Multidimensional spectral estimation

Methods of multidimensional power spectral estimation are reviewed. Seven types of estimators are discussed: Fourier, separable, data extension, MLM, MEM, AR, and Pisarenko estimators. Particular emphasis is given to MEM where current research is quite active. Theoretical developments are reviewed and computational algorithms are discussed.     

On nonparametric spectral estimation

In this paper the Cramér-Rao bound (CRB) for a general nonparametric spectral estimation problem is derived under a local smoothness condition (more exactly, the spectrum is assumed to be well approximated by a piecewise constant function). Further-more, it is shown that under the aforementioned condition the Thomson method (TM) and Daniell method (DM) for power spectral density (PSD) estimation can be interpreted as approximations of the maximum likelihood PSD estimator. Finally the statistical efficiency of the TM and DM as nonparametric PSD estimators is examined and also compared to the CRB for autoregressive moving-average (ARMA)-based PSD estimation. In particular for broadband signals, the TM and DM almost achieve the derived nonparametric performance bound and can therefore be considered to be nearly optimal.This work was supported in part by the Swedish Foundation for Strategic Research (SSF) through the Senior Individual Grant Program.     

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.     

Lattice methods for spectral estimation

Lattice forms provide convenient parametrization of rational spectra of stationary processes. A comprehensive summary of lattice algorithms for estimating spectral parameters of AR, MA, and ARMA processes is presented. It is shown that various well-known spectral estimation techniques, such as the Maximum Entropy Method (MEM) and Maximum Likelihood Method (MLM), can be efficiently computed from lattice parameters. Algorithms are presented for the autocorrelation, pre-windowed, and covariance methods of forming the sample covariance matrix.     

Minimum bias multiple taper spectral estimation

Two families of orthonormal tapers are proposed for multitaper spectral analysis: minimum bias tapers, and sinusoidal tapers {υ ^{(k/)}, where υ}sub n//sup (k/)=√(2/(N+1))sin(πkn/N+1), and N is the number of points. The resulting sinusoidal multitaper spectral estimate is Sˆ(f)=(1/2K(N+1))Σ_j=1^K |y(f+j/(2N+2))-y(f-j/(2N+2))|², where y(f) is the Fourier transform of the stationary time series, S(f) is the spectral density, and K is the number of tapers. For fixed j, the sinusoidal tapers converge to the minimum bias tapers like 1/N. Since the sinusoidal tapers have analytic expressions, no numerical eigenvalue decomposition is necessary. Both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth. The bandwidth of the jth taper is simply 1/N centered about the frequencies (±j)/(2N+2). Thus, the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers. The band limited spectral concentration, ∫_-w^w|V(f)|²df of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the Slepian (1978) tapers. In contrast, the Slepian tapers can have the local bias, ∫_-½^½f ²|V(f)|²df, much larger than of the minimum bias tapers and the sinusoidal tapers     

Minimum variance distortionless response spectral estimation

In this article, we concentrate on spectral estimation techniques that are useful in extracting the features to be used by automatic speech recognition (ASR) system. As an aid to understanding the spectral estimation process for speech signals, we adopt the source filter model of speech production as presented in X. Huang et al. (2001), wherein speech is divided into two broad classes: voiced and unvoiced. Voiced speech is quasi-periodic, consisting of a fundamental frequency corresponding to the pitch of a speaker, as well as its harmonics. Unvoiced speech is stochastic in nature and is best modeled as white noise convolved with an infinite impulse response filter.     

Two-dimensional minimum free energy spectral estimation

The author proposes a 2-D extension of the minimum free energy (MFE) parameter estimation method which may be used to determine autoregressive (AR) model parameters for 2-D spectral estimation. The performance of the technique for spectral estimation of 2-D sinusoids in white noise is demonstrated by numerical example. It is seen that MFE can provide superior spectral estimation over that which can be achieved with the multidimensional Levinson algorithm with equivalent computational burden. The performance of the technique in terms of computational expense and accuracy of spectral estimation over a number of simulation trials is compared with a modified covariance technique     

Optimal kernels for nonstationary spectral estimation

Current theories of a time-varying spectrum of a nonstationary process all involve, either by definition or by difficulties in estimation, an assumption that the signal statistics vary slowly over time. This restrictive quasistationarity assumption limits the use of existing estimation techniques to a small class of nonstationary processes. We overcome this limitation by deriving a statistically optimal kernel, within Cohen's (1989) class of time-frequency representations (TFR's), for estimating the Wigner-Ville spectrum of a nonstationary process. We also solve the related problem of minimum mean-squared error estimation of an arbitrary bilinear TFR of a realization of a process from a correlated observation. Both optimal time-frequency invariant and time-frequency varying kernels are derived. It is shown that in the presence of any additive independent noise, optimal performance requires a nontrivial kernel and that optimal estimation may require smoothing filters that are very different from those based on a quasistationarity assumption. Examples confirm that the optimal estimators often yield tremendous improvements in performance over existing methods. In particular, the ability of the optimal kernel to suppress interference is quite remarkable, thus making the proposed framework potentially useful for interference suppression via time-frequency filtering     

Asymptotically efficient estimation of spectral moments

The article studies parametric estimation of spectral moments of a zero-mean complex Gaussian stationary process immersed in independent Gaussian noise. With the merit of the maximum-likelihood (ML) approach as motivation, this work exploits a Whittle's (1953) type objective function that is able to capture the relevant features of the log-likelihood function while being much more manageable. The resulting estimates are strongly consistent and asymptotically efficient. As an example, application to Doppler weather radar data is considered     

基于几何估计的光谱解混方法

光谱解混是高光谱数据分析的重要技术之一。全约束(即非负性约束和归一化约束)最小二乘线性光谱混合模型(FCLS-LSMM)具有模型简单和物理意义明确等优点而得以广泛使用。然而,FCLS-LSMM的传统优化求解方法的迭代过程非常复杂。近年提出的几何方法为降低LSMM的求解复杂度提供了新思路,但是所获得的结果并非真正意义上的全约束最小二乘解。为此,建立一种完全符合FCLS要求的LSMM几何求解方法,具有复杂度低和可以获得理论最优解等优点。实验表明了所提出方法的有效性。     

High-resolution biosensor spectral peak shift estimation

In this paper, we present a maximum likelihood (ML) approach to high-resolution estimation of the shifts of a spectral signal. This spectral signal arises in application of optically based resonant biosensors, where high resolution in the estimation of signal shift is synonymous with high sensitivity to biological interactions. For the particular sensor of interest, the underlying signal is nonuniformly sampled and exhibits Poisson amplitude statistics. Shift estimation accuracies orders of magnitude finer than the sample spacing are sought. The new ML-based formulation leads to a solution approach different from typical resonance shift estimation methods based on polynomial fitting and peak (or ) estimation and tracking.     

High resolution two-dimensional ARMA spectral estimation

The authors present a practical algorithm for estimating the power spectrum of a 2-D homogeneous random field based on 2-D autoregressive moving average (ARMA) modeling. This algorithm is a two-step approach: first, the AR parameters are estimated by solving a version of the 2-D modified Yule-Walker equation, for which some existing efficient algorithms are available; then the MA spectrum parameters are obtained by simple computations. The potential capability and the high-resolution performance of the algorithm are demonstrated by using some numerical examples     

基于几何估计的光谱解混方法

光谱解混是高光谱数据分析的重要技术之一.全约束(即非负性约束和归一化约束)最小二乘线性光谱混合模型(FCLS-LSMM)具有模型简单和物理意义明确等优点而得以广泛使用.然而,FCLS-LSMM的传统优化求解方法的迭代过程非常复杂.近年提出的几何方法为降低LSMM的求解复杂度提供了新思路,但是所获得的结果并非真正意义上的全约束最小二乘解.为此,建立了一种完全符合FCLS要求的LSMM几何求解方法,具有复杂度低和可以获得理论最优解等优点.实验表明了所提出方法的有效性.     

Optimal data-based kernel estimation of evolutionary spectra

Complex demodulation of evolutionary spectra is formulated as a two-dimensional kernel smoother in the time-frequency domain. First, a tapered Fourier transform, y_v(f, t), is calculated. Then the log-spectral estimate, is smoothed. As the characteristic widths of the kernel smoother increase, the bias from the temporal and frequency averaging increases while the variance decreases. The demodulation parameters, such as the order, length, and bandwidth of spectral taper and the kernel smoother, are determined by minimizing the expected error. For well-resolved evolutionary, spectra, the optimal taper length is a small fraction of the optimal kernel halfwidth. The optimal frequency bandwidth, w, for the spectral window scales as w²~λ/τ, where τ is the characteristic time and λ_F is the characteristic frequency scalelength. In contrast, the optimal halfwidths for the second stage kernel smoother scales as h~1/(τλ_F )¹(p+2)/ where p is the order of the kernel smoother. The ratio of the optimal-frequency halfwidth to the optimal-time halfwidth is determined     

The application of spectral estimation methods to bearing estimation problems

The equivalence between the problem of determining the bearing of a radiating source with an array of sensors and the problem of estimating the spectrum of a signal is demonstrated. Modern spectral estimation algorithms are derived within the context of array processing using an algebraic approach. Emphasis is placed on the problem of determining the bearing of a sound source with an array. Special issues encountered in applying these estimates are discussed.     

Smoothed nonparametric spectral estimation via cepsturm thresholding - Introduction of a method for smoothed nonparametric spectral estimation

Cepstrum thresholding is shown to be an effective, automatic way of obtaining a smoothed nonparametric estimate of the spectrum of a stationary signal. In the process of introducing the cepstrum thresholding-based spectral estimator, we discuss a number of results on the cepstrum of a stationary signal, which might also be of interest to researchers in spectral analysis and allied topics, such as speech processing     

Data-adaptive algorithms for signal detection in sub-Gaussianimpulsive interference

We address the problem of coherent detection of a signal embedded in heavy-tailed noise modeled as a sub-Gaussian, alpha-stable process. We assume that the signal is a complex-valued vector of length L, known only within a multiplicative constant, while the dependence structure of the noise, i.e. the underlying matrix of the sub-Gaussian process, is not known. We implement a generalized likelihood ratio detector that employs robust estimates of the unknown noise underlying matrix and the unknown signal strength. The performance of the proposed adaptive detector is compared with that of an adaptive matched filter that uses Gaussian estimates of the noise-underlying matrix and the signal strength and is found to be clearly superior. The proposed new algorithms are theoretically analyzed and illustrated in a Monte-Carlo simulation     

Higher order spectral estimation for random fields

This paper is concerned with the nonparametric estimation of the higher order cumulant spectra of vector-valued stationary random fields onZ ^d by smoothing the periodograms, whereZ is the space of integers and the dimensiond1. We derive the asymptotic cumulant properties of the spectral estimates, and consider an application to multidimensional nonlinear systems identification. Numerical examples with simulated data are provided.     

Maximum likelihood filters in spectral estimation problems

This paper begins with a classification of power spectral estimates from the point of view of bank filter analysis. To reinforce the interest of such a classification, a review of the main and most familiar procedures for spectral estimation is included. Starting from the most general approach, due to Frost, we indicate why it is not appropriate to classify Capon's maximum likelihood method as a low resolution procedure.The second part of the paper deals with a modification of the so-called maximum likelihood estimate in order to obtain the resolution which corresponds to a power density estimate. The modification provided here consists in a bandwidth normalization. The resulting estimate shows how the area of application of ML filters (as the data depending filters reported some years ago by Capon and Lacoss could be named) is considerably extended to a reliable procedure for power level and power density level estimation.We also explain in this paper how to get cross-spectral estimates from ML filters. From our point of view, this approach is the only one, among currently reported methods, that enhances the adequate levels of quality in order to compete with classical Fourier analyzers.In addition, the interesting ideas of Pisarenko about power function estimates can also be applied to the new approach presented here. The resulting family of power function estimates can further improve resolution up to the quality provided by SVD like methods, but avoiding the computational burden associated with them.