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Recovering complex-valued images from multiple measurements of Fourier data is of interest in many application domains, such as synthetic aperture radar (SAR) and ultrasound imaging. While there are many algorithms designed to accurately reconstruct the magnitude, the phase information can be important for downstream processing such as coherent change detection. There is also increased interest in quantifying the recovery uncertainty. This investigation proposes a new empirical Bayesian inference method for recovering point estimates and samples of a complex-valued image posterior distribution from its corresponding multiple Fourier measurements. It also quantifies the uncertainty for both the magnitude and the phase. Our method uses a support-informed prior which we estimate directly from the multiple data acquisitions. In this way we allow for data compression, since only the sparse support information is required for the prior construction. Numerical examples demonstrate its ability to reduce the uncertainty in the magnitude of the recovery while also preserving phase information. [ABSTRACT FROM AUTHOR]

Complex-valued least mean kurtosis (CLMK) algorithm and its augmented version (ACLMK) have recently become popular as workhorse tools in the processing of complex-valued signals due to their superior performances. Unfortunately, they are not yet suitable for sparse system identification problems. In this paper, combining the well-known sparsity-promoting strategies into the cost function based on the negated kurtosis of the error signal, we introduce a suit of sparsity-aware CLMK algorithms, named l 0 -norm CLMK (l 0 -CLMK), l 0 -ACLMK, zero-attraction CLMK (ZA-CLMK), ZA-ACLMK, reweighted ZA-CLMK (RZA-CLMK), and RZA-ACLMK. Simulation results on synthetic and real-world sparse system identification scenarios in the complex domain show that the proposed algorithms outperform the existing sparsity-aware algorithms in terms of convergence rate, tracking, and steady-state error. • We design a novel family of sparsity-aware CLMK algorithms. • We leverage sparsity-promoting strategies and kurtosis cost function to design them. • Proposed sparsity-aware CLMK algorithms outperform their CLMS versions. [ABSTRACT FROM AUTHOR]

Interval-valued time series (ITS) has been widely concerned by the academic community due to its outstanding performance in dealing with the uncertainty of systems. Numerous ITS forecasting studies have emerged, while the most popular method is based on "divide and conquer". For ITS analysis, bivariate empirical mode decomposition (BEMD) is currently the main tool that can simultaneously consider the interrelationships between the upper and lower limits. The concern is that, like empirical mode decomposition (EMD), BEMD suffers from mode mixing and end-point effects, which leads to unsatisfactory results of decomposition. In this paper, a bivariate ensemble empirical mode decomposition (BEEMD) method is proposed and applied to ITS forecasting, which aims at alleviating the problems mentioned above. At the same time, dynamic time warping (DTW) is used to quantize and analyze the mode mixing. Then, the effects of complex-valued signal constructions on ITS forecasting are discussed. To overcome the uncertainty and instability that may arise from a single prediction method, an optimal prediction based on the model pool is designed for sub-modes, and the final result can be obtained by simple addition. Using carbon prices, West Texas Intermediate (WTI) crude oil prices, and short-term loads, as the research objects, the results indicate that BEEMD is effective in avoiding mode mixing in BEMD, the complex-valued signal construction based on center and radius method (CRM) is more conducive to forecasting and optimal prediction can further improve forecasting accuracy. [ABSTRACT FROM AUTHOR]

In this study, a new fully complex nonlinear adaptive finite impulse response (FIR) filter algorithm based on the complex-valued nonlinear least mean kurtosis (CNLMK) is proposed for nonlinear complex-valued signals. The performance of the proposed nonlinear adaptive FIR filter algorithm is assessed on the prediction of complex-valued signals. In addition, the performance of the proposed algorithm is compared with the complex-valued nonlinear gradient descent (CNGD) algorithm in terms of the mean square error (MSE) and the prediction gain. Simulation results clearly demonstrate that the proposed CNLMK algorithm outperforms the CNGD algorithm.

This paper deals with the optimization of kurtosis for complex-valued signals in the independent component analysis (ICA) framework, where source signals are linearly and instantaneously mixed. Inspired by the recently proposed reference-based contrast schemes, a similar contrast function is put forward, based on which a new fast fixed-point (FastICA) algorithm is proposed. The new optimization method is similar in spirit to the former classical kurtosis-based FastICA algorithm but differs in the fact that it is much more efficient than the latter in terms of computational speed, which is significantly striking with large number of samples. The performance of this new algorithm is confirmed through computer simulations.

Parameter estimation from noisy complex-valued measurements is a significant topic in various areas of science and engineering. In this aspect, an important goal is finding an unbiased estimator with minimum variance. Therefore, variance analysis of an estimator is desirable and of practical interest. In this paper, we concentrate on analyzing the complex-valued ℓ p -norm minimizer with p ≥ 1 . Variance formulas for the resultant nonlinear estimators in the presence of three representative bivariate noise distributions, namely, α -stable, Student's t and mixture of generalized Gaussian models, are derived. To guarantee attaining the minimum variance for each noise process, optimum selection of p is studied, in the case of known noise statistics, such as probability density function and corresponding density parameters. All our results are confirmed by simulations and are compared with the Cramér-Rao lower bound. [ABSTRACT FROM AUTHOR]

Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. The complex variational mode decomposition (CVMD) is proposed as a natural and a generic extension of the original VMD algorithm for the analysis of complex-valued data in this work. Moreover, the equivalent filter bank structure of the CVMD in the presence of white noise, and the effects of initialization of center frequency on the filter bank property are both investigated via numerical experiments. Benefiting from the advantages of CVMD algorithm, its bi-directional Hilbert time-frequency spectrum is developed as well, in which the positive and negative frequency components are formulated on the positive and negative frequency planes separately. Several applications in the real-world complex-valued signals support the analysis. [ABSTRACT FROM AUTHOR]

• Complex-valued adaptive filtering algorithms are proposed to reduce the cost of data processing in the complex domain. • The proposed algorithms are achieved by leveraging the advantages of the OC strategy and CASP.• They censor the less informative complex-valued data under certain rules and thus reduce the cost of data processing. • They censor the less informative complex-valued data under certain rules and thus reduce the cost of data processing. • To censor possible outliers in the complex domain, we also develop robust versions of the proposed algorithms. • This study paves the way for using the proposed algorithms to process streaming big data in the complex domain. A class of complex-valued adaptive filtering algorithms is proposed, with the aim to reduce the cost of data processing in the complex domain. This is achieved by leveraging the advantages of the online censoring (OC) strategy and complex-valued adaptive signal processing (ASP). The proposed algorithms, namely the OC based complex-valued least mean square (OC-CLMS), OC based augmented CLMS (OC-ACLMS), OC based hybrid (OC-Hybrid), OC based complex-valued recursive least square (OC-CRLS), and OC based augmented CRLS (OC-ACRLS) algorithms, censor the less informative complex-valued data under certain rules, that is, they use only the most informative data for updating weight vectors. This is shown to considerably reduce the cost of data processing for processing both circular and noncircular complex-valued signals. Moreover, to censor possible outliers in the complex domain, we also develop the robust versions of the proposed algorithms, called the ROC-CLMS, ROC-ACLMS, ROC-Hybrid, ROC-CRLS, and ROC-ACRLS algorithms. Simulation results over both system identification and real-world prediction scenarios verify the attractive properties of the proposed OC strategy based complex-valued algorithms. Moreover, this study paves the way for using the proposed algorithms to process streaming big data in the complex domain. [ABSTRACT FROM AUTHOR]

Blind source separation of complex-valued signals has been a vital issue especially in the field of digital communication signal processing. This paper proposes a novel method based on nonlinear autocorrelation to solve the problem. Relying on the temporal structure with nonlinear autocorrelation of the signals, the method has a potential capability of extracting non-stationary complex sources with Gaussian or non-Gaussian distribution. Most traditional methods would fail in separating this kind of sources. We also analyze the stability conditions of the method in theory. Numerical simulations on artificial complex Gaussian data and orthogonal frequency division multiplexing sources corroborate the validity and efficiency of the proposed method. Moreover, with respect to classical methods, including cumulant-based approach using the non-stationarity of variance and complexity pursuit, our method offers equally good results with lower computational cost and better robustness. Finally, experiments for the separation of real communication signals illustrate that our method has good prospects in real-world applications. [ABSTRACT FROM AUTHOR]

This paper deals with the optimization of kurtosisfor complex-valued signals in the independent component analysis (ICA) framework, where source signals are linearly and instantaneously mixed. Inspired by the recently proposed reference-based contrast schemes, a similar contrast function is put forward, based on which a new fast fixed-point (FastICA) algorithm is proposed. The new optimization method is similar in spirit to the former classical kurtosis-based FastICA algorithm but differs in the fact that it is much more efficient than the latter in terms of computational speed, which is significantly striking with large number of samples. The performance of this new algorithm is confirmed through computer simulations. [ABSTRACT FROM AUTHOR]

Abstract: Complex-valued data are encountered in many application areas of signal and image processing. In the context of the optimization of functions of real variables, subspace algorithms have recently attracted much interest, owing to their efficiency for solving large-size problems while simultaneously offering theoretical convergence guarantees. The goal of this paper is to show how some of these methods can be successfully extended to the complex case. More precisely, we investigate the properties of the proposed complex-valued Majorize-Minimize Memory Gradient (3MG) algorithm. Important practical applications of these results arise in inverse problems. Here, we focus on image reconstruction in Parallel Magnetic Resonance Imaging (PMRI). The linear operator involved in the observation model then includes a subsampling operator over the k-space (2D Fourier domain) the choice of which is analyzed through our numerical results. In addition, sensitivity matrices associated with the multiple channel coils come into play. Comparisons with existing optimization methods confirm the better performance of the proposed algorithm. [Copyright &y& Elsevier]

Complex-valued vector time series occur in diverse fields such as oceanography and meteorology, and scientifically interpretable parameters may be estimated from them. We show that it is possible to make inference such as confidence intervals on these parameters using a vector-valued circulant embedding simulation method, combined with bootstrapping. We apply the methodology to three parameters of interest in oceanography, and compare the resulting simulated confidence intervals with those computed using analytic results. We conclude that the simulation scheme offers an inference approach either in the absence of theoretical distributional results, or to check the effect of nuisance parameters where theoretical results are available. [ABSTRACT FROM AUTHOR]

The empirical mode decomposition (EMD) has been introduced quite recently to adaptively decompose nonstationary and/or nonlinear time series [1]. The method being initially limited to real-valued time series, we propose here an extension to bivariate (or complex-valued) time series that generalizes the rationale underlying the EMD to the bivariate framework. Where the EMD extracts zero-mean oscillating components, the proposed bivariate extension is designed to extract zero-mean rotating components. The method is illustrated on a real-world signal, and properties of the output components are discussed. Free Matlab/C codes are available at http://perso.ens-lyon.fr/patrick.flandrin. [ABSTRACT FROM AUTHOR]

A method for the empirical mode decomposition (EMD) of complex-valued data is proposed. This is achieved based on the filter bank interpretation of the EMD mapping and by making use of the relationship between the positive and negative frequency component of the Fourier spectrum. The so-generated intrinsic mode functions (IMFs) are complex-valued, which facilitates the extension of the standard EMD to the complex domain. The analysis is supported by simulations on both synthetic and real-world complex-valued signals. [ABSTRACT FROM AUTHOR]

This letter addresses the blind signal estimation problem in the so-called conjugate signal model, where the observed signal is a linear combination of the desired signal and its complex conjugate. It will be shown that blind signal recovery in this kind of signal model is feasible using only the second-order statistics of the observed signal, under the assumption of circular or proper complex signals. Furthermore, one practical example application in the field of communications receiver signal processing will be given, where the image signal interference caused by amplitude and phase mismatches of the receiver analog branches is digitally attenuated. In addition to the analytical results and practical implementation algorithms, the efficiency of the proposed estimation concepts in the digital image rejection application is evaluated using computer simulations, showing impressive performance results. [ABSTRACT FROM AUTHOR]

The paper deals with the autocorrelation equation on a finite interval and on the half-axis and with the triple correlation equation in the case of complex-valued signals. Via methods of Fourier transformation the equations are reduced to boundary value and continuation problems for holomorphic functions in the upper half-plane and to functional equations in two independent variables, respectively, and in this way solved in closed form. Thereby, on the one side, Cauchy integrals and Paley-Wiener conditions and, on the other side, the solution of a known functional equation in a single variable are applied. [ABSTRACT FROM AUTHOR]