Maximum likelihood estimation for array processing in colored noise

Direction of arrival estimation of multiple sources, using a uniform linear array, in noise with unknown covariance is considered. The noise is modeled as a spatial autoregressive process with unknown parameters. Both stochastic and deterministic signal models are considered. For the random signal case, an approximate maximum likelihood estimator of the signal and noise parameters is derived. It requires numerical maximization of a compressed likelihood function over the unknown arrival angles. Analytical expressions for the MLEs of the signal covariance and the AR parameters are given. Similar results for the case of deterministic signals are also presented     

Robust maximum likelihood bearing estimation in contaminatedGaussian noise

A robust maximum likelihood (ML) direction-of-arrival (DOA) estimation method that is insensitive to outliers and distributional uncertainties in Gaussian noise is presented. The algorithm has been shown to perform much better than the Gaussian ML algorithm when the underlying noise distribution deviates even slightly from Gaussian while still performing almost as well in pure Gaussian noise. As with the Gaussian ML estimation, it is still capable of handling correlated signals as well as single snapshot cases. Performance of the algorithm is analyzed using the unique resolution test procedure which determines whether a DOA estimation algorithm, at a given confidence level, can resolve two dominant sources with very close DOAs     

On maximum likelihood estimation for the two parameter Weibull distribution

This paper examines recent results presented on maximum likelihood estimation for the two parameter Weibull distribution. In particular, we seek to explain some recently reported values for estimator bias when the data for analysis contains both times to failure and censored times in operation; our discussion centres on the generation of sample data sets. We conclude that, under appropriate conditions, estimators are asymptotically unbiased, with relatively low bias in small to moderate samples. We then present the results of some further experiments which suggest that the previously reported values for estimator bias can be attributed to the method of generating sample data sets in simulation experiments.     

Maximum likelihood DOA estimation and asymptotic Cramer-Rao boundsfor additive unknown colored noise

This paper is devoted to the maximum likelihood estimation of multiple sources in the presence of unknown noise. With the spatial noise covariance modeled as a function of certain unknown parameters, e.g., an autoregressive (AR) model, a direct and systematic way is developed to find the exact maximum likelihood (ML) estimates of all parameters associated with the direction finding problem, including the direction-of-arrival (DOA) angles Θ, the noise parameters α, the signal covariance Φ_s, and the noise power σ². We show that the estimates of the linear part of the parameter set Φ_s and σ² can be separated from the nonlinear parts Θ and α. Thus, the estimates of Φ_s and σ² become explicit functions of Θ and α. This results in a significant reduction in the dimensionality of the nonlinear optimization problem. Asymptotic analysis is performed on the estimates of Θ and α, and compact formulas are obtained for the Cramer-Rao bounds (CRB's). Finally, a Newton-type algorithm is designed to solve the nonlinear optimization problem, and simulations show that the asymptotic CRB agrees well with the results from Monte Carlo trials, even for small numbers of snapshots     

Target position estimation in radar and sonar, and generalized ambiguity analysis for maximum likelihood parameter estimation

Target position estimation in radar and sonar means joint estimation of range and angle in the presence of noise and clutter. The global behavior of a maximum likelihood (ML) position estimator, and the clutter suppression capability of the system, can be written in terms of a range-angle ambiguity function. This function depends upon signal waveform and array configuration, i.e., upon both temporal and spatial characteristics of the system. Ambiguity and variance bound analysis indicates that system bandwidth can often be traded for array size, and direction-dependent signals can be used to obtain better angle resolution without increasing the size of the array. Wide-band direction-dependent signals (temporal diversity) can be traded for large real or synthetic arrays (spatial diversity). This tradeoff is apparently exploited by some animal echolocation systems. The above insights are obtained mostly from the properties of the range-angle ambiguity function. In general, an appropriate ambiguity function should be very useful for the design and evaluation of any ML parameter estimator. System identification methods and radio navigation systems, for example, can be optimized by minimizing the volume of a multiparameter ambiguity function.     

Minimised error criterion in recursive least-squares parameter estimation

In recursive least-squares parameter-estimation schemes, initial values are used to start the recursive procedure. It is demonstrated that these initial values result from another error criterion than the one which is commonly assumed to be minimised.     

Radio source parameter estimation by maximum likelihood processing of variable baseline correlation interferometer data

The accurate joint determination of the direction and strength of a point noise source when the mutual coherence function of its radiated field is spatially sampled atMbaselines by a correlation interferometer is considered. The measurements are corrupted by the combined effects of a) the additive background and receiver noises at the interferometer antennas and b) the finite integration time of a practical correlator. The problem is approached from a statistical point of view (as contrasted with beam forming techniques). First the probability density function of the measurements is derived. The source's two parameters (direction and strength) are then jointly estimated using the maximum likelihood (ML) method. Investigation of the estimates' properties shows that they are virtually unbiased with variances that effectively attain the standard Cramer-Rao (C-R) lower bound when the number of measurements exceeds a "threshold" which is a decreasing function of the measurements' signal-to-noise ratio (SNR). The empirically observed fact that such a threshold is quite small, even at low SNR's, as well as the unbiasedness of the estimates, makes the performance of these (ML) estimates optimum for most practical applications.     

Fast maximum likelihood DOA estimation in the two-target case with applications to automotive radar

Direction-of-arrival (DOA) estimation of two targets using a single snapshot plays an important role in automotive radar for advanced driver assistance systems. Conventional Fourier methods have a limited resolution and generally yield biased estimates. Subspace methods involve a numerically complex eigendecomposition and require multiple snapshots or a suboptimal pre-processing for reliable estimation. We therefore consider the maximum likelihood (ML) DOA estimator, which is applicable with a single snapshot and shows good statistical properties. To reduce the computational burden, we propose a grid search procedure with a simplified calculation of the objective function. The required projection operators are pre-calculated off-line and stored. To save storage space and computations, we further propose a rotational shift of the field-of-view such that the relevant angular sector, which has to be evaluated, is delimited and centered with respect to broadside. The final estimates are obtained using a quadratic interpolation. The developed method is demonstrated with an example. Simulations are designed to assess the performance of the considered ML estimator with grid search and interpolation, and to compare it among selected representative methods. We further present results obtained with experimental data from a typical application in automotive radar.     

Bootstrapping the generalized least-squares estimator in colored Gaussian noise with unknown covariance parameters

It is demonstrated that in problems involving the estimation of linear regression parameters in colored Gaussian noise, the simple least-squares estimator can be significantly suboptimal. When the noise covariance function can be described as a known function of a finite number of unknown nonrandom parameters, it is possible to take advantage of this information to improve upon the least-squares estimator by an appropriate bootstrapping technique. Two examples are given, and comments that may lead to other examples are presented.     

Maximum likelihood trend estimation in exponential noise

This paper considers the problem of estimating a linear trend in noise, where the noise is modeled as independent and identically distributed (i.i.d.) random process with exponential distribution. The corresponding maximum likelihood parameter estimator of the trend and noise parameters is derived, and its performance is analyzed. It turns out that the resulting maximum likelihood estimator has to solve a linear programming problem with number of constraints equal to the number of received data. A recursive form of the maximum likelihood estimator, which makes it suitable for implementation in real-time systems, is then proposed. The memory requirements of the recursive algorithm are data dependent and are investigated by simulations using both computer-generated and recorded data sets     

Bayesian detection and estimation of cisoids in colored noise

The problem of estimating the number of cisoids in colored noise is addressed. It is assumed that the noise can be modeled by an autoregression whose order has also to be estimated. A new criterion is proposed for estimating the number of cisoids and the autoregressive model order, as well as a new algorithm for estimating the cisoidal frequencies. In the derivation, a Bayesian methodology and subspace decomposition are employed. The proposed criterion significantly outperforms the popular MDL and AIC as applied in a paper by Nagesha and Kay. In addition, an algorithm that reduces the computational complexity of the solution is developed, computer simulations that demonstrate the performance of the criterion are included     

Mean-squared error and threshold SNR prediction of maximum-likelihood signal parameter estimation with estimated colored noise covariances

An interval error-based method (MIE) of predicting mean squared error (MSE) performance of maximum-likelihood estimators (MLEs) is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix; an intermediate step central to adaptive array detection and parameter estimation. The successful application of MIE requires good approximations of two quantities: 1) interval error probabilities and 2) asymptotic (SNRrarrinfin) local MSE performance of the MLE. Exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein, that in conjunction with the Union Bound provide accurate prediction of the required interval error probabilities. The Crameacuter-Rao Bound (CRB) often provides adequate prediction of the asymptotic local MSE performance of MLE. The signal parameters, however, are decoupled from the colored noise parameters in the Fisher Information Matrix for the deterministic signal model, rendering the CRB incapable of reflecting loss due to colored noise covariance estimation. A new modification of the CRB involving a complex central beta random variable different from, but analogous to the Reed, Mallett, and Brennan beta loss factor provides a working solution to this problem, facilitating MSE prediction well into the threshold region with remarkable accuracy     

Asymptotic maximum likelihood estimator performance for chaoticsignals in noise

The performance of the maximum likelihood estimator for a 1-D chaotic signal in white Gaussian noise is derived. It is found that the estimator is inconsistent and therefore the usual asymptotic distribution (large data record length) is invalid. However, for high signal-to-noise ratios (SNRs), the maximum likelihood estimator is asymptotically unbiased and attains the Cramer-Rao lower bound     

Maximum likelihood estimation of signals in autoregressive noise

Time series modeling as the sum of a deterministic signal and an autoregressive (AR) process is studied. Maximum likelihood estimation of the signal amplitudes and AR parameters is seen to result in a nonlinear estimation problem. However, it is shown that for a given class of signals, the use of a parameter transformation can reduce the problem to a linear least squares one. For unknown signal parameters, in addition to the signal amplitudes, the maximization can be reduced to one over the additional signal parameters. The general class of signals for which such parameter transformations are applicable, thereby reducing estimator complexity drastically, is derived. This class includes sinusoids as well as polynomials and polynomial-times-exponential signals. The ideas are based on the theory of invariant subspaces for linear operators. The results form a powerful modeling tool in signal plus noise problems and therefore find application in a large variety of statistical signal processing problems. The authors briefly discuss some applications such as spectral analysis, broadband/transient detection using line array data, and fundamental frequency estimation for periodic signals     

Maximum likelihood time delay estimation in non-Gaussian noise

In a non-Gaussian noise environment, it is theoretically possible to design a delay estimator that performs significantly better than the conventional linear correlator. We study the maximum likelihood estimator for passive time delay in non-Gaussian noise. We show that the form of the best estimator depends strongly on signal-to-noise ratio (SNR), and the estimator optimal at low SNR can fail catastrophically at high values of SNR. The paper uses simulations to examine this sensitivity problem and proposes an ad hoc instrumentation that is reasonably robust over a wide range of SNR values     

色噪声下估计信源数的方法及性能分析

分析了雷达谱估计信号源数的方法.在高频地波雷达中,当各阵元噪声功率非平稳时,提出了一种基于盖氏圆半径和Kullback对称散度的信号源数目估计方法.通过实验证明,该方法能够有效解决在小快拍数、阵元噪声功率不相等情况下信号源数目估计的问题,实现色噪声环境下信源数的估计.     

Conditional mean and maximum likelihood approaches to multiharmonicfrequency estimation

The performance of an extended Kalman filter (EKF) applied to the problem of estimating the (assumed constant) parameters (fundamental frequency, harmonic phases, and amplitudes) of a complex multiharmonic signal measured in noise is shown to be asymptotically (i.e., as the number of measurements tends to infinity) efficient. The Cramer-Rao (CR) bounds associated with the estimation problem are derived for the case where the measurements commence at an arbitrary time distinct from zero     

Fast evaluation of the likelihood of an HMM: ion channel currentswith filtering and colored noise

Hidden Markov models (HMMs) have been used in the study of single-channel recordings of ion channel currents for restoration of idealized signals from noisy recordings and for estimation of kinetic parameters. A key to their effectiveness from a computational point of view is that the number of operations to evaluate the likelihood, posterior probabilities and the most likely state sequence is proportional to the product of the square of the dimension of the state space and the length of the series. However, when the state space is quite large, computations can become infeasible. This can happen when the record has been lowpass filtered and when the noise is colored. In this paper, we present an approximate method that can provide very substantial reductions in computational cost at the expense of only a very small error. We describe the method and illustrate through examples the gains that can be made in evaluating the likelihood     

Maximum likelihood parameter and rank estimation in reduced-rankmultivariate linear regressions

This paper considers the problem of maximum likelihood (ML) estimation for reduced-rank linear regression equations with noise of arbitrary covariance. The rank-reduced matrix of regression coefficients is parameterized as the product of two full-rank factor matrices. This parameterization is essentially constraint free, but it is not unique, which renders the associated ML estimation problem rather nonstandard. Nevertheless, the problem turns out to be tractable, and the following results are obtained. An explicit expression is derived for the ML estimate of the regression matrix in terms of the data covariances and their eigenelements. Furthermore, a detailed analysis of the statistical properties of the ML parameter estimate is performed. Additionally, a generalized likelihood ratio test (GLRT) is proposed for estimating the rank of the regression matrix. The paper also presents the results of some simulation exercises, which lend empirical support to the theoretical findings     

Threshold parameter estimation in nonadditive non-Gaussian noise

Threshold or weak-signal locally optimum Bayes estimators (LOBEs) of signal parameters, where the observations are an arbitrary mixture of signal and noise, the latter being independent, are first derived for “simple” as well as quadratic cost functions under the assumption that the signal is present a priori. It is shown that the desired LOBEs are either a linear (simple cost function) or a nonlinear (quadratic cost function) functional of an associated locally optimum and asymptotically optimum Bayes detector. Second, explicit classes of (threshold) optimum estimators are obtained for both cost functions in the coherent as well as in the incoherent reception modes. Third, the general results are applied to amplitude estimation, where two examples are considered: (1) coherent amplitude estimation in multiplicative noise with simple cost function (SCF) and (2) incoherent amplitude estimation with quadratic cost function (QFC) of a narrowband signal arbitrarily mixed with noise. Moreover, explicit estimator structures are given together with desired properties (i.e. efficiency of the unconditional maximum likelihood (ML) estimator) and Bayes' risks. These properties are obtained by employing contiguity-a powerful concept in modern statistics-implied by the locally asymptotically normal character of the detection algorithms