uniqueness of positive solutions for semilinear elliptic systems abstract we prove uniqueness of positive solutions for the system u f v v g u in u v 0 on where is a bounded domain in r n f g r r and f x x p g x x q at for some positive numbers p q with p q 1

improved calibration of high dimensional precision matrices estimation of a precision matrix i e the inverse covariance matrix is a fundamental problem in statistical signal processing applications when the observation dimension is of the same order of magnitude as the number of samples the conventional estimators of covariance matrix and its inverse perform poorly in order to obtain well behaved estimators in high dimensional settings we consider a general class of estimators of covariance matrices and precision matrices based on weighted sampling and linear shrinkage the estimation error is measured in terms of both quadratic loss and stein s loss and these loss functions are used to calibrate the set of parameters defining our proposed estimator in an asymptotic setting where the observation dimension is comparable in magnitude to the number of samples we provide estimators of the precision matrix that are as good as their oracle counterparts we test our estimators with both synthetic data and financial market data and monte carlo simulations show the advantage of our precision matrix estimator compared with well known estimators in finite sample size settings

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

asymptotic mutual information statistics of mimo channels and clt of sample covariance matrices in this paper we consider the fluctuation of mutual information statistics of a multiple input multiple output channel communication systems without assuming that the entries of the channel matrix have zero pseudovariance to this end we also establish a central limit theorem of the linear spectral statistics for sample covariance matrices under general moment conditions by removing the restrictions imposed on the second moment and fourth moment on the matrix entries in bai and silverstein 2004

asymptotic performance of linear receivers in mimo fading channels linear receivers are an attractive low complexity alternative to optimal processing for multiple antenna multiple input multiple output mimo communications in this paper we characterize the information theoretic performance of mimo linear receivers in two different asymptotic regimes for fixed number of antennas we investigate the limit of error probability in the high signal to noise ratio snr regime in terms of the diversity multiplexing tradeoff dmt following this we characterize the error probability for fixed snr in the regime of large but finite number of antennas as far as the dmt is concerned we report a negative result we show that both linear zero forcing zf and linear minimum mean square error mmse receivers achieve the same dmt which is largely suboptimal even in the case where outer coding and deacircnot coding is performed across the antennas we also provide an apacircnot proximate quantitative analysis of the markedly different behavior of the mmse and zf receivers at finite rate and nonasymptotic snr and show that while the zf receiver achieves poor diversity at any finite rate the mmse receiver error curve slope flattens out progressively as the coding rate increases when snr is fixed and the number of antennas becomes large we show that the mutual information at the output of an mmse or zf linear receiver has fluctuations that converge in distribution to a gaussian random variable whose mean and variance can be characterized in closed form this analysis extends to the linear reacircnot ceiver case a well known result previously obtained for the optimal receiver simulations reveal that the asymptotic analysis captures accurately the outage behavior of systems even with a moderate number of antennas

fluctuations of an improved population eigenvalue estimator in sample covariance matrix models this paper provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on sample covariance matrices the focus is on limited sample size situations whereby the number of available observations is comparable in magnitude to the observation dimension an exact expression as well as an empirical asymptotically accurate approximation of the limiting variance is derived simulations are performed that corroborate the theoretical claims

uniqueness of positive solutions for a class of semilinear elliptic systems we prove existence and uniqueness of positive solutions for the system u f v v g u in b u v 0 on b where b is the open unit ball in rn f g r r and f x xp g x xq at for some p q 0 with pq 1

positive radial solutions to classes of nonlinear elliptic systems on the exterior of a ball abstract we study positive radial solutions to n n nonlinear elliptic systems of the form u i k i x f i u i 1 in x e u i 0 on x r 0 u i 0 as x where i 1 2 n u 1 u n 1 u div u 0 n 2 r 0 0 e x r n x r 0 k i c r 0 0 and f i c 0 r we establish existence multiplicity and uniqueness results for various nonlinearities f i

real periodic solutions of the liouville equation abstract real periodic solutions of the liouville equations 2 2 8 e that describe equilibria in two dimensional mhd and imcompressible fluids are obtained in the representation we approach this problem by taking a singular limit of the sinh poisson equation previously solved by ting chen and lee a unique and interesting property of the present system is the existence of internal degrees of freedom which appear as ghost terms in the evolution equations of the external degrees of freedom

a new approach for mutual information analysis of large dimensional multi antenna channels this paper adresses the behavior of the mutual information of correlated multiple input multiple output mimo rayleigh channels when the numbers of transmit and receive antennas converge to infin at the same rate using a new and simple approach based on poincare nash inequality and on an integration by parts formula it is rigorously established that the mutual information when properly centered and rescaled converges to a standard gaussian random variable simple expressions for the centering and scaling parameters are provided these results confirm previous evaluations based on the powerful but nonrigorous replica method it is believed that the tools that are used in this paper are simple robust and of interest for the communications engineering community

semiunitary precoding for spatially correlated mimo channels the focus of this paper is on spatial precoding in correlated multiantenna channels where the number of data streams is adapted independent of the number of transmit antennas towards the goal of a low complexity implementation a statistical semiunitary precoder is studied where the precoder matrix evolves fairly slowly with respect to the channel evolution while prior work on statistical precoding has focussed on information theoretic limits most of these computations result in complicated functional dependencies of the mutual information with the channel statistics that do not explicitly reveal the impact of statistics on performance in contrast estimates that are directly in terms of the channel statistics are obtained here for the relative mutual information loss of a semiunitary precoder with respect to a perfect channel information benchmark based on these estimates matching metrics are developed that capture the degree of matching of a channel to the precoder structure continuously and allow ordering two matrix channels in terms of their mutual information performance while these metrics are based on bounds numerical studies are used to show that the proposed metrics capture the performance tradeoffs accurately the main conclusion of this work is a simple to state fundamental principle in the context of signaling design for single user mimo systems the best channel for the statistical precoder is the channel that is matched to it

capacity scaling in mimo wireless systems under correlated fading previous studies have shown that single user systems employing n element antenna arrays at both the transmitter and the receiver can achieve a capacity proportional to n assuming independent rayleigh fading between antenna pairs we explore the capacity of dual antenna array systems under correlated fading via theoretical analysis and ray tracing simulations we derive and compare expressions for the asymptotic growth rate of capacity with n antennas for both independent and correlated fading cases the latter is derived under some assumptions about the scaling of the fading correlation structure in both cases the theoretic capacity growth is linear in n but the growth rate is 10 20 smaller in the presence of correlated fading we analyze our assumption of separable transmit receive correlations via simulations based on a ray tracing propagation model results show that empirical capacities converge to the limit capacity predicted from our asymptotic theory even at moderate n 16 we present results for both the cases when the transmitter does and does not know the channel realization

uniqueness of positive solutions for a class of semipositone elliptic systems abstract we consider the system u f v x v g u x u 0 v x where is a ball in r n n 1 and is its boundary are positive parameters and f g are smooth functions that are negative at the origin and f x x p and g x x q for large x for some p q 0 with p q 1 we establish the uniqueness of positive solutions when the parameters and are large

sinr statistics of correlated mimo linear receivers linear receivers offer a low complexity option for multiantenna communication systems therefore understanding the outage behavior of the corresponding sinr is important in a fading mobile environment in this paper we introduce a large deviation method valid nominally for a large number m of antennas which provides the probability density of the sinr of gaussian channel mimo minimum mean square error mmse and zero forcing zf receivers with arbitrary transmission power profiles and in the presence of receiver antenna correlations this approach extends the gaussian approximation of the sinr valid for large m asymptotically close to the center of the distribution to obtain the non gaussian tails of the distribution our methodology allows us to calculate the sinr distribution to next to leading order o 1 m and showcase the deviations from approximations that have appeared in the literature e g the gaussian or the generalized gamma distribution we also analytically evaluate the outage probability as well as the uncoded bit error rate we find that our approximation is quite accurate even for the smallest antenna arrays 2 2

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

a central limit theorem for the sinr at the lmmse estimator output for large dimensional signals this paper is devoted to the performance study of the linear minimum mean squared error lmmse 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 plus 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 multiple antenna 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

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

the shannon s mutual information of a multiple antenna time and frequency dependent channel an ergodic operator approach consider a random non centered multiple antenna radio transmission channel assume that the deterministic part of the channel is itself frequency selective and that the random multipath part is represented by an ergodic stationary vector process in the hilbert space l2 one can associate to this channel a random ergodic self adjoint operator having a so called integrated density of states ids shannon s mutual information per receive antenna of this channel coincides then with the integral of a log function with respect to the ids in this paper it is shown that when the numbers of antennas at the transmitter and at the receiver tend to infinity at the same rate the mutual information per receive antenna tends to a quantity that can be identified and in fact is closely related to that obtained within the random matrix approach i telatar eur trans telecommun 10 585 1999 this result can be obtained by analyzing the behavior of the stieltjes transform of the ids in the regime of the larg

on the precoder design of flat fading mimo systems equipped with mmse receivers a large system approach this paper is devoted to the design of precoders maximizing the ergodic mutual information emi of bi correlated flat fading mimo systems equipped with mmse receivers the channel state information and the second order statistics of the channel are assumed available at the receiver side and at the transmitter side respectively as the direct maximization of the emi needs the use of nonattractive algorithms it is proposed to optimize an approximation of the emi introduced recently obtained when the number of transmit and receive antennas t and converge to at the same rate it is established that the relative error between the actual emi and its approximation is a o 1 t2 term it is shown that the left singular eigenvectors of the optimum precoder coincide with the eigenvectors of the transmit covariance matrix and its singular values are solution of a certain maximization problem numerical experiments show that the mutual information provided by this precoder is close from what is obtained by maximizing the true emi but that the algorithm maximizing the approximation is much less computationally intensive

a deterministic equivalent for the analysis of non gaussian correlated mimo multiple access channels using large dimensional random matrix theory rmt we conduct mutual information analysis of a multiple input multiple output mimo multiple access channel mac our channel model reflects the characteristics in small cell networks where antenna correlations line of sight components and general type of fading distributions have to be included the mutual information expression can be expressed as functionals of the stieltjes transform through the so called shannon transform ideally if the stieltjes transform is known in the context of the large dimensional rmt then the problem is solved however it is difficult to derive the stieltjes transform of the considered channel models directly especially when the transmit correlation matrices are generally nonnegative definite and the channel entries are non gaussian to overcome this we use the generalized lindeberg principle to show that the stieltjes transforms of this class of random matrices with gaussian or non gaussian independent entries coincide in the large dimensional regime this result permits to derive the deterministic equivalents e g the stieltjes transform and the ergodic mutual information for non gaussian mimo channels from the known results developed for gaussian mimo channels as an application we determine the capacity achieving input covariance matrices for the mimo macs and prove that the capacity achieving input covariance matrices are asymptotically independent of the fading distribution

a uniqueness result for a semipositone p laplacian problem on the exterior of a ball abstract we consider steady state reaction diffusion equations on the exterior of a ball namely boundary value problems of the form p u k x f u in e u 0 on x r 0 u 0 when x where p z div z p 2 z 1 p n is a positive parameter r 0 0 and e x r n x r 0 here the weight function k c 1 r 0 satisfies k r 0 for r r 0 lim r k r 0 and the reaction term f c 0 c 1 0 is strictly increasing and satisfies f 0 0 semipositone lim sup s 0 s f s lim s f s lim s f s s p 1 0 and f s s q is nonincreasing on a for some a 0 and q 0 p 1 for a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball and exists for 1 we establish the uniqueness of this positive radial solution for 1

a clt on the snr of diagonally loaded mvdr filters this paper studies the fluctuations of the signal to noise ratio snr of minimum variance distorsionless response mvdr filters implementing diagonal loading in the estimation of the covariance matrix previous results in the signal processing literature are generalized and extended by considering both spatially as well as temporarily correlated samples specifically a central limit theorem clt is established for the fluctuations of the snr of the diagonally loaded mvdr filter under both supervised and unsupervised training settings in adaptive filtering applications our second order analysis is based on the nash poincare inequality and the integration by parts formula for gaussian functionals as well as classical tools from statistical asymptotic theory numerical evaluations validating the accuracy of the clt confirm the asymptotic gaussianity of the fluctuations of the snr of the mvdr filter

on the minimax capacity loss under sub nyquist universal sampling this paper investigates the information rate loss in analog channels when the sampler is designed to operate independent of the instantaneous channel occupancy specifically a multiband linear time invariant gaussian channel under universal sub nyquist sampling is considered the entire channel bandwidth is divided into n subbands of equal bandwidth at each time only k constant gain subbands are active where the instantaneous subband occupancy is not known at the receiver and the sampler we study the information loss through an information rate lossmetric that is the gap of achievable rates caused by the lack of instantaneous subband occupancy information we characterize the minimax information rate loss for the sub nyquist regime provided that the number n of subbands and the snr are both large the minimax limits depend almost solely on the band sparsity factor and the undersampling factor modulo some residual terms that vanish as n and snr grow our results highlight the power of randomized sampling methods i e the samplers that consist of random periodic modulation and low pass filters which are able to approach the minimax information rate loss with exponentially high probability

no eigenvalues outside the limiting support of generally correlated gaussian matrices this paper investigates the behaviour of the spectrum of generally correlated gaussian random matrices whose columns are zero mean independent vectors but have different correlations under the specific regime where the number of their columns and that of their rows grow at infinity with the same pace following the approach proposed by vallet et al we prove that under some mild conditions there is no eigenvalue outside the limiting support of generally correlated gaussian matrices as an outcome of this result we establish that the smallest singular value of these matrices is almost surely greater than zero from a practical perspective this control of the smallest singular value is paramount to applications from statistical signal processing and wireless communication in which this kind of matrices naturally arise

a clt for a band matrix model a law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on or above diagonal entries are independent but neither necessarily identically distributed nor necessarily all of the same variance the derivation is based on systematic combinatorial enumeration study of generating functions and concentration inequalities of the poincare type special cases treated with an explicit evaluation of limiting variances are generalized wigner and wishart matrices

lie group analysis and similarity solutions for the equation on determine le groupe des transformations ponctuelles pour l equation 2 u x 2 2 u 2 y 2 expu z 2 0

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

on the capacity achieving covariance matrix for frequency selective mimo channels using the asymptotic approach in this contribution an algorithm for evaluating the capacity achieving input covariance matrices for frequency selective rayleigh mimo channels is proposed in contrast with the flat fading rayleigh case no closed form expressions for the eigenvectors of the optimum input covariance matrix are available classically both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding in this paper it is proposed to optimize w r t the input covariance matrix a large system approximation of the average mutual information derived by moustakas and simon the validity of this asymptotic approximation is clarified thanks to gaussian large random matrices methods it is shown that the approximation is a strictly concave function of the input covariance matrix and that the average mutual information evaluated at the argmax of the approximation is equal to the capacity of the channel up to a o 1 t term where t is the number of transmit antennas an algorithm based on an iterative waterfilling scheme is proposed to maximize the average mutual information approximation and its convergence studied numerical simulation results show that even for a moderate number of transmit and receive antennas the new approach provides the same results as direct maximization approaches of the average mutual information

existence uniqueness and stability of positive solutions to sublinear elliptic systems reaction diffusion systems are used to model many chemical and biological phenomena in the natural world 25 26 and systems of coupled partial differential equations are also used in other physical models such as nonlinear schrodinger systems in multi component bose einstein condensates and nonlinear optics 19 23 the steady state solutions or standing wave solutions of such systems of nonlinear partial differential equations satisfy a nonlinear elliptic system with more than one equation much effort has been devoted to the existence of solutions of such systems see for example 4 8 10 11 13 15 17 24 30 33 34 but it is usually difficult to determine whether or not the solution is unique we consider the positive solutions of a semilinear elliptic system of the form

an existence result on positive solutions for a class of semilinear elliptic systems consider the system where is a positive parameter and is a bounded domain in r n we prove the existence of a large positive solution for large when lim x f mg x x 0 for every m 0 in particular we do not need any monotonicity assumptions on f g nor any sign conditions on f 0 g 0

on the capacity achieving covariance matrix for rician mimo channels an asymptotic approach in this paper the capacity achieving input covariance matrices for coherent block fading correlated multiple input multiple output mimo rician channels are determined in contrast with the rayleigh and uncorrelated rician cases no closed form expressions for the eigenvectors of the optimum input covariance matrix are available classically both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding in the asymptotic regime where the number of transmit and receive antennas converge to infinity at the same rate new results related to the accuracy of the approximation of the average mutual information are provided based on the accuracy of this approximation an attractive optimization algorithm is proposed and analyzed this algorithm is shown to yield an effective way to compute the capacity achieving matrix for the average mutual information and numerical simulation results show that even for a moderate number of transmit and receive antennas the new approach provides the same results as direct maximization approaches of the average mutual information

generalized consistent error estimator of linear discriminant analysis a classifier is epistemologically vacuous without an accurate estimate of its true error rate in situations where the number of sample points is of the same order of magnitude as the dimension of observations serious issues arise with respect to the performance of error estimators in this paper we place the problem of synthesizing an error rate estimator of a common linear classifier in an asymptotic setting in which the number of sample points is kept comparable in magnitude to the dimension of observations double asymptotic we construct a generalized consistent estimator of the true error rate for linear discriminant analysis in the multivariate gaussian model under the assumption of a common covariance matrix in other words the estimator converges to true error rate in the double asymptotic sense we employ simulations using both synthetic and real data to compare the performance of the new estimator to the classical consistent estimator of the true error plug in estimator as well as other well known estimators we observe that the constructed estimator can outperform other estimators of the true error in many situations in terms of bias and root mean square rms error

performance of mutual information inference methods under unknown interference in this paper the problem of fast point to point multiple input multiple output channel mutual information estimation is addressed in the situation where the receiver undergoes unknown colored interference whereas the channel with the transmitter is perfectly known the considered scenario assumes that the estimation is based on a few channel use observations during a short sensing period using large dimensional random matrix theory an estimator referred to as g estimator is derived this estimator is proved to be consistent as the number of antennas and observations grow large and its asymptotic performance is analyzed in particular the g estimator satisfies a central limit theorem with asymptotic gaussian fluctuations simulations are provided which strongly support the theoretical results even for small system dimensions

large system analysis of a glrt for detection with large sensor arrays in temporally white noise this paper addresses the behaviour of a classical multi antenna glrt test that allows to detect the presence of a known signal corrupted by a multi path propagation channel and by an additive temporally white gaussian noise with unknown spatial covariance matrix the paper is focused on the case where the number of sensors m is large and of the same order of magnitude as the sample size n a context which is modeled by the large system asymptotic regime m n in such a way that m n c for c 0 the purpose of this paper is to study the behaviour of a glrt statistics in this regime and to show that the corresponding theoretical analysis allows to accurately predict the performance of the test when m and n are of the same order of magnitude

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

limits of infinite interaction radius dimensionality and the number of components for random operators with off diagonal randomness we consider random one body operators that are analogs of the statistical mechanics hamiltonians with a varying interaction radiusr the dimensionality of spaced and the number of the field components orbitals n we prove that all the moments of the green functions for nonreal energies of these operators converge asr d n to the products of the average green functions just as in the mean field approximation of statistical mechanics we find in particular the selfconsistent equation for the limiting integrated density of states and the limiting form of the conductivity which is nonzero on the whole support of the integrated density of states

ergodic sum capacity of macrodiversity mimo systems in flat rayleigh fading the prospect of base station cooperation leading to joint combining at widely separated antennas has led to increased interest in macrodiversity systems where both sources and receive antennas are geographically distributed in this scenario analytical investigation of channel capacity is extremely challenging for finite size systems since the channel matrices have a very general form where each path may have a different power hence in this paper we consider the ergodic sum capacity of a macrodiversity multiple input multiple output system with arbitrary numbers of sources and receive antennas operating over rayleigh fading channels for this system we compute the exact ergodic capacity for a system with at most two transmit antennas and a compact approximation for the general system which is shown to be very accurate over a wide range of cases finally we compare our results with previous asymptotic results and bounds results are verified by monte carlo simulations and the impact on capacity of various channel power profiles is investigated

central limit theorem for mutual information of large mimo systems with elliptically correlated channels by considering elliptical correlations in the channel matrix of a large mimo system this paper investigates how the non linear dependency affects the asymptotic behaviors of shannon s mutual information as the numbers of antennas at transmitter and receiver both tend to infinity at the same rate beyond the almost sure convergence of the information per transmit antenna we also propose several central limit theorems for the information statistics under various scenarios the results show that the non linear correlation has a little impact on the almost sure convergence but has a tremendous effect on the fluctuation of the information statistics in particular the centralized information statistics could experience a phase transition in convergence rate when the strength of the non linear correlation increases as extensions several central limit theorems are established as well for general linear spectral statistics of large sample covariance matrices in elliptical distributions

on the resolution probability of conditional and unconditional maximum likelihood doa estimation after decades of research in direction of arrival doa estimation today maximum likelihood ml algorithms still provide the best performance in terms of resolution capabilities at the cost of a multidimensional search ml algorithms achieve a significant reduction of the outlier production mechanism in the threshold region where the number of snapshots per antenna and or the signal to noise ratio snr are low the objective of this paper is to characterize the resolution capabilities of ml algorithms in the threshold region both conditional and unconditional versions of the ml algorithms are investigated in the asymptotic regime where both the number of antennas and the number of snapshots are large but comparable in magnitude by using random matrix theory techniques the finite dimensional distributions of both cost functions are shown to be gaussian distributed in this asymptotic regime and a closed form expression of the corresponding asymptotic covariance matrices is provided these results allow to characterize the asymptotic behavior of the resolution probability which is defined as the probability that the cost function evaluated at the true doas is smaller than the values that it takes at the positions of the other asymptotic local minima

improved subspace estimation for multivariate observations of high dimension the deterministic signals case we consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude in this context traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones it has recently been suggested that this situation where the sample size is small compared to the observation dimension can be very accurately modeled by considering the asymptotic regime where the observation dimension m and the number of snapshots n converge to at the same rate using large random matrix theory results it can be shown that traditional subspace estimates are not consistent in this asymptotic regime furthermore new consistent subspace estimate can be proposed which outperform the standard subspace methods for realistic values of m and n the work carried out so far in this area has always been based on the assumption that the observations are random independent and identically distributed in the time domain the goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals in practice this allows to use the proposed approach regardless of the statistical properties of the source signals in order to construct the proposed estimators new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of information plus noise complex gaussian models are established these results are believed to be of independent interest

why does the kronecker model result in misleading capacity estimates many recent works that study the performance of multiple input multiple output mimo systems in practice assume a kronecker model where the variances of the channel entries upon decomposition on to the transmit and the receive eigenbases admit a separable form measurement campaigns however show that the kronecker model results in poor estimates for capacity motivated by these observations a channel model that does not impose a separable structure has been recently proposed and shown to fit the capacity of measured channels better in this paper we show that this recently proposed modeling framework can be viewed as a natural consequence of channel decomposition on to its canonical coordinates the transmit and or the receive eigenbases using tools from random matrix theory we then establish the theoretical basis behind the kronecker mismatch at the low and the high snr extremes 1 sparsity of the dominant statistical degrees of freedom dof in the true channel at the low snr extreme and 2 nonregularity of the sparsity structure disparities in the distribution of the dof across the rows and the columns at the high snr extreme

ber and outage probability approximations for lmmse detectors on correlated mimo channels this paper is devoted to the study of the performance of the linear minimum mean square error lmmse receiver for receive correlated multiple input multiple output mimo systems by the random matrix theory it is well known that the signal to noise ratio snr at the output of this receiver behaves asymptotically like a gaussian random variable as the number of receive and transmit antennas converge to infin at the same rate however this approximation being inaccurate for the estimation of some performance metrics such as the bit error rate ber and the outage probability especially for small system dimensions li proposed convincingly to assume that the snr follows a generalized gamma distribution which parameters are tuned by computing the first three asymptotic moments of the snr in this paper this technique is generalized to receive correlated channels and closed form expressions for the first three asymptotic moments of the snr are provided to obtain these results a random matrix theory technique adapted to matrices with gaussian elements is used this technique is believed to be simple efficient and of broad interest in wireless communications simulations are provided and show that the proposed technique yields in general a good accuracy even for small system dimensions

correlation tests and linear spectral statistics of the sample correlation matrix testing the independence of the entries of multidimensional gaussian observations is a very important problem in statistics with a number of applications in signal processing radar cognitive radio seismography and multiple other fields typically the problem is formulated as a binary hypothesis test whereby the presence of correlation is declared when the value of a certain statistic is higher than a certain predetermined threshold most of the statistics for correlation tests are constructed from the sample correlation matrix also known as sample coherence matrix in signal processing which is defined as a power normalized version of the sample covariance matrix in this paper correlation tests constructed from linear spectral statistics lss of the sample correlation matrix are analyzed under the asymptotic framework where both sample size and observation dimension become large but comparable in magnitude a central limit theorem clt is established on this class of statistics which is valid for generally correlated gaussian observations results show that lss asymptotically fluctuate as gaussian random variables under both the hypotheses with an asymptotic mean and variance that can be established for each particular test in particular this general clt can be used to establish the asymptotic behavior of two of the most important correlation test statistics namely the generalized likelihood ratio test and the frobenius norm test under both null and alternative hypotheses as a by product it is established that lss of sample covariance and sample correlation matrices have exactly the same first order behavior but quite different asymptotic fluctuations in the second order regime in both the cases the lss asymptotically behave as gaussian random variables although with quite different asymptotic means and variances

sublinear capacity scaling laws for sparse mimo channels recent attention on performance analysis of single user multiple input multiple output mimo systems has been on understanding the impact of the spatial correlation model on ergodic capacity in most of these works it is assumed that the statistical degrees of freedom dof in the channel can be captured by decomposing it along a suitable eigenbasis and that the transmitter has perfect knowledge of the statistical dof with an increased interest in large antenna systems in state of the art technologies these implicit channel modeling assumptions in the literature have to be revisited in particular multiantenna measurements have showed that large antenna systems are sparse where only a few dof are dominant enough to contribute towards capacity thus in this work it is assumed that the transmitter can only afford to learn the dominant statistical dof in the channel the focus is on understanding ergodic capacity scaling laws in sparse channels unlike classical results where linear capacity scaling is implicit sparsity of mimo channels coupled with a knowledge of only the dominant dof is shown to result in a new paradigm of sublinear capacity scaling that is consistent with experimental results and physical arguments it is also shown that uniform power signaling over all the antenna dimensions is wasteful and could result in a significant penalty over optimally adapting the antenna spacings in response to the sparsity level of the channel and transmit snr

a generalized spatial correlation model for 3d mimo channels based on the fourier coefficients of power spectrums previous studies have confirmed the adverse impact of fading correlation on the mutual information mi of two dimensional 2d multiple input multiple output mimo systems more recently the trend is to enhance the system performance by exploiting the channels degrees of freedom in the elevation which necessitates the derivation and characterization of three dimensional 3d channels in the presence of spatial correlation in this paper an exact closed form expression for the spatial correlation function scf is derived for 3d mimo channels the proposed method resorts to the spherical harmonic expansion she of plane waves and the trigonometric expansion of legendre and associated legendre polynomials the resulting expression depends on the underlying arbitrary angular distributions and antenna patterns through the fourier series fs coefficients of power azimuth and elevation spectrums the novelty of the proposed method lies in the scf being valid for any 3d propagation environment the developed scf determines the covariance matrices at the transmitter and the receiver that form the kronecker channel model in order to quantify the effects of correlation on system performance the information theoretic deterministic equivalents of the mi for the kronecker model are utilized in both mono user and multi user cases numerical results validate the proposed analytical expressions and elucidate the dependence of the system performance on azimuth and elevation angular spreads and antenna patterns some useful insights into the behavior of mi as a function of downtilt angles are provided the derived model will help evaluate the performance of correlated 3d mimo channels in the future

risk convergence of centered kernel ridge regression with large dimensional data this paper carries out a large dimensional analysis of a variation of kernel ridge regression that we call centered kernel ridge regression ckrr also known in the literature as kernel ridge regression with offset this modified technique is obtained by accounting for the bias in the regression problem resulting in the old kernel ridge regression but with centered kernels the analysis is carried out under the assumption that the data is drawn from a gaussian distribution and heavily relies on tools from random matrix theory rmt under the regime in which the data dimension and the training size grow infinitely large with fixed ratio and under some mild assumptions controlling the data statistics we show that both the empirical and the prediction risks converge to a deterministic quantities that describe in closed form fashion the performance of ckrr in terms of the data statistics and dimensions inspired by this theoretical result we subsequently build a consistent estimator of the prediction risk based on the training data which allows to optimally tune the design parameters a key insight of the proposed analysis is the fact that asymptotically a large class of kernels achieve the same minimum prediction risk this insight is validated with both synthetic and real data

the second order coding rate of the mimo quasi static rayleigh fading channel the second order coding rate of the multiple input multiple output mimo quasi static rayleigh fading channel is studied we tackle this problem via an information spectrum approach and statistical bounds based on recent random matrix theory techniques we derive a central limit theorem clt to analyze the information density in the regime where the block length n and the number of transmit and receive antennas k and n respectively grow simultaneously large this result leads to the characterization of closed form upper and lower bounds on the optimal average error probability when the coding rate is within mathcal o 1 sqrt nk of the asymptotic capacity

impact of antenna correlation on the capacity of multiantenna channels this paper applies random matrix theory to obtain analytical characterizations of the capacity of correlated multiantenna channels the analysis is not restricted to the popular separable correlation model but rather it embraces a more general representation that subsumes most of the channel models that have been treated in the literature for arbitrary signal to noise ratios snr the characterization is conducted in the regime of large numbers of antennas for the low and high snr regions in turn we uncover compact capacity expansions that are valid for arbitrary numbers of antennas and that shed insight on how antenna correlation impacts the tradeoffs among power bandwidth and rate

further results on the asymptotic mutual information of rician fading mimo channels rician fading multiple input multiple output mimo channels with kronecker decomposable covariance structure are considered in this study to derive information measures characterizing their achievable rate analytic results are obtained in the asymptotic setting corresponding to infinitely many transmit and receive antennas whose numbers approach a finite limiting ratio more specifically the asymptotic mean and variance of the mutual information are derived in this asymptotic regime and it is shown that the corresponding probability distribution converges to a gaussian distribution all these results are based on the replica method whose mathematical principles are discussed in this study the ergodic capacity is also addressed by deriving an iterative water filling algorithm yielding the capacity achieving input covariance matrix under an average power constraint resorting to the asymptotic gaussianity of the mutual information distribution an analytic expression of the outage capacity is derived based on the mean and variance of the mutual information finally numerical results are reported to assess the level of approximation of the analytic results in comparison with those obtained by monte carlo simulation