Two-dimensional spectrum estimation using noncausal autoregressive models

Two-dimensional (2-D) spectrum estimation from raw data is of interest in signal and image processing. A parametric technique for spectrum estimation using 2-D noncausal autoregressive (NCAR) models is given. The NCAR models characterize the statistical dependency of the observation at location s on its neighbors in all directions. This modeling assumption reduces the spectrum estimation problem to two subproblems: the choice of appropriate structure of the NCAR model and the estimation of parameters in NCAR models. By assuming that the true structure of the NCAR model is known, we first analyze the existence and uniqueness of Gaussian maximum likelihood (GML) estimates of NCAR model parameters. Due to the noncausal nature of the models, the computation of GML estimates is burdensome. By assuming specific boundary conditions, computationally tractable expressions are obtained for the likelihood function. Expressions for the asymptotic covariance matrix of the GML estimates as well as the simultaneous confidence bands for the estimated spectrum using GML estimates are derived. Finally, the usefulness of the method is illustrated by computer simulation results.     

Bearing estimation in the bispectrum domain

A new array processing method is presented for bearing estimation based on the cross bispectrum of the array output data. The method is based on the asymptotic distribution of cross-bispectrum estimates and uses maximum likelihood theory. It is demonstrated that, when the noise additive sources are spatially correlated Gaussian with unknown cross-spectral matrix (CSM), the cross-bispectrum method provides better bearing estimates than the stochastic maximum likelihood method with known CSM. Analytical studies and simulations are given to document the performance of the new method     

Maximum-likelihood bearing estimation with partly calibrated arraysin spatially correlated noise fields

The problem of using a partly calibrated array for maximum likelihood (ML) bearing estimation of possibly coherent signals buried in unknown correlated noise fields is shown to admit a neat solution under fairly general conditions. More exactly, this paper assumes that the array contains some calibrated sensors, whose number is only required to be larger than the number of signals impinging on the array, and also that the noise in the calibrated sensors is uncorrelated with the noise in the other sensors. These two noise vectors, however, may have arbitrary spatial autocovariance matrices. Under these assumptions the many nuisance parameters (viz., the elements of the signal and noise covariance matrices and the transfer and location characteristics of the uncalibrated sensors) can be eliminated from the likelihood function, leaving a significantly simplified concentrated likelihood whose maximum yields the ML bearing estimates. The ML estimator introduced in this paper, and referred to as MLE, is shown to be asymptotically equivalent to a recently proposed subspace-based bearing estimator called UNCLE and rederived herein by a much simpler approach than in the original work. A statistical analysis derives the asymptotic distribution of the MLE and UNCLE estimates, and proves that they are asymptotically equivalent and statistically efficient. In a simulation study, the MLE and UNCLE methods are found to possess very similar finite-sample properties as well. As UNCLE is computationally more efficient, it may be the preferred technique in a given application     

Maximum likelihood array processing for stochastic coherent sources

Maximum likelihood (ML) estimation in array signal processing for the stochastic noncoherent signal case is well documented in the literature. We focus on the equally relevant case of stochastic coherent signals. Explicit large-sample realizations are derived for the ML estimates of the noise power and the (singular) signal covariance matrix. The asymptotic properties of the estimates are examined, and some numerical examples are provided. In addition, we show the 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 noncoherent case (or its large-sample realizations) asymptotically achieves the lowest possible estimation error variance (corresponding to the coherent Cramer-Rao bound)     

Spectrum estimation of short-time stationary signals in additivenoise and channel distortion

In this work, spectrum estimation of a short-time stationary signal that is degraded by both channel distortion and additive noise is addressed. A maximum likelihood estimation (MLE) algorithm is developed to jointly identify the degradation system and estimate short-time signal spectra. The source signal is assumed to be generated by a hidden Markov model (HMM) with state-dependent short-time spectral distributions described by mixtures of Gaussian densities. The distortion channel is linear time-invariant, and the noise is Gaussian. The algorithm is derived by using the principle of expectation-maximization (EM), where the unknown parameters of channel and noise are estimated iteratively, and the short-time signal power spectra are obtained from the posterior sufficient statistics of the source signal. Other spectral representation parameters, such as autoregressive model parameters or cepstral parameters, are obtained by minimum mean-squared error (MMSE) estimation from the power spectral estimates. The estimation algorithm was evaluated on simulated signals at the signal-to-noise ratios (SNRs) of 20 dB down to 0 dB, where it produced convergent estimation and significantly reduced spectral distortion     

Analysis of subspace fitting and ML techniques for parameterestimation from sensor array data

It is shown that the multidimensional signal subspace method, termed weighted subspace fitting (WSF), is asymptotically efficient. This results in a novel, compact matrix expression for the Cramer-Rao bound (CRB) on the estimation error variance. The asymptotic analysis of the maximum likelihood (ML) and WSF methods is extended to deterministic emitter signals. The asymptotic properties of the estimates for this case are shown to be identical to the Gaussian emitter signal case, i.e. independent of the actual signal waveforms. Conclusions concerning the modeling aspect of the sensor array problem are drawn     

Approximate maximum likelihood estimators for array processing inmultiplicative noise environments

We consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise. This scenario is encountered, for example, in communications, owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media. Since the exact maximum likelihood (ML) estimator is computationally intensive, two approximate solutions are proposed, originating from the analysis of the high and low signal to-noise ratio (SNR) cases, respectively. First, starting with the no additive noise case, a very simple approximate ML (AML₁) estimator is derived. The performance of the AML₁ estimator in the presence of additive noise is studied, and a theoretical expression for its asymptotic variance is derived. Its performance is shown to be close to the Cramer-Rao bound (CRB) for moderate to high SNR. Next, the low SNR case is considered, and the corresponding AML₂ solution is derived. It is shown that the approximate ML criterion can be concentrated with respect to both the multiplicative and additive noise powers, leaving out a two-dimensional (2-D) minimization problem instead of a four-dimensional (4-D) problem required by the exact ML. Numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis     

An efficient code-timing estimator for receiver diversity DS-CDMAsystems

We propose an efficient algorithm for estimating the code timing of direct-sequence code-division multiple-access (DS-CDMA) systems that consist of an arbitrary antenna array at the receiver and work in a flat-fading and near-far environment. The algorithm is an asymptotic (for large number of data samples) maximum-likelihood (ML) estimator that is derived by modeling the known training sequence as the desired signal and all other signals including the interfering signals and the additive noise as unknown colored Gaussian noise. The algorithm does not require the search over a parameter space and the code timing is obtained by rooting a second-order polynomial, which is computationally very efficient. Simulation results show that the algorithm is quite robust against the near-far problem and channel fading. It requires a shorter training sequence than the single-antenna-based estimators     

A modified likelihood function approach to DOA estimation in thepresence of unknown spatially correlated Gaussian noise using a uniformlinear array

The problem of modified ML estimation of DOAs of multiple source signals incident on a uniform linear array (ULA) in the presence of unknown spatially correlated Gaussian noise is addressed here. Unlike previous work, the proposed method does not impose any structural constraints or parameterization of the signal and noise covariances. It is shown that the characterization suggested here provides a very convenient framework for obtaining an intuitively appealing estimate of the unknown noise covariance matrix via a suitable projection of the observed covariance matrix onto a subspace that is orthogonal complement of the so-called signal subspace. This leads to a formulation of an expression for a so-called modified likelihood function, which can be maximized to obtain the unknown DOAs. For the case of an arbitrary array geometry, this function has explicit dependence on the unknown noise covariance matrix. This explicit dependence can be avoided for the special case of a uniform linear array by using a simple polynomial characterization of the latter. A simple approximate version of this function is then developed that can be maximized via the-well-known IQML algorithm or its variants. An exact estimate based on the maximization of the modified likelihood function is obtained by using nonlinear optimization techniques where the approximate estimates are used for initialization. The proposed estimator is shown to outperform the MAP estimator of Reilly et al. (1992). Extensive simulations have been carried out to show the validity of the proposed algorithm and to compare it with some previous solutions     

Multiple target tracking using maximum likelihood principle

Proposes a method (tracking algorithm (TAL)) based on the maximum likelihood (ML) principle for multiple target tracking in near-field using outputs from a large uniform linear array of passive sensors. The targets are assumed to be narrowband signals and modeled as sample functions of a Gaussian stochastic process. The phase delays of these signals are expressed as functions of both range and bearing angle (“track parameters”) of respective targets. A new simplified likelihood function for ML estimation of these parameters is derived from a second-order approximation on the inverse of the data covariance matrix. Maximization of this likelihood function does not involve inversion of the M×M data covariance matrix, where M denotes number of sensors in the array. Instead, inversion of only a D×D matrix is required, where D denotes number of targets. In practice, D≪M and, hence, TAL is computationally efficient. Tracking is achieved by estimating track parameters at regular time intervals wherein targets move to new positions in the neighborhood of their previous positions. TAL preserves ordering of track parameter estimates of the D targets over different time intervals. Performance results of TAL are presented, and it is also compared with methods by Sword and by Swindlehurst and Kailath (1988). Almost exact asymptotic expressions for the Cramer-Rao bound (CRB) on the variance of angle and range estimates are derived, and their utility is discussed     

Preprocessing for direction finding with minimal variancedegradation

Numerous authors have advocated the use of preprocessing in high-resolution direction of arrival (DOA) algorithms. The benefits cited include reduced computation, improved performance in spatially colored noise, and enhanced resolution. The authors identify the preprocessing matrices that provide minimum variance estimates of DOA for a number of models and algorithms. They examine the Cramer-Rao bound (CRB) for Gaussian signals, the CRB for deterministic signals, and the asymptotic variance of the MUSIC estimator for preprocessed data. They also study the effect of array manifold errors on the direction estimates. As expected, the optimal preprocessor requires knowledge of the source directions. However, they show that performance that is close to optimal can be obtained with only approximate knowledge of the source directions (with an error not exceeding the array beamwidth) if the design rules outlined in this paper are used     

A model-based approach for estimation of two-dimensional maximum entropy power spectra

A stochastic model-based approach is presented for estimation of the two-dimensional maximum entropy power spectrum (MEPS) from given finite uniform array data. The method consists of fitting an appropriate two-dimensional noncausal Gaussian-Markov random field (GMRF) model to the given data using the maximum likelihood (ML) technique for parameter estimation. The nonlinear criterion function used for ML estimation is similar in structure to the function arising in the deterministic approach of Lang and McClellan. The model-based approach provides new insights into the two-dimensional MEPS estimation problem. For example, using the asymptotic normality of ML estimates, we derive simultaneous confidence bands for the estimated MEPS. It turns out that when the true correlations are generated by a noncausal GMRF model, the two-dimensional MEPS can be obtained by solving linear equations. This approach also suggests techniques for realizing two-dimensional GMRF models from the given correlation data. Several numerical examples are given to illustrate the usefulness of the approach.     

Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring

The mathematical theory for the point estimation of the parameters of the Burr Type XII distribution by maximum likelihood (ML) is developed for Type II censored samples. Also derived are necessary and sufficient conditions on the sample data that guarantee the existence, uniqueness and finiteness of the ML parameter estimates for all possible permissible parameter combinations. The asymptotic theory of ML is invoked to obtain approximate confidence intervals for the ML parameter estimates. An application to reliability data arising in a life test experiment is discussed.     

Performance analysis of direction finding with large arrays andfinite data

This paper considers analysis of methods for estimating the parameters of narrow-band signals arriving at an array of sensors. This problem has important applications in, for instance, radar direction finding and underwater source localization. The so-called deterministic and stochastic maximum likelihood (ML) methods are the main focus of this paper. A performance analysis is carried out assuming a finite number of samples and that the array is composed of a sufficiently large number of sensors. Several thousands of antennas are not uncommon in, e.g., radar applications. Strong consistency of the parameter estimates is proved, and the asymptotic covariance matrix of the estimation error is derived. Unlike the previously studied large sample case, the present analysis shows that the accuracy is the same for the two ML methods. Furthermore, the asymptotic covariance matrix of the estimation error coincides with the deterministic Cramer-Rao bound. Under a certain assumption, the ML methods can be implemented by means of conventional beamforming for a large enough number of sensors. We also include a simple simulation study, which indicates that both ML methods provide efficient estimates for very moderate array sizes, whereas the beamforming method requires a somewhat larger array aperture to overcome the inherent bias and resolution problem     

基于四阶累积量的最大似然测向方法研究

论述了最大似然（ML）算法测向以及四阶累积量阵列扩展的基本原理，在此基础上给出了一种基于最大似然算法和四阶累积量的DOA估计新方法。与普通的基于二阶矩的最大似然算法相比，本方法具有对阵列进行四阶扩展的能力，可以解决信号源数大于阵元数时的测向问题，并且由于四阶累积量自身的盲高斯性，还可以有效抑制高斯色噪声。     

Sensor array processing based on subspace fitting

Algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem. The methods considered are the deterministic maximum likelihood method (ML), ESPRIT, and a recently proposed multidimensional signal subspace method. These methods are formulated in a subspace-fitting-based framework, which provides insight into their algebraic and asymptotic relations. It is shown that by introducing a specific weighting matrix, the multidimensional signal subspace method can achieve the same asymptotic properties as the ML method. The asymptotic distribution of the estimation error is derived for a general subspace weighting, and the weighting that provides minimum variance estimates is identified. The resulting optimal technique is termed the weighted subspace fitting (WSF) method. Numerical examples indicate that the asymptotic variance of the WSF estimates coincides with the Cramer-Rao bound. The performance improvement compared to the other techniques is found to be most prominent for highly correlated signals     

A Spatiotemporal Framework for Estimating Trial-to-Trial Amplitude Variation in Event-Related MEG/EEG

A spatiotemporal framework for estimating trial-to-trial variability in evoked response (ER) data is presented. Spatial and temporal bases capture the aspects of the response that are consistent across trials, while the basis expansion coefficients represent the variable components of the response. We focus on the simplest case of constant spatiotemporal response shape and varying amplitude across trials. Two different constraints on the amplitude evolution are employed to effectively integrate the individual responses and improve robustness at low SNR. The linear dynamical system response constraint estimates the current trial amplitude as an unknown constant scaling of the estimate in the previous trial plus zero-mean Gaussian noise with unknown variance. The independent response constraint estimates response amplitudes across trials as independent Gaussian random variables having unknown mean and variance. We develop a generalized expectation-maximization algorithm to obtain the maximum-likelihood (ML) estimates of the signal waveform, noise covariance matrix, and unknown constraint parameters. ML source localization is achieved by scanning the likelihood over different sets of spatial bases. We demonstrate the variability estimation and source localization effectiveness of the proposed algorithms using both real and simulated ER data.     

Noncoherent maximum-likelihood block detection of orthogonalNFSK-LDPSK signals

This paper investigates the noncoherent block detection of orthogonal N frequency-shift keying (FSK)-L differential phase shift keying (DPSK) signals transmitted over the additive white Gaussian noise channel, based on the principle of maximum-likelihood (ML) sequence estimation. By virtue of a union bound argument, asymptotic upper bounds for the bit error probability of the developed ML block receiver are derived and verified by simulation. It is analytically shown that the noncoherent NFSK-LDPSK ML block receiver performs comparably with the ideal coherent NFSK-L phase shift keying (PSK) receiver for L = 2 and 4, as the observation block length is large enough. Furthermore, substantial performance improvement can be achieved by the ML block detection of the NFSK-LDPSK signal with L > 2 by increasing the observation block length     

Robust L-estimation based forms of signal transforms and time-frequency representations

The L-estimation based signal transforms and time-frequency (TF) representations are introduced by considering the corresponding minimization problems in the Huber (1981, 1998) estimation theory. The standard signal transforms follow as the maximum likelihood solutions for the Gaussian additive noise environment. For signals corrupted by an impulse noise, the median-based transforms produce robust estimates of the non-noisy signal transforms. When the input noise is a mixture of Gaussian and impulse noise, the L-estimation-based signal transforms can outperform other estimates. In quadratic and higher order TF analysis, the resulting noise is inherently a mixture of the Gaussian input noise and an impulse noise component. In this case, the L-estimation-based signal representations can produce the best results. These transforms and TF representations give the standard and the median-based forms as special cases. A procedure for parameter selection in the L-estimation is proposed. The theory is illustrated and checked numerically.     

Time delay estimation between two phase shifted signals via generalized cross-correlation methods

This paper examines the generalized cross correlation (GCC) method for estimating time delay between two phase shifted replicas of a common signal which are embedded in additive noise. The analysis applies for widesense jointly stationary narrow-band signal and noise and is independent of signal, noise, and phase distributions. The paper derives expressions for the estimation error, specifically bias and mean square error (MSE), by starting with the cross correlation method and then extending the results to the GCC method. Assuming symmetric spectra signals, we derive two optimal weigth functions (OWF) that minimize the above MSE. With the first OWF the maximum of the absolute value of the GCC function has to be found; this is identical to the Wax maximum likelihood (ML) method where Gaussian signals possessing uniformly distributed phase were assumed. With the second OWF the zero of the absolute value of the GCC function has to be found. For signals possessing non-symmetric spectra (e.g. single side band modulated signals) the OWF can be obtained numerically, by minimizing the MSE, e.g. by using variational techniques. Using the cross correlation method we derive a simplified expression for the MSE, which is to be used instead of a previously published result.