Complex-valued Signals Research Articles

A fast sparse Bayesian learning method with adaptive Laplace prior for space‐time adaptive processing

Space-time adaptive processing with finite samples is supposed to be a crucial technique for airborne radar systems. Inspired by the application of Gaussian prior in sparse Bayesian learning algorithm and the adaptive least absolute shrinkage and selection operator algorithm, a hierarchical Bayesian framework with adaptive Laplace priors is proposed. In this paper, a novel method is applied to avoid the high-dimension matrix inverse operation in the proposed algorithm. Moreover, in order to apply the method in the complex-valued domain, the complex-valued signal is split into two independent variables. Then, the sparse recovery problem in the complex-valued domain can be transformed into the real-value domain. Simulation experiments show that the proposed algorithm can achieve great clutter suppression performance and also ensure high computational efficiency.

Complex-Valued Neural Networks: A Comprehensive Survey

Complex-valued neural networks (CVNNs) have shown their excellent efficiency compared to their real counter-parts in speech enhancement, image and signal processing. Researchers throughout the years have made many efforts to improve the learning algorithms and activation functions of CVNNs. Since CVNNs have proven to have better performance in handling the naturally complex-valued data and signals, this area of study will grow and expect the arrival of some effective improvements in the future. Therefore, there exists an obvious reason to provide a comprehensive survey paper that systematically collects and categorizes the advancement of CVNNs. In this paper, we discuss and summarize the recent advances based on their learning algorithms, activation functions, which is the most challenging part of building a CVNN, and applications. Besides, we outline the structure and applications of complex-valued convolutional, residual and recurrent neural networks. Finally, we also present some challenges and future research directions to facilitate the exploration of the ability of CVNNs.

Generalized Maximum Complex Correntropy Augmented Adaptive IIR Filtering.

Augmented IIR filter adaptive algorithms have been considered in many studies, which are suitable for proper and improper complex-valued signals. However, lots of augmented IIR filter adaptive algorithms are developed under the mean square error (MSE) criterion. It is an ideal optimality criterion under Gaussian noises but fails to model the behavior of non-Gaussian noise found in practice. Complex correntropy has shown robustness under non-Gaussian noises in the design of adaptive filters as a similarity measure for the complex random variables. In this paper, we propose a new augmented IIR filter adaptive algorithm based on the generalized maximum complex correntropy criterion (GMCCC-AIIR), which employs the complex generalized Gaussian density function as the kernel function. Stability analysis provides the bound of learning rate. Simulation results verify its superiority.

High receiver skew-tolerant and hardware-efficient clock recovery for short-reach coherent transmission.

The performance of the high-baud-rate and high-order-modulation-format short-reach coherent transmission systems is sensitive to the in-phase and quadrature (IQ) skew. The conventional receiver IQ skew compensation schemes based on adaptive equalizers (AEQs) suffer from the IQ skew enhanced timing jitter incurred by the clock recovery algorithm (CRA), resulting in a serious sensitivity degradation. In this paper, we first propose a novel multiplication-free timing phase error detector (TPED) with the gain insensitive to the receiver IQ skew and the capability to deal with the complex-valued Nyquist signal with an arbitrary roll-off factor and its real-valued IQ tributaries. Based on the TPED, we then propose a new all-digital feedback CRA able to compensate for the receiver IQ skew. With the novel CRA, the IQ skew enhanced timing jitter is eliminated and the receiver sensitivity can be improved by more than 1 dB for the 61 GBaud dual-polarization Nyquist 16QAM system for an IQ skew of 5 ps. Furthermore, the proposed CRA can reduce the computation complexity of the AEQ by more than 25% compared with the existing schemes by relieving the AEQ from IQ skew compensation. Both numerical simulations and experiments are carried out to validate the advantages of the proposed algorithms. The high-skew-tolerant and low-complexity CRA is a strong candidate for the power-sensitive high-speed short-reach coherent transmission systems.

Charactering the Peak-to-Average Power Ratio of OTFS Signals: A Large System Analysis

Orthogonal time frequency space (OTFS) system constitutes an effective structure conceived for efficiently utilizing the channel information, which is capable of achieving a promising transmission performance in high-mobility environment. To extract enough channel diversity, a two-dimensional Fourier transformation combined with a pulse shape is designed at the OTFS transmitter. Consequently, the amplitude of OTFS signals may fluctuate drastically, owing to the combined dependency of the OTFS transformation and the pulse shape. To quantify the amplitude fluctuation, we investigate the peak-to-average power ratio (PAPR) of OTFS signals, for a large amount of data in the delay-Doppler domain. We first reveal that when the number of data points approaches to infinity, based on central limit theorems for dependent variables, the complex-valued OTFS signals weakly converge to a Gaussian distribution. Then, according to the extremal theory of the Chi-squared process for stationary OTFS signals, an accurate expression of the PAPR distribution is derived, depending on the transmit pulse and the number of data points. It is also demonstrated that upon modifying the exponential factor, the analytical PAPR expression is applicable for the non-stationary Gaussian distribution caused by the bandlimited pulse with a large roll-off factor. Simulation results confirm the accuracy of the analytical PAPR probability for practical conditions.

Online censoring based complex-valued adaptive filters

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.

Three-dimensional instantaneous orbit map for rotor-bearing system based on a novel multivariate complex variational mode decomposition algorithm

Full spectrum and holospectrum are homogeneous information fusion technology developed for the fault diagnosis of rotating machinery, which is extensively exploited in the analysis of the orbits of rotor-bearing systems. However, they are not adapted for non-stationary signals, nor can they be used for fusion analysis of vibrations of multiple bearing sections. By drawing inspiration from the multivariate variational mode decomposition (MVMD) and the complex-valued signal decomposition, we propose a method called multivariate complex variational mode decomposition (MCVMD). It can simultaneously extract the forward and backward components of multiple bearing sections and realize non-stationary complex signal decomposition of multiple bearing sections of the rotor. To achieve the visualization goal of condition monitoring, we propose the three-dimensional instantaneous orbit map (3D-IOM). It enables more features of shaft vibration of a rotor system to be displayed and offers a new way for the fusion analysis of vibration signals of multiple bearing sections of rotating machinery. Furthermore, making the most of the joint information, we also provide a high-resolution time-full spectrum (Time-FS) to display the forward and backward frequency components of multiple bearing sections. The effectiveness of the proposed method through both the simulated experiment and the real-life complex-valued signals is demonstrated in this paper.

Modified complex multitask Bayesian compressive sensing using Laplacian scale mixture prior

Bayesian compressive sensing (BCS) is an important sub-class of sparse signal reconstruction algorithms. In this paper, a modified complex multitask Bayesian compressive sensing (MCMBCS) algorithm using the Laplacian scale mixture (LSM) prior is proposed. The LSM prior is first introduced into the complex BCS framework by exploiting its better sparse characteristic and flexibility than traditional Laplacian prior. Furthermore, by integrating out the noise variance analytically, the MCMBCS algorithm significantly improves the signal recovery performance than the original CMBCS. More importantly, the authors not only present the iterative algorithm but also develop the sub-optimal fast implementation method based on the marginal likelihood maximisation, which dramatically reduce the computational complexity. Finally, sufficient numerical simulations validate the better performance of the proposed algorithm in reconstruction accuracy and computational effectiveness than existing work. It is revealed that the proposed algorithm has great potential in the complex-valued signal processing field.

The optimal [formula omitted]-Wigner distribution

This study devotes to the optimal parameter of the k-Wigner distribution (kWD) in terms of the highest time-frequency resolution. We first disclose a relation between spreads in time-kWD and time domains, as well as those in Fourier transform (FT)-kWD and FT domains. Then we use them to set up an equivalence relation between the uncertainty product in time-kWD and FT-kWD domains and that in time and FT domains. Finally we separately deduce Heisenberg’s uncertainty inequalities of all signals and complex-valued signals for the kWD. As a result, the optimal kWD is none other than the ordinary Wigner distribution. Examples are carried out to verify the correctness of the theoretical result. An application of the derived uncertainty inequalities in the estimation of bandwidths in kWD domains is also given.

Complex-Valued Random Fourier Geometric Algebra Adaptive Filtering

The complex-valued fixed dimensional adaptive filters (CFDAFs) can improve the performance and computational efficiency of complex-valued kernel adaptive filters (CKAFs), but have the challenge of over-coupling for the real and imaginary parts of complex-valued signals. To this end, this brief proposes a novel fixed dimensional adaptive filter for complex-valued signals named complex-valued random Fourier geometric algebra least mean square (CRFGALMS). On the basis of the framework of geometric algebraic adaptive filtering, the real and imaginary parts of complex-valued signals are mapped into random Fourier features space (RFFS) to improve the efficiency of the nonlinear mapping for complex-valued signals, respectively. The proposed random Fourier feature mapping based on the geometric algebra is endowed with superior presentation abilities in complex-valued domain as it can decouple the nonlinear mapping of the real and imaginary parts of complex-valued signals, efficiently. Simulations on nonlinear channel equalization are used to illustrate the superiority of the proposed CRFGALMS algorithm in terms of dimensionality and filtering performance.

A Fast Space-Time Adaptive Processing Algorithm Based on Sparse Bayesian Learning for Airborne Radar.

Space-time adaptive processing (STAP) plays an essential role in clutter suppression and moving target detection in airborne radar systems. The main difficulty is that independent and identically distributed (i.i.d) training samples may not be sufficient to guarantee the performance in the heterogeneous clutter environment. Currently, most sparse recovery/representation (SR) techniques to reduce the requirement of training samples still suffer from high computational complexities. To remedy this problem, a fast group sparse Bayesian learning approach is proposed. Instead of employing all the dictionary atoms, the proposed algorithm identifies the support space of the data and then employs the support space in the sparse Bayesian learning (SBL) algorithm. Moreover, to extend the modified hierarchical model, which can only apply to real-valued signals, the real and imaginary components of the complex-valued signals are treated as two independent real-valued variables. The efficiency of the proposed algorithm is demonstrated both with the simulated and measured data.

Improved Ultrasound Imaging Performance With Complex Cumulant Analysis.

Ultrasound localization microscopy (ULM) breaks the acoustic diffraction limit. However, the temporal resolution of ULM is relatively low because of a long data-acquisition time. Inspired by super-resolution optical fluctuation imaging (SOFI), in this paper, we propose a method for ultrasound imaging with improved imaging performance, which is achieved by using cumulant analysis. Specifically, to eliminate the axial oscillations, here, the cumulant analysis framework is extended, which is used to process the complex-valued analytic signals rather than the real-valued signals. The results from the numerical simulations and in vitro physical phantom experiments indicate that by generalizing cumulant analysis to complex-valued signals, a high imaging performance is achieved with an improvement of ∼35%-42% (lateral direction) and ∼41%-42% (axial direction) in the resolution compared with the temporal mean envelope image, in terms of full-width-half-max (FWHM). In particularly, the axial oscillations appearing in the real cumulant images are effectively eliminated by the complex cumulant analysis. Moreover, the proposed method can easily take advantage of SOFI. In the phantom experiment, a short data-acquisition time (∼2 sec) is enough to obtain the improved spatial resolution. The proposed method offers an implementation of US with high spatial resolution, fast data-acquisition speed, and axial oscillations removal characteristics. The method provides the potential in US imaging fast biological processes in vivo.

Complex Multisnapshot Sparse Bayesian Learning for Offgrid DOA Estimation

Direction of arrival (DOA) estimation has recently been developed based on sparse signal reconstruction (SSR). Sparse Bayesian learning (SBL) is a typical method of SSR. In SBL, the two-layer hierarchical model in Gaussian scale mixtures (GSMs) has been used to model sparsity-inducing priors. However, this model is mainly applied to real-valued signal models. In order to apply SBL to complex-valued signal models, a general class of sparsity-inducing priors is proposed for complex-valued signal models by complex Gaussian scale mixtures (CGSMs), and the special cases correspond to complex versions of several classical priors are provided, which is helpful to analyze the connections with different modeling methods. In addition, the expression of the SBL form of the real- and complex-valued model is unified by parameter values, which makes it possible to generalize and improve the properties of the SBL methods. Finally, the SBL complex-valued form is applied to the offgrid DOA estimation complex-valued model, and the performance between different sparsity-inducing priors is compared. Theoretical analysis and simulation results show that the proposed algorithm can effectively process complex-valued signal models and has lower algorithm complexity.

A Modified Complex Variational Mode Decomposition Method for Analyzing Nonstationary Signals with the Low-Frequency Trend

Complex variational mode decomposition (CVMD) has been proposed to extend the original variational mode decomposition (VMD) algorithm to analyze complex-valued data. Conventionally, CVMD divides complex-valued data into positive and negative frequency components using bandpass filters, which leads to difficulties in decomposing signals with the low-frequency trend. Moreover, both decomposition number parameters of positive and negative frequency components are required as prior knowledge in CVMD, which is difficult to satisfy in practice. This paper proposes a modified complex variational mode decomposition (MCVMD) method. First, the complex-valued data are upsampled through zero padding in the frequency domain. Second, the negative frequency component of upsampled data are shifted to be positive. Properties of analytical signals are used to get the real-valued data for standard variational mode decomposition and the complex-valued decomposition results after frequency shifting back. Compared with the conventional method, the MCVMD method gives a better decomposition of the low-frequency signal and requires less prior knowledge about the decomposition number. The equivalent filter bank structure is illustrated to analyze the behavior of MCVMD, and the MCVMD bi-directional Hilbert spectrum is provided to give the time–frequency representation. The effectiveness of the proposed algorithm is verified by both synthetic and real-world complex-valued signals.

Bayesian spatiotemporal modeling on complex-valued fMRI signals via kernel convolutions.

We propose a model-based approach that combines Bayesian variable selection tools, a novel spatial kernel convolution structure, and autoregressive processes for detecting a subject's brain activation at the voxel level in complex-valued functional magnetic resonance imaging (CV-fMRI) data. A computationally efficient Markov chain Monte Carlo algorithm for posterior inference is developed by taking advantage of the dimension reduction of the kernel-based structure. The proposed spatiotemporal model leads to more accurate posterior probability activation maps and less false positives than alternative spatial approaches based on Gaussian process models, and other complex-valued models that do not incorporate spatial and/or temporal structure. This is illustrated in the analysis of simulated data and human task-related CV-fMRI data. In addition, we show that complex-valued approaches dominate magnitude-only approaches and that the kernel structure in our proposed model considerably improves sensitivity rates when detecting activation at the voxellevel.

Unscented Kalman Filter With General Complex-Valued Signals

For the estimation of real-valued Gaussian signals, the unscented Kalman filter (UKF) can provide a state estimate with second-order accuracy. However, when a general complex-valued system is considered, a direct extension of UKF from the real domain to the complex domain is inadequate, since the complementary covariance information associated with general improper complex-valued signals has been systematically ignored. To this end, in this work, we propose a general complex-valued unscented Kalman filter (GCUKF) algorithm which can be applied for both proper and improper signals. This is achieved by first proposing a novel sigma points selection scheme for the general complex-valued case, followed by a modified state update method to fully utilize both the innovation and its conjugate. A rigorous MSE analysis illustrates the superiority of the proposed state update method, and simulations support the analysis.

On the Convergence of Non-Convex Phase Retrieval With Denoising Priors

Denoising priors have achieved empirical success in solving non-convex phase retrieval but still lack convergence guarantees in theory. In this paper, we provide a novel insight on the convergence guarantees for a general class of non-convex phase retrieval with denoising priors. Specifically, we propose a Wirtinger flow based framework, named DWF, that allows monotone and contractive iterative optimization. We demonstrate in theory that, under a milder demi-Lipschitz condition on denoisers, the proposed DWF framework converges to the actual signal (up to a global sign) at a geometric rate. The derived convergence guarantees and rates are general to accommodate to complex-valued and real-valued signals in the presence of noise perturbation. Furthermore, we exemplify the proposed framework with two prevailing denoising priors, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , plug-and-play priors (PnP) and regularization by denoising (RED), including specific conditions on denoisers and convergence rates. To our best knowledge, this paper is the first attempt to provide theoretical guarantees of convergence for non-convex phase retrieval with denoising priors. Numerical evaluations demonstrate the theoretical results for analytic denoiser like arithmetic mean filter and deep learning based denoiser DnCNN. Furthermore, extensive experiments under the Gaussian model and coded diffraction pattern show that the proposed framework outperforms existing denoising prior-based methods and evidently reduces the necessary sampling rate for stable reconstruction with a guarantee of convergence in theory.

Complex-Valued Frequency Estimation Network and Its Applications to Superresolution of Radar Range Profiles

Frequency estimation of multiple sinusoidals is fundamental in statistical signal processing. In this article, we focus on 1-D complex signals and propose a complex-valued network for frequency estimation, referred to as cResFreq, which overcomes the drawbacks of DeepFreq and 2-D ResFreq, that is, the performance loss associated with processing complex-valued signals by a real-valued network. First, weights of a hidden layer are optimized to approximate varieties of complex-valued complete basis vectors, and the locally uncorrelated noise suppression is realized by convolutional kernels. The feature maps/coarse frequency representations after a modulus operation are subsequently processed by real-valued residual blocks for obtaining a high-resolution frequency representation. Numerical experiments exhibit the superior performance of the proposed method in terms of resolution, estimation accuracy, and detection capability for weak components. Finally, the cResFreq is applied to realize the superresolution of range profiles in radar systems. The results based on the synthetic and real signals demonstrate that the HRRPs obtained by the cResFreq provide more target details, which shows promise in automatic target recognition (ATR). The codes can be found in <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/panpp-git/cResFreq</uri> .

IQ Symbols Processing Schemes With LSTMs in OFDM System

IQ modulation enjoys great popularity in wireless communication. Its main advantage is the symmetric convenience of combining independent signal components into a single composite signals in the transmitter. And it splits the composite signal into its independent components in the receiver. However, due to the limitations of communication facilities and imperfect channel environments, the symbols will be distorted. In this paper, we propose several signal processing schemes based on deep learning (DL) neural networks (i.e., long short-term memory (LSTM)) to process signal in an end-to-end manner, known as DLA, EDLA and PDNet. These schemes combine with advanced DL architectures and data-driven models for complex-valued signal processing. Enlightened by IQ demodulation ideas, we adopt LSTMs to develop several schemes to implement different functions (i.e., signal detection and peak to average power ratio (PAPR) reduction). The simulation results show that the schemes have significant performance improvement in bit error rate (BER) while reducing the probability of high PAPR.

The Diffusion Entropy Stochastic Gradient Descent Algorithm With Quasi-Optimal Combiners: Formulation and Analysis

We consider herein the diffusion entropy stochastic gradient descent (DE-SGD) algorithm for non-circular complex-valued signals. We theoretically analyze the mean stability and the steady-state network mean-square-deviation (MSD) of the DE-SGD. We develop both the quasi-optimal static and adaptive combination strategies to enhance the DE-SGD algorithm so as to utilize the spatial variations of the noise circularity coefficients, noise variances, the regressor powers and the step sizes across the entire network. We validate the proposed adaptive combiners theoretically and experimentally in small step size scenarios. We also demonstrate the consistency of the convergence and steady-state performance of both the static combiners and the proposed adaptive combiners. Illustrative simulations and real-world data evaluation validate the superior transient and steady-state performance of the DE-SGD algorithm with the proposed quasi-optimal combination strategies.