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     

Robust detection with the gap metric

In a multipath communication channel, the optimal receiver is matched to the maximum likelihood (ML) estimate of the multipath signal. In general, this leads to a computationally intensive multidimensional nonlinear optimization problem that is not feasible in most applications. We develop a detection algorithm that avoids finding the ML estimates of the channel parameters while still achieving good performance. Our approach is based on a geometric interpretation of the multipath detection problem. The ML estimate of the multipath signal is the orthogonal projection of the received signal on a suitable signal subspace S. We design a second subspace G, which is the representation subspace, that is close to S but whose orthogonal projection is easily computed. The closeness is measured by the gap metric. The subspace G is designed by using wavelet analysis tools coupled with a reshaping algorithm in the Zak transform domain. We show examples where our approach significantly outperforms the conventional correlator receiver (CR) and other alternative suboptimal detectors     

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     

On the robustness of parameter estimation of superimposed signalsby dynamic programming

We analyze a recently proposed dynamic programming algorithm (REDP) for maximum likelihood (ML) parameter estimation of superimposed signals in noise. We show that it degrades gracefully with deviations from the key assumption of a limited interaction signal model (LISMO), providing exact estimates when the LISMO assumption holds exactly. In particular, we show that the deviations of the REDP estimates from the exact ML are continuous in the deviation of the signal model from the LISMO assumption. These deviations of the REDP estimates from the MLE are further quantified by a comparison to an ML algorithm with an exhaustive multidimensional search on a lattice in parameter space. We derive an explicit expression for the lattice spacing for which the two algorithms have equivalent optimization performance, which can be used to assess the robustness of REDP to deviations from the LISMO assumption. The values of this equivalent lattice spacing are found to be small for a classical example of superimposed complex exponentials in noise, confirming the robustness of REDP for this application     

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     

Estimating parameters of multiple wideband polynomial-phase sourcesin sensor arrays

We consider the problem of estimating the parameters of multiple wideband polynomial-phase signal (PPS) sources in sensor arrays. A new maximum likelihood (ML) direction-of-arrival (DOA) estimator is introduced, and the exact Cramer-Rao bound (CRB) is derived for the general case of multiple constant-amplitude polynomial-phase sources. Since the proposed exact ML estimator is computationally intensive, an approximate solution is proposed, originating from the analysis of the log-likelihood (LL) function in the single chirp signal case. As a result, a new form of spatio-temporal matched filter (referred to as the chirp beamformer) is derived, which is applicable to "well-separated" sources that have distinct time-frequency or/and spatial signatures. This beamforming approach requires solving a three-dimensional (3-D) optimization problem and, therefore, enjoys essentially simpler implementation than that entailed by the exact ML. Simulation results are presented, illustrating the performance of the estimators and validating our theoretical CRB analysis     

Maximum likelihood hidden Markov modeling using a dominant sequenceof states

Approximate maximum likelihood (ML) hidden Markov modeling using the most likely state sequence (MLSS) is examined and compared with the exact ML approach that considers all possible state sequences. It is shown that for any hidden Markov model (HMM), the difference between the approximate and the exact normalized likelihood functions cannot exceed the logarithm of the number of states divided by the dimension of the output vectors (frame length). Furthermore, for Gaussian HMMs and a given observation sequence, the MLSS is typically the sequence of nearest neighbor states in the Itakura-Saito sense, and the posterior probability of any state sequence which departs from the MLSS in a single time instant, decays exponentially with the frame length. Hence, for a sufficiently large frame length the exact and approximate ML approach provide similar model estimates and likelihood values     

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     

Maximum Likelihood Estimate of the Arrival Time of Ultra-Wideband Quasi-Radio Signal with Unknown Amplitude and Phase

The maximum likelihood algorithm for estimating the arrival time of ultra-wideband quasi-radio signal with unknown amplitude and phase has been synthesized. The duration of the specified signal can amount to several periods or a fraction of harmonic oscillation period. The realization of maximum likelihood algorithm (ML) for estimating the arrival time of ultra-wideband quasi-radio signal is shown to be appreciably more complex than the realization of ML algorithm for estimating the arrival time of narrowband radio signal. The probability of reliable estimate, bias and scattering of ML estimate of the arrival time of ultra-wideband quasi-radio signal have been found with due regard for anomalous errors making it possible to investigate its threshold properties. The computer methods of statistical simulation were used to determine the performance efficiency of the synthesized algorithm of ML estimate and the limits of the application scope of obtained asymptotically exact (with the rise of signal-to-noise ratio) formulas for the characteristics of time arrival estimate of ultra-wideband quasi-radio signal with unknown amplitude and phase.     

A performance analysis of subspace-based methods in the presence ofmodel error. II. Multidimensional algorithms

For pt.I, see ibid., vol.40, no.7, p.1758-74 (1992). In pt.I the performance of the MUSIC algorithms for narrowband direction-of-arrival (DOA) estimation when the array manifold and noise covariance are not correctly modeled was investigated. This analysis is extended to multidimensional subspace-based algorithms including deterministic (or conditional) maximum likelihood, MD-MUSIC, weighted subspace fitting (WSF), MODE, and ESPRIT. A general expression for the variance of the DOA estimates that can be applied to any of the above algorithms and to any of a wide variety of scenarios is presented. Optimally weighted subspace fitting algorithms are presented for special cases involving random unstructured errors of the array manifold and noise covariance. It is shown that one-dimensional MUSIC outperforms all of the above multidimensional algorithms for random angle-independent array perturbations     

Iterative and recursive estimators for hidden Markoverrors-in-variables models

In this paper we propose maximum-likelihood (ML) estimation of errors in variables models with finite-state Markovian disturbances. Such models have applications in econometrics, speech processing, communication systems, and neurobiological signal processing. We derive the maximum likelihood (ML) model estimates using the expectation maximization (EM) algorithm. Then two recursive or “on-line” estimation schemes are derived for estimating such models. The first on-line algorithm is based on the EM algorithm and uses stochastic approximations to maximize the Kullback-Leibler (KL) information measure. The second on-line algorithm we propose is a gradient-based scheme and uses stochastic approximations to maximize the log likelihood     

On the partial MAP detection with applications to MIMO channels

We investigate a multidimensional detection problem with a partial information of the a posteriori probability, which is referred to as the partial maximum a posteriori probability (MAP) detection problem. We show that the maximum likelihood (ML) detection of a higher dimension can be reduced to the ML detection of a lower dimension with cancellation under a certain condition through the formulation of the partial MAP detection problem. Using this, we can propose a computationally efficient algorithm to apply to the detection problem for multiple input multiple output (MIMO) channels including multiple transmit and multiple receive antenna (MTMR) channels and intersymbol interference (ISI) channels. It is shown that the proposed method has less error propagation effect, and its performance is close to that of the full ML detection with a lower computational complexity.     

Approximate maximum likelihood hyperparameter estimation for Gibbspriors

The parameters of the prior, the hyperparameters, play an important role in Bayesian image estimation. Of particular importance for the case of Gibbs priors is the global hyperparameter, beta, which multiplies the Hamiltonian. Here we consider maximum likelihood (ML) estimation of beta from incomplete data, i.e., problems in which the image, which is drawn from a Gibbs prior, is observed indirectly through some degradation or blurring process. Important applications include image restoration and image reconstruction from projections. Exact ML estimation of beta from incomplete data is intractable for most image processing. Here we present an approximate ML estimator that is computed simultaneously with a maximum a posteriori (MAP) image estimate. The algorithm is based on a mean field approximation technique through which multidimensional Gibbs distributions are approximated by a separable function equal to a product of one-dimensional (1-D) densities. We show how this approach can be used to simplify the ML estimation problem. We also show how the Gibbs-Bogoliubov-Feynman (GBF) bound can be used to optimize the approximation for a restricted class of problems. We present the results of a Monte Carlo study that examines the bias and variance of this estimator when applied to image restoration.     

Efficient dipole parameter estimation in EEG systems with near-ML performance

Source signals that have strong temporal correlation can pose a challenge for high-resolution EEG source localization algorithms. In this paper, we present two methods that are able to accurately locate highly correlated sources in situations where other high-resolution methods such as multiple signal classification and linearly constrained minimum variance beamforming fail. These methods are based on approximations to the optimal maximum likelihood (ML) approach, but offer significant computational advantages over ML when estimates of the equivalent EEG dipole orientation and moment are required in addition to the source location. The first method uses a two-stage approach in which localization is performed assuming an unstructured dipole moment model, and then the dipole orientation is obtained by using these estimates in a second step. The second method is based on the use of the noise subspace fitting concept, and has been shown to provide performance that is asymptotically equivalent to the direct ML method. Both techniques lead to a considerably simpler optimization than ML since the estimation of the source locations and dipole moments is decoupled. Examples using data from simulations and auditory experiments are presented to illustrate the performance of the 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     

Estimating a covariance matrix from incomplete realizations of arandom vector

It is desired to estimate the mean and the covariance matrix of a Gaussian random vector from a set of independent realizations, with the complication that not every component of each realization of the random vector is observed. Subject to some restrictions, the authors obtained an exact, noniterative solution for the maximum likelihood (ML) estimates of the mean and the covariance matrix. The ML estimate of the covariance matrix that is obtained from the set of incomplete realizations is guaranteed to be positive definite, in contrast to ad hoc approaches based on averaging products of components from the same realization. The key to obtaining the ML estimates is a tractable expression for the likelihood function in terms of the Cholesky factors of the inverse covariance matrix. With this formulation, the ML estimates are found by fitting regression operators to appropriate subsets of the data. The Cholesky formulation also leads to a simple calculation by Cramer-Rao bounds     

Min-norm interpretations and consistency of MUSIC, MODE and ML

The multiple signal characterization (MUSIC) approach, its generalization to correlated signals known as the method of direction estimation (MODE), and the deterministic maximum likelihood (ML) approach for bearing estimation in array processing are shown to be signal subspace fitting approaches in a minimum norm sense. MODE, for example, is shown to be an approach in which the array manifold is linearly estimated from principal empirical eigenvectors in a minimum weighted Frobenius norm sense. Using the min-norm interpretations, a unified proof for strong consistency of the three approaches is provided for stationary and ergodic signals     

Blind Feedforward Frequency Estimation for OFDM Signals Transmitted over Multipath Fading Channels

This paper provides an approximate closed form solution to the problem of maximum likelihood (ML) estimation of the carrier frequency offset (CFO) in an orthogonal frequency division multiplexing (OFDM) signal transmitted over a multipath fading channel. This results in a novel feedforward frequency synchronizer, requiring only an approximate statistical knowledge of the communication channel. The performance of the proposed algorithm is assessed by computer simulations and is compared with that provided by other synchronizers and with Cramer-Rao bounds.     

平衰落信道下ML和MAP信道估计与导频符号长度的关系性能比较

信道估计是无线通信系统必须加以解决的关键技术之一，采用导频符号辅助的方法进行信道估计是目前各类无线通信系统常用的方法。本文针对平衰落信道提出了最大似然(ML)算法和最大后验概率(MAP)估计算法，给出了ML估计和MAP估计之间的关系，仿真了MAP估计和ML估计的方差与导频符号长度的关系，提出当导频符号长度的取值超过20个符号长度时，MAP信道估计明显优于ML信道估计。     

Signal waveform estimation in the presence of uncertainties about the steering vector

We consider the problem of signal waveform estimation using an array of sensors where there exist uncertainties about the steering vector of interest. This problem occurs in many situations, including arrays undergoing deformations, uncalibrated arrays, scattering around the source, etc. In this paper, we assume that some statistical knowledge about the variations of the steering vector is available. Within this framework, two approaches are proposed, depending on whether the signal is assumed to be deterministic or random. In the former case, the maximum likelihood (ML) estimator is derived. It is shown that it amounts to a beamforming-like processing of the observations, and an iterative algorithm is presented to obtain the ML weight vector. For random signals, a Bayesian approach is advocated, and we successively derive an (approximate) minimum mean-square error estimator and maximum a posteriori estimators. Numerical examples are provided to illustrate the performances of the estimators.