A closed electromagnetic resonant chamber (RC) is a highly favorable artificial environment for wireless communication. A pair of antennas within the chamber constitutes a two-port network described by an impedance matrix. We analyze communication between the two antennas when the RC has perfectly conducting walls and the impedance matrix is imaginary-valued. The transmit antenna is driven by a current source, and the receive antenna is connected to a load resistor whose voltage is measured by an infinite-impedance amplifier. There are a countably infinite number of poles in the channel, associated with resonance in the RC, which migrate towards the real frequency axis as the load resistance increases. There are two sources of receiver noise: the Johnson noise of the load resistor, and the internal amplifier noise. An application of Shannon theory yields the capacity of the link, subject to bandwidth and power constraints on the transmit current. For a constant transmit power, capacity increases without bound as the load resistance increases. Surprisingly, the capacity-attaining allocation of transmit power versus frequency avoids placing power close to the resonant frequencies.

This paper provides a deterministic channel model for a scenario where wireless connectivity is established through a reflection off a smooth planar surface of an infinite extent. The developed model is rigorously built upon the physics of wave propagation and is as precise as tight are the unboundedness and smoothness assumptions on the surface. This model allows establishing how line-of-sight multiantenna communication is altered by a reflection off an electrically large surface, a situation of high interest for mmWave and terahertz frequencies.

While the heat equation has been extensively studied, heat conduction has not been studied as a means of communication until recently. Recent literature focusing on covert channels shows the feasibility of using thermal conduction to communicate information. Since heat conduction is modelled by a linear partial differential equation, it can be analyzed as a linear system with an input heat source and output temperature distribution. The magnitude of the thermal channel's frequency response is an exponentially decaying function of frequency. The thermal channel's capacity increases when the total power increases, similar to typical communication based on electromagnetic waves. Uniquely however, the thermal channel's effective bandwidth is constrained by the total power since a water-filling algorithm determines a cutoff frequency. Additionally, the quadratic nature of the heat equation presents a novel quadratic scaling of the channel capacity. Scaling space by a factor of 2 and time by 4 improves the channel capacity by a factor of 4. This implies that scaling space down from centimeter to micrometer domain improves the channel capacity by a factor of 108• The thermal channel presents various novel qualities and a possible exciting application for intra-chip communication.

A resonant electromagnetic cavity - effectively the interior of a copper box - constitutes a highly favorable propa-gation environment for MIMO communications. A fundamental limitation on wireless communication within the box is thermal noise. Under conditions of thermal equilibrium the Equipartition Theorem of classical statistical mechanics states that every orthogonal normal mode has an independent random amplitude such that the expected energies of the modes are equal and proportional to absolute temperature. Collectively the randomly excited modes constitute a space-time stochastic process, station-ary in time but non-stationary in space because of boundary conditions. As the linear dimensions of the box grow relative to a wavelength the process is asymptotically stationary. At each temporal frequency the wavenumber spectral density is identical to that of the Clarke model - a superposition of random plane waves having no preferred direction of propagation. At a single point in space the spectral density is proportional to the square of temporal frequency - the Rayleigh-Jeans spectrum. We call our space-time noise model Rayleigh-Jeans-Clarke (RJC).

In most multiple-input multiple-output (MIMO) communication systems, e.g., Massive MIMO, the antenna spacing is generally no less than half a wavelength. It helps to reduce the mutual coupling and therefore facilitate the system design. The maximum array gain is the number of antennas in this settings. However, when the antenna spacing is made very small, the array gain of a compact array can be proportional to the square of the number of antennas - a value much larger than the traditional array. To achieve this so-called "superdirectivity" however, the calculation of the excitation coefficients (beamforming vector) is known to be a challenging problem. In this paper, we derive the beamforming vector of superdirective arrays based on a novel coupling matrix-enabled method. We also propose an approach to obtain the coupling matrix, which is derived by the spherical wave expansion method and active element pattern. The full-wave electromagnetic simulations are conducted to validate the effectiveness of our proposed method. Simulation results show that when the beamforming vector obtained by our method is applied, the directivity of the designed dipole antenna array has a good agreement with the theoretical values.

We provide a deterministic channel model for a scenario where wireless connectivity is established through a reflection from a planar smooth surface of an infinite extent. The developed model is rigorously built upon the physics of wave propagation, and is as precise as tight are the unboundedness and smoothness assumptions on the surface. This model allows establishing that line-of-sight spatial multiplexing can take place via reflection off an electrically large surface, a situation of high interest for mmWave and terahertz frequencies.

Imagine a MIMO communication system that fully exploits the propagation characteristics offered by an electromagnetic channel and ultimately approaches the limits imposed by wireless communications. This is the concept of Holographic MIMO communications. Accurate and tractable channel modeling is critical to understanding its full potential. Classical stochastic models used by communications theorists are derived under the electromagnetic far-field assumption, i.e. planar wave approximation over the array. However, such assumption breaks down when electromagnetically large (compared to the wavelength) antenna arrays are considered. In this paper, we start from the first principles of wave propagation and provide a Fourier plane-wave series expansion of the channel response, which fully captures the essence of electromagnetic propagation in arbitrary scattering and is also valid in the (radiative) near-field. The expansion is based on the Fourier spectral representation and has an intuitive physical interpretation, as it statistically describes the angular coupling between source and receiver. When discretized uniformly, it leads to a low-rank semi-unitarily equivalent approximation of the electromagnetic channel in the angular domain. The developed channel model is used to compute the ergodic capacity of a point-to-point Holographic MIMO system with different degrees of channel state information.

This paper proposes a communication model for multiuser multiple-input multiple-output (MIMO) systems based on large intelligent surfaces (LIS), where the LIS is modeled as a collection of tightly packed antenna elements. The LIS system is first represented in a circuital way, obtaining expressions for the radiated and received powers, as well as for the coupling between the distinct elements. Then, this circuital model is used to characterize the channel in a line-of-sight propagation scenario, rendering the basis for the analysis and design of MIMO systems. Due to the particular properties of LIS, the model accounts for superdirectivity and mutual coupling effects along with near field propagation, necessary in those situations where the array dimension becomes very large. Finally, with the proposed model, the matched filter transmitter and the weighted minimum mean square error precoding are derived under both realistic constraints: limited radiated power and maximum ohmic losses.

Non-cooperative cellular massive MIMO, combined with power control, is known to lead to significant improvements in per-user throughput compared with conventional LTE technology. In this paper, we investigate further refinements to massive MIMO, first, in the form of three-fold sectorization, and second, coordinated multi-point operation (with and without sectorization), in which the three base stations cooperate in the joint service of their users. For these scenarios, we analyze the downlink performance for both maximum-ratio and zero-forcing precoding and derive closed-form lower-bound expressions on the achievable rate of the users. These expressions are then used to formulate power optimization problems with two throughput fairness criteria:${i}$) network-wide max-min fairness, andii) per-cell max-min fairness. Furthermore, we provide centralized and decentralized power control strategies to optimize the transmit powers in the network. We demonstrate that employing sectorized antenna elements mitigates the detrimental effects of pilot contamination by rejecting a portion of interfering pilots in the spatial domain during channel estimation phase. Simulation results with practical sectorized antennas reveal that sectorization and multi-point coordination combined with sectorization lead to more than$1.7\times $and$2.6\times $improvements in the 95%-likely per-user throughput, respectively.

Imagine an array with a massive (possibly uncountably infinite) number of antennas in a compact space. We refer to a system of this sort as Holographic MIMO. Given the impressive properties of Massive MIMO, one might expect a holographic array to realize extreme spatial resolution, incredible energy efficiency, and unprecedented spectral efficiency. At present, however, its fundamental limits have not been conclusively established. A major challenge for the analysis and understanding of such a paradigm shift is the lack of mathematically tractable and numerically reproducible channel models that retain some semblance to the physical reality. Detailed physical models are, in general, too complex for tractable analysis. This paper aims to take a closer look at this interdisciplinary challenge. Particularly, we consider the small-scale fading in the far-field, and we model it as a zero-mean, spatially-stationary, and correlated Gaussian scalar random field. A physically-meaningful correlation is obtained by requiring that the random field be consistent with the scalar Helmholtz equation. This formulation leads directly to a rather simple and exact description of the three-dimensional small-scale fading as a Fourier plane-wave spectral representation. Suitably discretized, this yields a discrete representation for the field as a Fourier plane-wave series expansion, from which a computationally efficient way to generate samples of the small-scale fading over spatially-constrained compact spaces is developed. The connections with the conventional tools of linear systems theory and Fourier transform are thoroughly discussed.

Future wireless networks will be as pervasive as the air we breathe, not only connecting us but embracing us through a web of systems that support personal and societal well-being. That is, the ubiquity, speed and low latency of such networks will allow currently disparate devices and services to become a distributed intelligent communications, sensing, and computing platform.

We study joint unicast and multigroup multicast transmission in single-cell massive multiple-input-multiple-output (MIMO) systems, under maximum ratio transmission. For the unicast transmission, the objective is to maximize the weighted sum spectral efficiency (SE) of the unicast user terminals (UTs) and for the multicast transmission the objective is to maximize the minimum SE of the multicast UTs. These two problems are coupled to each other in a conflicting manner, due to their shared power resource and interference. To address this, we formulate a multiobjective optimization problem (MOOP). We derive the Pareto boundary of the MOOP analytically and determine the values of the system parameters to achieve any desired Pareto optimal point. Moreover, we prove that the Pareto region is convex, hence the system should serve the unicast and multicast UTs at the same time-frequency resource.

The spatially uncorrelated Rayleigh small-scale fading model is a useful stochastic tool for analyzing multiple-antenna wireless communication systems, and, as experiments have shown, it often is a good approximation to physical propagation. However, the assumption that the propagating field is uncorrelated from one point in space to another breaks down when, for example, antenna spacings are smaller than one-half wavelength - a model defect typically addressed by assuming some spatial correlation. Spatial correlation can have huge effects even in the absence of close spacing between antennas. While an ad-hoc correlation versus distance, such as exponential, may add an element of realism to the model, in general it does not capture the peculiar “action at a distance” phenomena associated with the wave equation. The very desirable property of spatial stationarity can be retained, provided the spatial autocorrelation is chosen such that the complex Gaussian small-scale fading random field satisfies the homogeneous wave equation. The fading model that is closest to iid Rayleigh fading, and that is still consistent with the wave equation, has an autocorrelation equal to sinc(2λR/λ, corresponding to planewaves arriving uniformly from all directions, and having independent, equal variance complex Gaussian amplitudes. The contribution of this paper is twofold: first, a Fourier planewave representation that provides a computationally efficient way to generate samples of the random field, second, an inverse representation that enables the efficient computation of the joint likelihood of noisy measurements of the field over continuous segments of lines, planes, and volumes.

Cell-free Massive MIMO (Multiple Input Multiple Output) employs a large number of AP's (Access Points) that are distributed throughout the intended coverage area to simultaneously serve a much smaller number of user AT's (Access Terminals). Conjugate beamforming is the simplest precoding method for the downlink transmission, and allows decentralized precoding processing. Max-min power control maximizes the minimal effective SINR (Signal-to-Interference-plus-Noise Ratio) among all the active users, therefore provides a uniform throughput to all users. Cell- free Massive MIMO with conjugate beamforming precoding and max-min power control is naturally radiated energy efficient due to two facts: a well-known fact that with high probability at least one AP is nearby every user, and a surprising fact that, with max-min power control, a large portion of the AP's does not transmit with full power. We find that a cell-free Massive MIMO can deliver more than 80 Mb/joule in urban, and more than 40 Mb/joule in suburban and rural scenarios. We compare its energy efficiency and spectral efficiency with single cell Massive MIMO systems and demonstrate that in suburban and rural scenarios, cell-free systems can more than double the radiated energy efficiency and at the same time dramatically increase the 95% likely per user throughput, while in an urban scenario, the gain in radiated energy efficiency can be moderate at less than 50% with a comparable 95% likely per user throughput.

A wireless massive multiple-input multiple-output (MIMO) system entails a large number of base station antennas serving a much smaller number of users, with large gains in spectral efficiency and energy efficiency compared with the conventional MIMO technology. Until recently, it was believed that as the number of base station antennas tends to infinity, the performance of such systems is limited by directed inter-cellular interference caused by unavoidable re-use of training sequences (pilot contamination) by users in different cells. We devise a new concept of large-scale fading precoding (LSFP) that leads to the effective elimination of inter-cell interference. The main idea of LSFP is that base stations linearly combine messages aimed at users from different cells that re-use the same training sequence. Crucially, the combining coefficients depend only on the large-scale fading coefficients between the users and the base stations. These coefficients change slowly and their number does not depend on the number of base station antennas. Thus, the traffic between base stations stays constant even if the number of antennas tends to infinity. Furthermore, we derive a capacity lower bound for massive MIMO systems with LSFP and a finite number of base station antennas. In this regime, mitigation of all types of interference, not only the pilot contamination, is required. We consider optimal and suboptimal LSFP precodings that take into account all sources of interference. Our simulations results show that LSFP provides significant gain even for the case of moderate number of base station antennas.

This paper considers the downlink precoding for physical layer multicasting in massive multiple-input multiple-output (MIMO) systems. We study the max-min fairness (MMF) problem, where channel state information at the transmitter is used to design precoding vectors that maximize the minimum spectral efficiency (SE) of the system, given fixed power budgets for uplink training and downlink transmission. Our system model accounts for channel estimation, pilot contamination, arbitrary path-losses, and multi-group multicasting. We consider six scenarios with different transmission technologies (unicast and multicast), different pilot assignment strategies (dedicated or shared pilot assignments), and different precoding schemes (maximum ratio transmission and zero forcing), and derive achievable spectral efficiencies for all possible combinations. Then, we solve the MMF problem for each of these scenarios, and for any given pilot length, we find the SE maximizing uplink pilot and downlink data transmission policies, all in closed forms. We use these results to draw a general guideline for massive MIMO multicasting design, where for a given number of base station antennas, number of users, and coherence interval length, we determine the multicasting scheme that shall be used.

We study the joint unicast and multi-group multicast transmission in massive multiple-input multiple-output systems. We consider a system model that accounts for channel estimation and pilot contamination and derive achievable spectral efficiencies (SEs) for unicast and multicast user terminals (UTs) under maximum ratio transmission and zero-forcing precoding. For unicast transmission, our objective is to maximize the weighted sum SE of the unicast UTs, and for the multicast transmission, our objective is to maximize the minimum SE of the multicast UTs. These two objectives are coupled in a conflicting manner, due to their shared power resource. Therefore, we formulate a multiobjective optimization problem (MOOP) for the two conflicting objectives. We derive the Pareto boundary of the MOOP analytically. As each Pareto optimal point describes a particular efficient tradeoff between the two objectives of the system, we determine the values of the system parameters (uplink training powers, downlink transmission powers, and so on) to achieve any desired Pareto optimal point. Moreover, we prove that the Pareto region is convex, and hence, the system should serve the unicast and multicast UTs at the same time-frequency resource. Finally, we validate our results using numerical simulations.

Optimal physical layer multicasting (PLM) is an NP-hard problem that for simplicity has been studied under idealistic assumptions, e.g., availability of perfect channel state information (CSI), both at the base station (BS) and at the user terminals (UTs). With the advent of massive multiple-input-multiple-output (MIMO), PLM has become more challenging, as the computational complexity of the precoder design is proportional to the number of BS antennas. In this paper, we address these issues by introducing computationally efficient precoders that account for practical CSI acquisition. We derive achievable spectral efficiencies for the proposed precoders. Then we introduce a novel problem formulation for the max-min fairness power control that accounts the CSI acquisition overhead, uplink training and downlink transmission powers. We solve this problem and find the optimal uplink and downlink power control policies in closed form. Using numerical simulations, we verify the effectiveness of our proposed schemes comparing them with the state of the art PLM schemes for massive MIMO systems.

Under LoS (line-of-sight) propagation and the assumption of perfect CSI (channel state information), for either MR (maximum-ratio) or ZF (zero-forcing) precoding/decoding, one can readily obtain Massive MIMO (multi-input multi-output) per-user effective SINR for single-cell scenarios. LoS channels are typically less correlated than IID (independent and identically distributed) Rayleigh channels, but the maximum correlation for LoS is typically much greater than for IID Rayleigh. This motivates an investigation of the dependence of max-min SINR on the maximum channel correlation. Perron-Frobenius theory and the classical Fischer inequality are used to establish some rigorous and explicit upper bounds on the effective max-min SINR (signal to interference plus noise ratio) that depend on the maximum channel correlation. These upper bounds provide an accurate description of this dependence relationship, and readily facilitate system performance analyses and scheduler designs without simulations. In high channel correlation environment, ZF can perform substantially better than MR in the downlink but the opposite is true for the uplink.

By calculating the effective max-min SINR (signal- to-interference-plus-noise ratio) and the corresponding power controls explicitly, and selectively dropping a small number of mobiles based on a simple and effective algorithm, we demonstrate that for both downlink and uplink, employing maximum-ratio or zero-forcing linear pre-coding and de-coding, Massive MIMO with max- min power control performs comparably in LoS (Line-of-Sight) and iid (independent and identically distributed) Rayleigh fading propagation environments.

A massive MIMO system entails a large number (tens or hundreds) of base station antennas serving a much smaller number of terminals. These systems demonstrate large gains in spectral and energy efficiency compared with the conventional MIMO technology. As the number of antennas grows, the performance of a massive MIMO system gets limited by the interference caused by pilot contamination. Ashikhmin and Marzetta proposed (under the name of Pilot Contamination Precoding) large scale fading precoding (LSFP) and large scale fading decoding (LSFD) based on limited cooperation between base stations. They showed that zero-forcing LSFP and LSFD eliminate pilot contamination entirely and lead to an infinite throughput as the number of antennas grows. In this paper, we focus on the uplink and show that even in the case of a finite number of base station antennas, LSFD yields a very large performance gain. In particular, one of our algorithms gives a more than 140 fold increase in the 5% outage data transmission rate! We show that the performance can be improved further by optimizing the transmission powers of the users. Finally, we present decentralized LSFD that requires limited cooperation only between neighboring cells.

Massive MIMO relies on the asymptotic orthogonality of channel vectors to different users. For M service antennas, the expected correlation between a pair of channel vectors under line-of-sight (LoS) conditions decreases at least as fast as log(M)/M, while in independent and identically distributed (iid) Rayleigh fading, it decreases much slower at 1/√M, but the variance is higher under LoS. This signifies that typically channel vectors are more nearly orthogonal under LoS, but with a non-negligible probability, they can have an anomalously large correlation. A single-cell analysis discloses that Massive MIMO with max-min power control performs comparably under LoS and iid Rayleigh, when a simple algorithm is applied under LoS to drop a small number of high-correlation users from service.

Cell-free Massive multiple-input multiple-output (MIMO) comprises a large number of distributed low-cost low-power single antenna access points (APs) connected to a network controller. The number of AP antennas is significantly larger than the number of users. The system is not partitioned into cells and each user is served by all APs simultaneously. The simplest linear precoding schemes are conjugate beamforming and zero-forcing. Max-min power control provides equal throughput to all users and is considered in this paper. Surprisingly, under max-min power control, most APs are found to transmit at less than full power. The zero-forcing precoder significantly outperforms conjugate beamforming. For zero-forcing, a near-optimal power control algorithm is developed that is considerably simpler than exact max-min power control. An alternative to cell-free systems is small-cell operation in which each user is served by only one AP for which power optimization algorithms are also developed. Cell-free Massive MIMO is shown to provide five- to ten-fold improvement in 95%-likely per-user throughput over small-cell operation.

A Cell-Free Massive MIMO (multiple-input multiple-output) system comprises a very large number of distributed access points (APs), which simultaneously serve a much smaller number of users over the same time/frequency resources based on directly measured channel characteristics. The APs and users have only one antenna each. The APs acquire channel state information through time-division duplex operation and the reception of uplink pilot signals transmitted by the users. The APs perform multiplexing/de-multiplexing through conjugate beamforming on the downlink and matched filtering on the uplink. Closed-form expressions for individual user uplink and downlink throughputs lead to max-min power control algorithms. Max-min power control ensures uniformly good service throughout the area of coverage. A pilot assignment algorithm helps to mitigate the effects of pilot contamination, but power control is far more important in that regard. Cell-Free Massive MIMO has considerably improved performance with respect to a conventional small-cell scheme, whereby each user is served by a dedicated AP, in terms of both 95%-likely per-user throughput and immunity to shadow fading spatial correlation. Under uncorrelated shadow fading conditions, the cell-free scheme provides nearly fivefold improvement in 95%-likely per-user throughput over the small-cell scheme, and tenfold improvement when shadow fading is correlated.

A heterogeneous system of a Massive multiple- input-multiple-output (MIMO) macro cell with low power ancillary small cells can achieve higher spectral and energy efficiency than a Massive MIMO macro cell alone. The performance of such heterogeneous system is examined in this paper. A few small cells are used to enhance the spectral and energy efficiency of the overall system. Macro Massive MIMO base station uses a large number of antennas, which enables accurate nulling of small cell users to suppress the interference between macro cell and small cells. We derive analytical expressions for the capacity and signal-to- interference-plus-noise-ratio (SINR) lower bounds for both the downlink (DL) and uplink (UL) of the heterogeneous Massive MIMO system. Our simulation results show that nulling from macro cell Massive MIMO is essential for small cells' good and stable performance.

For a general multicell Massive MIMO network, we obtain easily verifiable conditions for the existence of power controls that meet given down-link and up-link per access terminal SINR requirements. Surprisingly, some loosely constructed sufficient conditions that guarantee the existence of power controls become necessary in the single cell case, whose “max-min” SINR problems are then easily solved.

We provide explicit formulas for the optimal number of antennas per base station to maximize the cell total energy efficiency of a power- controlled multi-cell Massive MIMO. Furthermore, we show that equipping the same number of antennas in each base station results in virtually no loss in energy efficiency due to the flatness of energy efficiency function. Conjugate beamforming compares very competitively with zero-forcing and in fact, due to the loss in effective array gain and the additional computational cost in zero-forcing, conjugate beamforming can achieve better energy efficiency when per user throughput demand is moderate. In single cell case, we provide explicit algebraic formulas for the optimal number of antennas and optimal radiated power when a target per user throughput is given.

A Large Scale Antenna System (LSAS) entails a large number (tens or hundreds) of base station antennas serving a much smaller number of terminals, with large gains in spectral-efficiency and energy efficiency compared with conventional MIMO technology. As the number of antennas grows, the performance of an LSAS gets limited by pilot contamination, arising due to the use of same pilot sequences for channel estimation in neighboring cells. Recently A. Ashikhmin and T. Marzetta showed that using proper precoding/postcoding (PCP) and limited cooperation between cells, it is possible to eliminate pilot contamination entirely and get infinite throughput as M → ∞. In this paper, we focus on the uplink of an LSAS and show that even in the case of a finite number of base station antennas, PCP yields very significant performance gain in terms of data transmission rates. In particular, one of our algorithms gives a 140 fold increase in the 5% outage data transmission rates! We also show that the performance can be improved further by optimizing the transmission powers of the users, and present a simple decentralized algorithm in order to solve it.

Traditional macro-cellular wireless networks are not capable of delivering even throughputs to all the users due to large variations in slow fading and inter-cell and inter-user interferences, and the throughputs for cell edge users are necessarily sacrificed to achieve an acceptable level of cell spectral efficiency. We show that this is not the case for large-scale antenna systems (LSAS, also known as Massive MIMO). Specifically, we show that by means of its superior beamforming and frequency response flattening capabilities, simple noncooperative uplink and downlink power controls can be devised for an LSAS macro-cellular wireless network to provide intra-cell equalized, multi-Mbps throughputs to all users. Compared with current LTE, a 64-antenna LSAS can provide cell edge throughputs with at least a ten-fold increase in the uplink and a significant gain in the downlink, and at the same time provide a total spectral efficiency per cell that quintuples in the uplink and triples in the downlink.

In this paper we study the performance of cellular networks when their base stations have an unlimited number of antennas. In previous work, the asymptotic behavior of the signal to interference plus nose ratio (SINR) was obtained. We revisit these results by deriving the rigorous expression for the SINR of both downlink and uplink in the scenario of infinite number of antennas. We show that the contamination of the channel estimates happens whenever a pilot sequence is received at a base station simultaneously with non-orthogonal signals coming from other users. We propose a method to avoid such simultaneous transmissions from adjacent cells, thus significantly decreasing interference. We also investigate the effects of power allocation in this interference-limited scenario, and show that it results in gains of over 15dB in the signal to interference ratio for the scenario simulated here. The combination of these two techniques results in rate gains of about 18 times in our simulations.

The papers in this special issue focus on large-scale multiple antenna wireless systems and services.

Large-Scale Antenna Systems (LSAS) is a form of multi-user MIMO technology in which unprecedented numbers of antennas serve a significantly smaller number of autonomous terminals. We compare the two most prominent linear pre-coders, conjugate beamforming and zero-forcing, with respect to net spectral-efficiency and radiated energy-efficiency in a simplified single-cell scenario where propagation is governed by independent Rayleigh fading, and where channel-state information (CSI) acquisition and data transmission are both performed during a short coherence interval. An effective-noise analysis of the pre-coded forward channel yields explicit lower bounds on net capacity which account for CSI acquisition overhead and errors as well as the sub-optimality of the pre-coders. In turn the bounds generate trade-off curves between radiated energy-efficiency and net spectral-efficiency. For high spectral-efficiency and low energy-efficiency zero-forcing outperforms conjugate beamforming, while at low spectral-efficiency and high energy-efficiency the opposite holds. Surprisingly, in an optimized system, the total LSAS-critical computational burden of conjugate beamforming may be greater than that of zero-forcing. Conjugate beamforming may still be preferable to zero-forcing because of its greater robustness, and because conjugate beamforming lends itself to a de-centralized architecture and de-centralized signal processing.

A multiplicity of autonomous terminals simultaneously transmits data streams to a compact array of antennas. The array uses imperfect channel-state information derived from transmitted pilots to extract the individual data streams. The power radiated by the terminals can be made inversely proportional to the square-root of the number of base station antennas with no reduction in performance. In contrast if perfect channel-state information were available the power could be made inversely proportional to the number of antennas. Lower capacity bounds for maximum-ratio combining (MRC), zero-forcing (ZF) and minimum mean-square error (MMSE) detection are derived. An MRC receiver normally performs worse than ZF and MMSE. However as power levels are reduced, the cross-talk introduced by the inferior maximum-ratio receiver eventually falls below the noise level and this simple receiver becomes a viable option. The tradeoff between the energy efficiency (as measured in bits/J) and spectral efficiency (as measured in bits/channel use/terminal) is quantified for a channel model that includes small-scale fading but not large-scale fading. It is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single-antenna system.

We consider multicell multiuser MIMO systems with a very large number of antennas at the base station (BS). We assume that the channel is estimated by using uplink training. We further consider a physical channel model where the angular domain is separated into a finite number of distinct directions. We analyze the so-called pilot contamination effect discovered in previous work, and show that this effect persists under the finite-dimensional channel model that we consider. In particular, we consider a uniform array at the BS. For this scenario, we show that when the number of BS antennas goes to infinity, the system performance under a finite-dimensional channel model with P angular bins is the same as the performance under an uncorrelated channel model with P antennas. We further derive a lower bound on the achievable rate of uplink data transmission with a linear detector at the BS. We then specialize this lower bound to the cases of maximum-ratio combining (MRC) and zero-forcing (ZF) receivers, for a finite and an infinite number of BS antennas. Numerical results corroborate our analysis and show a comparison between the performances of MRC and ZF in terms of sum-rate.

An LSAS entails a large number (tens or hundreds) of base station antennas serving a much smaller number of terminals, with large gains in spectral-efficiency and energy-efficiency compared with conventional MIMO technology. Until recently it was believed that in multi-cellular LSAS, even in the asymptotic regime, as the number of service antennas tends to infinity, the performance is limited by directed inter-cellular interference. The interference results from unavoidable re-use of reverse-link pilot sequences (pilot contamination) by terminals in different cells. We devise a new concept that leads to the effective elimination of inter-cell interference in TDD LSAS systems. This is achieved by outer multi-cellular pre-coding, which we call pilot contamination pre-coding (PCP). The main idea of PCP is that each base station linearly combines messages aimed to terminals from different cells that re-use the same pilot sequence. Crucially, the combining coefficients depend only on the slow-fading coefficients between the terminals and the base stations. Each base station independently transmits its PCP-combined symbols using conventional linear pre-coding that is based on estimated fast-fading coefficients. Further we derive estimates for SINRs and a capacity lower bound for the case of LSASs with PCP and finite number of antennas M.

Multi-user multiple-input multiple-output theory predicts manyfold capacity gains by leveraging many antennas on wireless base stations to serve multiple clients simultaneously through multi-user beamforming (MUBF). However, realizing a base station with a large number antennas is non-trivial, and has yet to be achieved in the real-world. We present the design, realization, and evaluation of Argos, the first reported base station architecture that is capable of serving many terminals simultaneously through MUBF with a large number of antennas (M >> 10). Designed for extreme flexibility and scalability, Argos exploits hierarchical and modular design principles, properly partitions baseband processing, and holistically considers real-time requirements of MUBF. Argos employs a novel, completely distributed, beamforming technique, as well as an internal calibration procedure to enable implicit beamforming with channel estimation cost independent of the number of base station antennas. We report an Argos prototype with 64 antennas and capable of serving 15 clients simultaneously. We experimentally demonstrate that by scaling from 1 to 64 antennas the prototype can achieve up to 6.7 fold capacity gains while using a mere 1/64th of the transmission power.

We consider multicell multiuser MIMO systems with a very large number of antennas at the base station. We assume that the channel is estimated by using uplink training sequences, and we consider a physical channel model where the angular domain is separated into a finite number of directions. We analyze the so-called pilot contamination effect discovered in previous work, and show that this effect persists under the finite-dimensional channel model that we consider. We further derive closed-form bounds on the achievable rate of uplink data transmission with maximum-ratio combining, for a finite and an infinite number of base station antennas.

In many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set ofNindependent, identically distributed measurements of anMdimensional random vector the maximum likelihood estimate is the sample covariance matrix. Here we consider the case whereNMsuch that this estimate is singular (noninvertible) and therefore fundamentally bad. We present a radically new approach to deal with this situation based on the idea of dimensionality reduction through an ensemble of isotropically random unitary matrices. We obtain two estimates cov and invcov which are estimates for the covariance matrix and the inverse covariance matrix respectively. Both estimates retain the original eigenvectors while altering the eigenvalues. We have a closed form analytical expression for cov and invcov in terms of the eigenvector/eigenvalue decomposition of the sample covariance. We motivate the use of invcov through applications to linear estimation, supervised learning, and high-resolution spectral estimation. We also compare the performance of these estimators with other more conventional methods.

This paper considers a multi-cell multiple antenna system with precoding used at the base stations for downlink transmission. Channel state information (CSI) is essential for precoding at the base stations. An effective technique for obtaining this CSI is time-division duplex (TDD) operation where uplink training in conjunction with reciprocity simultaneously provides the base stations with downlink as well as uplink channel estimates. This paper mathematically characterizes the impact that uplink training has on the performance of such multi-cell multiple antenna systems. When non-orthogonal training sequences are used for uplink training, the paper shows that the precoding matrix used by the base station in one cell becomes corrupted by the channel between that base station and the users in other cells in an undesirable manner. This paper analyzes this fundamental problem of pilot contamination in multi-cell systems. Furthermore, it develops a new multi-cell MMSE-based precoding method that mitigates this problem. In addition to being linear, this precoding method has a simple closed-form expression that results from an intuitive optimization. Numerical results show significant performance gains compared to certain popular single-cell precoding methods.

This paper considers the problem of interference mitigation in multi-cell multi-antenna time division duplex (TDD) wireless systems for downlink transmission. An efficient way to obtain channel state information (CSI) at the base station is by using uplink pilots and reciprocity of the downlink channel. At the same time, it has been shown that pilots from different cells contaminate each other, resulting in corruption of precoding matrices used by base stations, and high inter-cell interference. This paper studies the effects of shifting the location of pilots in time frames used in neighboring cells, and its effectiveness in obtaining better channel estimates, and, thereby, inter-cell interference reduction.

A cellular base station serves a multiplicity of single-antenna terminals over the same time-frequency interval. Time-division duplex operation combined with reverse-link pilots enables the base station to estimate the reciprocal forward- and reverse-link channels. The conjugate-transpose of the channel estimates are used as a linear precoder and combiner respectively on the forward and reverse links. Propagation, unknown to both terminals and base station, comprises fast fading, log-normal shadow fading, and geometric attenuation. In the limit of an infinite number of antennas a complete multi-cellular analysis, which accounts for inter-cellular interference and the overhead and errors associated with channel-state information, yields a number of mathematically exact conclusions and points to a desirable direction towards which cellular wireless could evolve. In particular the effects of uncorrelated noise and fast fading vanish, throughput and the number of terminals are independent of the size of the cells, spectral efficiency is independent of bandwidth, and the required transmitted energy per bit vanishes. The only remaining impairment is inter-cellular interference caused by re-use of the pilot sequences in other cells (pilot contamination) which does not vanish with unlimited number of antennas.

This paper considers a multi-cell multiple antenna system with precoding at the base stations for downlink transmission. To enable precoding, channel state information (CSI) is obtained via uplink training. This paper mathematically characterizes the impact that uplink training has on the performance of multi-cell multiple antenna systems. When non-orthogonal training sequences are used for uplink training, it is shown that the precoding matrix used by the base station in one cell becomes corrupted by the channel between that base station and the users in other cells. This problem of pilot contamination is analyzed in this paper. A multi-cell MMSE-based precoding is proposed that, when combined with frequency/time/pilot reuse techniques, mitigate this problem.

The spectral efficiency achievable with joint processing of pilot and data symbol observations is compared with that achievable through the conventional (separate) approach of first estimating the channel on the basis of the pilot symbols alone, and subsequently detecting the data symbols. Studied on the basis of a mutual information lower bound, joint processing is found to provide a non-negligible advantage relative to separate processing, particularly for fast fading. It is shown that, regardless of the fading rate, only a very small number of pilot symbols (at most one per transmit antenna and per channel coherence interval) should be transmitted if joint processing is allowed.

The ubiquity of inexpensive sensors implies that we can measure vector-valued data of ever increasing dimension. But the number of independent measurements of the data vector is limited so the sample covariance matrix is usually singular. The traditional remedy for singularity is diagonal loading - the addition of a small identity matrix to make the covariance estimate invertible. An alternative to diagonal loading is to reduce the dimension of the data vectors to be smaller than the number of independent observations through an ensemble of isotropically random (Haar measure) unitary matrices. For every member of the unitary ensemble, the shortened data vectors yield a statistically meaningful, invertible covariance estimate from which we can compute an estimate for the ultimate desired quantity. The final step is to take the expectation of this estimate with respect to the unitary ensemble. For a class of applications that includes adaptive spectral estimation, the design of a linear estimator, and supervised learning the random matrix approach results in an estimate for the inverse covariance matrix which preserves the eigenvectors of the sample covariance matrix, but alters the eigenvalues in a nontrivial manner. A closed-form expression for the expectation over the unitary ensemble eludes us, but we have obtained a tractable asymptotic expression. Preliminary numerical results indicate considerable promise for this approach.

Differential unitary space–time modulation (DUSTM) and its earlier nondifferential counterpart, USTM, permit high-throughput multiple-input multiple-output (MIMO) communication entirely without the possession of channel state information by either the transmitter or the receiver. For an isotropically random unitary input we obtain the exact closed-form expression for the probability density of the DUSTM received signal, permitting the straightforward Monte Carlo evaluation of its mutual information. We compare the performance of DUSTM and USTM through both numerical computations of mutual information and through the analysis of low- and high-signal-to-noise ratio (SNR) asymptotic expressions. In our comparisons the symbol durations of the equivalent unitary space–time signals are equal to$T$. For DUSTM the number of transmit antennas is constrained by the scheme to be$M = T/2$, while USTM has no such constraint. If DUSTM and USTM utilize the same number of transmit antennas at high SNRs the normalized mutual information of the two schemes expressed in bits/s/Hz are asymptotically equal, with the differential scheme performing somewhat better. At low SNRs the normalized mutual information of DUSTM is asymptotically twice the normalized mutual information of USTM. If, instead, USTM utilizes the optimum number of transmit antennas then USTM can outperform DUSTM at sufficiently low SNRs.

The singular value decomposition (SVD), a standard tool for theoretical and computational matrix analysis, represents a matrix as an ordered product of a unitary matrix, a nonnegative, real diagonal matrix, and a second unitary matrix. Our contribution is to extend the SVD representation to a matrix-valued impulse response. The representation involves a countably infinite number of vector-valued eigenfunctions and scalar singular values, and it provides the most concise and elegant description of the action of a matrix-valued filter (for example, a space-time communication channel) when driven by a finite-duration input signal.

Abstract Coexistence of different access technologies, hierarchical cellular deployment, a wide variety of data services, requirements for transparent operation across different technologies, adaptivity to varying network conditions and mobility and quality of service (QoS) constraints introduce a number of challenges in the design of future generation systems and the specification of new air interfaces, such as efficiency and flexibility in the utilization of spectrum, dynamic resource allocation and exploitation of the multiuser diversity and reconfigurable interference management and inter‐cell coordination. In this paper, three critical issues for the design of next generation systems are addressed: (i) duplexing, (ii) scheduling and resource allocation and (iii) interference and inter‐cell coordination. A number of research directions are presented, which constitute promising potential candidates for next generation systems specification. Copyright © 2005 John Wiley & Sons, Ltd.

Wireless data traffic is expected to grow over the next few years and the technologies that will provide data services are still being debated. One possibility is to use multiple antennas at base stations and terminals to get very high spectral efficiencies in rich scattering environments. Such multiple-input/multiple-output (MIMO) channels can then be used in conjunction with scheduling and rate-feedback algorithms to further increase channel throughput. This paper provides an analysis of the expected gains due to scheduling and bits needed for rate feedback. Our analysis requires an accurate approximation of the distribution of the MIMO channel mutual information. Because the exact distribution of the mutual information in a Rayleigh-fading environment is difficult to analyze, we prove a central limit theorem for MIMO channels with a large number of antennas. While the growth in average mutual information (capacity) of a MIMO channel with the number of antennas is well understood, it turns out that the variance of the mutual information can grow very slowly or even shrink as the number of antennas grows. We discuss implications of this "channel-hardening" result for data and voice services, scheduling, and rate feedback. We also briefly discuss the implications when shadow fading effects are included.

We analyze the capacity of a multiple-antenna wireless link with M antennas at the transmitter and N antennas at the receiver in a Rician fading channel when the channel is unknown at both the transmitter and the receiver. The Rician model is a nonstandard model with a Rayleigh component and an isotropically random rank-one specular component. The Rayleigh and specular components remain constant for T symbol periods, after which they change to completely independent realizations, and so on. To maximize mutual information over the joint density of T/spl middot/M complex transmitted signals it is sufficient to maximize over a joint density of min{T,M} real transmitted signal magnitudes. The capacity-achieving signal matrix is equal to the product of two independent matrices, a T/spl times/T isotropically random unitary matrix and a T/spl times/M real nonnegative diagonal matrix. If M>T, optimum signaling uses only T out of the M transmit antennas. We derive a novel lower bound on capacity which enables us to compute achievable rate regions for many cases. This lower bound is also valid for the case of purely Rayleigh-fading channels, where it has not been feasible, in general, to compute capacity, or mutual information. Our numerical results also indicate that the Rayleigh model is surprisingly robust: under our Rician model, up to half of the received energy can arrive via the specular component without significant reduction in capacity compared with the purely Rayleigh case.

[Guest Editors introduction to: Special issue on space-time transmission, reception, coding and signal processing] \n \nEvery episode of the classic 1966–1969 television series Star Trek begins with Captain Kirk’s (played by William Shatner) famous words : “Space: The final frontier….” While space may not be the final frontier for the information and communication theory community, it is proving to be an important and fruitful one. \n \nIn the information theory community, the notion of space can be broadly defined as the simultaneous use of multiple, possibly coupled, channels. The notions of space–time and multiple-input multiple-output (MIMO) channels are therefore often used interchangeably. The connection between space and MIMO is most transparent when we view the multiple channels as created by two or more spatially separated antennas at a wireless transmitter or receiver. \n \nA large component of the current interest in space–time methods can be attributed to discoveries in the late 1980s and early 1990s that a rich wireless scattering environment can be beneficial when multiple antennas are used on a point-to-point link. We now know that adding antennas in a rich environment provides proportional increases in point-to-point data rates, without extra transmitted power or bandwidth.

An important open problem in multiple-antenna communications theory is to compute the capacity of a wireless link subject to flat Rayleigh block-fading, with no channel-state information (CSI) available either to the transmitter or to the receiver. The isotropically random (i.r.) unitary matrix-having orthonormal columns, and a probability density that is invariant to premultiplication by an independent unitary matrix-plays a central role in the calculation of capacity and in some special cases happens to be capacity-achieving. We take an important step toward computing this capacity by obtaining, in closed form, the probability density of the received signal when transmitting i.r. unitary matrices. The technique is based on analytically computing the expectation of an exponential quadratic function of an i.r. unitary matrix and makes use of a Fourier integral representation of the constituent Dirac delta functions in the underlying density. Our formula for the received signal density enables us to evaluate the mutual information for any case of interest, something that could previously only be done for single transmit and receive antennas. Numerical results show that at high signal-to-noise ratio (SNR), the mutual information is maximized for M=min(N, T/2) transmit antennas, where N is the number of receive antennas and T is the length of the coherence interval, whereas at low SNR, the mutual information is maximized by allocating all transmit power to a single antenna.

We previously showed that arbitrarily reliable communication is possible within a single coherence interval in Rayleigh flat fading as the symbol duration of the coherence interval and the number of transmit antennas grow simultaneously. This effect, where the space-time signals act as their own channel codes, is called autocoding. For relatively short (e.g., 16-symbol) coherence intervals, a codebook of independent isotropically random unitary space-time signals theoretically supports transmission rates that are a significant fraction of autocapacity with an extremely low probability of error. The exploitation of space-time autocoding requires the creation and decoding of extraordinarily large constellations-typically L = 2/sup 80/. We make progress on the first part of the problem through a random, but highly structured, constellation that is completely specified by log/sub 2/ L independent isotropically distributed unitary matrices. The distinguishing property of this construction is that any two signals in the constellation are pairwise statistically independent and isotropically distributed. Thus, the pairwise probability of error, and hence the union bound on the block probability of error, of the structured constellation is identical to that of a fully random constellation of independent signals. We establish the limitations of an earlier construction through a subsidiary result that is interesting in its own right: the square (or for that matter, any integer power greater than one) of an isotropically random unitary matrix is not isotropically random, with the sole exception of the one-by-one unitary matrix.

Consider M independent users, each user having his own transmit antenna, that transmit simultaneously to a receiver equipped with N antennas through a Rayleigh block-fading channel having a coherence interval of T symbols, with no channel state information (CSI) available to either the transmitters or to the receiver. The total transmitted power is independent of the number of users. For a given coherence time T, we wish to identify the best multiaccess strategy that maximizes the total throughput. If perfect CSI were available to the receiver, it is known that the total capacity would increase monotonically with the number of users. If the CSI is available to both the receiver and to all transmitters, the throughput maximizing strategy implies for N=1 that only the single user who enjoys the best channel condition transmits. In the absence of CSI one is forced to a radically different conclusion. In particular, we show that if the propagation coefficients take on new independent values for every symbol (e.g., T=1) then the total capacity for any M > 1 users is equal to the capacity for M=1 user, in which case time division multiple access (TDMA) is an optimal scheme for handling multiple users. This result follows directly from a recent treatment of the single-user multiple antenna block-fading channel. Again, motivated by the single-user results, one is lead to the following conjecture for the multiple-user case: for any T > 1, the maximum total capacity can be achieved by no more than M = T users. The conjecture is supported by establishing the asymptotic result that, for a fixed N and a constant M/T for large T, the total capacity is maximized when M/T/spl rarr/0, which yields a total capacity per symbol of N log(1 + /spl rho/), where /spl rho/ is the expected signal-to-noise ratio (SNR) at the receiver. We further support the conjecture by examining the asymptotic behavior with large to for fixed M, T, and N /spl les/ T.

We present a transmit diversity technique for the downlink of (wideband) direct-sequence (DS) code division multiple access (CDMA) systems. The technique, called space-time spreading (STS), improves the downlink performance by using a small number of antenna elements at the base and one or more antennas at the handset, in conjunction with a novel spreading scheme that is inspired by space-time codes. It spreads each signal in a balanced way over the transmitter antenna elements to provide maximal path diversity at the receiver. In doing so, no extra spreading codes, transmit power or channel information are required at the transmitter and only minimal extra hardware complexity at both sides of the link. Both our analysis and simulation results show significant performance gains over conventional single-antenna systems and other open-loop transmit diversity techniques. Our approach is a practical way to increase the bit rate and/or improve the quality and range in the downlink of either mobile or fixed CDMA systems. A STS-based proposal for the case of two transmitter and single-receiver antennas has been accepted and will be included as an optional diversity mode in release A of the IS-2000 wideband CDMA standard.

We consider the computational cutoff rate and its implications on signal design for the complex quasi-static Rayleigh flat-fading spatio-temporal channel under a peak-power constraint where neither transmitter nor receiver know the channel matrix. The cutoff rate has an integral representation which is an increasing function of the distance between pairs of complex signal matrices. When the analysis is restricted to finite-dimensional sets of signals, interesting characterizations of the optimal rate-achieving signal constellation can be obtained. For an arbitrary finite dimension, the rate-optimal constellation must admit an equalizer distribution, i.e., a positive set of signal probabilities which equalizes the average distance between signal matrices in the constellation. When the number N of receive antennas is large, the distance-optimal constellation is nearly rate-optimal. When the number of matrices in the constellation is less than the ratio of the number of time samples to the number of transmit antennas, the rate-optimal cutoff rate attaining constellation is a set of equiprobable mutually orthogonal unitary matrices. When the signal-to-noise ratio (SNR) is below a specified threshold, the matrices in the constellation are rank one and the cutoff rate is achieved by applying all transmit power to a single antenna and using orthogonal signaling. Finally, we derive recursive necessary conditions and sufficient conditions for a constellation to lie in the feasible set.

Prior treatments of space-time communications in Rayleigh flat fading generally assume that channel coding covers either one fading interval-in which case there is a nonzero "outage capacity"-or multiple fading intervals-in which case there is a nonzero Shannon capacity. However, we establish conditions under which channel codes span only one fading interval and yet are arbitrarily reliable. In short, space-time signals are their own channel codes. We call this phenomenon space-time autocoding, and the accompanying capacity the space-time autocapacity. Let an M-transmitter antenna, N-receiver antenna Rayleigh flat fading channel be characterized by an M×N matrix of independent propagation coefficients, distributed as zero-mean, unit-variance complex Gaussian random variables. This propagation matrix is unknown to the transmitter, it remains constant during a T-symbol coherence interval, and there is a fixed total transmit power. Let the coherence interval and number of transmitter antennas be related as T=βM for some constant β. A T×M matrix-valued signal, associated with R·T bits of information for some rate R is transmitted during the T-symbol coherence interval. Then there is a positive space-time autocapacity Ca such that for all R

Motivated by information-theoretic considerations, we propose a signaling scheme, unitary space-time modulation, for multiple-antenna communication links. This modulation is ideally suited for Rayleigh fast-fading environments, since it does not require the receiver to know or learn the propagation coefficients. Unitary space-time modulation uses constellations of T/spl times/M space-time signals (/spl Phi//sub i/, l=1, ..., L), where T represents the coherence interval during which the fading is approximately constant, and M

We propose a systematic method for creating constellations of unitary space-time signals for multiple-antenna communication links. Unitary space-time signals, which are orthonormal in time across the antennas, have been shown to be well-tailored to a Rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients. The signals can achieve low probability of error by exploiting multiple-antenna diversity. Because the fading coefficients are not known, the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum-Euclidean-distance norm. Our construction begins with the first signal in the constellation-an oblong complex-valued matrix whose columns are orthonormal-and systematically produces the remaining signals by successively rotating this signal in a high-dimensional complex space. This construction easily produces large constellations of high-dimensional signals. We demonstrate its efficacy through examples involving one, two, and three transmitter antennas.

An elliptical region of the plane supports a positive-valued function whose Radon transform depends only on the slope of the integrating line. Any two parallel lines that intersect the ellipse generate equal line integrals of the function. We prove that this peculiar property is unique to the ellipse; no other convex, compact region of the plane supports a nonzero-valued function whose Radon transform depends only on slope. We motivate this problem by considering the detection of a constant-velocity moving object in a sequence of images. In the presence of additive, white, Gaussian noise. The intensity distribution of the object is known, but the velocity is only assumed to lie in some known set, for example, an ellipse or a rectangle. The object is to find a space-time linear filter, operating on the image sequence, whose minimum output signal-to-noise ratio (SNR) for any velocity in the set is maximized. For an ellipse (and its special cases, the disk and the line-segment) the special Radon transform property of the ellipse enables us to obtain a closed-form, analytical solution for the minimax filter, which significantly outperforms the conventional three-dimensional (3-D) matched filter. This analytical solution also suggests a constrained minimax filter for other velocity sets, obtainable in closed form, whose SNR can be very close to the minimax SNR.

We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flat-fading environment. The propagation coefficients between pairs of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence interval of T symbol periods, after which they change to new independent values which they maintain for another T symbol periods, and so on. Computing the link capacity, associated with channel coding over multiple fading intervals, requires an optimization over the joint density of T/spl middot/M complex transmitted signals. We prove that there is no point in making the number of transmitter antennas greater than the length of the coherence interval: the capacity for M>T is equal to the capacity for M=T. Capacity is achieved when the T/spl times/M transmitted signal matrix is equal to the product of two statistically independent matrices: a T/spl times/T isotropically distributed unitary matrix times a certain T/spl times/M random matrix that is diagonal, real, and nonnegative. This result enables us to determine capacity for many interesting cases. We conclude that, for a fixed number of antennas, as the length of the coherence interval increases, the capacity approaches the capacity obtained as if the receiver knew the propagation coefficients.

We combine certain results from two disparate areas, kinematics and geophysics, to obtain a convenient representation for the class of convex compact planar sets, in terms of a sequence of complex valued reflection coefficients. This gives a one-to-one relation between any convex compact planar set S and any set of parameters comprising: a) the coordinates of a reference point in /spl Sscr/, b) the circumference of the set, and c) a complex reflection coefficient sequence, {k/sub 1/, k/sub 2/,...}, such that 1) k/sub 1/=0, 2) |k/sub n/|/spl les/1, /spl forall/n, 3) if |k/sub N/|=1 for some N then k/sub n/=0, /spl forall/n>N. For a finite duration reflection coefficient sequence, where k/sub n/=0, /spl forall/n>N, if 0<|k/sub N/|<1 then the boundary of S is an infinitely differentiable convex curve, and if |k/sub N/|=1, then the boundary is an N-sided convex polygon.

The Barankin (1949) bound is the greatest lower bound on the variance of any unbiased estimate for a nonrandom parameter. Computing this bound yields, as a byproduct, an unbiased estimator that is at least locally best in the following sense. The estimator formula contains a reference parameter, when the unknown parameter happens to be equal to the reference, the variance of the estimate achieves the Barankin bound. If the dependence of the Barankin estimate on the reference parameter vanishes, then the estimate is also uniformly minimum variance. We obtain a simple derivation of the Barankin bound as the solution of an unconstrained convex quadratic optimization problem. In contrast the standard form of the Barankin bound involves the maximization of a ratio of quadratic quantities. For the case of PET inversion and natural gamma ray spectrometry, the Barankin estimate is only locally minimum variance, but it can be a viable alternative to the maximum likelihood estimate.

A convex elliptical region of the plane supports a positive-valued function of two variables whose Radon transform depends only on the slope of the integrating line: any two parallel lines that intersect the ellipse generate equal line integrals of the function. It is somewhat surprising that such a function exists. It is even more surprising, as we show in this paper, that the elliptical region is the only convex compact planar set that supports such a function. We discovered this result while attempting to extend an analytical solution technique for a 2-D integral equation that figures in a certain problem of motion detection in a noisy image sequence.

A polarimetric synthetic aperture radar (SAR) forms a complex vector-valued image where each pixel comprises the polarization-dependent reflectivity of a portion of a target or scene. The most common statistical model for this type of image is the zero-mean, circularly-symmetric, multivariate, complex Gaussian model. A logical generalization of this model is a circularly-symmetric, multivariate, complex Rician model which results from having a nonzero-mean complex target reflectivity. Direct maximum-likelihood estimation of the Rician model parameters is infeasible, since setting derivatives equal to zero results in an intractable system of coupled nonlinear equations. The contribution of the paper is a complete iterative solution to the Rician parameter estimation problem by means of the EM (expectation-maximization) algorithm.

The optimal algorithm for detecting a moving object in an image sequence that is corrupted by additive, white, Gaussian noise depends on the available prior knowledge of the object's velocity and its intensity distribution. This paper addresses the optimal detection problem where the object intensity distribution in known, but the object velocity is assumed only to be contained in some set, V. We formulate an optimal 3-D detection filter as the solution of a convex mini-max optimization problem, such that for all velocities contained in V, the minimum signal-to-noise ratio (SNR) improvement of the filter is maximized. The optimal filter is expressed in terms of a 2-D Lagrange multiplier function, that in turn satisfies a 2-D integral equation. For the case where V is a circular disk one can obtain a closed-form analytical solution for the optimal filter that gives a greater than 6.7 dB improvement in mini-max SNR compared with an optimized 3-D matched filter that is designed for the center velocity of the disk. Equivalently, for the same guaranteed SNR the optimal filter can cover a region of the velocity plane having an area 22 times greater than that covered by the 3-D matched filter. Put still another way, one optimal filter gives the same performance as a bank of 22 3-D matched filters.>

This paper develops a theory for the application of fan filters to moving objects. In contrast to previous treatments of the subject based on the 3-D Fourier transform, simplicity and insight are achieved by using the 3-D Radon transform. With this point of view, the Radon transform decomposes the image sequence into a set of plane waves that are parameterized by a two-component slowness vector. Fan filtering is equivalent to a multiplication in the Radon transform domain by a slowness response function, followed by an inverse Radon transform. The plane wave representation of a moving object involves only a restricted set of slownesses such that the inner product of the plane wave slowness vector and the moving object velocity vector is equal to one. All of the complexity in the application of fan filters to image sequences results from the velocity-slowness mapping not being one-to-one; therefore, the filter response cannot be independently specified at all velocities. A key contribution of this paper is to elucidate both the power and the limitations of fan filtering in this new application. A potential application of 3-D fan filters is in the detection of moving targets in clutter and noise. For example, an appropriately designed fan filter can reject perfectly all moving objects whose speed, irrespective of heading, is less than a specified cut-off speed, with only minor attenuation of significantly faster objects. A simple geometric construction determines the response of the filter for speeds greater than the cut-off speed.

A new approach to designing fan filters for moving objects that is based on mapping from velocity sets to slowness sets is presented. All of the complexity in the application of fan filters to image sequences results from the velocity-slowness mapping not being one-to-one, so the filter response cannot be independently specified at all velocities. Despite this limitation, the fan filter promises to be a powerful tool in image sequence analysis. Both the power and the limitations of the fan filter in this new application are elucidated.>

The up/down convolutional velocity filter is commonly used in vertical seismic profiling to separate an upgoing from a downgoing wave field. In the neighborhood of a velocity discontinuity one expects to see transient effects. For a simple medium containing a single velocity discontinuity it is possible to derive an explicit formula for these transients. This analysis predicts that the transients decay at a rate inversely proportional to the distance from the velocity discontinuity. Despite the slow decay rate, these transients are found to be quite insignificant.>

The frequency response of a finite-duration impulse response (FIR) filter is degraded by unknown variations in the values of the filter coefficients. If the filter coefficients can only be specified to within a certain percentage accuracy, the solution of a nested optimization problem gives a filter design such that over all possible coefficient variations, the worst frequency response is as good as possible in the Chebyshev sense. The size of the nested optimization problem grows exponentially with the length of the filter. However a tractable optimization problem can be solved whose solution gives a suboptimal filter design that is likely to be nearly as good as the globally optimal solution.

The determination of a power density spectrum from a finite set of correlation samples is an ill-posed problem. Furthermore. it is not possible even to bound the values that consistent power density spectra can take on at a particular point. A more reasonable problem is to try to determine the total spectral power in some frequency interval. Although this power cannot be determined exactly, upper and lower bounds on its possible values can be determined. This observation leads to a unified treatment of certain classical and modern spectral estimation techniques and to new interpretations for two data adaptive spectral estimators. maximum likelihood method (MLM) and data adaptive spectral estimator (DASE). According to these new interpretations. MLM and DASE provide upper bounds on spectral power in a specified frequency region subject to the assumption that the spectral density is constant in that region. These methods make no use of an extendibility constraint that can be used to obtain tight upper bounds, as well as nontrivial lower bounds on power. Cybenko has studied a related problem of bounding windowed power, for an arbitrary window, with no assumptions about the form of the spectral density. A new type of classical resolution limit for these bounds is derived and a numerical example is presented.

The mapping from a finite set of correlation samples to a power density spectrum is not unique. Furthermore, power density spectra exist that take on arbitrary values at a particular frequency and yet are consistent with the correlation samples. Thus values of the spectral density function at a particular frequency cannot be determined without further prior information. In a recent paper, Cybenko comments that tight upper and lower bounds on linear functionals of the spectral density can be obtained as solutions of semi-infinite linear programming problems. In this paper, the primal linear programming problem is interpreted as a search for extremal spectra and the dual linear programming problem is interpreted as a data-adaptive window design procedure. The effect of discretization on both the primal and dual problems is noted. Finally, it is shown how the dual window design problem can be used to design fixed classical type windows for the computation of suboptimal bounds.

This paper provides new interpretations for two modern spectral estimators, the Data Adaptive Spectral Estimator (DASE) of Davis and Regier, and the earlier Maximum Likelihood Method (MLM) of Capon, a special case of DASE. These methods provide estimates for spectral power in some region of frequency space, in terms of an estimate for a correlation matrix. They are conventionally interpreted as window-type spectral estimates, where the window is a function of the estimated correlation matrix. Assuming that the estimated correlation matrix is correct, it is shown that the problem of determining the spectral power is ill-posed. Specifically it is shown that DASE and MLM provide upper bounds on spectral power in some region of frequency space where the spectral density is assumed constant. Furthermore, it is shown that the assumptions and constraints that determine these upper bounds yield trivial lower bounds of zero.

Some recent results concerning the evaluation of autocorrelation functions associated with 2-D autoregressive (AR) spectra are reviewed. In contrast to the 1-D case, 2-D AR autocorrelation functions can, in general, only be evaluated by means of a numerical integration. However, if the minimum-phase whitening filter for the AR process has finite reflection coefficient support, then the autocorrelation function can be evaluated algebraically, either by means of a "backward" 2-D Levinson algorithm, or by means of a partial fraction expansion. The special properties of 2-D AR spectra of this class make them potentially useful in the problem of finding correlation-matching spectral estimates. The possibility of performing 2-D covariance extension by fitting AR models with finite reflection coefficient Support iS partially explored.

In this paper the reflection (or partial correlation) coefficient representation for 2-D minimum-phase filters is discussed. The representation is the basis for a new approach to designing 2-D all-pole recursive filters: the filters are designed by choosing appropriately a finite set of 2-D reflection coefficients, and stability is guaranteed merely by constraining the reflection coefficient magnitudes to be less than one.