Necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the Cramer-Rao lower bound. Part I. General approach with an application to time-delay and Doppler shift estimation.

Analytic expressions for the first order bias and second order covariance of a general maximum likelihood estimate (MLE) are presented. These expressions are used to determine general analytic conditions on sample size, or signal-to-noise ratio (SNR), that are necessary for a MLE to become asymptotically unbiased and attain minimum variance as expressed by the Cramer-Rao lower bound (CRLB). The expressions are then evaluated for multivariate Gaussian data. The results can be used to determine asymptotic biases. variances, and conditions for estimator optimality in a wide range of inverse problems encountered in ocean acoustics and many other disciplines. The results are then applied to rigorously determine conditions on SNR necessary for the MLE to become unbiased and attain minimum variance in the classical active sonar and radar time-delay and Doppler-shift estimation problems. The time-delay MLE is the time lag at the peak value of a matched filter output. It is shown that the matched filter estimate attains the CRLB for the signal's position when the SNR is much larger than the kurtosis of the expected signal's energy spectrum. The Doppler-shift MLE exhibits dual behavior for narrow band analytic signals. In a companion paper, the general theory presented here is applied to the problem of estimating the range and depth of an acoustic source submerged in an ocean waveguide.     

相干激光雷达中最大似然离散谱峰值估计及Monte Carlo仿真

相干测风激光雷达中一个核心的问题是从微弱的气溶胶后向散射信号中估计出风速。基于零均值复高斯随机过程协方差矩阵统计模型的后向散射信号,首先讨论了最大似然(ML)离散谱峰值(DSP)风速估计算法的克拉美-罗下界(CRLB)与由Fisher信息矩阵论得到的精确CRLB之间的关系。其次,对于ML DSP估计应用于相干测风激光雷达中协方差矩阵统计模型的后向散射信号时,使用计算机Monte Carlo仿真的方法研究了风速估计的概率密度函数。分别讨论了信噪比、激光脉冲累积发数和发射激光脉冲宽度对ML DSP风速估计性能的影响。计算仿真结果表明:ML DSP风速估计的CRLB低于精确的CRLB;在信噪比为-20dB,100发激光脉冲累积和信噪比为-30dB,10 000发激光脉冲累积条件下,ML DSP风速估计中"坏"的估计值所占的比例都为0,"好"的估计值的标准差分别为0.62m/s和0.50m/s。     

Impacts of polarization mode dispersion on the pulse-width with considering initial pulse chirp and fiber GVD

The impacts of polarization mode dispersion (PMD) on the pulse-width in linear systems have been investigated. The analytical solutions, including the effects of initial frequency chirp and group velocity dispersion (GVD), are derived. Analyses show that the pulse broadening effects induced by the second-order PMD depend on GVD and chirp parameter, which are different from those induced by the first-order PMD. An initially chirped Gaussian pulse is taken as an example, upon which analytical solutions of rms pulse-width are derived before and after the first-order PMD compensation. The first-order PMD compensator is also evaluated based on these solutions. The results show that the pulse broadening effects will be resisted efficiently by choosing appropriate GVD and chirp parameter; in general, the post-transmission compensation method will be less efficient than the PSP-transmission method.     

Neumann边界条件的处理对差分解逼近精度的影响

本文以一维对流扩散方程为例,较系统地论证了Neumann边界的差分处理对差分解逼近精度的影响。并从数值上考察了Neumann边界的差分处理对二维Poisson方程差分解的影响。结果表明:O(h)格式可能导致一阶精度的差分解,也可能导致二阶精度的差分解;而O(h²)和O(h³)格式产生的差分解只有二阶精度。     

Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

Second-order conditional moment closure modeling of a turbulent CH4/H2/N2 jet diffusion flame

The second-order CMC model for a detailed chemical mechanism is used to model a turbulent CH₄/H₂/N₂ jet diffusion flame. Second-order corrections are made to the three rate limiting steps of methane–air combustion, while first-order closure is employed for all the other steps. Elementary reaction steps have a wide range of timescales with only a few of them slow enough to interact with turbulent mixing. Those steps with relatively large timescales require higher-order correction to represent the effect of fluctuating scalar dissipation rates. Results show improved prediction of conditional mean temperature and mass fractions of OH and NO. Major species are not much influenced by second-order corrections except near the nozzle exit. A parametric study is performed to evaluate the effects of the variance parameter in log-normal scalar dissipation PDF and the constants for the dissipation term in conditional variance and covariance equations.     

单模光纤偏振模色散统计特性的分析

用琼斯矩阵本征分析法对偏振模色散各参量的统计特性进行了分析 ,得到了一阶和二阶偏振模色散的统计规律 ,并给出了二阶偏振模色散各参量与一阶偏振模色散各参量之间的比例关系 ,对二阶偏振模色散的获取、补偿和系统设计均有指导意义。     

Signal processing of laser Doppler self-velocimeter

A frequency spectrum refinement and correction algorithm has been put forward to improve the accuracy of measurements, and the Cramer–Rao lower bound (CRLB) of Gaussian group Doppler signal was given based on the introduction of acceleration. Results of simulations and experiments showed that the Goertzel refinement algorithm could improve the resolution of the spectrum of the Doppler signal; the ratio correction algorithm made the results closer to the real value. The CRLBs of the estimated parameters were related to the sampling data, the signal to noise ratio (SNR) and the broadness of Gaussian group, and it can be decreased by increasing the length of the sampling data or improving the SNR; the gap between the variances of the measuring results and the CRLBs narrowed when the SNR of the signal was improved, and was almost eliminated when the SNR was higher than 6 dB.     

Ultrafast all-optical first- and higher-order differentiators based on interferometers

We demonstrate that a conventional two-arm interferometer can implement first-order temporal differentiation of ultrafast arbitrary optical waveforms. Straightforward extension of this technique to nth-order optical differentiation is also suggested. This approach is experimentally demonstrated by an efficient and accurate first- and second-order temporal differentiation of (sub-)picosecond Gaussian optical pulses.     

Minimum-Covariance Estimates of Signal Parameters

We show the existence of the only procedure ensuring the absolute minimum of an arithmetic mean covariance in a certain parameter-variation range and coinciding identically with the algorithm for calculating the estimate minimizing the mean risk for a quadratic loss function and a uniform a priory distribution of parameters. This procedure is compared with the standard method for obtaining maximum likely estimates. In particular, it is shown that the procedure ensures a significantly smaller random scatter of estimates during the estimation of deterministic-signal parameters against the background of Gaussian interference. In the case where the estimate covariance is independent of the parameter estimated for physical reasons, the use of the above procedure with an arbitrary sample size allows us to reach the actual attainable variance bound.     

Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system

The propagation properties of decentered twisted Gaussian Schell-model (DTGSM) beams passing through a misaligned first-order optical system are studied. The explicit expressions for the cross-spectral density function and Wigner distribution function of the output beam are derived, which retain their form unchanged. It is shown that the DTGSM beams preserve their closed property. The second-order moments matrix and the Wigner distribution function evolve with the usual laws, whereas the first-order moments matrix varies, as if a ray passes through such system. The propagation of DTGSM beams through an aligned first-order optical system is treated as the limiting case that corresponds to the vanishing misalignment parameters.     

High accuracy measurement of unsteady flows using digital particle image velocimetry

The analysis of digital PIV data, either derived from CCD technology or through film and then scanned, typically involves two quantization steps: spatial and intensity quantization. The all-optical systems do not introduce these sources of error. For systems which make use of digital technology however, it is of crucial importance to have reliable error bounds and a sufficiently accurate estimate of particle position, taking into consideration both types of quantization. The accuracy demanded by aerodynamicists from PIV has been a major barrier to its practical application in the past. The more recent approach of using the Gaussian profile of the particle images to yield sub-pixel accurate position estimates has resulted in robust measurements being taken to an accuracy of 1/10th pixel and 1% in velocity for the in-plane velocity, in hostile industrial environments. A major problem for 3D PIV estimation has historically been that the out-of-plane velocity error was of the order of 3–4 times larger than in-plane. The out-of-plane velocity estimate can be derived from the change in the ratio of amplitude to variance—known as the depth factor—of the Gaussian form, as a particle traverses the beam profile. However, such measurements are crucially dependent not only on an accurate position estimate but also on an equally accurate estimate of the amplitude and variance. The accuracy of the Gaussian profile fit using a Nelder–Meade optimisation method as developed until now however, is not capable of providing the required accuracies. Therefore, this paper presents a development of the “locales” approach to position estimation to achieve the desired objective of high accuracy PIV measurements. This approach makes use of the fact that by considering all the possible digital representations of the Gaussian particle profile, regions of indistinguishable position can be derived. These positions are referred to as “locales”. By considering the density, distribution, and shape of these locales, the available precision can be estimated and an accurate (no worse than 0.5% error for a typical PIV image) in-plane velocity, accuracy can be obtained; while at the same time providing estimates of the depth factor with an error of approximately 0.8%. This paper describes the implementation of an efficient algorithm to provide velocity estimates to an accuracy of at least 0.5% in-plane, together with a discussion of the required constraints imposed on the imaging. The method was validated by creating a synthetic PIV image with CCD-type noise. The flow being analysed is that of flow past the near wake of a cylinder at a Reynolds Number of 140,000. This image was then analysed with the new method and the velocity estimates compared to the CFD data for a range of signal-to-noise ratios (SNR). For a realistic SNR of 5, the accuracy of the method is confirmed as being at least 0.5% in-plane. Finally, the algorithm was used to map an experimental transonic flow field of the stator trailing edge region of a full-size annular cascade with an estimated error of 0.5%. The experimental results are found to be in good agreement with a previously reported steady state viscous calculation and PIV mapping.     

Statistical Inference of the Generalized Inverted Exponential Distribution under Joint Progressively Type-II Censoring

In this paper, we study the statistical inference of the generalized inverted exponential distribution with the same scale parameter and various shape parameters based on joint progressively type-II censored data. The expectation maximization (EM) algorithm is applied to calculate the maximum likelihood estimates (MLEs) of the parameters. We obtain the observed information matrix based on the missing value principle. Interval estimations are computed by the bootstrap method. We provide Bayesian inference for the informative prior and the non-informative prior. The importance sampling technique is performed to derive the Bayesian estimates and credible intervals under the squared error loss function and the linex loss function, respectively. Eventually, we conduct the Monte Carlo simulation and real data analysis. Moreover, we consider the parameters that have order restrictions and provide the maximum likelihood estimates and Bayesian inference.     

Stochastic Resonance of a Periodically Driven Bistable System with Two Different Kinds of Time Delays

In this paper, we study the phenomenon of stochastic resonance (SR) in a periodically driven bistable system with correlations between multiplicative and additive white noise terms when there are two different kinds of time delays existed in the deterministic and fluctuating forces, respectively. Using the small time delay approximation and the theory of signal-to-noise ratio (SNR) in the adiabatic limit, the expression of SNR is obtained. The effects ofthe delay time τ in the deterministic force, and the delay time θ in the fluctuating force on SNR are discussed. Based on the numerical computation, it is found that: (i) There appears a reentrant transition between one peak and two peaks and then to one peak again in the curve of SNR when the value of the time delay θ is increased. (ii) SR can be realized by tuning thetime delay τ or θ with fixed noise, i.e., delay-inducedstochastic resonance (DSR) exists.     

Removal of extracerebral tissues in dual-echo magnetic resonance images via linear scale-space features

The location and masking of the brain and surrounding cerebrospinal fluid (CSF) in two-dimensional (2D) dual-echo fast spin-echo (FSE) magnetic resonance (MR) images of the head is achieved by an automated procedure with a voxel-based computational algorithm. Linear scale-space features are derived from the short-echo, proton-density (PD)-weighted images. The second-order Gaussian derivative (the Laplacian) operator is applied at three different spatial scales as a measure of image convexity/concavity with a first-order Gaussian derivative measure (the squared gradient) at a single scale used to circumscribe cortical regions. A mask obtained from the long-echo, T2-weighted image is used to remove extracerebral components of the visual system. A three-dimensional (3D) connectivity analysis then identifies the largest connected volume as the brain. Five dual-echo fast spin-echo images acquired by repeated scanning of the same normal volunteer were used to verify reproducibility; and coronal and axial acquisitions from another normal volunteer to demonstrate the method's robustness to data collected with non-cubic voxels. Images acquired from five individuals with Alzheimer's disease are also presented to show that the algorithm can be used in cases of non-normative anatomy. Validity is affirmed by demonstrating that cerebral volumes estimated by this method for all 12 images are highly correlated (R = 0.98) with estimates obtained by an expert human operator.     

Resolving Lambertian surface orientation from fluctuating radiance

A maximum likelihood method for estimating remote surface orientation from multi-static acoustic, optical, radar, or laser images is presented. It is assumed that the images are corrupted by signal-dependent noise, known as speckle, arising from complex Gaussian field fluctuations, and that the surface properties are effectively Lambertian. Surface orientation estimates for a single sample are shown to have biases and errors that vary dramatically depending on illumination direction. This is due to the signal-dependent nature of speckle noise and the nonlinear relationship between surface orientation, illumination direction, and fluctuating radiance. The minimum number of independent samples necessary for maximum likelihood estimates to become asymptotically unbiased and to attain the lower bound on resolution of classical estimation theory are derived, as are practical design thresholds.     

Electromagnetic scattering from two parallel 2D targets arbitrarily located in a Gaussian beam

In this paper based on the equivalence principle and the reciprocity theorem, the scattered field up to second-order by two parallel 2D targets arbitrarily located in a Gaussian beam is considered. The first-order solution can easily be obtained by calculating the scattered field from isolated targets when illuminated by a Gaussian beam. However, because of the difficulty in formulating the couple scattering field, it is almost impossible to find an analytical solution for the second-order scattered field if the shapes of 2D targets are not canonical geometries. In order to overcome this problem, in this paper, the second-order solution is derived by using the technique based on the reciprocity theorem and the equivalence principle. Meanwhile, the relation between the secondary scattered field from target #1 and target {\#}2 is obtained. Specifically, the bi- and mono-static scattering of Gaussian beam by two parallel adjacent inhomogeneous plasma-coated conducting circular cylinders is calculated and the dependence of attenuation of the scattering width on the thickness of the coated layer, electron number density, collision frequency and radar frequency is discussed in detail.     

Inference for Inverse Power Lomax Distribution with Progressive First-Failure Censoring

This paper investigates the statistical inference of inverse power Lomax distribution parameters under progressive first-failure censored samples. The maximum likelihood estimates (MLEs) and the asymptotic confidence intervals are derived based on the iterative procedure and asymptotic normality theory of MLEs, respectively. Bayesian estimates of the parameters under squared error loss and generalized entropy loss function are obtained using independent gamma priors. For Bayesian computation, Tierney–Kadane’s approximation method is used. In addition, the highest posterior credible intervals of the parameters are constructed based on the importance sampling procedure. A Monte Carlo simulation study is carried out to compare the behavior of various estimates developed in this paper. Finally, a real data set is analyzed for illustration purposes.     

两个相邻目标对平面波、高斯波束的光散射

基于等效原理和互易性定理研究了两个靠近目标对平面波、高斯波束的光散射问题,给出了这一复合光散射模型的二阶散射结果。通常一阶散射结果容易求解,但由于耦合效应的复杂性,很难给出二阶散射结果的解析形式。为了解决这一问题,应用互易性定理给出了求解任意相邻介质目标二阶散射场的公式,同时借助等效原理将求解散射场公式中的体积分简化为面积分的形式,从而降低了求解难度。求解了两相邻球形粒子的复合散射场,并将求解结果与应用时域积分方程法求得的结果进行了比较。同时,还讨论了束腰半径、目标位置对散射截面及偏振度的影响。