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Abstract

Information extraction is a frequent and relevant problem in digital signal processing. In the past few years, different methods have been utilized for the parameterization of signals and the achievement of efficient descriptors. When the signals possess statistical cyclostationary properties, the Cyclic Autocorrelation Function (CAF) and the Spectral Cyclic Density (SCD) can be used to extract second-order cyclostationary information. However,second-order statistics tightly depends on the assumption of gaussianity, as the cyclostationary analysis in this case should comprise higher-order statistical information. This paper proposes a new mathematical formulation for the higher-order cyclostationary analysis based on the correntropy function. In particular, we prove that the CCF contains information regarding second- and higher-order cyclostationary moments, being a generalization of the CAF. The cyclostationary analysis is revisited focusing on the information theory, while the Cyclic Correntropy Function (CCF) and Cyclic Correntropy Spectral Density (CCSD) are also defined.The CCF has different properties compared with CAF that can be very useful in nongaussian signal processing, especially in the impulsive noise environment which implies in the expansion of the class of problems addressed by the second-order cyclostationary analysis. The performance of the aforementioned functions in the extraction of higher-order cyclostationary characteristics is analyzed in a wireless communication system in which nongaussian noise is present. The results demonstrate the advantages of the proposed method over the second-order cyclostationary.

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... AMC of QAM in time-varying channels [96]. High order cyclostationary features can be further extended to kernel-based ones which can subsume both second and higher order features in a computationally efficient manner [97]. ...
... The main research on applying cyclic correntropy to AMC has been by Fontes et al. [95], [97] and Câmara et al. [100]. They showed mathematically that CCF contains more information other than CAF [97], including high order information, and a sinusoidal function which controls the phase shift of the response regardless of the type of random process. ...
... The main research on applying cyclic correntropy to AMC has been by Fontes et al. [95], [97] and Câmara et al. [100]. They showed mathematically that CCF contains more information other than CAF [97], including high order information, and a sinusoidal function which controls the phase shift of the response regardless of the type of random process. These properties serve to make the cyclic correntropy relatively insensitive to non-Gaussian noise, e.g. ...
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In this paper, we present a comprehensive survey and detailed comparison of techniques that have been applied to the problem of identifying the type of modulation contained within received wireless signals. Known as automatic modulation classification (AMC), the problem has been studied for many decades. AMC plays a significant role in both military and civilian scenarios and is the main step in smart receivers. With the development of software-defined radios and automatic communication systems, IoT technology and the spread of 5G technology, there has been exponential growth in the number of spectrum-using equipment making the issue of scarce spectrum resources more prominent. Although AMC techniques can be optimized from the classifier’s point of view, signal pre-processing also plays a critical role. Relevant data representation approaches include time-frequency analysis, cyclostationary transforms, and hybrid techniques. We provide a taxonomy of common approaches based on order and dimensionality along with an overall analysis of signal pre-processing algorithms for AMC. Furthermore, we reproduce the major existing schemes under uniform conditions, allowing an objective comparison among different methodologies. Finally, we create an open-source reproducible Python library to simulate these techniques, ensuring the usefulness for future research.
... Recently, a generalized cyclic correlation analysis technique named cyclic correntropy (CCE) analysis was proposed to address the problem of impulsive noise [31] [32] [33]. Combining the cyclostationary modeling and correntropy theory, this technique performed well when processing the communication signals in the presence of impulsive noise. ...
... Then, it is reasonable to define the CCE function () x V   as (10) [33]: ...
... Therefore, a wellselected kernel size can provide an effective mechanism to suppress the impulsive noise. Currently, the kernel size of a CCE function is mainly obtained through random tests [31] [32] [33], and is usually not stable. It is essential to determine a kernel size selection method. ...
Article
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Rolling element bearings are important components in various types of industrial equipment. It is necessary to develop advanced fault diagnosis techniques to prevent unexpected accidents caused by bearing failures. However, impulsive background noise in industrial fields also presents a similar fault-excited characteristic, which brings interference to the fault diagnosis of rolling element bearings. Focusing on this issue, this paper proposes a new feature extraction method based on the cyclic correntropy spectrum (CCES) for intelligent fault identification.. In this study, the cyclic correntropy (CCE) function is introduced to suppress the impulsive noise. A simplified frequency spectrum named CCES is obtained for the feature extraction. Then, narrowband kurtosis vectors are extracted from the CCES. Finally, these extracted features are used to train the least squares support vector machine (LSSVM) for the fault pattern identification. Analyses of two bearing datasets, including train axle bearing data that are contaminated by impulsive noise are used as case studies for the validation of the proposed method. To illustrate the advancement of the new method, performance comparisons with two recently developed methods are conducted. The experimental results verify that the proposed method not only outperforms these two methods but also exhibits a stable self-adaptation ability.
... Recently, a new cyclostationary analysis technology named cyclic correntropy (CCE) analysis has emerged to suppress impulsive noise [21][22][23]. CCE is a kernel-based similarity measure of cyclostationary modeling signals. Related research has shown that it has a good suppression performance when dealing with the binary phase-shift keying signal under an impulsive noise environment [21,23]. ...
... CCE is a kernel-based similarity measure of cyclostationary modeling signals. Related research has shown that it has a good suppression performance when dealing with the binary phase-shift keying signal under an impulsive noise environment [21,23]. However, to the best of our knowledge, only several works related to CCE analysis have been published in the field of communication and its applications need to be developed and reinforced. ...
... Combined with Equation (2), the CCE function can be rewritten as [23]: ...
Article
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Rolling element bearings are widely used in various industrial machines. Fault diagnosis of rolling element bearings is a necessary tool to prevent any unexpected accidents and improve industrial efficiency. Although proved to be a powerful method in detecting the resonance band excited by faults, the spectral kurtosis (SK) exposes an obvious weakness in the case of impulsive background noise. To well process the bearing fault signal in the presence of impulsive noise, this paper proposes a fault diagnosis method based on the cyclic correntropy (CCE) function and its spectrum. Furthermore, an important parameter of CCE function, namely kernel size, is analyzed to emphasize its critical influence on the fault diagnosis performance. Finally, comparisons with the SK-based Fast Kurtogram are conducted to highlight the superiority of the proposed method. The experimental results show that the proposed method not only largely suppresses the impulsive noise, but also has a robust self-adaptation ability. The application of the proposed method is validated on a simulated signal and real data, including rolling element bearing data of a train axle.
... In 2017, Fontes et al. [23] introduced correlation entropy [24] from information theory into cyclic stationary analysis. They defined the CCF as a generalization of the CAF. ...
... As a generalized correlation function, the correlation entropy is widely used in nonlinear detection [23] for any random process {x t , t ∈ T }. It is defined as: ...
... The spectra from the NACCF demodulation of the signals for these five patterns are shown in Fig. 13. NACCF uses the CCF, which is a generalization of the CAF [23]. It retains the demodulation performance of CAF, as long as the outer ring fails, regardless of the damage mechanism. ...
Article
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Rolling bearings are critical in industrial mining machinery. Due to strong Gaussian noise, frequent random shocks, and disordered loads in industrial settings, it is usually difficult to detect weak fault symptoms in vibration signals from a bearing. To detect incipient bearing faults, this paper proposes a new multi-domain kernel extreme learning machine (MKELM) based on variational modal decomposition (VMD) and a cyclic correntropy function. A normalized approximation algorithm for a cyclic correntropy function (NACCF) was first built to suppress the impulsive background noise. This approach is suitable for machine learning. To eliminate the Gaussian noise effectively, genetic mutation particle swarm optimization (GMPSO) with cyclic information entropy (CIE) was used to optimize the VMD parameters. The CIE was created as a fitness function in GMPSO to search for the best hyperparameters. It can be used to select effective intrinsic mode functions (IMFs) to reconstruct denoised signals. Then, statistical functions based on NACCF were used to extract the cyclic frequency-domain characteristics of the denoised signal, and the singular values of the IMFs were obtained as time-domain features of the signal. Finally, the multi-dimensional features from the two domains were input into MKELM to classify the health of the bearing. Experimental studies were carried out to investigate the proposed method in bearing fault detection and identification. The results demonstrated the effectiveness of the proposed method in motor-bearing failure detection and its robustness to noise when analyzing bearing vibration signals under different working loads.
... The CCF is a generalization of the CAF with information regarding the second and higher-order cyclostationary moments. Compared with the CAF, the CCF has different properties, which can be very useful in non-Gaussian signal processing, especially in the impulsive noise environment [20]. ...
... The cyclic correntropy estimator in (5) is asymptotically unbiased and consistent that can be derived following the same lines that are used for the cycle correlation function [20], [23]. ...
... Furthermore, in order to verify the estimated accuracy of the parameters γ (k) i , we show the case of divergence and convergence of γ (k) i with respect to k iterations and the theoretically optimal regularization parameter γ * i for reference in Fig. 3. In addition, the theoretical parameter γ * i is calculated using (40), where the theoretical LP parameter q can be obtained in (20) with the simulation condition K ε = 2, and the theoretical variance σ 2 can be given in (25) with the theoretical parameter q and the true matrix in (17). As expected, the estimated parameters Next, we examine the performance of the DOA estimation via the proposed algorithm. ...
Article
In this paper, a new direction of arrival (DOA) and the number of signals of interest (SOIs) estimation method is proposed for wideband sources in impulsive noise environments. By evaluating the cyclic correntropy function (CCF) of the received signals at a certain cycle frequency, the impulsive noise and all co-channel interferences with different cycle frequencies can be suppressed. In this approach, a theorem of the CCF is proposed, and a linear prediction (LP) model of the CCF of the array data, which is important for both DOA and number of SOIs estimations, is built by the theorem. Additionally, we introduce multiple regularization parameters into the LP model and derive an analytical expression of the maximum correntropy criterion (MCC) of the parameters for obtaining the optimal regularization parameters expression. Furthermore, by the expression of regularization parameters and the estimation error of the CCF in a limited number of snapshots, an iterative algorithm is proposed for joint estimation of the DOA and the number of SOIs with only knowledge of the cycle frequency of the SOIs.
... Classical statistical methods used in signal processing, when applied to cyclostationary signals, only allow performing a limited analysis since they consider the analyzed signal is stationary. However, techniques based on cyclostationary analysis are more suitable to process communication signals [21]. ...
... The prevalent techniques for the information processing in impulsive non-Gaussian environments are the fractional lower-order statistical analysis [35]- [38], correntropy measure [39], [40], spatial sign cyclic correlation estimation [41]- [43], fractional lower-order cyclostationary analysis [44], [45], and cyclic correntropy [21], [46], [47]. ...
... On the other hand, cyclic correntropy is capable of extracting cyclostationary signatures from modulated signals contaminated by impulsive noise [21], [46], with application in spectral sensing [21]. However, further investigations still must to be performed in order to evaluate the performance of this technique in recognition of digital modulations in non-Gaussian impulsive environments, such as M-PSK (Mary phase-shift keying) and M-QAM (M-ary quadrature amplitude modulation), which are widely employed in several communication systems [48]- [50]. ...
Article
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Cyclostationary analysis has several applications in communications, e.g., spectral sensing, signal parameter estimation, and modulation classification. Most of them consider the additive white Gaussian noise (AWGN) channel model, although wireless communication systems may also be subject to non-Gaussian interference and impulsive noise. In this context, the communication channel can be better modeled by heavy-tailed distributions, such as the non-Gaussian alpha-stable one. Some applications of the cyclostationary approach based on the spatial sign cyclic correlation function (SSCCF), fractional lower-order cyclic autocorrelation function (FLOCAF), and cyclic correntropy function (CCF) demonstrate that these are promising solutions for the analysis of signals in the presence of impulsive non-Gaussian noise. However, the investigation of functions above applied to digital modulation recognition in impulsive environments, and the comparison among them are topics that did not adequately explore yet. This work demonstrates that SSCCF is a particular case of the FLOCAF. Besides, a detailed analysis of the use of the FLOCAF and CCF is presented to obtain cyclostationary descriptors for the recognition of digital modulations BPSK, QPSK, 8-QAM, 16-QAM, and 32-QAM. Automatic modulation classification (AMC) architectures, based on the functions mentioned above, are also proposed. Besides, another contribution showed is that both the FLOCAF and CCF allow the symbol rate parameter estimation. The performances of AMC architectures were evaluated in the scenario with modulated signals contaminated with additive non-Gaussian alpha-stable noise. The results demonstrate that both architectures can classify signals in different contamination scenarios. However, the architecture based on the CCF is more efficient than the FLOCAF-based one.
... Correntropy is a similarity measure that is capable of extracting high-order statistical information from data [15]. As a nonlinear similarity measure, correntropy has been successfully used as an efficient optimization cost function in signal processing and machine learning, being specially robust in the presence of non-Gaussian noise and successful in different applications such as cognitive radio [16]- [19], adaptive filtering [20]- [22], principal component analysis (PCA) [23], deep learning [24], [25], and state estimation [26]. ...
... In fact, accuracy is one of the strengths of NESTA [11], which is the method that achieves the highest SER level in the noiseless environment, as can be seen in Table 1. Because 0 -LMS and 0 -MCCC use the same strategy to approximate the 0 gradient, they achieve similar SER levels that are numerically limited by the relation β 2 w ± β that composes (16). ...
Article
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Correntropy is a similarity function capable of extracting high-order statistical information from data. It has been used in different kinds of applications as a cost function to overcome traditional methods in non-Gaussian noise environments. One of the recent applications of correntropy was in the theory of compressive sensing, which takes advantage of sparsity in a transformed domain to reconstruct the signal from a few measurements. Recently, an algorithm called ℓ0–MCC was introduced. It applies the Maximum Correntropy Criterion (MCC) in order to deal with a non-Gaussian noise environment in a compressive sensing problem. However, because correntropy was only defined for real-valued data, it was not possible to apply the ℓ0–MCC algorithm in a straightforward way to compressive sensing problems dealing with complex-valued measurements. This paper presents a generalization of the ℓ0–MCC algorithm to complex-valued measurements. Simulations show that the proposed algorithm can outperform traditional minimization algorithms such as Nesterov’s algorithm (NESTA) and the ℓ0–Least Mean Square (ℓ0–LMS) in the presence of non-Gaussian noise.
... It can be understood as a generalization of the correlation concept [9]. Several works have proposed the use of correntropy in adaptive system training, thus achieving excellent performance in practical applications where the errors are typically nonGaussian [10], [11], [12], [13], [14]. ...
... Recent studies have demonstrated that correntropy is an efficient tool for analyzing higher-order statistical moments in machine learning and sig-nal processing applications (Liu et al., 2007;Fontes et al., 2014Fontes et al., , 2015Fontes et al., , 2017. In particular, the maximum correntropy criterion (MCC) has been successfully used in filtering, robust regression, and signal processing applications. ...
Article
The use of correntropy as a similarity measure has been increasing in different scenarios due to the well-known ability to extract high-order statistic information from data. Recently, a new similarity measure between complex random variables was defined and called complex correntropy. Based on a Gaussian kernel, it extends the benefits of correntropy to complex-valued data. However, its properties have not yet been formalized. This paper studies the properties of this new similarity measure and extends this definition to positive-definite kernels. Complex correntropy is applied to a channel equalization problem as good results are achieved when compared with other algorithms such as the complex least mean square (CLMS), complex recursive least squares (CRLS), and least absolute deviation (LAD).
... Consider a 0-mean signal ( ) x t . The time-varying correlation function is shown as [8]: ...
Conference Paper
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Planetary gearbox is a widely used transmission machine with complex and dynamic internal mechanical structures and it usually operates under non-stationary hostile operational conditions. The complex internal gears dynamic behaviors and the resultant vibrations make the fault diagnosis of planetary gears problematic. Effective diagnostic methods for planetary gear system are in demand. In this paper, an order analysis based second-order cyclic function method is proposed to extract the damage caused amplitude modulation under non-stationary operational conditions. The damage caused fault information is emphasized and being extracted and calculated as a distinctive indicator for fault diagnosis of planetary gear system. The experimental studies with 3 types of typical sun gear faults are demonstrated to validate the effectiveness of the proposed method.
... orrentropy is an effective tool for the similarity measure [1] between two random variables, especially in the case when the noise is non-Gaussian. It has been widely used in many machine learning and signal processing scenarios [2][3][4]. Compared with the mean square error (MSE) criterion based algorithms, such as least mean square (LMS) [5] and its variants [6], correntropy has shown its superiority in adaptive filtering. Generally, correntropy employs a Gaussian function as kernel function [1,7] since it is smooth and strictly positive definite. ...
Article
Adaptive filtering for complex data has received more attentions recently. As a similarity measure for the complex random variables, complex correntropy has been shown robustness in the design of adaptive filter. However, existing works using complex correntropy are limited to a Gaussian kernel function, which is not always the optimal choice. In this paper, we propose a class of new adaptive filtering algorithm for complex data using complex correntropy, which employs the complex generalized Gaussian density (CGGD) function as kernel function. Stability analysis provides the bound for learning rate and the steady-state excess mean square error (EMSE) is derived for theoretical analysis. Simulation results show that the proposed algorithm has zero probability of divergence (POD) and verify its superiority.
... The concept and framework of cyclic correntropy were first proposed in 2016 [35]. The pseudocode of computing cyclic correntropy spectrum and related simulations are given in 2017 [36]. Similar to the fusion model of FLOCC, cyclic correntropy is a combination of cyclostationarity and correntropy that is used to address co-channel interference and impulsive noise concurrently. ...
Article
Over the past several decades, cyclostationarity has been regarded as one of the most significant theories in the research of non-stationary signal processing; therefore, it has been widely used to solve a large variety of scientific problems, such as weak signal detection, parameter estimation, pattern recognition, and mechanical signature analysis. Despite offering a feasible solution, cyclostationarity-based methods suffer from performance degradation in the presence of impulsive noise, so the methods are less adaptable and practicable. To improve the effectiveness of these algorithms, a nonlinear similarity measurement, referred to as cyclic correntropy or cyclostationary correntropy, was recently proposed that innovatively combines the cyclostationarity technology and the concept of correntropy and successfully changes the signal analysis from the finite dimensional space (Euclidean space) to the infinite dimensional space (Hilbert space). However, to date, the study of cyclic correntropy has been limited, and it needs to be explored further. In this paper, the foundations and theories of cyclic correntropy are elucidated rigorously to complete and develop the methodology, including basic definitions, statistical formalisms, mathematical derivations, convergence theorem, spectrum analysis and kernel length estimation. It is believed that cyclic correntropy, a novel methodology equipped with the precise framework of cyclostationarity, can address the problem of impulsive noise in mechanical and communication signals and that its algorithmic idea of crossing spaces will have a far-reaching impact on the development of signal processing.
... Correntropy can be applied as a cost function for system identification with the advantage that it is a local criterion of similarity. Correntropy has been used in several applications including system identification problems (Liu and Chen 2013;Linhares et al. 2015;Guimaraes et al. 2016;Peng et al. 2017;Kulikova 2017;Fontes et al. 2015Fontes et al. , 2017, with good performance in non-Gaussian noise environments. ...
Article
In past years, the system identification area has emphasized the identification of nonlinear dynamic systems. In this field, polynomial nonlinear autoregressive with exogenous (NARX) models are widely used due to flexibility and prominent representation capabilities. However, the traditional identification algorithms used for model selection and parameter estimation with NARX models have some limitation in the presence of non-Gaussian noise, since they are based on second-order statistics that tightly depend on the assumption of Gaussianity. In order to solve this dependence, a novel identification method called simulation correntropy maximization with pruning (SCMP) based on information theoretic learning is introduced by this paper. Results obtained in non-Gaussian noise environment in three experiments (numerical, benchmark data set and measured data from a real plant) are presented to validate the performance of the proposed approach when compared to other similar algorithms previously reported in the literature, e.g., forward regression with orthogonal least squares and simulation error minimization with pruning. The proposed SCMP method has shown increased accuracy and robustness for three different experiments.
... The correntropy with suitable kernel function is an efficient method for exploiting higher-order statistics in impulsive noise environment [24][25][26][27]. The correntropy based correlation (CRCO) [28] treats the maximum correntropy criterion (MCC) as a weighting factor to construct the covariance matrix and retrieves DOA with MUSIC algorithm. ...
Article
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Most existing methods for direction-of-arrival (DOA) estimation are severely influenced by impulsive noise due to their Gaussian noise assumption. As a typical non-linear similarity measure, the maximum correntropy criterion (MCC) has been considered to restrain impulsive noise, because of its ability to exploit the high-order statistics of signal. However, Gaussian kernel-based MCC method is not always the optimal choice and is only suitable for the real-valued signal, which certainly limits its applications. To solve the aforementioned problems, in this study, the authors proposed a novel generalised maximum complex correntropy criterion (GMCCC)-based complex-valued quasi-Newton method to restrain impulsive noise. GMCCC adopts the generalised complex Gaussian density function as the kernel function with more flexible parameters. Besides, it can extend the benefit to the complex-valued signal. Furthermore, its properties are formalised. The complex-valued quasi-Newton method guarantees the positive definite Hessian matrix to achieve the alternate minimisation of signal subspace and signal matrix. GMCCC achieves the accurate DOA estimation from the received data which does not require the covariance matrix. Stability performance and convergence are analysed. Experiment results show that the proposed GMCCC algorithm possesses the robustness and outperforms the state-of-the-art algorithms.
... Correntropy is a kernel-based similarity measure capable of extracting infinite even-order statistical moments from data, being a generalization of the correlation concept [16]. Another important characteristic of this measure is that it provides improved performance when compared with second-order methods when dealing with non-Gaussian noise such as impulsive noise environments [17]- [26]. Owing to such characteristics, correntropy has been successfully applied to many practical problems, e.g., extraction of higher-order temporal characteristics [27], [28], estimation of impulsive noise [29] and active noise control [30], [31]- [33], Kalman filter [19], [34], compressive sensing problems in impulsive noise environments [35], [36]. ...
Article
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Circular statistics has been applied to several areas of knowledge in which the input data is circular or directional. Noisy measurements are still a problem in circular data applications and, like non-circular data, second-order statistics have some limitations to deal with non-Gaussian noise. Recently, a similarity function called correntropy has been successfully employed in applications involving impulsive noise for being capable of extracting more information than second-order methods. However, correntropy has not been studied from the perspective of circular data so far. This paper defines a novel statistical measure called circular correntropy (CC). It uses the von Mises density function in order to redefine correntropy in this domain. In particular, it is proved analytically that the CC contains information regarding second-order and higher-order moments, being a generalization of the circular correlation measure. The performance of this novel similarity measure is evaluated as a cost function in a nonlinear regression problem, where the signals are contaminated with additive impulsive noise. The simulations demonstrate that the CC is more robust than circular correlation in impulsive noise environments.
... The kernelized correlation is proved to be an efficient tool in feature extraction. Algorithms evolved from the kernelized correlation, such as correntropy (Liu et al. 2007) and cyclic correntropy (Fontes et al. 2017) are thriving in recent years. More importantly, kenelized correlation methods are also more effective when signals are contaminated by strong background noise or outliers, which is a common phenomenon in the early bearing fault diagnosis. ...
Article
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The incipient bearing fault diagnosis is crucial to the industrial machinery maintenance. Developed based on the blind source separation, blind source extraction (BSE) has recently become the focus of intensive research work. However, owing to certain industrial restrictions, the number of sensors is usually less than that of the source signals, which is defined as an underdetermined BSE problem to identify the fault signals. The kernelized methods are found to be robust to the noise, especially in the presence of outliers, which makes it a suitable tool to extract fault signatures submerged in the strong environment noise. Thus, this paper proposes a new underdetermined BSE method based on the empirical mean decomposition and kernelized correlation. The experimental results indicate that the extracted fault signature presents more obvious periodicity. Two important parameters of this method, including the multi-shift number and the kernel size are investigated to improve the algorithm performance. Furthermore, performance comparisons with underdetermined BSE based on the second order correlation are made to emphasize the advantage of the presented method. The application of the proposed method is validated using the simulated signal and the rolling element bearing signal of the train vehicle axle.
... In the literature, the special attention is paid to the fractional lower-order correlation analysis, where the fractional moments and fractional dependency measures are examined as the alternative statistics to the classical auto-correlation function [11][12][13][14][15]. The other interesting approach is based on the entropy in time and spectral domain [16][17][18]. This approach was successfully used in the impulsive signals analysis and also in the technical diagnostics [19][20][21][22], see also [23][24][25]. ...
Article
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The problem of the informative frequency band (IFB) selection for local fault detection is considered in the paper. There are various approaches that are very effective in this issue. Most of the techniques are vibration-based and they are related to the cyclic impulses detection (associated with the local fault) in the background noise. However, when the background noise in the vibration signal has non-Gaussian impulsive behavior, the classical methods seem to be insufficient. Recently, new techniques were proposed by several authors and interesting approaches were tested for different non-Gaussian signals. We demonstrate the comparative analysis related to the results for three selected techniques proposed in recent years, namely the Alpha selector, Conditional Variance-based selector, and Spearman selector. The techniques seem to be effective for the IFB selection for the non-Gaussian distributed vibration signals. The main purpose of this article is to investigate how spectral overlapping of informative and non-informative impulsive components will affect diagnostic procedures. According to our knowledge, this problem was not considered in the literature for the non-Gaussian signals. Nevertheless, as we demonstrated by the simulations, the level of overlapping and the location of a center frequency of the mentioned frequency bands have a significant influence on the behavior of the considered selectors. The discussion about the effectiveness of each analyzed method is conducted. The considered problem is supported by real-world examples.
... Correntropy is a similarity measure based on Rényi entropy capable of extracting highorder statistical information from data, thus generalizing the correlation concept [1]. In the past few years, this concept been successfully applied in the solution of various engineering problems, such as the kernel Kalman filtering [2], adaptive time delay estimation [3], automatic modulation classification [4,5], nonlinear auto-regressive with exogenous (NARX) for identification system [6], compressive sensing problem in an impulsive noise environment [7], signal detector in MIMO systems [8], and fuzzy neural system [9]. Training adaptive systems with cost functions such as the error correntropy criterion requires the selection of a proper kernel width. ...
Article
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Recently, the maximum correntropy criterion (MCC) has been successfully applied in numerous applications regarding nonGaussian data processing. MCC employs a free parameter called kernel width, which affects the convergence rate, robustness, and steady-state performance of the adaptive filtering. However, determining the optimal value for such parameter is not always a trivial task. Within this context, this paper proposes a novel method called adaptive convex combination maximum correntropy criterion (ACCMCC), which combines an adaptive kernel algorithm with convex combination techniques. ACCMCC takes advantage from a convex combination of two adaptive MCC-based filters, whose kernel widths are adjusted iteratively as a function of the minimum error value obtained in a predefined estimation window. Results obtained in impulsive noise environment have shown that the proposed approach achieves equivalent convergence rates but with increased accuracy and robustness when compared with other similar algorithms reported in literature.
... It can be understood as a generalization of the correlation concept [9]. Several works have proposed the use of correntropy in adaptive system training, thus achieving excellent performance in practical applications where the errors are typically nonGaussian [10], [11], [12], [13], [14]. ...
Article
Recent studies have demonstrated that correntropy is an efficient tool for analyzing higher-order statistical moments in nonGaussian noise environments. Although correntropy has been used with complex data, no theoretical study was pursued to elucidate its properties, nor how to best use it for optimization. By using a probabilistic interpretation, this work presents a novel similarity measure between two complex random variables, which is defined as complex correntropy. A new recursive solution for the maximum complex correntropy criterion (MCCC) is introduced based on a fixed-point (FP) solution. This technique is applied to a system identification and the results demonstrate prominent advantages when compared against three other algorithms: the complex least mean square (CLMS), complex recursive least squares (RLS) and least absolute deviation (LAD). By the aforementioned probabilistic interpretation, correntropy can now be applied to solve several problems involving complex data in a more straightforward way.
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In array signal processing, many methods of handling cases of impulsive noise with an alpha-stable distribution have been studied. By introducing correntropy with a robust statistical property, this paper proposes a novel fractional lower order correntropy (FLOCR) method. The FLOCR-based estimator for array outputs is defined and applied with multiple signal classification (MUSIC) to estimate the direction of arrival (DOA) in alpha-stable distributed noise environments. Comprehensive Monte Carlo simulation results demonstrate that FLOCR-MUSIC outperforms existing algorithms in terms of root mean square error (RMSE) and the probability of resolution, especially in the presence of highly impulsive noise.
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The Internet of Things (IoT) pervades every aspect of our daily lives and industrial productions since billions of interconnected devices are deployed everywhere of the globe. However, the seamless IoT unveils a number of physical-layer threats such as jamming and spoofing that decrease the communication performance and the reliability of the IoT systems. As the process of identifying the modulation format of signals corrupted by noise and fading, automatic modulation classification (AMC) plays a vital role in physical layer security as it can detect and identify the pilot jamming, deceptive jamming and sybil attacks. In this paper, we propose a novel cyclic correntropy vector (CCV) based AMC method using long short-term memory densely connected network (LSMD). Specifically, cyclic correntropy model-driven feature CCV is firstly extracted using the received signals as it contains both the second-order and the higher-order characteristics of cyclostationary. Then, the extracted CCV feature is put into the data-driven LSMD which mainly consists of long short-term memory (LSTM) network and dense network (DenseNet). Moreover, an additive cosine loss is utilized to train the LSMD for maximizing the inter-class feature differences and minimizing the intra-class feature variations. Simulations demonstrate that the proposed CCV-LSMD method yields superior performance than other recent schemes.
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Non-stationary, non-Gaussian signal processing is a challenging topic in signal processing research. Over the past decade, due to effectively addressing co-channel interference, cyclostationarity-based methodologies have found a wide range of applications, such as wireless communication, cognitive radio, and mechanical vibration monitoring. Despite offering a feasible scheme, the second and higher-order cyclostationarity-based methodologies suffer under non-Gaussian noise environments, particularly impulsive noise environments. In this paper, through studying the similarity measurement, nonlinear function, and mapping mode, we propose a novel methodology named hyperbolic-tangent-function-based cyclic correlation (HTCC) to address both Gaussian and non-Gaussian noises with a uniform expression. The idea is inspired by the fact that hyperbolic tangent function is not only a bounded function but also achieves a differential compression. In addition, the theoretical foundations of this novel method are introduced step by step, including the definition, property, and spectrum. A number of numerical experiments are carried out to compare the algorithm performance with existing competitive methods. The proposed method generally shows good effectiveness and robustness and can be utilized for denoising problems in signal processing.
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Non-Gaussianity of noises and non-stationarity of signals have been the two crucial considerations in the fields of signal processing and communications. Correspondingly, many denoising and cyclostationary methods have been published to deal with the relevant problems, respectively. Recently, a novel method named cyclic correntropy or cyclostationary correntropy was proposed to deal with the two problems simultaneously. Thanks to the symmetry and sparsity of cyclic correntropy spectrum, the compressed spectrum obtained by compressive sensing can be used to complete the task in certain situations. In this paper, a novel method to estimate the cyclic frequency by compressed cyclic correntropy spectrum is proposed to reduce computational complexity and storage cost. To reveal the proposed method's effectiveness and robustness to impulsive noise, a number of numerical experiments are carried out to compare with existing cyclostationary methods. As cyclostationary signal processing and compressive sensing are the two great theories, through in-depth study, they can have more collaborative work opportunities and meanings.
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Automatic modulation classification (AMC) plays an important role in many military and civilian communication applications. However, it remains a challenging task to support such AMC mechanisms under impulsive noise environments. Aiming at improving the classification performance in impulsive noise, in this paper, a novel modulation classification method is proposed by using the cyclic correntropy spectrum (CCES). In the proposed method, CCES is introduced into AMC for effectively suppressing impulsive noise. Specifically, it is verified that modulation types can be distinguished through CCES. Then, Multi-slices are extracted at different cycle-frequencies from CCES as the original features for AMC. Following the extraction, the principal component analysis (PCA) is applied to these slices to further optimize the original features. Finally, the radial basis function (RBF) neural network is used as a classifier to perform modulation classification. Monte Carlo simulations demonstrate that the proposed algorithm outperforms other existing schemes in impulsive noise cases, especially with a low generalized signal to noise ratio (GSNR).
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This paper addresses the issue of time difference of arrival (TDOA) estimation of cyclostationary signal under impulsive noise environments modeled by α-stable distribution. Since α-stable distribution has not finite second-order statistics, the conventional cyclic correlation based signal-selective TDOA estimation algorithm does not work effectively. To resolve this problem, we define the generalized cyclic correntropy (GCCE) which is a robust cyclic correlation and can be reviewed as an extension of the generalized correntropy for cyclostationary signal. A robust signal-selective TDOA estimation algorithm based on GCCE is proposed. The computer simulation results demonstrate that the proposed algorithm outperforms the conventional cyclic correlation and the fractional lower order cyclic correlation based algorithms.
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Over the past decades, cyclostationary signal processing has been considered one of the most critical theories in the field of nonstationary signal processing. Moreover, it has been widely adopted to handle practical problems, including signal parameter estimation, weak signal detection and pattern recognition. To improve the robustness of conventional methods under non-Gaussian noise, cyclic correntropy was recently proposed as the extension of correntropy in the cycle frequency domain. Due to the advantages inherited from both cyclostationary statistics and correntropy, cyclic correntropy has been employed in various applications involving co-channel interference and impulsive noise, such as carrier frequency estimation, direction of arrival (DOA) estimation, time difference of arrival (TDOA) estimation, automatic modulation classification and bearing fault diagnosis. Although cyclic correntropy has gained much attention, the related theoretical study is still insufficient, prohibiting further applications in other areas. To complete the theoretical framework, the existence condition of cyclic correntropy is first discussed in detail. Then, the relations among the cyclic correntropy spectrum, the symbol rate and the carrier frequency are addressed and proved mathematically. Besides, other properties of cyclic correntropy are also studied, involving symmetry, conjugation and time-shifting. These properties are studied for the first time and they fulfill the void of cyclic correntropy theory. Furthermore, a novel method is proposed for symbol rate estimation under non-Gaussian noise. Simulations are executed to validate the proposed method's superior robustness compared with other methods based on different cyclic spectra under the framework of cyclostationary signal processing. This method contributes to the application of cyclic correntropy, further highlighting the significance of the properties mentioned above.
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Automatic Modulation Classification (AMC) is a essential component in Cognitive Radio (CR) for recognizing the modulation scheme. Many modulated signals manifest the property of cyclostationarity as a feature so it can be exploited for classification. In this paper, we study the performance of digital modulation classification technique based on the cyclostationary features and different classifiers such as Neural Network, Support Vector Machine, k-Nearest Neighbor, Naive Bayes, Linear Discriminant Analysis and Neuro-Fuzzy classifier. In this study we considered modulations i.e. BPSK, QPSK, FSK and MSK for classification. All classification methods studied using performance matrix including classification accuracy and computational complexity (time). The robustness of these methods are studied with SNR ranging from 0 to 20dB. Based upon the result we found that combining cyclostationary features with Naive Bayes and Linear Discriminant Analysis classifiers leads to provide better classification accuracy with less computational complexity.
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Since the second-order statistics based methods rely heavily on Gaussianity assumption and the fractional lower-order statistics based methods depend on a priori knowledge of non-Gaussian noise, there remains a void in stochastic signal processing. In this paper, a novel signal analysis method referred to as cyclic correntropy is proposed to deal with cyclostationary signals under impulsive noise environment based on kernel methods. Furthermore, the cyclic correntropy spectrum is also defined. The application in frequency estimation is presented to illustrate the advantages of the cyclic correntropy over the second-order and the fractional lower-order cyclic statistics based methods in the presence of α-stable impulsive noise.
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In this paper, a novel algorithm for fractional time delay estimation is proposed. It is based on the maximum correntropy criterion and the Lagrange fractional delay filter (FDF). The instantaneous correntropy is introduced to measure the similarity between received signals and estimated ones in the proposed algorithm. It leads to an effective performance under both Gaussian and impulsive noises. The performances, including the convergence of the algorithm and the variance of the time delay estimation, are theoretically analyzed. Simulations demonstrate that the time delay estimation precision of the proposed algorithm is higher than that utilizing the mixed modulated Lagrange explicit time delay estimation (MMLETDE) especially under low generalized signal-to-noise ratio (GSNR).
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We propose a blind spectrum sensing method using signal cyclostationarity. Often, signal characteristics of the primary user (PU), such as carrier frequency, data rate, modulation and coding may not be known to cognitive users. This uncertainty introduces difficulties in searching for spectrum holes in cognitive radios. At a low signal-to-noise ratio, it has been understood that monitoring the presence of the PU's signal is hardly possible without knowing its cycle frequencies. The proposed sensing method makes it possible to detect the PU's signal without the relevant information of the signal attributes. Blind spectrum sensing is also much simpler than the conventional cyclostationary spectral correlation detection.
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Correntropy, as an adaptive criterion of Information Theoretic Learning (ITL), has been successfully used in signal processing and machine learning. How to appropriately select the kernel width of correntropy is a crucial problem in correntropy applications. Existing kernel width selection methods are not suitable enough for this problem. In this paper, we develop an adaptive method for kernel width selection in correntropy. Based on the Middleton's non-Gaussian models, this method utilizes the kurtosis as a ratio to adjust the standard deviation of the prediction error to obtain the kernel width online. The superior performance of the new method has been demonstrated by simulation examples in the noisy frequency doubling and echo cancelation problems.
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This paper addresses the problem of semi-blindly extracting one single desired signal using a priori information about its higher order temporal structure. Our approach is based on the maximization of the autocorrentropy function for a given time delay. The a priori information is quantified as a time delay and a size for a Gaussian kernel to set the free parameters in the correntropy function. Those values provide information which allows the proposed method to adapt a demixing vector to extract the desired signal without the indeterminacy of the permutation problem in blind source separation. Moreover, this method is different from those for Independent Component Analysis that separate all the available sources, which, in some problems, is not desirable or computationally possible. Since correntropy can be interpreted as a generalization of correlation, we demonstrate that it is a suitable measure for studying the temporal behavior of higher order statistics of a signal. Also, the flexibility brought by the kernel size selection allows the user to choose the range of statistics he is interested in. We show in simulations that correntropy achieve better or equal separation than other linear methods proposed in the literature for source extraction based on temporal structures.
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A 50-year time series of monthly stratospheric ozone readings from Arosa, Switzerland, is analyzed. The time series exhibits the properties of a periodically correlated (PC) random sequence with annual periodicities. Spectral properties of PC random sequences are reviewed and a test to detect periodic correlation is presented. An autoregressive moving-average (ARMA) model with periodically varying coefficients (PARMA) is fitted to the data in two stages. First, a periodic autoregressive model is fitted to the data. This fit yields residuals that are stationary but non-white. Next, a stationary ARMA model is fitted to the residuals and the two models are combined to produce a larger model for the data. The combined model is shown to be a PARMA model and yields residuals that have the correlation properties of white noise.
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Cross-validation with Kullback-Leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation. A framework for this problem is constructed and used to derive an alternative method of cross-validation, based on integrated squared error, recently also proposed by Rudemo (1982). Hall (1983) has established the consistency and asymptotic optimality of the new method. For small and moderate sized samples, the performances of the two methods of cross-validation are compared on simulated data and specific examples.
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This paper proposes a novel pitch determination algorithm (PDA) based on the newly introduced concept of a generalized correlation function called correntropy. Correntropy is a positive definite kernel function which implicitly transforms the original signal into a high-dimensional reproducing kernel Hilbert space (RKHS) in a nonlinear way, and calculates very efficiently the generalized correlation in that RKHS. By incorporating the kernel function, correntropy is able to utilize higher order statistics to enhance the resolution of pitch estimation. The proposed PDA computes the summary of correntropy functions from the outputs of an equivalent rectangular bandwidth (ERB) filter bank. We present simulations on pitch determination for a single vowel, double vowels, and a benchmark database test. Simulations show that correntropy exhibits much better resolution than conventional autocorrelation in pitch determination and outperforms other PDAs in the benchmark database test.
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Nonlinearity tests have become an essential step in system analysis and modeling due to the computational demands and complexity of analysis involved in nonlinear modeling. Standard nonlinear measures are either too complicated to estimate accurately (such as Lyapunov exponents and correlation dimension), or not able to capture sufficient but not necessary conditions of nonlinearity (such as time asymmetry). Correntropy is a kernel-based similarity measure which contains the information of both statistical and temporal structure of the underlying dataset. The capability of preserving nonlinear characteristics makes correntropy a suitable candidate as a measure for determining nonlinear dynamics. Moreover, since correntropy makes use of kernel methods, its estimation is computationally efficient. Using correntropy as the test statistic, nonlinearity tests based on the null hypothesis that signals of interest are realizations of linear Gaussian stochastic processes are carried out via surrogate data methods. Experiments performed on linear Gaussian, linear non-Gaussian, and nonlinear systems with varying in-band noise levels, data lengths, and kernel sizes confirm that correntropy can be employed as a discriminative measure for detecting nonlinear characteristics in time series. Results of tests performed on data collected from natural systems are in agreement with findings in time series analysis literature.
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This paper presents a comprehensive characterization of cyclostationary feature detectors through theoretical analysis, hardware implementation, and real-time performance measurements. Results of our study show that feature detectors are highly susceptible to sampling clock offsets. We propose a new detector that overcomes this limitation, and characterize its performance through experiments. In addition, the comparison with a conventional energy detector shows that feature detectors are more robust to adjacent channel interference.
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In modern car engines the angle of ignition has to be controlled adaptively to achieve high efficiency of the engine. This control process is usually based on evaluation of cylinder pressure or structural vibration signals. Due to the nature of the combustion process these signals are highly nonstationary. Therefore signal processing methods well suited to this special kind of signal has to be used. Based on real data experiments we show that a cyclostationary model is appropriate for constant rotation speed of the engine. Use of this model enables us to get a consistent estimate of the Wigner-Ville spectrum of the signals. This time-frequency representation has very good resolution and can give us information about the quality of sensor positions or about important parameters that can be used for adaptive control of the engine
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The presence of kth-order cyclostationarity is defined in terms of nonvanishing cyclic-cumulants or polyspectra. Relying upon the asymptotic normality and consistency of kth-order cyclic statistics, asymptotically optimal χ<sup>2</sup> tests are developed to detect the presence of cycles in the kth-order cyclic cumulants or polyspectra, without assuming any specific distribution on the data. Constant false alarm rate tests are derived in both time- and frequency-domain and yield consistent estimates of possible cycles present in the kth-order cyclic statistics. Explicit algorithms for k&les;4 are discussed. Existing approaches are rather empirical and deal only with k&les;2 case. Simulation results are presented to confirm the performance of the given tests
The periodic correlation-random field as a model for bidimensional ocean waves
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  • I N Yavorskii
Dragan, Y. P., & Yavorskii, I. N. (1982). The periodic correlation-random field as a model for bidimensional ocean waves. Otbor Peredacha Inform, 51, 15-21.
Implementation of cyclic periodogram detection on vee for cognitive radio. Cognitive Radio Models for Agilent VEE
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Zhang, Z. (2007). Implementation of cyclic periodogram detection on vee for cognitive radio. Cognitive Radio Models for Agilent VEE.