Minimax MSE-ratio estimation with signal covariance uncertainties

In continuation to an earlier work, we further consider the problem of robust estimation of a random vector (or signal), with an uncertain covariance matrix, that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix. While, in the earlier work, we developed and proposed a competitive minimax approach of minimizing the worst-case mean-squared error (MSE) difference regret criterion, here, we study, in the same spirit, the minimum worst-case MSE ratio regret criterion, namely, the worst-case ratio (rather than difference) between the MSE attainable using a linear estimator, ignorant of the exact signal covariance, and the minimum MSE (MMSE) attainable by optimum linear estimation with a known signal covariance. We present the optimal linear estimator, under this criterion, in two ways: The first is as a solution to a certain semidefinite programming (SDP) problem, and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure. We then show that the linear minimax ratio regret estimator can also be interpreted as the MMSE estimator that minimizes the MSE for a certain choice of signal covariance that depends on the uncertainty region. We demonstrate that in applications, the proposed minimax MSE ratio regret approach may outperform the well-known minimax MSE approach, the minimax MSE difference regret approach, and the "plug-in" approach, where in the latter, one uses the MMSE estimator with an estimated covariance matrix replacing the true unknown covariance.     

Blind NLLS Carrier Frequency-Offset Estimation for QAM, PSK, and PAM Modulations: Performance at Low SNR

We address the problem of blind carrier frequency-offset (CFO) estimation in quadrature amplitude modulation, phase-shift keying, and pulse amplitude modulation communications systems. We study the performance of a standard CFO estimate, which consists of first raising the received signal to the$M$th power, where$M$is an integer depending on the type and size of the symbol constellation, and then applying the nonlinear least squares (NLLS) estimation approach. At low signal-to noise ratio (SNR), the NLLS method fails to provide an accurate CFO estimate because of the presence of outliers. In this letter, we derive an approximate closed-form expression for the outlier probability. This enables us to predict the mean-square error (MSE) on CFO estimation for all SNR values. For a given SNR, the new results also give insight into the minimum number of samples required in the CFO estimation procedure, in order to ensure that the MSE on estimation is not significantly affected by the outliers.     

Ambiguity resolution for permanent scatterer interferometry

In the permanent scatterer technique of synthetic aperture radar interferometry, there is a need for an efficient and reliable nonlinear parameter inversion algorithm that includes estimation of the phase cycle ambiguities. Present techniques make use of a direct search of the solution space, treating the observations as deterministic and equally weighted, and which do not yield an exact solution. Moreover, they do not describe the quality of the estimated parameters. Here, we use the integer least squares estimator, which has the highest probability of correct integer estimation for problems with a multivariate normal distribution. With this estimator, the propagated variance-covariance matrix of the estimated parameters can be obtained. We have adapted the LAMBDA method, part of an integer least squares estimator developed for the ambiguity resolution of carrier phase observations in global positioning systems, to the problem of permanent scatterers. Key elements of the proposed method are the introduction of pseudo-observations to regularize the system of equations, decorrelation of the ambiguities for an efficient estimation, and the combination of a bootstrap estimator with an integer least squares search to obtain the final integer estimates. The performance of the proposed algorithm is demonstrated using simulated and real data.     

基于频域模型的宽带信号子空间谱估计方法

宽带阵列信号是频率的函数,因此其阵列流型及协方差矩阵都随频率变化.本文基于宽带信号的频域模型,通过分析宽带阵列信号协方差矩阵的特征分解结构,证明了宽带阵列信号噪声子空间不随频率变化的特性,并根据这一特性,提出了基于频域模型的宽带子空间谱估计(SSEFD)方法.应用K.Buckley的BASS-ALE方法解决了该方法用于均匀线阵时存在的频率-方位模糊问题.计算机仿真结果验证了SSEFD方法的有效性,与H.Wang的CSS方法的统计性比较表明,新方法具有更高的估计精度.     

Linear and nonlinear techniques for the deconvolution of hormonetime-series

Pulsatile hormone secretion is usually investigated by measuring hormone concentration in samples of peripheral plasma. Here, the deconvolution of hormone time series to reconstruct the instantaneous secretion rate of glands is considered. Various techniques are discussed and compared in order to overcome the ill-conditioning of the problem and reduce the computational burden. In particular, linear techniques based on least squares, maximum a posteriori (MAP) estimation, and Wiener filtering are compared. A new nonlinear MAP estimator that keeps into account the non-Gaussian distribution of the unknown signal is worked out and shown to yield the best results. The performances of the algorithms are tested on simulated time series as well as on series of luteinizing hormone     

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     

Joint estimation of carrier offset and code timing for DS-CDMA with performance analysis

This paper considers the problem of joint carrier offset and code timing estimation for code-division multiple-access (CDMA) systems. In contrast to existing schemes that require nonlinear iterative searches over the multidimensional parameter space, this paper proposes a blind estimator that provides an algebraic solution to the joint parameter estimation problem. By exploiting the subspace structure of the observed signal, the multiuser estimation is first decoupled into a series of single-user estimation problems, and then analytical tools of polynomial matrices are invoked for joint carrier and code timing estimation of a single user. The proposed estimator is near-far resistant. It can deal with frequency-selective and time-varying channels. The performance of the proposed scheme is examined analytically by a first-order perturbation analysis. The authors also derive an unconditional Crame/spl acute/r-Rao bound (CRB) that is conditioned neither on fading coefficients nor information symbols; as such, the CRB is considered a suitable lower bound for blind methods. Numerical examples are presented to evaluate and compare the proposed and a multidimensional search (MD-search)-based scheme.     

Adaptive antenna arrays for OFDM systems with cochannelinterference

Orthogonal frequency-division multiplexing (OFDM) is one of the promising techniques for future mobile wireless data systems. For OFDM systems with cochannel interference, adaptive antenna arrays can be used for interference suppression. This paper focuses on a key issue for adaptive antenna arrays, that is, parameter estimation for the minimum mean square error (MMSE) diversity combiner (DC). Using the instantaneous correlation estimation approach developed in the paper, an original parameter estimator for the MMSE-DC is derived. Based on the original estimator, we propose an enhanced parameter estimator. Extensive computer simulation demonstrates that the MMSE-DC using the proposed parameter estimators can effectively suppress both synchronous and asynchronous interference in OFDM systems for packet and continuous data transmission     

Nonlinear autoregressive and nonlinear autoregressive moving average model parameter estimation by minimizing hypersurface distance

The least squares (LS) can be used for nonlinear autoregressive (NAR) and nonlinear autoregressive moving average (NARMA) parameter estimation. However, for nonlinear cases, the LS results in biased parameter estimation due to its assumption that the independent variables are noise free. The total least squares (TLS) is another method that can used for nonlinear parameter estimation to increase the accuracy of the LS because it specifically accounts for the fact that the independent variables are noise corrupted. TLS has its own limitations, however, mainly because it is difficult for the method to isolate noise from the signal components. We present a new method that is based on minimization of hypersurface distance for accurate parameter estimation for NAR and NARMA models. Computer simulation examples show that the new method results in far more accurate NAR and NARMA model parameter estimates than do either the LS and TLS, with noise that is either white or colored, and retains its high accuracy even when the signal-to-noise ratio (SNR) is as low as 10 dB.     

A Blind Frequency Offset Estimator for Coherent M-PSK System in Autonomous Radio

This paper presents a novel blind frequency offset estimator for coherent M-PSK systems in an autonomous radio. The proposed estimator is based on the spectrum of the signal’s argument. A data removal block is developed. We derive the distribution of the instantaneous phase, which is applied to indicate that the proposed estimator can be considered as a class of nonlinear least-squares estimator. We provide a method to analyze the asymptotic performance of the proposed estimator. This enable us to predict the mean-square error on frequency offset estimation for all signal-to-noise ratio (SNR) values. Computer simulations indicate that the proposed estimator achieves better performance than the original estimator. The performance of the proposed estimator as a blind estimator is also illustrated.     

低信噪比情况下调频信号的瞬时频率估计

提出一种基于互Wigner-Ville分布(XWVD)的瞬时频率迭代估计方法.理论分析了该方法的收敛性,通过仿真比较了各种瞬时频率估计方法在噪声下的估计方差,证明此方法在低信噪比情况下对估计线性调频信号的瞬时频率有较好的效果.并采用加窗的方法改进了此算法,仿真结果证明,改进的方法对非线性调频信号的瞬时频率进行了有效估计.     

The stability of nonlinear least squares problems and the Cramer-Rao bound

A number of problems of interest in signal processing can be reduced to nonlinear parameter estimation problems. The traditional approach to studying the stability of these estimation problems is to demonstrate finiteness of the Cramer-Rao bound (CRB) for a given noise distribution. We review an alternate, deterministic notion of stability for the associated nonlinear least squares (NLS) problem from the realm of nonlinear programming (i.e., that the global minimizer of the least squares problem exists and varies smoothly with the noise). Furthermore, we show that under mild conditions, identifiability of the parameters along with a finite CRB for the case of Gaussian noise is equivalent to the deterministic stability of the NLS problem. Finally, we demonstrate the application of our result, which is general, to the problems of multichannel blind deconvolution and sinusoid retrieval to generate new stability results for these problems with little additional effort.     

WVD瞬时频率无偏估计研究

通过检测Ｗｉｇｎｅｒ－Ｖｉｌｌｅ分布的峰值估计信号瞬时频率的方法，对常幅、线性高超尖信号是无偏的。本文利用该估计器估计调幅－调频信号瞬时频率时发现：如果信号满足Ｂｅｄｒｏｓｉａｎ定理，那第调幅对瞬时频率估计没有影响。从而，扩展了ＷＶＤ频率估计器的无偏条件。     

A new approach for nonlinear distortion correction in endoscopicimages based on least squares estimation

Images captured with a typical endoscope show spatial distortion, which necessitates distortion correction for subsequent analysis. In this paper, a new methodology based on least squares estimation is proposed to correct the nonlinear distortion in the endoscopic images. A mathematical model based on polynomial mapping is used to map the images from distorted image space onto the corrected image space. The model parameters include the polynomial coefficients, distortion center, and corrected center. The proposed method utilizes a line search approach of global convergence for the iterative procedure to obtain the optimum expansion coefficients. A new technique to find the distortion center of the image based on curvature criterion is presented. A dual-step approach comprising token matching and integrated neighborhood search is also proposed for accurate extraction of the centers of the dots contained in a rectangular grid, used for the model parameter estimation. The model parameters were verified with different grid patterns. The distortion-correction model is applied to several gastrointestinal images and the results are presented. The proposed technique provides high-speed response and forms a key step toward online camera calibration, which is required for accurate quantitative analysis of the images.     

Mean likelihood frequency estimation

Estimation of signals with nonlinear as well as linear parameters in noise is studied. Maximum likelihood estimation has been shown to perform the best among all the methods. In such problems, joint maximum likelihood estimation of the unknown parameters reduces to a separable optimization problem, where first, the nonlinear parameters are estimated via a grid search, and then, the nonlinear parameter estimates are used to estimate the linear parameters. We show that a grid search can be avoided by using the mean likelihood estimator for estimating the unknown nonlinear parameters and how its performance can be made equivalent to that of the maximum likelihood estimator (MLE). The mean likelihood estimator requires computation of a multidimensional integral. However, using the concepts of importance sampling, we obtain the mean likelihood estimate without using integration. The technique is computationally far less burdensome than the direct maximum likelihood method but performs just as well. Simulation examples for estimating frequencies of multiple sinusoids in noise are given. The general technique can be applied to a large class of nonlinear regression problems     

Quasi maximum likelihood estimator of polynomial phase signals for compressed sensed data

Several papers in the literature cover parameter estimation of frequency modulated (FM) signals under reduced number of signal samples with respect to the Nyquist/Shannon criterion, i.e., within the compressive sensing (CS) framework. However, scope of these papers is mainly limited to sinusoids or sum of sinusoids. In this paper, the CS framework is extended to parameter estimation of higher order polynomial phase signals (PPSs) using the quasi-maximum likelihood (QML) estimator and robust short-time Fourier transform (STFT). The considered signal is assumed to be non-uniformly sampled PPS with smaller number of samples with respect to the Nyquist/Shannon criterion. However, the proposed technique can also be generalized to uniformly sampled signals with missing or unreliable samples.     

MA estimation in polynomial time

The parameter estimation of moving-average (MA) signals from second-order statistics was deemed for a long time to be a difficult nonlinear problem for which no computationally convenient and reliable solution was possible. We show how the problem of MA parameter estimation from sample covariances can be formulated as a semidefinite program that can be solved in a time that is a polynomial function of the MA order. Two methods are proposed that rely on two specific (over) parametrizations of the MA covariance sequence, whose use makes the minimization of a covariance fitting criterion a convex problem. The MA estimation algorithms proposed here are computationally fast, statistically accurate, and reliable. None of the previously available algorithms for MA estimation (methods based on higher-order statistics included) shares all these desirable properties. Our methods can also be used to obtain the optimal least squares approximant of an invalid (estimated) MA spectrum (that takes on negative values at some frequencies), which was another long-standing problem in the signal processing literature awaiting a satisfactory solution     

基于子空间跟踪的宽带短波探测低失真窄带干扰抑制算法

短波信道中窄带干扰分布密集,导致宽带短波探测系统接收信噪比恶化。针对传统子空间跟踪干扰抑制后信号失真较大而严重影响电离层信道参数精确提取的问题,论文基于自适应子空间跟踪提出了一种新颖的双子空间频谱标识算法,在抑制干扰的同时有效减小了信号失真。该算法利用基带探测信号频谱对称特性,通过干扰子空间标识得到非对称干扰频带内的有用信号;且使用信号子空间标识将干扰频带与信号频带有效分离实现干扰抑制。仿真分析与实测数据处理效果表明,双子空间频谱标识算法较大程度地保持了接收信号的原始状态,处理后数据较传统算法可以获得更高的相关后信噪比,大增加了探测信号的捕获概率,对于短波电离层信道探测具有特殊的重要意义。      

Linear prediction approach to oversampling parameter estimation for multiple complex sinusoids

The problem of oversampling parameter estimation for noisy sinusoidal signals is addressed. We first extend the weighted least squares (WLS) approach to the complex sinusoids. Then the oversampling weighted least squares (OSWLS) estimator is proposed based on data decimation. Estimation performance of the OSWLS method is analyzed via theoretical and simulation studies. Results are also compared to those of the WLS and decimative unitary ESPRIT methods as well as Cramér-Rao lower bound.     

改进多尺度Chirplet路径追踪算法在非线性调频信号估计中的应用

针对在低信噪比下雷达非线性调频信号(Non-linear Frequency Modulation,NLFM)瞬时频率估计精度不足的问题,提出一种基于多尺度线调频小波路径追踪算法(Multi-scale Chirplet Path Pursuit,MCPP)的拟合方法估计信号的瞬时频率。首先,该方法将时频平面分成二进制动态时间支撑区并形成多尺度线调频(Chirp)原子函数库;然后在每一个时间支撑区选择一个投影系数最大的Chirp原子,按照连接原则进行最佳路径连接;最后按时间支撑区顺序提取每个Chirp原子的起始和终止频率,采用最小二乘法进行瞬时频率拟合。仿真表明该方法在低信噪比下可以提高信号瞬时频率估计的精度。