Robust and efficient design of uniform linear arrays (ULA) with non-uniform weights for multi-user massive MIMO systems is formulated as a constrained min-max optimization problem to minimize the number of antennas while meeting performance target. Chebyshev beamforming weights are shown to be optimal for this problem in line-of-sight (LoS) channels when precise channel state information (CSI) of interfering users is not available. Based on this, explicit closed-form solutions are obtained for sum as well as per-user interference constraints, which reveal the scaling of the minimal number of antennas with target signal-to-interference plus noise ratio and the number of interfering users. Comparison with widely-used uniformweighting arrays highlights the superior scalability and efficiency of the proposed Chebyshev-based design, particularly in complex interference environments.

The wiretap channel with additive (possibly non-Gaussian) noise and rate-limited help, available at the legitimate receiver (Rx) or/and transmitter (Tx), is studied under various channel configurations (degraded, reversely degraded and non-degraded) and power/amplitude constraints. For all channel configurations, the rate-limited Rx help results in a (weak or strong) secrecy capacity boost equal to the help rate. This holds irrespective of whether the help is secure or not, or whether the helper is aware of the message being transmitted or not; the secrecy of help or helper’s knowledge of the message does not provide any extra capacity boost. The secrecy capacity is positive for the reversely-degraded channel (where the no-help secrecy capacity is zero) and no wiretap coding is needed to achieve it under weak secrecy. The same capacity boost also holds if non-secure help is available to the transmitter (encoder), in addition to or instead of the same Rx help, so that, in the case of the joint Tx/Rx help, one help link can be omitted without affecting the capacity. If Rx/Tx help links are independent of each other, the capacity boost is the sum of help rates and no link can be omitted without loss in the capacity. Non-singular correlation of the receiver and eavesdropper noises does not affect the secrecy capacity and non-causal help does not bring in any capacity increase over the causal one. The choice of the secrecy criterion (weak/strong) affects the complexity of implementation but not the secrecy capacity. Stronger noise at the legitimate receiver can sometimes result in higher secrecy capacity.

The impact of random errors (implementation inaccuracies) in element locations and beamforming phases on favorable propagation (FP) in massive multiple-input multiple-output (MIMO) line-of-sight (LOS) channels is studied. For arbitrary array geometry and under independent (possibly non-Gaussian) errors, the FP property is shown to hold for the perturbed array as long as it holds for the unperturbed one. This means that small errors do not have catastrophic impact on the FP, even for a large number of antennas. The negative impact of random errors is to slow down the convergence to the asymptotic value so that more antennas are needed under random errors to achieve the same low inter-user interference as without errors. For large but finite number of antennas, the distribution and an analytically-tractable approximation of the inter-user interference power are obtained. Practical design guidelines are given that quantify the accuracy level needed to make the impact of random errors negligible. The analytical results are validated via numerical simulations and are in agreement with measurement-based studies.

The impact of random errors in element locations and beamforming phases on the performance of massive multipleinput multiple-output (MIMO) systems and their ability to cancel inter-user interference (IUI) are studied. For an arbitrary array geometry, user orthogonality, also known as “favorable propagation” (FP), is shown to hold asymptotically for the perturbed array as long as it holds for the unperturbed one, for independent (possibly non-Gaussian) errors. This means that small errors do not have catastrophic impact on the FP, even for a large number of antennas, and IUI can be reduced to any desired level. The negative impact of random errors is to slow down the convergence to the asymptotic value so that more antennas are needed under random errors to achieve the same low IUI as without errors. Practical design guidelines are given as to what implementation accuracy is needed to make the impact of random errors negligible and a closed-form estimate of IUI under random errors is presented. The analytical results are validated via numerical simulations and are in agreement with measurement-based studies.

The Gaussian wiretap channel (WTC) with rate-limited help, available at the transmitter/encoder (Tx), in addition to or instead of the same help at the legitimate receiver, is studied under various channel configurations. For the degraded or reversely-degraded WTC, rate-limited non-secure Tx help results in a secrecy capacity boost equal to the help rate irrespective of whether the help is causal or not. For the non-degraded WTC, the secrecy capacity boost is lower bounded by the help rate. A capacity-achieving signaling is two-phase time sharing, where wiretap coding without help is used in Phase 1 and help without wiretap coding is used in Phase 2. The secrecy capacity with Tx help is positive for the reversely-degraded channel (where the no-help secrecy capacity is zero) and no Phase 1 is needed to achieve it. Unlike the no-help case, more noise at the legitimate receiver can sometimes result in higher secrecy capacity with Tx help. In the case of the joint Tx/Rx non-secure help, one help link can be omitted without affecting the capacity.

Multi-user massive MIMO (mMIMO) systems with uniform linear arrays (ULA) are considered in the non-asymptotic regime. To reduce the complexity of implementation, ULA’s design is optimized to minimize the number of antennas subject to signal-to-noise plus interference (SINR) constraints. Both per-user and sum interference constraints are considered. While the resulting optimization problems are not convex (so that the standard tools of convex optimization cannot be used), a novel analytical approach is proposed and a number of globally-optimal closed-form solutions/designs are obtained. They reveal the scalings of the optimal number of antennas with the target SINR, the total number of users and their angular separation. The proposed designs are robust, since they do not require a precise knowledge of directions of arrival of interfering users, and allow for distributed implementation.

Intelligent reflective surfaces (IRS) have recently emerged as a significant enhancement to 5/6G systems to improve their energy and spectral efficiencies at reasonable cost. IRS-assisted single-input multiple-output (SIMO) or multiple-input single-output (MISO) systems are considered in this paper. While no globally-optimal solutions are known to the IRS phase shift optimization problem in the general case (in part, due to its non-convex nature), a number of closed-form solutions are obtained here for some special cases, which show that the globally-optimal IRS gain scaling with the number of its elements can be either linear or quadratic. Upper/lower bounds to the globally-optimal IRS gain are established in the general case, which are tight for many channels. Based on this, a global optimality gap is obtained for the alternating optimization algorithm (which is less than 1 dB for many channels). Extensive numerical experiments validate the analytical results and demonstrate the tightness of the proposed bounds.

The Gaussian wiretap channel with rate-limited help available at the legitimate receiver (decoder) is studied under various channel configurations (degraded, reversely degraded and non-degraded). In all considered cases but one, the rate-limited help results in a secrecy capacity boost equal to the help rate. This holds irrespective of whether the help is secure or not, so that secure help does not provide any advantage over non-secure one. The secrecy capacity is positive for the reversely-degraded channel (where the no-help secrecy capacity is zero) and no wiretap coding is needed to achieve it. More noise at the legitimate receiver can sometimes result in higher secrecy capacity. The same secrecy capacity boost also holds if non-secure help is available to the transmitter (encoder), in addition to or instead of the receiver help.

We show that the nonfeedback capacity of multiple-input multiple-output (MIMO) additive Gaussian noise (AGN) channels, when the noise is nonstationary and unstable, is characterized by an asymptotic optimization problem–the per unit time limit of the characterization of a finite block or transmission without feedback information (FTwFI) capacity, that involves two generalized matrix difference Riccati equations (DREs) of filtering theory, and a matrix difference Lyapunov equation of stability theory, of Gaussian systems. Further, we identify conditions and prove, that the characterization of nonfeedback capacity is the uniform asymptotic per unit time limit, over all initial distributions. The asymptotic characterization of capacity involves two generalized matrix algebraic Riccati equations (AREs) and a matrix algebraic Lyapunov equation. We also present an example to illustrate that our characterization of capacity produces a known closed-form expression of the water-filling solution of capacity (for power levels above a minimum power).

The problem of optimal power allocation for parallel Gaussian channels under modulation order constrains, in addition to the total transmit power constraint, is considered. It is motivated by coded-modulation systems using powerful capacity-approaching codes. While only analytically-intractable solution is known to this problem, an explicit closed-form solution is obtained here using a sphere-packing-based approximation for modulation-constrained rates. It can be interpreted as waterfilling with variable water level, which is also expressed in a closed-form. The obtained power allocation also solves the dual problem of minimizing the total transmit power subject to the sum rate and modulation order constraints. More insightful analytical solutions are obtained in some special cases. While the new power allocation is similar to the well-known waterfilling procedure at low SNR, it is dramatically different at moderate to high SNR. Proportional cardinality allocation is shown to be optimal at high SNR under the uniform power allocation.

In this paper we characterize the feedback capacity of Additive Gaussian Noise (AGN) channels driven by stable and unstable autoregressive noise, for time-invariant feedback codes (channel input distributions). For stable (resp. unstable) channel noise we identify necessary and sufficient conditions for the optimal input process to induce asymptotic stationarity and ergodicity of the channel output (resp. innovations) process. We call this the ergodic feedback capacity. From our characterization follows the surprising result: for a time-invariant unit memory Gaussian autoregressive noise AR(c), c ∈ (-∞, ∞), (i) feedback does not increase capacity for the region with c ∈ (-1, 1) and certain unstable c, and total transmit power κ ∈ [0,%), and (ii) feedback increases capacity for the compliment of the region of values of (c, κ), not covered in (i).

In this paper we employ the information structures of optimal channel inputs of feedback capacity of additive Gaussian noise (AGN) channels, driven by stable and unstable autoregressive noise, to derive achievable rates for nofeedback capacity, and the corresponding channel input processes. The expressions of rates are derived using time-domain methods, they hold for stable and unstable noise, for all values of transmit power. The method avoids power spectral techniques and their limitations. The lower bounds are evaluated against the classical nonfeedback capacity obtained via water-filling frequency-domain techniques.

Gaussian MIMO channel under total transmit and multiple interference power constraints (TPC and IPCs) is considered. A closed-form solution for its optimal transmit covariance matrix is obtained in the general case (up to dual variables). A number of more explicit closed-form solutions are obtained in some special cases, including full-rank and rank-1 (beamforming) solutions, which differ significantly from the well-known water-filling solutions (e.g. signaling on the channel eigenmodes is not optimal anymore and the capacity can be zero for non-zero transmit power). A whitening filter is shown to be an important part of optimal precoding under interference constraints. Capacity scaling with transmit power is studied: its qualitative behaviour is determined by a natural linear-algebraic structure induced by MIMO channels of multiple users. A simple rank condition is given to characterize the cases where spectrum sharing is possible. An interplay between the TPC and IPCs is investigated, including the transition from power-limited to interference-limited regimes. A number of unusual properties of an optimal covariance matrix under IPCs are pointed out and a bound on its rank is established. Partial null forming known in the adaptive antenna array literature is shown to be optimal from the information-theoretic perspective as well in some cases.

Gaussian MIMO channel under the joint transmit (Tx) and interference power constraints (TPC and IPC) is studied. A closed-form solution for the optimal Tx covariance matrix in the general case is obtained using the KKT-based approach, up to dual variables. A number of more explicit closed-form solutions are given with optimal dual variables, including full-rank and rank-1 (beamforming) cases as well as the case of identical eigenvectors (typical for massive MIMO settings), which differer significantly from the standard water-filling solution. A “whitening” filter is shown to be an important part of optimal precoding under interference constraints. Sufficient and necessary conditions for each constraint to be redundant are given. Capacity scaling with the SNR is shown to be determined by a natural linear-algebraic structure of sub-spaces induced by channel matrices of multiple users. A number of unusual properties of optimal Tx covariance matrix under the joint TPC and IPC are pointed out and a bound on its rank is established. An interplay between the TPC and IPC is studied, including the transition from power-limited to interference-limited regimes as the Tx power increases. While closed-from solutions for optimal dual variables are given in some special cases, an iterative bisection algorithm (IBA) is proposed to find optimal dual variables in the general case and its convergence is proved for some special cases. Numerical experiments illustrate its efficient performance. Bounds for the optimal dual variables are given.

Secure signaling over multiple-input multiple-output (MIMO) wiretap channel (WTC) is studied under interference and transmit power constraints. The classical MIMO WTC model is extended to interference-limited scenarios, so that interference to other users does not exceed a given threshold while ensuring simultaneously no information leakage to an eavesdropper. The operational secrecy capacity of the Gaussian MIMO WTC under interference and transmit power constraints is rigorously established in two forms (a non-convex max problem and a convex-concave max-min problem), to which per-antenna power constraints can be added as well. Optimal signaling directions are characterized in the general case, from which (tight) upper bounds to the rank of optimal transmit covariance matrix are derived. A sufficient condition for the optimality of beamforming and a necessary condition for optimal full-rank signaling are given. Closed-form rank-1 and high-rank solutions are obtained in the case of zero interference constraints. Sufficient and necessary conditions for non-zero secrecy capacity are established. The results are extended to multi-user scenarios. A sufficient and necessary condition for the unbounded growth of the secrecy capacity with transmit power is obtained. The interplay between transmit and interference power constraints is studied, and its significant impact on optimal signaling is demonstrated (so that neither constraint can be absorbed into the other one in general, as was sometimes suggested in the literature). Overall, these results provide insights into fundamental information-theoretic limits and optimal signaling strategies for secure communications under interference constraints.

A method is developed to realize optimal channel input conditional distributions, which maximize the finite transmission feedback information (FTFI) capacity, often called $n$ -block length feedback capacity, by information lossless randomized strategies. The method is applied to compute closed form expressions for the FTFI capacity and feedback capacity, of nonstationary, nonergodic, unstable, multiple input multiple output Gaussian channels with memory on past channel outputs, subject to average transmission cost constraints of quadratic form in the channel inputs and outputs. It is shown that randomized strategies decompose into two orthogonal parts-an deterministic part, which controls the channel output process, and an innovation part, which transmits new information over the channel. Then a separation principle is shown between the computation of the optimal deterministic part and the random part of the optimal randomized strategies. Finally, the ergodic theory of linear-quadratic-Gaussian stochastic optimal control theory, is applied to identify sufficient conditions, expressed in terms of solutions to matrix difference and algebraic Riccati equations, so that the optimal control part of randomized strategies induces asymptotic stationarity and ergodicity, and feedback capacity is characterized by the per unit time limit of the FTFI capacity. The method reveals an interaction of the control and the information transmission parts of the optimal randomized strategies, and that whether feedback increases capacity, is directly related to the channel parameters and the transmission cost function, through the solutions of the matrix Riccati equations. For unstable channels, it is shown that feedback capacity exists and it is strictly positive, provided the power exceeds a critical threshold.

Feedback capacity is extended beyond classical communication channels, to stochastic dynamical systems, which may correspond to unstable control systems or unstable communication channels, subject to average cost constraints of total power κ ∈ [0, ∞). It is shown that optimal conditional distributions or randomized strategies, have a dual role, to simultaneously control the output process and to encode information. The dual role is due to the interaction of control and information transmission; it states that encoders in communication channels operate as encoders-controllers, while controllers in control systems operate as controllers-encoders. The concepts are illustrated through the analysis of Gaussian control systems with randomized strategies, which are equivalent to Additive Gaussian Noise channels, Stable or Unstable, with arbitrary memory on past outputs, with an average constraint of quadratic form. It is shown that such unstable dynamical systems have Control-Coding Capacity which is operational, precisely as in Shannon's operational definition. However, the control-coding capacity is zero, unless the power κ allocated to the system, exceeds a threshold Kmin, where Kminis the minimum cost of ensuring asymptotic stability and ergodicity. The excess power κ - Kminis turned into an achievable rate of information transmission over the dynamical system.

The capacity of a fixed Gaussian multiple-input multiple-output (MIMO) channel and the optimal transmission strategy under the total power (TP) constraint and full channel state information are well known. This problem remains open in the general case under individual per-antenna (PA) power constraints, while some special cases have been solved. These include a full-rank solution for the MIMO channel and a general solution for the multiple-input single-output (MISO) channel. In this paper, the fixed Gaussian MISO channel is considered and its capacity and optimal transmission strategies are determined in a closed form under the joint total and PA power constraints in the general case. In particular, the optimal strategy is hybrid and includes two parts: first is equal-gain transmission and second is maximum-ratio transmission, which are responsible for the PA and TP constraints, respectively. The optimal beamforming vector is given in a closed form and an accurate yet simple approximation to the capacity is proposed. Finally, the above results are extended to the MIMO case by establishing the ergodic capacity of fading MIMO channels under the joint power constraints when the fading distribution is right unitary-invariant (of which i.i.d. and semi-correlated Rayleigh fading are special cases). Unlike the fixed MISO case, the optimal signaling is shown to be isotropic in this case.

The capacity of a fixed Gaussian MIMO channel and the optimal transmission strategy under the total power (TP) constraint and full channel state information are well-known. This problem remains open in the general case under individual per-antenna (PA) power constraints, while some special cases have been solved. These include a full-rank solution for the MIMO channel and a general solution for the MISO channel. In this paper, the Gaussian MISO channel is considered and its capacity as well as optimal transmission strategies are determined in a closed form under the joint total and per-antenna power constraints in the general case. In particular, the optimal strategy is hybrid and includes two parts: first is equal-gain transmission and second is maximum-ratio transmission, which are responsible for the PA and TP constraints respectively. The optimal beamforming vector is given in a closed-form and an accurate yet simple approximation to the capacity is proposed.

A discrete compound channel with memory is considered, where no stationarity, ergodicity or information stability is required, and where the uncertainty set can be arbitrary. When the discrete noise is additive but otherwise arbitrary and there is no cost constraint on the input, it is shown that the causal feedback does not increase the capacity. This extends the earlier result obtained for general channels with full transmitter (Tx) channel state information (CSI). It is further shown that, for this compound setting and under a mild technical condition on the additive noise, the addition of the full Tx CSI does not increase the capacity either, so that the worst-case and compound channel capacities are the same, thus revealing a saddle-point property.

Capacity-achieving signaling strategies for the Gaussian wiretap multiple-input multple-output (MIMO) channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of a weak eavesdropper is considered in detail, and the optimal covariance is established in an explicit, closed form with no extra assumptions. This provides lower and upper bounds to the secrecy capacity in the general case with a bounded gap, which are tight for a weak eavesdropper or/and low SNR. Closed-form solutions are also obtained for isotropic and omnidirectional eavesdroppers, based on which lower and upper bounds to the secrecy capacity are established in the general case. Sufficient and necessary conditions for the optimality of three popular transmission techniques, namely, the zero-forcing (ZF), the standard water-filling over the channel eigenmodes, and the isotropic signaling (IS), are established for the MIMO wiretap channel. These solutions are appealing due to their lower complexity. In particular, no wiretap codes are needed for the ZF transmission, and no precoding or feedback is needed for the isotropic signaling.

Optimal signaling over the Gaussian multiple-input multiple-output wire-tap channel is studied under the total transmit power constraint. A closed-form solution for an optimal transmit covariance matrix is obtained when the channel is strictly degraded. In combination with the rank-1 solution, this provides the complete characterization of the optimal covariance for the case of two transmit antennas. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the high-SNR regime. Necessary optimality conditions and a tight upper bound on the rank of an optimal covariance matrix are established for the general case, along with a lower bound to the secrecy capacity, which is tight in a number of scenarios.

A general formula for the capacity of arbitrary compound channels, which are not necessarily ergodic, stationary or information-stable, is obtained using the information density approach. A direct (constructive) proof is given. To prove achievability, we generalize Feinstein Lemma to the compound channel setting, and to prove converse, we generalize Verdu-Han Lemma to the same compound setting. This extends the general formula for channel capacity in [8] to arbitrary compound channels (not necessarily finite-state or countable).

Optimal signaling for secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While this channel has attracted a significant attention recently and a number of results have been obtained, including the proof of the optimality of Gaussian signalling, an optimal transmit covariance matrix is known for some special cases only and the general case remains an open problem. An iterative custom-made algorithm to find a globally-optimal transmit covariance matrix in the general case is developed in this paper, with guaranteed convergence to a global optimum. While the original optimization problem is not convex and hence difficult to solve, its minimax reformulation can be solved via the convex optimization tools, which is exploited here. The proposed algorithm is based on the barrier method extended to deal with a minimax problem at hand. Its convergence to a global optimum is proved for the general case (degraded or not) and a bound for the optimality gap is given for each step of the barrier method. The performance of the algorithm is demonstrated via numerical examples. In particular, 20 to 40 Newton steps are already sufficient to solve the sufficient optimality conditions with very high precision (up to the machine precision level), even for large systems. Even fewer steps are required if the secrecy capacity is the only quantity of interest. The algorithm can be significantly simplified for the degraded channel case and can also be adopted to include the per-antenna power constraints (instead or in addition to the total power constraint). It also solves the dual problem of minimizing the total power subject to the secrecy rate constraint.

Novel matrix singular value inequalities are established for a sum/product of three matrices. Their application to the uncertain (compound) multiple-input multiple-output (MIMO) channel subject to normed additive uncertainty establishes the saddle-point property for a wide range of performance metrics monotonic in the channel singular values, including, among others, the mutual information, MMSE, error exponent, and pairwise error probability. This, in turn, implies that the transmission on the eigenmodes of the nominal (or worst case) channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity. The worst case channel turns out to be antiparallel of the nominal one for all these performance metrics. An application of these results to beamforming over compound MIMO channels is discussed. An optimal robust precoder for the uncertain MIMO channel is obtained in a closed-form under the sum-MSE criterion and the total power constraint. The saddle-point property is shown to hold and the optimal strategy is to diagonalize the nominal (or worst case) channel.

Strong secrecy capacity of compound wiretap channels is studied. The known lower bounds for the secrecy capacity of compound finite-state memoryless channels under discrete alphabets are extended to arbitrary uncertainty sets and continuous alphabets under the strong secrecy criterion. The conditions under which these bounds are tight are given. Under the saddle-point condition, the compound secrecy capacity is shown to be equal to that of the worst-case channel. Based on this, the compound Gaussian multiple-input multiple-output wiretap channel is studied under the spectral norm constraint and without the degradedness assumption. First, it is assumed that only the eavesdropper channel is unknown, but is known to have a bounded spectral norm (maximum channel gain). The compound secrecy capacity is established in a closed form and the optimal signaling is identified. The compound capacity equals the worst-case channel capacity and thus establishing the saddlepoint property; the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it. More general uncertainty sets are considered and the existence of a maximum element is shown to be sufficient for a saddle-point to exist, so that signaling on the worst-case channel achieves the compound capacity of the whole class of channels. The case of rank-constrained eavesdropper is considered and the respective compound secrecy capacity is established. Subsequently, the case of additive uncertainty in the legitimate channel, in addition to the unknown eavesdropper channel, is studied. Its compound secrecy capacity and the optimal signaling are established in a closed form as well, revealing the same saddle-point property. When a saddle-point exists under strong secrecy, strong and weak secrecy compound capacities are equal.

Capacity-achieving signaling strategies for the Gaussian wiretap MIMO channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of weak eavesdropper is considered in details and the optimal covariance is established in an explicit, closed-form with no extra assumptions. The conditions for optimality of zero-forcing signaling are established, and the standard water-filling is shown to be optimal under those conditions. No wiretap codes are needed in this case. The case of identical right singular vectors for the required and eavesdropper channels is studied and the optimal covariance is established in an explicit closed form. As a by-product of this analysis, we establish a generalization of celebrated Hadamard determinantal inequality using information-theoretic tools.

Optimum ordering strategies for the coded Vertical Bell Labs Layered Space-Time (V-BLAST) architecture with capacity achieving temporal codes on each stream are analytically studied, including 4 different power/rate allocation strategies among data streams. Compact closed-form solutions are obtained for the case of zero-forcing (ZF) V-BLAST with two transmit antennas and necessary optimality conditions are found for the general case. The optimal rate allocation is shown to have a major impact (stronger streams are detected last) while the optimal power allocation does not alter the original Foschini ordering (stronger streams are detected first). Sufficient conditions for the optimality of the greedy ordering are established: it is optimal for the ZF V-BLAST under an optimal rate allocation with two transmit antennas at any SNR and with any number of antennas in the low and high SNR regimes. It satisfies the necessary optimality conditions for larger systems at any SNR and is nearly-optimal in many cases. An SNR gain of ordering is introduced and studied, including closed-form expressions as well as lower and upper bounds and the conditions for their achievability. For the minimum mean square error (MMSE) V-BLAST under an optimal rate allocation, any ordering is shown to deliver the same system capacity. All the results also apply to a multiple-access channel with the successive interference cancelation receiver.

Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. The recent results are extended in several directions, including a rank-deficient solution for the optimal covariance, lower and upper capacity bounds for the general case, and characterization of optimality of the isotropic signaling. An isotropic eavesdropper model is studied, which provides (tight) upper and lower capacity bounds for the non-isotropic case and also serves as the worst-case scenario. The optimal signaling for this model is obtained in an explicit form and its properties are studied, including the high and low-SNR behavior, the conditions for the eavesdropper to be negligible and the capacity saturation effect.

The compound MIMO Gaussian wiretap channel is studied, where the channel to the legitimate receiver is known and the eavesdropper channel is not known to the transmitter but is known to have a bounded spectral norm (channel gain). The compound secrecy capacity is established without the de-gradedness assumption and the optimal signaling is identified: the compound capacity equals the worst-case channel capacity thus establishing the saddle-point property, the optimal signaling is Gaussian and on the eigenvectors of the legitimate channel and the worst-case eavesdropper is isotropic. The eigenmode power allocation somewhat resembles the standard water-filling but is not identical to it.

Convexity properties of error rates of a class of decoders, including the maximum-likelihood/min-distance one as a special case, are studied for arbitrary constellations, bit mapping, and coding. Earlier results obtained for the additive white Gaussian noise channel are extended to a wide class of noise densities, including unimodal and spherically invariant noise. Under these broad conditions, symbol and bit error rates are shown to be convex functions of the signal-to-noise ratio (SNR) in the high-SNR regime with an explicitly determined threshold, which depends only on the constellation dimensionality and minimum distance, thus enabling an application of the powerful tools of convex optimization to such digital communication systems in a rigorous way. It is the decreasing nature of the noise power density around the decision region boundaries that ensures the convexity of symbol error rates in the general case. The known high/low-SNR bounds of the convexity/concavity regions are tightened and no further improvement is shown to be possible in general. The high-SNR bound fits closely into the channel coding theorem: all codes, including capacity-achieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem), satisfy this high-SNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacity-achieving codes have convex error rates. Convexity properties in signal amplitude and noise power are also investigated. Some applications of the results are discussed. In particular, it is shown that fading is convexity-preserving and is never good in low dimensions under spherically invariant noise, which may also include any linear diversity combining.

The aggregate interference distribution in cognitive radio networks is studied in a rigorous analytical way using the popular Poisson point process model. While a number of results are available for this model of regular (non-cognitive) networks, cognitive ones present an extra level of difficulty for the analysis, mainly due to the exclusion region around the primary receiver, which are typically addressed via various ad-hoc approximations (e.g. based on the interference cumulants) or via the large-deviation analysis. Unlike the previous studies, here we do not use ad-hoc approximations but rather obtain the asymptotic interference distribution in a systematic and rigorous way. This is in contrast to the large deviation analysis, which provides only the (exponential) order of scaling but not the outage probability itself. Unlike the cumulant-based analysis, our approach provides a guaranteed level of accuracy at the distribution tail. Additionally, our analysis also provides a number of novel insights. In particular, we demonstrate that there is a critical transition point below which the outage probability decays only polynomially but above which it decays exponentially. This provides a solid analytical foundation to the earlier empirical observations in the literature and also reveals how typical outage events occur in different regimes. In addition, the proposed asymptotic expressions are also shown to be accurate in the non-asymptotic regimes.

Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. A direct proof of the necessary condition of optimality (signaling on the positive directions of the difference channel) is given using the necessary KKT conditions. Based on it, an explicit, closed-form solution for the optimal transmit covariance matrix is given when the latter is of the full rank. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the latter regime. A refined estimate of the rank of an optimal covariance matrix is given for the general case.

The aggregate interference distribution in cognitive radio networks is studied in a rigorous analytical way using the popular Poisson point process model. While a number of results are available for this model of regular (non-cognitive) networks, cognitive ones present an extra level of difficulties for the analysis, mainly due to the exclusion region around the primary receiver, which are typically addressed via various ad-hoc approximations (e.g. based on the interference cumulants) or via the large-deviation analysis. Unlike the previous studies, we do not use here ad-hoc approximations but rather obtain the asymptotic interference distribution in a systematic and rigorous way, which also has a guaranteed level of accuracy at the distribution tail. This is in contrast to the large deviation analysis, which provides only the (exponential) order of scaling but not the outage probability itself. Unlike the cumulant-based analysis, our approach provides a guaranteed level of accuracy at the distribution tail. Additionally, our analysis provides a number of novel insights. In particular, we demonstrate that there is a critical transition point below which the outage probability decays only polynomially but above which it decays super-exponentially. This provides a solid analytical foundation to the earlier empirical observations in the literature and also reveals what are the typical ways outage events occur in different regimes. The analysis is further extended to include interference cancelation and fading (from a broad class of distributions). The outage probability is shown to scale down exponentially in the number of canceled nearest interferers in the below-critical region and does not change significantly in the above-critical one. The proposed asymptotic expressions are shown to be accurate in the non-asymptotic regimes.

Outage probability and capacity of a class of block fading MIMO channels are considered under partial channel distribution information. Specifically, the channel or its distribution is not known but the latter is known to belong to a class of distributions where each member is within a certain distance (uncertainty) from a nominal distribution. Relative entropy is used as a measure of distance between distributions. Compound outage probability defined as min (over the transmitted signal distribution) -max (over the channel distribution class) outage probability is introduced and investigated. This generalizes the standard outage probability to the case of partial channel distribution information. Compound outage probability characterization (via 1-D convex optimization and in a closed form), its properties, and approximations are given. It is shown to have two-regime behavior: when the nominal outage probability decreases (e.g., by increasing the SNR), the compound outage first decreases linearly down to a certain threshold (related to the relative entropy distance; this is the nominal outage-dominated regime) and then only logarithmically (i.e., very slowly; this is the uncertainty-dominated regime) so that no significant further decrease is possible. This suggests the following design guideline: the outage probability is decreased by increasing the SNR or optimizing the transmitted signal distribution (both decrease nominal outage) in the first regime and by reducing the channel distribution uncertainty (e.g., via better estimation) in the second one. The compound outage depends on the relative entropy distance and the nominal outage only, all other details (nominal fading and noise distributions) being irrelevant. The transmit signal distribution optimized for the nominal channel distribution is shown to be also optimal for the whole class of distributions. The effect of swapping the distributions in relative entropy is investigated and an error floor effect is established. The compound outage probability under Lpdistance constraint is also investigated. The obtained results hold in full generality, i.e., for the general channel model with arbitrary nominal fading and noise distributions.

The compound capacity of uncertain multiple-input multiple-output channels is considered, when the channel is modeled by a class described by a (known) nominal channel and a constrained-norm (unknown) uncertainty. Within this framework, two types of classes are investigated with additive and multiplicative uncertainties subject to a spectral norm constraint, using the singular value decomposition and related singular value inequalities as the main tools. The compound capacity is a maxmin mutual information, representing the capacity of the class, in which the minimization is done over the class of channels while the maximization is done over the transmit covariance. Closed-form solutions for the compound capacity of the classes are obtained and several properties related to transmit and receive eigenvectors are presented. It is shown that, under certain conditions, the compound capacity of the class is equal to the worst-case channel capacity, thus establishing a saddle-point property. Explicit closed-form solutions are given for the worst-case channel uncertainty and the capacity-achieving transmit covariance matrix: the best transmission strategy achieving the compound capacity is a multiple beamforming on the nominal (known) channel eigenmodes with the beam power distribution via the water filling at a degraded SNR. As the uncertainty increases, fewer eigenmodes are used until only the strongest one remains active so that transmit beamforming is an optimal robust transmission strategy in this large-uncertainty regime, for which explicit conditions are given. Using these results, upper and lower bounds of the compound capacity are constructed for other bounded uncertainties and some generic properties are pointed out. The results are extended to compound multiple-access and broadcast channels. In all considered cases, the price to pay for channel uncertainty is an SNR loss (or, equivalently, the nominal channel degradation) commensurate with the uncertainty set radius measured by the spectral norm and the optimal signaling strategy is the transmission on the degraded nominal channel.

Outage probability of a class of block-fading (MIMO) channels is considered under channel distribution uncertainty, when the channel or its distribution are not known but the latter is known to belong to a class of distributions where each member is within a certain distance from a nominal distribution. Relative entropy is used as a measure of distance between distributions. Compound outage probability defined as min (over the input distribution) -max (over the channel distribution class) outage probability is introduced and investigated, which generalizes the standard outage probability to the case of partial channel distribution information. Compound outage probability characterization via one-dimensional convex optimization, its properties and approximations are given. It is shown to have a two-regime behavior: when the nominal outage probability decreases, the compound outage first decreases linearly down to a certain threshold and then only logarithmically (i.e. very slowly), so that no significant further decrease is possible. The input distribution optimized for the nominal channel distribution is shown to be also optimal for the whole class of distributions. The effect of swapping the distributions in relative entropy is investigated and an error floor effect is established. The obtained results hold for a generic channel model (arbitrary nominal fading and noise distributions).

Several optimization strategies for instantaneous rate and/or power allocation in the coded V-BLAST are studied analytically. Outage probabilities and system capacities of these strategies in a spatial multiplexing system are compared under generic settings. Since the conventional waterfilling algorithm is suboptimal for the coded V-BLAST, a recently-proposed “fractional waterfilling” algorithm is studied, which simultaneously maximizes the system capacity and minimizes the outage probability. A comparative, closed-form performance analysis of this and other algorithms is presented, including bounds on the outage probability and its low-outage approximations. The fractional waterfilling algorithm attains the full MIMO channel diversity and outperforms the other algorithms by a wide margin.

An analytical framework for performance analysis and optimization of coded V-BLAST is developed. Average power and/or rate allocations to minimize the outage probability as well as their robustness and dual problems are investigated. Compact, closed-form expressions for the optimum allocations and corresponding system performance are given. The uniform power allocation is shown to be near optimum in the low outage regime in combination with the optimum rate allocation. The average rate allocation provides the largest performance improvement (extra diversity gain), and the average power allocation offers a modest SNR gain limited by the number of transmit antennas but does not increase the diversity gain. The dual problems are shown to have the same solutions as the primal ones. All these allocation strategies are shown to be robust. The reported results also apply to coded multiuser detection and channel equalization systems relying on successive interference cancellation.

Several instantaneous optimization strategies for rate and/or power allocation in the coded V-BLAST are studied analytically. Outage probabilities and system capacities of these strategies in a spatial multiplexing system are compared under generic settings. The conventional waterfilling algorithm is shown to be suboptimal for the coded V-BLAST and a new algorithm ("fractional water-filling") is proposed, which simultaneously maximizes the system capacity and minimizes the outage probability. Closed-form performance analysis of the considered algorithms is given, and the fractional water-filling algorithm is shown to attain the full MIMO channel diversity, significantly outperforming other strategies. Many of the results also apply to generic multi-stream transmission systems (e.g. spatial multiplexing on the channel eigenmodes, OFDM) or the systems relying on successive interference cancelation (multi-user detection, channel equalization).

Fading MIMO relay channels are studied analytically, when the source and destination are equipped with multiple antennas and the relays have a single one. Compact closed-form expressions are obtained for the outage probability under i.i.d. and correlated Rayleigh-fading links. Low-outage approximations are derived, which reveal a number of insights, including the impact of correlation, of the number of antennas, of relay noise and of relaying protocol. The effect of correlation is shown to be negligible, unless the channel becomes almost fully correlated. The SNR loss of relay fading channels compared to the AWGN channel is quantified. The SNR-asymptotic diversity-multiplexing tradeoff (DMT) is obtained for a broad class of fading distributions, including, as special cases, Rayleigh, Rice, Nakagami, Weibull, which may be non-identical, spatially correlated and/or non-zero mean. The DMT is shown to depend not on a particular fading distribution, but rather on its polynomial behavior near zero, and is the same for the simple amplify-and-forward protocol and more complicated decode-and-forward one with capacity achieving codes, i.e. the full processing capability at the relay does not help to improve the DMT. There is however a significant difference between the SNR-asymptotic DMT and the finite-SNR outage performance: while the former is not improved by using an extra antenna on either side, the latter can be significantly improved and, in particular, an extra antenna can be traded-off for a full processing capability at the relay. The results are extended to the multi-relay channels with selection relaying and typical outage events are identified.

An information-theoretic analysis of a multi-keyhole channel, which includes a number of statistically independent keyholes with possibly different correlation matrices, is given. When the number of keyholes or/and the number of Tx/Rx antennas is large, there is an equivalent Rayleigh-fading channel such that the outage capacities of both channels are asymptotically equal. In the case of a large number of antennas and for a broad class of fading distributions, the instantaneous capacity is shown to be asymptotically Gaussian in distribution, and compact, closed-form expressions for the mean and variance are given. Motivated by the asymptotic analysis, a simple, full-ordering scalar measure of spatial correlation and power imbalance in MIMO channels is introduced, which quantifies the negative impact of these two factors on the outage capacity in a simple and well-tractable way. It does not require the eigenvalue decomposition, and has the full-ordering property. The size-asymptotic results are used to prove Telatar's conjecture for semi-correlated multi-keyhole and Rayleigh channels. Since the keyhole channel model approximates well the relay channel in the amplify-and-forward mode in certain scenarios, these results also apply to the latter.

MIMO single-relay fading channels are studied, where the source and destination are equipped with multiple antennas and the relay has a single one. Compact closed-form expressions are obtained for the outage probability under i.i.d. and correlated Rayleigh-fading links. Insightful high-SNR approximations are derived, which show the impact of the number of antennas, correlation, relay noise, relaying protocol, etc. Diversity-multiplexing tradeoff (DMT) is obtained for a broad class of fading distributions, including, as special cases, Rayleigh, Rice, Nakagami, Weibull, which may be non-identical, spatially correlated and/or non-zero mean. The DMT is shown to depend not on a particular fading distribution, but rather on its polynomial behavior near zero. It turns out to be the same for the simple “amplify-and-forward” protocol and more complicated “decode-and-forward” one with capacity achieving codes, i.e. the full processing capability at the relay does not help to improve the DMT. However, we also emphasize significant difference between the SNR-asymptotic DMT and the finite-SNR outage performance: while the former is not improved by using an extra antenna on either side, the latter can be significantly improved and, in particular, an extra antenna can be traded-off for a full processing capability at the relay.

Motivated by a wide-spread use of convex optimization techniques, convexity properties of bit error rate of the maximum likelihood detector operating in the AWGN channel are studied for arbitrary constellations and bit mappings, which also includes coding under maximum-likelihood decoding. Under this generic setting, the pairwise probability of error and bit error rate are shown to be convex functions of the SNR and noise power in the high SNR/low noise regime with explicitly-determined boundary. Any code, including capacity-achieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem) satisfies this high SNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacity-achieving codes have convex error rates.

This paper provides a partial answer to the question: "what is the best angular density of multipath in MIMO channels?" using the size-asymptotic theory of Toeplitz matrices for uniform 2-D and 3-D antenna arrays. A Kronecker-type approximation of the array correlation structure is proposed and used to find the angular densities that completely eliminate correlation between any elements of antenna arrays and thus maximize the asymptotic MIMO capacity for a broad class of fading distributions. At half-wavelength spacing, the best angular density is shown to be non-uniform, which implies that the popular Clarke's (Jake's) model does not represent the best case scenario. The asymptotic results are validated via Monte-Carlo simulations, and a number of practical guidelines for antenna design and optimal orientation are provided.

This paper analyzes the outage probability in cognitive radio networks, based on the Poisson point process model of node spatial distribution and the standard propagation path loss model, including Rayleigh and log-normal fading. To make the analysis tractable, all possible scenarios are classified into three cases based on typical outage events. When the average number of nodes in the forbidden region is much smaller than unity, the aggregate interference can be well approximated by the nearest node for both non-fading and fading scenarios (the nearest node dominates the outage performance). When the average number of nodes in the forbidden region is greater than unity, the aggregate interference can be well approximated by a Gaussian random variable for non-fading scenario (many nodes contribute to outage events, rather than a single dominant one). This approximation also applies to the fading scenario, but its accuracy is a bit worse at the transition region. An alternative approximation is proposed, which is accurate for any outage probability. When the average number of nodes in the forbidden region is slightly smaller than unity, neither the nearest node approximation nor the Gaussian one is accurate for the non-fading scenario (since only a few near-by nodes are dominant), and finding an accurate approximation for the outage probability in this case is an open problem. The alternative approximation above is accurate for the fading scenario. All approximations are validated via Monte-Carlo simulations.

Motivated by a recent surge of interest in convex optimization techniques, convexity/concavity properties of error rates of the maximum likelihood detector operating in the AWGN channel are studied and extended to frequency-flat slow-fading channels. Generic conditions are identified under which the symbol error rate (SER) is convex/concave for arbitrary multidimensional constellations. In particular, the SER is convex in SNR for any one- and two-dimensional constellation, and also in higher dimensions at high SNR. Pairwise error probability and bit error rate are shown to be convex at high SNR, for arbitrary constellations and bit mapping. Universal bounds for the SER first and second derivatives are obtained, which hold for arbitrary constellations and are tight for some of them. Applications of the results are discussed, which include optimum power allocation in spatial multiplexing systems, optimum power/time sharing to decrease or increase (jamming problem) error rate, an implication for fading channels (¿fading is never good in low dimensions¿) and optimization of a unitary-precoded OFDM system. For example, the error rate bounds of a unitary-precoded OFDM system with QPSK modulation, which reveal the best and worst precoding, are extended to arbitrary constellations, which may also include coding. The reported results also apply to the interference channel under Gaussian approximation, to the bit error rate when it can be expressed or approximated as a nonnegative linear combination of individual symbol error rates, and to coded systems.

Diversity-multiplexing tradeoff (DMT) was characterized asymptotically (SNR- > infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse . The SNR-asymptotic DMT overestimates the finite-SNR one . This paper outlines a number of additional limitations and difficulties of the DMT framework and discusses their implications. Using the recent results on the size-asymptotic (in the number of antennas) outage capacity distribution, the finite-SNR, size-asymptotic DMT is derived for a broad class of fading distributions. The SNR range over which the finite-SNR DMT is accurately approximated by the SNR-asymptotic one is characterized. The multiplexing gain definition is shown to affect critically this range and thus should be carefully selected, so that the SNR-asymptotic DMT is an accurate approximation at realistic SNR values and thus has operational significance to be used as a design criterion. The finite-SNR diversity gain is shown to decrease with correlation and power imbalance in a broad class of fading channels, and such an effect is described in a compact, closed form. Complete characterization of the outage probability (or outage capacity) requires not only the finite-SNR DMT, but also the SNR offset, which is introduced and investigated as well. This offset, which is not accounted for in the DMT framework, is shown to have a significant impact on the outage probability for a broad class of fading channels, especially when the multiplexing gain is small. The analytical results and conclusions are validated via extensive Monte Carlo simulations. Overall, the size-asymptotic DMT represents a valuable alternative to the SNR-asymptotic one.

An analytical framework for minimizing the outage probability of a coded spatial multiplexing system while keeping the rate close to the capacity is developed. Based on this framework, specific strategies of optimum power and rate allocation for the coded V-BLAST architecture are obtained and its performance is analyzed. A fractional waterfilling algorithm, which is shown to optimize both the capacity and the outage probability of the coded V-BLAST, is proposed. Compact, closed-form expressions for the optimum allocation of the average power are given. The uniform allocation of average power is shown to be near optimum at moderate to high SNR for the coded V-BLAST with the average rate allocation (when per-stream rates are set to match the per-stream capacity). The results reported also apply to multiuser detection and channel equalization relying on successive interference cancelation.

A statistical model of interference in wireless networks is considered, which is based on the traditional propagation channel model and a Poisson model of random spatial distribution of nodes in 1-D, 2-D and 3-D spaces with both uniform and non-uniform densities. The power of nearest interferer is used as a major performance indicator, instead of a traditionally-used total interference power, since at the low outage region, they have the same statistics so that the former is an accurate approximation of the latter. This simplifies the problem significantly and allows one to develop a unified framework for the outage probability analysis, including the impacts of complete/partial interference cancellation, of different types of fading and of linear filtering, either alone or in combination with each other. When a given number of nearest interferers are completely canceled, the outage probability is shown to scale down exponentially in this number. Three different models of partial cancellation are considered and compared via their outage probabilities. The partial cancellation level required to eliminate the impact of an interferer is quantified. The effect of a broad class of fading processes (including all popular fading models) is included in the analysis in a straightforward way, which can be positive or negative depending on a particular model and propagation/system parameters. The positive effect of linear filtering (e.g. by directional antennas) is quantified via a new statistical selectivity parameter. The analysis results in formulation of a tradeoff relationship between the network density and the outage probability, which is a result of the interplay between random geometry of node locations, the propagation path loss and the distortion effects at the victim receiver.

Various normalizations of the MIMO channel matrix are discussed from a physical perspective. It is demonstrated that the physics of antenna arrays and propagation channel should be taken into account when normalization is chosen, so that SNR has proper physical meaning, the conclusions are physical and correspond to realistic systems. The antenna array geometry and the transmission strategy (coherent/non-coherent) limits the choice of normalization and determines how the capacity and other performance metrics scale with the number of antennas, which is more pronounced for densely-populated antenna arrays. This is especially important for an asymptotic analysis, when the number of antennas increases to infinity. Limitations of such analysis from the physical perspective are pointed out.

The compound capacity of uncertain MIMO channels is considered, when the channel is modeled by a class described by an induced norm constraint. Within this framework, two types of classes are investigated, namely, additive and multiplicative uncertainties subject to a spectral norm constraint, using partial channel state information at the transmitter side. The compound capacity is defined as a maxmin of the mutual information, corresponding to the capacity of the class, in which the minimization is done over the class of channels while the maximization is done over the transmit covariance. Closed form solutions for the compound capacity of the classes are obtained while several properties related to transmit and received eigenvectors are presented. It is also shown that capacity of the class of channels is equal to the worst-case channel capacity, while establishing a saddle-point property. Additionally, explicit closed-from solutions are given for the capacity-achieving Tx covariance matrix and the worst-case channel uncertainty. The effect of uncertainty is shown to be equivalent to an SNR loss which is proportional to the size of the uncertainty of the channel matrix measured by the spectral norm.

A statistical model of interference in wireless networks is considered, which is based on the traditional propagation channel model, a Poisson model of random spatial distribution of the nodes in 1-D, 2-D and 3-D spaces (with both uniform and non-uniform densities), and a threshold-based model of the receiver performance. The power of the dominant interferer is used as a major performance indicator, instead of a traditionally-used aggregate interference power, since the former is an accurate approximation of the latter. This simplifies the problem significantly so that compact closed-form expressions are obtained for the outage probability, including the case when a given number of strongest interferers are suppressed: the outage probability is shown to scale down exponentially in this number. The effect of Rayleigh and log-normal fading can also be included in the analysis. The positive effect of linear filtering (e.g. by directional antennas) is quantified via a new statistical selectivity parameter. The analysis culminates in formulation of an explicit tradeoff relationship between the network density and the outage probability, which is a result of the interplay between random geometry of node locations, the propagation path loss and the distortion effects at the victim receiver.

A unified analytical framework for optimum power allocation in the unordered V-BLAST algorithm and its comparative performance analysis are presented. Compact closed-form approximations for the optimum power allocation are derived, based on average total and block error rates. The choice of the criterion has little impact on the power allocation and, overall, the optimum strategy is to allocate more power to lower step transmitters and less to higher ones. High-SNR approximations for optimized average block and total error rates are given. The SNR gain of optimization is rigorously defined and studied using analytical tools, including lower and upper bounds, high and low SNR approximations. The gain is upper bounded by the number of transmit antennas, for any modulation format and type of fading channel. While the average optimization is less complex than the instantaneous one, its performance is almost as good at high SNR. A measure of robustness of the optimized algorithm is introduced and evaluated. The optimized algorithm is shown to be robust to perturbations in individual and total transmit powers. Based on the algorithm robustness, a pre-set power allocation is suggested as a low-complexity alternative to the other optimization strategies, which exhibits only a minor loss in performance over the practical SNR range.

Keyhole multiple-input-multiple-output (MIMO) channels have recently received significant attention since they can model, to a certain extend, some practically important propagation scenarios and also relay channels in the amplify-and-forward mode. This paper investigates instantaneous signal-to-noise ratio (SNR) and outage capacity distributions of spatially correlated keyhole MIMO channels with perfect channel state information (CSI) at the receive end and with or without CSI at the transmit end. For a small number of antennas, the impact of correlation on the capacity distribution can be characterized by the effective average SNR. This SNR, as well as the outage capacity, decreases with correlation. For a large number of transmit (receive) antennas, the keyhole channel is asymptotically equivalent (in terms of capacity) to the Rayleigh diversity channel with a single transmit (receive) antenna and multiple receive (transmit) antennas. The outage capacity of the keyhole channel is upper-bounded by that of the equivalent Rayleigh diversity channel. When the number of both transmit and receive antennas is large, the outage capacity distribution of the keyhole channel is asymptotically Gaussian. In some cases, the asymptotic Gaussian approximation is accurate already for a reasonably small number of antennas. The perfect transmit CSI is shown to bring a fixed SNR gain. A more general channel model with multiple keyholes is proposed. For a large number of antennas, the capacity of a multikeyhole channel is a normally distributed sum of the capacities of single keyhole channels. The fact that, despite the strong degenerate nature of the keyhole channel, its outage capacity distribution is asymptotically normal indicates that Gaussian distribution has a high degree of universality for the capacity analysis of MIMO channels.

A stronger and general sufficient condition for the asymptotic normality of MIMO channel eigenvalues and its capacity is given. Physical interpretation of this condition is discussed. Simple alternative conditions, which do not require eigenvalue decomposition, are proposed. It is demonstrated that some popular correlation matrix models satisfy these conditions. In many cases, the convergence to the asymptotic normality is at least as 1/radic(nt), wherentis the number of Tx antennas.

Outage and error rate performance of the ordered BLAST with more than 2 transmit antennas is evaluated for i.i.d. Rayleigh fading channels. A number of lower and upper bounds on the 1st step outage probability at any SNR are derived, which are further used to obtain accurate approximations to average block and total error rates. For m Tx antennas, the effect of the optimal ordering at the first step is an m-fold SNR gain. As m increases to infinity, the BLER decreases to zero, which is a manifestation of the space-time autocoding effect in the V-BLAST. While the sub-optimal ordering (based on the before-projection SNR) suffers a few dB SNR penalty compared to the optimal one, it has a lower computational complexity and a 3 dB SNR gain compared to the unordered V-BLAST and can be an attractive solution for low-complexity/low-energy systems. Uncoded D-BLAST exhibits the same outage and error rate performance as that of the V-BLAST. An SNR penalty of the linear receiver interfaces compared to the BLAST is also analytically evaluated.

Comprehensive performance analysis of the unordered V-BLAST algorithm with various power allocation strategies is presented, which makes use of analytical tools and resorts to Monte-Carlo simulations for validation purposes only. High-SNR approximations for the optimized average block and total error rates are given. The SNR gain of optimization is rigorously defined and studied using analytical tools, including lower and upper bounds, high and low SNR approximations. The gain is upper bounded by the number of transmitters, for any modulation format and any type of fading This upper bound is achieved at high SNR by the considered optimization strategies. While the average optimization is less complex than the instantaneous one, its performance is almost as good at high SNR. A measure of robustness of the optimized algorithm is introduced and evaluated, including compact closed-form approximations. The optimized algorithm is shown to be robust to perturbations in individual and total transmit powers. Based on the algorithm robustness, a pre-set power allocation is suggested as a low-complexity alternative to the other optimization strategies, which exhibits only a minor loss in performance over the practical SNR range.

Diversitymultiplexing tradeoff (DMT) presents a compact framework to compare various MIMO systems and channels in terms of the two main advantages they provide (i.e. high data rate and/or low error rate). This tradeoff was characterized asymptotically (SNR-> infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse (2003). The SNR-asymptotic DMT overestimates the finite-SNR one (R. Narasimhan, 2006). In this paper, using the recent results on the size-asymptotic (in the number of antennas) outage capacity distribution, we derive and analyze the finite-SNR DMT for a broad class of channels (not necessarily Rayleigh fading). Systems with unequal number of Tx and Rx antennas exhibit qualitatively-different behavior from those with equal number of antennas: while the size-asymptotic DMT of the latter converges to the SNR- asymptotic DMT as the SNR grows, that of the former does not. However, the size-asymptotic DMT does provide an accurate approximation of the true DMT at low to moderately-high SNR, even for modest number of antennas, and hence is complementary of the SNR-asymptotic DMT of Zheng and Tse. Combining these two, a new DMT is obtained that is accurate over the whole SNR range. A number of generic properties of the DMT that hold at any SNR, for any number of antennas (i.e. not only asymptotic, either in size or in SNR) and for any fading channel are given. In particular, we demonstrate that the linear interpolation of the DMT for fractional multiplexing gain in (Zheng and Tse 2003) does not hold at finite SNR. Extensive Monte-Carlo simulations validate the analysis and the conclusions.

Convexity/concavity properties of symbol error rates (SER) of the maximum likelihood detector operating in the AWGN channel (non-fading and fading) are studied. Generic conditions are identified under which the SER is a convex/concave function of the SNR. Universal bounds for the SER 1st and 2nd derivatives are obtained, which hold for arbitrary constellations and are tight for some of them. Applications of the results are discussed, which include optimum power allocation in spatial multiplexing systems, optimum power/time sharing to decrease or increase (jamming problem) error rate, and implication for fading channels.

Diversity-multiplexing tradeoff (DMT) presents a compact framework to compare various MIMO systems and channels in terms of the two main advantages they provide (i.e. high data rate and/or low error rate). This tradeoff was characterized asymptotically (SNR-> infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse [1]. The asymptotic DMT overestimates the finite-SNR one [2]. In this paper, using the recent results on the asymptotic (in the number of antennas) outage capacity distribution, we derive and analyze the finite- SNR DMT for a broad class of channels (not necessarily Rayleigh fading). Based on this, we give the convergence conditions for the asymptotic DMT to be approached by the finite-SNR one. The multiplexing gain definition is shown to affect critically the convergence point: when the multiplexing gain is defined via the mean (ergodic) capacity, the convergence takes place at realistic SNR values. Furthermore, in this case the diversity gain can also be used to estimate the outage probability with reasonable accuracy. The multiplexing gain definition via the high-SNR asymptote of the mean capacity (as in [1]) results in very slow convergence for moderate to large systems (as l/ln(SNR) circ2) and, hence, the asymptotic DMT cannot be used at realistic SNR values. For this definition, the high-SNR threshold increases exponentially in the number of antennas and in the multiplexing gain. For correlated keyhole channel, the diversity gain is shown to decrease with correlation and power imbalance of the channel. While the SNR-asymptotic DMT of Zheng and Tse does not capture this effect, the size-asymptotic DMT does.

Keyhole MIMO channels were predicted theoretically and also observed experimentally. However, they are not often encountered in practice since the assumption of a single propagation eigenmode is only a rough approximation of real propagation environments. To overcome this problem, the paper extends the single-keyhole channel model by introducing a "multi-keyhole channel", which includes a number of statistically independent keyholes. Correlated full-rank and rank-deficient multi-keyhole channels are considered in detail. It is shown that under some general conditions the full-rank multi-keyhole channel is asymptotically Rayleigh fading, if the number of keyholes is large. When the number of both Tx and Rx antennas is large, the asymptotic capacity of a rank-deficient multi-keyhole channel is a sum of the capacities of the equivalent single-keyhole channels. The outage capacity distribution of both full-rank and rank-deficient multi-keyhole channels is asymptotically Gaussian. Based on the asymptotic capacity analysis, full ordering scalar measure of MIMO channel correlation and power imbalance is introduced

The peak factor of a continuous digitally- modulated signal is often analyzed from its samples taken at the Nyquist rate. This, however, may involve a significant error. It has been claimed, based on an illustrative example, that the peak factor of a continuous signal may be arbitrary large while the peak factor of the corresponding sampled signal is limited [Wulich, D., 2000]. A validity of this example has been questioned in [Ermolova, N., 2001; Minn, E., et al., 2001] based on a flaw in [Wulich, D., 2000]. In this paper, we demonstrate that the original illustrative example requires a small modification only to remove the flaw. It is also demonstrated that the continuous peak factor, in its traditional definition, may be arbitrary large while the sampled peak factor and the signal energy are bounded. An upper bound on the continuous peak factor of a BPSK sequence is derived.

Keyhole MIMO channels, which were predicted theoretically and also observed experimentally, have recently received significant attention as they may appear in some practically-important propagation scenarios. This paper concentrates on a capacity study of such channels. Closed-form expressions for the instantaneous SNR and outage capacity distributions of a spatially correlated keyhole MIMO channel are given. The case of non-singular correlation matrices with distinct eigenvalues is considered in detail. When the number of Tx (Rx) antennas is large, the correlated keyhole channel tends asymptotically to the Rayleigh diversity channel with a single Tx (Rx) and multiple Rx (Tx) antennas. The outage capacity at low outage probabilities and the diversity order of the keyhole channel is upper-bounded by that of the equivalent Rayleigh diversity channel. The asymptotic outage capacity distribution, when the numbers of Tx and Rx antennas are both large, is Gaussian under general conditions on the correlation (the average SNR affects the mean and the correlation affects the variance). The Gaussian approximation is accurate already for a reasonably small number of antennas. Using the single-parameter exponential correlation matrices, we show that the outage capacity at low outage probabilities decreases with correlation

Abstract — The BLAST algorithm is simple and, hence, popular solution for a signal processing at the MIMO receiver. In this paper, we present a closed-form analytical analysis of the V-BLAST without optimal ordering. A result on the zero-forcing maximum ratio combining weights at each detection step is derived to obtain a number of results: independence of noise, distribution of signal to noise ratio and block or bit error rates. We present a detailed analytical analysis and closed-form expressions for instantaneous and average BER at each detection step, which account for the error propagation and hold true for any modulation format and take simple form in some cases (BPSK). Asymptotic form, for large average SNR, of these expressions is especially simple; the effect of the error propagation in this mode is to increase the total average BER by about 20, which is not catastrophic at all. It is demonstrated that the conventional V-BLAST and QR-decomposition based V-BLAST are essentially identical and have the same performance. Extensive Monte-Carlo simulations validate the analytical results and conclusions. Index Terms—MIMO, V-BLAST, multi-antenna system, BER, outage, fading,

The Bell Labs layered space-time (BLAST) algorithm is simple, and hence, a popular choice for a multiple-input multiple-output (MIMO) receiver. Its bit-error rate (BER) performance has been studied mainly using numerical (Monte Carlo) techniques, since exact analytical evaluation presents serious difficulties. Close examination of the problem of BLAST BER performance analysis reveals that the major difficulty for analytical evaluation is due to the optimal ordering procedure. Hence, we analyze the algorithm performance without optimal ordering. While this is a disadvantage of the analysis, there are certain advantages as well. Exact closed-form analytical evaluation is possible for arbitrary number of transmit and receive antennas in an independent, identically distributed Rayleigh fading channel, which provides deep insight and understanding that cannot be gained using the Monte Carlo approach alone. A result on the maximum ratio combining weights, which is used at each detection step, is derived to obtain a number of results: independence of noise, distribution of signal-to-noise ratio (SNR), and block- or bit-error rates. We present a detailed analysis and expressions for uncoded error rates at each detection step, which hold true for any modulation format and take simple closed form in some cases. Asymptotic form of these expressions for large SNRs is particularly simple. Extensive Monte Carlo simulations validate the analytical results and conclusions

The V-BLAST algorithm is an attractive simple solution for the receiver processing of a MIMO system. The paper presents an analytical analysis of some aspects of the algorithm performance over a flat-fading Rayleigh channel. A closed-form rigorous analytical expression for the joint outage probability and its PDF at the 1/sup st/ detection step are derived for the case of the 2/spl times/n system (i.e., 2 transmit and n receive antenna system). Corresponding distribution moments are also evaluated, and asymptotic expressions are given. The analytical results are validated through extensive Monte-Carlo simulations.

The performance of hybrid microdiversity, in the form of generalized selection combining (GSC), and macrodiversity is presented for lognormal shadowed Rayleigh fading channels. The GSC-augmented macrodiversity consists of K ports in a cell, each port carrying N microscopic diversity antennas. The macroscopic diversity involves selecting the port with the highest long-term local mean SNR among the K ports, and the GSC uses n strongest signals of the N branch received signals from that port for processing. We derive analytical expressions for error probability and outage for systems employing this hybrid scheme. The expressions are valid for any configurations of K,N, n. In microcell systems substantial correlation could exist among the ports in contrast to macrocell; results are also shown for correlated lognormal shadowed Rayleigh channels. Extensive simulations are carried out to validate the analytical expressions derived.

Fading correlation has a profound impact on the MlMO system performance. Salz-Winters model is a popular tool to study this effect. However, it is limited to one cluster only. Measurements indicate that multipath often arrives in several clusters. We extend the Salz-Winters model to the case of multi-cluster channels and study it in detail. Closed-form expression for correlations are derived and applied to MIMO capacity/diversity gain analysis. The maximum gain/capacity are achieved provided that a minimum element spacing (derived in the paper) is respected. It is shown that the correlation has an oscillatory behavior as antenna spacing increases; the envelope of correlation is determined by a single cluster angular spread while the oscillations within the envelope are determined by the inter-cluster angular spread. It is demonstrated that the correlation depends significantly on the power distribution among the clusters. In the case of 2 widely-separated clusters, it is possible to orient the antenna array in such a way that the correlation is minimized and, hence, the capacity/gain are maximized. We study this optimization problem and derive the optimum array orientation. Overall, the paper presents a new insight on correlation properties of multipath clustered channels, and on the MIMO system performance over such channels.

A geometrically based analytical approach to the performance analysis of the V-BLAST algorithm is presented in this paper, which is based on the analytical model of the Gramm-Schmidt process. This approach presents a new geometrical view of the V-BLAST and explains some of its properties in a complete and rigorous form, including a statistical analysis of postprocessing signal-to-noise ratios for a 2/spl times/n system (where n is the number of receive antennas). Closed-form analytical expressions of the vector signal at ith processing step and its power are presented. A rigorous proof that the diversity order at ith step (without optimal ordering) is (n-m+i) is given (where m is the number of transmit antennas). It is shown that the optimal ordering is based on the least correlation criterion and that the after-processing signal power is determined by the channel correlation matrices in a fashion similar to the channel capacity. Closed-form analytical expressions are derived for outage probabilities and average BER of a 2/spl times/n system. The effect of the optimal ordering is shown to be to increase the first step SNR by 3 dB (rather than to increase the diversity order as one might intuitively expect based on the selection combining argument) and to increase the second step outage probability twice.

It has been demonstrated that zero correlation of a random multiple-input multiple-output channel is not a guarantee of its high capacity. Degenerate channels exist, which have zero correlation and still low capacity. We provide a statistical analysis of this phenomenon, formulate the general condition for a channel to be degenerate, and propose a method to estimate its capacity.

A universal upper bound on the MIMO architecture capacity, which is not limited to a particular scenario, is derived using the correlation matrix approach and the Jensen's inequality. This bound accounts for both transmit and receive branch correlation in such a way that the impact of these branches can be estimated separately, which simplifies the procedure substantially and also allows one to decide which site is responsible for capacity reduction, which is not easy to do using traditional approaches. Further, using the results above and the Salz-Winters (1994) model of fading spatial correlation, it is demonstrated that the correlation has no impact on the MIMO capacity provided that the two-element antenna array beamwidth is smaller than the angle spread of the incoming multipath signals. A fundamental tradeoff between MIMO capacity and diversity order is also pointed out.

Fade depth prediction on airborne line-of-sight communication links is considered. There is no specific model for this scenario at the moment. The two ray multipath model, adapted to a realistic scenario of hilly or mountainous terrain, was used to account for flight and terrain geometry and for antenna parameters. Surprisingly, it was found that (i) in many practically important cases the fade depth depends on the path clearance angle only and (ii) the two-ray model predicts roughly the same fade depth dependence on the path clearance angle as the well-known Olsen-Segal model. This may be considered as a theoretical justification, to the best of our knowledge for the first time, of the path elevation angle factor in that model.