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Anomaly detection in broadband networks: Using normalizing flows for multivariate time series
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-01-02 DOI: 10.1016/j.sigpro.2024.109874
Tobias Engelhardt Rasmussen, Facundo Esteban Castellá Algán, Andreas Baum
Hybrid Fiber-Coaxial (HFC) networks are a popular infrastructure for delivering internet to consumers, however, they are complex and susceptible to various errors. Internet service providers currently rely on manual operations for network monitoring, underscoring the need for automated fault detection. We propose a novel framework for estimating the density of multivariate time series, tailored for anomaly detection in broadband networks. Our framework comprises two phases. In the first phase, we employ an autoencoder based on one-dimensional convolutions to learn a latent representation of time series windows, thereby preserving context. In the second phase, we utilize a Normalizing Flow (NF) to model the distribution within this latent space, enabling subsequent anomaly detection. For efficient separation, we propose an iterative weighing algorithm allowing the NF to model only the systematic behavior, thereby separating outlying behavior. We validated our methodology using a publically available synthetic dataset and real-world data from TDC NET, Denmark’s leading provider of digital infrastructure. Initial experiments with the synthetic dataset demonstrated that our density-based estimator effectively distinguishes anomalies from normal behavior. When applied to the unlabeled TDC NET dataset, our framework exhibits promising performance, identifying outliers clustering themselves away from the high-density region, thus enabling subsequent root cause analysis.
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引用次数: 0
Analysis of interference suppression performance in MR-FDA-MIMO radar
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-01-01 DOI: 10.1016/j.sigpro.2024.109873
Zhixia Wu , Shengqi Zhu , Jingwei Xu , Lan Lan , Ximin Li , Zijing Zhang
In this paper, we primarily analyze three situations affecting mainlobe interference suppression performance in minimum redundancy frequency diverse array multiple-input multiple-output (MR-FDA-MIMO) radar. The MR-FDA-MIMO radar breaks through the degrees of freedom (DOF) in the transmit dimension, the number of mainlobe interference suppression beyond the number of transmit elements. Utilizing a two-step beamforming technique, MR-FDA-MIMO mitigates sidelobe interference in the receive domain and counteract multiple false targets in transmit virtual domain. This paper provides a detailed analysis of the mainlobe interference suppression performance of MR-FDA-MIMO radar in the transmit virtual domain. First, the optimal output signal-to-interference-plus-noise ratio (SINR) based on the virtual array is derived, ultimately resulting in SINRout=SNR+10log(2MM+1). Second, considering the presence of a true target in the virtual samples, the virtual covariance matrix is derived, showing that the power of the true target doubles in the virtual covariance matrix. Finally, the properties of the cross terms in the virtual covariance matrix for the multiple virtual samples algorithm are analyzed, and it was found that they are located between the false targets, and their number is related to the number of false targets Q, specifically, the number of cross terms is Q(Q-1)/2. The effectiveness of the analysis is verified through simulation examples.
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引用次数: 0
Image denoising based on fractional anisotropic diffusion and spatial central schemes
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-31 DOI: 10.1016/j.sigpro.2024.109869
Milorad P. Paskaš
Fractional-order anisotropic diffusion realized in the Fourier domain is a widely used model for image denoising. While fractional differentiation in the Fourier domain introduces a complex component, differentiation with central schemes in the spatial domain is preferred in image processing applications. This paper presents numerical solution to the fractional anisotropic diffusion equation in the spatial domain, using novel central fractional difference schemes. The proposed central schemes assume a two-part differentiation approach: an integer order, defined by the integer part of the order of differentiation, and a non-integer order, defined by the non-integer part. This approach allows the proposed schemes to incorporate integer-order calculus. The conducted stability analysis of the numerical schemes yields optimistic results regarding convergence conditions, demonstrating that the schemes are unconditionally stable for orders of differentiation greater than 0.5. The parameters of the proposed model are adjusted through a set of experiments that illustrate its performance. The proposed model is tested against counterpart models from the literature using an image dataset, and the obtained qualitative and quantitative results favor the proposed model across various noise levels.
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引用次数: 0
Visible light positioning with intelligent reflecting surfaces under mismatched orientations
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-30 DOI: 10.1016/j.sigpro.2024.109867
Issifu Iddrisu, Sinan Gezici
Accurate localization can be performed in visible light systems in non-line-of-sight (NLOS) scenarios by utilizing intelligent reflecting surfaces (IRSs), which are commonly in the form of mirror arrays with adjustable orientations. When signals transmitted from light emitting diodes (LEDs) are reflected from IRSs and collected by a receiver, the position of the receiver can be estimated based on power measurements by utilizing the known parameters of the LEDs and IRSs. Since the orientation vectors of IRS elements (mirrors) cannot be adjusted perfectly in practice, it is important to evaluate the effects of mismatches between desired and true orientations of IRS elements. In this study, we derive the misspecified Cramér–Rao lower bound (MCRB) and the mismatched maximum likelihood (MML) estimator for specifying the estimation performance and the lower bound in the presence of mismatches in IRS orientations. We also provide comparisons with the conventional maximum likelihood (ML) estimator and the CRB in absence of orientation mismatches for quantifying the effects of mismatches. It is shown that orientation mismatches can result in significant degradation in localization accuracy at high signal-to-noise ratios.
{"title":"Visible light positioning with intelligent reflecting surfaces under mismatched orientations","authors":"Issifu Iddrisu,&nbsp;Sinan Gezici","doi":"10.1016/j.sigpro.2024.109867","DOIUrl":"10.1016/j.sigpro.2024.109867","url":null,"abstract":"<div><div>Accurate localization can be performed in visible light systems in non-line-of-sight (NLOS) scenarios by utilizing intelligent reflecting surfaces (IRSs), which are commonly in the form of mirror arrays with adjustable orientations. When signals transmitted from light emitting diodes (LEDs) are reflected from IRSs and collected by a receiver, the position of the receiver can be estimated based on power measurements by utilizing the known parameters of the LEDs and IRSs. Since the orientation vectors of IRS elements (mirrors) cannot be adjusted perfectly in practice, it is important to evaluate the effects of mismatches between desired and true orientations of IRS elements. In this study, we derive the misspecified Cramér–Rao lower bound (MCRB) and the mismatched maximum likelihood (MML) estimator for specifying the estimation performance and the lower bound in the presence of mismatches in IRS orientations. We also provide comparisons with the conventional maximum likelihood (ML) estimator and the CRB in absence of orientation mismatches for quantifying the effects of mismatches. It is shown that orientation mismatches can result in significant degradation in localization accuracy at high signal-to-noise ratios.</div></div>","PeriodicalId":49523,"journal":{"name":"Signal Processing","volume":"230 ","pages":"Article 109867"},"PeriodicalIF":3.4,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143132713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quadratic filtering for linear stochastic non-Gaussian systems under false data injection attacks
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-28 DOI: 10.1016/j.sigpro.2024.109855
Zhijian Kuang, Shiyuan Wang, Yunfei Zheng, Yinhong Liao, Dongyuan Lin, Sanshan Liu, Shungang Peng
This paper addresses the quadratic filtering issue for a class of linear discrete-time systems in the presence of non-Gaussian noise under the false data injection attacks. The false data injection attacks are modeled with the simultaneously established additive and multiplicative false data. The majority of measurements are consistently inaccurate due to the presence of these two types of false data. To this end, a recursive quadratic filtering algorithm is proposed by constructing a quadratic system that combines the original system states with their second-order Kronecker powers. An upper bound for the filtering error covariance is derived recursively, and can be minimized by appropriately choosing the gain parameters. In addition, a sufficient condition is obtained to guarantee the mean-square boundedness of the upper bound. Finally, simulations are provided to validate the efficacy of the proposed quadratic filtering algorithm.
{"title":"Quadratic filtering for linear stochastic non-Gaussian systems under false data injection attacks","authors":"Zhijian Kuang,&nbsp;Shiyuan Wang,&nbsp;Yunfei Zheng,&nbsp;Yinhong Liao,&nbsp;Dongyuan Lin,&nbsp;Sanshan Liu,&nbsp;Shungang Peng","doi":"10.1016/j.sigpro.2024.109855","DOIUrl":"10.1016/j.sigpro.2024.109855","url":null,"abstract":"<div><div>This paper addresses the quadratic filtering issue for a class of linear discrete-time systems in the presence of non-Gaussian noise under the false data injection attacks. The false data injection attacks are modeled with the simultaneously established additive and multiplicative false data. The majority of measurements are consistently inaccurate due to the presence of these two types of false data. To this end, a recursive quadratic filtering algorithm is proposed by constructing a quadratic system that combines the original system states with their second-order Kronecker powers. An upper bound for the filtering error covariance is derived recursively, and can be minimized by appropriately choosing the gain parameters. In addition, a sufficient condition is obtained to guarantee the mean-square boundedness of the upper bound. Finally, simulations are provided to validate the efficacy of the proposed quadratic filtering algorithm.</div></div>","PeriodicalId":49523,"journal":{"name":"Signal Processing","volume":"230 ","pages":"Article 109855"},"PeriodicalIF":3.4,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143132715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Poisson2Poisson-Sparse: Unsupervised Poisson noise image denoising based on sparse modeling
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-28 DOI: 10.1016/j.sigpro.2024.109870
Lingzhi Xiao , Shengbiao Wang , Jun Zhang , Jiuzhe Wei , Shihua Yang
In fields such as low-light photography, astronomical imaging, and low-dose computed tomography scanning, Poisson noise severely degrades image quality due to extremely low photon counts (averaging below one) and their Poisson-distributed statistical characteristics. A recent self-supervised Poisson denoising method uses only a single noisy image to improve image quality. However, it struggles under high Poisson noise due to its denoising model based on Gaussian distribution and suffers from long inference times. To address these issues, we propose an unsupervised Poisson denoising method based on sparse representation. Specifically, we first establish a more accurate sparse representation model based on Poisson distribution to enhance denoising performance. Given the difficulty of solving this model directly, we develop an iterative optimization algorithm using convolutional sparse coding and the alternating direction method of multipliers. Inspired by the unfolding technique, we further reduce computational cost by unfolding the iterative process into a finite-cycle learning network. To overcome the reliance on paired datasets and accelerate inference times, we employ a Poisson loss function, a Neighbor2Neighbor training strategy, and incorporate total variation loss, which together enable unsupervised learning. Experimental results demonstrate that our proposed method significantly outperforms existing unsupervised Poisson denoising methods and achieves high computational efficiency.
{"title":"Poisson2Poisson-Sparse: Unsupervised Poisson noise image denoising based on sparse modeling","authors":"Lingzhi Xiao ,&nbsp;Shengbiao Wang ,&nbsp;Jun Zhang ,&nbsp;Jiuzhe Wei ,&nbsp;Shihua Yang","doi":"10.1016/j.sigpro.2024.109870","DOIUrl":"10.1016/j.sigpro.2024.109870","url":null,"abstract":"<div><div>In fields such as low-light photography, astronomical imaging, and low-dose computed tomography scanning, Poisson noise severely degrades image quality due to extremely low photon counts (averaging below one) and their Poisson-distributed statistical characteristics. A recent self-supervised Poisson denoising method uses only a single noisy image to improve image quality. However, it struggles under high Poisson noise due to its denoising model based on Gaussian distribution and suffers from long inference times. To address these issues, we propose an unsupervised Poisson denoising method based on sparse representation. Specifically, we first establish a more accurate sparse representation model based on Poisson distribution to enhance denoising performance. Given the difficulty of solving this model directly, we develop an iterative optimization algorithm using convolutional sparse coding and the alternating direction method of multipliers. Inspired by the unfolding technique, we further reduce computational cost by unfolding the iterative process into a finite-cycle learning network. To overcome the reliance on paired datasets and accelerate inference times, we employ a Poisson loss function, a Neighbor2Neighbor training strategy, and incorporate total variation loss, which together enable unsupervised learning. Experimental results demonstrate that our proposed method significantly outperforms existing unsupervised Poisson denoising methods and achieves high computational efficiency.</div></div>","PeriodicalId":49523,"journal":{"name":"Signal Processing","volume":"230 ","pages":"Article 109870"},"PeriodicalIF":3.4,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143132712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Kalman filter for dynamic source power and steering vector estimation based on empirical covariances
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-28 DOI: 10.1016/j.sigpro.2024.109868
Cyril Cano , Mohammed Nabil El Korso , Éric Chaumette , Pascal Larzabal
Interferometric measurements correspond to sample covariance matrices of signals received by multiple sensors. In dynamic scenarios, such as radio astronomy imaging, the properties of these signals can vary over time, posing a significant challenge for study. This work addresses the issue of estimating the stochastic power and steering vector of signal sources from sample covariance measurements. A novel approach is proposed, introducing a non-standard Kalman filter designed to accommodate any noise and signal distribution, thereby broadening the Kalman filter’s applicability to situations with unknown measurement models. The effectiveness of this method is highlighted in the case of joint estimation of source power and direction of arrival through simulations using synthetic data.
{"title":"Kalman filter for dynamic source power and steering vector estimation based on empirical covariances","authors":"Cyril Cano ,&nbsp;Mohammed Nabil El Korso ,&nbsp;Éric Chaumette ,&nbsp;Pascal Larzabal","doi":"10.1016/j.sigpro.2024.109868","DOIUrl":"10.1016/j.sigpro.2024.109868","url":null,"abstract":"<div><div>Interferometric measurements correspond to sample covariance matrices of signals received by multiple sensors. In dynamic scenarios, such as radio astronomy imaging, the properties of these signals can vary over time, posing a significant challenge for study. This work addresses the issue of estimating the stochastic power and steering vector of signal sources from sample covariance measurements. A novel approach is proposed, introducing a non-standard Kalman filter designed to accommodate any noise and signal distribution, thereby broadening the Kalman filter’s applicability to situations with unknown measurement models. The effectiveness of this method is highlighted in the case of joint estimation of source power and direction of arrival through simulations using synthetic data.</div></div>","PeriodicalId":49523,"journal":{"name":"Signal Processing","volume":"230 ","pages":"Article 109868"},"PeriodicalIF":3.4,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143132718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Noise robust HRRP sequence recognition based on a deep unfolded go decomposition network
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-28 DOI: 10.1016/j.sigpro.2024.109876
Mei Liu, Xunzhang Gao, Zhiwei Zhang
The performance of most high-resolution range profile (HRRP) recognition methods degrades dramatically under low-SNR conditions. Based on the scattering center model, an HRRP sequence power matrix consisted of multiple sequential power HRRPs can be decomposed into three components: a low-rank self term, a sparse cross term and a noise term. Notably, the low-rank self term contains most target structure signatures that are essential for recognition. This paper aims to achieve noise robust recognition by separating the self term from the HRRP sequence power matrix based on the low-rank decomposition theroy. Since the performance of the traditional low-rank decomposition algorithms heavily depends on manually selected parameters and needs numerous iterations, a deep unfolded Go Decomposition Network (GoDecNet) is proposed. Specifically, we improve and unfold the Go Decomposition (GoDec) algorithm into a network to estimate the self term and suppress the noise over the range profile. Additionally, to obtain more robust temporal-spatial features, we design a CNN-based module to extract fast-time structure features and introduce the diagonally-structured state space model to explore slow-time temporal correlations. Finally, a hybrid loss function is designed to train the network end-to-end and facilitate interaction between the modules. Experiments conducted on measured data and simulatied HRRP data demonstrate the superior performance of the proposed method under low-SNR conditions.
{"title":"Noise robust HRRP sequence recognition based on a deep unfolded go decomposition network","authors":"Mei Liu,&nbsp;Xunzhang Gao,&nbsp;Zhiwei Zhang","doi":"10.1016/j.sigpro.2024.109876","DOIUrl":"10.1016/j.sigpro.2024.109876","url":null,"abstract":"<div><div>The performance of most high-resolution range profile (HRRP) recognition methods degrades dramatically under low-SNR conditions. Based on the scattering center model, an HRRP sequence power matrix consisted of multiple sequential power HRRPs can be decomposed into three components: a low-rank self term, a sparse cross term and a noise term. Notably, the low-rank self term contains most target structure signatures that are essential for recognition. This paper aims to achieve noise robust recognition by separating the self term from the HRRP sequence power matrix based on the low-rank decomposition theroy. Since the performance of the traditional low-rank decomposition algorithms heavily depends on manually selected parameters and needs numerous iterations, a deep unfolded <strong>G</strong>o <strong>Dec</strong>omposition <strong>Net</strong>work (GoDecNet) is proposed. Specifically, we improve and unfold the Go Decomposition (GoDec) algorithm into a network to estimate the self term and suppress the noise over the range profile. Additionally, to obtain more robust temporal-spatial features, we design a CNN-based module to extract fast-time structure features and introduce the diagonally-structured state space model to explore slow-time temporal correlations. Finally, a hybrid loss function is designed to train the network end-to-end and facilitate interaction between the modules. Experiments conducted on measured data and simulatied HRRP data demonstrate the superior performance of the proposed method under low-SNR conditions.</div></div>","PeriodicalId":49523,"journal":{"name":"Signal Processing","volume":"230 ","pages":"Article 109876"},"PeriodicalIF":3.4,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143132716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Variational Bayesian based robust nonlinear filter for systems with unknown measurement loss and multi-step delay
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-25 DOI: 10.1016/j.sigpro.2024.109871
Zhaoxu Tian, Hongpo Fu, Yongmei Cheng
For nonlinear state estimation of the systems with randomly occurring measurement loss and multi-step delay (MLaMD), this paper investigates a variational Bayesian (VB) based robust cubature Kalman filter (VBRCKF), which does not require prior knowledge of the probabilities or delay steps. The proposed filter is to incorporate the VB framework into the CKF algorithm. Firstly, the randomly occurring MLaMD is modeled by using Bernoulli and categorical variables, thereby formulating a modified measurement model. Subsequently, the joint prior distribution of the system state along with the unknown variables associated with MLaMD is formulated. The joint posterior distribution is then approximately calculated by VB method. The resulting VBRCKF innovatively considers randomly occurring MLaMD without prior information and carries out adaptive estimation of these unknown variables. Finally, two simulation experiments for target tracking demonstrate the effectiveness of the proposed VBRCKF.
{"title":"Variational Bayesian based robust nonlinear filter for systems with unknown measurement loss and multi-step delay","authors":"Zhaoxu Tian,&nbsp;Hongpo Fu,&nbsp;Yongmei Cheng","doi":"10.1016/j.sigpro.2024.109871","DOIUrl":"10.1016/j.sigpro.2024.109871","url":null,"abstract":"<div><div>For nonlinear state estimation of the systems with randomly occurring measurement loss and multi-step delay (MLaMD), this paper investigates a variational Bayesian (VB) based robust cubature Kalman filter (VBRCKF), which does not require prior knowledge of the probabilities or delay steps. The proposed filter is to incorporate the VB framework into the CKF algorithm. Firstly, the randomly occurring MLaMD is modeled by using Bernoulli and categorical variables, thereby formulating a modified measurement model. Subsequently, the joint prior distribution of the system state along with the unknown variables associated with MLaMD is formulated. The joint posterior distribution is then approximately calculated by VB method. The resulting VBRCKF innovatively considers randomly occurring MLaMD without prior information and carries out adaptive estimation of these unknown variables. Finally, two simulation experiments for target tracking demonstrate the effectiveness of the proposed VBRCKF.</div></div>","PeriodicalId":49523,"journal":{"name":"Signal Processing","volume":"230 ","pages":"Article 109871"},"PeriodicalIF":3.4,"publicationDate":"2024-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A class of widely linear quaternion blind equalisation algorithms
IF 3.4 2区 工程技术 Q2 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2024-12-24 DOI: 10.1016/j.sigpro.2024.109863
Min Zhang , Min Xiang , Zhi Zheng , Sayed Pouria Talebi , Danilo P. Mandic
Quaternion adaptive filters have been applied extensively to model three- and four-dimensional phenomena in signal processing, but most of them require a known reference signal. In this paper, a class of widely linear quaternion-valued Godard (WL-QGodard) algorithms is derived, which include the widely linear quaternion-valued constant modulus algorithm (WL-QCMA) as a special case. The derived filter allows for signal recovery operations in the absence of reference signals to be performed directly in the quaternion domain, eliminating the need for transformation to real-valued vector algebras and preserving the advantages of the quaternion division algebra. Compared to state-of-the-art quaternion blind equalisation algorithms, the proposed algorithm models the signal transmission channel using the widely linear quaternion framework, which has more extensive applicability and can better represent real-world scenarios. Furthermore, aided by GHR calculus, for the first time, we present a performance analysis framework for the QGodard algorithm and WL-QGodard algorithms, which depicts the dynamic and their static convergence behaviours, overcoming the challenges posed by the noncommutative quaternion algebra and non-isomorphism between the quaternion equalisers and real-valued equalisers. Finally, simulation results over physically meaningful wireless communication signals indicate the effectiveness and superiority of the proposed WL-QCMA, and the validity of the theoretical performance analysis.
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Signal Processing
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