Deployments of millimeter wave systems suffer from increased propagation and blockage losses that lead to a focus on small cell coverage. With an increased densification of such systems, cooperative transmissions are possible at a user equipment (UE) from multiple transmission reception points (TRPs) and with repeater nodes (e.g., relay UEs, intelligent reflecting surfaces, reflectarrays) in the vicinity of a certain UE. In this context, this work proposes and studies the performance of a class of directional beam steering and co-phasing schemes that leverage the existing set of beam steering solutions for point-to-point transmissions. To benchmark the performance of the proposed schemes, an unrealizable upper bound to the SNR is first established. We then quantify the SNR gap between the proposed directional schemes and the unrealizable upper bound via theory and numerical studies and show that the proposed solutions are of low complexity and comparable performance to the unrealizable upper bound.

Hand and body blockage are known to result in significant performance degradation at millimeter wave carrier frequencies. A number of prior works have proposed statistical models to capture the spatial region lost due to blockage and the associated losses incurred in signal strength over this region. In particular, it is understood that while a hard hand grip can lead to substantial losses, a more relaxed/looser hand grip can lead to moderate losses. However, all these works consider only the signal strength degradation with blockage mitigation mechanisms consisting of densification of the network and the use of more antenna modules at the user equipment end. In practice, the number of antenna modules in a user equipment may be limited leading to the need for more novel blockage mitigation strategies. Towards this goal, in this work, we show that the presence of the hand on top/near an antenna module can distort the observed amplitude and phase of the array response. We also show that a non-directional analog beamforming codebook enhancement can de-randomize some of the phase distortions and recover some of the gains lost due to the hand with a directional beamforming codebook.

The ongoing commercial deployment of millimeter wave systems brings into focus a number of practical issues in form factor user equipment (UE) design. With wavelengths becoming smaller, antenna gain patterns becoming directional, and link budgets critically dependent on beamforming, it becomes imperative to use a number of antenna modules at different locations of the UE for good performance. While more antennas/modules can enhance beamforming array gains, it comes with the trade-off of higher component cost, power consumption of the associated radio frequency circuitry, and a beam management overhead in learning the appropriate beam weights. Thus, the goal of a good UE design is to provide robust spherical coverage corresponding to good array gains over the entire sphere around the UE with a low beam management overhead, complexity, and cost. The scope of this paper is to study the implications of two popular commercial millimeter-wave UE designs (a face and an edge design) on spherical coverage. We show that the analog beam codebooks can result in good performance for both the designs, and the edge design provides a better trade-off in terms of robust performance (with hand blockage), beam management overhead, implementation complexity from an antenna placement standpoint, and cost.

There has been a strong interest in understanding hybrid precoding tradeoffs for millimeter wave (mmW) multi-input multi-output (MIMO) systems. A common assumption in most of these works is that the analog part of the hybrid precoder can only be designed with phase shifters. The consequent search for analog and digital precoding matrices is solved with different black box-type optimization algorithms. In contrast, this work motivates an analog precoding structure at the base-station end that can be realized with both phase shifters and gain controls. Such a structure is necessary for interference management in multi- user transmissions and is easily realized with low complexity and cost. We then propose a feedback framework of the top-P beams over a beam alignment phase from each user. This framework allows the base-station to reconstruct the channel matrix between it and each user, and to manage interference with a simple zeroforcing solution. Such a structured solution is implemented with the amplitude and phase control of the analog part of the hybrid precoder. We finally illustrate the performance improvement with the proposed solution over simpler constructions such as beam steering that can be implemented with phase shifters alone.

The focus of this paper is on the activity profiles of certain networks, such as terrorist networks, that show frequent spurts and downfalls. Of particular interest is the quick detection of changes in two specific activity patterns corresponding to resilience and level of coordination in the network. Understanding changes in resilience and coordination could provide insights into the underlying organizational dynamics and aid in more informed decision-making. Prior work in tackling this problem is based on parametric approaches and relies on models developed with time-series analysis techniques, self-exciting hurdle models, or hidden Markov models. While these approaches detect spurts and downfalls reasonably accurately, they are all based on model learning-a task that is computationally difficult in practice because of the “rare” nature of terrorist attacks from a model learning perspective. In this paper, we pursue an alternate statistical nonparametric approach for spurt detection in activity profiles. Our approach is based on binning the count data of activity to form observation vectors that can be compared with each other. Motivated by a majorization theory framework, these vectors are then transformed via certain functionals and used in spurt detection and classification. While the parametric approaches often result in either a large number of missed detections of real changes or false alarms, the proposed approach is shown to result in a small number of missed detections and false alarms. Furthermore, since spurt detection is a problem of importance across multiple applications, the nonparametric nature of the approach makes it attractive in these applications.

We propose two novel, low-complexity techniques for beam alignment in time division duplexing (TDD) multiple-input multiple-output (MIMO) systems. The techniques are inspired by the power method, an iterative algorithm to determine eigenvalues and eigenvectors through repeated matrix multiplications and improve upon this simple idea by providing a better performance in the low-SNR regime. The first technique sequentially constructs a least-squares estimate of the channel matrix, which is then used to calculate the optimal beamformer/combiner pair. The second technique aims to mitigate the effect of additive noise by using a linear combination of the previously tried beams to calculate the next beam in the iteration. Simulation results provide insight on the performance of both algorithms in the presence of noise and compare them with similar techniques from the literature.

This paper studies the broadcast channel with M antennas at the base-station and M single antenna users. Most of the works in the literature assume perfect channel state information at both ends of each link. In this work, we assume that these links are spatially correlated and only the long-term statistical information corresponding to the covariance matrices of the links are available at both ends. We are interested in the design of beamforming vectors for data transmission to the M users to maximize the ergodic sum-rate obtained by treating interference as noise. In the simpler M = 2 setting, our prior work has shown the optimality of a generalized eigenvector beamformer structure for this problem. However, these results are not easily extendable to the general M user setting. We overcome these difficulties by first establishing a tractable approximation for the ergodic sum-rate in terms of the beamforming vectors and covariance matrices that is asymptotically tight in M. We then cast the sum-rate approximation maximization problem as a manifold optimization and illustrate the optimality of the generalized eigenvector structure in the general M user setting.

Given the cost and complexity associated with the deployment of a large number of radio frequency (RF) chains in millimeter wave (mmW) multi-input multi-output (MIMO) systems, this paper addresses the question of network efficiency considerations for hybrid precoding. We first establish the relevance of directional precoding structures as a low-complexity and robust solution to meet the data rate demands of single-user MIMO systems relative to the more complex and less robust eigen-mode-based precoding structures. Key to the relevance of the directional precoding structures is the sparsity of the mmW channel coupled with higher antenna dimensionality affordable due to smaller wavelengths. We then leverage the directional structure of the channel to propose a simple class of directional schedulers that offers a low-complexity and yet approximately fair separation plane in the user space. We then compare the performance of single-user precoding schemes with multi-user precoding schemes and show that from network efficiency considerations, it would be more worthwhile to expend the RF chain resource on multi-user transmissions.

The focus of this work is on cost and complexity- induced rank-constrained precoding for millimeter- wave (mmW) multi-input multi-output (MIMO) systems. The higher antenna dimensionality and sparse channel structure of mmW systems motivate directional precoding schemes over more complex and less robust eigen-mode-based precoding schemes. Building on a prior work that illustrated the superiority in design, robustness and performance of a directional rank-1 beamformer, this work considers higher-rank directional precoding structures for mmW systems. It is shown that directional precoders continue to enjoy similar properties in single-user MIMO (SU-MIMO) systems. But from a network efficiency consideration, it would be more worthwhile to expend the radio frequency (RF) chain resource on multi-user transmissions. Towards this goal, a directional scheduler and different directional beamforming structures are proposed here. These structures are shown to result in significant performance improvement over higher-rank SU-MIMO schemes.

The focus of this paper is on beamforming in a millimeter-wave (mmW) multi-input multi-output (MIMO) set-up that has gained increasing traction in meeting the high data-rate requirements of next- generation wireless systems. For a given MIMO channel matrix, the optimality of beamforming with the dominant right-singular vector (RSV) at the transmit end and with the matched filter to the RSV at the receive end has been well-understood. When the channel matrix can be accurately captured by a physical (geometric) scattering model across multiple clusters/paths as is the case in mmW MIMO systems, we provide a physical interpretation for this optimal structure: beam steering across the different paths with appropriate power allocation and phase compensation. While such an explicit physical interpretation has not been provided hitherto, practical implementation of such a structure in a mmW system is fraught with considerable difficulties (complexity as well as cost) as it requires the use of per-antenna gain and phase control. This paper characterizes the loss in received SNR with an alternate low-complexity beamforming solution that needs only per-antenna phase control and corresponds to steering the beam to the dominant path at the transmit and receive ends. While the loss in received SNR can be arbitrarily large (theoretically), this loss is minimal in a large fraction of the channel realizations reinforcing the utility of directional beamforming as a good candidate solution for mmW MIMO systems.

A linear beamformer design for the broadcast channel where the base station is equipped with Ntantennas and signals to M = 2 users (each with a single antenna) and where the vector channels are spatially correlated is considered. This problem is of relevance in wireless standardization efforts where two users are simultaneously scheduled (instead of the theoretically feasible Ntuser scheduling) to minimize signaling overhead. The users are assumed to have perfect channel state information (CSI), whereas the base station has statistical information of the channels. In the first part of this work, the role of the relevance of feedback is studied via the quantification of the gap in ergodic sum-rate between the perfect CSI and statistics-only extremes. In this direction, the importance of orthogonality of the dominant eigenmodes of the two users in maximizing the gap is established. In the second part of this work, in scenarios with a large gap, limited feedback beamforming codebooks and a codeword selection metric are designed for the low-rate feedback setting. The main contribution here is the proposal of a generalized eigenvector codebook and feedback of the codeword index that maximizes an estimate of the signal-to-interference-and-noise ratio (SINR) of each user. Extensions of this scheme are also proposed for the general Ntantenna case with M users, where M ≤ Nt. It is shown via numerical studies that this scheme leads to significant performance improvement across a large family of channels over schemes such as Grassmannian/Random Vector Quantization/single-user codebooks with the channel projection metric for codeword selection.

The focus of this work is on developing probabilistic models for temporal activity of users in social networks (e.g., posting and tweeting) by incorporating the social network influence as perceived by the user. Although prior work in this area has developed sophisticated models for user activity, these models either ignore social network influence completely or incorporate it in an implicit manner. We overcome the nontransparency of the network in the model at the individual scale by proposing a coupled hidden Markov model (HMM), where each user's activity evolves according to a Markov chain with a hidden state that is influenced by the collective activity of the friends of the user. We develop generalized Baum-Welch and Viterbi algorithms for parameter learning and state estimation for the proposed framework. We then validate the proposed model using a significant corpus of user activity on Twitter. Our numerical studies show that with sufficient observations to ensure accurate model learning, the proposed framework explains the observed data better than either a renewal process-based model or a conventional (uncoupled) HMM. We also demonstrate the utility of the proposed approach in predicting the time to the next tweet. Finally, clustering in the model parameter space is shown to result in distinct natural clusters of users characterized by the interaction dynamic between a user and his network.

The broadcast problem with multiple antennas at the transmitter end and a single antenna at each user is studied in this work. A low-complexity transmitter architecture where linear beamforming is used to convey information to the users, and multi-user interference is treated as noise at the receiver unit of each user is considered. The vector channels that connect the transmitter to the users are assumed to be spatially correlated. The goal of this work is the design of a low-rate limited feedback framework to improve the achievable sum-rate of the system relative to a baseline scenario where only correlation information is known at the transmitter. While prior work considers either a codebook structure that is designed for spatially uncorrelated channels or for spatially correlated point-to-point systems, it is argued here that a “good” codebook design should incorporate a statistical version of the interference structure that is seen at each user. Further, instead of feeding back the codeword index that is best aligned to the user's channel vector, the index that maximizes an estimate of the signal-to-interference-and-noise ratio as perceived by the user is fed back. It is shown via numerical results that the proposed framework results in a substantial performance improvement over existing schemes across a large family of spatial correlation and across a practically relevant operating regime.

A Rayleigh fading spatially correlated broadcast setting with$M = 2$antennas at the transmitter and two users (each with a single antenna) is considered. It is assumed that the users have perfect channel information about their links, whereas the transmitter has only statistical information of each user's link (covariance matrix of the vector channel). A low-complexity linear beamforming strategy that allocates equal power and one spatial eigenmode to each user is employed at the transmitter. Beamforming vectors on the Grassmann manifold that depend only on statistical information are to be designed at the transmitter to maximize the ergodic sum-rate delivered to the two users. Toward this goal, the beamforming vectors are first fixed and a closed-form expression is obtained for the ergodic sum-rate in terms of the covariance matrices of the links. This expression is nonconvex in the beamforming vectors ensuring that the classical Lagrange multiplier technique is not applicable. Despite this difficulty, the optimal solution to this problem is shown to be the same as the solution to the maximization of an appropriately defined average signal-to-interference and noise ratio metric for each user. This solution is the dominant generalized eigenvector of a pair of positive-definite matrices where the first matrix is the covariance matrix of the forward link and the second is an appropriately designed “effective” interference covariance matrix. In this sense, our work is a generalization of optimal signalling along the dominant eigenmode of the transmit covariance matrix in the single-user case. Finally, the ergodic sum-rate for the general broadcast setting with$M$antennas at the transmitter and$M$-users (each with a single antenna) is obtained in terms of the covariance matrices of the links and the beamforming vectors.

The ensemble properties of random vector quantization (RVQ) codebooks for limited-feedback beamforming in multiinput multioutput (MIMO) systems are studied with the metrics of interest being the received \ssr SNR loss and mutual information loss, both relative to a perfect channel state information (CSI) benchmark. The simplest case of unskewed codebooks is first studied in the correlated MIMO setting and these loss metrics are computed as a function of the number of bits of feedback ( B), transmit antenna dimension ( Nt), and spatial correlation. In particular, it is established that: 1) the loss metrics are a product of two components-a quantization component and a channel-dependent component; 2) the quantization component, which is also common to analysis of channels with i.i.d. fading, decays as B increases at the rate 2-B/(Nt-1); 3) the channel-dependent component reflects the condition number of the channel. Further, the precise connection between the received \ssr SNR loss and the squared singular values of the channel is shown to be a Schur-convex majorization relationship. Finally, the ensemble properties of skewed codebooks that are generated by skewing RVQ codebooks with an appropriately designed fixed skewing matrix are studied. Based on an estimate of the loss expression for skewed codebooks, it is established that the structure of a good skewing matrix is critically dependent on the condition numbers of the effective channel (product of the true channel and the skewing matrix) and the skewing matrix.

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.

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.

The focus of this work is on the two user interference channel with two antennas at each transmitter and a single antenna at both users. We assume that both the transmitters use a linear beamforming scheme for signaling and the users treat interference as noise. The links are assumed to be spatially correlated and the focus is on understanding the structure of the beamforming vectors that maximize the average sum-rate as a function of the statistics of all the channels and the SNR level at both the transmitters. Building on recent work that studies Pareto optimality in a similar context, we show that (from the class of linear beamforming schemes) a rank-1 beamformer is always optimal when both the transmitters are of low-SNR. In general, the rank of the optimal beamforming matrix is a complicated function of the SNRs and channel statistics. Nevertheless, it is shown that the high-SNR optimal rank-1 beamformer is the generalized eigenvector of the two links as seen from each transmitter.

A spatially correlated broadcast setting with M antennas at the base station and M users (each with a single antenna) is considered. We assume that the users have perfect channel information about their links and the base station has only statistical information about each user's link. The base station employs a linear beamforming strategy with one spatial eigen-mode allocated to each user. The goal of this work is to understand the structure of the beamforming vectors that maximize the ergodic sum-rate achieved by treating interference as noise. In the M = 2 case, we first fix the beamforming vectors and compute the ergodic sum-rate in closed-form as a function of the channel statistics. We then show that the optimal beamforming vectors are the dominant generalized eigenvectors of the covariance matrices of the two links. It is difficult to obtain intuition on the structure of the optimal beamforming vectors for M > 2 due to the complicated nature of the sum-rate expression. Nevertheless, in the case of asymptotic M, we show that the optimal beamforming vectors have to satisfy a set of fixed-point equations.

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.

Recent attention in quickest change detection in the multisensor setting has been on the case where the densities of the observations change at the same instant at all the sensors due to the disruption. In this work, a more general scenario is considered where the change propagates across the sensors, and its propagation can be modeled as a Markov process. A centralized, Bayesian version of this problem is considered, with a fusion center that has perfect information about the observations anda prioriknowledge of the statistics of the change process. The problem of minimizing the average detection delay subject to false alarm constraints is formulated in a dynamic programming framework. Insights into the structure of the optimal stopping rule are presented. In the limiting case of rare disruptions, it is shown that the structure of the optimal test reduces to thresholding thea posterioriprobability of the hypothesis that no change has happened. Under a certain condition on the Kullback-Leibler (K-L) divergence between the post- and the pre-change densities, it is established that the threshold test is asymptotically optimal (in the vanishing false alarm probability regime). It is shown via numerical studies that thislow-complexitythreshold test results in a substantial improvement in performance overnaivetests such as a single-sensor test or a test that incorrectly assumes that the change propagates instantaneously.

Codebooks based on random vector quantization (RVQ) are popular in limited feedback beamforming applications over MIMO channels because of their low-complexity design properties. The goal of this work is on understanding the performance of an ensemble of RVQ codebooks as a function of the number of bits of feedback (B), antenna dimensions, and spatial correlation. We analyze the case of correlated MIMO channels and arbitrary choice of B. Towards this goal, we first study the distribution function of weighted norms of isotropically distributed beamforming vectors. From this, we compute the received SNR loss and mutual information loss of a B-bit RVQ scheme relative to a perfect channel information benchmark. Our computation reveals the following: (i) The loss terms are a product of two factors. The first factor, which is also common to analysis of i.i.d. channels, decays as B increases at the rate 2-B/(Nt-1)where Ntis the number of transmit antennas; (ii) The second factor reflects the condition number of the channel. A channel that minimizes/maximizes the condition number on average also minimizes/maximizes the performance loss, respectively. Such behavior is typical of channels that correspond to rich (i.i.d.) spatial scattering, and poor (rank-1 channels) spatial scattering, respectively.

Attention in quickest change detection in the multi-sensor setting has been on the case where the densities of observations change at the same instant at all the sensors due to the disruption. In this work, a more general change process scenario is studied where the change propagates across the sensors, and its propagation can be modeled as a Markov process. A centralized, Bayesian version of this problem is considered, with a fusion center that has perfect information about the observations and a priori knowledge of the statistics of the change process. Insights into the structure of the optimal stopping rule are presented, that minimizes average detection delay subject to false alarm constraints. In the limiting case of rare disruptions, it is shown that this test structure reduces to a threshold rule. The asymptotic optimality (in the vanishing false alarm regime) of this threshold test is established under a certain condition on the Kullback-Leibler (K-L) divergence between the post- and the pre-change densities. In the special case of near-instantaneous change propagation across the sensors, this condition reduces to the mild condition that the K-L divergence be positive. Numerical studies show that this low-complexity threshold test results in a substantial improvement in performance over naive tests such as a single-sensor test or a test that incorrectly assumes that the change propagates instantaneously.

Recent attention on correlated multi-input multi-output systems has centered around the case of imperfect channel or statistical information at the transmitter. The focus of this work is on correlated channels with arbitrary antenna array geometry and spacings, a coherent receiver, and imperfect statistical knowledge at the transmitter. Leveraging a recently proposed channel modeling paradigm that exploits processing in the angular domain, we first elucidate the structure of the optimal schemes when 'genie-aided' perfect statistical information is available at the transmitter. In the low-S N R case, we then show that the beamforming scheme that is optimal when perfect statistical information is available does not degrade smoothly with imperfections in the statistical information. We then go on to show that there exists a certain low-complexity beamforming approach which, while being sub-optimal in the genie-aided case, is robust to statistical feedback as well as the dynamics of its evolution.

Many recent works that study the impact of spatial correlation on the performance of multi-input multi-output (MIMO) systems assume a separable (also known as the Kronecker) model where the variances of channel entries, upon decomposition onto the transmit and the receive eigen-bases, admit a separable form. If the true statistics of the channel coefficients are non-separable, the separability assumption leads to a flattening and spreading of the degrees of freedom (DoF) in the channel, and hence results in misleading estimates of capacity - an observation consistent with many measurement campaigns. Towards understanding this observation, we first elucidate the importance of channel power normalization, an often-ignored notion, in the capacity analysis of correlated channels. Using tools from random matrix theory, we characterize the mismatch in estimating capacity with the separable model, and thus provide a theoretical underpinning behind many measurement-based observations.

Recent results in the wideband/low-SNR regime show that even one bit of perfect feedback is sufficient to achieve the same rate of capacity scaling as in the benchmark case of perfect CSI at the transmitter. However, the capacity achieving signals are peaky in the non-coherent scenario when CSI is estimated at the receiver. Signal peakiness is related to channel coherence and recent measurement campaigns show that, in contrast to previous assumptions, wideband channels exhibit a sparse multipath structure that naturally leads to coherence in time and frequency. With perfect receiver CSI, we show that limited feedback, even with an instantaneous power constraint, is sufficient to achieve the benchmark capacity in sparse multipath channels. Our analysis reveals the benefits of channel sparsity in the non-coherent scenario, where we employ a training-based signaling scheme. With an average power constraint, it is shown that the benchmark is achievable, provided the channel coherence scales at a sufficiently fast rate with signal space dimension. Furthermore, in contrast to peaky signaling schemes that violate instantaneous power constraints, we show that the benchmark is attainable in sparse channels even with finite instantaneous transmit power. We present rules of thumb on choosing the signaling parameters as a function of the channel parameters so that the full benefits of sparsity can be realized.

The focus of this paper is on spatial precoding in correlated multi-antenna channels where the number of independent data-streams can be adapted to trade off the data rate with the transmitter complexity. A structured precoding scheme is proposed, where the precoder structure evolves fairly slowly at a rate comparable with the statistical evolution of the channel, and in addition, enjoys low-complexity. A particular case of the proposed scheme, semiunitary precoding, is shown to be near- optimal in matched channels where the dominant eigenvalues of the transmit covariance matrix are well-conditioned and their number equals the number of independent data-streams, and the receive covariance matrix is also well-conditioned. In mismatched channels where the above conditions do not hold, it is shown that the loss in performance with semiunitary precoding when compared with a perfect channel information benchmark is substantial. This loss can be mitigated via limited feedback techniques that provide partial channel information to the transmitter. We also develop matching metrics 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 or error probability performance.

Space-time codes leverage the availability of multiple antennas to enhance the reliability of communication over wireless channels. While space-time codes have initially been designed with a focus on open-loop systems, recent technological advances have enabled the possibility of low-rate feedback from the receiver to the transmitter. The focus of this paper is on the implications of this feedback in a single-user multi-antenna system with a general model for spatial correlation. We assume a limited feedback model, that is, a coherent receiver and statistical knowledge at both the ends, along with B bits of error-free quantized channel information at the transmitter. We study space-time coding with a family of linear dispersion (LD) codes that meet an additional orthogonality constraint so as to ensure low-complexity decoding. Our results show that, when the number of bits of feedback (B) is small, a space-time coding scheme that is equivalent to beamforming and does not code across time is optimal in a weak sense in that it maximizes the average received SNR. As B increases, this weak optimality transitions to optimality in a strong sense that is characterized by the maximization of average mutual information. Thus, from a system designer's perspective, our work suggests that beamforming may not only be attractive from a low-complexity viewpoint, but also from an information-theoretic viewpoint.

The design of spatio-temporal power allocation schemes is considered for space-time coding over spatially correlated multiple-input multiple-output (MIMO) channels. The focus is on linear dispersion (LD) space-time codes that are constructed to maximize the mutual information between the input of the space-time encoder and the output of the channel. While perfect channel state information (CSI) is assumed at the receiver, three cases are considered for the CSI at the transmitter: 1) perfect CSI is available, 2) only statistical CSI is available, and 3) partial CSI in the form of a B-bit quantized channel information along with the statistical information is available. In all the three cases, it is shown that the optimal temporal power allocation is uniform. The optimal spatial power allocation for the case where only statistical CSI is available was studied previously in [1] where it was shown to be a nontrivial function of the spatial correlation. Here, the cases of perfect and partial CSI are studied. For the perfect CSI case, it is shown that it is optimal to excite only one spatial mode. For the partial and statistical CSI cases, the optimal allocation excites multiple modes, in general. However, it is attractive to use a low-complexity scheme that excites only the dominant spatial mode. We show that this low-complexity scheme is near-optimal in two settings: 1) large receive antenna asymptotics, and 2) for fixed antenna dimensions, when the transmit and receive covariance matrices are ill- and well-conditioned, respectively. Based on the optimal schemes for the extreme cases of perfect and statistical CSI, low-complexity spatial power allocation for the case of partial CSI is considered. Simulation results indicate that even in this case, exciting one spatial mode leads to a minimal loss in performance over the optimal spatial power allocation scheme.

Technological advances in radio-frequency (RF) front-ends, such as reconfigurable antenna arrays, afford a new "hardware" dimension for dynamic spectrum access in cognitive wireless networks. In this paper, we study the impact of reconfigurable antenna arrays on maximizing the spectral efficiency of multiple input multiple output (MIMO) wireless communication links. Physical wireless channels exhibit spatial correlation and there is growing experimental evidence that sparse multipath is a key source of such correlation. We propose a model for sparse multipath channels and show that sparsity affords a new dimension over which MIMO capacity can be optimized: the distribution or configuration of the sparse statistically independent degrees of freedom (DoF) in the available spatial signal space dimensions. We provide an explicit characterization of the optimal capacity-maximizing channel configurations as a function of the operating SNR. We then develop a framework for realizing the optimal channel configuration at any SNR by systematically adapting the antenna spacings at the transmitter and the receiver. In contrast to a fixed-array MIMO system in which the multiplexing gain is lost at low SNRs, such adaptively reconfigurable MIMO systems deliver the maximum multiplexing gain at all SNRs. Surprisingly, three canonical array configurations are sufficient for near-optimum performance over the entire SNR range. Numerical results based on a realistic physical model are presented and implications of this work for cognitive radio and dynamic spectrum access are discussed.

We consider a spatially correlated multi-user communication system with Ntantennas at the base station (BS), U users (Ntof whom are serviced), and each user with a single antenna. While the users are assumed to know their channels perfectly, the BS has neither instantaneous channel state information (CSI) nor the spatial statistics corresponding to any of the users. We consider a limited feedback model where each user conveys B bits of partial channel information to the BS. Due to the lack of statistical information at the BS, a codebook based on random vector quantization (RVQ) and tailored to uncorrected Rayleigh fading is used in signaling. Our first contribution is to characterize the impact of spatial correlation (in particular, the eigenvalues of the transmit covariance matrix) on the performance of an RVQ scheme. We also show that spatial correlation has significant consequences on practical aspects of limited feedback systems like storage and construction of codebooks. While independent RVQ codebooks have to be generated for the different users in the i.i.d. case, under certain relaxed assumptions on the correlation, the same codebook can be reused across all the users. This is because the spatial signature induced by the correlation can be exploited to distinguish the quantizers of different users.

The optimal input covariance matrix Q that achieves the ergodic capacity under the coherent assumption in a point-to-point, multi-antenna setting is a function of the spatial correlation and the transmit SNR. While the eigenvectors of Q can be characterized in closed-form for many realistic correlation models, the eigenvalues have to be determined numerically. However, it is well-known that the rank of Q is a non-decreasing function of SNR. Motivated by this fact, in this work, we study communication with a low-complexity family of input covariance matrices that are characterized by their rank, assuming uniform power allocation over the smaller-dimensional eigen-space. We quantify the impact of spatial correlation on the M-th transition-SNR which is defined as the smallest SNR at which rank-M transmission becomes optimal.

The full diversity gain provided by a multi-antenna channel can be achieved by transmit beamforming and receive combining. This requires the knowledge of channel state information (CSI) at the transmitter which is difficult to obtain in practice. Quantized beamforming where fixed codebooks known at both the transmitter and the receiver are used to quantize the CSI has been proposed to solve this problem. Most recent works focus attention on limited feedback codebook design for the uncorrelated Rayleigh fading channel. Such designs are sub-optimal when used in correlated channels. In this paper, we propose systematic codebook design for correlated channels when channel statistical information is known at the transmitter. This design is motivated by studying the performance of pure statistical beamforming in correlated channels and is implemented by maps that can rotate and scale spherical caps on the Grassmannian manifold. Based on this study, we show that even statistical beamforming is near-optimal if the transmitter covariance matrix is ill-conditioned and receiver covariance matrix is well-conditioned. This leads to a partitioning of the transmit and receive covariance spaces based on their conditioning with variable feedback requirements to achieve an operational performance level in the different partitions. When channel statistics are difficult to obtain at the transmitter, we propose a universal codebook design (also implemented by the rotation-scaling maps) that is robust to channel statistics. Numerical studies show that even few bits of feedback, when applied with our designs, lead to near perfect CSI performance in a variety of correlated channel conditions.

Transmit beamforming with receive combining is a low-complexity solution that achieves the full diversity afforded by a multi-antenna channel. Building on our recent result which shows that even channel statistics are sufficient to achieve perfect feedback performance (in the limit of antenna dimensions) with beamforming and combining, we propose near-optimal codebook designs for correlated channels with a focus on few antennas at the transmitter and the receiver. In the process, we refine the answer to the question: When are channel statistics sufficient to achieve near perfect feedback performance? We show that the condition number of the transmit and receive covariance matrices hold the key to this question. We partition the transmit and receive covariance spaces into 4 regions based on well and ill-conditioning of the covariance matrices and show that the number of bits required for near perfect feedback performance is dependent on the condition numbers of these matrices

While the intense research on multi-antenna (MIMO) wireless communication channels was pioneered by results based on an i.i.d. channel model representing a rich multipath environment, there is growing experimental evidence that physical wireless channels exhibit a sparse multipath structure, even at relatively low antenna dimensions. In this paper, we propose a model for sparse multipath channels and study coherent MIMO capacity as a function of SNR for a fixed number of antennas. In a recent work, we had shown that the spatial distribution of the sparse multipath has a significant impact on capacity and had also characterized the optimal distribution (the ideal MIMO channel) that maximizes capacity at any operating SNR. In this paper, we refine these results and develop a framework for maximizing MIMO capacity at any SNR by systematically adapting the array configurations (antenna spacings) at the transmitter and receiver to the level of sparsity. Surprisingly, three canonical array configurations are sufficient for near-optimum performance over the entire SNR range. In a scattering environment with randomly distributed paths, the capacity gain due to optimal configuration is directly proportional to the number of antennas at low SNR's. Numerical results based on a realistic physical model are presented to illustrate capacity gains with reconfigurable antenna arrays

Recent works on the distribution function of the capacity of independent and identically distributed (i.i.d.) and semicorrelated narrowband channels show that the outage capacity computed using a Gaussian approximation is close to the true outage capacity even for few antennas. Motivated by physical scattering considerations, we study a multi-antenna channel model with independent entries that are not necessarily identically distributed and show the weak convergence of capacity to a Gaussian random variable. The channel model considered in this paper subsumes well-studied cases like the i.i.d. and separable correlation models and thus we generalize previous results on weak convergence of multi-antenna capacity. Using recent results from random matrix theory, we also study the rate of convergence of ergodic capacity of i.i.d. channels to its limit value. Employing a well-known conjecture from random matrix theory, we establish tight results for the rate of convergence and show a dependence of this rate on signal-to-noise ratio

Existing works on MIMO channel capacity assume that the channel power scales quadratically with antenna dimensions. In a recent work, we introduced a framework for sub-quadratic channel power scaling to model physical MIMO channels, and studied a family of channels with different spatial distributions of channel power. In particular, we introduced the notion of a beamforming channel, realizable in practice via the use of reconfigurable antenna arrays, that yields the highest capacity and maximal spectral efficiency in the low SNR regime. In this paper, we show that the MIMO beamforming channel requires a significantly smaller coherence requirement to achieve the wideband capacity limit at finite bandwidth than has been known in the literature. Contrasting the recent claims that training-based signaling schemes are strictly sub-optimal over communication schemes that do not perform explicit training, we show that training offers significant advantages in achieving wideband capacity. The price of explicitly training the channel over a non-explicit training scheme is a bandwidth penalty, albeit at a significantly less stringent coherence requirement on the channel

Transmit beamforming and receive combining are low complexity techniques that help in achieving the full diversity afforded by the multi-antenna channel. Complete channel state information at the transmitter may be possible only under rare instances. Thus quantized beamforming with limited feedback on a reverse link has been a topic that has attracted great attention recently. But almost all of the work to date has focussed on modeling the channel with independent, identically distributed (i.i.d.) Rayleigh fading between antenna pairs. In this paper, we consider the correlated channel case, and show that Grassmannian line packing is an artificial artifact of the i.i.d. assumption. We show that there are dominant peaks in the eigen-domain when correlation is imposed and the codebook construction should be matched to the correlation in the channel, which renders the problem in this case easier than for i.i.d. channels

Existing results on MIMO channel capacity assume a rich scattering environment in which the channel power scales quadratically with the number of antennas, resulting in linear capacity scaling with the number of antennas. However, such scaling in channel power is physically impossible indefinitely. We thus address the following fundamental question: for a given channel power scaling law, what is the best achievable capacity scaling law? For a channel power scaling, /spl rho//sub c/(N) = O(N/sup /spl gamma//), /spl gamma/ /spl isin/ (0,2], we argue that the channel capacity cannot scale faster than C(N) = O(/spl radic/(/spl rho//sub c/(N))) = O(N/sup /spl gamma//2/). Our approach is based on a family of space-time channels corresponding to different distributions of channel power in the spatial signal space dimensions. We develop the concept of an ideal MIMO channel that achieves the optimal scaling law for a given /spl rho//sub c/(N). For a given number of antennas, unlike existing results that either emphasize the low or high SNR regimes, we propose a methodology for capacity-optimal signaling at any SNR. The methodology is based on creating the ideal channel from any given physical scattering environment via adaptive-resolution array configurations.

Multi input multi output (MIMO) channels under a particular set of correlated channel assumptions, viz. the product correlation, have been shown to differ from i.i.d. channels essentially in the scaling factor of capacity increase [Chen-Nee Chuah, et al. (2002)]. Using a scattering environment based channel representation, we extend the paradigm of capacity scaling to a regime where it saturates or scales depending on how the environment grows with the number of antennas. We also show how the capacity per dB at low SNR's depends on the channel parameters. Based on some (preliminary) results on the convergence rate of the empirical spectral distribution function of an i.i.d. channel we present a rigorous understanding of the notion of 'fast convergence'.

The capacity of MIMO systems under ideal (i.i.d.) channel conditions has been shown to increase linearly with the number of antennas. Capacity scaling in a correlated channel environment as well but the constant of scaling is shown in Chee-Nee Chuah, D.N. C. Tse, J.M. Khan, R.A. Valanzuela [2002] to be less than that of the i.i.d. scaling factor. Using a channel representation motivated by physical scattering considerations, the virtual channel representation, we show this effect does occur provided the number of effective scattering paths keeps asymptotically increasing with the increase in the number of antennas. Otherwise we show that saturation of capacity occurs. Using random banded matrix theory results, we also show that the effect of correlation is to reduce the effective received SNR according to the degree of correlation. This also yields a closed form expression for the asymptotic capacity of correlated MIMO channels.

The dramatic linear increase in ergodic capacity with the number of antennas promised by multiple-input multiple-output (MIMO) wireless communication systems is based on idealized channel models representing a rich scattering environment. Is such scaling sustainable in realistic scattering scenarios? Existing physical models, although realistic, are intractable for addressing this problem analytically due to their complicated nonlinear dependence on propagation path parameters, such as the angles of arrival and delays. In this paper, we leverage a recently introduced virtual representation of physical models that is essentially a Fourier series representation of wide-band MIMO channels in terms of fixed virtual angles and delays. Motivated by physical considerations, we propose a D-connected model for correlated channels defined by a virtual spatial channel matrix consisting of D nonvanishing diagonals with independent and identically distributed (i.i.d.) Gaussian entries. The parameter D provides a meaningful and tractable measure of the richness of scattering. We derive general bounds for the coherent ergodic capacity and investigate capacity scaling with the number of antennas and bandwidth. In the large antenna regime, we show that linear capacity scaling is possible if D scales linearly with the number of antennas. This, in turn, is possible if the number of resolvable paths grows quadratically with the number of antennas. The capacity saturates for linear growth in the number of paths (fixed D). The ergodic capacity does not depend on frequency selectivity of the channel in the wide-band case. Increasing bandwidth tightens the bounds and hastens the convergence of scaling behavior. For large bandwidth, the capacity scales linearly with the signal-to-noise ratio (SNR) as well. We also provide an explicit characterization of the wide-band slope recently proposed by Verdu. Numerical results are presented to illustrate the key theoretical results.