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A novel analytical framework for evaluating the statistics of the product of independent random variables is proposed. Compared with other methods, which use either infinite series or special functions, the new method provides simple and efficient closed-form approximations, in terms of elementary functions, such as powers and exponentials. Numerical results, which are used to check the accuracy of the new approximations, show that it is quite accurate in most regions of interest. As an application, the proposed analytical results can be efficiently used in wireless communications theory to evaluate in closed form the outage probability of cascaded fading channels, as well as the rate offset of the hybrid automatic repeat request (H-ARQ) transmission. Numerical examples show that the derived expressions provide significant insights on the behaviors of important system parameters as the outage probability and the rate offset. [ABSTRACT FROM PUBLISHER]

Channel estimation at the receiver side is essential to adaptive modulation schemes, prohibiting low complexity systems from using variable rate and/or variable power transmissions. Towards providing a solution to this problem, we introduce a variable-rate (VR) M-PSK modulation scheme, for communications over fading channels, in the absence of channel gain estimation at the receiver. The choice of the constellation size is based on the signal-plus-noise (S+N) sampling value rather than on the signal-to-noise ratio (S/N). It is analytically shown that S+N can serve as an attractive simpler criterion, alternative to S/N, for determining the modulation order in VR systems. In this way, low complexity transceivers can use VR transmissions in order to increase their spectral efficiency under an error performance constraint. As an application, we utilize the proposed VR modulation scheme in equal gain combining (EGC) diversity receivers. [ABSTRACT FROM AUTHOR]

Ascertaining the importance of the dual selection combining (SC) receivers and the suitability of the Weibull model to describe mobile fading channels, we study the performance of a dual SC receiver over correlated Weibull fading channels with arbitrary parameters. Exact closed-form expressions are derived for the probability density function, the cumulative distribution function, and the moments of the output signal-to-noise ratio (SNR). Important performance criteria, such as average output SNR, amount of fading, outage probability, and average bit-error probability for several modulation schemes are studied. Furthermore, for these performance criteria, novel closed-form analytical expressions are derived. The proposed analysis is complemented by various performance evaluation results, including the effects of the input SNR's unbalancing, fading severity, and fading correlation on the overall system's performance. Computer simulation results have verified the validity and accuracy of the proposed analysis. [ABSTRACT FROM AUTHOR]

In this letter, an alternative moments-based approach for the performance analysis of an L-branch predetection equal gain combiner (EGC) over independent or correlated Nakagami-m fading channels is presented. Exact closed-form expressions are derived for the moments of the EGC output signal-to-noise (SNR), while the corresponding moment-generating function (MGF) is accurately approximated with the aid of Padé approximants theory. Important performance criteria are studied; the average output SNR, which is expressed in closed form both for independent and correlative fading and for arbitrary system parameters, the average symbol-error probability for several coherent, noncoherent, and multilevel modulation schemes, and the outage probability, which are both accurately approximated using the well-known MGF approach. The proposed mathematical analysis is illustrated by various numerical results, and computer simulations have been performed to verify the validity and the accuracy of the theoretical approach. [ABSTRACT FROM AUTHOR]

In this letter, an approach to the evaluation of the error performance in dual predetection equal-gain combining (EGC) systems over correlated Nakagami-m fading channels is presented. Deriving an infinite series representation for the characteristic function of the sum of two correlated Nakagami-m variables, a closed-form formula is extracted for binary phase-shift keying and coherent binary frequency-shift keying, while several other modulation schemes are studied, capitalizing on a Parsevals's theorem-based approach, previously published. Numerical results and simulations are also presented to illustrate the proposed mathematical analysis and to point out the effect of the input signal-to-noise ratio unbalancing, the fading severity, and the fading correlation on the system's error performance. [ABSTRACT FROM AUTHOR]

In this letter, an approach to the performance analysis of a triple selection-diversity system over exponentially correlated Nakagami-m fading channels is presented. Closed-form expressions of converged sums for both outage and average error probabilities are derived. Numerical results are presented to point out the effect of the fading correlation, the fading severity, as well as the improvement achieved by the triple selection combining compared with the corresponding dual diversity case. [ABSTRACT FROM AUTHOR]

The Gamma-Gamma (Γ Γ ) distribution has recently attracted the interest of the research community due to its involvement in various communication systems. In the context of RF wireless communications, Γ Γ distribution accurately models the power statistics in composite shadowing/fading channels as well as in cascade multipath fading channels, while in optical wireless (OW) systems, it describes the fluctuations of the irradiance of optical signals distorted by atmospheric turbulence. Although Γ Γ channel model offers analytical tractability in the analysis of single input single output (SISO) wireless systems, difficulties arise when studying multiple input multiple output (MIMO) systems, where the distribution of the sum of independent Γ Γ variates is required. In this paper, we present a novel and simple closed-form approximation for the distribution of the sum of independent, but not necessarily identically distributed Γ Γ variates. It is shown that the probability density function (PDF) of the Γ Γ sum can be efficiently approximated either by the PDF of a single Γ Γ distribution, or by a finite weighted sum of PDFs of Γ Γ distributions. To reveal the importance of the proposed approximation, the performance of RF wireless systems in the presence of composite fading, as well as MIMO OW systems impaired by atmospheric turbulence, are investigated. Numerical results and simulations illustrate the accuracy of the proposed approach. [ABSTRACT FROM AUTHOR]

This is the second part of a two-part series of papers, where the error performance of multidimensional lattice constellations with signal space diversity (SSD) is investigated. In Part I, following a novel combinatorial geometrical approach which is based on parallelotope geometry, we have presented an exact analytical expression and two closed-form bounds for the symbol error probability (SEP) in Additive White Gaussian Noise (AWGN). In the present Part II, we extend the analysis and present a novel analytical expression for the Frame Error Probability (FEP) of multidimensional lattice constellations over Nakagami-m fading channels. As the FEP of infinite lattice constellations is lower bounded by the Sphere Lower Bound (SLB), we propose the Sphere Upper Bound (SUB) for block fading channels. Furthermore, two novel bounds for the FEP of multidimensional lattice constellations over block fading channels, named Multiple Sphere Lower Bound (MSLB) and Multiple Sphere Upper Bound (MSUB), are presented. The expressions for the SLB and SUB are given in closed form, while the corresponding ones for MSLB and MSUB are given in closed form for unitary block length. Numerical and simulation results illustrate the tightness of the proposed bounds which can be efficiently used to set the performance limits on the FEP of lattice constellations of arbitrary structure, dimension and rank. [ABSTRACT FROM PUBLISHER]

In this paper, we analytically investigate Gallager's exponent for space-time block codes over multiple-input multiple-output block-fading channels with Gaussian input distribution. As a suitable metric of the fundamental tradeoff between communication reliability and information rate, Gallager's exponent can be used to determine the required codeword length to achieve a prescribed error probability at a given rate below the channel capacity. We assume that the receiver has full channel state information (CSI), while the transmitter has no CSI and performs equal power allocation across all transmit antennas. In the following, novel exact expressions for Gallager's exponent are derived for two well-known channel fading models, namely η-μ and κ-μ fading models. More importantly, the implications of fading parameters and channel coherence time on Gallager's exponent are investigated. In addition, we present new expressions for the Shannon capacity, cutoff rate and expurgated exponent for the above mentioned fading models, while in the high signal-to-noise ratio regime, simplified closed-form expressions are also derived. Finally, we highlight the fact that the presented analysis encompasses all previously known results on Nakagami-m, Rician, Rayleigh and Hoyt fading channels, as special cases. [ABSTRACT FROM AUTHOR]

This paper elaborates on the ergodic capacity of fixed-gain amplify-and-forward (AF) dual-hop systems, which have recently attracted considerable research and industry interest. In particular, two novel capacity bounds that allow for fast and efficient computation and apply for nonidentically distributed hops are derived. More importantly, they are generic since they apply to a wide range of popular fading channel models. Specifically, the proposed upper bound applies to Nakagami-m, Weibull, and generalized-K fading channels, whereas the proposed lower bound is more general and applies to Rician fading channels. Moreover, it is explicitly demonstrated that the proposed lower and upper bounds become asymptotically exact in the high signal-to-noise ratio (SNR) regime. Based on our analytical expressions and numerical results, we gain valuable insights into the impact of model parameters on the capacity of fixed-gain AF dual-hop relaying systems. [ABSTRACT FROM PUBLISHER]

In this paper, the decremental effect of beamforming with feedback delay on the performance of a two-hop amplify-and-forward (AF) relay network over Rayleigh-fading channels is investigated. An antenna configuration in which the source and the destination are equipped with multiple antennas, whereas the relay is equipped with a single antenna, is assumed. We derive new expressions for the outage probability and the average bit error rate (BER), which are useful for a large number of modulation schemes. To gain further insights, simple outage probability and average BER approximations at high signal-to-noise ratio (SNR) are also presented. It is shown that, whenever a feedback delay exists, the network is not capable of offering diversity gains. Furthermore, source and relay power allocation results show significantly different behavior with feedback delay. Numerical results supported by simulations are provided to show that feedback delay can severely degrade the performance of the considered AF relay system. [ABSTRACT FROM PUBLISHER]

We study a parametric quadrature amplitude modulation (QAM) family, called θ-QAM, which includes other known constellations, such as square QAM (SQAM) and triangular QAM (TQAM), as special cases. The versatile structure of the θ-QAM signal constellation, which occurs from the varying angle between the signal points, results in achieving the minimum symbol error rate (SER) or bit error rate (BER), under an average power constraint. The theoretical study aims at providing insight into the trade-off between error performance and complexity of this parametric modulation scheme. Exact analytical expressions are obtained for the SER in additive white Gaussian Noise (AWGN) and Nakagami-m fading channels, while highly accurate BER approximations are presented. Finally, we find the optimum angles, in a minimal SER or BER sense, for a specific signal-to-noise ratio (SNR) and modulation order, M. This serves as an indicator for the appropriate constellation with respect to channel conditions and SER or BER requirements. The presented theoretical analysis is validated via extensive computer simulations. [ABSTRACT FROM AUTHOR]

We study the error performance of the binary phase-shift keying (BPSK)-modulated distributed selection combining (SC) and distributed switched-and-stay combining (DSSC) schemes that utilize a single fixed-gain amplify-and-forward (AF) relay under Rayleigh fading conditions. The error performance of the distributed maximal ratio combining (MRC) is also investigated, which serves here as a benchmark. The numerical examples confirm that SC and DSSC offer a simpler substitute for MRC without much loss in performance. Moreover, we interestingly note that the performance of DSSC with a single fixed-gain AF relay is similar to that of DSSC with a single decode-and-forward relay, despite its relative simplicity. [ABSTRACT FROM AUTHOR]

Motivated by combinatorial optimization theory, we propose an algorithmic power allocation method that minimizes the total transmitting power in transmitter diversity systems, provided that the instantaneous Bit-Error-Rate (BER) is not greater than a predetermined value. This method applies to many practical applications where the power transmitted by each antenna is constrained. We also provide closed-form expressions for the average total transmitted power for the case of two transmitting antennas operating in Rayleigh fading, and the average number of active antennas at the transmitter assuming Nakagami-m fading channels. Simulations and numerical results show that, compared to the conventional equi-power scheme, the proposed model offers a considerable reduction in the total transmitting power and the average number of active antennas, without loss in error performance. [ABSTRACT FROM AUTHOR]

A novel approach to the performance analysis of switched-and-stay combining diversity receivers over independent Weibull fading channels is presented. Closed-form expressions are extracted for important performance parameters, such as the average output signal-to-noise ratio, the amount of fading, the outage probability, and the switching rate. [ABSTRACT FROM AUTHOR]

In this paper, we consider a radio-frequency identification (RFID) backscatter system with two scenarios: single input multiple output (SIMO) and multiple input single output (MISO). We investigate the impact of channel correlation between the forward and backscatter links on the symbol error rate (SER), and we compare the performance of SIMO and MISO RFID systems with maximum ratio combining reception over arbitrarily correlated Nakagami- $m$ fading channels. In order to gain an insight for the system design, closed-form expressions for the asymptotic SER and an upper bound are derived for $M$ -ary phase-shift keying and quadrature amplitude modulation. The diversity order in the fully correlated cases, revealed from these expressions, is half of that in the partially correlated and uncorrelated cases. Finally, simulations are performed to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

We present a novel framework for analyzing and optimizing the outage performance of a multiple amplify-and-forward relay network, where $N$ relays, each with finite cache capacity, are deployed to assist the transmission from a base station equipped with $L$ antennas to the destination. In this network, if the requested file is cached at the relays and the associated relay channels are in good condition, then the data are directly transmitted from the relays to the destination; otherwise, traditional two-hop transmission is performed. Based on this concept, we propose a relay-selection criterion to choose the best relay, which maximizes the received signal-to-noise ratio at the destination. For this criterion, we derive exact and asymptotic analytical expressions for the system outage probability in Nakagami- $m$ fading. To further improve the network outage performance, we optimize the cache placement, which minimizes the outage probability. Numerical and simulation experiments are performed to validate the analysis and demonstrate the conditions where the proposed cache placement strategy outperforms the existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we consider a point-to-point multi-input multi-output wireless-powered communication system, where the source S is powered by a dedicated power beacon (PB) with multiple antennas. Employing the time splitting protocol, the energy constrained source S first harvests energy through the radio-frequency signals sent by the PB and then uses this energy to transmit information to the destination D. Unlike several prior works, we assume that the energy transfer link is subjected to Rician fading, which is a real fading environment, due to relatively short range power transfer distance and the existence of a strong line of sight path. We present a comprehensive analysis of the achievable ergodic capacity in two scenarios, depending on the availability of channel state information (CSI) at PB, namely, the absence of CSI and partial CSI. For the former case, equal power allocation is used, while for the later one, energy beamforming is used to enhance the energy transfer efficiency. For both the cases, closed-form expressions for the upper and lower bounds of the ergodic capacity are derived. Furthermore, the optimal time split is discussed, and the capacity in the low and high signal-to-noise ratio regimes is studied through simple closed-form expressions. Numerical results and simulations are provided to validate the theoretical analysis. The results show that the Rician factor $K$ has a significant impact on the ergodic capacity performance, and this impact strongly depends on the availability of the CSI at the PB. [ABSTRACT FROM AUTHOR]

This paper quantifies the impact of correlated fading on secure communication of multiple amplify-and-forward (AF) relaying networks. In such a network, the base station (BS) is equipped with multiple antennas and communicates with the destination through multiple AF relays, while the message from the relays can be overheard by an eavesdropper. We focus on the practical communication scenario, where the main and eavesdropper’s channels are correlated. In order to enhance the transmission security, transmit antenna selection is performed at the BS, and the best relay is chosen according to the full- or partial-relay selection criterion, which relies on the dual-hop relay channels or the second-hop relay channels, respectively. For these criteria, we study the impact of correlated fading on the network secrecy performance, by deriving an analytical approximation for the secrecy outage probability and an asymptotic expression for the high main-to-eavesdropper ratio. From these results, it is concluded that the channel correlation is always beneficial to the secrecy performance of full relay selection. However, it deteriorates the secrecy performance if partial-relay selection is used, when the number of antennas at the BS is less than the number of relays. [ABSTRACT FROM PUBLISHER]

In this paper, we consider a unified approach to model wireless channels by the mixture of Gaussian (MoG) distribution. The proposed approach provides an accurate approximation for the envelope and the signal-to-noise ratio (SNR) distributions of wireless fading channels. Simulation results have shown that the proposed model can accurately characterize multipath and composite fading channels. We utilize the well-known expectation–maximization (EM) algorithm to estimate the parameters of the MoG distribution and further utilize the Bayesian information criterion (BIC) to determine the number of mixture components automatically. We employ the Kullback–Leibler (KL) divergence and the mean-square-error (MSE) criteria to demonstrate that the proposed distribution provides both high accuracy and low computational complexity. Additionally, we provide closed-form expressions or approximations for several performance metrics used in wireless communication systems, including the moment generating function (MGF), the raw moments, the amount of fading (AF), the outage probability, the average channel capacity, and the probability of energy detection for cognitive radio (CR). Numerical analysis and Monte Carlo simulation results are presented to corroborate the analytical results. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the performance of cooperative spectrum sensing (CSS) with multiple-antenna nodes over generalized and composite fading channels. To this end, we approximate the probability density function (pdf) of the signal-to-noise ratio (SNR) of various fading channels using the mixture Gamma (MG) distribution. Based on this, we derive an exact closed-form expression and a generic infinite series representation for the corresponding probability of energy detection, along with a finite upper bound for the involved truncation error. Both expressions have a relatively simple algebraic form that gives them convenience in handling both analytically and numerically. Furthermore, the composite effect of multipath fading and shadowing scenarios in CSS is mitigated by applying an optimal fusion rule that minimizes the total error rate (TER), where the optimal number of nodes is derived under the Bayesian criterion, assuming erroneous feedback channels. We also extend the derived average detection probability to include diversity reception techniques, namely, maximal-ratio combining, square-law combining, and square-law selection (SLS). For the SLS, we demonstrate the existence of an error rate floor as the number of antennas of the cognitive radio nodes increases in erroneous decision feedback channels. Accordingly, we derive the optimal rule for the number of antennas that minimizes the TER in the SLS framework. Monte Carlo simulations are presented to corroborate the analytical results and to provide illustrative performance comparisons and insights between different composite fading channels. [ABSTRACT FROM PUBLISHER]

Energy efficiency and its optimization constitute critical tasks in the design of low-power wireless networks. This paper is devoted to the error rate analysis and energy efficiency optimization of regenerative cooperative networks in the presence of multipath fading under spatial correlation. To this end, exact and asymptotic analytic expressions are first derived for the symbol error rate (SER) of $M$-ary quadrature amplitude and $M$-ary phase-shift keying modulations ( $M$-QAM and $M$- PSK), respectively, assuming a dual-hop decode-and-forward (DF) relay system, spatially correlated Nakagami- $m$ multipath fading, and maximum ratio combining (MRC) at the destination. The derived expressions are subsequently employed in quantifying the energy consumption of the considered system, incorporating both transmit energy and the energy consumed by the transceiver circuits, and in deriving the optimal power allocation (OPA) formulation for minimizing energy consumption under certain quality-of-service (QoS) requirements. A relatively harsh path-loss (PL) model, which also accounts for realistic device-to-device communications, is adopted in numerical evaluations, and various useful insights are provided for the design of future low-energy wireless networks deployments. Indicatively, it is shown that, depending on the degree of spatial correlation, severity of fading, transmission distance, relay location, and power allocation strategy, target performance can be achieved with much overall energy reduction compared with direct transmission (DT) reference. [ABSTRACT FROM AUTHOR]

We study the problem of private key generation in a cooperative pairwise independent network (PIN), with $M+2$ terminals (Alice, Bob, and $M$ relays), $M\geq 2$ . In the PIN, the correlated source observed by every pair of terminals is independent of the sources observed by any other pairs of terminals. Moreover, all terminals can communicate with each other over a public channel, which is also observed by Eve, noiselessly. The objective is to generate a private key between Alice and Bob with the help of the $M$ relays; such a private key needs to be protected not only from Eve but also from all relays. A single-letter expression for the private key capacity of this PIN model is obtained, where the achievability part is established by proposing a random binning (RB)-based key generation algorithm, and the converse part is established by deriving upper bounds of $M$ enhanced source models. Next, we consider a cooperative wireless network and use the estimates of fading channels to generate private keys. It has been shown that the proposed RB key generation algorithm can achieve a multiplexing gain of $M-1$ , which is an improvement compared with the existing XOR algorithm, whose achievable multiplexing gain is $\lfloor M /2\rfloor $ . [ABSTRACT FROM PUBLISHER]

Cooperative communication has been shown to provide significant increase of transmission reliability and network capacity while expanding coverage in cellular networks. The present work is devoted to the investigation of the end-to-end performance and power allocation of a maximum-ratio-combining based regenerative multi-relay cooperative network over non-homogeneous scattering environment, which is the realistic case in many practical wireless communication scenarios. Novel analytic expressions are derived for the end-to-end symbol-error-rate of both M-ary phase-shift keying and M-ary quadrature amplitude modulation over independent and non-identically distributed generalized fading channels are given by exact analytic expressions that involve the Lauricella function and can be readily evaluated with the aid of a proposed computing algorithm. Simple analytic expressions are also derived for the corresponding symbol-error-rate at asymptotically high signal-to-noise ratios. The derived expressions are corroborated with respective results from computer simulations and are subsequently employed in formulating a sum-power optimization problem that enhances the system performance under total sum-power constraint within the multi-relay cooperative system. It is also shown that asymptotically optimum power allocation provides substantial performance gains over the corresponding equal power allocation, particularly, when the source-relay and relay-destination paths are highly unbalanced. [ABSTRACT FROM AUTHOR]

This paper is devoted to the derivation of novel analytic expressions and bounds for a family of special functions that are useful in wireless communication theory. These functions are the well-known Nuttall Q- function, incomplete Toronto function, Rice Ie- function, and incomplete Lipschitz–Hankel integrals. Capitalizing on the offered results, useful identities are additionally derived between the above functions and Humbert, \Phi _{1} , function as well as for specific cases of the Kampéde Fériet function. These functions can be considered as useful mathematical tools that can be employed in applications relating to the analytic performance evaluation of modern wireless communication systems, such as cognitive radio, cooperative, and free-space optical communications as well as radar, diversity, and multiantenna systems. As an example, new closed-form expressions are derived for the outage probability over nonlinear generalized fading channels, namely, \alpha {-}\eta {-}\mu , \alpha -\lambda -\mu , and \alpha -\kappa -\mu as well as for specific cases of the \eta -\mu and \lambda -\mu fading channels. Furthermore, simple expressions are presented for the channel capacity for the truncated channel inversion with fixed rate and corresponding optimum cutoff signal-to-noise ratio for single-antenna and multiantenna communication systems over Rician fading channels. The accuracy and validity of the derived expressions is justified through extensive comparisons with respective numerical results. [ABSTRACT FROM AUTHOR]

We study the effect of outdated channel state information on the outage and error rate performance of amplify-and-forward (AF) relay selection, where only one out of the set of available relays is activated. We consider two variations of AF relay selection, namely best relay selection and partial relay selection, when the selection is based upon outdated channel estimates. For both these variations, closed-form expressions for the outage probability are obtained, along with approximate expressions for the symbol error rate in the medium to high signal-to-noise-ratio (SNR) regime. The diversity gain and coding gain of the above schemes are also explicitly derived. Numerical results manifest that the outage performance of AF relay selection is highly dependent on the level of correlation between the actual channel conditions and their corresponding (outdated) estimates. This result has a significant impact on the deployment of relay selection in practical applications, implying that a high feedback rate may be required in practice in order to attain the full benefits of relay selection. It is further shown that it may be preferable, in terms of outage and symbol error probability, not to include links in the relay selection process that experience a sufficiently high maximum Doppler shift, since in those cases partial relay selection outperforms best relay selection. [ABSTRACT FROM AUTHOR]

This paper provides a systematic characterization of Zero-Forcing (ZF) detectors over multiple-input multiple-output (MIMO) channels that experience both small and large-scale fading. In particular, we consider the generic K distribution (Rayleigh/gamma distribution) to model the composite fading fluctuations and also assume the general case of semi-correlated small-scale fading. In the following, novel exact analytical expressions for the achievable sum rate are derived, followed by asymptotic expressions in the high and low Signal-to-Noise ratio (SNR) regimes. In these limiting cases, two common and insightful affine expansions are studied followed by new, closed-form upper and lower bounds on the sum rate that remain tight for all SNRs. In the second part of the paper, we present exact tractable expressions along with first-order expansions for the symbol error rate (SER) and outage probability; we also quantify the performance of ZF detectors in terms of diversity order and array (or coding) gain. The implications of the model parameters on the ZF detector performance are investigated via Monte-Carlo simulations which also validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

We propose an adaptive transmission technique for free space optical (FSO) systems, operating in atmospheric turbulence and employing subcarrier phase shift keying (S-PSK) intensity modulation. Exploiting the constant envelope characteristics of S-PSK, the proposed technique offers efficient utilization of the FSO channel capacity by adapting the modulation order of S-PSK, according to the instantaneous state of turbulence induced fading and a pre-defined bit error rate (BER) requirement. Novel expressions for the spectral efficiency and average BER of the proposed adaptive FSO system are presented and performance investigations under various turbulence conditions, turbulence models, and target BER requirements are carried out. Numerical results indicate that significant spectral efficiency gains are offered without increasing the transmitted average optical power or sacrificing BER requirements, especially in moderate-to-strong turbulence conditions. Furthermore, the proposed variable rate transmission technique is applied to multiple input multiple output (MIMO) FSO systems, providing additional improvement in the achieved spectral efficiency as the number of the transmit and/or receive apertures increases. [ABSTRACT FROM AUTHOR]