Characterization of sleep spindles using higher order statistics and spectra

This work characterizes the dynamics of sleep spindles, observed in electroencephalogram (EEG) recorded from humans during sleep, using both time and frequency domain methods which depend on higher order statistics and spectra. The time domain method combines the use of second- and third-order correlations to reveal information on the stationarity of periodic spindle rhythms to detect transitions between multiple activities. The frequency domain method, based on normalized spectrum and bispectrum, describes frequency interactions associated with nonlinearities occurring in the observed EEG.     

Motion estimation using higher order statistics

The objective of this paper is to introduce a fourth-order cost function of the displaced frame difference (DFD) capable of estimating motion even for small regions or blocks. Using higher than second-order statistics is appropriate in case the image sequence is severely corrupted by additive Gaussian noise. Some results are presented and compared to those obtained from the mean kurtosis and the mean square error of the DFD.     

Time delay estimation using higher order statistics

The problem of time delay estimation in spatially correlated Gaussian noises is addressed. New normal equations based upon higher order statistics of the sensor measurements are established, from which a new adaptive algorithm for time delay estimation is proposed. Simulation examples are presented to illustrate the effectiveness of this algorithm     

Detection and classification of spectrally equivalent processesusing higher order statistics

This paper addresses the problem of detecting two spectrally equivalent processes, which are modeled by a noisy AR process and an ARMA process. Higher order statistics (HOS) are shown to be efficient tools for detection. Two HOS-based detectors are derived for the binary hypothesis testing problem (i.e., known signal spectrum): The first detector exploits the asymptotic Gaussianity of the sample estimates of the cumulants. The second detector exploits the singularity of a certain matrix based on HOS. The more general composite hypothesis testing problem (i.e., with unknown signal spectrum) is then considered and a detector proposed. The performances of the different detectors are compared in terms of receiver operating characteristic (ROC) curves. Approximate closed-form expressions are derived for the threshold and the ROCs of the three detectors     

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)     

An analysis of the Wigner higher order spectra of multicomponent signals

The analysis of multicomponent signal representation by the Wigner higher order spectra is done. It is shown that the cross-terms can be removed only for an odd order, having equal number of conjugated and nonconjugated terms in the local multidimensional moment function. A method for higher order time-multifrequency analysis that eliminates cross-terms is proposed. This method turns out to be a dual definition of the L-Wigner distribution. The theory is illustrated by the numerical example.     

Harmonic retrieval using higher order statistics: a deterministicformulation

Given a single record, the authors consider the problem of estimating the parameters of a harmonic signal buried in noise. The observed data are modeled as a sinusoidal signal plus additive Gaussian noise of unknown covariance. The authors define novel higher order statistics-referred to as “mixed” cumulants-that can be consistently estimated using a single record and are insensitive to colored Gaussian noise. Employing fourth-order mixed cumulants, they estimate the sinusoid parameters using a consistent, nonlinear matching approach. The algorithm requires an initial estimate that is obtained from a consistent, linear estimator. Finally, the authors examine the performance of the proposed method via simulations     

General formula for conditional mean using higher order statistics

A general formula is given for conditional mean, which is the well-known minimum variance estimator, in terms of cross-cumulants and locally optimum nonlinearities. This formula will greatly facilitate the solution of various problems emerging in nonlinear system identification, non-Gaussian estimation theory, and nonlinear signal processing     

System reconstruction from higher order spectra slices

We consider the problem of system reconstruction from higher order spectra (HOS) slices. We establish that the impulse response of a complex system can be reconstructed up to a scalar and a shift based on any pair of HOS slices, as long as the distance between the two slices satisfies a certain condition. One slice is sufficient for the reconstruction in the case of a real system. We propose a cepstrum-based method for system reconstruction. We also propose a new method for the reconstruction of the system Fourier phase based on the phase of any odd-indexed bispectrum slice. Being able to choose the slices to be used in the reconstruction allows us to avoid bispectrum regions dominated by noise     

Adaptive interference canceler for narrowband and widebandinterferences using higher order statistics

A new higher order statistics-based adaptive interference canceler is introduced to mitigate narrowband and wideband interferences in environments where the interference is non-Gaussian and a reference signal, which is highly correlated with the interference, is available. The new scheme uses higher order statistics (HOS) of the primary and reference inputs and employs a gradient-type algorithm for updating the adaptive filter coefficients. The update equation of the HOS-based adaptive filter is independent of uncorrelated Gaussian noises and can mitigate the interference more effectively than adaptive filters based on second-order statistics. The performance of the. HOS-based adaptive filter is much less sensitive to the choice of the step size parameter than the adaptive filters based on the LMS algorithm. It is demonstrated, by means of extensive simulations, that the HOS-based filter can mitigate both narrowband and wideband interferences effectively. Comparisons with adaptive filters based on the LMS algorithm and second-order statistics are also presented in the paper     

Low-rank estimation of higher order statistics

Low-rank estimators for higher order statistics are considered in this paper. The bias-variance tradeoff is analyzed for low-rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants. In general, the low-rank estimators have a larger bias and smaller variance than the corresponding full-rank estimator, and the mean-squared error can be significantly smaller. This makes the low-rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics. The low-rank estimators also offer considerable reductions in the computational complexity of such algorithms. The design of subspaces to optimize the tradeoffs between bias, variance, and computation is discussed, and a noisy input, noisy output system identification problem is used to illustrate the results     

Texture decomposition by harmonics extraction from higher order statistics

In this paper, a method of harmonics extraction from Higher Order Statistics (HOS) is developed for texture decomposition. We show that the diagonal slice of the fourth-order cumulants is proportional to the autocorrelation of a related noiseless sinusoidal signal with identical frequencies. We propose to use this fourth-order cumulants slice to estimate a power spectrum from which the harmonic frequencies can be easily extracted. Hence, a texture can be decomposed into deterministic components and indeterministic components as in a unified texture model through a Wold-like decomposition procedure. The simulation and experimental results demonstrated that this method is effective for texture decomposition and it performs better than traditional lower order statistics based decomposition methods.     

Detection of multiplicative noise in stationary random processesusing second- and higher order statistics

This paper addresses the problem of detecting the presence of colored multiplicative noise, when the information process can be modeled as a parametric ARMA process. For the case of zero-mean multiplicative noise, a cumulant based suboptimal detector is studied. This detector tests the nullity of a specific cumulant slice. A second detector is developed when the multiplicative noise is nonzero mean. This detector consists of filtering the data by an estimated AR filter. Cumulants of the residual data are then shown to be well suited to the detection problem. Theoretical expressions for the asymptotic probability of detection are given. Simulation-derived finite-sample ROC curves are shown for different sets of model parameters     

Estimation of linear parametric models using inverse filtercriteria and higher order statistics

Considers the problem of estimating the parameters of a stable, scalar ARMA (p, q) signal model (causal or noncausal, minimum phase or mixed phase) driven by an i.i.d. non-Gaussian sequence. The driving noise sequence is not observed. The Wiggins-Donoho (1978, 1991) class of inverse filter criteria for estimation of model parameters are analyzed and extended. These criteria have been considered in the past only for moving average inverse filters. These criteria are extended to general ARMA inverses. Computer simulation examples are presented to illustrate the proposed approaches     

Detection of COPD’s auscultative symptoms using higher order statistics in the analysis of respiratory sounds

In this paper we present the method for determination of the specific auscultatory diagnostic signs in patients with chronic obstructive pulmonary disease (COPD), which is based upon the utilization of the polyspectral analysis and the calculation of higher order statistics. The main stages of the method are the calculation and construction of the bicoherence function of the lung sound signal in order to find its maximal value. The visual and numerical estimations of the obtained maximum allow us to conclude the presence or absence in this lung’s audio signal of the artifact, which indicates the pathology. For more accurate results one needs to determine asymmetry coefficient and to perform the estimation of bifrequency corresponding to the maximal value of the bicoherence coefficient. The calculation of skewness and kurtosis coefficients of cross-correlation functions of lung sound signals, which were recorded simultaneously in four channels, allows us to reduce the sensitivity of the method to noise components. Therefore, by analyzing all proposed calculated characteristics and parameters one can conclude the presence or absence of the pathology in this audio signal.     

Blind identification of quadratic nonlinear models using neuralnetworks with higher order cumulants

A novel approach to blindly estimate kernels of any discrete- and finite-extent quadratic models in higher order cumulants domain based on artificial neural networks is proposed in this paper. The input signal is assumed an unobservable independently identically, distributed random sequence which is viable for engineering practice. Because of the properties of the third-order cumulant functions, identifiability of the nonlinear model holds, even when the model output measurement is corrupted by a Gaussian random disturbance. The proposed approach enables a nonlinear relationship between model kernels and model output cumulants to be established by means of neural networks. The approximation ability of the neural network with the weights-decoupled extended Kalman filter training algorithm is then used to estimate the model parameters. Theoretical statements and simulation examples together with practical application to the train vibration signals modeling corroborate that the developed methodology is capable of providing a very promising way to identify truncated Volterra models blindly     

A nonparametric phase estimation method for SIMO systems based onsecond-order and higher order statistics

We present a nonparametric phase estimation algorithm for linear single-input multiple-output (SIMO) channels. Given an unknown stationary input signal with known statistics, our approach is to obtain the joint minimum mean square phase estimation based on the polyspectra and the cross-spectra of the SIMO channel outputs. By utilizing both higher order and second-order statistics of the channel outputs, our approach is shown to be more accurate and reliable than methods based on higher order statistics alone. It can be applied to SIMO channels with common zeros     

Identification and deconvolution of multichannel linearnon-Gaussian processes using higher order statistics and inverse filtercriteria

This paper is concerned with the problem of estimation and deconvolution of the matrix impulse response function of a multiple-input multiple-output (MIMO) system given only the measurements of the vector output of the system. The system is assumed to be driven by a temporally i.i.d. and spatially independent non-Gaussian vector sequence (which is not observed). An iterative, inverse filter criteria-based approach is developed using the third-order or the fourth-order normalized cumulants of the inverse filtered data at zero lag. Stationary points of the proposed cost functions are investigated. The approach is input iterative, i.e., the input sequences are extracted and removed one by one. The matrix impulse response is then obtained by cross correlating the extracted inputs with the observed outputs. Identifiability conditions are analyzed. The strong consistency of the proposed approach is also briefly discussed. Computer simulation examples are presented to illustrate the proposed approaches     

Application of higher order statistics techniques to EMG signals to characterize the motor unit action potential

The electromyographic (EMG) signal provides information about the performance of muscles and nerves. At any instant, the shape of the muscle signal, motor unit action potential (MUAP), is constant unless there is movement of the position of the electrode or biochemical changes in the muscle due to changes in contraction level. The rate of neuron pulses, whose exact times of occurrence are random in nature, is related to the time duration and force of a muscle contraction. The EMG signal can be modeled as the output signal of a filtered impulse process where the neuron firing pulses are assumed to be the input of a system whose transfer function is the motor unit action potential. Representing the neuron pulses as a point process with random times of occurrence, the higher order statistics based system reconstruction algorithm can be applied to the EMG signal to characterize the motor unit action potential. In this paper, we report results from applying a cepstrum of bispectrum based system reconstruction algorithm to real wired-EMG (wEMG) and surface-EMG (sEMG) signals to estimate the appearance of MUAPs in the Rectus Femoris and Vastus Lateralis muscles while the muscles are at rest and in six other contraction positions. It is observed that the appearance of MUAPs estimated from any EMG (wEMG or sEMG) signal clearly shows evidence of motor unit recruitment and crosstalk, if any, due to activity in neighboring muscles. It is also found that the shape of MUAPs remains the same on loading.     

On higher order statistics of the Nakagami-m distribution

This paper proposes a fading model that leads to a formal, but simple method to obtain the exact formula of the Nakagami-m published in 1960 distribution for m=n/2, with n a nonzero integer number. Based on such a model, the joint distribution of the envelope and its time derivative are accomplished, and exact formulas for the level crossing rate (closed-form formula) and for the average fade duration are derived. Simulation curves and exact formulas are checked against each other and a very good agreement between them is attained. Although derived for discrete values of m (m being an integer multiple of 1/2), there are no mathematical constraints for these expressions to be used for any real value of m⩾1/2