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     

Multiscale deconvolution of sensor array signals viasum-of-cumulants

This correspondence presents a solution to a multiscale deconvolution problem using higher order spectra where the data to be deconvolved consist of noise-corrupted sensor array measurements. We assume that the data are generated as a convolution of an unknown wavelet with reflectivity sequences that are linearly time-scaled versions of an unknown reference reflectivity sequence. This type of data occurs in many signal processing applications, including sonar and seismic processing. Our approach relies on exploiting the redundancy in the measurements due to time scaling and does not require knowledge of the wavelet or the reflectivity sequences. We formulate and solve the deconvolution problem as a quadratic minimization subject to a quadratic constraint in the sum-of-cumulants (SOC) domain. The formulation using the SOC approach reduces the effect of additive Gaussian noise on the accuracy of the results when compared with the standard time-domain formulation. We demonstrate this improvement using a simulation example     

A fast neural net training algorithm and its application to speech classification

This paper describes a fast training algorithm for feedforward neural nets, as applied to a two-layer neural network to classify segments of speech as voiced, unvoiced, or silence. The speech classification method is based on five features computed for each speech segment and used as input to the network. The network weights are trained using a new fast training algorithm which minimizes the total least squares error between the actual output of the network and the corresponding desired output. The iterative training algorithm uses a quasi-Newtonian error-minimization method and employs a positive-definite approximation of the Hessian matrix to quickly converge to a locally optimal set of weights. Convergence is fast, with a local minimum typically reached within ten iterations; in terms of convergence speed, the algorithm compares favorably with other training techniques. When used for voiced-unvoiced-silence classification of speech frames, the network performance compares favorably with current approaches. Moreover, the approach used has the advantage of requiring no assumption of a particular probability distribution for the input features.     

A geometric characterization of positive definite sequences and ofthe Fourier transform

If every positive-semidefinite normalized sequence of p autocorrelation coefficients is represented as a point in p-dimensional real space, then the set of such points forms a convex region: the positive semidefinite region. It is shown that the positive semidefinite region can be generated completely as the convex hull of a finite-length, one-dimensional curve that lies on the surface of the region. The curve is specified, several of its properties are given, and it is shown that its length is on the order of p^3/2. The curve represents geometrically the kernel of the Fourier transform; computing the inverse Fourier transform of the spectrum then corresponds to taking the convex linear combination of points on this curve. It is shown that the surface of the positive semidefinite region can then be characterized by a set of polytopes with [p/2]+1 or fewer vertices     

The generalized evolutionary spectrum

Evolutionary spectral theory defines a decomposition of signal energy jointly over time and frequency by assuming that processes are composed of sinusoidal components with time-dependent amplitudes. We expand this theory by allowing time variations in the frequency of the sinusoidal components. Examples are presented illustrating the advantages of this generalization     

Evolutionary periodogram for nonstationary signals

Presents a novel estimator for the time-dependent spectrum of a nonstationary signal. By modeling the signal, at any given frequency, as having a time-varying amplitude accurately represented by an orthonormal basis expansion, the authors are able to compute a minimum mean-squared error estimate of this time-varying amplitude. Repeating the process over all frequencies, they obtain a power distribution as a function of time and frequency that is consistent with the Wold-Cramer evolutionary spectrum. Based on the model assumptions, the authors develop the evolutionary periodogram (EP) for nonstationary signals, an estimator analogous to the periodogram used in the stationary case. They also derive the time-frequency resolution of the new estimator. The approach is free of some of the drawbacks of the bilinear distributions and of the short-time Fourier transform spectral estimates. It is guaranteed to produce nonnegative spectra without the cross-term behavior of the bilinear distributions, and it does not require windowing of data in the time domain. Examples illustrating the new estimator are given     

Evolutionary Maximum Entropy Spectral Estimation and Heart Rate Variability Analysis

Spectral analysis has been used extensively in heart rate variability (HRV) studies. The spectral content of HRV signals is useful in assessing the status of the autonomic nervous system. Although most of the HRV studies assume stationarity, the statistics of HRV signals change with time due to transients caused by physiological phenomena. Therefore, the use of time-frequency analysis to estimate the time-dependent spectrum of these non-stationary signals is of great importance. Recently, the spectrogram, the Wigner distribution, and the evolutionary periodogram have been used to analyze HRV signals. In this paper, we propose the application of the evolutionary maximum entropy (EME) spectral analysis to HRV signals. The EME spectral analysis is based on the maximum entropy method for stationary processes and the evolutionary spectral theory. It consists in finding an EME spectrum that matches the Fourier coefficients of the evolutionary spectrum. The spectral parameters are efficiently calculated by means of the Levinson algorithm. The EME spectral estimator provides very good time-frequency resolution, sidelobe reduction and parametric modeling of the evolutionary spectrum. With the help of real HRV signals we show the superior performance of the EME over the earlier methods.     

Discrete all-pole modeling

A method for parametric modeling and spectral envelopes when only a discrete set of spectral points is given is introduced. This method, called discrete all-pole (DAP) modeling, uses a discrete version of the Itakura-Saito distortion measure as its error criterion. One result is an autocorrelation matching condition that overcomes the limitations of linear prediction and produces better fitting spectral envelopes for spectra that are representable by a relatively small discrete set of values, such as in voiced speech. An iterative algorithm for DAP modeling that is shown to converge to a unique global minimum is presented. Results of applying DAP modeling to real and synthetic speech are also presented. DAP modeling is extended to allow frequency-dependent weighting of the error measure, so that spectral accuracy can be enhanced in certain frequency regions     

Partitioning capabilities of two-layer neural networks

It has been observed that feedforward neural nets with a single hidden layer are capable of forming either convex decision regions or nonconvex but connected decision regions in the input space. In this correspondence, it is shown that two-layer nets with a single hidden layer are capable of forming disconnected decision regions as well. In addition to giving examples of the phenomenon, it is explained why and how disconnected decision regions are formed. Through the hypothesization of the existence of additional virtual cells formed by the first layer, it is shown how the decision regions formed by the second layer can indeed be disconnected. It is shown that the number of such disconnected regions can be very large. Using a recent theoretical result about the sufficiency of two layers to approximate arbitrary decision regions in a finite portion of the space, an example is given of how that is possible with the use of virtual cells     

The application of time-frequency methods to the analysis ofpostural sway

We apply time-frequency (TF) spectral analysis techniques, namely evolutionary spectral estimators, to postural sway data gathered during quiet standing and in response to external visual stimuli. These techniques provide insight into the time-varying properties of the human balance control systems during standing. We demonstrate by means of individual and group examples that the results of the TF methods can be used to characterize the behavior of the balance system for groups of patients and controls. Specifically we show that, for healthy control subjects, sway at a visual stimulus frequency toward and away from the subject shows an amplitude which decays in time. On the other hand, patients display a response whose amplitude at the stimulus frequency increases with time. Thus TF analysis yields insights into the time-varying nature of the postural control system