Analytical Analysis of Bayesian CramÉr–Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System

In this paper, we consider the Bayesian Cramer-Rao bound (BCRB) for the dynamical estimation of multipath Rayleigh channel complex gains in data-aided (DA) and non-data-aided (NDA) OFDM systems. This bound is derived in an online and offline scenarios for time-invariant and time-varying complex gains within one OFDM symbol, assuming the availability of prior information. In NDA context, whereas this true BCRB is hard to evaluate, we present a closed-form expression of a BCRB, i.e., the asymptotic BCRB (ABCRB) or the modified BCRB (MBCRB). We discuss, based on the theoretical and simulation results, the interest of using some past and future observations in terms of Doppler spread for the complex gains estimation.     

Discrete spectral peak estimation in incoherent backscatterheterodyne lidar. I. Spectral accumulation and the Cramer-Rao lowerbound

The theory of the Cramer-Rao lower bound (CRLB) and maximum-likelihood (ML) estimators is summarized in the context of a heterodyne lidar. Numerical experiments are described that indicate the scaling of this CRLB with parameters such as the signal bandwidth and the level of noise. This CRLB is also compared with the CRLB of a highly idealized noiseless direct detection system using photon counting. It is found that the asymptotic bounds developed in the radar literature for the heterodyne CRLB should not be used as approximations for the correct expression in lidar applications at intermediate signal levels. Moreover, the variance of the ML estimator may be greater or even less than the heterodyne CRLB, depending on the mechanism leading to the departure from the bound     

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     

Unified analysis of a class of blind feedforward symbol timing estimators employing second-order statistics

In this letter, all the previously proposed digital blind feedforward symbol timing estimators employing second-order statistics are casted into a unified framework. The finite sample mean-square error (MSE) expression for this class of estimators is established. Simulation results are also presented to corroborate the analytical results. It is found that the feedforward conditional maximum likelihood (CML) estimator and the square law nonlinearity (SLN) estimator with a properly designed prefilter perform the best and their performances coincide with the asymptotic conditional Cramer-Rao bound (CCRB), which is the performance lower bound for the class of estimators under consideration.     

Cramer-Rao bounds for wavelet transform-based instantaneous frequency estimates

We use an asymptotic integral approximation of a wavelet transform as a model for the estimation of instantaneous frequency (IF). Our approach allows the calculation of the Cramer-Rao bound for the IF variance at each time directly, without the need for explicit phase parameterization. This is in contrast to other approaches where the Cramer-Rao bounds rely on a preliminary decomposition of the IF with respect to a (usually polynomial) basis. Attention is confined to the Morlet wavelet transform of single-component signals corrupted with additive Gaussian noise. Potential computationally and statistically efficient IF extraction algorithms suggested by the analysis are also discussed.     

Direction estimation in partially unknown noise fields

The problem of direction of arrival estimation in the presence of colored noise with unknown covariance is considered. The unknown noise covariance is assumed to obey a linear parametric model. Using this model, the maximum likelihood directions parameter estimate is derived, and a large sample approximation is formed. It is shown that a priori information on the source signal correlation structure is easily incorporated into this approximate ML (AML) estimator. Furthermore, a closed form expression of the Cramer-Rao bound on the direction parameter is provided. A perturbation analysis with respect to a small error in the assumed noise model is carried out, and an expression of the asymptotic bias due to the model mismatch is given. Computer simulations and an application of the proposed technique to a full-scale passive sonar experiment is provided to illustrate the results     

Explicit Ziv-Zakai lower bound for bearing estimation

The extended Ziv-Zakai bound for vector parameters is used to develop a lower bound on the mean square error in estimating the 2-D bearing of a narrowband planewave signal using planar arrays of arbitrary geometry. The bound has a simple closed-form expression that is a function of the signal wavelength, the signal-to-noise ratio (SNR), the number of data snapshots, the number of sensors in the array, and the array configuration. Analysis of the bound suggests that there are several regions of operation, and expressions for the thresholds separating the regions are provided. In the asymptotic region where the number of snapshots and/or SNR are large, estimation errors are small, and the bound approaches the inverse Fisher information. This is the same as the asymptotic performance predicted by the local Cramer-Rao bound for each value of bearing. In the a priori performance region where the number of snapshots or SNR is small, estimation errors are distributed throughout the a priori parameter space and the bound approaches the a priori covariance. In the transition region, both small and large errors occur, and the bound varies smoothly between the two extremes. Simulations of the maximum likelihood estimator (MLE) demonstrate that the bound closely predicts the performance of the MLE in all regions     

Optimal blind nonlinear least-squares carrier phase and frequency offset estimation for general QAM modulations

This paper introduces a family of blind feedforward nonlinear least-squares (NLS) estimators for joint estimation of the carrier phase and frequency offset of general quadrature amplitude modulated (QAM) transmissions. As an extension of the Viterbi and Viterbi (1983) estimator, a constellation-dependent optimal matched nonlinear estimator is derived such that its asymptotic (large sample) variance is minimized. A class of conventional monomial estimators is also proposed. The asymptotic performance of these estimators is established in closed-form expression and compared with the Cramer-Rao lower bound. A practical implementation of the optimal matched estimator, which is a computationally efficient approximation of the latter and exhibits negligible performance loss, is also derived. Finally, computer simulations are presented to corroborate the theoretical performance analysis and indicate that the proposed optimal matched nonlinear estimator improves significantly the performance of the classic fourth-power estimator.     

Two-Dimensional Polynomial Phase Signals: Parameter Estimation and Bounds

This paper considers the problem of parametric modeling and estimation of nonhomogeneous two-dimensional (2-D) signals. In particular, we focus our study on the class of constant modulus polynomial-phase 2-D nonhomogeneous signals. We present two different phase models and develop computationally efficient estimation algorithms for the parameters of these models. Both algorithms are based on phase differencing operators. The basic properties of the operators are analyzed and used to develop the estimation algorithms. The Cramer-Rao lower bound on the accuracy of jointly estimating the model parameters is derived, for both models. To get further insight on the problem we also derive the asymptotic Cramer-Rao bounds. The performance of the algorithms in the presence of additive white Gaussian noise is illustrated by numerical examples, and compared with the corresponding exact and asymptotic Cramer-Rao bounds. The algorithms are shown to be robust in the presence of noise, and their performance close to the CRB, even at moderate signal to noise ratios.     

On the cramer-rao bound for carrier frequency estimation in the presence of phase noise

We consider the carrier frequency offset estimation in a digital burst-mode satellite transmission affected by phase noise. The corresponding Cramer-Rao lower bound is analyzed for linear modulations under a Wiener phase noise model and in the hypothesis of knowledge of the transmitted data. Even if we resort to a Monte Carlo average, from a computational point of view the evaluation of the Cramer-Rao bound is very hard. We introduce a simple but very accurate approximation that allows to carry out this task in a very easy way. As it will be shown, the presence of the phase noise produces a remarkable performance degradation of. the frequency estimation accuracy. In addition, we provide asymptotic expressions of the Cramer-Rao bound, from which the effect of the phase noise and the dependence on the system parameters of the frequency offset estimation accuracy clearly result. Finally, as a by-product of our derivations and approximations, we derive a couple of estimators specifically tailored for the phase noise channel that will be compared with the classical Rife and Boorstyn algorithm, gaining in this way some important hints on the estimators to be used in this scenario     

On the computation of an asymptotic bound for estimating autoregressive signals in white noise

The Cramer-Rao lower bound (CRLB) provides a useful reference for evaluating the performance of parameter estimation techniques. This paper considers the problem of estimating the parameters of an autoregressive signal corrupted by white noise. An explicit formula is derived for computing the asymptotic CRLB for the signal and noise parameters. Formulas for the asymptotic CRLB for functions of the signal and noise parameters are also presented. In particular, the center frequency, bandwidth and power of a second order process are considered. Some numerical examples are presented to illustrate the usefulness of these bounds in studying estimation accuracy.     

Barankin bounds for source localization in an uncertain oceanenvironment

Ambiguity surfaces for underwater acoustic matched-field processing are prone to having high secondary peaks. This leads to anomalous source localization estimates below a threshold signal-to-noise ratio (SNR) at which the performance rapidly departs from that predicted by the Cramer-Rao lower bound (CRLB). In this paper, Barankin bounds are used to predict the threshold SNR under two different models including known or uncertain shallow-water environments and monochromatic or random narrowband sources. Evaluation of the Barankin bound suggests that although asymptotic localization performance degrades with increasing environmental uncertainty, the threshold SNR is relatively unaffected     

On the Cramer-Rao bound of finite-length cepstrum spectral estimators

Spectral estimators based on finite-length cepstrum modelling are useful in several applications. The Cramer-Rao lower bound of the asymptotic variance of their logarithmic spectral estimates can be obtained in explicit form. This does not depend on the specific underlying spectrum.<>     

交替分离算法的Cramer-Rao界

该文推导了交替分离算法的Cramer-Rao界。交替分离算法的Cramer-Rao界涉及到矩阵的减逆,而矩阵减逆具有比通常Moore-Penrose广义逆更为宽松的定义条件,在理论上,一个确定矩阵有无数个减逆。为了建立分离算法的Cramer-Rao界,该文求出了一个确定矩阵的一个特定减逆矩阵。根据任一确定性定理,得到分离数据的密度分布函数,从而获得交替分离算法的Cramer-Rao界。交替分离算法的Cramer-Rao界将多信号对信号参数估计的影响能更直观反映出来。通过对交替分离算法的Cramer-Rao界的讨论,该文还给出了有关矩阵分离算子一些重要的性质。     

The asymptotic Cramer- Rao bound for Gaussian ARMA processes with periodically missing data (Corresp.)

An asymptotic formula is derived for the Cramer-Rao lower bound on unbiased estimates of the parameters of Gaussian autoregressive moving-average (ARMA) processes, in the case where the measurements are not contiguous, but follow a periodic pattern of misses. The formula is then used to illustrate the behavior of the bound for some specific examples.     

On the accuracy of estimating the parameters of a regularstationary process

Any regular stationary random processes can be represented as the sum of a purely indeterministic process and a deterministic one. This paper considers the achievable accuracy in the joint estimation of the parameters of these two components, from a single observed realization of the process. An exact form of the Cramer-Rao bound (CRB) is derived, as well as a conditional CRB. The relationships between these bounds, and their relations to the previously derived asymptotic bound, are explored by analysis and numerical examples     

Simple and accurate direction of arrival estimator in the case ofimperfect spatial coherence

We consider the direction-finding problem in the imperfect spatial coherence case, i.e., when the amplitude and phase of the wavefront vary randomly along the array aperture. This phenomenon can originate from propagation through an inhomogeneous medium. It is also encountered in the case of spatially dispersed sources. We derive a fast and accurate estimator for the direction of arrival of a single source using a uniform linear array (ULA) of sensors. The estimator is based on a reduced statistic obtained from the subdiagonals of the covariance matrix of the array output. It only entails computing the Fourier transform of an (m-1)-length sequence where m is the number of array sensors. A theoretical analysis is carried out, and an expression for the asymptotic variance of the estimator is derived. Numerical simulations validate the theoretical results and show that the estimator has an accuracy very close to the Cramer-Rao bound     

MIMO signal separation for FIR channels: a criterion andperformance analysis

Signal separation for a general system of an arbitrary number of signals is investigated. The signal separation research area deals with the problem of separating unknown source signals that are mixed in an unknown way when only these mixtures are available. A criterion, based on second-order statistics, is formulated to be used in estimating the mixing system. This estimate of the mixing system is used in a separation structure with a parameterization that minimizes the number of parameters to be estimated. Formulae for the gradient and Hessian of the criterion are derived. A formula for the lower bound for the variance of the estimated parameters of the mixing matrix is derived. This lower bound is the asymptotic (assuming the number of data samples to be large) Cramer-Rao lower bound (CRLB). The proposed algorithm is tested with simulations and compared with the CRLB     

Compact CramÉr–Rao Bound Expression for Independent Component Analysis

Despite of the increased interest in independent component analysis (ICA) during the past two decades, a simple closed form expression of the Cramer-Rao bound (CRB) for the demixing matrix estimation has not been established in the open literature. In the present paper we fill this gap by deriving a simple closed-form expression for the CRB of the demixing matrix directly from its definition. A simulation study comparing ICA estimators with the CRB is given.     

Arma spectral estimation based on partial autocorrelations — Part II: Statistical analysis

In a previous paper [1], a new algorithm for ARMA spectral estimation of stationary time series has been presented. The algorithm is based on nonlinear least squares fit of the sample partial autocorrelations to the partial autocorrelations generated by the assumed ARMA model. This paper explores the statistical properties of the above algorithm, including some numerical examples of the asymptotic variance of the estimated parameters, as compared to the Cramer-Rao bound. The results confirm the good performance of the algorithm and suggest an improvement in its implementation.