Fractional Lower Order Statistics Research Articles

PolInSAR Statistical Analysis and Coherence Optimization Using Fractional Lower Order Statistics

In this letter, the polarimetric synthetic aperture radar (SAR) interferometry (PolInSAR) statistical model is studied using the alpha-stable distribution. Based on this model, a vector coherence formulation using fractional lower order statistics is proposed. This vector coherence is the generalization of the vector coherence in the paper of Cloude and Papathanassiou when the PolInSAR data are non-Gaussian and conform to the symmetrical form of the alpha-stable distribution. The standard coherence optimization method in the aforementioned paper is modified based on this generalized vector coherence. Results were demonstrated using a small L-band PolInSAR data set and suggest that this proposed coherence optimization method reduces artifacts in the optimized phases in certain areas.

Radar echo frequency spectrum estimation based on α stable distribution covariation

When radar is working in a complex environment,the performance of conventional algorithms degenerates. New frequency spectrum estimation methods based on Fractional Lower Order Statistics (FLOS) covariation about radar were proposed: FLOS-direct method,FLOS-indirect method,FLOS-welch method,FLOS-music method. By estimating the sinusoidal signals embedded in the stable noise,the method in this paper is robust in both Gauss and non-Gauss environments.

Constant Modulus Algorithm for Cochannel Signal Separation in Non-Gaussian Impulsive Noise Environments

The separation of cochannel signals is of interest in communication community. Some algorithms based on constant modulus (CM) have been previously developed to separate cochannel signals with the assumption of Gaussian channel noise. The mismatches of noise models between the assumed channel noise and the practical noise may occur. These mismatches will inevitably lead the performance of cochannel signals separation to degrade. In this paper the alpha-stable distribution is employed as noise model to simulate impulsive noise occurring in wireless channel. A constant modulus algorithm is proposed to separate the cochannel signals based on fractional lower-order statistics (FLOS). The convergence of the CM array is analyzed. Numerical simulations are presented to verify the accuracy of the analytical results.

New HB-weighted time delay estimation algorithm under impulsive noise environment

The traditional HB-weighted time-delay estimation (TDE) method degenerates under the impulsive noise environment. Two new time-delay estimation methods are proposed based on fractional lower order statistics (FLOS) according to the impulsive characteristics of fractional lower order α-stable noises. Theoretic analysis and computer simulations indicate that the proposed covariation based HB weighted (COV-HB) algorithm can suppress impulsive noises in one received signal for 1 ≤ α ≤ 2, whereas the other proposed fractional lower order covariance-based HB weighted (FLOC-HB) algorithm has robust performance under arbitrary impulsive noise conditions for the whole range of 0 <α ≤ 2.

New blind estimation method of evoked potentials based on minimum dispersion criterion and fractional lower order statistics

Evoked potentials (EPs) have been widely used to quantify neurological system properties. Tra-ditional EP analysis methods are developed under the condition that the background noises in EP are Gaussian distributed. Alpha stable distribution, a generalization of Gaussian, is better for modeling impulsive noises than Gaussian distribution in biomedical signal proc-essing. Conventional blind separation and es-timation method of evoked potentials is based on second order statistics or high order Statis-tics. Conventional blind separation and estima-tion method of evoked potentials is based on second order statistics (SOS). In this paper, we propose a new algorithm based on minimum dispersion criterion and fractional lower order statistics. The simulation experiments show that the proposed new algorithm is more robust than the conventional algorithm.

Performance of RBF neural networks for array processing in impulsive noise environment

This paper addresses the array processing problems (mainly focuses on direction of arrival estimation and beamforming) of mobile communication system using linear antenna arrays in high impulsive noise environment. One possible way to simulate the impulsive noise is to introduce alpha-stable distribution as the noise model. In order to reduce the computational complexity, the problems of DOA and beamforming are approached as a nonlinear mapping which can be modeled using a suitable radial-basis function neural network (RBFNN) trained with input-output pairs. This paper discusses the application of a three-layer RBFNN to perform the DOA estimation and beamforming in presence of impulsive noise. The performance of the network is compared to that of the algorithms based fractional lower-order statistics. Simulations show that the RBFNN is appropriate to approach the DOA estimation and beamforming. At the same time, the RBFNN substantially reduces the computation complexity.

Data block adaptive filtering algorithms for α-stable random processes

The least mean p-norm (LMP) algorithm is an effective algorithm for processing the signal of @a-stable distribution. This paper proposes data block adaptive filtering algorithms for the @a-stable random processes based on the fractional lower order statistics (FLOS). The data block algorithms change the direction of coefficient increment vector by introducing a matrix which includes the information of more past input signal vectors than which are used in the LMP algorithm during the iteration process, taking full advantage of the past values of the gradient vector during the adaptation. Simulations studies indicate that the proposed algorithms increase convergence rate in non-Gaussian stable distribution noise environments compared to the existing algorithms based on FLOS summarized in this paper.

High Resolution Multipath Time Delay Estimation Based on FLOCCS-ESPRIT

Abstract The non-Gaussian α-stable distribution is introduced to model impulsive noise. Based on the theory of fractional lower order statistics (FLOS), the fractional lower order cross-covariance (FLOCC) sequence for two re-ceived signals is obtained and the fractional lower order cross-covariance spectrum (FLOCCS) can be approached by taking a Fourier transform for the FLOCC sequence. When the FLOCCS is treated as a sequence in the time domain, the problem of multipath time delay estimation (TDE) may be converted into one on multi-frequencies estimation or di-rections of arrival estimation. Accordingly, the high resolution multipath TDE can be realized with the ESPRIT tech-nology. This idea on multipath TDE is referred to as FLOCCS-ESPRIT in this paper. Computer simulations show that this method has good performance both in a Gaussian noise and in an impulsive noise environment.

The robustness analysis of DLMP algorithm under the fractional lower-order α-stable distribution environments

The direct least mean p-norm (DLMP) algorithm is a robust algorithm for time delay estimation under both Gaussian and fractional lower-order α-stable noise conditions. This paper briefly reviews the fundamental theory of α-stable distribution, abstracts two important theorems of fractional lower-order statistics, and under the Gaussian signal assumption, analyzes the convergence property of the DLMP algorithm, including the convergence property of the cost function, the unbiasness of the estimator and the appropriate selection of the convergence parameter μ. The improved performance of DLMP algorithm is clearly demonstrated through comprehensive simulations and detailed data analysis for estimating the latency changes of evoked potential (EP) signals. In the case of p=2, it is presented that DLMP algorithm keeps consistent with DLMS algorithm, which is based on second-order statistics, involving the cost function, the unbiasness of the estimator and the selection of the convergence parameter. Therefore, this paper proves that DLMP algorithm is a generalization of DLMS algorithm in α-stable distribution noisy environments.

Robust subspace-based algorithms for joint angle/Doppler estimation in non-Gaussian clutter

We consider the problem of joint angle and Doppler estimation for space-time adaptive processing (STAP) airborne radar in non-Gaussian clutter which is modeled as a complex symmetric alpha stable S α S process. We investigate the performance of the subspace algorithms based on the sign covariance matrix estimate (SCM) which has almost robust performance in heavy tailed noise [S. Visuri, H. Oja, V. Koivunen, Subspace-based direction-of-arrival estimation using non-parametric statistics, IEEE Trans. Signal Process. 49 (September 2001) 2060–2073]. We propose a new class of covariance estimation based on phased fractional lower-order moment (PFLOM). Performance of the proposed technique is assessed through simulations and it is shown that this algorithm reveals better performance than fractional lower order moment (FLOM) and SCM based algorithms [T. Liu, J.M. Mendel, A subspace-based direction finding algorithm using fractional lower order statistics, IEEE Trans. Signal Process. 49 (8) (August 2001) 1605–1613; P. Tsakalides, et al., Angle/Doppler estimation in heavy tailed clutter backgrounds, IEEE Trans. Aerospace Electron. Syst. 35 (2) (April 1999) 419–436; S. Visuri, H. Oja, V. Koivunen, Subspace-based direction-of-arrival estimation using non-parametric statistics, IEEE Trans. Signal Process. 49 (September 2001) 2060–2073].

Zero-Order Statistics: A Mathematical Framework for the Processing and Characterization of Very Impulsive Signals

Impulsive or heavy-tailed processes with infinite variance appear naturally in a variety of practical problems that include wireless communications, teletraffic, hydrology, geology, and economics. Most signal processing and statistical methods available in the literature have been designed under the assumption that the processes possess finite variance, and they usually break down in the presence of infinite variance. Although methods based on fractional lower-order statistics (FLOS) have proven successful in dealing with infinite variance processes, they fail in general when the noise distribution has very heavy algebraic tails. In this paper, we introduce a new approach to statistical moment characterization which is well defined over all processes with algebraic or lighter tails. Unlike FLOS, these zero-order statistics (ZOS), as we will call them, provide a common ground for the analysis of basically any distribution of practical use known today. Three new parameters, namely the geometric power, the zero-order location and the zero-order dispersion, constitute the foundation of ZOS. They play roles similar to those played by the power, the expected value and the standard deviation, in the theory of second-order processes. We analyze the properties of the new parameters, and derive a ZOS framework for location estimation that gives rise to a novel mode-type estimator with important optimality properties under very impulsive noise. Several simulations are presented to illustrate the potential of ZOS methods

Direction finding in non-Gaussian impulsive noise environments

The class of alpha-stable distributions is better for modeling impulsive noise than Gaussian distribution in array signal processing. This paper briefly introduces the statistical characteristics of stable distribution and its fractional lower-order statistics, covariation and fractional-order correlation, and proposes a new minimum norm method (FOC-MinNorm) of direction finding based on the fractional-order correlation and subspace technique under alpha-stable noise conditions. We analyze the performances of the FOC-MinNorm, including the accuracy in the estimation, the capability and probability of resolution, and the pseudo peaks of the FOC-MinNorm method. The analysis is based on the assumption that the additive noise can be modeled as a complex alpha-stable process. Simulation and analysis show that the proposed method is robust in a wide range of characteristic exponent values of stable distribution. Its resolution capability and probability of resolution are better than those of the conventional second-order statistics based MinNorm algorithm and covariation based ROC-MUSIC method, furthermore, fractional-order correlation is more suitable than covariation in practical applications.

Underwater sources location in non-Gaussian impulsive noise environments

The class of alpha-stable distributions is better for modeling impulsive noise than Gaussian distribution in array signal processing. After briefly introducing the statistical characteristics of stable distribution and the fractional lower order statistics, including the covariation and the fractional order correlation, this paper proposes a new FOC-ESPRIT method of 2-D direction finding based on the fractional order correlation and subspace technique for underwater 2-D source localization using a vector hydrophones array under alpha-stable noise conditions. A vector hydrophone comprises two or three spatially co-located, orthogonally oriented identical velocity hydrophones (each of which measures one Cartesian component of the underwater acoustical particle velocity vector-field) plus an optional pressure hydrophone. Simulation experiments show that the proposed method is robust in a wide range of characteristic exponent values of stable distribution. Its performances are better than those of the conventional second-order statistics based ESPRIT algorithm, furthermore, the fractional order correlation is more suitable than the covariation in practical applications.

Constant modulus blind equalization based on fractional lower-order statistics

The presence of non-Gaussian ambient channel noise in wireless systems can degrade the performance of existing equalizers and signal detectors. In this paper, we investigate the problem of blind equalization in noisy communication channels by addressing the negative effects of heavy-tailed noise to the original constant modulus algorithm (CMA). We propose a new CM criterion employing fractional lower-order statistics (FLOS) of the equalizer input. The associated FLOS-CM blind equalizer, based on a stochastic gradient descent algorithm, is able to mitigate impulsive channel noise while restoring the constant modulus character of the transmitted communication signal. We perform an analytic study of the lock and capture properties of the proposed adaptive filter and we illustrate its improved convergence behavior and lower bit error rate through computer simulations with various types of noise environments that include the Gaussian and the alpha-stable.

Moving object detection using adaptive subband decomposition and fractional lower-order statistics in video sequences

In this paper, a moving object detection method in video sequences is described. In the first step, the camera motion is eliminated using motion compensation. An adaptive subband decomposition structure is then used to analyze the motion compensated image. In the “low–high” and “high–low” subimages moving objects appear as outliers and they are detected using a statistical detection test based on fractional lower-order statistics. It turns out that the distribution of the subimage pixels is almost Gaussian in general. On the other hand, at the object boundaries the distribution of the pixels in the subimages deviates from Gaussianity due to the existence of outliers. By detecting the regions containing outliers the boundaries of the moving objects are estimated. Simulation examples are presented.

Subspace-based frequency estimation of sinusoidal signals in alpha-stable noise

In the frequency estimation of sinusoidal signals observed in impulsive noise environments, techniques based on Gaussian noise assumption are unsuccessful. One possible way to find better estimates is to model the noise as an alpha-stable process and to use the fractional lower order statistics (FLOS) of the data to estimate the signal parameters. In this work, we propose a FLOS-based statistical average, the generalized covariation coefficient (GCC). The GCCs of multiple sinusoids for unity moment order in S αS noise attain the same form as the covariance expressions of multiple sinusoids in white Gaussian noise. The subspace-based frequency estimators FLOS–multiple signal classification (MUSIC) and FLOS–Bartlett are applied to the GCC matrix of the data. On the other hand, we show that the multiple sinusoids in S αS noise can also be modeled as a stable autoregressive moving average process approximated by a higher order stable autoregressive (AR) process. Using the GCCs of the data, we obtain FLOS versions of Tufts–Kumaresan (TK) and minimum norm (MN) estimators, which are based on the AR model. The simulation results show that techniques employing lower order statistics are superior to their second-order statistics (SOS)-based counterparts, especially when the noise exhibits a strong impulsive attitude. Among the estimators, FLOS–MUSIC shows a robust performance. It behaves comparably to MUSIC in non-impulsive noise environments, and both in impulsive and non-impulsive high-resolution scenarios. Furthermore, it offers a significant advantage at relatively high levels of impulsive noise contamination for distantly located sinusoidal frequencies.

A subspace-based direction finding algorithm using fractional lower order statistics

We propose several classes of fractional lower order moment (FLOM)-based matrices that can be used with MUSIC to estimate the DOAs of independent circular signals embedded in additive S/spl alpha/S (symmetric /spl alpha/ stable) noise (e.g., sea clutter). We run simulations with different choices of the FLOM parameter p for our FLOM-based matrices and conclude that when the noise is S/spl alpha/S with unknown /spl alpha//spl ne/2, FLOM-multiple signal classification (MUSIC) with p close to unity yields good performance. The performance of FLOM-MUSIC and robust covariation-based (ROC)-MUSIC are similar. Three scenarios that contain circular signals (phase modulation (PM), circularly symmetrical Gaussian, and quaternary phase-shift keying (QPSK)) and one scenario that contains noncircular signals (binary phase-shift keying (BPSK)), all embedded in the same S/spl alpha/S noise, are tested. These simulation results reveal that the scenario containing BPSK signals leads to poor performance, indicating that FLOM-MUSIC is presently limited to circular signals.

High-resolution autofocus techniques for SAR imaging based on fractional lower-order statistics

The authors address the autofocusing problem in synthetic aperture radar imagery by introducing techniques that achieve robust image formation in the presence of severe heavy-tailed clutter and noise. Current state-of-the-art methods are extended, which are based on second-order moment theory, by employing fractional lower-order statistics (FLOS) of the phase-history data. The introduced FLOS-based methods perform a nonlinear transformation of the radar measurements, can mitigate the effects of impulsive additive clutter, and include the conventional algorithms as special cases. The benefits of the proposed approach are quantified by means of simulations, and the new FLOS-based methods are compared to current state-of-the-art processing with real synthetic aperture radar imagery data.

Robust spatial filtering of coherent sources for wireless communications

This paper addresses the inter-signal coherence problem in mobile communications applications where multipath parameters, such as path angle of arrival, must be estimated with an array of sensors operating in impulsive interference. We reduce the measured coherence by using the spatial smoothing approach to the structure of the measurement matrix while at the same time we mitigate the effects of the heavy-tailed background noise by employing a signed-power nonlinearity to the array data. The combination of these two processing modules gives rise to a family of robust smoothed subspace array processing methods based on fractional lower-order statistics (FLOS), which are able to identify all incident angles regardless of their correlation structure. The improved performance of the proposed techniques is demonstrated via Monte Carlo simulations

Alpha-stable modeling of noise and robust time-delay estimation in the presence of impulsive noise

A new representation of audio noise signals is proposed, based on symmetric /spl alpha/-stable (S/spl alpha/S) distributions in order to better model the outliers that exist in real signals. This representation addresses a shortcoming of the Gaussian model, namely, the fact that it is not well suited for describing signals with impulsive behavior. The /spl alpha/-stable and Gaussian methods are used to model measured noise signals. It is demonstrated that the /spl alpha/-stable distribution, which has heavier tails than the Gaussian distribution, gives a much better approximation to real-world audio signals. The significance of these results is shown by considering the time delay estimation (TDE) problem for source localization in teleimmersion applications. In order to achieve robust sound source localization, a novel time delay estimation approach is proposed. It is based on fractional lower order statistics (FLOS), which mitigate the effects of heavy-tailed noise. An improvement in TDE performance is demonstrated using FLOS that is up to a factor of four better than what can be achieved with second-order statistics.