Passive time delay estimation in non-Gaussian noise

This article deals with the structure of the maximum-likelihood (ML) estimator for time delay with arbitrary signal and noise statistics. At high signal-to-noise ratios (SNRs), the ML estimation performs a nonlinear operation on the delayed difference of the two received waveshapes. The required nonlinearity depends only on the noise statistics. At low SNR, a closed-form simple expression for the ML, which depends only on the noise statistics and on the second-order statistics of the signal, is provided. With statistically independent noise processes, the estimator correlates two vectors generated by separate nonlinear operations on the two received waveshapes     

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     

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 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     

Exact maximum likelihood time delay estimation for shortobservation intervals

An exact solution is presented to the problem of maximum likelihood time delay estimation for a Gaussian source signal observed at two different locations in the presence of additive, spatially uncorrelated Gaussian white noise. The solution is valid for arbitrarily small observation intervals; that is, the assumption T≫τ _c, |d| made in the derivation of the conventional asymptotic maximum likelihood (AML) time delay estimator (where τ _c is the correlation time of the various random processes involved and d is the differential time delay) is relaxed. The resulting exact maximum likelihood (EML) instrumentation is shown to consist of a finite-time delay-and-sum beamformer, followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one-dimensional integral equation with nonconstant weight. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities. Finally, the performance of the EML and AML estimators is compared by means of computer simulations     

Maximum likelihood direction-of-arrival estimation in unknown noise fields using sparse sensor arrays

We address the problem of maximum likelihood (ML) direction-of-arrival (DOA) estimation in unknown spatially correlated noise fields using sparse sensor arrays composed of multiple widely separated subarrays. In such arrays, intersubarray spacings are substantially larger than the signal wavelength, and therefore, sensor noises can be assumed to be uncorrelated between different subarrays. This leads to a block-diagonal structure of the noise covariance matrix which enables a substantial reduction of the number of nuisance noise parameters and ensures the identifiability of the underlying DOA estimation problem. A new deterministic ML DOA estimator is derived for this class of sparse sensor arrays. The proposed approach concentrates the ML estimation problem with respect to all nuisance parameters. In contrast to the analytic concentration used in conventional ML techniques, the implementation of the proposed estimator is based on an iterative procedure, which includes a stepwise concentration of the log-likelihood (LL) function. The proposed algorithm is shown to have a straightforward extension to the case of uncalibrated arrays with unknown sensor gains and phases. It is free of any further structural constraints or parametric model restrictions that are usually imposed on the noise covariance matrix and received signals in most existing ML-based approaches to DOA estimation in spatially correlated noise.     

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     

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     

Maximum likelihood estimation of the parameters of discretefractionally differenced Gaussian noise process

A maximum-likelihood estimation procedure is constructed for estimating the parameters of discrete fractionally differenced Gaussian noise from an observation set of finite size N. The procedure does not involve the computation of any matrix inverse or determinant. It requires N²/2+O(N) operations. The expected value of the loglikelihood function for estimating the parameter d of fractionally differenced Gaussian noise (which corresponds to a parameter of the equivalent continuous-time fractional Brownian motion related to its fractal dimension) is shown to have a unique maximum that occurs at the true value of d. A Cramer-Rao bound on the variance of any unbiased estimate of d obtained from a finite-sized observation set is derived. It is shown experimentally that the maximum-likelihood estimate of d is unbiased and efficient when finite-size data sets are used in the estimation procedure. The proposed procedure is extended to deal with noisy observations of discrete fractionally differenced Gaussian noise     

基于逼近噪声子空间的求根时延估计算法

多重信号分类(MUSIC)时延估计算法需要多径数估计,且其特征分解和谱峰搜索的计算复杂度较高。针对此问题,给出了一种基于逼近噪声子空间的求根时延估计算法。该算法利用协方差矩阵逆的高次幂逼近噪声子空间与其自身共轭转置的积,并构造多项式等式,以多项式求根的方式避免谱峰搜索,从而降低了计算复杂度。仿真结果表明,在无需多径数估计和复杂度低于MUSIC算法的条件下,所提算法的性能与MUSIC算法的性能相当,并且逼近克拉美罗界。     

Maximum likelihood filters in spectral estimation problems

This paper begins with a classification of power spectral estimates from the point of view of bank filter analysis. To reinforce the interest of such a classification, a review of the main and most familiar procedures for spectral estimation is included. Starting from the most general approach, due to Frost, we indicate why it is not appropriate to classify Capon's maximum likelihood method as a low resolution procedure.The second part of the paper deals with a modification of the so-called maximum likelihood estimate in order to obtain the resolution which corresponds to a power density estimate. The modification provided here consists in a bandwidth normalization. The resulting estimate shows how the area of application of ML filters (as the data depending filters reported some years ago by Capon and Lacoss could be named) is considerably extended to a reliable procedure for power level and power density level estimation.We also explain in this paper how to get cross-spectral estimates from ML filters. From our point of view, this approach is the only one, among currently reported methods, that enhances the adequate levels of quality in order to compete with classical Fourier analyzers.In addition, the interesting ideas of Pisarenko about power function estimates can also be applied to the new approach presented here. The resulting family of power function estimates can further improve resolution up to the quality provided by SVD like methods, but avoiding the computational burden associated with them.     

Maximum pseudo likelihood estimation in network tomography

Network monitoring and diagnosis are key to improving network performance. The difficulties of performance monitoring lie in today's fast growing Internet, accompanied by increasingly heterogeneous and unregulated structures. Moreover, these tasks become even harder since one cannot rely on the collaboration of individual routers and servers to measure network traffic directly. Even though the aggregative nature of possible network measurements gives rise to inverse problems, existing methods for solving inverse problems are usually computationally intractable or statistically inefficient. A pseudo likelihood approach is proposed to solve a group of network tomography problems. The basic idea of pseudo likelihood is to form simple subproblems and ignore the dependences among the subproblems to form a product likelihood of the subproblems. As a result, this approach keeps a good balance between the computational complexity and the statistical efficiency of the parameter estimation. Some statistical properties of the pseudo likelihood estimator, such as consistency and asymptotic normality, are established. A pseudo expectation-maximization (EM) algorithm is developed to maximize the pseudo log-likelihood function. Two examples, with simulated or real data, are used to illustrate the pseudo likelihood proposal: 1) inference of the internal link delay distributions through multicast end-to-end measurements; 2) origin-destination matrix estimation through link traffic counts.     

脉冲噪声环境下的SCOT加权时间延迟估计新方法

针对平滑相干变换(SCOT)加权广义相关时间延迟估计方法在脉冲噪声环境下的退化现象,依据分数低阶α稳定分布噪声的尖峰脉冲特性和分数低阶统计量理论,提出了基于分数低阶协方差的SCOT加权时间延迟估计方法,并进一步提出了不依赖于分数低阶α稳定分布噪声参数估计的基于非线性变换(Sigmoid变换和反正切变换)的SCOT加权时间延迟估计方法。理论分析和计算机仿真结果表明,新方法在高斯和非高斯脉冲噪声环境下都具有良好的顽健性。     

Maximum likelihood estimation of object location in diffractiontomography

The problem is formulated within the context of diffraction tomography, where the complex phase of the diffracted wavefield is modeled using the Rytov approximation and the measurements consist of noisy renditions of this complex phase at a single frequency. The log likelihood function is computed for the case of additive zero mean Gaussian white noise and shown to be expressible in the form of the filtered backpropagation algorithm of diffraction tomography. In this form however, the filter function is no longer the rho filter appropriate to least square reconstruction but is now the generalized projection (propagation) of the object (centered at the origin) onto the line(s) parallel to the measurement line(s), but passing through the origin. This result allows the estimation problem to be solved via a diffraction tomographic imaging procedure where the noisy data is filtered and backpropagated in a first step, and the point of maximum value of the resulting image is then the maximum likelihood (ML) estimate of the object's location. The authors include a calculation of the Cramer-Rao bound for the estimation error and a computer simulation study illustrating the estimation procedure     

脉冲噪声环境下改进的顽健循环时延估计算法

研究了脉冲噪声环境下循环平稳信号的时延估计问题,针对脉冲噪声环境中基于传统二阶谱相关函数的时延估计方法性能退化问题,提出了基于分数低阶循环谱的改进顽健算法。相对于传统算法,新算法对脉冲噪声、高斯噪声、干扰信号都具有较好的抑制作用。仿真结果证明了算法的有效性和顽健性。     

Maximum likelihood signal estimation for polarization sensitivearrays

The authors consider the problem of separating and estimating the waveforms of superimposed signals received by a polarization-sensitive array. Signal estimation is accomplished by a two-step maximum likelihood procedure: (i) The directions of arrival and polarization parameters of all the signals are estimated. (ii) The estimated signal is obtained as a linear combination of the array outputs, with weights which are computed from the estimated direction/polarization parameters. The objective of this study is to analyze the quality of the estimated signal in terms of the output signal-to-interference ratio (SIR) and output signal-to-noise ratio (SNR). Closed-form expressions are derived for the output SIR and SNR of a general diversely polarized array. By evaluating these expressions for selected test cases it is shown that polarization-sensitive arrays can provide significantly higher output SIR and SNR than uniformly polarized arrays. The performance improvement is especially significant for closed spaced sources with sufficiently different polarization characteristics     

Maximum likelihood estimation for a multivariate autoregressivemodel

The paper provides an analytical expression for the exact log likelihood function and its first derivatives for a multivariate autoregressive model. Based on these results, two algorithms for constructing the maximum likelihood estimate, using the Fisher's scoring technique, are proposed. The estimated model is guaranteed to be stable. Simulation examples show that this algorithm has good convergence properties and the resulting maximum likelihood estimator could perform better than earlier methods, in cases where the record length is short and the autoregressive polynomial has roots near the unit circle     

Maximum likelihood estimation for InSAR image co-registration

This paper presents a closed-form robust phase correlation based algorithm for performing image registration to subpixel accuracy. The subpixel translational shift information is directly ob- tained from the phase of the normalized cross power spectrum by using Maximum Likelihood Esti- mation （MLE）. The proposed algorithm also has slighter time complexity. Experimental results show that the proposed algorithm yields superior registration precision on the Cram@r-Rao Bound （CRB） in the presence of aliasing and noise.     

一种空间相关高斯噪声背景下的时变时延估计算法

在空间相关高斯噪声的背景下,基于二阶统计量的时延估计方法会失效,该文提出了一种基于三阶统计量的自适应时变时延估计算法,并分析了算法的收敛性,最后的仿真结果表明该算法可以有效地抑制相关高斯噪声。     

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