Stable Distribution Noise Research Articles

Mixture generalized minimum error entropy-based distributed lattice Kalman filter

The Gaussian kernel function-based Minimum Error Entropy (MEE) criterion is effective for special types non-Gaussian noise. However, non-Gaussian noise distributions and shapes are diverse in practice, the traditional MEE methods are difficult to fit non-Gaussian effectively due to the shape parameters of MEE cannot be adjusted. In this paper, the Mixture Generalized Minimum Error Entropy (MGMEE) criterion is proposed by a mixture generalized Gaussian kernel function. Then, a new Mixture Generalized Minimum Error Entropy-based Distributed Lattice Kalman Filter (MGMEE-DLKF) is proposed for multi-sensor nonlinear systems with non-Gaussian noise. The complexity analysis and convergence condition of proposed MGMEE-DLKF algorithm are derived. In the end, the target tracking simulations are verified for systems with mixture Gaussian noise, Rayleigh distribution noise and α−stable distribution noise. The simulation results demonstrate that the proposed filter has the smallest root mean square error.

Variable step size VLF/ELF nonlinear channel adaptive filtering algorithm based on Sigmoid function

The signals received by very low-frequency/extremely low-frequency nonlinear receivers are frequently affected by intense atmospheric pulse noise stemming from thunderstorms and global lightning activity. Current noise processing algorithms designed for nonlinear channels within these frequency ranges, which are predicated on fractional p-order moment alpha stable distribution criteria (where 0 < p < α < 2, and p and α denote distinct characteristic indices of alpha stable distribution noise), are constrained by their reliance on limited p-order moment statistics. As a result, the performance of low-frequency nonlinear channel receivers experiences significant degradation when confronted with robust pulse noise interference (0 < p < α < 2). To tackle this challenge, the present study introduces a novel variable step robust mixed norm (RMN) adaptive filtering algorithm, designated as SVS-RMN, which is based on the Sigmoid function. Leveraging the nonlinearity of the Sigmoid function and building upon the power function Hammerstein nonlinear channel model, the algorithm aims to enhance the RMN algorithm by deriving new cost functions and adaptive iteration formulas. The performance of the proposed algorithm is evaluated in comparison to conventional RMN algorithms based on fractional low-order moment (FLOM) criteria (0 < p < 2), as well as other algorithms employing variable step sizes and either FLOM or radial basis function (RBF) criteria, across various intensities of pulse noise and mixed signal-to-noise ratios. The experimental results reveal the following: (1) The proposed algorithm effectively mitigates strong pulse noise interference and significantly enhances the tracking performance of the RMN algorithm compared to conventional RMN algorithms based on FLOM criteria. (2) In terms of computational efficiency, simplicity of structure, convergence speed, and stability, the proposed algorithm surpasses other algorithms based on FLOM or RBF criteria.

Applications of improved linear chirplet time frequency representation to Machine bearing fault analysis

Time-frequency analysis is a widely used detection tool for the rotating machinery fault signals. The mechanical bearing fault vibration signals are a non-stationary and non-Gaussian process, it is confirmed that they are α stable distribution, and the characteristic index1<α<2, even the noises belong to α stable distribution process in special circumstances. The existing linear chirplet transform(LCT) methods degenerate, even fail in α stable distribution noise. This paper proposes several robust time–frequency analysis methods to overcome the influence of the impulse noise including.fractional lower order continuous wavelet transform (FLOCWT), fractional low order linear chirplet transform(FLOLCT), fractional low order general linear chirplet transform(FLOGLCT) and fractional low order velocity synchronous linear chirplet transform(FLOVSLCT). The improved methods is applied to compare with the existing methods based on second order statistics in Gaussian and α distribution environments, the simulation results demonstrate performance advantages of the new methods. The features, deficiencies and application scenarios of the improved time–frequency methods have been summarized. Finally, the new methods are applied to analyze the bearing outer race DE fault data contaminated by α stable distribution noise and extract their fault signature, the results illustrate their performances.

Hopping time estimation algorithm of frequency hopping signal under impulse noise

Aiming at the hopping time estimation problem of frequency hopping (FH) signals under alpha stable distributed noise, a hopping time estimation algorithm based on time-frequency pattern gradient sequence is proposed in this paper. Firstly, the FH signal model and the alpha stable distribution noise model are established according to the received signals; Secondly the fractional lower order SPWVD (FLOSPWVD) transform is used to suppress the impulse noise in the time frequency pattern, and a clear and reliable time frequency pattern is obtained by using the morphological filtering. Finally, the hopping time estimation of the frequency hopping signal is completed according to the jump of the gradient sequence of the time frequency graph.

Fractional lower order linear chirplet transform and its application to bearing fault analysis.

The amplitude and frequency of the mechanical bearing fault vibration signals vary with time, and which are non-stationary and non-Gaussian process. The fault signals belong to α stable distribution, and the characteristic index 1 < α < 2, even the noises are α stable distribution in extreme cases. The existing linear chirplet transform (LCT) degenerates, even fails under α stable distribution environment. A fractional low order linear chirplet transform (FLOLCT) which takes advantage of fractional p order moment is presented for α stable distribution noise environment, and the corresponding FLOLCT time-frequency representation (FLOLCTTFR) is developed in this paper. By employing a series of polynomial chirp rate parameters instead of a single chirp rate of the FLOLCT method, a fractional low order polynomial linear chirplet transform (FLOPLCT) is developed to improve time frequency concentration of the signals. The improved FLOLCT and FLOPLCT methods are used to compare with the existing LCT and PLCT methods based on second order statistics, the results reveal performance advantages of the proposed methods. Finally, the FLOLCT and FLOPLCT methods are applied to analyze the fault signature of the bearing ball fault data in the position of DE (Drive end accelerometer) and extract their fault signature, the result illustrates their performances.

Kernel least logarithm absolute difference algorithm based on P-norm

The kernel adaptive filtering is an efficient and nonlinear approximation method which is developed in reproducing kernel Hilbert space (RKHS). Kernel function is used to map input data from original space to RKHS space, thus solving nonlinear problems is efficient.Impulse noise and non-Gaussian noise exist in the real application environment, and the probability density distribution of these noise characteristics shows a relatively heavy trailing phenomenon in the statistical sense. α stable distribution can be used to model this kind of non-Gaussian noise well. The kernel least mean square(KLMS) algorithms usually perform well in Gaussian noise, but the mean square error criterion only captures the second-order statistics of the error signal, this type of algorithm is very sensitive to outliers, in other words, it lacks robustness in α stable distribution noise. The kernel least logarithm absolute difference(KLLAD) algorithm can deal with outliers well, but it has the problem of slow convergence.In order to further improve the convergence speed of nonlinear adaptive filtering algorithm in α stable distributed noise background, a new kernel least logarithm absolute difference algorithm based on p-norm (P-KLLAD) is presented in this paper. The algorithm combining least logarithm absolute difference algorithm and p norm, on the one hand, the least logarithm difference criteria is ensure the algorithm to have good robustness in α stable distribution noise environment, and on the other hand, add p norm on the absolute value of error.The steepness of the cost function is controlled by p norm and a posititive constant ɑ to improve the convergence speed of the algorithm.The computer simulation results of Mackey-Glass chaotic time series prediction and nonlinear system identification show that this algorithm improves the convergence speed with good robustness,and the convergence speed and robustness better than the kernel least mean square algorithm,the kernel fractional lower power algorithm, the kernel least logarithm absolute difference algorithm and the kernel least mean p-norm algorithm.

Applications of Fractional Lower Order Synchrosqueezing Transform Time Frequency Technology to Machine Fault Diagnosis

Synchrosqueezing transform (SST) is a high resolution time frequency representation technology for nonstationary signal analysis. The short time Fourier transform-based synchrosqueezing transform (FSST) and the S transform-based synchrosqueezing transform (SSST) time frequency methods are effective tools for bearing fault signal analysis. The fault signals belong to a non-Gaussian and nonstationary alpha (α) stable distribution with 1&lt;α&lt;2 and even the noises being also α stable distribution. The conventional FSST and SSST methods degenerate and even fail under α stable distribution noisy environment. Motivated by the fact that fractional low order STFT and fractional low order S-transform work better than the traditional STFT and S-transform methods under α stable distribution noise environment, we propose in this paper the fractional lower order FSST (FLOFSST) and the fractional lower order SSST (FLOSSST). In addition, we derive the corresponding inverse FLOSST and inverse FLOSSST. The simulation results show that both FLOFSST and FLOSSST perform better than the conventional FSSST and SSST under α stable distribution noise in instantaneous frequency estimation and signal reconstruction. Finally, FLOFSST and FLOSSST are applied to analyze the time frequency distribution of the outer race fault signal. Our results show that FLOFSST and FLOSSST extract the fault features well under symmetric stable (SαS) distribution noise.

2-D DOA Robust Estimation of Echo Signals Based on Multiple Satellites Passive Radar in the Presence of Alpha Stable Distribution Noise

In this paper, the low-performance problem of two-dimensional (2-D) direction of arrival (DOA) estimation with non-Gaussian noise in low signal-to-noise ratio is addressed. For echo signals of the multiple satellites’ passive radar, a robust 2-D DOA estimation method based on improved zero-order statistics fractional low-order cyclic correlation (IZOS-FLOCC) under the alpha-stable distribution noise environment is proposed. First, we employ a uniform plane array to establish the model of received signals. Then, the IZOS-FLOCC of the echo signals is constructed based on the cyclostationarity. Finally, the IZOS-FLOCC of signals subspace is obtained by the optimization method. Moreover, we derive the Cramer–Rao low bound of the 2-D DOA estimation of echo signals based on multiple satellites’ passive radar in the presence of alpha-stable distribution noise. The simulation results show that the proposed method can effectively estimate the 2-D DOA of echo signals based on multi-satellite radiation sources in the alpha-stable distribution noise environment and achieve better performance than the existing methods.

Metaheuristic optimization-based identification of fractional-order systems under stable distribution noises

This research investigates the identification problem of fractional-order chaotic systems under stable distribution noises. A powerful metaheuristic optimization method called composite differential evolution is used for the identification of the fractional-order Lorenz and Chen systems in the noisy environment, where the structure, parameters, orders and initial values of the systems are all unknown. The identification accuracy is examined when the noise follows the three special cases of stable distributions, i.e., Gaussian, Cauchy and Lévy distributions. In addition, the impact of the four parameters of stable distributions on the identification accuracy is discussed. The experimental results show that the identification error becomes larger when the noise switches from Gaussian to Cauchy and Lévy distributions. The results also turn out that the location of the stable distribution noise plays the most substantial role in the identification accuracy.

A Robust DOA Estimation Based on Sigmoid Transform in Alpha Stable Noise Environment

To suppress alpha-stable noise and co-channel interference, this paper defines a novel cyclic correlation function, and proposes a new MUSIC algorithm based on sigmoid cyclic correlation function. Furthermore, the proposed algorithm (SCC-MUSIC) is applied to estimate direction of arrival (DOA) in alpha stable distribution noise. Simulation results demonstrate that the proposed SCC-MUSIC can get a better performance than several existing algorithms especially in the highly impulsive noise environments.

A WHT Signal Detection-Based FLO-TF-UBSS Algorithm Under Impulsive Noise Environment

Linear frequency modulated (LFM) signal is used to describe radar echo signal, and Wigner–Hough transform (WHT) is a useful detection tool for LMF signals. The noise in radar echo signal has strong pulse under complex environment, which belongs to $$\alpha $$ stable distribution process. The WHT method degenerates under $$\alpha $$ stable distribution environment. Hence, fractional lower-order pseudo-Wigner–Ville distribution (FLO-PWVD) time–frequency method is introduced, and fractional lower-order pseudo-Wigner–Hough transform (FLO-PWHT) method based on the FLO-PWVD method and Hough transform is proposed for the detection of LFM signals. When the LFM signals are overlapped in time domain and generalized signal-to-noise ratio (GSNR) is relatively low, the FLO-PWHT method degenerate, even which cannot work. Therefore, fractional lower-order spacial time–frequency matrix is applied to substitute spacial time–frequency distribution matrix, and a new fractional lower-order spacial time–frequency underdetermined blind source separation (FLO-TF-UBSS) algorithm employing the time–frequency underdetermined blind source separation algorithm (TF-UBSS) is proposed in this paper. Also, we combine the FLO-PWHT method with the FLO-TF-UBSS algorithm and propose a fractional lower-order spatial time–frequency underdetermined blind source separation pseudo-Wigner–Hough transform (FLO-TF-UBSS-PWHT) method. The simulation results show that the FLO-PWHT method is obviously better than the existing PWHT algorithm under $$\alpha $$ stable distribution noise or Gaussian noise environment, which is robust. The FLO-TF-UBSS algorithm can effectively downgrade mean square error of the reconstructed LFM signals, and its performance is better than the existing TF-UBSS and minimum dispersion BSS algorithms. The FLO-TF-UBSS-PWHT algorithm can work well in the cases of time domain overlapping and relatively low GSNR; its performance is better than the FLO-PWHT algorithm, which has certain toughness. Finally, we apply the FLO-PWVD, FLO-TF-UBSS, and FLO-PWHT methods to analyze and extract fault features of the bearing outer race fault signal in DE; the experimental results illustrate their performance superiority.

The Fractional Order Duffing Oscillator Detection Method for Nanovolt Level Weak Signal

The traditional weak signal detection makes it difficult to detect nV level signal under strong noise background, so it proposes a new detection method of improving fractional order Duffing oscillator (IMFR-Duffing). Using variable substitution method for improving conventional fractional order Duffing equation obtained a mathematical model of IMFR-Duffing oscillator. It studied anti-noise performance of applying this mathematical model to low frequency nV level weak signal detection in the Gaussian and non-Gaussian noise background. The detection results of contrasting the IMFR-Duffing with the integer order Duffing oscillator verify the effectiveness of the proposed method, and the minimum signal-to-noise ratio (SNR) threshold is reduced by 10 dB. The simulation results show that the lowest SNR threshold of the IMFR-Dufffing oscillator reaches -73 dB in Gaussian white noise, -57 dB in Gaussian colored noise and -153 dB in symmetric Alpha stable distribution noise, and the amplitude detection has higher precision.

Applications of fractional lower order S transform time frequency filtering algorithm to machine fault diagnosis.

Stockwell transform(ST) time-frequency representation(ST-TFR) is a time frequency analysis method which combines short time Fourier transform with wavelet transform, and ST time frequency filtering(ST-TFF) method which takes advantage of time-frequency localized spectra can separate the signals from Gaussian noise. The ST-TFR and ST-TFF methods are used to analyze the fault signals, which is reasonable and effective in general Gaussian noise cases. However, it is proved that the mechanical bearing fault signal belongs to Alpha(α) stable distribution process(1 < α < 2) in this paper, even the noise also is α stable distribution in some special cases. The performance of ST-TFR method will degrade under α stable distribution noise environment, following the ST-TFF method fail. Hence, a new fractional lower order ST time frequency representation(FLOST-TFR) method employing fractional lower order moment and ST and inverse FLOST(IFLOST) are proposed in this paper. A new FLOST time frequency filtering(FLOST-TFF) algorithm based on FLOST-TFR method and IFLOST is also proposed, whose simplified method is presented in this paper. The discrete implementation of FLOST-TFF algorithm is deduced, and relevant steps are summarized. Simulation results demonstrate that FLOST-TFR algorithm is obviously better than the existing ST-TFR algorithm under α stable distribution noise, which can work better under Gaussian noise environment, and is robust. The FLOST-TFF method can effectively filter out α stable distribution noise, and restore the original signal. The performance of FLOST-TFF algorithm is better than the ST-TFF method, employing which mixed MSEs are smaller when α and generalized signal noise ratio(GSNR) change. Finally, the FLOST-TFR and FLOST-TFF methods are applied to analyze the outer race fault signal and extract their fault features under α stable distribution noise, where excellent performances can be shown.

Parameter estimator based on the log-cumulant and its performance analysis for heavy-tailed-distributed impulsive interference}{Parameter estimator based on the log-cumulant and its performance analysis for heavy-tailed-distributed impulsive interference

The impulsive interference in many wireless communication networks can be modeled as a symmetric $\alpha$-stable distribution noise. The probability density function of the interference needs to be obtained in advance in signal detection, channel decoding, wireless network outage probability, and bit error rate analysis application scenarios. The method of log-cumulant is used to estimate the characteristic exponent and dispersion of the interference. Furthermore, the probability distribution of the estimated parameters is derived in detail, which can be used for quantitative analysis to evaluate the estimation reliability. In addition, the noise in the receiver of the actual system is bivariate mixture noise including the independent Gaussian noise and S$\alpha$S interference. We suggest that the bivariate noise can be approximated from its univariate counterpart. We prove that this type of approximation is reasonable by means of simulation and numerical calculation. The relationship between the parameters of the mixture noise and the geometric power signal-to-noise ratio are provided in this paper based on this assumption. The estimator based on the log-cumulant and performance analysis is therefore still valid in the bivariate noise environment.

Interference Suppression of VLF Communications in Atmospheric Noise

This paper focuses on the weak signal reception and atmospheric noise suppression in VLF Chirp spread spectrum communication system. Due to interference of atmospheric noise generated by lightning phenomena, it makes the chirp signal detection under low SNR condition becoming more difficult. Therefore, we select alpha stable distribution noise model to generate noise to simulate the real atmospheric noise. Then, we introduced a new technique of Chirp Spread Spectrum (CSS) communication based on Fractional Fourier Transform (FRFT). Finally, we study a nonlinear processing interference suppression method of clipping in time domain. Simulation results verified its feasibility.

Frequency Spectrum Estimate Based on Decomposed FLOC Matrix under Stable Distribution Noise

The radar work in a complex environment where the noise has very strong pulse, the noise can be described by stable distribution. The conventional spectrum estimation method based on second order statistics is reasonable in many cases, however, the performance of the conventional algorithm degenerate in stable distribution environment. We propose three new frequency spectrum estimate method with Fractional lower order covariance (FLOC), FLOC-Pisarenko method, FLOC-Esprit method and FLOC multiple signal classification (FLOC-MUSIC) method. The conventional spectrum estimation methods with three proposed methods under Gaussian noise and stable distribution are compared in this paper, simulations show that the conventional methods degenerate, but the proposed algorithms can work better in stable distribution environment, and are robust.

Study of VLF Communication Technique Based on FRFT

The atmospheric noise generated by lightning phenomena is the main source of interference in VLF communications, it is rather difficult to detect the communication signal that not only merged in strong channel noise, but probably interfered by high-tension impulse caused by thunder. This paper, Alpha stable distribution noise model is proposed to simulate atmospheric noise firstly, then, we study a new technique of Chirp Spread Spectrum (CSS) communication based on Fractional Fourier Transform (FRFT), This paper studies a new VLF communication technique of the Chirp Spread Spectrum (CSS) communication based on Fractional Fourier Transform (FRFT), and it demodulates message utilizing energy gathering property of chirp signal in FRFT domain.

Simulation of VLF CSS System in Atmospheric Noise

Very Low Frequency (VLF) communication is designed to operate in the frequency between 3 kHz and 30 kHz. The atmospheric noise generated by lightning phenomena is the main source of interference in VLF communications, it is rather difficult to detect the communication signal that not only merged in strong channel noise, but probably interfered by high-tension impulse caused by thunder. This paper, Alpha stable distribution noise model is proposed to simulate atmospheric noise firstly, then, we study a new technique of Chirp Spread Spectrum (CSS) communication based on Fractional Fourier Transform (FRFT), This paper studies a new VLF communication technique of the Chirp Spread Spectrum (CSS) communication based on Fractional Fourier Transform (FRFT), and it demodulates message utilizing energy gathering property of chirp signal in FRFT domain.

Pseudo Cohen Time-Frequency Distributions in Infinite Variance Noise Environment

stable distribution has been suggested as a more appropriate model in impulsive noise environment.The performance of conventional time-frequency distributions (TFDs) degenerate in stable distribution noise environment. Hence, three improved methods are proposed based on Fractional Low Order statistics, Fractional Low Order Wigner-Ville Distribution (FLO-WVD), Fractional Low Order Statistic pseudo Wigner-Ville Distribution (FLO-PWVD), Fractional Low Order Statistic Cohen class distribution (FLO-Cohen). In order for real-time, on-line operation and fairly long signals processing, a new smoothed pseudo Fractional Low Order Cohen class distribution (PFLO-Cohen) is proposed.Simulations show that the methods demonstrate the advantages in this paper, are robust.

Fractals in trade duration: capturing long-range dependence and heavy tailedness in modeling trade duration

Several studies that have investigated a few stocks have found that the spacing between consecutive financial transactions (referred to as trade duration) tend to exhibit long-range dependence, heavy tailedness, and clustering. In this study, we empirically investigate whether a larger sample of stocks exhibit those characteristics. We do so by comparing goodness of fit in modeling trade duration data for stable distribution and fractional stable noise based on a procedure applying bootstrap methods developed by the authors with several alternative distributional assumptions in modeling trade duration data. The empirical results suggest that the autoregressive conditional duration model with stable distribution fits better than other combinations, while fractional stable noise itself fits better for the time series of trade duration. Our result is consistent with the general findings in the literature that trade duration is informative and that short trade durations move prices more than long trade duration. In addition, our result confirms the advantage of fractal models in the study of roughness in trade duration and provides some evidence for duration dependence.