We consider the problem of tracking moving targets using mobile wireless sensors (of possibly different types). This is a joint estimation and control problem in which a tracking system must take into account both target and sensor dynamics. We make minimal assumptions about the target dynamics, namely only that their accelerations are bounded. We develop a control law that determines the sensor motion control signals so as to maximize target resolvability as the target dynamics evolve. The method is given a tractable formulation that is amenable to an efficient search method and is evaluated in a series of experiments involving both round-trip time based ranging and Doppler frequency shift measurements. • A unified tracking-control framework for heterogeneous mobile sensors. • The control law improves target resolvability under bounded acceleration. • Sensor control accounts for target uncertainty over a receding horizon. • A tractable formulation enables efficient solution with first-order methods.

We consider the joint problem of online experiment design and parameter estimation for identifying nonlinear system models, while adhering to system constraints. We utilize a receding horizon approach and propose a new adaptive input design criterion, which is tailored to continuously updated parameter estimates, along with a new sequential estimator. We demonstrate the ability of the method to design informative experiments online, while steering the system within operational constraints.

In this letter we present a framework for estimating sparse covariance matrices, wherein we solve the$\ell _{0}-$norm penalized maximum likelihood estimation problem using the extended Bayesian information criterion (EBIC), a high dimensional model selection rule. The framework combines choosing the sparsity pattern and estimating the covariance matrix in a single step, eliminating the need for any hyper-parameter tuning. Using the framework we propose a cyclic majorization-minimization based technique and apply it to synthetic data to evaluate its performance in terms of normalized root mean square error (NRMSE) and Kullback Leibler (KL) divergence.

Multistatic localization techniques employ noisy range measurements collected via multiple transmitters and receivers to localize a target. However, in many realistic scenarios the data are corrupted by outliers which may be due to the failure of or malicious attack on one or more sensors. The presence of outliers leads to performance degradation in terms of target localization accuracy. In this letter, we address the problem of multistatic target localization when the measurements contain outliers. We employ a multi-hypothesis testing method based on the false discovery rate (FDR) to detect the outliers. More specifically, we consider a penalized maximum likelihood problem for joint estimation of the number and positions of the outliers as well as the target position, and the noise variance. To solve this problem, an iterative algorithm employing the majorization-minimization technique that minimizes the objective in a monotonic manner is developed. Through numerical simulations, we compare the proposed algorithm with other robust state-of-the-art algorithms and show that the proposed algorithm has superior performance.

The paper considers the problem of multi-objective decision support when outcomes are uncertain. We extend the concept of Pareto-efficient decisions to take into account the uncertainty of decision outcomes across varying contexts. This enables quantifying trade-offs between decisions in terms of tail outcomes that are relevant in safety-critical applications. We propose a method for learning efficient decisions with statistical confidence, building on results from the conformal prediction literature. The method adapts to weak or nonexistent context covariate overlap and its statistical guarantees are evaluated using both synthetic and real data.

In this letter we revisit the problem of smoothed nonparametric spectral estimation via cepstrum thresholding. We formulate the problem of cepstrum thresholding as a multiple hypothesis testing problem and use the false discovery rate (FDR) and familywise error rate (FER) procedures to threshold the cepstral coefficients. We compare the FDR and FER approaches with a previously proposed individual hypothesis testing approach and show that the cepstrum thresholding based on FDR and FER can yield spectral estimates with lower mean square error (MSE).

In this letter, we propose an algorithm for learning a sparse weighted graph by estimating its adjacency matrix under the assumption that the observed signals vary smoothly over the nodes of the graph. The proposed algorithm is based on the principle of majorization-minimization (MM), wherein we first obtain a tight surrogate function for the graph learning objective and then solve the resultant surrogate problem which has a simple closed form solution. The proposed algorithm does not require tuning of any hyperparameter and it has the desirable feature of eliminating the inactive variables in the course of the iterations - which can be used to speed up the algorithm. The numerical simulations conducted using both synthetic and real world (brain-network) data show that the proposed algorithm converges faster, in terms of the average number of iterations, than several existing methods in the literature.

In this letter we propose a new method to estimate the covariance matrix under the constraint that its off-diagonal elements are non-negative, which has applications to portfolio selection in finance. We incorporate the non-negativity constraint in the maximum likelihood (ML) estimation problem and propose an algorithm based on the block coordinate descent method to solve for the ML estimate. To study the effectiveness of the proposed algorithm, we perform numerical simulations on both synthetic and real-world financial data, and show that our proposed method has better performance than that of a state-of-the-art method.

In this letter we address the problem of multistatic target localization using time-delay measurements corrupted by noise with unknown non-uniform variances. More concretely, we consider the problem of joint maximum likelihood (ML) estimation of the target position and the noise variances, for which we propose a majorization-minimization based algorithm. The proposed approach is compared with state-of-the-art algorithms, and the simulation results show the excellent accuracy of our algorithm.

We address the problem of learning a decision policy from observational data of past decisions in contexts with features and associated outcomes. The past policy maybe unknown and in safety-critical applications, such as medical decision support, it is of interest to learn robust policies that reduce the risk of outcomes with high costs. In this paper, we develop a method for learning policies that reduce tails of the cost distribution at a specified level and, moreover, provide a statistically valid bound on the cost of each decision. These properties are valid under finite samples -- even in scenarios with uneven or no overlap between features for different decisions in the observed data -- by building on recent results in conformal prediction. The performance and statistical properties of the proposed method are illustrated using both real and synthetic data.

Predictors are learned using past training data which may contain features that are unavailable at the time of prediction. We develop an approach that is robust against outlying missing features, based on the optimality properties of an oracle predictor which observes them. The robustness properties of the approach are demonstrated on both real and synthetic data.

Spatial smoothing is a common preprocessing scheme for subspace methods that resolves their sensitivity to coherent sources. The source resolvability problem of spatial-smoothing-based subspace methods has been extensively investigated using different analysis techniques. In this paper, a unified Hadamard product technique is provided to recover these results. This is done by answering a long-standing question in linear algebra as to under what conditions the Hadamard product of two singular positive-semidefinite matrices is positive definite.

A spatial point process can be characterized by an intensity function which predicts the number of events that occur across space. In this paper, we develop a method to infer predictive intensity intervals by learning a spatial model using a regularized criterion. We prove that the proposed method exhibits out-of-sample prediction performance guarantees which, unlike standard estimators, are valid even when the spatial model is misspecified. The method is demonstrated using synthetic as well as real spatial data.

In many applications, different populations are compared using data that are sampled in a biased manner. Under sampling biases, standard methods that estimate the difference between the population means yield unreliable inferences. Here, we develop an inference method that is resilient to sampling biases and is able to control the false positive errors under moderate bias levels in contrast to the standard approach. We demonstrate the method using synthetic and real biomarker data.

Radio frequency interference (RFI) causes serious problems to ultrawideband (UWB) radar operations due to severely degrading radar imaging capability and target detection performance. This paper formulates proper data models and proposes novel methods for effective RFI mitigation. We first apply the single-snapshot Sparse Iterative Covariance-based Estimation (SPICE) algorithm to data from each pulse repetition interval for RFI mitigation and discuss the connection of SPICE to the l1-penalized least absolute deviation (l1-PLAD) approach. Then, we devise a modified group SPICE algorithm and we prove that it is equivalent to a special case of the l1,2-PLAD method. The modified group SPICE algorithm can be applied to data from a coherent processing interval for effective RFI mitigation. Both the single-snapshot SPICE and the modified group SPICE methods simultaneously exploit the sparsity properties of both RFI spectrum and UWB radar target echoes. Unlike the existing sparsity-based RFI suppression methods, such as the robust principal component analysis algorithm, the proposed methods are hyperparameter-free and therefore easier to use in practical applications. Furthermore, the fast implementation of the SPICE methods is considered by exploiting the special structures of both single-snapshot and multiple-snapshot covariance matrices. Finally, the results obtained from applying the SPICE methods to simulated data as well as measured data collected by the U.S. Army Research Laboratory synthetic aperture radar system are presented to demonstrate the effectiveness of the proposed methods.

The problem of model order selection from signed measurements obtained via 1-bit sampling is considered. The extension of the Bayesian information criterion (BIC) to the case of 1-bit sampling, referred to as 1bBIC, is proposed to determine the signal model order. A detailed analysis of and an application to sinusoidal signals are presented to demonstrate the performance of 1bBIC when the One-Bit RELAX algorithm is used to estimate the parameters of sinusoidal signals.

We derive a phase retrieval algorithm using the alternating direction method of multipliers. For the cost function obtained from the maximum likelihood criterion, we introduce auxiliary amplitude and phase variables to avoid the absolute value operator and decouple the determination of the auxiliary phase variables from that of the auxiliary amplitude variables. As a result, a phase retrieval algorithm consisting only of two least-squares steps is derived. The performance of the proposed algorithm is investigated via numerical examples, as well as an application to flat-spectrum periodic unimodular sequence design.

This paper considers the quantification of the prediction performance in Gaussian process regression. The standard approach is to base the prediction error bars on the theoretical predictive variance, which is a lower bound on the mean square-error (MSE). This approach, however, does not take into account that the statistical model is learned from the data. We show that this omission leads to a systematic underestimation of the prediction errors. Starting from a generalization of the Cramér-Rao bound, we derive a more accurate MSE bound which provides a measure of uncertainty for prediction of Gaussian processes. The improved bound is easily computed and we illustrate it using synthetic and real data examples.

In this paper, we propose a new approach to the design of training sequences that can be used for an accurate estimation of multi-input multi-output channels. The proposed method is particularly instrumental in training sequence designs that deal with three key challenges: 1) arbitrary channel and noise statistics that do not follow specific models, 2) limitations on the properties of the transmit signals, including total power, per-antenna power, having a constant-modulus, discrete-phase, or low peak-to-average-power ratio, and 3) signal design for large-scale or massive antenna arrays. Several numerical examples are provided to examine the proposed method.

Clock synchronization is ubiquitous in wireless systems for communication, sensing, and control. In this paper, we design a scalable system in which an indefinite number of passively receiving wireless units can synchronize to a single master clock at the level of discrete clock ticks. Accurate synchronization requires an estimate of the node positions to compensate the time-of-flight transmission delay in line-of-sight environments. If such information is available, the framework developed here takes position uncertainties into account. In the absence of such information, as in indoor scenarios, we propose an auxiliary localization mechanism. Furthermore, we derive the Cramer-Rao bounds for the system, which show that it enables synchronization accuracy at sub-nanosecond levels. Finally, we develop and evaluate an online estimation method, which is statistically efficient.

We consider the problem of sum-rate maximization in massive MIMO two-way relay networks with multiple (communication) operators employing the amplify-and-forward (AF) protocol. The aim is to design the relay amplification matrix (i.e., the relay beamformer) to maximize the achievable communication sum-rate through the relay. The design problem for the case of single-antenna users can be cast as a non-convex optimization problem, which in general, belongs to a class of NP-hard problems. We devise a method based on the minorization-maximization technique to obtain quality solutions to the problem. Each iteration of the proposed method consists of solving a strictly convex unconstrained quadratic program; this task can be done quite efficiently such that the suggested algorithm can handle the beamformer design for relays with up to ∼ 70 antennas within a few minutes on an ordinary PC. Such a performance lays the ground for the proposed method to be employed in massive MIMO scenarios.

The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic $\ell _{0}$ norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods.

We consider the problem of sum-rate maximization in multiple-input multiple-output (MIMO) amplify-and-forward relay networks with multi-operator. The aim is to design the MIMO relay amplification matrix (i.e., the relay beamformer) to maximize the achievable communication sum rate through the relay. The design problem for the case of single-antenna users can be cast as a non-convex optimization problem, which, in general, belongs to a class of NP-hard problems. We devise a method based on the minorization–maximization technique to obtain quality solutions to the problem. Each iteration of the proposed method consists of solving a strictly convex unconstrained quadratic program. This task can be done quite efficiently, such that the suggested algorithm can handle the beamformer design for relays with up to$\sim 70$antennas within a few minutes on an ordinary personal computer. Such a performance lays the ground for the proposed method to be employed in medium-scale (or lower regime massive) MIMO scenarios.

The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1D frequency estimation. Many related numerical tools have also been developed for multi-dimensional (MD), especially 2D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided under appropriate conditions. The new result is also used to study MD frequency estimation from compressive data within the recent super-resolution framework. A systematic approach is proposed and a numerical simulation is provided to demonstrate its effectiveness compared to the existing atomic norm method.

In this paper we present an algorithm for sequence design with magnitude constraints. We formulate the design problem in a general setting, but also illustrate its relevance to parallel excitation MRI. The formulated non-convex design optimization criterion is minimized locally by means of a cyclic algorithm, consisting of two simple algebraic sub-steps. Since the algorithm truly minimizes the criterion, the obtained sequence designs are guaranteed to improve upon the estimates provided by a previous method, which is based on the heuristic principle of the Iterative Quadratic Maximum Likelihood algorithm. The performance of the proposed algorithm is illustrated in two numerical examples.

In this paper, we study the problem of unimodular code design to improve the detection performance of statistical multiple-input multiple-output (MIMO) radar systems. To this end, we consider a system transmitting arbitrary unimodular signals and a discrete-time formulation of the problem. Due to the complicated form of the performance metric of the optimal detector, we resort to the Bhattacharyya distance for code design. We devise a novel method based on the majorization of matrix functions to obtain solutions to the constrained design problem. Simulation results show the effectiveness of the proposed method.

In this paper, we study the joint design of Doppler robust transmit sequence and receive filter to improve the performance of an active sensing system dealing with signal-dependent interference. The signal-to-interference-plus-noise ratio (SINR) of the filter output is considered as the performance measure of the system. The design problem is cast as a max-min optimization problem to robustify the system SINR with respect to the unknown Doppler shifts of the targets. To tackle the design problem, we devise a novel method to obtain optimized pairs of transmit sequence and receive filter sharing the desired robustness property.

In this paper, we study the problem of approaching peak periodic or aperiodic correlation bounds for complex-valued sets of sequences. In particular, novel algorithms based on alternating projections are devised to approach a given peak periodic or aperiodic correlation bound. Several numerical examples are presented to assess the tightness of the known correlation bounds as well as to illustrate the effectiveness of the proposed methods for meeting these bounds.

The NP-hard problem of optimizing a quadratic form over the unimodular vector set arises in radar code design scenarios as well as other active sensing and communication applications. To tackle this problem, a monotonically error-bound improving technique (MERIT) is proposed to obtain the global optimum or a local optimum of UQP with good sub-optimality guarantees. The provided sub-optimality guarantees are case-dependent and may outperform the π/4 approximation guarantee of semi-definite relaxation.

Due to its long-standing importance, the problem of designing the receive filter and transmit sequence for clutter/interference rejection in active sensing has been studied widely in the last decades. In this letter, we propose a cyclic optimization of the transmit sequence and the receive filter. The proposed approach can handle arbitrary peak-to-average-power ratio (PAR) constraints on the transmit sequence, and can be used for large dimension designs (with ~ 103variables) even on an ordinary PC.

In certain applications involving direction of arrival (DOA) estimation the operator may have a-priori information on some of the DOAs. This information could refer to a target known to be present at a certain position or to a reflection. In this study, the authors investigate a methodology for array processing that exploits the information on the known DOAs for estimating the unknown DOAs as accurately as possible. Algorithms are presented that can efficiently handle the case of both correlated and uncorrelated sources when the receiver is a uniform linear array. The authors find a major improvement in estimator accuracy in feasible scenarios, and they compare the estimator performance to the corresponding theoretical stochastic Cramér–Rao bounds as well as to the performance of other methods capable of exploiting such prior knowledge. In addition, real data from an ultra-sound array is applied to the investigated estimators.

Model selection is an important task in many signal processing applications. In this letter, we present a generalized likelihood ratio (GLR)-based derivation of the recently proposed EEF rule in an attempt to cast EEF in the main stream of model order selection approaches and provide further insights into its theoretical foundations. We also show that EEF can be expected to behave asymptotically (in the number of data samples) similarly to the Bayesian information criterion (BIC). To evaluate the finite sample performance we consider two numerical examples, including the selection of the number of components in a Gaussian mixture model (GMM), by means of which we show that EEF behaves similarly to BIC.

In this paper we introduce a new technique for estimating the parameters of the Keplerian model commonly used in radial velocity data analysis for extrasolar planet detection. The un known parameters in the Keplerian model, namely eccentricity e, orbital frequency f, periastron passage time T, longitude of periastron ω, and radial velocity amplitude K are estimated by a new approach named SPICE (a semi-parametric iterative covariance-based estimation technique). SPICE enjoys global convergence, does not require selection of any hyperparameters, and is computationally efficient (indeed computing the SPICE estimates boils down to solving a numerically efficient linear program (LP)). The parameter estimates obtained from SPICE are then refined by means of a relaxation-based maximum likelihood algorithm (RELAX) and the significance of the resultant estimates is determined by a generalized likelihood ratio test (GLRT). A real-life radial velocity data set of the star HD 9446 is analyzed and the results obtained are compared with those reported in the literature.

The cross ambiguity function (CAF) arises in many areas such as radar/sonar and communications when correlation processing is performed in the presence of a Doppler frequency shift. In this paper, the CAF synthesis problem is tackled: a pair of waveforms are jointly designed so that their CAF approximates a desired one. The so-generated waveforms have relatively low peak-to-average power ratios and in certain cases can be constant modulus. Numerical examples are provided to show the effectiveness of the proposed algorithm in synthesizing different types of CAF.

In this paper, Perfect Root-of-Unity Codes (PRUCs) with entries in αp= {x ∈ C | xp= 1} where p is a prime are studied. A lower bound on the number of distinct phases in PRUCs over αpis derived. We show that PRUCs of length L ≥ p(p - 1) must use all phases in αp. It is also shown that if there exists a PRUC of length L over αpthen p divides L. We derive equations (which we call principal equations) that give possible lengths of a PRUC over αptogether with their phase distribution. Using these equations, we prove for example that the length of a 3-phase perfect code must be of the form L = 1/4 (9h12+ 3h22) for (h1, h2) ∈ Z2and we also give the exact number of occurrences of each element from α3in the code. Finally, all possible lengths (≤100) of PRUCs over α5and α7together with their phase distributions are provided.

In this paper we present a new sparse iterative covariance-based estimation approach, called SPICE, to the direction of arrival estimation problem. SPICE is obtained by the minimization of a statistically well motivated covariance matrix fitting criterion and can be used in both single and multiple-snapshot cases. Some of the unique features enjoyed by SPICE are: it takes account of the noise in the data in a natural manner, it does not require selection of any hyper-parameters, and it has global convergence properties.

In a number of direction of arrival (DOA) estimation applications there exists prior knowledge about the sources whose bearings are to be determined. We study the case when this prior information concerns some of the source positions and their correlation state, which is a relevant case in, for example, RADAR scenarios where stationary objects exists in the regions of interest. Traditional DOA methods are not designed to exploit such information, and thus cannot obtain the highest theoretical accuracy. We present a method that can utilize in an asymptotically efficient manner both knowledge on some source positions and that the source signals are uncorrelated.

We consider a fast implementation of the weighted least-squares based iterative adaptive approach (IAA) for spectral estimation of uniformly sampled sequences. IAA is a robust, user parameter-free and nonparametric adaptive algorithm that can work with a single data sequence or snapshot. Compared with the conventional periodogram, IAA can be used to significantly increase the resolution and suppress the sidelobe levels. However, due to its high computational complexity, IAA can only be used in applications with short data sequences. We present herein a novel fast implementation of IAA using a Gohberg-Semencul (G-S)-type factorization of the IAA covariance matrix. By exploiting the Toeplitz structure of the said matrix, we are able to reduce the computational cost by two orders of magnitudes even for sequences with moderate lengths.

Two computationally efficient approaches are proposed for aeroacoustic source power estimation, under the assumption of a single dominant source contaminated by uncorrelated noise. The proposed methods are evaluated using both simulated and measured data. The numerical examples show that the proposed algorithms yield good power estimates even under low signal-to-noise ratio (SNR) conditions, and the experimental results show that the power estimates obtained via the proposed approaches are consistent with each other. Furthermore, the approaches are computationally much more efficient than the existing nonlinear least-squares (NLS) algorithm.

Sparse Bayesian learning (SBL) has been used as a signal recovery algorithm for compressed sensing. It has been shown that SBL is easy to use and can recover sparse signals more accurately than the well-known Basis Pursuit (BP) algorithm. However, the computational complexity of SBL is quite high, which limits its use in large-scale problems. We propose herein an efficient Gibbs sampling approach, referred to as GS-SBL, for compressed sensing. Numerical examples show that GS-SBL can be faster and perform better than the existing SBL approaches.

In this paper, we show that Sudoku puzzles can be formulated and solved as a sparse linear system of equations. We begin by showing that the Sudoku ruleset can be expressed as an underdetermined linear system: Ax = b, where A is of size m times n and n > m. We then prove that the Sudoku solution is the sparsest solution of Ax = b, which can be obtained by lo norm minimization, i.e. min ||x:||0s.t. Ax = b. Instead of this minimization SB problem, inspired by the sparse representation literature, we solve the much simpler linear programming problem of minimizing the l1norm of x, i.e. min ||x||1s.t. Ax = b, and show numerically that this approach solves representative Sudoku puzzles.

In this note, we show that the iterative frequency estimation technique proposed in is nothing but an approximate Gauss-Newton (GN) algorithm for minimizing the standard nonlinear least squares fitting criterion. Starting from the complex sinusoidal data model used in , we derive the GN algorithm for frequency estimation and show that by approximating some steps in the GN algorithm we arrive at the estimator proposed in . We then numerically compare the performances of GN and approximate GN algorithms with that of zero padded FFT.

We present a new derivation of a lower bound for anaperiodiccorrelation metric: the integrated sidelobe level (ISL) of a set of sequences under the energy constraint. Sequences (or sequence sets) with low aperiodic correlations are widely demanded in many applications, including radar/sonar range compression, medical imaging, channel estimation and multi-user spread-spectrum communications. While the lower bound has been implicitly discussed in the literature before, here we adopt a different framework to derive the bound. In particular, we make use in the derivation of our recently proposed cyclic algorithm framework, which can also be used to efficiently synthesize unimodular sequences with low correlations. We also show that by relaxing the unimodular constraint, the ISL lower bound can be approached closely.

Elfving theorem is a fundamental result in the area of optimal experiment design, and yet its available proofs require a number of somewhat indirect geometrical arguments that might detract a potential user from its full understanding and exploitation. In this letter, we provide a direct algebraic proof of this theorem. Furthermore, we make some connections with the ℓ1- norm minimization approach commonly used for sparse estimation, which suggest importation of algorithms and results from the latter area into that of optimal experiment design.

Sequence sets with low periodic correlations are used in many areas, such as asynchronous code-division multiple access (CDMA) systems, medical imaging, radar and sonar. Lower bounds on the integrated sidelobe level (ISL) and the peak sidelobe level (PSL) of periodic sequence sets, under a power constraint, have been previously derived in the literature. In this letter, we obtain the ISL and PSL lower bounds using a different framework. The main contribution of the letter consists in using this framework to derive closed-form expressions forallpower constrained periodic sequence sets that meet the ISL lower bound.

Square-root (in particular, Cholesky) factorization of Toeplitz matrices and of their inverses is a classical area of research. The Schur algorithm yields directly the Cholesky factorization of a symmetric Toeplitz matrix, whereas the Levinson algorithm does the same for the inverse matrix. The objective of this letter is to use results from the theory of rational orthonormal functions to derive square-root factorizations of the inverse of an ntimesn positive definite Toeplitz matrix. The main result is a new factorization based on the Takenaka-Malmquist functions, that is parameterized by the roots of the corresponding auto-regressive polynomial of order n . We will also discuss briefly the connection between our analysis and some classical results such as Schur polynomials and the Gohberg-Semencul inversion formula.

Unimodular (i.e., constant modulus) sequences with good autocorrelation properties are useful in several areas, including communications, radar and sonar. The integrated sidelobe level (ISL) is often used to express the goodness of the autocorrelation properties of a given sequence. In this paper, we present several cyclic algorithms for the local minimization of ISL-related metrics. To illustrate the performance of the proposed algorithms, we present a number of examples including the design of sequences that have virtually zero autocorrelation sidelobes in a specified lag interval, and of long sequences that could hardly be handled by means of other algorithms previously suggested in the literature.

We introduce a missing data recovery methodology based on a weighted least squares iterative adaptive approach (IAA). The proposed method is referred to as the missing-data IAA (MIAA) and it can be used for uniform or non-uniform sampling as well as for arbitrary data missing patterns. MIAA uses the IAA spectrum estimates to retrieve the missing data, based on a spectral least squares criterion similar to that used by IAA. Numerical examples are presented to show the effectiveness of MIAA for missing data recovery. We also show that MIAA can outperform an existing competitive approach, and this at a much lower computational cost.

Sequences with impulse-like correlations are at the core of several radar and communication applications. Two criteria that can be used to design such sequences, and which lead to rather different results in the aperiodic correlation case, are shown to be identical in the periodic case. Furthermore, two simplified versions of these two criteria, which similarly yield completely different sequences in the aperiodic case, are also shown to be equivalent. A corollary of these unexpected equivalences is that the periodic correlations of an arbitrary sequence must satisfy an intriguing identity, which is also presented in this letter.

We introduce a missing data recovery methodology based on a weighted least squares iterative adaptive approach (IAA). The proposed method is referred to as the missing-data IAA (MIAA) and it can be used for uniform or nonuniform sampling as well as for arbitrary data missing patterns. MIAA uses the IAA spectrum estimates to retrieve the missing data, by means of either a frequency domain or a time domain approach. Numerical examples are presented to show the effectiveness of MIAA for missing data reconstruction. In particular, we show that MIAA can outperform an existing competitive approach, and this at a much lower computational cost.

Penalized least-squares (PELS) rules for signal denoising can be obtained via the use of various information criteria (AIC, BIC, etc.) or various minmax LS approaches. Let S denote the set of "significant" parameters in the denoising problem (which is to be determined), let nsbe the dimension of S, and let nspdenote the penalty term of a PELS criterion. We show that, depending on the expression for p, the following cases can occur: type-1) If p does not depend on S, then denoising via the corresponding PELS rule is equivalent to simple thresholding; and type-2) If p depends on ns only, then the equivalence to thresholding no longer holds but the PELS rule can still be implemented quite efficiently. We also show that the use of BIC leads to an existing PELS rule of type-1 when the noise variance in the denoising problem is known, and to a novel PELS rule of type-2 when the noise variance is unknown.

The advance of cognitive radio (CR) technology put in evidence the need of new spectral estimation methods for proper labeling of licensed and un-licensed users. We present a new spectral estimation procedure for monitoring the radio spectrum. The estimate is derived from a different view point of traditional filter bank approach. The resulting method is able to detect a predetermined spectral shape forming part or contributing to a given data record, providing at the same time an estimate of its power level and its frequency location. We prove that traditional filter-bank spectral estimation reduces to a particular case of our procedure. The specific spectral shape to detect is named hereafter as the candidate spectrum. The major motivation for this procedure was the proper spectrum labeling of licensed users in cognitive radio scenarios. The performance of the spectral monitoring procedure is demonstrated in the detection of a BPSK primary user in a wireless scenario containing DVB-T emissions.

An interesting but apparently forgotten auto-regressive moving-average (ARMA) parameter estimation method, introduced by one of us in 1984, and refined later on by others, is revisited. In the process, we provide a new simpler derivation of the method as well as an enhanced version of its cepstrum-based step. We argue that this method has an appealing advantage over Durbin's method, which is probably the most frequently used non-iterative method for ARMA parameter estimation.

One of the most well-known robust adaptive beamforming approaches is diagonal loading. However, there are usually no clear guidelines on how to choose the diagonal loading level reliably. In this paper, we present algorithms that can compute the diagonal loading level fully automatically from the given data without the need of specifying any user parameters. The proposed diagonal loading algorithms use shrinkage-based covariance matrix estimates, instead of the conventional sample covariance matrix, in the standard Capon beamforming formulation. The performance of the resulting beamformers is illustrated via numerical examples and compared with other adaptive beamforming techniques.

Pulse compression radar systems make use of transmit code sequences and receive filters that are specially designed to achieve good range resolution and target detection capability at practically acceptable transmit peak power levels. The present paper is a contribution to the literature on the problem of designing transmit codes and receive filters for radar. In a nutshell: the main goal of this paper, which considers the cases of both negligible and non-negligible Doppler shifts, is to show how to design the receive filter (including its length) and the transmit code sequence via the optimization of a number of relevant metrics considered separately or in combination. The paper also contains several numerical studies whose aim is to illustrate the performance of the proposed designs.

In array processing, when the available snapshot number is comparable with or even smaller than the sensor number, the sample covariance matrix R is a poor estimate of the true covariance matrix R. To estimate R more accurately, we can make use of prior environmental knowledge, which is manifested as knowing an a priori covariance matrix R0. In this paper, we consider both modified general linear combinations (MGLC) and modified convex combinations (MCC) of the a priori covariance matrix R0, the sample covariance matrix R, and an identity matrix I to get an enhanced estimate of R, denoted as R. Numerical examples are provided to demonstrate the type of achievable performance by using R instead of R in the standard Capon beamformer.

The so-called merit factor approach (MFA) to radar binary sequence design has led to several theoretical contributions in fairly diverse research areas including information theory, computer science, combinatorial optimization, and analytical number theory. However, the MFA-which basically aims at minimizing the clutter effect on radar performance-implicitly assumes the use of a least squares (LS) receiver that is optimal only when there is no clutter. This problem can be eliminated by using a more general optimal instrumental-variables (IV) receiver in lieu of the LS receiver. The IV receiver can reject clutter more efficiently than the LS receiver. Additionally, the binary sequence design problem associated with the IV approach has an interesting form.

In this paper a number of covariance matrix estimators suggested in the literature are compared in terms of their performance in the context of array signal processing. More specifically they are applied in adaptive beamforming which is known to be sensitive to errors in the covariance matrix estimate and where often only a limited amount of data is available for estimation. As many covariance matrix estimators have the form of diagonal loading or eigenvalue adjustments of the sample covariance matrix and as they sometimes offer robustness to array imperfections and finite sample error, they are compared to a recent robustified adaptive Capon beamforming (RCB) method which also has a diagonal loading interpretation. Some of the covariance estimators show a significant improvement over the sample covariance matrix and in some cases they match the performance of the RCB even when a priori knowledge, which is not available in practice, is used for choosing the user parameter of RCB.

In this paper, we formulate the multi-pitch estimation problem and propose a number of methods to estimate the set of fundamental frequencies. The methods, which are based on nonlinear least-squares, multiple signal classification (MUSIC) and the Capon principles, have in common the fact that the multiple fundamental frequencies are estimated by means of a one-dimensional search. The statistical properties of the methods are evaluated via Monte Carlo simulations.

In this paper we derive a class of new parameter free robust adaptive beamformers using the generalized sidelobe canceler reparameterization of the Capon beamformer. In this parameterization the minimum variance beamformer is obtained as the solution of a linear least squares problem. In the case of an inaccurate steering vector and/or few data snapshots this marginally overdetermined system gives an ill fit causing signal cancellation in the standard minimum variance solution. By regularizing the problem using ridge regression techniques we get a whole class of robust adaptive beamformers, none of which requires the choice of a user parameter. We also propose a novel empirical Bayes-based ridge regression technique. The performance is compared to other robust adaptive beamformers.

The estimation of signal covariance matrices is a crucial part of many signal processing algorithms. In some applications, the structure of the problem suggests that the underlying, true, covariance matrix is the Kronecker product of two matrices. Examples of such problems are channel modelling for MIMO communications and signal modelling of EEG data. In applications it may also be that the Kronecker factors in turn can be assumed to possess additional, linear, structure. The maximum likelihood (ML) estimator for the problem has been proposed previously. It is asymptotically efficient but has the drawback of requiring an iterative search. Two methods that are both non-iterative and asymptotically efficient are proposed in this paper. The first method is derived from a well-known iterative maximization technique for the likelihood function. It performs on par with ML in simulations, but has the drawback of not allowing for extra structure in addition to the Kronecker structure. The second method is based on covariance matching principles, and does not suffer from this drawback. However, while the large sample performance is shown to be identical to ML, it performs somewhat worse in small samples than the first estimator. In addition, the Cramer-Rao lower bound (CRB) for the problem is derived in a compact form.

A MIMO (multi-input multi-output) radar system, unlike standard phased-array radar, can transmit via its antennas multiple probing signals. This waveform diversity offered by MIMO radar enables superior capabilities compared with a standard phased-array radar. 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. Numerical examples are provided to demonstrate the effectiveness of the approaches we consider herein.

The so-called merit factor approach (MFA) to radar binary sequence design has led to several theoretical contributions in fairly diverse research areas including information theory, computer science, combinatorial optimization, and analytical number theory. However, the pragmatic motivation of this approach is shown here to be questionable. Specifically, the MFA - which basically aims at minimizing thecluttereffect on radar performance - implicitly assumes the use of a least-squares (LS) receiver that is optimal only when there isno clutter. This apparent inconsistency in the problem formulation can be eliminated by using a more general optimal instrumental-variables (IV) receiver in lieu of the LS receiver. The IV receiver proposed here is shown to reject clutter much more efficiently than the LS receiver. Additionally, the binary sequence design problem associated with the IV receiver is shown to have an interesting form that is likely to attract the attention of the readers previously interested in the MFA.

A multi-input multi-output (MIMO) radar system, unlike a standard phased-array radar, can transmit multiple linearly independent probing signals via its antennas. We show herein that this waveform diversity enables the MIMO radar to significantly improve its parameter identifiability. Specifically, we show that the maximum number of targets that can be uniquely identified by the MIMO radar is up to$M_{t}$times that of its phased-array counterpart, where$M_{t}$is the number of transmit antennas.

Thresholding the cepstrum associated with the periodogram is a smoothing technique that appears to be very useful for variance reduction. Here the thresholding is performed via the methods SThresh and EbayesThresh. They both work fine in the broadband spectra case, even if some of the data is missing. The SThresh method appears to be more efficient as it shows a smaller variance and is faster computationally. The smoothing methods are also shown to perform well on a real-life broadband signal

Consider a vector of independent normal random variables with unknown means but known variances. Our problem is to reduce the total variance of these random variables by exploiting the prior information that a significant proportion of them have "small" means. We show that thresholding is an effective means of solving this problem, and propose two schemes for threshold selection: one based on a uniformly most powerful unbiased test, the other on a Bayesian information criterion selection rule. As an example application we consider cepstral analysis and we show via numerical simulation that the simple thresholding scheme proposed herein can achieve significant reductions of total variance

We present two improved Multi-static Adaptive Microwave Imaging (MAMI) methods: MAMI-2 and MAMI-C, for early breast cancer detection. MAMI is one of the microwave imaging modalities based the significant contrast between the di-electric properties of normal and malignant breast tissues and employs multiple antennas that take turns to transmit ultra wideband (UWB) pulses while all antennas are used to receive the reflected signals. The MAMI methods we investigate herein utilize the data-adaptive robust Capon beamformer (RCB) to achieve high resolution and interference suppression. We will demonstrate the effectiveness of our proposed methods for breast cancer detection via numerical examples with data simulated using the finite difference time domain (FDTD) method based on a 3-D realistic breast model.

We consider the problem of smoothed nonparametric estimation of two-dimensional (2-D) spectra. In the one-dimensional (1-D) scenario, several methods have been developed in the past for the computation of nonparametric spectral estimates with lower variance than that of the standard periodogram. Some of these techniques can also be extended to the 2-D case. However, such methods usually require a careful selection of certain design parameters, which can be hard to make. The spectral estimator proposed here is based on cepstrum thresholding and is shown to have significantly lower variance than the standard 2-D periodogram. Moreover, the thresholding is performed in a simple and practically automatic manner without the requirement of extensive prior knowledge about the signal of interest.

In wireless communication systems, channel state information is often assumed to be available at the receiver. Traditionally, a training sequence is used to obtain the estimate of the channel. Alternatively, the channel can be identified using known properties of the transmitted signal. However, the computational effort required to find the joint ML solution to the symbol detection and channel estimation problem increases exponentially with the dimension of the problem. To significantly reduce this computational effort, we formulate the joint ML estimation and detection as an integer least-squares problem, and show that for a wide range of signal-to-noise ratios (SNR) and problem dimensions it can be solved via sphere decoding with expected complexity comparable to the complexity of heuristic techniques

The expectation-maximization (EM) algorithm is often used in maximum likelihood (ML) estimation problems with missing data. However, EM can be rather slow to converge. In this paper, we introduce a new algorithm for parameter estimation problems with missing data, which we call equalization-maximization (EqM) (for reasons to be explained later). We derive the EqM algorithm in a general context and illustrate its use in the specific case of a Gaussian autoregressive time series with a varying amount of missing observations. In the presented examples, EqM outperforms EM in terms of computational speed, at a comparable estimation performance.

We consider two-dimensional (2D) nonparametric complex spectral estimation (with its 1D counterpart as a special case) of data matrices with missing samples occurring in arbitrary patterns. Previously, the MAPES-EM algorithms were developed for the general 1D missing-data problem and shown to have excellent spectral estimation performance. In this paper, we present 2D extensions of MAPES-EM and develop another 2D MAPES algorithm, referred to as MAPES-CM, which solves a maximum likelihood problem iteratively via cyclic maximization (CM). Compared with MAPES-EM, MAPES-CM has similar spectral estimation performance but is computationally much more efficient.

We consider the relationship between several single-symbol detectors for double differential modulation. We have shown in a previous paper that a simple heuristic detector is identical to the generalized likelihood ratio test (GLRT) detector. In this correspondence, we introduce a maximum a posteriori probability (MAP) detector and show that it is identical to the heuristic and GLRT detectors. Consequently, neither GLRT nor MAP can offer any gain over the simple heuristic detector, which means that the latter should be the detector of choice.

In this paper, we address the problem of identifying convolutive channels using a Gaussian maximum-likelihood (ML) approach when short training sequences (possibly shorter than the channel impulse-response length) are periodically inserted in the transmitted signal. We consider the case where the channel is quasi-static (i.e., the sampling period is several orders of magnitude smaller than the coherence time of the channel). Several training sequences can thus be used in order to produce the channel estimate. The proposed method can be classified as semiblind and exploits all channel-output samples containing contributions from the training sequences (including those containing contributions from the unknown surrounding data symbols). Experimental results show that the proposed method closely approaches the Cramer-Rao bound and outperforms existing training-based methods (which solely exploit the channel-output samples containing contributions from the training sequences only). Existing semiblind ML methods are tested as well and appear to be outperformed by the proposed method in the considered context. A major advantage of the proposed approach is its computational complexity, which is significantly lower than that of existing semiblind methods.

This paper presents a new method for adaptive equalization of communication channels with intersymbol interference, for which the impulse response has an unknown length. Our approach is based on a parameterization of the impulse response as a Gaussian mixture model, whose parameters are estimated from a training sequence. It is shown that the new method can outperform conventional adaptive equalizers using training to estimate the channel.

We consider the problem of soft-detection for communication systems employing double differential modulation and forward error correction codes, for example, a convolutional code. We propose a soft-detector (based on the multiple symbol heuristic detector) with low computational complexity. Simulation results are provided to show the superior performance of the new soft-detector.

The information criterion (IC) rule and the generalized likelihood ratio test (GLRT) have been usually considered to be two rather different approaches to model order selection. However, we show here that a natural implementation of the GLRT is, in fact, equivalent to the IC rule. A consequence of this equivalence is that a specific IC rule, such as Akaike IC or Bayesian IC, can be viewed as a more direct way of implementing a GLRT with a specific threshold. Another consequence of the equivalence, which is emphasized herein, is a possibly original way of exploiting the information provided by the local behavior of an IC for selecting the structure of sparse models (the parameter vectors of which comprise "many" elements equal to zero).

Two algorithms for double-differential detection of multiple phase-shift keying modulation are proposed, based on a least-squares criterion and a generalized-likelihood ratio test, respectively. While both algorithms take advantage of the performance gain obtained by observing the received signal over an observation interval longer than that required for symbol-by-symbol detection, the former has the important advantage of reduced implementation complexity, whereas the latter offers better performance.

The maximum-likelihood detector (MLD), also called the generalized likelihood ratio test (GLRT) detector, for single differential modulation is easy to derive. On the other hand, the MLD problem associated with double differential modulation is much more complicated and solving it was deemed to require a computationally unattractive nonlinear search. Consequently, a simple heuristic detector was usually preferred to the MLD, on computational grounds. In this correspondence, we prove the somewhat unexpected result that the aforementioned heuristic detector coincides with the exact MLD.

In this paper, we address the general problem of identifying convolutive channels when several training sequences are inserted in the transmitted data symbols stream. We analyze the general situation where the training sequences differ from each other. We consider quasi-static channels (i.e. the sampling period is several orders of magnitude below the coherence time of the channel). There are no requirements on the length of the training sequence and all the received symbols that contain contributions from the training symbols are used for the identification. We first propose an iterative method that quickly converges to the maximum likelihood (ML) channel estimate. We also derive a simple closed form expression that approximates the ML channel estimate.

The amplitude estimation of a signal whose waveform is known (up to an unknown scaling factor) in the presence of interference and noise is of interest in several applications including using the emerging quadrupole resonance (QR) technology for explosive detection. In such applications a sensor array is often deployed for interference suppression. This paper considers the complex amplitude estimation of a known waveform signal whose array response is also known a priori. We study a practical scenario where the interference and noise is both spatially and temporally correlated. We model the interference and noise vector as a multichannel autoregressive (AR) random process. A cyclic iterative ML (IML) method is presented. We show that in most cases the IML method is superior to its simple ML counterpart that ignores the temporal correlation of the interference and noise.

Whenever the knowledge of the array steering vector is imprecise (as is often the case in practice), the performance of the Capon beamformer may become worse than that of the standard beamformer. Diagonal loading (including its extended versions) has been a popular approach to improve the robustness of the Capon beamformer. In this paper we show that a natural extension of the Capon beamformer to the case of uncertain steering vectors also belongs to the class of diagonal loading approaches but the amount of diagonal loading can be precisely calculated based on the uncertainty set of the steering vector. The proposed robust Capon beamformer can be efficiently computed at a comparable cost with that of the standard Capon beamformer. Its excellent performance is demonstrated via a number of numerical examples.

In wireless communication systems, channel state information is often assumed to be available at the receiver. Traditionally, a training sequence is used to obtain the estimate of the channel. Alternatively, the channel can be identified using known properties of the transmitted signal. However, the computational effort required to find the joint ML solution to the symbol detection and channel estimation problem increases exponentially with the dimension of the problem. To significantly reduce this computational effort, we formulate the aforementioned problem in a way that makes it possible to solve it via the use of sphere decoding, an algorithm that has polynomial expected complexity. We also provide simulation results and a complexity discussion.

Linear space-time block coding (STBC) is a conceptually simple transmission technique for channels with multiple transmit and receive antennas. In this paper, we study a subclass of linear STBC, namely orthogonal STBC (OSTBC), with the primary goal of contributing towards a complete understanding of OSTBC. In particular, we prove that OSTBC is optimal in an important minimum mean-square error (MSE) sense. As a by-product, we also obtain a concise characterization of linear and orthogonal STBC's as well as the relationship between them. The MSE optimality of OSTBC shown herein can be used as a way of introducing this coding scheme in a formal manner.

Linear space-time block coding (STBC) is a conceptually simple transmission technique for channels with multiple transmit and receive antennas. We study a subclass of linear STBC, namely orthogonal STBC (OSTBC), with the primary goal of contributing toward a complete understanding of OSTBC. In particular, we prove that OSTBC is optimal in a minimum mean-square-error (MSE) sense, provided that a zero-forcing detector is used at the receiver. As a by-product, we also obtain a concise characterization of linear and orthogonal STBCs as well as the relationship between them. The MSE optimality of OSTBC can be used as a way of introducing this coding scheme from first principles.

The Capon beamformer has better resolution and much better interference rejection capability than the standard (data-independent) beamformer, provided that the array steering vector corresponding to the signal of interest (SOI) is accurately known. However, whenever the knowledge of the SOI steering vector is imprecise (as is often the case in practice), the performance of the Capon beamformer may become worse than that of the standard beamformer. We present a natural extension of the Capon beamformer to the case of uncertain steering vectors. The proposed robust Capon beamformer can no longer be expressed in a closed form, but it can be efficiently computed. Its excellent performance is demonstrated via a number of numerical examples.

We consider the problem of data-aided frequency-offset and channel estimation in the case of frequency-selective channels. More precisely, we address the problem of training sequence selection with the goal of providing accurate frequency offset and channel estimates. Toward this end, we consider the Crame/spl acute/r-Rao bound (CRB), for which we derive a closed-form expression. Since the CRB is a complicated function of the training sequence and the channel parameters, a much simpler asymptotic CRB is derived. Two criteria for training sequence design based on the asymptotic CRB are proposed, and a minmax approach is presented to optimize them. Our main contribution is to show that a white sequence is minmax optimal for both criteria considered, and that the quest for a generally optimal sequence is hardly motivated.

We consider the problem of frequency-offset estimation in frequency selective channels in a data-aided context. More specifically, we address the training sequence selection issue with the goal of providing the most accurate frequency offset estimates. Towards this end, we examine the Cramér-Rao bound (CRB) for the problem at hand. Since it is hardly feasible to derive the training sequence that results in a minimum CRB, an expression for the asymptotic CRB is derived which depends in a simple way on the channel impulse response and the training sequence correlation. Based on the asymptotic CRB, two methods are presented to select an optimal training sequence. Numerical simulations illustrate the estimation performance obtained with these training sequences.

We describe the so-called time-reversal space-time block coding (TR-STBC) transmission scheme for communication systems with multiple transmit antennas operating over frequency-selective channels. TR-STBC can be seen as an extension of the orthogonal space-time block codes for flat fading channels. We show that using TR-STBC, a linear filtering at the receiver can achieve an approximate decoupling of the space-time channel into scalar and independent frequency-selective channels and hence any standard maximum-likelihood sequence detector (MLSD) can be used for equalization. Numerical examples are provided to illustrate the performance of the TR-STBC transmission scheme.

In this paper we derive and compare designs for the optimal linear precoderes to be used in transmissions over frequency selective multiple-input multiple-output (MIMO) channels. We assume as alternative design constraints the average transmit power and the peak power. The design criteria are scalable with respect to the number of antennas, size of the coding block and transmit average/peak power. The solutions are shown to convert in both cases the MIMO channel with memory into a set of parallel independent fiat fading subchannels, regardless of the design criterion, while appropriate power/bits loading on the sub-channels is the specific signature of the different designs.

Space-Time Block Coding (STBC) requires knowledge of the channel at the receiver. In practice the channel has to be estimated. In this paper we consider three estimation/detection schemes and compare their performances. In particular we introduce a blind scheme which does not require the transmission of pilot symbols to estimate the channel.

We consider a space-time differential modulation scheme where neither the transmitter nor the receiver has to know the channel. Our scheme is based on the theory of unitary space-time block codes. Compared to the existing differential modulation schemes for multiple antennas our scheme has a much smaller computational complexity. Moreover, our codes have a higher coding gain and lower bit error rate than the codes proposed by other researchers.

Stoica, Li and Li, (see IEEE Trans. Signal Processing, vol.48, p.338-52, 2000) introduced an amplitude estimation based scheme for one-dimensional (1-D) system identification that overcomes several drawbacks (e.g., computational complexity, local convergence, and statistical inefficiency when spectrally colored noise is present) suffered by the conventional output error method (OEM). Along the same line, we herein propose a two-dimensional (2-D) system identification scheme that makes use of 2-D amplitude estimation. In particular, we consider the recently introduced 2-D amplitude and phase estimation (APES) amplitude estimator, which has been shown to yield superior performance over its competitors. To benchmark the proposed scheme, we also derive the Cramer-Rao bound (CRB) for the 2-D system identification problem.

We derive the Cramer-Rao bound for the parameters of a general time series model whose parameterization is dependent upon an unknown integer model order. To illustrate the usefulness of the theoretical results, the example of autoregressive spectral density estimation using Akaike (1974) order selection criterion is presented.

We consider the direction-of-arrival (DOA) problem for a wavefront whose amplitude and phase vary randomly along the array aperture. This phenomenon can for instance originate from propagation through an inhomogeneous medium. A simple and accurate DOA estimator is derived in the case of an uniform linear array of sensors. The estimator is based upon a reduced statistic obtained from the sub-diagonals of the covariance matrix of the array output. It only entails computing the Fourier transform of an (m-1)-length sequence where m is the number of array sensors. A theoretical expression for the asymptotic variance of the estimator is derived. Numerical simulations validate the theoretical results and show that the estimator has an accuracy very close to the Cramer-Rao bound.

The stochastic Cramer-Rao bound (CRB) plays an important role in array processing because several high-resolution direction-of-arrival (DOA) estimation methods are known to achieve! this bound asymptotically In this paper, we study the stochastic CRB on DOA estimation accuracy in the general case of arbitrary unknown noise field parametrized by a vector of unknowns. We derive explicit closed-form expressions for the CRB and examine its properties theoretically and by representative numerical examples.

The performance breakdown of subspace-based parameter estimation methods can be naturally related to a switch of vectors between the estimated signal and noise subspaces (a "subspace swap"). We derive a lower bound for the probability of such an occurrence and use it to obtain a simple data-based indicator of whether or not the probability of a performance breakdown is significant. We also present a conceptually simple technique to determine from the data whether or not a subspace swap has actually occurred, and to extend the range of SNR values or data samples in which a given subspace method produces accurate estimates.

The matched-filterbank spectral estimators APES and CAPON have received considerable attention in a number of applications. Unfortunately, their computational complexity tends to limit their usage in several cases-a problem that has previously been addressed by different authors. In this paper, we introduce a novel method to the computation of the 1D and 2D APES and CAPON spectra, which is considerably faster than all existing techniques. Numerical examples are provided to demonstrate the application of APES to synthetic aperture radar (SAR) imaging, and to illustrate the reduction in computational complexity provided by our implementation.

We previously studied amplitude estimation of one-dimensional (1D) sinusoidal signals from measurements corrupted by possibly colored observation noise (see Stoica, P. et al., IEEE Trans. on Sig. Proc., vol. 48, p.338-52, 2000). We extend those results for two-dimensional (2D) amplitude estimation. In particular, we investigate the 2D sinusoidal amplitude estimation within the general frameworks of least squares (LS), weighted least squares (WLS), and MAtched FIlterbank (MAFI) estimation. A variety of 2D amplitude estimators are presented, which are all asymptotically statistically efficient. The performances of these estimators in finite samples are compared numerically with one another. Making use of amplitude estimation techniques, we introduce a new scheme for 2D system identification, which is shown to be computationally simpler and statistically more accurate than the conventional output error method (OEM), when the observation noise is colored.

We consider a communication scenario involving an m /spl times/ n MIMO linear channel whose input is a symbol stream multiplied prior to transmission by an n x n space-time coding matrix X, and whose output is fed into an m /spl times/ n linear combiner Z. We show how to choose the matrices X and Z to maximize the SNR of the linear combiner output data that are used for detection, under total power constraint (TPC), elemental power constraint (EPC), or total and elemental power constraint (TEPC). The TEPC design (considered here for the first time) is shown to include the TPC and EPC designs (previously considered by the authors) as special cases, and hence to provide a theoretically and practically interesting unifying framework. We make use of this framework to discuss various tradeoffs of the three space-time designs considered, such as transmission rate and requirements for channel status information at the transmission side.

We propose a method for estimating the amplitude spectrum of a 2D-signal from frequency domain (or phase history) data that contain gaps. This is a key problem in synthetic aperture radar (SAR) imaging with angular diversity. Our method is an extension of the well-known amplitude and phase estimation (APES) algorithm to the incomplete data case, and is called gapped-data APES (GAPES). It has recently been shown that APES minimizes a certain least-squares (LS) criterion, and our extension of APES is based on minimizing this criterion with respect to the missing data as well. We provide an example of the algorithm applied to SAR imaging from phase history data with angular diversity; and we also show the advantages of 2D data processing over 1D (row or column-wise) data processing that are enabled by our new algorithm.

The stochastic Cramer-Rao bound (CRB) for direction estimation in array processing applications was indirectly derived some ten years ago as the (asymptotic) covariance matrix of the maximum likelihood (ML) estimator. Attempts to obtain the stochastic CRB directly via the CRB theory fell short of providing a simple derivation and consequently, no direct derivation of this useful performance bound was available in the open literature. we correct this situation by providing a textbook-like direct derivation of the stochastic CRB.

We address the problem of designing jointly optimum linear precoder and decoder for a MIMO channel possibly with delay-spread, using a weighted minimum mean-squared error (MMSE) criterion subject to a transmit power constraint. We show that the optimum linear precoder and decoder diagonalize the MIMO channel into eigen subchannels, for any set of error weights. Furthermore, we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights. We show how to obtain: (1) the maximum information rate design; (2) QoS-based design (we show how to achieve any set of relative SNRs across the subchannels); and (3) the (unweighted) MMSE and equal-error design for fixed rate systems.

In Tarokh et al. (1999) space-time block codes were introduced to obtain coded diversity for a multiple-antenna communication system, in this work, we cast space-time codes in an optimal signal-to-noise ratio (SNR) framework and show that they achieve the maximum SNR and, in fact, they correspond to a generalized maximal ratio combiner. The maximum SNR framework also helps in calculating the distribution of the SNR and in deriving explicit expressions for bit error rates. We bring out the connection between the theory of amicable orthogonal designs and space-time codes. Based on this, we give a much simpler proof to one of the main theorems on space-time codes for complex symbols. We present a rate 1/2 code for complex symbols which has a smaller delay than the code already known. We also present another rate 3/4 code which is simpler than the one already known, in the sense it does not involve additions or multiplications. We also point out the connection between generalized real designs and generalized orthogonal designs.

In this correspondence, we consider the case of multiple-antenna transmission diversity. We outline a recent result on the maximum possible performance (in terms of mutual information) and show that it can be achieved using scalar codes instead of vector codes. The scalar coding schemes we propose are based on space-time block codes and do not involve any reduction in data rate for two transmitter antennas. For the case of three and four transmitter antennas we achieve 3/4 of the rate achieved using vector codes. For more than four transmitter antennas we achieve 1/2 of the maximum possible rate. In all cases, the diversity performance is the same as that corresponding to vector codes. In addition, the receiver has a simple structure which makes implementation easier and keeps down the computational complexity.

We consider the utilization of multiple transmitter and receiver antennas for space-time diversity. The optimal SNR scheme, which also provides the best diversity, is outlined. This scheme however involves a reduction in the data rate. Coding schemes are then presented which not only achieve the optimal SNR but also mitigate the reduction of data rate. The proposed schemes are based on the theory of orthogonal designs and amicable orthogonal designs.

We present an efficient implementation of the 2-D amplitude spectrum capon (ASC) estimator, denoted the 2-D Burg-based ASC (BASC) estimator. The algorithm, which will depend only on the (forward) linear prediction matrices and the (forward) prediction error covariance matrices, can be implemented using the 2-D fast Fourier transform. To compute the needed prediction matrices, we make use of a recently proposed 2-D lattice algorithm, which computes the linear prediction matrices directly from the multichannel data without first computing the autocorrelation sequence.

We consider the problem of estimating the direction-of-arrival (DOA) of one or more signals using an array of sensors, where some of the sensors fail to work before the measurement is completed. Methods for estimating the array output covariance matrix are discussed and used for DOA estimation together with the MUSIC algorithm and with a covariance matching technique. In contrast to MUSIC, the covariance matching technique can utilize information on the estimation accuracy of the array covariance matrix, and it is shown that this can yield a significant performance gain.

In a paper on spectroscopic imaging Spielman et al. (1988) made the important point that apriori information about the compounds present can and should be incorporated into the estimation of spectroscopic signal parameters. They proposed using the maximum likelihood (ML) approach for parameter estimation, but failed to incorporate properly the full apriori information that was assumed to be available. Consequently they ended-up with a spectroscopic imaging method that is only a suboptimal approximation of the ML method. In this paper we derive the exact ML method, present a computationally efficient implementation of it and illustrate numerically the performance gain that can be achieved over the method of Spielman et al.

This paper describes generalizations of the MUSIC and root-MUSIC algorithms for direction of arrival (DOA) estimation to arrays composed of multiple translated subarrays. The advantage of these new approaches is that the DOAs can be estimated using either a one-dimensional search or by rooting a polynomial, as opposed to a multidimensional search as required by the multiple invariance (MI)-ESPRIT algorithm. While MI-MUSIC and root-MI-MUSIC are not statistically efficient like MI-ESPRIT, they do perform better than a single invariance implementation of ESPRIT, and are thus better suited for finding the initial conditions required by the MI-ESPRIT search.

We propose a new MODE-based direction of arrival (DOA) estimation algorithm with an improved SNR threshold as compared to the conventional MODE technique. Our algorithm preserves all good properties of MODE, such as asymptotic efficiency, excellent performance in scenarios with coherent sources, as well as a reasonable computational cost. Similarly to root-MODE, the proposed method does not require any global multidimensional optimization since it is based on a combination of polynomial rooting and a simple combinatorial search. Our technique is referred to as MODEX (MODE with EXtra roots) because it makes use of a certain polynomial with a larger degree than that of the conventional MODE-polynomial. The source DOAs are estimated via checking a certain (enlarged) number of candidate DOAs using either the stochastic or the deterministic maximum likelihood function. To reduce the computational cost of MODEX, a priori information about source localization sectors can be exploited.

In some applications the covariance matrix of the observations is not only symmetric with respect to its main diagonal but also with respect to the anti-diagonal. The standard forward-only sample covariance estimate does not impose this extra symmetry. In such cases one often uses the so-called forward-backward sample covariance estimate. In this paper, a direct comparative study of the relative accuracy of the two sample estimates is performed. An explicit expression for the difference between the estimation error covariance matrices of the two sample estimates is given. The presented results are also useful in the analysis of estimators based on either of the two sample covariances. As an example, spatial power estimation by means of the Capon method is considered. It is shown that Capon based on the forward-only sample covariance (F-Capon) underestimates the power spectrum, and also that the bias for Capon based on the forward-backward sample covariance is half that of F-Capon.

We investigate herein the problem of amplitude estimation of sinusoidal signals from observations corrupted by colored noise. A relatively large number of amplitude estimators are described which encompass least squares (LS) and weighted least squares (WLS) methods. Additionally, filterbank approaches, which are widely used for spectral analysis, are extended to amplitude estimation. Specifically, we consider the matched-filterbank (MAFI) approach and show that, by appropriately designing the prefilters, the MAFI approach includes the WLS approach. The amplitude estimation techniques discussed in this paper do not model the noise, and yet they are all asymptotically statistically efficient. It is their different finite-sample properties that are of particular interest to this study. Numerical examples are provided to illustrate the differences among the various estimators. Though amplitude estimation applications are numerous, we focus on system identification using sinusoidal probing signals.

After reviewing the main existing methods for determining the maximum-likelihood (ML) estimates of the direction-of-arrival (DOA) parameters in array signal processing applications, we introduce a new conceptually simple and computationally effective approach that consists of maximizing the likelihood function (LF) over a set of points derived from the data. We show that the data-supported grid search of the LF provides a performance similar to that achieved by a genetic algorithm, but at a significantly lower computational cost. We use an ESPRIT-like algorithm to obtain the grid points with support in the data, although our approach is not limited to this choice.

We show that the main result on eigenpolynomials for two-dimensional (2-D) sinusoidal signals in a recent letter by Li and Cheng (see ibid., vol.5, p.71-3, March 1998) is incorrect, and hence the characterization of 2-D eigenpolynomials is still an open problem. Furthermore, we state a related problem that is also open and hence awaiting a (satisfactory) solution.

We introduce a novel design criterion for data-dependent narrowband filters that are of interest in temporal or spatial spectral analysis applications. The solution to the design problem considered is shown to coincide with the previously introduced amplitude and phase estimation (APES) filter. The new derivation of APES in this article sheds more light on the properties of APES and provides some intuitive explanation of the performance superiority of the APES filter over the Capon filter.

Nonlinear least-squares (NLS) fitting of rational transfer functions to frequency response data yields the maximum likelihood estimator (MLE) of the transfer function coefficient vector under mild conditions on the observation noise. Furthermore, the NLS approach is robust to errors in the modeling of data. However the NLS criterion is in general difficult to minimize. Here we show that an asymptotic realization of the NLS estimator can be obtained from the elemental set parameter estimates by simple linear operations. The latter estimates are derived by matching the frequency response data on many "elemental sets" comprising a number of frequencies equal to half the number of unknown parameters.

In this paper, we consider the problem of estimating the frequency of a sinusoidal signal whose amplitude could be either constant or time-varying. We present a nonlinear least-squares (NLS) approach when the envelope is time-varying. We show that the NLS estimator can be efficiently implemented using a FFT. A statistical analysis shows that the NLS frequency estimator is nearly efficient. The problem of detecting amplitude time variations is next addressed. A statistical test is formulated, based on the statistics of the difference between two frequency estimates. The test is computationally efficient and yields as a by-product consistent frequency estimates under either hypothesis (i.e. constant or time-varying amplitude). Numerical examples are included to show the performance in terms of both estimation and detection.

This paper considers the problem of estimating the time delays and Doppler shifts of a known waveform received via several distinct paths by an array of antennas. The general maximum likelihood estimator is presented, and is shown to require a 2d-dimensional non-linear minimization, where d is the number of received signal reflections. Two alternative solutions based on signal and noise subspace fitting are proposed, requiring only a d-dimensional minimization. In particular, we show how to decouple the required search into a two-step procedure, where the delays are estimated and the Dopplers solved for explicitly. Initial conditions for the time delay search can be obtained by applying generalizations of the MUSIC and ESPRIT algorithms.

A computationally efficient method for structured covariance matrix estimation is presented. The proposed method provides an asymptotic (for large samples) maximum likelihood estimate of a structured covariance matrix and is referred to as AML. A closed-form formula for estimating Hermitian Toeplitz covariance matrices is derived which makes AML computationally much simpler than most existing Hermitian Toeplitz matrix estimation algorithms. The AML covariance matrix estimator can be used in a variety of applications. We focus on array processing and show that AML enhances the performance of angle estimation algorithms, such as MUSIC, by making them attain the corresponding Cramer-Rao bound (CRB) for uncorrelated signals.

We consider robust and computationally efficient maximum likelihood algorithms for estimating the parameters of a radar target whose signal is observed by an array of sensors in interference with unknown second-order statistics. Two data models are described: one that uses the target direction of arrival and signal amplitude as parameters and one that is a simpler, unstructured model that uses a generic target "spatial signature". An extended invariance principle is invoked to show how the less accurate maximum likelihood estimates obtained from the simple model may be refined to achieve asymptotically the performance available using the structured model. The resulting algorithm requires two one-dimensional (1-D) searches rather than a two-dimensional search, as with previous approaches for the structured case. If a uniform linear array is used, only a single 1-D search is needed. A generalized likelihood ratio test for target detection is also derived under the unstructured model. The principal advantage of this approach is that it is computationally simple and robust to errors in the model (calibration) of the array response.

This paper presents a simple expression for the Cramer-Rao bound (CRB) for parametric estimation under differentiable, deterministic constraints on the parameters. In contrast to previous works, the constrained CRB presented does not require that the Fisher information matrix (FIM) for the unconstrained problem be of full rank. This is a useful extension because, for several signal processing problems (such as blind channel identification), the unconstrained problem is unidentifiable. Our expression for the constrained CRB depends only on the unconstrained FIM and a basis of the nullspace of the constraint's gradient matrix. We show that our constrained CRB formula reduces to the known expression when the FIM for the unconstrained problem is nonsingular. A necessary and sufficient condition for the existence of the constrained CRB is also derived.

Sinusoidal signals with random time-varying amplitude show up in many signal processing applications. Amplitude modulation results in degeneracy of the signal subspace, i.e. the signal subspace corresponding to one amplitude modulated sinusoid is no longer spanned by one vector. In this paper, we propose modifications of two subspace-based techniques, namely ESPRIT and MODE for estimating the center frequency of a sinusoidal signal with random time-varying ARMA amplitude. Numerical simulations illustrate the good performance of the methods. Finally, a robust scheme of the proposed methods is described and successfully applied to real radar data.

We present a comparative study of using the IQML (iterative quadratic maximum likelihood) algorithm and the MODE (method of direction estimation) algorithm for direction-of-arrival estimation with a uniform linear array. The consistent condition and the theoretical mean-squared error for the parameter estimates of IQML are presented. The computational complexities of both algorithms are also compared. We show that the frequency estimates obtained via MODE are asymptotically statistically efficient, while those obtained via IQML are almost always inconsistent and hence inefficient. We also show that the amount of computations required by IQML is usually much larger than that required by MODE, especially for low signal-to-noise ratio and large number of snapshots.

This paper describes how the computationally efficient one-dimensional MODE (1D-MODE) algorithm can be used to estimate the frequencies of two-dimensional complex sinusoids. We show that the 1D-MODE algorithm is computationally more efficient than the asymptotically statistically efficient 2D-MODE algorithm, especially when the numbers of spatial measurements are large. We find that the 1D-MODE algorithm is asymptotically statistically efficient for high signal-to-noise ratio. We also show that although the 1D-MODE is no longer statistically efficient when the number of temporal snapshots is large, the performance of the 1D-MODE can still be very close to that of the 2D-MODE under mild conditions. Numerical examples comparing the performance of the 1D-MODE and 2D-MODE algorithms are also presented.

In this paper, we consider the best asymptotic accuracy that can be achieved when estimating the parameters of a random-amplitude sinusoid from its sample covariances. An estimator, based upon matching in a weighted least-squares sense the sample correlation sequence to the theoretical sequence is presented. The asymptotic properties of the estimator are analyzed. A lower bound on the estimation of the parameters from sample covariances is derived. This bound is shown to be attainable by appropriately choosing the weighting matrix. Numerical simulations illustrate the performance of the proposed estimator and the validity of the theoretical analysis. Finally, a comparison with Yule-Walker methods is given.

The problem of estimating the parameters of several wavefronts from the measurements of multiple sensors is often referred to as array signal processing. The maximum likelihood (ML) estimator in array signal processing for the case of non-coherent signals has been studied extensively. The focus here is on the ML estimator for the case of stochastic coherent signals which arises due to, for example, specular multipath propagation. We show the very surprising fact that the ML estimates of the signal parameters obtained by ignoring the information that the sources are coherent, coincide in large samples with the ML estimates obtained by exploiting the coherent source information. Thus, the ML signal parameter estimator derived for the non-coherent case (or its large-sample realizations such as MODE os WSF) asymptotically achieves the lowest possible estimation error variance (corresponding to the coherent Cramer-Rao bound).

For a data model consisting of deterministic signals in additive Gaussian noise, we prove that the Cramer-Rao bound (CRB) corresponding to the signal parameters decreases as the number of data samples increases provided that the number of new observations is larger than the number of additional unknowns required to parameterize these observations. We also show that the CRB theory is not applicable whenever the aforementioned condition does not hold true.

This paper considers the estimation of the parameters of a linear discrete-time system from noisy input and output measurements. The conditions imposed on the system are quite general. The proposed method makes use of an instrumental variable (IV) vector whose cross-covariance with the system's regression vector is pre- and post-multiplied by some prechosen weights. The singular vectors of this matrix possess complete information on the system parameters. A weighted sub-space fitting (WSF) method is then applied to these singular vectors to consistently estimate the parameters of the system. The proposed method is non-iterative, easy to implement and has a small computational burden. The asymptotic distribution of its estimation errors is derived and the result is used to motivate the choice of the weighting matrix in the WSF step and also to predict the estimation accuracy. A numerical example is included to illustrate the performance.

We perform an analysis of constrained and unconstrained MUSIC demonstrating that (asymptotically) improved subspace estimates always result from the use of constraints, and (asymptotically) the variance of constrained MUSIC is less than that of unconstrained MUSIC under either high coherence, large numbers of sensors, or high SNR conditions. As part of this analysis, we study the effects of coherence on MUSIC and derive best/worst case coherences in terms of the variance of MUSIC. We also demonstrate that those conditions where the variance of MUSIC is predicted to be less than that of constrained MUSIC generally correspond to conditions where MUSIC is in breakdown (and constrained MUSIC is not). So, unconstrained MUSIC does not achieve its predicted advantage in those cases.

The paper describes and analyses, in a unifying manner, the spatial and temporal IV-SSF (instrumental variable-signal subspace fitting) approaches recently proposed for array signal processing in colored noise fields. We derive a general, optimally-weighted, IV-SSF direction estimator and show that this estimator encompasses the UNCLE estimator of Wong and Wu (see IEEE Trans.SP, vol.42, Sept. 1994), which is a spatial IV-SSF method; and the temporal IV-SSF estimator of Viberg, Stoica and Ottersten (see IEEE Trans.SP, May 1995). The latter two estimators have seemingly different forms, so their asymptotic equivalence shown in this paper comes as a surprising unifying result.

Least-squares equation-error models are widely used as a simple means of estimating an input-output transfer function in a system identification context.. Although the models furnished by the least-squares method are not always stable, some recent works have shown that an autoregressive constraint on the input is sufficient to ensure stability of the furnished model. Here we provide a simple proof of this property for multivariable system estimation.>

The column-unitary matrix whose range space is closest, in the Frobenius norm metric, to the range of a given matrix and whose columns belong to the null space of another specified matrix is obtained. An application of the matrix optimization result derived herein to constrained beamformer design in array signal processing is briefly described.>

The paper makes a number of remarks on the convergence properties of a time-varying recursion that appears in several signal/system parameter estimation problems. Among others, it is shown by means of an example that the recursion in question can be divergent.>

This paper considers the problem of statistically efficient estimation of the parameters of partially polarized electromagnetic (EM) waves with a uniform linear array of crossed dipoles. We consider the maximum likelihood (ML) estimation of incident angles and the degrees of polarization. We present a computationally efficient large sample ML estimator that avoids the multidimensional search over the parameter space required by the exact ML estimator.>

Most methods for sensor array signal processing require the covariance matrix of the background noise to be known. Various techniques for overcoming this limitation have recently been proposed. While most of these are based on assumptions on the noise, we present herein an alternative approach based on partial knowledge of the signals. Methods yielding minimum variance estimates for the model in question are presented and analyzed.>

Signal parameter estimation from sensor array data is of great interest in a variety of applications, including radar, sonar, and radio communication. A large number of high-resolution (i.e., model-based) techniques have been suggested in the literature. The vast majority of these require knowledge of the spatial noise correlation matrix, which constitutes a significant drawback. A novel instrumental variable (IV) approach to the sensor array problem is proposed. By exploiting temporal correlatedness of the source signals, knowledge of the spatial noise covariance is not required. The asymptotic properties of the IV estimator are examined, and an optimal IV method is derived. Simulations are presented examining the properties of the IV estimators in data segments of realistic lengths.>

The asymptotic properties of the high-order Yule-Walker (HOYW) estimators of the frequencies of complex sine waves are investigated. An explicit formula is derived for the covariance matrix of the corresponding estimation errors. An analytical study of the HOYW error covariance matrix shows that its elements are inversely proportional to the squared signal-to-noise ratio, enjoy a certain invariance to frequency shifts, and decrease significantly as the number of YW equations and the model order increases. These properties are also shared by the MUSIC and ESPRIT methods. A numerical study of the performance of these three methods shows that the HOYW method usually gives the best tradeoff between statistical accuracy and computational complexity.>

Two forms for retrieving sinusoidal frequencies from an estimated characteristic polynomial, the root form and the spectral form, are examined. It is proved that both forms give frequency estimates with the same asymptotic covariance matrix, regardless of the (consistent) method used to estimate the characteristic polynomial. The finite sample case is examined by means of Monte Carlo simulations for a specific method used to estimate the characteristic polynomial. It is shown that the root form has a distinctly smaller failure rate than the spectral form. However, it is shown that in the successful realizations the forms have nearly identical mean square errors.>

The large-sample second-order properties of multiple signal classification (MUSIC) and subspace rotation methods such as ESPRIT for sinusoidal frequency estimation are analyzed. Both MUSIC and ESPRIT are based on the eigendecomposition of a sample data covariance matrix. Explicit expressions for the covariance elements of the estimation errors associated with either method are derived. These expressions of covariances are used to analyze and compare the statistical performance of the MUSIC and ESPRIT methods. It is shown that ESPRIT is usually slightly more accurate than MUSIC. Since MUSIC is computationally more demanding than ESPRIT, it appears that the ESPRIT method for frequency estimation should be preferred to MUSIC in most cases.>

An analysis is made of the consistency of two direction-of-arrival estimation algorithms used in the presence of multipath propagation and with very few snapshots. The conditional maximum-likelihood (CML) algorithm and the method of direction estimation (MODE) are discussed. The cost functions of these algorithms are shown to coincide for a very large number of snapshots or very large signal-to-noise ratio. Necessary and sufficient conditions are derived for the algorithms to yield unique estimates. It is shown that their uniqueness conditions coincide with fundamental uniqueness conditions for the array that are independent of the algorithm used.>

Two different types of data model used in estimating the direction-of-arrival (DOA) of narrowband signals using sensor arrays are considered: the conditional model (CM), which assumes the signals to be nonrandom, and the unconditional model (UM), which assumes the signals to be random. These models leased to different maximum-likelihood (ML) methods (termed CML and UML, respectively) and different Cramer-Rao bounds (CRB) on DOA estimation accuracy (B/sub c/ and B/sub u/, respectively). An explicit expression is derived for the covariance matrix of the UML and for B/sub u/. It is shown that CML, UML, and a recently introduced method of direction estimation (MODE), as well as many other DOA estimation methods, have the same asymptotic statistical properties under CM as under UM. It is proven that: CML is statistically less efficient then UNL; MODE is asymptotically equivalent to UML; UML and MODE achieve the unconditional CRB, B/sub u/; and B/sub u/ is a lower bound on the asymptotic statistical accuracy of any (consistent) DOA estimate based on the data sample covariance matrix; B/sub c/ cannot be attained. It is also proven that B/sub u/ and B/sub c/ decrease monotonically as the number of sensors or snapshots increases and increase monotonically as the number of sources increases.>

A number of results have been presented recently on the statistical performance of the multiple signal characterization (MUSIC) and the maximum-likelihood (ML) estimators for determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements. It is shown that in the class of weighted MUSIC estimators, the unweighted MUSIC achieves the best performance (i.e. the minimum variance of estimation errors) in large samples. The covariance matrix of the ML estimator is derived, and detailed analytic studies of the statistical efficiency of MUSIC and ML estimators are presented. These studies include performance comparisons of MUSIC and MLE with each other as well as with the ultimate performance corresponding to the Cramer-Rao bound (CRB).>

The authors analyze the convergence of several adaptive notch filter algorithms for sine waves in colored noise, which were recently proposed in the literature. After pointing out the previous algorithms' potential convergence problems, an algorithm is proposed for this problem that does not have convergence problems and provides accurate estimation results.>

The authors consider methods for solving the problem of finding the directions of multiple plane waves with linear arrays of sensors and the related one of estimating the parameters of multiple superimposed exponential signals in noise. Specifically, the MUSIC and maximum-likelihood (ML) methods have been proposed for solving these problems. The authors study the performance of the MUSIC and ML methods, and analyze their statistical efficiency. They also derive the Cramer-Rao bound (CRB) for the estimation problems mentioned above, and establish some useful properties of the CRB covariance matrix. The relationship between the MUSIC and ML estimators is investigated as well.>

The authors establish the large-sample accuracy properties of two nonlinear least-squares estimators (NLSEs) of sine waves parameters: the basic NLSE, which ignores the possible correlation of the noise; and the optimal NLSE, which, besides the sine-wave parameters, also estimates the noise correlation (appropriately parameterized). It is shown that these two NLSEs have the same accuracy in large samples. This result provides complete justification for preferring the computationally less-expensive basic NLSE over the optimal NLSE. Both estimators are shown to achieve the Cramer-Rao bound (CRB) as the sample size increases. A simple explicit expression for the CRB matrix is provided which should be useful in studying the performance of sine-wave parameter estimators designed to work in the colored noise case.>

A three step method for obtaining nearly maximum likelihood ARMA spectral estimates is presented. The computational complexity of the algorithm is comparable to Yule-Walker methods, but the method gives asymptotically efficient estimates. The implementation of the algorithm is discussed, and numerical examples are presented to illustrate its performance.

A new family of algorithms is developed for adaptive parameter estimation of constrained autoregressive moving-average (ARMA) signals in the presence of noise. These algorithms utilize a priori known information concerning the signal's properties, such as its spectral shape or a spatial domain characteristic. Special cases include autoregressive (AR) and band-pass spectrum signals in the presence of noise, signal deconvolution and image deblurring.

The asymptotic properties of the high-order Yule-Walker (HOYW) estimator of sinusoidal frequencies are analyzed. An explicit formula for the covariance matrix of the HOYW frequency estimation error is derived. The effects of the number of equations, the model order and a certain weighting matrix are investigated analytically. The optimally weighted HOYW estimator is described. The analysis motivates using an overdetermined and high-order Yule-Walker estimator.

Adaptive notch filters are used to eliminate narrow-band or sine wave components with unknown or slowly time varying frequencies from observed time series. This paper derives the asymptotic bias and variance of the sine wave frequency estimate obtained by the adaptive notch filter recently proposed in [4].