One of the primary goals in spectrum occupancy mapping is to create a system that is robust to assumptions about the number of sensors, occupancy threshold (in dBm), sensor noise, number of emitters and the propagation environment. We show that such a system may be designed with neural networks using a process of aggregation to allow a variable number of sensors during training and testing. This process transforms the sensor measurements into log-likelihood ratios (LLRs), which are fed as a fixed-resolution image into a neural network. The use of LLRs provides robustness to the effects of noise and occupancy threshold, allowing a system to be trained for a nominal number of sensors, threshold and noise level, and still operate well at other levels without retraining. Our system operates without knowledge of the number of emitters and does not explicitly attempt to estimate their number or power. It is shown that even a sparse sensor distribution can render a high-resolution occupancy map. Receiver operating curves with realistic propagation environments using topographic maps with commercial network design tools show how performance of the neural network varies with the environment. Finally, we show that the use of noisy sensors with low dynamic range in this system can still yield good performance.

Low-resolution transceivers are being considered for millimeter-wave and higher frequency communications because of their simplicity and low power consumption. However, the non-linearities introduced by low-resolution digital-to-analog con- verters at the transmitters can cause significant out-of-band emissions since traditional bandwidth-limited pulse-shaping is not generally available. We model the performance of a low- resolution transmitter in terms of its spectral efficiency under out- of-band emission constraints. We show that the spectral efficiency can increase linearly with the symbol rate while satisfying out- of-band constraint. This implies that in order to achieve a given spectral efficiency under the bandwidth constraint, the symbol rate of the transmitter should be larger than a threshold. We derive an upper bound on this threshold.

We provide a mutual information lower bound that can be used to analyze the effect of training in models with unknown parameters. For large-scale systems, we show that this bound can be calculated using the difference between two derivatives of a conditional entropy function. We provide a step-by-step process for computing the bound, and apply the steps to a quantized large-scale multiple-antenna wireless communication system with an unknown channel. Numerical results demonstrate the interplay between quantization and training.

Future wearable and portable devices with multiple transmit antennas operating below 6 GHz are constrained by regulatory limitations on the level of electromagnetic radiation a user can be exposed to, measured using the specific absorption rate (SAR). Signaling designs that are optimized to include SAR constraints can improve the performance of uplink transmission. These signaling schemes could include closed-loop beamforming, closed-loop precoding, and space-time coding, which have all been shown to achieve increased rates when optimized as a function of SAR. Previous research addressed SAR constrained optimization only within a single coherence time block. In this paper, we present transmit policies that dynamically allocate user electromagnetic radiation exposure over time. We propose three exposure allocation methods - optimal, uniform, and asymptotic - in the practical case with causal channel state information (CSI), and an on-off transmission approach for the low SAR-to-noise ratio regime. Our results demonstrate that the performance of SAR-aware transmission can be further improved by exploiting frequency and time diversity.

We analyze the channel capacity of a system with a large number of one-bit transceivers in a classical Rayleigh environment with perfect channel information at the receiver. With M transmitters and N =αM receivers, we derive an expression of the capacity per transmitter C, where C ≤ min(1,α), as a function of a and signal-to-noise ratio (SNR) ρ, when M → ∞. We show that our expression is a good approximation for small M, and provide simple approximations of C for various ranges of α and ρ. We conclude that at high SNR, C reaches its upper limit of one only if α > 1.24. Expressions for determining when C “saturates” as a function of α and ρ are given.

To aggregate or not to aggregate, that is the question. Analog beamforming with phased arrays is a promising technique for 5G wireless communication in millimeter wave bands. However, beam squint degrades the performance of analog beamforming for wideband systems with a large number of antennas, because the array response varies with frequency. In this paper, we show that carrier aggregation for phased-array analog beamforming should take beam squint into consideration. Specifically, we study the optimal beam alignment to maximize channel capacity, and demonstrate that, with sufficient band separation, focusing on only one band outperforms carrier aggregation. Approximations are developed for a system with two bands to determine the critical values of system parameters like band separation, angle of arrival, and signal-to-noise ratio beyond which it is preferable not to aggregate.

Analog beamforming with phased arrays is a promising technique for 5G wireless communication in millimeter wave bands. A beam focuses on a small range of angles of arrival or departure and corresponds to a set of fixed phase shifts across frequency due to practical hardware constraints. In switched beamforming, a discrete codebook consisting of multiple beams is used to cover a larger angle range. However, for sufficiently large bandwidth, the gain provided by the phased array is frequency dependent even if the radiation pattern of the antenna elements is frequency independent, an effect called beam squint. This paper shows that the beam squint reduces channel capacity of a uniform linear array (ULA). The beamforming codebook is designed to compensate for the beam squint by imposing a channel capacity constraint. For example, our codebook design algorithm can improve the channel capacity by 17.8% for a ULA with 64 antennas operating at bandwidth of 2.5 GHz and carrier frequency of 73 GHz. Analysis and numerical examples suggest that a denser codebook is required to compensate for the beam squint compared to the case without beam squint. Furthermore, the effect of beam squint is shown to increase as bandwidth increases, and the beam squint limits the bandwidth given the number of antennas in the array.

Fifth generation (5G) and beyond cellular systems are expected to support multiple uplink transmit antennas. Previous research demonstrates that designing waveforms satisfying near field user exposure constraints affects the farfield data rates achievable by portable devices using multiple transmit antennas. Therefore, user exposure constraints need to be taken into account in the uplink transmission covariance matrix design (e.g., precoder design) for 5G. Specific absorption rate (SAR) is a widely accepted user exposure measurement used in wireless communication regulations throughout the world. In this paper, we perform sum-rate analysis for a multi-user multiple-input multiple-output (MIMO) system with SAR constraints enforced at each user. The maximum achievable sum rates for various channel state information at the transmitter scenarios are studied in this paper. The SAR-aware MIMO transmission methods are based on the modified waterfilling algorithm. Simulation results show our proposed methods outperform the conventional transmission strategy for the two user case.

Analog beamforming with phased arrays is a promising technique for 5G wireless communication at millimeter wave frequencies. Using a discrete codebook consisting of multiple analog beams, each beam focuses on a certain range of angles of arrival or departure and corresponds to a set of fixed phase shifts across frequency due to practical hardware considerations. However, for sufficiently large bandwidth, the gain provided by the phased array is actually frequency dependent, which is an effect called beam squint, and this effect occurs even if the radiation pattern of the antenna elements is frequency independent. This paper examines the nature of beam squint for a uniform linear array (ULA) and analyzes its impact on codebook design as a function of the number of antennas and system bandwidth normalized by the carrier frequency. The criterion for codebook design is to guarantee that each beam's minimum gain for a range of angles and for all frequencies in the wideband system exceeds a target threshold, for example 3 dB below the array's maximum gain. Analysis and numerical examples suggest that a denser codebook is required to compensate for beam squint. For example, 54% more beams are needed compared to a codebook design that ignores beam squint for a ULA with 32 antennas operating at a carrier frequency of 73 GHz and bandwidth of 2.5 GHz. Furthermore, beam squint with this design criterion limits the bandwidth or the number of antennas of the array if the other one is fixed.

We present a tracking algorithm to maintain the communication link between a base station (BS) and a mobile station (MS) in a millimeter wave (mmWave) communication system, where antenna arrays are used for beamforming in both the BS and MS. Downlink transmission is considered, and the tracking is performed at the MS as it moves relative to the BS. Specifically, we consider the case that the MS rotates quickly due to hand movement. The algorithm estimates the angle of arrival (AoA) by using variations in the radiation pattern of the beam as a function of this angle. Numerical results show that the algorithm achieves accurate beam alignment when the MS rotates in a wide range of angular speeds. For example, the algorithm can support angular speeds up to 800 degrees per second when tracking updates are available every 10 ms.

The Bode-Fano bounds on the integral over all frequency of the logarithm of the reflection coefficient between a source and a load, connected to each other through a matching network, are well-known measures of the bandwidth of the matching network and load. When there are multiple coupled loads being driven by independent sources, establishing the maximum possible bandwidth of the matching network and loads requires the development of a broadband matching theory that applies to coupled systems. We define a bandwidth measure for coupled loads based on the integral logarithm of their collective power reflection, and develop upper bounds on this measure. These upper bounds can be used to evaluate the broadband performance of matching networks for an arbitrary number of coupled loads.

Mobile handsets, classified as portable devices, are regulated on the amount of user electromagnetic exposure. The widely accepted exposure measurement is the specific absorption rate (SAR). Despite the prevalence of SAR constraints throughout the world, there has been barely any work on the design and analysis of communication signals for SAR-constrained wireless systems. In this paper, we show that multiple-antenna systems greatly reduce the SAR measurements when proper precoders are used. Fifth-generation (5G) and beyond cellular systems will be expected to support high rate uplinks, making multiple transmit antennas on user equipment a necessity. Our proposed SAR-aware transmission for multiple-antenna systems can be applied in 5G handsets to reduce the SAR and increase the rate. Assuming that channel knowledge is available at the transmitter and the receiver, we perform capacity analysis for multiple-antenna systems under both transmit power and SAR constraints. Analytical and numerical results demonstrate substantial performance improvements over schemes that ignore the SAR constraint. Our work shows that SAR-constrained precoders have structures similar to precoders designed for spatially correlated channels.

With the continuous evolution of cellular systems, portable wireless communication devices, such as mobile phones, are becoming more and more powerful, complex, and essential to our daily life. However, the debate over the health effects from using portable devices in close proximity has still not been settled. To limit the amount of user exposure, regulatory agencies in most countries set exposure threshold in terms of the specific absorption rate (SAR), measured in Watts per kilogram. SAR is a measure of the rate of electromagnetic energy absorption by the human body. Surprisingly, portable devices are often designed with little attention to the SAR thresholds, instead focusing on transmit power constraints. In this paper, we propose SAR-aware transmission that considers constraints on both transmit power and SAR with multiple antennas. We discover that the effect of a SAR constraint can be regarded in a way similar to a transmitter-side spatial correlation. Numerical results demonstrate the proposed method increases the capacity by 1.75 folds under the regulatory SAR limitation of 1.6 W/kg over schemes that only consider the power constraint.

We examine the design of space-time codes that allow simple encoding and decoding of high and low-priority streams of data. This paper comprises two parts. In the first part we introduce the system model, establish performance and complexity criteria, and introduce "direct-sum" codes that combine existing space-time codes with hierarchical modulation. In this second part, we show that the direct-sum codes of the first part can be greatly improved upon by non-direct-sum designs. We demonstrate unitary-transform\/ codes for two and four antennas. In particular, one such code performs 4 dB better than the direct-sum Alamouti code, with per-bit decoding complexity on one stream that is a bounded function of the rate of the other stream.

Multi-stream codes are used in wireless broadcast services where two or more priority classes of data are transmitted simultaneously, and a receiving terminal may decode one or more of the streams as a function of its receive signal to noise ratio. Present-day commercial cellular-based broadcast services generally utilize hierarchical modulation, a clever technique for embedding high and low-priority streams in a single modulated quadrature symbol. With many cellular standards moving to multiple-transmit antenna configurations, there are opportunities for designing new multi-stream encoding techniques that exploit these antennas. We examine the design of space-time codes that allow simple encoding and decoding of high and low-priority streams of data. A desirable multi-stream space-time code has a combination of good performance and low complexity. Performance is generally measured as coded bit-error rate, assuming max-log maximum aposteriori decoding. Complexity is measured as the effort needed to compute the aposteriori probabilities for either stream. Hierarchical modulation, for single-antenna transmissions, allows each stream to be decoded simply and independently. The paper comprises two parts. In this first part, we establish a general performance criterion for two-stream space-time codes and derive a formula for the complexity of max-log maximum aposteriori decoding of either stream. We also show how existing space-time codes may be combined with hierarchical modulation in a "direct sum". The direct-sum codes have low complexity, but we show in the second part of this paper that these codes can be significantly outperformed by non-direct-sum codes. One of our proposed two-stream codes performs 4 dB better than the Alamouti direct-sum code; it also has the benefit of decoding complexity in one stream that is a bounded function of the rate of the other.

In this paper, we show that the use of cyclic delay diversity (CDD) on the downlink (DL) can improve the performance of uplink (UL) diversity methods such as antenna selection in a time-division duplexing (TDD) system. An analytical framework for showing why CDD transmission on the DL improves UL diversity transmission in TDD system is provided. The analysis is used to quantify the gain in the UL received signal-to-noise ratio (SNR) when DL CDD is used. The analytical results show that for a 2times2 MIMO system that uses DL CDD, UL transmit diversity based on antenna selection improves the average UL SNR by 0.41 dB relative to the case when CDD is not being applied at the BS transmitter and improves the 1% CDF UL SNR by 1.58 dB. Measurement-based analysis validates the analytical results using measured frequency response of a MIMO system using a wideband channel sounder.

We analyze the sensitivity of the capacity of a multi-antenna multi-user system to the number of users being served. We show analytically that, for a given desired sum-rate, the extra power needed to serve a subset of the users at low SNR (signal-to-noise ratio) can be very small, and is generally much smaller than the extra power needed to serve the same subset at high SNR. The advantages of serving only subsets of the users are many: multi-user algorithms have lower complexity, reduced channel-state information requirements, and, often, better performance. We provide guidelines on how many users to serve to get near-capacity performance with low complexity. For example, we show how in an eight-antenna eight-user system we can serve only four users and still be approximately 2 dB from capacity at very low SNR.

Channel inversion is one of the simplest techniques for multiuser downlink systems with single-antenna users. In this paper, we extend the regularized channel inversion technique developed for the single-antenna user case to multiuser multiple- input multiple-output (MIMO) channels with multiple-antenna users. We first employ the multiuser preprocessing to project the multiuser signals near the null space of the unintended users based on the MMSE criterion, and then the single-user preprocessing is applied to the decomposed MIMO interference channels. In order to reduce the complexity, we focus on non- iterative solutions for the multiuser transmit beamforming and use a linear receiver based on an MMSE criterion. Simulation results show that the proposed scheme outperforms existing joint iterative algorithms in most multiuser configurations.

In this paper, we investigate spatial multiplexing schemes for closed-loop multiple-input multiple-output (MIMO) systems. The performance of the singular value decomposition (SVD) scheme is limited by the smallest singular value. When all the subchannels are utilized, uniform channel decomposition (UCD) was recently proposed to obtain a performance gain by making subchannels have equal gains. The UCD requires a successive interference cancellation (SIC) receiver, and thus it suffers from the error propagation inherent in the SIC receiver. We propose the blockwise UCD (BL-UCD) scheme which increases the minimum subchannel gain by pairing two singular values. The proposed scheme allows single-symbol decodable maximum-likelihood detection (MLD) instead of the SIC receiver. The simulation results demonstrate that the proposed BL-UCD scheme outperforms the SVD scheme and the conventional UCD at full spatial multiplexing for four transmit antennas and four receive antennas by 8 dB and 5 dB, respectively.

In this paper, we present a new transmission technique for the downlink of a multiuser multiple-input multiple- output (MIMO) system. Recently, the block-diagonalization (BD) algorithm has been proposed for the multiuser MIMO downlink where each user has multiple antennas. The key idea of the BD scheme is to eliminate all multiuser interference by transmitting each user's data along the nullspace of the other users' channel matrix. However, the BD algorithm based on the zero- interference condition is not a proper choice when the interference is known at the transmitter noncausally. In this paper, we propose a block-echelonization (BE) algorithm for utilizing the noncausally known interference at the transmitter. The sum rate analysis and simulation results show the effectiveness of the proposed multiuser scheme.

A vector-precoding technique known as "vector-perturbation" achieves near-capacity on a multi- antenna multi-user wireless channel. However, the technique currently requires choices for certain regularization and perturbation parameters and these parameters are currently chosen in a rather ad-hoc manner. We analyze the statistical properties of the vector- perturbation scheme and provide guidelines for choosing these parameters in a more systematic manner. These properties suggest that a large perturbation-parameter can give better performance at low SNR. We also propose a receiver that incorporates the statistical properties of the perturbation, thereby improving receiver performance over known methods. Finally, we incorporate an iteration loop between the demodulator and inner decoder. The net result of these improvements puts us approximately 1.5 dB closer to capacity.

A network of nodes in which pairs communicate over a shared wireless medium is analyzed. We consider the maximum total aggregate traffic flow possible as given by the number of users multiplied by their data rate. The model in this paper differs substantially from the many existing approaches in that the channel connections in this network are entirely random: rather than being governed by geometry and a decay-versus-distance law, the strengths of the connections between nodes are drawn independently from a common distribution. Such a model is appropriate for environments where the first-order effect that governs the signal strength at a receiving node is a random event (such as the existence of an obstacle), rather than the distance from the transmitter. It is shown that the aggregate traffic flow as a function of the number of nodes n is a strong function of the channel distribution. In particular, for certain distributions the aggregate traffic flow is at least n/(logn)/sup d/ for some d>0, which is significantly larger than the O(/spl radic/n) results obtained for many geometric models. The results provide guidelines for the connectivity that is needed for large aggregate traffic. The relation between the proposed model and existing distance-based models is shown in some cases.

We analyze a network of nodes in which pairs communicate over a shared wireless medium. We are interested in the maximum total aggregate traffic flow that is possible through the network. Our model differs substantially from the many existing approaches in that the channel connections in our network are entirely random: we assume that, rather than being governed by geometry and a decay law, the strength of the connections between nodes is drawn independently from a common distribution. Such a model is appropriate for environments where the first order effect that governs the signal strength at a receiving node is a random event (such as the existence of an obstacle), rather than the distance from the transmitter. We show that the aggregate traffic flow is a strong function of the channel distribution. In particular, we show that for certain distributions, the aggregate traffic flow scales at least as n/(log n)vfor some fixed v > 0, which is significantly larger than the O(radic/n) results obtained for many geometric models

Recent theoretical results describing the sum-capacity when using multiple antennas to communicate with multiple users in a known rich scattering environment have not yet been followed with practical transmission schemes that achieve this capacity. We introduce a simple encoding algorithm that achieves near-capacity at sum-rates of tens of bits/channel use. The algorithm is a variation on channel inversion that regularizes the inverse and uses a "sphere encoder" to perturb the data to reduce the energy of the transmitted signal. The paper is comprised of two parts. In this second part, we show that, after the regularization of the channel inverse introduced in the first part, a certain perturbation of the data using a "sphere encoder" can be chosen to further reduce the energy of the transmitted signal. The performance difference with and without this perturbation is shown to be dramatic. With the perturbation, we achieve excellent performance at all signal-to-noise ratios. The results of both uncoded and turbo-coded simulations are presented.

Recent theoretical results describing the sum capacity when using multiple antennas to communicate with multiple users in a known rich scattering environment have not yet been followed with practical transmission schemes that achieve this capacity. We introduce a simple encoding algorithm that achieves near-capacity at sum rates of tens of bits/channel use. The algorithm is a variation on channel inversion that regularizes the inverse and uses a "sphere encoder" to perturb the data to reduce the power of the transmitted signal. This work is comprised of two parts. In this first part, we show that while the sum capacity grows linearly with the minimum of the number of antennas and users, the sum rate of channel inversion does not. This poor performance is due to the large spread in the singular values of the channel matrix. We introduce regularization to improve the condition of the inverse and maximize the signal-to-interference-plus-noise ratio at the receivers. Regularization enables linear growth and works especially well at low signal-to-noise ratios (SNRs), but as we show in the second part, an additional step is needed to achieve near-capacity performance at all SNRs.

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.

Recent advancements in iterative processing of channel codes and the development of turbo codes have allowed the communications industry to achieve near-capacity on a single-antenna Gaussian or fading channel with low complexity. We show how these iterative techniques can also be used to achieve near-capacity on a multiple-antenna system where the receiver knows the channel. Combining iterative processing with multiple-antenna channels is particularly challenging because the channel capacities can be a factor of ten or more higher than their single-antenna counterparts. Using a "list" version of the sphere decoder, we provide a simple method to iteratively detect and decode any linear space-time mapping combined with any channel code that can be decoded using so-called "soft" inputs and outputs. We exemplify our technique by directly transmitting symbols that are coded with a channel code; we show that iterative processing with even this simple scheme can achieve near-capacity. We consider both simple convolutional and powerful turbo channel codes and show that excellent performance at very high data rates can be attained with either. We compare our simulation results with Shannon capacity limits for ergodic multiple-antenna channel.

Multiple-antenna wireless communication links promise very high data rates with low error probabilities, especially when the wireless channel response is known at the receiver. In practice, knowledge of the channel is often obtained by sending known training symbols to the receiver. We show how training affects the capacity of a fading channel-too little training and the channel is improperly learned, too much training and there is no time left for data transmission before the channel changes. We compute a lower bound on the capacity of a channel that is learned by training, and maximize the bound as a function of the received signal-to-noise ratio (SNR), fading coherence time, and number of transmitter antennas. When the training and data powers are allowed to vary, we show that the optimal number of training symbols is equal to the number of transmit antennas-this number is also the smallest training interval length that guarantees meaningful estimates of the channel matrix. When the training and data powers are instead required to be equal, the optimal number of symbols may be larger than the number of antennas. We show that training-based schemes can be optimal at high SNR, but suboptimal at low SNR.

[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.

Multiple antenna differential modulation using unitary matrices requires no channel knowledge at the receiver, and so is ideal for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Although this basic principle is well understood, it is not known how to generate good-performing constellations of unitary matrices, for any number of transmit and receive antennas and especially at high rates. We propose a class of Cayley codes that works with any number of antennas, and allows for polynomial-time near-maximum-likelihood decoding based on either successive nulling/cancelling or sphere decoding. The codes use the Cayley transform, which maps the highly nonlinear Stiefel manifold of unitary matrices to the linear space of skew-Hermitian matrices. This leads to a simple linear constellation structure in the Cayley transform domain and to an information-theoretic design criterion based on emulating a Cauchy random matrix. Simulations show that Cayley codes allow efficient and effective high-rate data transmission in multi-antenna communication systems without knowing the channel.

For an (N/sub 1/, N/sub 2/) interleaver of block size L, we prove that L/spl ges/N/sub 1/N/sub 2/. Conversely, we prove that whenever L/spl ges/N/sub 1/N/sub 2/, an (N/sub 1/, N/sub 2/) interleaver of block size L exists. Explicit algebraic constructions are provided. It is also proven that rectangular interleavers are not optimal.

One method for communicating with multiple antennas is to encode the transmitted data differentially using unitary matrices at the transmitter, and to decode differentially without knowing the channel coefficients at the receiver. Since channel knowledge is not required at the receiver, differential schemes are ideal for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Although this basic principle is well understood, it is not known how to generate good-performing constellations of unitary matrices, for any number of transmit and receive antennas and for any rate. This is especially true at high rates where the constellations must be rapidly encoded and decoded. We propose a class of Cayley codes that works with any number of antennas, and has efficient encoding and decoding at any rate. The codes are named for their use of the Cayley transform, which maps the highly nonlinear Stiefel manifold of unitary matrices to the linear space of skew-Hermitian matrices. This transformation leads to a simple linear constellation structure in the Cayley transform domain and to an information-theoretic design criterion based on emulating a Cauchy random matrix. Moreover, the resulting Cayley codes allow polynomial-time near-maximum-likelihood (ML) decoding based on either successive nulling/canceling or sphere decoding. Simulations show that the Cayley codes allow efficient and effective high-rate data transmission in multiantenna communication systems without knowing the channel.

Multiple-antenna systems that operate at high rates require simple yet effective space-time transmission schemes to handle the large traffic volume in real time. At rates of tens of bits per second per hertz, Vertical Bell Labs Layered Space-Time (V-BLAST), where every antenna transmits its own independent substream of data, has been shown to have good performance and simple encoding and decoding. Yet V-BLAST suffers from its inability to work with fewer receive antennas than transmit antennas-this deficiency is especially important for modern cellular systems, where a base station typically has more antennas than the mobile handsets. Furthermore, because V-BLAST transmits independent data streams on its antennas there is no built-in spatial coding to guard against deep fades from any given transmit antenna. On the other hand, there are many previously proposed space-time codes that have good fading resistance and simple decoding, but these codes generally have poor performance at high data rates or with many antennas. We propose a high-rate coding scheme that can handle any configuration of transmit and receive antennas and that subsumes both V-BLAST and many proposed space-time block codes as special cases. The scheme transmits substreams of data in linear combinations over space and time. The codes are designed to optimize the mutual information between the transmitted and received signals. Because of their linear structure, the codes retain the decoding simplicity of V-BLAST, and because of their information-theoretic optimality, they possess many coding advantages. We give examples of the codes and show that their performance is generally superior to earlier proposed methods over a wide range of rates and signal-to-noise ratios (SNRs).

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.

Multiple-antenna systems that operate at high rates require simple yet effective space-time transmission schemes to handle the large traffic volume in real time. V-BLAST, where every antenna transmits its own independent substream of data, has been shown to have good performance and simple encoding and decoding. Yet its drawbacks include its inability to work with fewer receive antennas than transmit antennas, and its absence of built-in spatial coding. On the other hand, there are many previously-proposed space-time codes that have good fading resistance and simple decoding, but generally poor performance at high data rates or with many antennas. We propose a high-rate coding scheme that can handle any configuration of transmit and receive antennas and that subsumes both V-BLAST (Vertical Bell Labs Layered Space-Time) and many proposed space-time codes as special cases. The scheme transmits substreams of data in linear combinations over space and time and the codes are designed to optimize the mutual information between the transmitted and received signals. Because of their linear structure, the codes retain the decoding simplicity of V-BLAST, and because of their information-theoretic optimality, they possess many coding advantages.

Many communications systems employ training, ie, the transmission of known signals, so that the channel parameters may be learned at the receiver. This has a dual effect: too little training and the channel is improperly learned, too much training and there is no time left for data transmission before the channel changes. We use an information-theoretic approach to find the optimal amount of training for frequency selective channels described by a block-fading model. When the training and data powers are allowed to vary, we show that the optimal number of training symbols is equal to the length of the channel impulse response. When the training and data powers are instead required to be equal, the optimal number of symbols may be larger. We further show that at high SNR training-based schemes are capable of capturing most of the channel capacity, whereas at low SNR they are highly suboptimal.

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.

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

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels.

We construct signal constellations for differential transmission with multiple basestation antennas. The signals are derived using the theory of fixed-point-free groups and are especially suitable for mobile cellular applications because they do not require the handset to have more than one antenna or to know the time-varying propagation environment. Yet we achieve full transmitter diversity and excellent performance gains over a single-antenna system.

We present a framework for differential modulation with multiple antennas across a continuously fading channel, where neither the transmitter nor the receiver knows the fading coefficients. The framework can be seen as a natural extension of standard differential phase-shift keying commonly used in single-antenna unknown-channel systems. We show how our differential framework links the unknown-channel system with a known-channel system, and we develop performance design criteria. As a special ease, we introduce a class of diagonal signals where only one antenna is active at any time, and demonstrate how these signals may be used to achieve full transmitter diversity and low probability of error.

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.

Hidden Markov processes such as the Gilbert-Elliott (1960) channel have an infinite dependency structure. Therefore, entropy and channel capacity calculations require knowledge of the infinite past. In practice, such calculations are often approximated with a finite past. It is commonly assumed that the approximations require an unbounded amount of the past as the memory in the underlying Markov chain increases. We show that this is not necessarily true. We derive an exponentially decreasing upper bound on the accuracy of the finite-past approximation that is much tighter than existing upper hounds when the Markov chain mixes well. We also derive an exponentially decreasing upper bound that applies when the Markov chain does not mix at all. Our methods are demonstrated on the Gilbert-Elliott channel, where we prove that a prescribed finite-past accuracy is quickly reached, independently of the Markovian memory. We conclude that the past can be used either to learn the channel state when the memory is high, or wait until the states mix when the memory is low. Implications fur computing and achieving capacity on the Gilbert-Elliott channel are discussed.

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.

Consider a system that quantizes and encodes analog data for transmission across an additive noise Gaussian channel. To minimize distortion, the channel code rate must be chosen to optimally allocate the available transmission rate between lossy source coding and block channel coding. We establish tight upper and lower bounds on the channel code rate that minimizes the average distortion of a vector quantizer cascaded with a channel coder and a Gaussian channel, thus extending some recently obtained results for the binary-symmetric channel. The upper hounds are obtained by averaging, whereas the lower bounds are uniform, over all possible index assignments. Analytic expressions are derived for large and small signal-to-noise ratios, and also for large source vector dimension. As in the binary-symmetric channel, the optimal channel code rate is often substantially smaller than the channel capacity and the distortion decays exponentially with the number of channel uses. Exact exponents are derived.

A fundamental problem in the transmission of analog information across a noisy discrete channel is the choice of channel code rate that optimally allocates the available transmission rate between lossy source coding and block channel coding. We establish tight bounds on the channel code rate that minimizes the average distortion of a vector quantizer cascaded with a channel coder and a binary-symmetric channel. Analytic expressions are derived in two cases of interest: small bit-error probability and arbitrary source vector dimension; arbitrary bit-error probability and large source vector dimension. We demonstrate that the optimal channel code rate is often substantially smaller than the channel capacity, and obtain a noisy-channel version of the Zador (1982) high-resolution distortion formula.

We present a method to simplify the analytical computation of the Cramer-Rao bound. The method circumvents bound calculations for so-called nuisance parameters. Under mild regularity conditions the technique, which replaces expectations with almost sure limits, can significantly lower the analytical complexity as compared to traditional methods. The dimension of a matrix that requires computation and inversion is reduced to the length of the parameter vector of interest. We give applications to random variables having densities in the exponential family. For normal distributions the resulting expressions take on particularly simple closed forms.>