An Efficient Formula for Estimating the Generalized Moments of the Power Spectral Density (PSD) Without Computing the Fourier Transform

Estimates of the moments of the power spectral density (PSD) are derived from a rapidly converging series of weighted samples of the autocorrelation function (ACF). As a result of the rapid convergence of the weighting coefficients of this series, accurate PSD moment estimates are obtained from a small number of ACF samples without computing the Fourier transform of the time series, thereby greatly reducing the computational requirements for generatiing the generalized moments of the PSD. The general expression for the moment generating series is presented along with explicit expressions for computing the first four moments from which mean, variance, skewness, and kurtosis of the PSD are obtained.     

低信噪比下雷达回波谱矩快速估计方法

针对在低信噪比下机载气象雷达回波多普勒参数（谱矩）估计不准确的问题，本文在气象目标的雷达回波频谱服从高斯分布的基础上，给出了一种利用协方差矩阵分解的快速参数化谱矩估计算法。通过理论分析，推导出雷达回波的协方差矩阵具有范德蒙结构特性，进而将用于谱矩估计的代价函数转化为类傅里叶变换结构，然后进一步通过快速傅里叶变换和高斯加权滑窗计算代价函数，实现快速的谱矩估计。仿真实验结果表明，该方法在信噪比低于5 dB时仍可以有效估计雷达回波的谱矩参数，同时运算复杂度大大降低，而且在谱宽值较大情况下仍能保持较好的估计性能。      

Statistics on exponential averaging of periodograms

The algorithm of exponential averaging applied to subsequent periodograms of a stochastic process is used to estimate the power spectral density (PSD). For an independent process, assuming the periodogram estimates to be distributed according to a χ² distribution with two degrees of freedom, the probability density function (PDF) of the PSD estimate is derived. A closed expression is obtained for the moments of the distribution. Surprisingly, the proof of this expression features some new insights into the partitions and Euler's infinite product. For large values of the time constant of the averaging process, examination of the cumulant generating function shows that the PDF approximates the Gaussian distribution. Although restrictions for the statistics are seemingly tight, simulation of a real process indicates a wider applicability of the theory     

Capon estimation of covariance sequences

Estimating the covariance sequence of a wide-sense stationary process is of fundamental importance in digital signal processing (DSP). A new method, which makes use of Fourier inversion of the Capon spectral estimates and is referred to as theCapon method, is presented in this paper. It is shown that the Capon power spectral density (PSD) estimator yields an equivalent autoregressive (AR) or autoregressive moving-average (ARMA) process; hence, theexact covariance sequence corresponsing to the Capon spectrum can be computed in a rather convenient way. Also, without much accuracy loss, the computation can be significantly reduced via an approximate Capon method that utilizes the fast Fourier transform (FFT). Using a variety of ARMA signals, we show that Capon covariance estimates are generally better than standard sample covariance estimates and can be used to improve performances in DSP applications that are critically dependent on the accuracy of the covariance sequence estimates.This work was supported in part by National Science Foundation Grant MIP-9308302, Advanced Research Project Agency Grant MDA-972-93-1-0015, the Senior Individual Grant Program of the Swedish Foundation for Strategic Research and the Swedish Research Council for Engineering Sciences (TFR).     

A covariance fitting approach to parametric localization of multiple incoherently distributed sources

In this paper, a new algorithm for parametric localization of multiple incoherently distributed sources is presented. This algorithm is based on an approximation of the array covariance matrix using central and noncentral moments of the source angular power densities. Based on this approximation, a new computationally simple covariance fitting-based technique is proposed to estimate these moments. Then, the source parameters are obtained from the moment estimates. Compared with earlier algorithms, our technique has lower computational cost and obtains the parameter estimates in a closed form. In addition, it can be applied to scenarios with multiple sources that may have different angular power densities, while other known methods are not applicable to such scenarios.     

Spectral moments from the ACF or directly from the signal

The moments of a bandlimited power spectrum can be estimated from samples of the autocorrelation function without computing the spectrum. Further, short-time estimates of the moments can be continually formed directly from the signal, circumventing ACF computation. Such fast moment computation is of interest in EEG analysis.     

Quadratic system identification using higher order spectra ofi.i.d. signals

The properties of higher order moment sequences and higher order spectral moments of an i.i.d. (independent, identically distributed) process up to fourth-order are discussed. These properties are utilized to develop algorithms to identify time-invariant nonlinear systems, which can be represented by second-order Volterra series and which are subjected to an i.i.d. input. A relatively simple solution for estimating the linear and quadratic transfer functions, which requires neither the calculation of the higher order spectral moments of the input for various frequencies nor the calculation of the inverse of matrix, is shown to exist, even though the second-order Volterra series is not an orthogonal model for an i.i.d. input (unless the input is a white Gaussian process)     

苯、吡啶及吡嗪分子的超拉曼和表面增强超拉曼光谱

用量子化学从头算法计算了苯、吡啶及吡嗪分子的超粒曼和表面增强的超拉曼光谱，并比较了理论计算与实验测量的结果，用Gaussian98中的密度泛函的方法计算分子的偶极矩、极化率和超极化率以及偶极矩、极化率的导数，而超极化率的导数则有限差分的方法来计算，为了检验有限差分法的准确性，用该方法计算了上述分子的红外和拉曼光谱，其结果与Gaussian98的计算结果高度一致，建立了基于有限差分法计算分子红外，拉曼，表面增强拉曼。超拉曼和表面增强超拉曼的光谱强度的方法，并编写了计算程序。     

Time-domain procedures for testing that a stationary time-series isGaussian

A class of time-domain procedures for testing that a stationary time-series is Gaussian is presented and analyzed. These tests are based on the deviations of the sample value of finite memory nonlinear transformations of the process from their ensemble averaged counterparts. Asymptotic distributions of these tests are derived under the null hypothesis of Gaussianity and under a class of local and fixed alternatives. Specific tests are then developed, based, respectively, on higher order moments and on the characteristic functions. Practical construction of the test statistics is discussed, with a special emphasis on the estimation of the covariance of the sample statistics, which appears to play a key role in the performance of the tests when dealing with `small' samples     

Decorrelation of Wavelet Coefficients for Long-Range Dependent Processes

We consider a discrete-time stationary long-range dependent process (X_k)_kisinZ such that its spectral density equals phi(|lambda|)^-2d, where phi is a smooth function such that phi(0)=phi'(0)=0 and phi(lambda)gesclambda for lambdaisin[0,pi]. Then for any wavelet psi with N vanishing moments, the lag k within-level covariance of wavelet coefficients decays as O(k^2d-2N-1) when krarrinfin. The result applies to fractionally integrated autoregressive moving average (ARMA) processes as well as to fractional Gaussian noise     

Multi-dimensional parametric spectral estimation

The approach taken toward estimating the power spectral density (PSD) function of multi-dimensional (m-d) data fields is sometimes too general considering that in many areas of application, the PSD is not arbitrary but is low order parametric. However, if the parametric form of such fields is not taken into account, then much is lost in estimating their power spectra. In this paper a new m-d (m = 3, 4) parametric spectrum estimation approach is introduced based on the minimum variance representations of m-d data fields. These representations are defined in integrated and compact linear predictive forms with their PSD interpretations being generally ARMA. For example, in the 3-d case it is shown that there are four possible models: causal, semicausal I, semicausal II, and noncausal. The selected model parameters for spectral estimation achieve the minimum covariance recursion error of a given finite length m-d data field. Spectra computed from short, long, noisy, narrow-band and wide-band data fields are compared with spectra computed by standard techniques and show improvement in resolution and in accuracy of spectrum matching.     

Statistics of nonstationary shot processes and the fluctuations of detected signals

Campbell's theorems on the first and second order moments are extended to shot processes that are generated by Poisson processes of time-varying parameter. The covariance function of the limiting Gaussian process is derived and illustrated by a simple example.     

Performance Analysis of Covariance Matrix Estimates in Impulsive Noise

This paper deals with covariance matrix estimates in impulsive noise environments. Physical models based on compound noise modeling [spherically invariant random vectors (SIRV), compound Gaussian processes] allow to correctly describe reality (e.g., range power variations or clutter transitions areas in radar problems). However, these models depend on several unknown parameters (covariance matrix, statistical distribution of the texture, disturbance parameters) that have to be estimated. Based on these noise models, this paper presents a complete analysis of the main covariance matrix estimates used in the literature. Four estimates are studied: the well-known sample covariance matrix M_SCM and a normalized version M_N, the fixed-point (FP) estimate M_FP, and a theoretical benchmark M_TFP. Among these estimates, the only one of practical interest in impulsive noise is the FP. The three others, which could be used in a Gaussian context, are, in this paper, only of academic interest, i.e., for comparison with the FP. A statistical study of these estimates is performed through bias analysis, consistency, and asymptotic distribution. This study allows to compare the performance of the estimates and to establish simple relationships between them. Finally, theoretical results are emphasized by several simulations corresponding to real situations.     

Structured estimation, II: Multivariate probability density estimation

We continue the research begun in 1975 on structured estimation. The original work in 1976 by Morgera and Cooper dealt with the Gaussian two-category classification problem when the common covariance matrix is unknown and must be estimated in order to approximate the hyperplane for decisionmaking, which is optimum for the true covariance matrix. We formulate the probability density function (pd0 estimation problem as a multivariate extension of the Rosenblatt-Parzen kernel method in which the multivariate characteristic function (cf) is estimated. A Gaussian form is assumed for the underlying probability distribution, and two methods are presented for the estimation of the covariance matrix in the cf: 1) a maximum-likelihood (MLE) general sample covariance matrix estimate, and 2) a constrained Toeplitz form estimate which takes full advantage of the structure imposed by weak stationarity of the underlying probability distribution. It is shown that both resulting cf estimates are asymptotically unbiased and consistent, albeit the structured covariance matrix estimate is itself only a {em first approximation to the MLE} and may not be positive definite. It is, however, apparently this difference in the estimators which gives rise to a considerable difference in finite sample sire performance. Typical calculations show that the effective sample size increase of the structured estimate can be considerable, a fact very important in nonparametric problems in which data are limited, or in which the sample size-to-dimensionality ratio is small. Applications of this research to the areas of nonparametric pattern recognition and communications theory are discussed.     

On the reconstruction of the covariance of stationary Gaussian processes observed through zero-memory nonlinearities

The problem of reconstructing the normalized covariance functionR(t)of a zero-mean stationary Gaussian process observed through a zero-memory nonlinearityf(x)is considered, when the nonlinearity and the correlation function or the second-order distribution of the output process are known. Three kinds of results are established. (i) Arbitrary covariances can be reconstructed for certain nonlinearities, including monotonicf, appropriate interval windows, and certain quite generalf. (ii) Certain covariances can be reconstructed for arbitrary nonlinearities: included here are positive covariances(geq 0), covariances with rational spectral densities, and bandlimited covariances. (iii) Certain covariances, satisfying rather weak conditions, that can easily be checked in terms of the output correlation function, can be reconstructed for certain nonlinearities that include symmetric as well as nonsymmetricf.     

Analytical Solution to the n -nth Moment Equation of Wave Propagation in Continuous Random Media

Higher order symmetrical moments play an important role in wave propagation and scattering in random media, however it remains to be solved under strong fluctuations. In this paper, a modified Gaussian solution method is proposed for analytically solving the n-nth moment. After propagating through a random medium in the fully saturated regime, the higher order symmetrical moment of the received wave is the sum of products of the second moments, i.e., the Gaussian solution. In strong scattering regimes, the higher order symmetrical moment can be considered as a sum of the Gaussian solution and a non-Gaussian correction term, where the key issue is how to solve the derived equation of the correction term. Two methods are proposed, i.e., Green's function method and the Rytov approximation approach. Green's function method leads to a rigorous solution form, but it is complicated due to an integral equation. The approach using the Rytov approximation is found to be reasonable, as the correction is relatively small     

Performance of normalized correlation estimators for complexprocesses

Direct and fast techniques for estimating normalized second-order moments of complex processes are discussed. First, the accuracy of the direct estimate is explicitly given in terms of bias and covariance for the Gaussian processes. Then, a class of estimators based on the complex invariance property is considered. Their theoretical accuracy is given for some cases of paramount interest from a computational point of view in the Gaussian case. In particular, applications in the areas of autoregressive spectrum analysis and spectral centroid estimation are presented. General analytic results and simple expression for immediate evaluation are provided     

Ocean wave extraction from RADARSAT synthetic aperture radarinter-look image cross-spectra

This study is concerned with the extraction of directional ocean wave spectra from synthetic aperture radar (SAR) image spectra. The statistical estimation problem underlying the wave-SAR inverse problem is examined in detail in order to properly quantify the wave information content of SAR. As a concrete focus, a data set is considered comprising six RADARSAT SAR images co-located with a directional wave buoy off the east coast of Canada. These SAR data are transformed into inter-look image cross-spectra based on two looks at the same ocean scene separated by 0.4 s. The general problem of wave extraction from SAR is cast in terms of a statistical estimation problem that includes the observed SAR spectra, the wave-SAR transform, and prior spectral wave information. The central role of the weighting functions (inverse of the error covariances) is demonstrated, as well as the consequence of approximate (based on the quasilinear wave-SAR transform) versus exact linearizations on the convergence properties of the algorithm. Error estimates are derived and discussed. This statistical framework is applied to the extraction of spectral wave information from observed RADARSAT SAR image cross-spectra. A modified wave-SAR transform is used to account for case-specific geophysical and imaging effects. Analysis of the residual error of simulated and observed SAR spectra motivates a canonical form for the SAR observation error covariance. Wave estimates are then extracted from the SAR spectra, including wavenumber dependent error estimates and explicit identification of spectral null spaces where the SAR contains no wave information. Band-limited SAR wave information is also combined with prior (buoy) spectral wave estimates through parameterization of the wave spectral shape and use of regularization     

Statistical characterization of the MUSIC null spectrum

Statistical characterization of the MUSIC sample null spectrum is presented for an arbitrary number of (possibly) correlated emitters observed by a general array. A remarkably simple expression is developed for the spectral covariance matrix in the vicinity of the emitters. The covariance result together with prior mean spectral results support a Gaussian characterization in the neighborhood of the emitters for many scenarios. It is shown that the sample spectrum is exactly Wishart distributed in the emitter directions. The results can be utilized to quantify a variety of performance metrics. As an illustration, models based on physical parameters are constructed for the curve of probability of resolution versus signal-to-noise ratio. The models predict the curve with a high degree of accuracy for example scenarios