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     

Harmonics in multiplicative and additive noise: parameterestimation using cyclic statistics

The problem of concern here is parameter estimation of harmonics in the presence of multiplicative and additive noise. Cyclic statistics are employed to estimate the frequencies and phases, after which the time series is demodulated and cumulants of the noise processes are estimated. The latter are then supplied to linear or nonlinear cumulant-based algorithms to identify ARMA model parameters for the noises. Cyclic statistics and higher order spectra-based approaches are shown to yield the same frequency estimates. Simulation examples illustrate the algorithms     

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     

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.     

Spectral density of random signals corrupted by multiplicative noise

A simple method of evaluating the spectral density of stationary random signals corrupted by multiplicative noise is presented. It is assumed that the multiplicative noise is also stationary, and that it is statistically independent of the signal.     

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 of quadratic phase coupling from human EEG signals using higher order statistics and spectra

Interactions among neural signals in different frequency components have become a focus of strong interest in biomedical signal processing. The bispectrum is a method to detect the presence of quadratic phase coupling (QPC) between different frequency bands in a signal. The traditional way to quantify phase coupling is by means of the bicoherence index (BCI), which is essentially a normalized bispectrum. The main disadvantage of the BCI is that the determination of significant QPC becomes compromised with noise. To mitigate this problem, a statistical approach that combines the bispectrum with an improved surrogate data method to determine the statistical significance of the phase coupling is introduced. The method was first tested on two simulation examples. It was then applied to the human EEG signal that has been recorded from the scalp using international 10–20 electrodes system. The frequency domain method, based on normalized spectrum and bispectrum, describes frequency interactions associated with nonlinearities occurring in the observed EEG.     

Performance analysis of cyclic statistics for the estimation ofharmonics in multiplicative and additive noise

The problem of estimating the frequencies of harmonics in multiplicative and additive noise is addressed. The cyclic mean (CM) can be used if the multiplicative noise has nonzero mean; the cyclic variance (CV) can be used whether or not the multiplicative noise has zero mean. This paper answers the following question: under what conditions should we use the CV instead of the CM? The criteria used are the ease of detection and the accuracy of estimation. The CV is preferable to the CM if the coherent to noncoherent harmonic power ratio is less than a threshold that depends on the first four cumulants; when the noises are colored, this threshold becomes frequency dependent. Third- and fourth-order cyclic statistics are also studied, and it is shown that they will always be outperformed either by the CM or CV when the multiplicative noise is symmetric     

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     

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.     

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     

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.     

Application of higher order statistics to surface electromyogram signal classification

We propose a novel approach for surface electromyogram (sEMG) signal classification. This approach utilizes higher order statistics of sEMG signal to classify four primitive motions, i.e., elbow flexion, elbow extension, forearm supination, and forearm pronation. In documented research, the sEMG signal generated during isometric contraction is modeled by a stationary process whose probability density function (pdf) is assumed to be either Gaussian or Laplacian. In this paper, using Negentropy, we demonstrate that the level of non-Gaussianity of sEMG signal recorded in muscular forces below 25% of maximum voluntary contraction (MVC) is significant. Therefore, application of higher order statistics in sEMG signal processing is justified, due to the fact that more useful information can be extracted from the corresponding higher order statistics. An accurate classification is achieved by using the sequential forward selection (SFS) method for reducing of the dimensionality of feature space and the K-nearest neighbor (KNN) classifier. The results indicate that the proposed approach provides higher sEMG correct classification rates as compared to the existing methods.     

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     

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 weak random signals in IID non-Gaussian noise

This paper considers the detection of weak random signals in circularly symmetric, independent, identically distributed noise. Locally optimum detectors and ad hoc nonlinearities are considered, with asymptotic expressions provided for evaluation of detection performance. The analytical expressions are used to evaluate the robustness of detectors to mismatch in the noise models. Finite-sample Monte Carlo simulation results indicate the reliability of these asymptotic measures in cases of practical interest. The results show that, as has been found for detection of weak known signals in non-Gaussian noise, reasonably configured ad hoc nonlinearities are nearly optimum and robust to modest errors in the noise statistics     

Performance of blind equalization with higher order statistics inindoor radio environments

This paper analyzes the performance of blind equalization using the complex cepstrum of third-order moments applied to 4-QAM time division multiple access (TDMA) indoor radio communication systems. In particular, we have modeled a dispersive indoor channel with Rice statistics. We used the blind algorithms to estimate the channel-impulse response, and from this, we computed the equalizer coefficients using a classical minimum mean square error (MMSE) algorithm. In order to evaluate the system performance, we calculated the bit error rate (BER) of a decision feedback equalizer (DFE) that uses a tricepstrum algorithm to estimate the channel-impulse response. The results are compared with those obtained using a least sum of square errors (LSSE) algorithm as a channel estimator and considering the exact channel response. The results obtained show that this kind of blind equalizer performs better than the more classically trained equalizer when Rice channels with a strong direct path and signal-to-noise ratios (SNRs) lower than 20 dB are taken into account. However, some problems relating to the length of time needed for convergence must be solved     

Real-time noise suppression in 3D ultrasound imaging based on order statistics

An approach based on order statistics for speckle and impulsive noise suppression in 3D ultrasound images is presented. The proposed technique uses the rank M-type estimators and these are adapted to 3D image processing applications. The results from known techniques are compared with the proposed method to demonstrate its performance in terms of noise suppression and detail preservation. The DSP TMS320C6711 has been used to realise 3D image processing and for comparison of the processing time values.     

Adaptive blind equalization using second- and higher orderstatistics

This paper presents two classes of adaptive blind algorithms based on second- and higher order statistics. The first class contains fast recursive algorithms whose cost functions involve second and third- or fourth-order cumulants. These algorithms are stochastic gradient-based but have structures similar to the fast transversal filters (FTF) algorithms. The second class is composed of two stages: the first stage uses a gradient adaptive lattice (GAL) while the second stage employs a higher order-cumulant (HOC) based least mean squares (LMS) filter. The computational loads for these algorithms are all linearly proportional to the number of taps used. Furthermore, the second class, as various numerical examples indicate, yields very fast convergence rates and low steady state mean square errors (MSE) and intersymbol interference (ISI). MSE convergence analyses for the proposed algorithms are also provided and compared with simulation results