, Naples, Italy, in 2001 and the Ph.D. degree in information engineering from the University of Benevento, Benevento, Italy, in October 2007. From 2002 to 2004, she was involved in radar and satellite signal processing studies at the National Research Council and University of Naples. She is a Postdoctorate Researcher at the Politecnico di Torino, Torino, Italy. Her research lies mainly in the field of random matrix theory applications to MIMO wireless communications and sensor networks, and to the characterization of physical layers of random networks
Antonio De Maio (S’01–AM’02–M’03–SM’07) was born in Sorrento, Italy, on June 20, 1974. He received the Dr.Eng. degree (Hons.) and the Ph.D. degree in information engineering, both from the University of Naples Federico II, Naples, Italy, in 1998 and 2002, respectively. From October to December 2004, he was a Visiting Researcher at the U.S. Air Force Research Laboratory, Rome, NY. From November to December 2007, he was a visiting researcher at the Chinese University of Hong Kong, Hong Kong. Currently, he is an Associate Professor at the University of Naples Federico II. His research interest lies in the field of statistical signal processing, with emphasis on radar detection, convex optimization applied to radar signal processing, and multiple-access communications. Dr. De Maio is the recipient of the IEEE 2010 Fred Nathanson Award.
Antonia Maria Tulino (M’00–SM’05) received the Ph.D. degree from the Electrical Engineering Department, Seconda Universitá degli Studi di Napoli, Naples, Italy, in 1999. She has served as Associate Professor at the Department of Electrical and Telecommunications Engineering at the Universitá degli Studi di Napoli “Federico II” since 2002. She is currently with the Department of Wireless Communications, Bell Laboratories, Alcatel-Lucent, Holmdel, NJ. She held research positions at the Center for Wireless Communications, Oulu, Finland and at the Department of Electrical Engineering, Princeton University, Princeton, NJ. She has served on the Faculty of Engineering, Universitá degli Studi del Sannio, Benevento, Italy. Her research interests lay in the broad area of communication systems approached with the complementary tools provided by signal processing, information theory, and random matrix theory. Dr. Tulino received the 2009 Stephen O. Rice Prize in the Field of Communications Theory for the best paper published in the IEEE TRANSACTIONS ON COMMUNICATIONS in 2008. She is a frequent contributor to the IEEE TRANSACTIONS ON INFORMATION THEORY, the IEEE TRANSACTIONS ON COMMUNICATIONS, and the IEEE TRANSACTIONS ON SIGNAL PROCESSING.
