Showing papers in "IEEE Transactions on Signal Processing in 2008"

Bayesian Compressive Sensing

Abstract: The data of interest are assumed to be represented as N-dimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately using only a small number M Lt N of basis-function coefficients associated with B. Compressive sensing is a framework whereby one does not measure one of the aforementioned N-dimensional signals directly, but rather a set of related measurements, with the new measurements a linear combination of the original underlying N-dimensional signal. The number of required compressive-sensing measurements is typically much smaller than N, offering the potential to simplify the sensing system. Let f denote the unknown underlying N-dimensional signal, and g a vector of compressive-sensing measurements, then one may approximate f accurately by utilizing knowledge of the (under-determined) linear relationship between f and g, in addition to knowledge of the fact that f is compressible in B. In this paper we employ a Bayesian formalism for estimating the underlying signal f based on compressive-sensing measurements g. The proposed framework has the following properties: i) in addition to estimating the underlying signal f, "error bars" are also estimated, these giving a measure of confidence in the inverted signal; ii) using knowledge of the error bars, a principled means is provided for determining when a sufficient number of compressive-sensing measurements have been performed; iii) this setting lends itself naturally to a framework whereby the compressive sensing measurements are optimized adaptively and hence not determined randomly; and iv) the framework accounts for additive noise in the compressive-sensing measurements and provides an estimate of the noise variance. In this paper we present the underlying theory, an associated algorithm, example results, and provide comparisons to other compressive-sensing inversion algorithms in the literature.

A Consistent Metric for Performance Evaluation of Multi-Object Filters

Abstract: The concept of a miss-distance, or error, between a reference quantity and its estimated/controlled value, plays a fundamental role in any filtering/control problem. Yet there is no satisfactory notion of a miss-distance in the well-established field of multi-object filtering. In this paper, we outline the inconsistencies of existing metrics in the context of multi-object miss-distances for performance evaluation. We then propose a new mathematically and intuitively consistent metric that addresses the drawbacks of current multi-object performance evaluation metrics.

Diffusion Least-Mean Squares Over Adaptive Networks: Formulation and Performance Analysis

Abstract: We formulate and study distributed estimation algorithms based on diffusion protocols to implement cooperation among individual adaptive nodes. The individual nodes are equipped with local learning abilities. They derive local estimates for the parameter of interest and share information with their neighbors only, giving rise to peer-to-peer protocols. The resulting algorithm is distributed, cooperative and able to respond in real time to changes in the environment. It improves performance in terms of transient and steady-state mean-square error, as compared with traditional noncooperative schemes. Closed-form expressions that describe the network performance in terms of mean-square error quantities are derived, presenting a very good match with simulations.

Consensus in Ad Hoc WSNs With Noisy Links— Part I: Distributed Estimation of Deterministic Signals

Abstract: We deal with distributed estimation of deterministic vector parameters using ad hoc wireless sensor networks (WSNs). We cast the decentralized estimation problem as the solution of multiple constrained convex optimization subproblems. Using the method of multipliers in conjunction with a block coordinate descent approach we demonstrate how the resultant algorithm can be decomposed into a set of simpler tasks suitable for distributed implementation. Different from existing alternatives, our approach does not require the centralized estimator to be expressible in a separable closed form in terms of averages, thus allowing for decentralized computation even of nonlinear estimators, including maximum likelihood estimators (MLE) in nonlinear and non-Gaussian data models. We prove that these algorithms have guaranteed convergence to the desired estimator when the sensor links are assumed ideal. Furthermore, our decentralized algorithms exhibit resilience in the presence of receiver and/or quantization noise. In particular, we introduce a decentralized scheme for least-squares and best linear unbiased estimation (BLUE) and establish its convergence in the presence of communication noise. Our algorithms also exhibit potential for higher convergence rate with respect to existing schemes. Corroborating simulations demonstrate the merits of the novel distributed estimation algorithms.

Quality of Service and Max-Min Fair Transmit Beamforming to Multiple Cochannel Multicast Groups

Abstract: The problem of transmit beamforming to multiple cochannel multicast groups is considered, when the channel state is known at the transmitter and from two viewpoints: minimizing total transmission power while guaranteeing a prescribed minimum signal-to-interference-plus-noise ratio (SINR) at each receiver; and a "fair" approach maximizing the overall minimum SINR under a total power budget. The core problem is a multicast generalization of the multiuser downlink beamforming problem; the difference is that each transmitted stream is directed to multiple receivers, each with its own channel. Such generalization is relevant and timely, e.g., in the context of the emerging WiMAX and UMTS-LTE wireless networks. The joint problem also contains single-group multicast beamforming as a special case. The latter (and therefore also the former) is NP-hard. This motivates the pursuit of computationally efficient quasi-optimal solutions. It is shown that Lagrangian relaxation coupled with suitable randomization/cochannel multicast power control yield computationally efficient high-quality approximate solutions. For a significant fraction of problem instances, the solutions generated this way are exactly optimal. Extensive numerical results using both simulated and measured wireless channels are presented to corroborate our main findings.

Novel Low-Density Signature for Synchronous CDMA Systems Over AWGN Channel

Abstract: Novel low-density signature (LDS) structure is proposed for transmission and detection of symbol-synchronous communication over memoryless Gaussian channel. Given N as the processing gain, under this new arrangement, users' symbols are spread over N chips but virtually only dv < N chips that contain nonzero-values. The spread symbol is then so uniquely interleaved as the sampled, at chip rate, received signal contains the contribution from only dc < K number of users, where K denotes the total number of users in the system. Furthermore, a near-optimum chip-level iterative soft-in-soft-out (SISO) multiuser decoding (MUD), which is based on message passing algorithm (MPA) technique, is proposed to approximate optimum detection by efficiently exploiting the LDS structure. Given beta = K/N as the system loading, our simulation suggested that the proposed system alongside the proposed detection technique, in AWGN channel, can achieve an overall performance that is close to single-user performance, even when the system has 200% loading, i.e., when beta = 2. Its robustness against near-far effect and its performance behavior that is very similar to optimum detection are demonstrated in this paper. In addition, the complexity required for detection is now exponential to dc instead of K as in conventional code division multiple access (CDMA) structure employing optimum multiuser detector.

Diffusion recursive least-squares for distributed estimation over adaptive networks

Abstract: We study the problem of distributed estimation over adaptive networks where a collection of nodes are required to estimate in a collaborative manner some parameter of interest from their measurements. The centralized solution to the problem uses a fusion center, thus, requiring a large amount of energy for communication. Incremental strategies that obtain the global solution have been proposed, but they require the definition of a cycle through the network. We propose a diffusion recursive least-squares algorithm where nodes need to communicate only with their closest neighbors. The algorithm has no topology constraints, and requires no transmission or inversion of matrices, therefore saving in communications and complexity. We show that the algorithm is stable and analyze its performance comparing it to the centralized global solution. We also show how to select the combination weights optimally.

Zero-Forcing Precoding and Generalized Inverses

Abstract: We consider the problem of linear zero-forcing precoding design and discuss its relation to the theory of generalized inverses in linear algebra. Special attention is given to a specific generalized inverse known as the pseudo-inverse. We begin with the standard design under the assumption of a total power constraint and prove that precoders based on the pseudo-inverse are optimal among the generalized inverses in this setting. Then, we proceed to examine individual per-antenna power constraints. In this case, the pseudo-inverse is not necessarily the optimal inverse. In fact, finding the optimal matrix is nontrivial and depends on the specific performance measure. We address two common criteria, fairness and throughput, and show that the optimal generalized inverses may be found using standard convex optimization methods. We demonstrate the improved performance offered by our approach using computer simulations.

One or Two Frequencies? The Empirical Mode Decomposition Answers

Abstract: This paper investigates how the empirical mode decomposition (EMD), a fully data-driven technique recently introduced for decomposing any oscillatory waveform into zero-mean components, behaves in the case of a composite two-tones signal. Essentially two regimes are shown to exist, depending on whether the amplitude ratio of the tones is greater or smaller than unity, and the corresponding resolution properties of the EMD turn out to be in good agreement with intuition and physical interpretation. A refined analysis is provided for quantifying the observed behaviors and theoretical claims are supported by numerical experiments. The analysis is then extended to a nonlinear model where the same two regimes are shown to exist and the resolution properties of the EMD are assessed.

Exact and Approximate Solutions of Source Localization Problems

Abstract: We consider least squares (LS) approaches for locating a radiating source from range measurements (which we call R-LS) or from range-difference measurements (RD-LS) collected using an array of passive sensors. We also consider LS approaches based on squared range observations (SR-LS) and based on squared range-difference measurements (SRD-LS). Despite the fact that the resulting optimization problems are nonconvex, we provide exact solution procedures for efficiently computing the SR-LS and SRD-LS estimates. Numerical simulations suggest that the exact SR-LS and SRD-LS estimates outperform existing approximations of the SR-LS and SRD-LS solutions as well as approximations of the R-LS and RD-LS solutions which are based on a semidefinite relaxation.

The Kernel Least-Mean-Square Algorithm

Abstract: The combination of the famed kernel trick and the least-mean-square (LMS) algorithm provides an interesting sample-by-sample update for an adaptive filter in reproducing kernel Hilbert spaces (RKHS), which is named in this paper the KLMS. Unlike the accepted view in kernel methods, this paper shows that in the finite training data case, the KLMS algorithm is well posed in RKHS without the addition of an extra regularization term to penalize solution norms as was suggested by Kivinen [Kivinen, Smola and Williamson, ldquoOnline Learning With Kernels,rdquo IEEE Transactions on Signal Processing, vol. 52, no. 8, pp. 2165-2176, Aug. 2004] and Smale [Smale and Yao, ldquoOnline Learning Algorithms,rdquo Foundations in Computational Mathematics, vol. 6, no. 2, pp. 145-176, 2006]. This result is the main contribution of the paper and enhances the present understanding of the LMS algorithm with a machine learning perspective. The effect of the KLMS step size is also studied from the viewpoint of regularization. Two experiments are presented to support our conclusion that with finite data the KLMS algorithm can be readily used in high dimensional spaces and particularly in RKHS to derive nonlinear, stable algorithms with comparable performance to batch, regularized solutions.

Gradient Pursuits

Abstract: Sparse signal approximations have become a fundamental tool in signal processing with wide-ranging applications from source separation to signal acquisition. The ever-growing number of possible applications and, in particular, the ever-increasing problem sizes now addressed lead to new challenges in terms of computational strategies and the development of fast and efficient algorithms has become paramount. Recently, very fast algorithms have been developed to solve convex optimization problems that are often used to approximate the sparse approximation problem; however, it has also been shown, that in certain circumstances, greedy strategies, such as orthogonal matching pursuit, can have better performance than the convex methods. In this paper, improvements to greedy strategies are proposed and algorithms are developed that approximate orthogonal matching pursuit with computational requirements more akin to matching pursuit. Three different directional optimization schemes based on the gradient, the conjugate gradient, and an approximation to the conjugate gradient are discussed, respectively. It is shown that the conjugate gradient update leads to a novel implementation of orthogonal matching pursuit, while the gradient-based approach as well as the approximate conjugate gradient methods both lead to fast approximations to orthogonal matching pursuit, with the approximate conjugate gradient method being superior to the gradient method.

Distributing the Kalman Filter for Large-Scale Systems

Abstract: This paper presents a distributed Kalman filter to estimate the state of a sparsely connected, large-scale, n -dimensional, dynamical system monitored by a network of N sensors. Local Kalman filters are implemented on nl-dimensional subsystems, nl Lt n, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an Lth order Gauss-Markov approximation to the centralized filter. We quantify the information loss due to this Lth-order approximation by the divergence, which decreases as L increases. The order of the approximation L leads to a bound on the dimension of the subsystems, hence, providing a criterion for subsystem selection. The (approximated) centralized Riccati and Lyapunov equations are computed iteratively with only local communication and low-order computation by a distributed iterate collapse inversion (DICI) algorithm. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter. Nowhere in the network, storage, communication, or computation of n-dimensional vectors and matrices is required; only nl Lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors. In other words, knowledge of the state is itself distributed.

Distributed Beamforming for Relay Networks Based on Second-Order Statistics of the Channel State Information

Abstract: In this paper, the problem of distributed beamforming is considered for a wireless network which consists of a transmitter, a receiver, and relay nodes. For such a network, assuming that the second-order statistics of the channel coefficients are available, we study two different beamforming design approaches. As the first approach, we design the beamformer through minimization of the total transmit power subject to the receiver quality of service constraint. We show that this approach yields a closed-form solution. In the second approach, the beamforming weights are obtained through maximizing the receiver signal-to-noise ratio (SNR) subject to two different types of power constraints, namely the total transmit power constraint and individual relay power constraints. We show that the total power constraint leads to a closed-form solution while the individual relay power constraints result in a quadratic programming optimization problem. The later optimization problem does not have a closed-form solution. However, it is shown that using semidefinite relaxation, this problem can be turned into a convex feasibility semidefinite programming (SDP), and therefore, can be efficiently solved using interior point methods. Furthermore, we develop a simplified, thus suboptimal, technique which is computationally more efficient than the SDP approach. In fact, the simplified algorithm provides the beamforming weight vector in a closed form. Our numerical examples show that as the uncertainty in the channel state information is increased, satisfying the quality of service constraint becomes harder, i.e., it takes more power to satisfy these constraints. Also our simulation results show that when compared to the SDP-based method, our simplified technique suffers a 2-dB loss in SNR for low to moderate values of transmit power.

Complete Characterization of the Pareto Boundary for the MISO Interference Channel

Abstract: In this correspondence, we study the achievable rate region of the multiple-input single-output (MISO) interference channel, under the assumption that all receivers treat the interference as additive Gaussian noise. Our main result is an explicit parametrization of the Pareto boundary for an arbitrary number of users and antennas. The parametrization describes the boundary in terms of a low-dimensional manifold. For the two-user case we show that a single real-valued parameter per user is sufficient to achieve all points on the Pareto boundary and that any point on the Pareto boundary corresponds to beamforming vectors that are linear combinations of the zero-forcing (ZF) and maximum-ratio transmission (MRT) beamformers. We further specialize the results to the MISO broadcast channel (BC). A numerical example illustrates the result.

Filter Bank Spectrum Sensing for Cognitive Radios

Abstract: The primary task in any cognitive radio (CR) network is to dynamically explore the radio spectrum and reliably determine portion(s) of the frequency band that may be used for the communication link(s). Accordingly, each CR node in the network has to be equipped with a spectrum analyzer. In this paper, we propose filter banks as a tool for spectrum sensing in CR systems. Various choices of filter banks are suggested and their performance are evaluated theoretically and through numerical examples. Moreover, the proposed spectrum analyzer is contrasted with the Thomson's multitaper (MT) method - a method that in the recent literature has been recognized as the best choice for spectrum sensing in CR systems. A novel derivation of the MT method that facilitates our comparisons as well as reveals an important aspect of the MT method that has been less emphasized in the recent literature is also presented.

Opportunistic Spectrum Access via Periodic Channel Sensing

Abstract: The problem of opportunistic access of parallel channels occupied by primary users is considered. Under a continuous-time Markov chain modeling of the channel occupancy by the primary users, a slotted transmission protocol for secondary users using a periodic sensing strategy with optimal dynamic access is proposed. To maximize channel utilization while limiting interference to primary users, a framework of constrained Markov decision processes is presented, and the optimal access policy is derived via a linear program. Simulations are used for performance evaluation. It is demonstrated that periodic sensing yields negligible loss of throughput when the constraint on interference is tight.

Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors

Abstract: The rapid developing area of compressed sensing suggests that a sparse vector lying in a high dimensional space can be accurately and efficiently recovered from only a small set of nonadaptive linear measurements, under appropriate conditions on the measurement matrix. The vector model has been extended both theoretically and practically to a finite set of sparse vectors sharing a common sparsity pattern. In this paper, we treat a broader framework in which the goal is to recover a possibly infinite set of jointly sparse vectors. Extending existing algorithms to this model is difficult due to the infinite structure of the sparse vector set. Instead, we prove that the entire infinite set of sparse vectors can be recovered by solving a single, reduced-size finite-dimensional problem, corresponding to recovery of a finite set of sparse vectors. We then show that the problem can be further reduced to the basic model of a single sparse vector by randomly combining the measurements. Our approach is exact for both countable and uncountable sets, as it does not rely on discretization or heuristic techniques. To efficiently find the single sparse vector produced by the last reduction step, we suggest an empirical boosting strategy that improves the recovery ability of any given suboptimal method for recovering a sparse vector. Numerical experiments on random data demonstrate that, when applied to infinite sets, our strategy outperforms discretization techniques in terms of both run time and empirical recovery rate. In the finite model, our boosting algorithm has fast run time and much higher recovery rate than known popular methods.

Covariance Structure Maximum-Likelihood Estimates in Compound Gaussian Noise: Existence and Algorithm Analysis

Abstract: Recently, a new adaptive scheme [Conte (1995), Gini (1997)] has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non-Gaussian noise. This latter has been modeled by compound-Gaussian noise, which is the product c of the square root of a positive unknown variable tau (deterministic or random) and an independent Gaussian vector x, c=radictaux. Because of the implicit algebraic structure of the equation to solve, we called the corresponding solution, the fixed point (FP) estimate. When tau is assumed deterministic and unknown, the FP is the exact maximum-likelihood (ML) estimate of the noise covariance structure, while when tau is a positive random variable, the FP is an approximate maximum likelihood (AML). This estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness. The major contribution of this paper is to fill these gaps. Our derivation is based on some likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts. Moreover, the corresponding iterative algorithm used for the FP estimate practical determination is also analyzed and we show the convergence of this recursive scheme, ensured whatever the initialization.

MIMO Radar Space–Time Adaptive Processing Using Prolate Spheroidal Wave Functions

Abstract: In the traditional transmitting beamforming radar system, the transmitting antennas send coherent waveforms which form a highly focused beam. In the multiple-input multiple-output (MIMO) radar system, the transmitter sends noncoherent (possibly orthogonal) broad (possibly omnidirectional) waveforms. These waveforms can be extracted at the receiver by a matched filterbank. The extracted signals can be used to obtain more diversity or to improve the spatial resolution for clutter. This paper focuses on space-time adaptive processing (STAP) for MIMO radar systems which improves the spatial resolution for clutter. With a slight modification, STAP methods developed originally for the single-input multiple-output (SIMO) radar (conventional radar) can also be used in MIMO radar. However, in the MIMO radar, the rank of the jammer-and-clutter subspace becomes very large, especially the jammer subspace. It affects both the complexity and the convergence of the STAP algorithm. In this paper, the clutter space and its rank in the MIMO radar are explored. By using the geometry of the problem rather than data, the clutter subspace can be represented using prolate spheroidal wave functions (PSWF). A new STAP algorithm is also proposed. It computes the clutter space using the PSWF and utilizes the block-diagonal property of the jammer covariance matrix. Because of fully utilizing the geometry and the structure of the covariance matrix, the method has very good SINR performance and low computational complexity.

A Theory for Sampling Signals From a Union of Subspaces

Abstract: One of the fundamental assumptions in traditional sampling theorems is that the signals to be sampled come from a single vector space (e.g., bandlimited functions). However, in many cases of practical interest the sampled signals actually live in a union of subspaces. Examples include piecewise polynomials, sparse representations, nonuniform splines, signals with unknown spectral support, overlapping echoes with unknown delay and amplitude, and so on. For these signals, traditional sampling schemes based on the single subspace assumption can be either inapplicable or highly inefficient. In this paper, we study a general sampling framework where sampled signals come from a known union of subspaces and the sampling operator is linear. Geometrically, the sampling operator can be viewed as projecting sampled signals into a lower dimensional space, while still preserving all the information. We derive necessary and sufficient conditions for invertible and stable sampling operators in this framework and show that these conditions are applicable in many cases. Furthermore, we find the minimum sampling requirements for several classes of signals, which indicates the power of the framework. The results in this paper can serve as a guideline for designing new algorithms for various applications in signal processing and inverse problems.

MIMO Relaying With Linear Processing for Multiuser Transmission in Fixed Relay Networks

Abstract: In this paper, a novel relaying strategy that uses multiple-input multiple-output (MIMO) fixed relays with linear processing to support multiuser transmission in cellular networks is proposed. The fixed relay processes the received signal with linear operations and forwards the processed signal to multiple users creating a multiuser MIMO relay. This paper proposes upper and lower bounds on the achievable sum rate for this architecture assuming zero-forcing dirty paper coding at the base station, neglecting the direct links from the base station to the users, and with certain structure in the relay. These bounds are used to motivate an implementable multiuser precoding strategy that combines Tomlinson-Harashima precoding at the base station and linear signal processing at the relay, adaptive stream selection, and QAM modulation. Reduced complexity algorithms based on the sum rate lower bounds are used to select a subset of users. We compare the sum rates achieved by the proposed system architecture and algorithms with the sum rate upper bound and the sum rate achieved by the decode-and-forward relaying.

Higher-Order SVD-Based Subspace Estimation to Improve the Parameter Estimation Accuracy in Multidimensional Harmonic Retrieval Problems

Abstract: Multidimensional harmonic retrieval problems are encountered in a variety of signal processing applications including radar, sonar, communications, medical imaging, and the estimation of the parameters of the dominant multipath components from MIMO channel measurements. R-dimensional subspace-based methods, such as R-D Unitary ESPRIT, R-D RARE, or R-D MUSIC, are frequently used for this task. Since the measurement data is multidimensional, current approaches require stacking the dimensions into one highly structured matrix. However, in the conventional subspace estimation step, e.g., via an SVD of the latter matrix, this structure is not exploited. In this paper, we define a measurement tensor and estimate the signal subspace through a higher-order SVD. This allows us to exploit the structure inherent in the measurement data already in the first step of the algorithm which leads to better estimates of the signal subspace. We show how the concepts of forward-backward averaging and the mapping of centro-Hermitian matrices to real-valued matrices of the same size can be extended to tensors. As examples, we develop the R-D standard Tensor-ESPRIT and the R-D Unitary Tensor-ESPRIT algorithms. However, these new concepts can be applied to any multidimensional subspace-based parameter estimation scheme. Significant improvements of the resulting parameter estimation accuracy are achieved if there is at least one of the R dimensions, which possesses a number of sensors that is larger than the number of sources. This can already be observed in the two-dimensional case.

Optimal Linear Precoding Strategies for Wideband Noncooperative Systems Based on Game Theory—Part I: Nash Equilibria

Abstract: In this two-part paper, we propose a decentralized strategy, based on a game-theoretic formulation, to find out the optimal precoding/multiplexing matrices for a multipoint-to-multipoint communication system composed of a set of wideband links sharing the same physical resources, i.e., time and bandwidth. We assume, as optimality criterion, the achievement of a Nash equilibrium and consider two alternative optimization problems: 1) the competitive maximization of mutual information on each link, given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies; and 2) the competitive maximization of the transmission rate, using finite order constellations, under the same constraints as above, plus a constraint on the average error probability. In this first part of the paper, we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies. Then, we prove that the optimal precoding/multiplexing scheme for both games leads to a channel diagonalizing structure, so that both matrix-valued problems can be recast in a simpler unified vector power control game, with no performance penalty. Thus, we study this simpler game and derive sufficient conditions ensuring the uniqueness of the Nash equilibrium. Interestingly, although derived under stronger constraints, incorporating for example spectral mask constraints, our uniqueness conditions have broader validity than previously known conditions. Finally, we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a Pareto-optimal centralized scheme. To reach the Nash equilibria of the game, in Part II, we propose alternative distributed algorithms, along with their convergence conditions.

Range Compression and Waveform Optimization for MIMO Radar: A CramÉr–Rao Bound Based Study

Abstract: A multi-input multi-output (MIMO) radar system, unlike standard phased-array radar, can transmit via its antennas multiple probing signals that may be correlated or uncorrelated with each other. This waveform diversity offered by MIMO radar enables superior capabilities compared with a standard phased-array radar. One of the common practices in radar has been range compression. We first address the question of ldquoto compress or not to compressrdquo by considering both the Cramer-Rao bound (CRB) and the sufficient statistic for parameter estimation. Next, we consider MIMO radar waveform optimization for parameter estimation for the general case of multiple targets in the presence of spatially colored interference and noise. We optimize the probing signal vector of a MIMO radar system by considering several design criteria, including minimizing the trace, determinant, and the largest eigenvalue of the CRB matrix. We also consider waveform optimization by minimizing the CRB of one of the target angles only or one of the target amplitudes only. Numerical examples are provided to demonstrate the effectiveness of the approaches we consider herein.

Signal Synthesis and Receiver Design for MIMO Radar Imaging

Abstract: Multiple-input-multiple-output (MIMO) radar is an emerging technology that has significant potential for advancing the state-of-the-art of modern radar. When orthogonal waveforms are transmitted, with M+N (N transmit and M receive) antennas, an MN-element filled virtual array can be obtained. To successfully utilize such an array for high-resolution MIMO radar imaging, constant-modulus transmit signal synthesis and optimal receive filter design play critical roles. We present in this paper a computationally attractive cyclic optimization algorithm for the synthesis of constant-modulus transmit signals with good auto- and cross-correlation properties. Then we go on to discuss the use of an instrumental variables approach to design receive filters that can be used to minimize the impact of scatterers in nearby range bins on the received signals from the range bin of interest (the so-called range compression problem). Finally, we present a number of numerical examples to demonstrate the effectiveness of the proposed approaches.

Distributed Average Consensus With Dithered Quantization

Abstract: In this paper, we develop algorithms for distributed computation of averages of the node data over networks with bandwidth/power constraints or large volumes of data. Distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted. We propose a distributed algorithm in which the nodes utilize probabilistically quantized information, i.e., dithered quantization, to communicate with each other. The algorithm we develop is a dynamical system that generates sequences achieving a consensus at one of the quantization values almost surely. In addition, we show that the expected value of the consensus is equal to the average of the original sensor data. We derive an upper bound on the mean-square-error performance of the probabilistically quantized distributed averaging (PQDA). Moreover, we show that the convergence of the PQDA is monotonic by studying the evolution of the minimum-length interval containing the node values. We reveal that the length of this interval is a monotonically nonincreasing function with limit zero. We also demonstrate that all the node values, in the worst case, converge to the final two quantization bins at the same rate as standard unquantized consensus. Finally, we report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios.

Linear Coherent Decentralized Estimation

Abstract: We consider the distributed estimation of an unknown vector signal in a resource constrained sensor network with a fusion center. Due to power and bandwidth limitations, each sensor compresses its data in order to minimize the amount of information that needs to be communicated to the fusion center. In this context, we study the linear decentralized estimation of the source vector, where each sensor linearly encodes its observations and the fusion center also applies a linear mapping to estimate the unknown vector signal based on the received messages. We adopt the mean squared error (MSE) as the performance criterion. When the channels between sensors and the fusion center are orthogonal, it has been shown previously that the complexity of designing the optimal encoding matrices is NP-hard in general. In this paper, we study the optimal linear decentralized estimation when the multiple access channel (MAC) is coherent. For the case when the source and observations are scalars, we derive the optimal power scheduling via convex optimization and show that it admits a simple distributed implementation. Simulations show that the proposed power scheduling improves the MSE performance by a large margin when compared to the uniform power scheduling. We also show that under a finite network power budget, the asymptotic MSE performance (when the total number of sensors is large) critically depends on the multiple access scheme. For the case when the source and observations are vectors, we study the optimal linear decentralized estimation under both bandwidth and power constraints. We show that when the MAC between sensors and the fusion center is noiseless, the resulting problem has a closed-form solution (which is in sharp contrast to the orthogonal MAC case), while in the noisy MAC case, the problem can be efficiently solved by semidefinite programming (SDP).

Sample Eigenvalue Based Detection of High-Dimensional Signals in White Noise Using Relatively Few Samples

Abstract: The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a mathematically justifiable, computationally simple, sample-eigenvalue-based procedure for estimating the number of high-dimensional signals in white noise using relatively few samples. The main motivation for considering a sample-eigenvalue-based scheme is the computational simplicity and the robustness to eigenvector modelling errors which can adversely impact the performance of estimators that exploit information in the sample eigenvectors. There is, however, a price we pay by discarding the information in the sample eigenvectors; we highlight a fundamental asymptotic limit of sample-eigenvalue-based detection of weak or closely spaced high-dimensional signals from a limited sample size. This motivates our heuristic definition of the effective number of identifiable signals which is equal to the number of ldquosignalrdquo eigenvalues of the population covariance matrix which exceed the noise variance by a factor strictly greater than . The fundamental asymptotic limit brings into sharp focus why, when there are too few samples available so that the effective number of signals is less than the actual number of signals, underestimation of the model order is unavoidable (in an asymptotic sense) when using any sample-eigenvalue-based detection scheme, including the one proposed herein. The analysis reveals why adding more sensors can only exacerbate the situation. Numerical simulations are used to demonstrate that the proposed estimator, like Wax and Kailath's MDL-based estimator, consistently estimates the true number of signals in the dimension fixed, large sample size limit and the effective number of identifiable signals, unlike Wax and Kailath's MDL-based estimator, in the large dimension, (relatively) large sample size limit.

Waveform Synthesis for Diversity-Based Transmit Beampattern Design

Abstract: Transmit beampattern design is a critically important task in many fields including defense and homeland security as well as biomedical applications. Flexible transmit beampattern designs can be achieved by exploiting the waveform diversity offered by an array of sensors that transmit probing signals chosen at will. Unlike a standard phased-array, which transmits scaled versions of a single waveform, a waveform diversity-based system offers the flexibility of choosing how the different probing signals are correlated with one another. Recently proposed techniques for waveform diversity-based transmit beampattern design have focused on the optimization of the covariance matrix of the waveforms, as optimizing a performance metric directly with respect to the waveform matrix is a more complicated operation. Given an , obtained in a previous optimization stage or simply pre-specified, the problem becomes that of determining a signal waveform matrix whose covariance matrix is equal or close to , and which also satisfies some practically motivated constraints (such as constant-modulus or low peak-to-average-power ratio constraints). We propose a cyclic optimization algorithm for the synthesis of such an , which (approximately) realizes a given optimal covariance matrix under various practical constraints. A numerical example is presented to demonstrate the effectiveness of the proposed algorithm.