EM/SAGE Based ML Channel Estimation for Uplink DS-CDMA Systems over Time-Varying Fading Channels

The matrix inversion for the maximum likelihood (ML) channel estimation requires high complexity for the direct-sequence code-division multiple-access (DS-CDMA) systems. The prime motivation of the paper is to propose channel estimators that achieve mean square error (MSE) performance of ML channel estimator in an iterative manner without any matrix inversion. Therefore, two computationally efficient solutions to the problem of ML channel estimation are proposed.We compare the both algorithms in terms of the number of used iteration and show that the proposed algorithms converge the same MSE performance of the ML estimator as the increasing number of iterations.     

Joint Frequency Offset and Channel Estimation for MIMO Systems in the Presence of Timing Error

This paper deals with the problem of joint frequency offset (FO) and channel estimation for multi-input multi-output (MIMO) systems in the presence of a timing error. Two equivalent signal models with FO and a timing error are given, and then a joint estimation method is derived. The proposed estimation method consists of two steps. Firstly, a maximum likelihood (ML) FO estimator is proposed based on the second signal model. Secondly, based on the FO estimate, we formulate the timing error and channel estimation as a problem of composite hypothesis testing according to the first signal model, and then solve the problem using a composite hypothesis testing approach. Simulation results are performed to show the effectiveness of the proposed method.     

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.     

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     

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。     

Maximum Likelihood Localization of a Diffusive Point Source Using Binary Observations

In this paper, we investigate the problem of localization of a diffusive point source of gas based on binary observations provided by a distributed chemical sensor network. We motivate the use of the maximum likelihood (ML) estimator for this scenario by proving that it is consistent and asymptotically efficient, when the density of the sensors becomes infinite. We utilize two different estimation approaches, ML estimation based on all the observations (i.e., batch processing) and approximate ML estimation using only new observations and the previous estimate (i.e., real time processing). The performance of these estimators is compared with theoretical bounds and is shown to achieve excellent performance, even with a finite number of sensors     

线性调频信号参数快速估计

线性调频(LFM)信号参数检测是对SAR对抗的一个重要问题,本文在对用Radon-Wign er变换、快速解线调和最大似然(ML)估计和分析LFM信号的基础上,给出一种消除ML估计带来旁瓣的方法,进一步提出了一种局部快速搜索的最大似然估计法,并用于LFM信号的起始频率和调频斜率等参数的估计。最后给出了三种快速算法的计算机模拟结果。     

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     

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.     

Frequency domain identification methods

Methods for estimating linear dynamical models from frequency data are studied, including the properties of frequency domain data generated by the discrete Fourier transform. The stochastic characteristics of the frequency domain data lead to a maximum likelihood (ML) formulation of the frequency domain estimation problem. Both discretetime and continuous time models are discussed. Consistency and variance of the ML estimate are described, and the connection with simpler frequency domain estimation schemes as well as the time domain ML method is pointed out.Supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT). This work was partly completed while the author was visiting the Department of Electrical and Computer Engineering, University of Newcastle, Australia.     

High-resolution direction finding: the missing data case

This paper considers the problem of estimating the direction-of-arrival (DOA) of one or more signals using an array of sensors, where some of the sensors fail to work before the measurement is completed. Methods for estimating the array output covariance matrix are discussed. In particular, the maximum-likelihood (ML) estimate of this covariance matrix and its asymptotic accuracy are derived and discussed. Different covariance matrix estimates are used for DOA estimation together with the MUSIC algorithm and with a covariance matching technique. In contrast to MUSIC, the covariance matching technique can utilize information on the estimation accuracy of the array covariance matrix, and it is demonstrated that this yields a significant performance gain     

Computationally efficient angle estimation for signals with knownwaveforms

This paper presents a large sample decoupled maximum likelihood (DEML) angle estimator for uncorrelated narrowband plane waves with known waveforms and unknown amplitudes arriving at a sensor array in the presence of unknown and arbitrary spatially colored noise. The DEML estimator decouples the multidimensional problem of the exact ML estimator to a set of 1-D problems and, hence, is computationally efficient. We shall derive the asymptotic statistical performance of the DEML estimator and compare the performance with its Cramer-Rao bound (CRB), i.e., the best possible performance for the class of asymptotically unbiased estimators. We will show that the DEML estimator is asymptotically statistically efficient for uncorrelated signals with known waveforms. We will also show that for moderately correlated signals with known waveforms, the DEML estimator is no longer a large sample maximum likelihood (ML) estimator, but the DEML estimator may still be used for angle estimation, and the performance degradation relative to the CRB is small. We shall show that the DEML estimator can also be used to estimate the arrival angles of desired signals with known waveforms in the presence of interfering or jamming signals by modeling the interfering or jamming signals as random processes with an unknown spatial covariance matrix. Finally, several numerical examples showing the performance of the DEML estimator are presented in this paper     

On nonexistence of the maximum likelihood estimate in blind multichannel identification

In this paper, a blind multichannel identification problem for which the maximum likelihood estimate (MLE) does not exist is considered. More specifically, the likelihood function associated with this problem turns out to have no maximum but only saddle points. This interesting instance of nonexistence of the MLE for a practically relevant problem was first presented in the statistical literature on errors-in-variables regression (M. Solari, 1969). New insights into this result are presented in this paper, along with a direct proof based on the indefiniteness of the Hessian matrix.     

Linear Regression With Gaussian Model Uncertainty: Algorithms and Bounds

In this paper, we consider the problem of estimating an unknown deterministic parameter vector in a linear regression model with random Gaussian uncertainty in the mixing matrix. We prove that the maximum-likelihood (ML) estimator is a (de)regularized least squares estimator and develop three alternative approaches for finding the regularization parameter that maximizes the likelihood. We analyze the performance using the Cramer-Rao bound (CRB) on the mean squared error, and show that the degradation in performance due the uncertainty is not as severe as may be expected. Next, we address the problem again assuming that the variances of the noise and the elements in the model matrix are unknown and derive the associated CRB and ML estimator. We compare our methods to known results on linear regression in the error in variables (EIV) model. We discuss the similarity between these two competing approaches, and provide a thorough comparison that sheds light on their theoretical and practical differences.     

Joint estimation of symbol timing and carrier frequency offset of OFDM signals over fast time-varying multipath channels

In this paper, we present a novel joint algorithm to estimate the symbol timing and carrier frequency offsets of wireless orthogonal frequency division multiplexing (OFDM) signals. To jointly estimate synchronization parameters using the maximum likelihood (ML) criterion, researchers have derived conventional models only from additive white Gaussian noise (AWGN) or single-path fading channels. We develop a general ML estimation algorithm that can accurately calculate symbol timing and carrier frequency offsets over a fast time-varying multipath channel. To reduce overall estimation complexity, the proposed scheme consists of two estimation stages: coarse and fine synchronizations. A low complexity coarse synchronization based on the least-squares (LS) method can rapidly estimate the rough symbol timing and carrier frequency offsets over a fast time-varying multipath channel. The subsequent ML fine synchronization can then obtain accurate final results based on the previous coarse synchronization. Simulations demonstrate that the coarse-to-fine method provides a good tradeoff between estimation accuracy and computational complexity.     

Covariance Structure Maximum-Likelihood Estimates in Compound Gaussian Noise: Existence and Algorithm Analysis

Recently, a new adaptive scheme [Conte (1995), Gini (1997)] has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non-Gaussian noise. This latter has been modeled by compound-Gaussian noise, which is the product c of the square root of a positive unknown variable tau (deterministic or random) and an independent Gaussian vector x, c=radictaux. Because of the implicit algebraic structure of the equation to solve, we called the corresponding solution, the fixed point (FP) estimate. When tau is assumed deterministic and unknown, the FP is the exact maximum-likelihood (ML) estimate of the noise covariance structure, while when tau is a positive random variable, the FP is an approximate maximum likelihood (AML). This estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness. The major contribution of this paper is to fill these gaps. Our derivation is based on some likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts. Moreover, the corresponding iterative algorithm used for the FP estimate practical determination is also analyzed and we show the convergence of this recursive scheme, ensured whatever the initialization.     

Relationships between adaptive minimum variance beamforming andoptimal source localization

For many years, the popular minimum variance (MV) adaptive beamformer has been well known for not having been derived as a maximum likelihood (ML) estimator. This paper demonstrates that by use of a judicious decomposition of the signal and noise, the log-likelihood function of source location is, in fact, directly proportional to the adaptive MV beamformer output power. In the proposed model, the measurement consists of an unknown temporal signal whose spatial wavefront is known as a function of its unknown location, which is embedded in complex Gaussian noise with unknown but positive definite covariance. Further, in cases where the available observation time is insufficient, a constrained ML estimator is derived here that is closely related to MV beamforming with a diagonally loaded data covariance matrix estimate. The performance of the constrained ML estimator compares favorably with robust MV techniques, giving slightly better root-mean-square error (RMSE) angle-of-arrival estimation of a plane-wave signal in interference. More importantly, however, the fact that such optimal ML techniques are closely related to conventional robust MV methods, such as diagonal loading, lends theoretical justification to the use of these practical approaches     

Performance of robust symbol‐timing and carrier‐frequency estimation for OFDM systems

In recent years, many maximum likelihood (ML) blind estimators have been proposed to estimate timing and frequency offsets for orthogonal frequency division multiplexing (OFDM) systems. However, the previously proposed ML blind estimators utilizing cyclic prefix do not fully characterize the random observation vector over the entire range of the timing offset and will significantly degrade the estimation performance. In this paper, we present a global ML blind estimator to compensate the estimation error. Moreover, we extend the global ML blind estimator by accumulating the ML function of the estimation parameters to achieve a better accuracy without increasing the hardware or computational complexity. The simulation results show that the proposed algorithm can significantly improve the estimation performance in both additional white Gaussian noise and ITU‐R M.1225 multipath channels. Copyright © 2008 John Wiley & Sons, Ltd.     

Blind and Semiblind Channel and Carrier Frequency-Offset Estimation in Orthogonally Space-Time Block Coded MIMO Systems

In this paper, the problem of joint channel and carrier frequency offset (CFO) estimation is studied in the context of multiple-input multiple-output (MIMO) communications based on orthogonal space-time-block codes (OSTBCs). A new blind approach is proposed to jointly estimate the channel matrix and the CFO parameters using a relaxed maximum likelihood (ML) estimator that, for the sake of simplicity, ignores the finite alphabet constraint. Although the proposed technique can be applied to the majority of OSTBCs, there are, however, a few codes that suffer from an intrinsic ambiguity in the joint channel, CFO, and symbol estimates. For such specific OSTBCs, a semiblind modification of the proposed approach is developed that resolves the aforementioned estimation ambiguity. Our simulation results demonstrate that although the finite alphabet constraint is relaxed, the performance of the proposed techniques approaches that of the informed (fully frequency-synchronized and coherent) receiver, provided that a sufficient number of data blocks is available for each channel realization.