We address the problem of detecting a multidimensional subspace signal among colored noise with unknown covariance matrix using a second-order model, i.e. assuming that the presence of the signal of interest results in a rank- R modification of the noise covariance matrix. The problem is tackled through the prism of invariance. We derive the group of transformations that leave the problem invariant and subsequently derive the maximal invariant statistic (MIS) along with the induced maximal invariant (IMI). A statistical representation of the MIS is obtained which is quite different from that of a first-order model. Hints at possible test statistics based on the MIS are also provided.

We consider adaptive radar detection of Rician targets which, when present, result in a shift of the mean of the observations as well as a rank-one modification of the covariance matrix. In the paper, we first derive the Fisher information matrix for the statistical model under consideration. This enables us to obtain the expressions of Rao, Wald, Durbin and gradient tests statistics whose distributions are parameter-free under the null hypothesis. Numerical simulations illustrate their probability of detection in both matched and mismatched cases, i.e., when the signature under test coincides or not with the actual target signature. They indicate that under some scenarios Rao, Durbin and gradient tests may be valuable alternatives to the GLRT.

We consider adaptive detection of multidimensional second-order subspace signals in colored noise with a view to provide simpler alternatives to the generalized likelihood ratio test which requires alternate maximization over two sets of matrices. Since the main problem comes from maximum likelihood estimation of the unknown parameters under the alternative hypothesis we consider detection schemes that do not need it. Towards this end we derive closed-form expressions of Rao and Durbin tests for the problem at hand. These detectors are computationally simple and provide a constant false alarm rate. Through numerical simulations we illustrate the trade-off between computational complexity and detection performance.

We consider a modification of the minimum variance distortionless response (MVDR) filter using Stein unbiased risk estimation (SURE). The starting point of this modification lies in the observation that the component of the MVDR filter in the subspace orthogonal to the signal of interest is the maximum likelihood estimate (MLE) of the location of a certain multivariate distribution. This draws us to consider James-Stein type estimates which have been shown to outperform MLE for minimization of some risks. In this letter we propose two kinds of modifications inspired by Stein's approach. A natural risk is defined and we derive a loss function which results in an unbiased estimate of this risk, then proceed to its minimization. Numerical simulations compare the so-modified MVDR filter to its original version.

We address the problem of detecting a Gaussian rank-one signal using training samples to learn the covariance matrix of the noise present in the samples under test. Towards this end, we propose to use the latter after they have been whitened by the sample covariance matrix of the training samples. As an alternative to the generalized likelihood ratio test, we investigate three simpler alternatives, namely the Rao, gradient and Durbin tests. Closed-form expressions of the corresponding test statistics are derived and the detectors are shown to have a constant false alarm rate. Their performance is assessed via numerical simulations.

Maximum Likelihood (ML) frequency estimation of a single tone in noise is known to be a computationally intensive task that does not cope with many real-time and embedded hardware architectures. Thereby, many sub-optimal techniques, based on approximations, have been proposed in the literature. In this paper, we show that the ML criterion can be solved directly, using an appropriate two-step procedure. The closed-form solution is shown to be asymptotically equivalent to the ML. Moreover, its formulation is very close to the popular Fitz's expression, with a slight correction. Numerical simulations show that the proposed scheme is very close to the ML.

Satellite altimeters provide sea-level measurements along satellite track. A mean profile based on the measurements averaged over a time period is then subtracted to estimate the sea-level anomaly (SLA). In the spectral domain, SLA is characterized by a power spectral density (PSD) whose slope in a log–log scale is a parameter of great interest for ocean monitoring. Estimation of this spectral slope is usually done through a cumulated periodogram using a large number of signal samples. The location and dates of the data induce the spatial and temporal resolution of the slope estimates. To improve this resolution, this article studies a new parametric method based on an autoregressive model combined with a warping of the frequency scale (denoted as ARWARP). This ARWARP model provides a PSD estimate, with a lower variance than the classical Fourier-based ones and is reliable in the case of a small sample number. To give a reference in the performance of the SLA slope estimation, the corresponding Cramér–Rao bound is derived. Then, rather than performing linear regression on the spectral estimates, a new estimator of the slope is suggested, based on a model fitting of the PSD. A statistical validation is proposed on simulated SLA signals, showing the performance of slope estimation using this ARWARP spectral estimator, compared to classical Fourier-based methods. Application to Sentinel-3 real data highlights the main advantage of the ARWARP model, making possible SLA slope estimation on a short signal segment, i.e., with a high spatial and/or temporal resolution.

Satellite altimeters provide sea level measurements along satellite track. A mean profile based on the measurements averaged over a time period is then subtracted to estimate the sea level anomaly (SLA). In the spectral domain, SLA is characterized by a power spectral density of the form f-αwhere the slope α is a parameter of great interest for ocean monitoring. However, this information lies in a narrow frequency band, located at very low frequencies, which calls for some specific spectral analysis methods. This paper studies a new parametric method based on an autoregressive model combined with a warping of the frequency scale (denoted as ARWARP). A statistical validation is proposed on simulated SLA signals, showing the performance of slope estimation using this ARWARP spectral estimator, compared to classical Fourier-based methods. Application to Sentinel-3 real data highlights the main advantage of the ARWARP model, making possible SLA slope estimation on a short signal segment, i.e., with a high spatial resolution.

We consider adaptive detection of a signal of interest when two sets of training samples are available, one sharing the same covariance matrix as the data under test, the other set being mismatched. The approach proposed in this letter is to whiten both the data under test and the matched training samples using the sample covariance matrix of the mismatched training samples. The distribution of the whitened data is then derived and subsequently the generalized likelihood ratio test is obtained. Numerical simulations show that it performs well and is rather robust.

We develop Adaptive Cosine Estimator (ACE) type detector for non-zero mean Gaussian interference specifically for the replacement and additive target models of the hyperspectral imaging problem. We consider the case where the data under test and the training samples differ from one scaling factor on the mean and one scaling factor on the covariance matrix. We derive two-step generalized likelihood ratio tests for both the additive model and the replacement model and show that the new detectors differ in the way the mean value is removed. A real data experiment shows that they outperform the standard version.

One of the main objectives of hyperspectral image processing is to detect a given target among an unknown background. The standard data to conduct such detection is a reflectance map, where the spectral signatures of each pixel's components, known as endmembers, are associated with their abundances in the pixel. Due to the low spatial resolution of most hyperspectral sensors, such a target occupies a fraction of the pixel. A widely used model in the case of subpixel targets is the replacement model. Among the vast number of possible detectors, algorithms matched to the replacement model are quite rare. One of the few examples is the finite target matched filter (MF), which is an adjustment of the well-known MF. In this article, we derive the exact generalized likelihood ratio test for this model. This new detector can be used both with a local covariance estimation window or a global one. It is shown to outperform the standard target detectors on real data, especially for small covariance estimation windows.

The adaptive matched filter, like many other adaptive detection schemes, uses in its test statistic the data under test whitened by the sample covariance matrix S of the training samples. Actually, it is a generalized likelihood ratio test (GLRT) based on the conditional (i.e., for given S) distribution of the adaptively whitened data. In this letter, we investigate detection of a Gaussian rank-one signal using the marginal (unconditional) distribution of the adaptively whitened data. A first contribution is to derive the latter and to show that it only depends on a scalar parameter, namely the signal to noise ratio. Then, a GLRT is formulated from this unconditional distribution and shown to have the constant false alarm rate property. We show that it bears close resemblance with the plain GLRT based on the whole data set (data under test and training samples). The new detector performs as well as the plain GLRT and even better with multiple cells under test and low training sample support.

This paper considers adaptive signal detection in stochastic homogeneous environments where the disturbance covariance matrix of both test and training signals, R, is assumed to be a random matrix with a priori knowledge of R. Unlike existing detectors assuming a known hyperparameter associated with R, a knowledge-aided detector with the capability of automatic weighting is considered by accounting for the uncertainty of the prior knowledge. Specifically, the generalized likelihood ratio test (GLRT) is utilized to develop the test statistic, along with the maximum marginal likelihood (MML) estimation of the hyperparameter. The proposed KA-MML-GLRT detector is evaluated by numerical simulations and the results show improved detection performance over conventional and knowledge-aided detectors, especially in the case of limited training signals and inaccurate prior knowledge.

Regularization, which consists in shrinkage of the sample covariance matrix to a target matrix, is a commonly used and effective technique in low sample support covariance matrix estimation. Usually, a target matrix is chosen and optimization of the shrinkage factor is carried out, based on some relevant metric. In this letter, we rather address the choice of the target matrix. More precisely, we aim at evaluating, from observation of the data matrix, whether a given target matrix is a good regularizer. Towards this end, the expected likelihood (EL) approach is investigated. At a first step, we re-interpret the regularized covariance matrix estimate as the minimum mean-square error estimate in a Bayesian model where the target matrix serves as a prior. The likelihood function of the data is then derived, and the EL principle is subsequently applied. Over-sampled and under-sampled scenarios are considered.

This letter deals with the problem of fluctuating target detection in heavy-tailed K-distributed clutter over a number T of independent coherent intervals, e.g., in the case of a long observation interval (“stare mode”), or that of independent (range) resolution cells as per the track before detect techniques. The generalized likelihood ratio test for the problem at hand is derived, as well as an approximation of it, whose distribution under the null hypothesis is derived. We also show some significant differences as compared to the usual Gaussian case, in particular the influence of T and of the shape parameter of the K distribution.

The generalized likelihood ratio test (GLRT) is a very widely used technique for detecting signals of interest amongst noise, when some of the parameters describing the signal (and possibly the noise) are unknown. The threshold of such a test is set from a desired probability of false alarm Pfa and hence this threshold depends on the statistical assumptions made about noise. In practice however, the noise statistics are seldom known and it becomes crucial to characterize Pfa under a mismatched distribution. In this letter, we address this problem in the case of a simple binary composite hypothesis testing problem (matched filter) when the threshold is designed under a Gaussian assumption while the noise actually follows an elliptically contoured distribution. We also consider the inverse situation. Generic expressions for the assumed and actual Pfa are derived and illustrated on the particular case of Student distributions for which simple, closed-form expressions are obtained. The latter show that the GLRT based on Gaussian assumption is not robust while that based on Student assumption is.

In a recent letter we addressed the problem of detecting a fluctuating target in K distributed noise using multiple coherent processing intervals. It was shown through simulations that the performance of the likelihood ratio test is dominated by the snapshot which corresponds to the minimal value of the texture. However, for this particular snapshot the clutter to thermal noise ratio is not large and hence thermal noise cannot be neglected. In the present letter, we continue our investigation with a view to consider detection in a mixture of K distributed and Gaussian noise. Towards this end we study the sensitivity of our previously derived detectors. First, we provide stochastic representations that allow to evaluate their performance in K distributed noise only or Gaussian noise only. Then, their robustness to a mixture is assessed.

We consider Direction of Arrival (DoA) estimation in the case of two closely spaced sources. In this case, most high resolution techniques fail to estimate the two DoAs if the waveforms are highly correlated. Maximum Likelihood Estimators (MLE) are known to be more robust, but their excessive computational load limits their use in practice. In this paper, we propose an asymptotic approximation of the Unconditional Maximum Likelihood (UML) procedure in the case of a Uniform Linear Array (ULA) and two closely spaced targets. This approximation is based on an asymptotically (in the number of observations) equivalent formulation of the UML criterion, and on its Taylor series approximation for small DoA separation. This simplified procedure, which requires solving a 1D-optimization problem only, is shown to be accurate for source separation lower than half the mainlobe. Furthermore, it outperforms conventional high resolution algorithms in the case of two correlated sources.

The problem of adaptive detection of a signal of interest embedded in elliptically distributed noise with unknown scatter matrix R is addressed, in the specific case where the number of training samples T is less than the dimension M of the observations. In this under-sampled scenario, whenever R is treated as an arbitrary positive definite Hermitian matrix, one cannot resort directly to the generalized likelihood ratio test (GLRT) since the maximum likelihood estimate (MLE) of R is not well-defined, the likelihood function being unbounded. Indeed, inference of R can only be made in the subspace spanned by the observations. In this letter, we present a modification of the GLRT which takes into account the specific features of under-sampled scenarios. We come up with a test statistic that, surprisingly enough, coincides with a subspace detector of Scharf and Friedlander: the detector proceeds in the subspace orthogonal to the training samples and then compares the energy along the signal of interest to the total energy. Moreover, this detector does not depend on the density generator of the noise elliptical distribution. Numerical simulations illustrate the performance of the test and compare it with schemes based on regularized estimates of R.

We consider the problem of estimating the scatter matrix in complex elliptically symmetric (CES) distributions using the expected likelihood (EL) approach. The latter, originally derived in the Gaussian case, is based on the fact that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual covariance matrix does not depend on this matrix, and is fully specified by the matrix dimension M and the number of independent training samples T. We extend this result to CES distributions as well as to angular central Gaussian (ACG) distributions. More precisely, we prove that for CES distributions, the p.d.f. of the LR, evaluated at the true scatter matrix Σ0, does not depend on the latter but depends on the density generator of the CES distribution. As for the ACG case, we demonstrate that the LR for Σ0is distribution-free. This invariance property paves the way to derivation of regularized covariance matrix estimates, where the regularization parameters are chosen from the EL principle. The relevance of such a choice for the regularization parameters is illustrated on an example with fixedpoint diagonally loaded estimates.

We consider robust adaptive beamforming in the presence of steering vector uncertainties. A Bayesian approach is presented where the steering vector of interest is treated as a random vector with a Bingham prior distribution. Moreover, in order to also improve robustness against low sample support, the interference plus noise covariance matrix R is assigned a non informative prior distribution which enforces shrinkage to a scaled identity matrix, similarly to diagonal loading. The minimum mean square distance estimate of the steering vector as well as the minimum mean square error estimate of R are derived and implemented using a Gibbs sampling strategy. The new beamformer is shown to converge within a limited number of snapshots, despite the presence of steering vector errors.

The Slepian-Bangs formula provides a very convenient way to compute the Fisher information matrix (FIM) for Gaussian distributed data. The aim of this letter is to extend it to a larger family of distributions, namely elliptically contoured (EC) distributions. More precisely, we derive a closed-form expression of the FIM in this case. This new expression involves the usual term of the Gaussian FIM plus some corrective factors that depend only on the expectations of some functions of the so-called modular variate. Hence, for most distributions in the EC family, derivation of the FIM from its Gaussian counterpart involves slight additional derivations. We show that the new formula reduces to the Slepian-Bangs formula in the Gaussian case and we provide an illustrative example with Student distributions on how it can be used.

Subspace estimation using relatively few samples is a frequently encountered problem in numerous applications, including hyperspectral imagery the target application of this paper. We address this problem in a Bayesian framework assuming that some rough prior knowledge about the subspace is available. Our approach is based on the CS decomposition of an orthogonal matrix whose columns span the subspace of interest. This parametrization only involves mild assumptions about the distribution of the angles between the actual subspace and the prior subspace, and is intuitively appealing. We derive the posterior distribution for the matrices involved in the CS decomposition and the angles between subspaces, and we propose a Gibbs sampling scheme to compute the minimum mean-square distance estimator of the subspace of interest. The estimator accuracy is evaluated through numerical simulations and tested against real hyperspectral data.

Conventional multibeam satellite communications systems ensure coverage of wide areas through multiple fixed beams where all users inside a beam share the same bandwidth. The authors consider a new and more flexible system where each user is assigned his own beam, and the users can be very geographically dispersed. This is achieved through the use of a large direct radiating array coupled with adaptive beamforming so as to reject interferences and to provide a maximal gain to the user of interest. New fast-converging adaptive beamforming algorithms are presented, which allow one to obtain good signal to interference and noise ratio with a number of snapshots much lower than the number of antennas in the array. These beamformers are evaluated on reference scenarios.

We address the problem of estimating a covariance matrix R using K samples zkwhose covariance matrices are τkR, where τkare random variables. This problem naturally arises in radar applications in the case of compound-Gaussian clutter. In contrast to the conventional approach which consists in considering R as a deterministic quantity, a knowledge-aided (KA) approach is advocated here, where R is assumed to be a random matrix with some prior distribution. The posterior distribution of R is derived. Since it does not lead to a closed-form expression for the minimum mean-square error (MMSE) estimate of R, both R and τkare estimated using a Gibbs-sampling strategy. The maximum a posteriori (MAP) estimator ofR is also derived. It is shown that it obeys an implicit equation which can be solved through an iterative procedure, similarly to the case of deterministic τks, except that KA is now introduced in the iterative scheme. The new estimators are shown to improve over conventional estimators, especially in small sample support.

The auxiliary-vector beamformer is an algorithm that generates iteratively a sequence of beamformers which, under the assumption of a positive definite covariance matrixR, converges to the minimum variance distortionless response beamformer, without resorting to any matrix inversion. In the case whereRis rank-deficient, e.g., whenRis substituted for the sample covariance matrix and the number of snapshots is less than the number of array elements, the behavior of the AV beamformer is not known theoretically. In this letter, we derive a new convergence result and show that the AV beamformer weights converge whenRis rank-deficient, and that the limit belongs to the class of reduced-rank beamformers..

We propose a two-stage detector consisting of a subspace detector followed by the whitened adaptive beamformer orthogonal rejection test. The performance analysis shows that it possesses the constant false alarm rate property with respect to the unknown co-variance matrix of the noise and that it guarantees a wider range of directivity values with respect to previously proposed two-stage detectors. The probability of false alarm and the probability of detection (for both matched and mismatched signals) have been evaluated by means of numerical integration techniques.

The adaptive coherence estimator (ACE) is known to be the generalized likelihood ratio test (GLRT) in partially homogeneous environments, i.e., when the covariance matrix Msof the secondary data is proportional to the covariance matrix Mpof the vector under test (or Ms= gamma/Mp). In this letter, we show that ACE is indeed the GLRT for a broader class of nonhomogeneous environments, more precisely when Msis a random matrix, with inverse complex Wishart prior distribution whose mean only is proportional to Mp. Furthermore, we prove that, for this class of heterogeneous environments, the ACE detector satisfies the constant false alarm rate (CFAR) property with respect to gamma and Mp.

In many applications, it is required to detect, from a primary vector, the presence of a signal of interest embedded in noise with unknown statistics. We consider a situation where the training samples used to infer the noise statistics do not share the same covariance matrix as the vector under test. A Bayesian model is proposed where the covariance matrices of the primary and the secondary data are assumed to be random, with some appropriate joint distribution. The prior distributions of these matrices reflect a rough knowledge about the environment. Within this framework, the minimum mean-square error (MMSE) estimator and the maximum a posteriori (MAP) estimator of the primary data covariance matrix are derived. A Gibbs sampling strategy is presented for the implementation of the MMSE estimator. Numerical simulations illustrate the performances of these estimators and compare them with those of the sample covariance matrix estimator.

A synthetic aperture radar scale model has been developed to improve signal processing teaching. Based on low frequency ultrasound transmission, it is a low cost demonstration kit. The overall software is directly running on Matlab® and allows easy and realtime modifications. This educational tool can be used to test different waveforms and show the effects of a real scene on the final image. It can also be used in a more advanced way to test different signal processing in order to improve image focusing or to reduce computation burden.

We consider the problem of detecting a signal of interest in the presence of colored noise, in the case of a covariance mismatch between the test cell and the training samples. More precisely, we consider a situation where an interfering signal (e.g., a sidelobe target or an undernulled interference) is present in the test cell and not in the secondary data. We show that the adaptive coherence estimator (ACE) is the generalized likelihood ratio test for such a problem, which may explain the previously observed fact that the ACE has excellent sidelobe rejection capability, at the price of low mainlobe target sensitivity

This letter considers adaptive detection of a signal in a nonhomogeneous environment, more precisely under a covariance mismatch between the test vector and the training samples, due to an interference that is not accounted for by the training samples, e.g., a sidelobe target or an under-nulled interference. We assume that the covariance matrices of the test vector and the training samples verify the so-called generalized eigenrelation. Under this assumption, we derive the generalized likelihood ratio test and show that it coincides with Kelly's detector.

We address the problem of detecting a signal of interest in the presence of noise with unknown covariance matrix, using a set of training samples. We consider a situation where the environment is not homogeneous, i.e., when the covariance matrices of the primary and the secondary data are different. A knowledge-aided Bayesian framework is proposed, where these covariance matrices are considered as random, and some information about the covariance matrix of the training samples is available. Within this framework, the maximum a priori (MAP) estimate of the primary data covariance matrix is derived. It is shown that it amounts to colored loading of the sample covariance matrix of the secondary data. The MAP estimate is in turn used to yield a Bayesian version of the adaptive matched filter. Numerical simulations illustrate the performance of this detector, and compare it with the conventional adaptive matched filter

In this paper we propose decision schemes to distinguish between the H0hypothesis that range cells under test contain disturbance only (i.e., noise plus interference) and the H1hypothesis that they also contain signal components along a direction which is a priori unknown, but constrained to belong to a given subspace (H) of the observables. The disturbance is modeled in terms of complex normal noise vectors plus deterministic interference assumed to belong to a known subspace (J) of the observables. At the design stage we resort to either the plain generalized likelihood ratio test (GLRT) or the two-step GLRT-based design procedure. Moreover, we assume that a set of noise only (secondary) data is available. A preliminary performance analysis, conducted by resorting to simulated data, shows that the one-step GLRT performs better than the two-step GLRT-based design procedure

In this paper, we address the problem of detecting a signal whose associated spatial signature is subject to uncertainties, in the presence of subspace interference and broadband noise, and using multiple snapshots from an array of sensors. To account for steering vector uncertainties, we assume that the spatial signature of interest lies in a given linear subspacewhile its coordinates in this subspace are unknown. The generalized likelihood ratio test (GLRT) for the problem at hand is formulated. We show that the GLRT amounts to searching for the best direction in the subspaceafter projecting out the interferences. The distribution of the GRLT under both hypotheses is derived and numerical simulations illustrate its performance.

Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. Recently, worst-case approaches which consist of protecting the array's response in an ellipsoid centered around the nominal steering vector have been proposed. They amount to generalized (i.e. non necessarily diagonal) loading of the covariance matrix. In this paper, we present a theoretical analysis of the signal to interference plus noise ratio (SINR) for this class of robust beamformers, in the presence of random steering vector errors. A closed-form expression for the SINR is derived which is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR.

Partial-band interferences are known to have a deleterious effect on the receiver performance in frequency-hopping communication systems. We consider the rejection of such interferences using an array of sensors. A simple yet effective method is proposed based on a constant power algorithm. The principle behind this method is that partial-band interferences contribute to power variations in the received signal, hence, the idea to constrain the output power of the array to be constant in order to reject interferences. It is shown that the algorithm converges in a reasonably low number of hops. Additionally, it achieves a nearly optimal signal-to-interference-plus-noise ratio without requiring any information about the location of the desired user. Finally, its robustness to synchronization errors is illustrated via numerical simulations.

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 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.

This paper considers the problem of estimating the parameters of chirp signals with randomly time-varying amplitude. Two methods for solving this problem are presented. First, a nonlinear least-squares approach (NLS) is proposed. It is shown that by minimizing the NLS criterion with respect to all samples of the time-varying amplitude, the problem reduces to a two-dimensional maximization problem. A theoretical analysis of the NLS estimator is presented and an expression for its asymptotic variance is derived. It is shown that the NLS estimator has a variance very close to the Cramer-Rao bound. The second approach combines the principles behind the high-order ambiguity function (HAF) and the NLS approach. It provides a computationally simpler but suboptimum estimator. A statistical analysis of this estimator is also carried out. Numerical examples attest to the validity of the theoretical analysis and establish a comparison between the two proposed methods.

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.

Laser anemometers have become a promising technique for estimating velocities in a flow. In this paper we study their use for on-board aircraft speed of flight estimation. More specifically, this paper addresses the problem of simultaneous detection of the arrival of aerosol particles in a laser anemometer and estimation of their velocity. A joint detection-estimation scheme is proposed. A likelihood ratio test is presented and considerations about the specificities of the problem are used to calculate the threshold. Computationally efficient algorithms for estimating the parameters of interest are derived and on-line implementation issues are addressed. Numerical examples attest for the performance of the method, on both simulated and real data recorded during a flight test.

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.

This paper addresses the problem of estimating particle's velocity in the vicinity of an aircraft by means of a laser velocimeter. A model for the signal generated by a particle of air passing through a probe volume consisting of equidistant bright and dark fringes is given. From this model, a frequency estimator based on the phase of the correlation sequence of the signal is proposed. A theoretical analysis of the frequency estimator is presented. In particular, a formula for the variance of the estimate is derived under the assumption of small estimation errors. Numerical examples confirm the validity of the analysis. Finally, the effectiveness of the proposed algorithm is demonstrated on real data.

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.

A new method for estimating the frequency of a random amplitude sinusoid is proposed. It is based upon solving overdetermined Yule-Walker equations using constrained least-squares techniques. A Gauss-Newton algorithm is derived for proceeding to the constrained minimization. Simulation results prove the superiority of the new method over the unconstrained method, specially for a small number of equations.

Some insights are provided into the analysis of a recently proposed Doppler frequency estimator. The analysis of a multiplicative model and the algorithm used are reviewed. An optimal and a modified version of this algorithm are studied. The Cramer-Rao bounds for pole estimation of an equivalent autoregressive moving-average (ARMA) process are presented. Numerical simulations that described the statistical behavior of these estimators were performed. It is verified that the Doppler frequency is exactly the centroid of the set of ARMA pole frequencies. A novel algorithm (ARCOS), an optimally weighted version of ARCOS, and ARCOS modified by the Newton-Raphson (NR) technique are derived. Although a direct optimal estimator cannot be derived, it is shown that the basic and the optimal centroid are equivalent and below the ARMA bound. NRARCOS is shown to improve the variance, practically reaching the performance of the optimal centroid.>

A novel realistic model for Doppler signals is investigated. The intended application is the accurate estimation of a train's speed using an onboard Doppler radar. The signal is shown to be conveniently modeled by a sine wave amplitude modulated by an autoregressive process. The spectral properties of this model are derived and used to define a relevant frequency estimation algorithm based on conventional ARMA (autoregressive moving average) spectral estimators. Statistical properties of the algorithm (covariances and cross covariances of estimated frequencies) are presented to show there could be a considerable improvement in the estimation of the frequency. The performance of this algorithm is evaluated by means of simulation and compared with a conventional estimator.>