Autoregressive modelling in vector spaces: An application to narrow-bandwidth spectral estimation

A generalisation of the Burg algorithm to vector spaces is described. This can be used to subsample time series that have narrow frequency bandwidth, thereby reducing the computation time and the complexity of the autoregressive model required. The technique is preferable to conventional subsampling.     

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.     

Autoregressive estimation using residual energy ratios (Corresp.)

The use of least squares estimates and their residual energies for obtaining autoregressive estimates satisfying a stability property are investigated. The partial correlation coefficients are used to provide an appropriate parametrization for this purpose. An algorithm is presented for efficient calculation of the estimates. Recursive versions of the estimate and maximum entropy properties are briefly 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.     

Data-adaptive evolutionary spectral estimation

We present a novel data-adaptive estimator for the evolutionary spectrum of nonstationary signals. We model the signal at a frequency of interest as a sinusoid with a time-varying amplitude, which is accurately represented by an orthonormal basis expansion. We then compute a minimum mean-squared error estimate of this amplitude and use it to estimate the time-varying spectrum at that frequency, all while minimizing the interference from the signal components at other frequencies. Repeating the process over all frequencies, we obtain a power distribution that is consistent with the Wold-Cramer evolutionary spectrum and reduces to Capon's (1969) method for the stationary case. Our estimator possesses desirable properties in terms of time-frequency resolution and positivity and is robust in the spectral estimation of noisy nonstationary data. We also propose a new estimator for the autocorrelation of nonstationary signals. This autocorrelation estimate is needed in the data-adaptive spectral estimation. We illustrate the performance of our estimator using simulation examples and compare it with the recently presented evolutionary periodogram and the bilinear time-frequency distribution with exponential kernels     

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.     

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     

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     

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     

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     

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     

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     

Stable spectral estimates with maximum entropy estimation

The maximum-entropy-estimation technique has been found to give stable spectra at low and high signal/noise ratios with an unstable middle region.     

Spatial spectral estimation using multiple beam antennas

The processing simplifications which result in using a multiple beam antenna (MBA) as a spatial sensor for performing spectral estimation are considered. Sources are presumed to be located over a two-dimensional field of view characterized by the two angular coordinatesthetaandphi. The MBA configuration consists of an aperture (usually either a reflector or lens) illuminated by a collection of feeds located in its focal plane (see Fig. 1), followed by a switch network for selecting the outputs of any desired feed port. Using the MBA as the spatial sensor for performing spectral estimation, as contrasted to the array antenna configuration, has a distinct advantage: for a given collection of source wavefronts incident on the aperture, a crude estimate of each source position is obtained simply by monitoring the power output of each feed port. This is to be contrasted to the array configuration, where the average output power of each element port is the same, so long as the wavefronts incident on the aperture emanate from uncorrelated sources. As shall be developed further, this initial crude estimate of source location can be used to develop refined estimates using processing algorithms which significantly reduce processing requirements when compared to those required using a comparable array when the number of anticipated sources existing over the field of view (FOV) is large. Finally, since the spectral estimate of the source location is essentially an "open-loop" estimate, involving a priori measured quantities such as the antenna port radiation patterns, we consider the effects of measurement errors on the estimate. The results are normalized so as to be generally applicable to both the array antenna configuration as well as for the MBA.     

Power spectral density estimation via wavelet decomposition

A soft decision algorithm for wavelet decomposition, in which a probability measure is assigned to each frequency band bearing energy, is presented. This soft decision algorithm is used as an approximate estimator of power spectral density. A staircase approximation of power spectral density (PSD) is obtained by plotting the 2/sup m/ probabilities after an m-stage decomposition. Different wavelet filters are used for estimating the PSD of a speech segment. The type of the wavelet filter used can be selected as a compromise between accuracy and complexity.