Fourth-order Statistics Research Articles

Simulation models with correct statistical properties for rayleigh fading channels

In this paper, new sum-of-sinusoids statistical simulation models are proposed for Rayleigh fading channels. These new models employ random path gain, random initial phase, and conditional random Doppler frequency for all individual sinusoids. It is shown that the autocorrelations and cross correlations of the quadrature components, and the autocorrelation of the complex envelope of the new simulators match the desired ones exactly, even if the number of sinusoids is as small as a single-digit integer. Moreover, the probability density functions of the envelope and phase, the level crossing rate, the average fade duration, and the autocorrelation of the squared fading envelope which contains fourth-order statistics of the new simulators, asymptotically approach the correct ones as the number of sinusoids approaches infinity, while good convergence is achieved even when the number of sinusoids is as small as eight. The new simulators can be directly used to generate multiple uncorrelated fading waveforms for frequency selective fading channels, multiple-input multiple-output channels, and diversity combining scenarios. Statistical properties of one of the new simulators are evaluated by numerical results, finding good agreements.

Noise-source turbulence statistics and the noise from a Mach 0.9 jet

Turbulence statistics that are relevant to jet noise modeling but difficult to measure in experiments are computed using a previously validated simulation database of a Mach 0.9 cold jet. Initial focus is on fourth-order statistics that are at the core of acoustic analogy based models built on both the Lilley and Lighthill equations. Common simplifications of fourth-order correlations based on normal statistics are found to be accurate. We see that although two-point correlations are well fitted by exponential functions, as is typical of turbulence at all but the lowest Reynolds numbers, the spatially integrated fourth-order space/retarded-time covariances, which are used in the models, are instead very well fitted by Gaussian functions of different widths for different components, which is counter to conventional modeling practice. We also examine the components of Lighthill’s analogous noise source that are linear and quadratic in velocity fluctuations, as well as components that are deviations from p′=a∞2ρ′. The spectrum from the linear components is more peaked and more direction dependent than the spectral shape of the quadratic component’s noise, which is relatively independent of angle. These two components are also correlated, especially at small angles where their mutual correlation coefficient reaches as low as −0.4, which casts doubt on models that treat these so-called shear noise (linear) and self-noise (quadratic) terms as distinct. The p′−a∞2ρ′ contribution is relatively small, but not negligible as might be expected for this nearly isothermal jet. The total radiated power of the quadratic terms is nearly the same as that of all components combined. It is shown that the standard Lighthill framework does not lead to a straight forward designation of what noise comes from what region of the jet.

On the second-order statistics of the weighted sample covariance matrix

The second-order statistics of the sample covariance are encountered in many covariance based processing algorithms. This paper derives closed-form expressions for the covariance of the weighted sample covariance matrix with an arbitrary weight for both a real system and complex system. Given a system model, the results explicitly rely on the second-order and fourth-order statistics of the channel noise and inputs. They are shown to coincide with the existing results when the channel inputs and noise are Gaussian distributed. Our results can be directly applied to analyze the statistical properties of subspace-based channel estimation methods for single-input multiple-output (SIMO) systems and code-division multiple access (CDMA) systems. Numerical examples are provided to further verify analyses.

４次の統計量に基づく脳磁図の逆問題解析

In recent years, several inverse solutions of magnetoencephalography (MEG) have been proposed. One of them, the multiple signal classification (MUSIC) method, utilizes spatiotemporal information obtained from magnetic fields. The conventional MUSIC method is, however, sensitive to Gaussian noise, and a sufficiently large signal-to-noise ratio (SNR) is required to estimate the number of sources and to specify the precise locations of electrical neural activities. In this paper, a universal fourth order MUSIC (UFO-MUSIC) method, which is based on fourth order statistics, is proposed. This method is shown to be more robust against Gaussian noise than the conventional MUSIC method. It is an algebraic approach to independent component analysis (ICA) . Although ICA and the analysis of the MEG inverse problem have been separately discussed, the proposed method incorporates ICA into the MEG inverse solution. The results of numerical simulations demonstrate the validity of the proposed method.

Universal behavior of the resistance noise across the metal-insulator transition in silicon inversion layers.

Studies of low-frequency resistance noise show that the glassy freezing of the two-dimensional (2D) electron system in the vicinity of the metal-insulator transition occurs in all Si inversion layers. The size of the metallic glass phase, which separates the 2D metal and the (glassy) insulator, depends strongly on disorder, becoming extremely small in high-mobility samples. The behavior of the second spectrum, an important fourth-order noise statistic, indicates the presence of long-range correlations between fluctuators in the glassy phase, consistent with the hierarchical picture of glassy dynamics.

Blind identification of non-stationary MA systems

A new adaptive algorithm for blind identification of time-varying MA channels is derived. This algorithm proposes the use of a novel system of equations derived by combining the third- and fourth-order statistics of the output signals of MA models. This overdetermined system of equations has the important property that it can be solved adaptively because of their symmetries via an overdetermined recursive instrumental variable-type algorithm. This algorithm shows good behaviour in arbitrary noisy environments and good performance in tracking time-varying systems.

Blind equalization of MIMO-FIR channels driven by white but higher order colored source signals

The goal of the article is to find a minimal amount of statistical information (or a much weaker condition) about source signals for which blind equalization is possible for multiple-input multiple-output finite-impulse response (MIMO-FIR) channels. First, a sufficiently broad framework is set up within which such a theoretical problem is well posed. Within this framework, it is shown that second-order statistics (SOS) alone are not sufficient for blind equalization when the source signals are white. Additional higher order statistics (HOS) are needed. Then we show that the only additional higher order statistical information needed is spatial fourth-order cumulants. Though it has not yet been proved to be minimal, it is interesting to note that this is the same as the weakest known condition on the source signals even for an instantaneous mixture. We then show a necessary and sufficient condition for blind equalization when the source signals are white and spatially fourth-order uncorrelated. Based on this condition, criterion (A) for blind equalization of MIMO-FIR channels is developed by exploiting the temporal fourth-order statistics. Finally, based on this criterion, a new necessary and sufficient condition in terms of cumulants for the blind equalization of MIMO-FIR channels is obtained.

Locating wide band acoustic sources using higher order statistics

The conventional array processing algorithms are effective when the propagation model and the array geometry are known and when the additive noise is white or with a known covariance matrix. In several real situations these assumptions are not verified, hence a degradation of their performances. To cope with this problem, we resort to the higher order statistics of the received signals. The interest of the published papers was the separation of the narrow band and uncorrelated acoustic sources. In this paper, we propose a cumulant based algorithm to characterize the wide band and fully correlated signals. This algorithm removes the additive Gaussian noise and then improves the existing narrow band array processing techniques. This method consists in combining the different information contained in the analysis band by using the optimal transformation matrices. The basic idea is based on both the fourth order statistics and the whitening processing of the received data. This leads to cleaned data and then to improve the source separation. The performance of the proposed algorithm is obtained by simulations for several test cases of interest.

Fourth-order velocity statistics

An investigation of several fourth-order velocity statistics is described. Whereas local isotropy and local scaling are applicable to structure functions expressible as averages of differences of velocity, local isotropy and local scaling are inapplicable to the structure functions that we study. Data from wind tunnel grid turbulence show the behavior of the fourth-order statistics for nearly isotropic turbulence. The scaling relations predicted by the joint Gaussian assumption (JGA) are considered, as are those from the statistical independence assumption (SIA). The basis of the JGA is that velocities at several points are joint Gaussian random variables, whereas the basis of the SIA is that locally averaged velocity is statistically independent of velocity difference. The JGA and SIA relate fourth-order statistics to the second-order velocity structure function, as well as to the velocity covariance. For various fourth-order statistics, the predictions of the JGA and SIA are compared with data. These comparisons quantify how accurately our fourth-order statistics follow the scaling dependence on second-order velocity structure functions and on velocity covariance as predicted by the JGA and SIA. Our measured structure functions are in agreement with the scaling predicted by the SIA with no exceptions and with that predicted by the JGA with two exceptions. As is known, one exception is the structure function that obeys local isotropy [e.g., ⟨(ui−ui′)4⟩] (which the SIA does not predict). The other exceptions are called anomalous components [e.g., ⟨(ui2−ui′2)(uj2−uj′2)⟩]. These anomalous components are shown to be sensitive indicators of intermittency for locally isotropic turbulence, whereas they are indicators of anisotropy for the case of anisotropic turbulence. The anomalous components are in reasonable agreement with the scaling predicted by the SIA but not with the JGA.

Combined estimation scheme for blind source separationwitharbitrary source pdfs

An alternative closed-form estimator for blind source separation based on fourth-order statistics is presented. In contrast to other estimators, the new estimator works well when the source kurtosis sum is zero. Arbitrary source PDFs are successfully treated through a combined estimation scheme based on a heuristic decision rule for choosing between the new estimator and an existing estimator.

Beats, kurtosis and visual coding

Techniques adapted from standard higher-order statistical methods are applied to natural-image data in an attempt to discover exactly what makes `wavelet' representations of natural scenes sparse. Specifically, this paper describes a measure known as the phase-only second spectrum, a fourth-order statistic which quantifies harmonic beat interactions in data, and uses it to show that there are statistical consistencies in the phase spectra of natural scenes. The orientation-averaged phase-only second spectra of natural images appear to show power-law behaviour rather like image power spectra, but with a spectral exponent of approximately -1 instead of -2. They also appear to display a similar form of scale-invariance. Further experimental results indicate that the form of these spectra can account for the observed sparseness of bandpass-filtered natural scenes. This implies an intimate relationship between the merits of sparse neural coding and the exploitation of non-Gaussian `beats' structures by the visual system.

Blind phase recovery in general QAM communication systems using alternative higher order statistics

Blind phase recovery for square QAM communication systems using higher order statistics is well established. It is demonstrated that this method can be adapted for cross QAM constellations, if alternative fourth-order statistics are used. However, a four quadrant inverse tangent function must now be used. Monte Carte simulation provides evidence of the usefulness of the approach.

Linear system blind identification based on fourth order spectral analysis

This paper describes algorithms for blind identification of linear time invariant systems. The measured data are the system complex-valued output fourth order statistics. The originality of the contribution is in the development of methods in the frequency domain applied to complex signals instead of techniques in the time domain for processing real data proposed by other authors. Frequency domain analysis is of interest because the validity of the model and the accuracy of the higher order spectrum estimates are directly checked in this domain. The algorithms are extensions of methods applied earlier for the analysis of third order spectra. The first method is recursive and applies when the measurements are accurate. The second minimizes a quadratic criterion. It requires a prior phase unwrapping. Methods performing third order spectra phase unwrapping are extended to fourth order spectra. The validity of the algorithms is checked with simulations.

SNR estimation of speech signals using subbands and fourth-order statistics

This article addresses the problem of instantaneous signal-to-noise ratio (SNR) estimation during speech activity for the purpose of improving the performance of speech enhancement algorithms. It is shown that the kurtosis of noisy speech may be used to individually estimate speech and noise energies when speech is divided into narrow bands. Based on this concept, a novel method is proposed to continuously estimate the SNR across the frequency bands without the need for a speech detector. The derivations are based on a sinusoidal model for speech and a Gaussian assumption about the noise. Experimental results using recorded speech and noise show that the model and the derivations are valid, though not entirely accurate across the whole spectrum; it is also found that many noise types encountered in mobile telephony are not far from Gaussianity as far as higher statistics are concerned, making this scheme quite effective.

A blind fractionally spaced equalizer using higher order statistics

In this paper, we introduce a new blind fractionally spaced equalizer based on the fourth-order statistics of the input symbol sequence. The input symbol sequence is assumed to come from an independent identically distributed finite alphabet with nonzero fourth-order cumulants. We formulate the equalizer as a column vector and compute it by simultaneously diagonalizing a set of matrices obtained from the fourth-order cross-cumulants of the input and output of the equalizer. Simulation results show that this equalizer works well with a short symbol sequence, even if the channel time span is not accurately estimated.

Performance analysis of cyclic statistics for the estimation of harmonics 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.

Model validation and order selection for linear model fitting using thirdand fourth-order cumulants

Given a linear stationary non-Gaussian signal, suppose that we fit a linear model using higher order statistics and one of several existing methods. The model is fitted under certain assumptions on the data and the underlying (true) model. Having obtained a model, how do we know if the fitted model is "good"? This paper is devoted to the problem of model diagnostics and validation. We propose frequency-domain tests that are applicable to both third- and fourth-order statistics based model fitting unlike existing tests. Model order selection is a byproduct of the model validation approach. Two computer simulation examples are presented to illustrate the proposed tests.

Combining an adapted wavelet analysis with fourth-order statistics for transient detection

In this paper, we present a new detector of transient acoustic signals. It combines two powerful detection tools: a local wavelet analysis and higher-order statistical properties of the signals. Using both techniques makes detection possible in low signal-to-noise ratio conditions, when other means of detection are no longer sufficient. The proposed algorithm uses the adapted Malvar wavelet transform. It leads to a partition of the signal which is ‘optimal’ according to a criterion that tests the Gaussian nature of the segments. A statistic based on the fourth-order cumulant is computed from this segmentation. The second part of this paper is devoted to the study of performance of the detector. This study is made in terms of experimental ROC curves. We show that in general the detector performs better than the energy detector. Actually, the performance depends on the nature of the transients; our detector is especially well adapted for oscillatory transients.

Eigenvector algorithm for blind MA system identification

We present a novel approach to the blind estimation of a linear time-invariant possibly mixed-phase moving average (MA) system (channel) based on second and fourth order statistics of the stationary received signal. As the algorithm incorporates the solution of an eigenvector problem, it is termed E igenV ector approach to blind I dentification (EVI). One of EVI’s main features is its ability to obtain reliable estimates of the channel’s MA parameters on the basis of very short records of received data samples. It is also robust with respect to an overestimation of the channel order. Furthermore, we demonstrate that, if independent additive white Gaussian noise is present, the degradation of the MA parameter estimates is minor even at low signal-to-noise ratios. By simulation results, we finally show the potential applicability of EVI to mobile radio communication channels under time-invariance conditions typically assumed in GSM receivers.

Order determination of MA models using third- and fourth-order statistics

Most of the existing methods for blind equalisation and system identification attempt to model non-minimun phase FIR or MA channels assuming the model order is known. In this letter we propose an algorithm to estimate the FIR system order using only the third- and fourth-order cumulants of the observed process. Based on equations relating several 1-D slices of third- and fourth-order cumulants, the proposed algorithm performs a search for the minimum of a cumulant-reconstruction-based function. To test the behaviour of the model-order selection algorithm, simulation results are also presented at different SNR values and data lengths.