Complex-valued Signals Research Articles

FBSE-EWT Technique-based Complex-valued Signal Analysis

FBSE-EWT Technique-based Complex-valued Signal Analysis

Fully Complex-Valued Gated Recurrent Neural Network for Ultrasound Imaging.

Ultrasound imaging is widely used in medical diagnosis. It has the advantages of being performed in real time, cost-efficient, noninvasive, and nonionizing. The traditional delay-and-sum (DAS) beamformer has low resolution and contrast. Several adaptive beamformers (ABFs) have been proposed to improve them. Although they improve image quality, they incur high computation cost because of the dependence on data at the expense of real-time performance. Deep-learning methods have been successful in many areas. They train an ultrasound imaging model that can be used to quickly handle ultrasound signals and construct images. Real-valued radio-frequency signals are typically used to train a model, whereas complex-valued ultrasound signals with complex weights enable the fine-tuning of time delay for enhancing image quality. This work, for the first time, proposes a fully complex-valued gated recurrent neural network to train an ultrasound imaging model for improving ultrasound image quality. The model considers the time attributes of ultrasound signals and uses complete complex-number calculation. The model parameter and architecture are analyzed to select the best setup. The effectiveness of complex batch normalization is evaluated in training the model. The effect of analytic signals and complex weights is analyzed, and the results verify that analytic signals with complex weights enhance the model performance to reconstruct high-quality ultrasound images. The proposed model is finally compared with seven state-of-the-art methods. Experimental results reveal its great performance.

An augmented complex-valued gradient-descent total least-squares algorithm for noncircular signals

In this paper, we propose a novel augmented complex-valued gradient-descent total least-squares (ACGDTLS) adaptive filter for processing noisy input and output noncircular complex-valued signals. First, a Rayleigh quotient cost function is formulated by incorporating augmented complex-valued statistics and the output-to-input-noise-ratio within the widely linear error-in-variable model, whereby the ACGDTLS is developed using the gradient-descent approach. Next, rigorous analysis is conducted to establish a conservative step-size bound guaranteeing mean convergence, a closed-form expression for the steady-state mean-squared deviation, and the algorithm’s computational complexity. Finally, through simulations conducted in system identification, wind/speech prediction, and stereophonic acoustic echo cancellation, the analytical findings are validated, and the proposed ACGDTLS filter demonstrates superior estimation accuracy compared to the augmented complex-valued least-mean-square algorithm and two state-of-the-art bias-compensated methods. Remarkably, this performance advantage persists across a wide range of step-sizes, input noise variances, and output noise variances.

Sparsity-aware complex-valued least mean kurtosis algorithms

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 l0-norm CLMK (l0-CLMK), l0-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.

A new transfer entropy method for measuring directed connectivity from complex-valued fMRI data.

Inferring directional connectivity of brain regions from functional magnetic resonance imaging (fMRI) data has been shown to provide additional insights into predicting mental disorders such as schizophrenia. However, existing research has focused on the magnitude data from complex-valued fMRI data without considering the informative phase data, thus ignoring potentially important information. We propose a new complex-valued transfer entropy (CTE) method to measure causal links among brain regions in complex-valued fMRI data. We use the transfer entropy to model a general non-linear magnitude-magnitude and phase-phase directed connectivity and utilize partial transfer entropy to measure the complementary phase and magnitude effects on magnitude-phase and phase-magnitude causality. We also define the significance of the causality based on a statistical test and the shuffling strategy of the two complex-valued signals. Simulated results verified higher accuracy of CTE than four causal analysis methods, including a simplified complex-valued approach and three real-valued approaches. Using experimental fMRI data from schizophrenia and controls, CTE yields results consistent with previous findings but with more significant group differences. The proposed method detects new directed connectivity related to the right frontal parietal regions and achieves 10.2-20.9% higher SVM classification accuracy when inferring directed connectivity using anatomical automatic labeling (AAL) regions as features. The proposed CTE provides a new general method for fully detecting highly predictive directed connectivity from complex-valued fMRI data, with magnitude-only fMRI data as a specific case.

Unique wavelet sign retrieval from samples without bandlimiting

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multiwavelet frame coefficients \[ { | W ϕ i f ( α m β n , α m ) | : i ∈ { 1 , 2 , 3 } , m , n ∈ Z } \{\lvert \mathcal {W}_{\phi _i} f(\alpha ^{m}\beta n,\alpha ^{m}) \rvert : i\in \{1,2,3\}, m,n\in \mathbb {Z}\} \] for every α > 1 , β > 0 \alpha >1,\beta >0 with β ln ⁡ ( α ) ≤ 4 π / ( 1 + 4 p ) \beta \ln (\alpha )\leq 4\pi /(1+4p) , p > 0 p>0 , when the three wavelets ϕ i \phi _i are suitable linear combinations of the Poisson wavelet P p P_p of order p p and its Hilbert transform H P p \mathscr {H}P_p . For complex-valued signals we find that this is not possible for any choice of the parameters α > 1 , β > 0 \alpha >1,\beta >0 , and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients.

Estimation of complete mutual information exploiting nonlinear magnitude-phase dependence: Application to spatial FNC for complex-valued fMRI data

BackgroundReal-valued mutual information (MI) has been used in spatial functional network connectivity (FNC) to measure high-order and nonlinear dependence between spatial maps extracted from magnitude-only functional magnetic resonance imaging (fMRI). However, real-valued MI cannot fully capture the group differences in spatial FNC from complex-valued fMRI data with magnitude and phase dependence. MethodsWe propose a complete complex-valued MI method according to the chain rule of MI. We fully exploit the dependence among magnitudes and phases of two complex-valued signals using second and fourth-order joint entropies, and propose to use a Gaussian copula transformation with a lower bound property to avoid inaccurate estimation of joint probability density function when computing the joint entropies. ResultsThe proposed method achieves more accurate MI estimates than the two histogram-based (normal and symbolic approaches) and kernel density estimation methods for simulated signals, and enhances group differences in spatial functional network connectivity for experimental complex-valued fMRI data. Comparison with existing methodsCompared with the simplified complex-valued MI and real-valued MI, the proposed method yields higher MI estimation accuracy, leading to 17.4 % and 145.5 % wider MI ranges, and more significant connectivity differences between healthy controls and schizophrenia patients. A unique connection between executive control network (EC) and right frontal parietal areas, and three additional connections mainly related to EC are detected than the simplified complex-valued MI. ConclusionsWith capability in quantifying MI fully and accurately, the proposed complex-valued MI is promising in providing qualified FNC biomarkers for identifying mental disorders such as schizophrenia.

Robust DOA Estimation Using Multi-Scale Fusion Network with Attention Mask

To overcome the limitations of traditional methods in reverberant and noisy environments, a robust multi-scale fusion neural network with attention mask is designed to improve direction-of-arrival (DOA) estimation accuracy for acoustic sources. It combines the benefits of deep learning and complex-valued operations to effectively deal with the interference of reverberation and noise in speech signals. The unique properties of complex-valued signals are exploited to fully capture inherent features and rich information is preserved in the complex field. An attention mask module is designed to generate distinct masks for selectively focusing and masking based on the input. After that, the multi-scale fusion block efficiently captures multi-scale spatial features by stacking complex-valued convolutional layers with small size kernels, and reduces the module complexity through special branching operations. Experimental results demonstrate that the model achieves significant improvements over other methods for speaker localization in reverberant and noisy environments. It provides a new solution for DOA estimation for acoustic sources in different scenarios, which has significant theoretical and practical implications.

Filter-assisted direct detection-based field recovery scheme for complex-valued double-sideband signals

Double-sideband self-coherent detection (DSB-SCD) can achieve optical field recovery of complex-valued double-sideband (CV-DSB) signals without a local oscillator laser. To date, various proposed DSB-SCD schemes require a high carrier-to-signal power ratio (CSPR) to mitigate the inherent signal-signal beat interference (SSBI) induced by direct detection. Aiming at relaxing the high CSPR requirement for the DSB-SCD, filter-assisted direct detection (FADD) is proposed to retrieve the CV-DSB signals. In this paper, we clarify the concept of the FADD, which uses optical filters in the receiver to achieve optical field recovery of CV-DSB signals via direct detection. The CV-DSB signals can be retrieved at low CSPR with the aid of optical filters. We also propose a novel FADD receiver that uses an optical bandstop filter (OBSF) to obtain CV-DSB signals from self-coherent signals, called OBSF-DD with 90° hybrid. Considering the high complexity of the OBSF-DD with 90° hybrid, a simplified OBSF-DD receiver with 2 × 2 coupler is put forward as well, which uses a 2 × 2 coupler instead of optical hybrid, and three single-ended photodiodes are required to achieve signal reception. By using cost-effective FADD receiver, the proposed FADD scheme provides an effective approach for low CSPR field recovery.

CTV-Net: Complex-Valued TV-Driven Network With Nested Topology for 3-D SAR Imaging.

The regularization-based approaches offer promise in improving synthetic aperture radar (SAR) imaging quality while reducing system complexity. However, the widely applied l1 regularization model is hindered by their hypothesis of inherent sparsity, causing unreal estimations of surface-like targets. Inspired by the edge-preserving property of total variation (TV), we propose a new complex-valued TV (CTV)-driven interpretable neural network with nested topology, i.e., CTV-Net, for 3-D SAR imaging. In our scheme, based on the 2-D holography imaging operator, the CTV-driven optimization model is constructed to pursue precise estimations in weakly sparse scenarios. Subsequently, a nested algorithmic framework, i.e., complex-valued TV-driven fast iterative shrinkage thresholding (CTV-FIST), is derived from the theory of proximal gradient descent (PGD) and FIST algorithm, theoretically supporting the design of CTV-Net. In CTV-Net, the trainable weights are layer-varied and functionally relevant to the hyperparameters of CTV-FIST, which aims to constrain the algorithmic parameters to update in a well-conditioned tendency. All weights are learned by end-to-end training based on a two-term cost function, which bounds the measurement fidelity and TV norm simultaneously. Under the guidance of the SAR signal model, a reasonably sized training set is generated, by randomly selecting reference images from the MNIST set and consequently synthesizing complex-valued label signals. Finally, the methodology is validated, numerically and visually, by extensive SAR simulations and real-measured experiments, and the results demonstrate the viability and efficiency of the proposed CTV-Net in the cases of recovering 3-D SAR images from incomplete echoes.

Class-Incremental Learning for Recognition of Complex-Valued Signals

Class-Incremental Learning for Recognition of Complex-Valued Signals

A fully Bayesian approach for comprehensive mapping of magnitude and phase brain activation in complex-valued fMRI data

Functional magnetic resonance imaging (fMRI) plays a crucial role in neuroimaging, enabling the exploration of brain activity through complex-valued signals. These signals, composed of magnitude and phase, offer a rich source of information for understanding brain functions. Traditional fMRI analyses have largely focused on magnitude information, often overlooking the potential insights offered by phase data. In this paper, we propose a novel fully Bayesian model designed for analyzing single-subject complex-valued fMRI (cv-fMRI) data. Our model, which we refer to as the CV-M&P model, is distinctive in its comprehensive utilization of both magnitude and phase information in fMRI signals, allowing for independent prediction of different types of activation maps. We incorporate Gaussian Markov random fields (GMRFs) to capture spatial correlations within the data, and employ image partitioning and parallel computation to enhance computational efficiency. Our model is rigorously tested through simulation studies, and then applied to a real dataset from a unilateral finger-tapping experiment. The results demonstrate the model's effectiveness in accurately identifying brain regions activated in response to specific tasks, distinguishing between magnitude and phase activation.

Complex-Valued Convolutional Gated Recurrent Neural Network for Ultrasound Beamforming.

Ultrasound detection is a potent tool for the clinical diagnosis of various diseases due to its real-time, convenient, and noninvasive qualities. Yet, existing ultrasound beamforming and related methods face a big challenge to improve both the quality and speed of imaging for the required clinical applications. The most notable characteristic of ultrasound signal data is its spatial and temporal features. Because most signals are complex-valued, directly processing them by using real-valued networks leads to phase distortion and inaccurate output. In this study, for the first time, we propose a complex-valued convolutional gated recurrent (CCGR) neural network to handle ultrasound analytic signals with the aforementioned properties. The complex-valued network operations proposed in this study improve the beamforming accuracy of complex-valued ultrasound signals over traditional real-valued methods. Further, the proposed deep integration of convolution and recurrent neural networks makes a great contribution to extracting rich and informative ultrasound signal features. Our experimental results reveal its outstanding imaging quality over existing state-of-the-art methods. More significantly, its ultrafast processing speed of only 0.07 s per image promises considerable clinical application potential. The code is available at https://github.com/zhangzm0128/CCGR.

RFF Template Design: Adaptively Decreasing Both Doppler Shifts and Noise for Complex-Valued Signals

RFF Template Design: Adaptively Decreasing Both Doppler Shifts and Noise for Complex-Valued Signals

Variational multi-harmonic duality mode pursuit method for extracting repetitive transient components from vibration signals

The repetitive transient component (RTC) in vibration signals is a typical symptom reflecting the localised defects of rotating machinery, which can be mathematically defined by feature parameters, such as the inter-arrival interval, maximum amplitude, resonance frequency, and attenuation ratio. In particular, the inter-arrival interval and maximum amplitude reveal the location and severity of failure and provide critical information for condition monitoring and fault diagnosis. However, there are some shortcomings of applicability, anti-noise, and fidelity, which are unfavourable for pursuing fault-induced RTC and enabling reliable recognition of key feature parameters. Hence, a novel method called variational multi-harmonic duality mode pursuit (VMHDMP) is proposed to extract the RTC accurately. A multi-harmonic duality mode function (MHDMF) that comprehensively depicts the critical feature parameters of the RTC was first defined in the frequency domain. Based on the defined MHDMF, the pursuit of RTC is formulated as a frequency-domain variational optimisation problem with a local compact time duration constraint, which optimally searches for the mode with an explicit MHDMF from the complex-valued frequency signal. A strategy involving the mutual conversion of real- and complex-valued signals was incorporated into the solution algorithm to satisfy the decomposability of complex-valued frequency signals. Finally, an inverse Fourier transform is performed on the analytical form of the extracted MHDMF to obtain a faithful RTC. Essentially, VMHDMP provides an optimal ‘transient-pass’ inverse filter bank with compact passage only near the occurrence moment of each transient in RTC. Thus, the vital parameters closely related to the faults are fully retained, whereas the frequency-overlap noise and large-amplitude interference occurring at different moments can be effectively eliminated. Simulated and experimental examples were used to test the effectiveness of VMHDMP.

A Novel Complex-Valued Hybrid Neural Network for Automatic Modulation Classification

Currently, dealing directly with in-phase and quadrature time series data using the deep learning method is widely used in signal modulation classification. However, there is a relative lack of methods that consider the complex properties of signals. Therefore, to make full use of the inherent relationship between in-phase and quadrature time series data, a complex-valued hybrid neural network (CV-PET-CSGDNN) based on the existing PET-CGDNN network is proposed in this paper, which consists of phase parameter estimation, parameter transformation, and complex-valued signal feature extraction layers. The complex-valued signal feature extraction layers are composed of complex-valued convolutional neural networks (CNN), complex-valued gate recurrent units (GRU), squeeze-and-excite (SE) blocks, and complex-valued dense neural networks (DNN). The proposed network can improve the extraction of the intrinsic relationship between in-phase and quadrature time series data with low capacity and then improve the accuracy of modulation classification. Experiments are carried out on RML2016.10a and RML2018.01a. The results show that, compared with ResNet, CLDNN, MCLDNN, PET-CGDNN, and CV-ResNet models, our proposed complex-valued neural network (CVNN) achieves the highest average accuracy of 61.50% and 62.92% for automatic modulation classification, respectively. In addition, the proposed CV-PET-CSGDNN has a significant improvement in the misjudgment situation between 64QAM, 128QAM, and 256QAM compared with PET-CGDNN on RML2018.01a.

Towards practical single-shot phase retrieval with physics-driven deep neural network.

Phase retrieval (PR), a long-established challenge for recovering a complex-valued signal from its Fourier intensity-only measurements, has attracted considerable attention due to its widespread applications in optical imaging. Recently, deep learning-based approaches were developed and allowed single-shot PR. However, due to the substantial disparity between the input and output domains of the PR problems, the performance of these approaches using vanilla deep neural networks (DNN) still has much room to improve. To increase the reconstruction accuracy, physics-informed approaches were suggested to incorporate the Fourier intensity measurements into an iterative estimation procedure. Since the approach is iterative, they require a lengthy computation process, and the accuracy is still not satisfactory for images with complex structures. Besides, many of these approaches work on simulation data that ignore some common problems such as saturation and quantization errors in practical optical PR systems. In this paper, a novel physics-driven multi-scale DNN structure dubbed PPRNet is proposed. Similar to other deep learning-based PR methods, PPRNet requires only a single Fourier intensity measurement. It is physics-driven that the network is guided to follow the Fourier intensity measurement at different scales to enhance the reconstruction accuracy. PPRNet has a feedforward structure and can be end-to-end trained. Thus, it is much faster and more accurate than the traditional physics-driven PR approaches. Extensive simulations and experiments on an optical platform were conducted. The results demonstrate the superiority and practicality of the proposed PPRNet over the traditional learning-based PR methods.

A Constellation Diagram Learning-Based Adaptive Sparse Nonorthogonal Wavelet Division Multiplexing for Sonar Image Underwater Acoustic Transmission

In underwater Internet of Things (IoT) applications, efficient and reliable transmission is demanding. Recently, the non-orthogonal multicarrier (MC) modulation is utilized as a promising spectrum-efficient solution in the UWA channel, where generally, the constellation mapping (CM) methods (such as 8-PSK and 16-QAM) are utilized to further increase the transmission rate. In this paper, we focus on the transmission of sonar images among IoT nodes and a constellation diagram learning-based adaptive sparse non-orthogonal wavelet division multiplexing (CDLA-SNOWDM) system is proposed to combat complex channel states between IoT nodes. We found that different CM methods contribute to unique time-frequency characteristics in constellation diagrams. We first utilize the time-frequency characteristics in designing the non-orthogonal subcarriers. The complex-valued CM signals are regarded as a new type of image. We introduce dictionary learning (DL) to non-orthogonal subcarrier designing, with the idea that designing CM signal adaptive non-orthogonal subcarriers can be regarded as designing an adaptive subcarrier dictionary. The CDLA-SNOWDM modulation is performed as a projection of the CM image to the subcarriers with an adaptive dictionary. The learned adaptive subcarriers can represent constellation diagrams more efficiently and improve transmission reliability. Both simulations and experiments show that the proposed scheme improved the reliability in transmitting sonar images over other orthogonal and non-orthogonal modulation schemes in UWA scenarios.

Weighted Decision Enabled Phase Retrieval Receiver With Adaptive Intensity Transformation

Phase-retrieval (PR) receivers can reconstruct complex-valued signals from two de-correlated intensity measurements by the use of iterative PR algorithm, without the assistance of any optical carriers. However, both the computational complexity with hundreds of iterations and the limited PR accuracy prevent it from being applied to high-speed photonic interconnections. In current submission, we propose a weighted decision enhanced PR receiver with adaptive intensity transformation (WD_AIT_PR). The weighted decision based on a compressed sigmoid nonlinear function is helpful to avoid the hard-decision (HD) errors, leading to the performance enhancement of the PR receiver, in terms of both convergence speed and PR accuracy. When the 56 GBaud 16QAM signal after transmission over 80 km standard single mode fiber (SSMF) is reconstructed, the WD_AIT_PR receiver is numerically verified to possess the faster convergency speed with only 70 iterations and the best steady BER performance by reducing the required optical signal-to-noise ratio (OSNR) by 0.8 dB, 1.8 dB and 1.8 dB in order to reach 7% HD forward error correction (HD-FEC), in comparison with the HD_AIT_PR receiver by replacing WD with HD, the PR receiver with the phase reset (PR_PR) and the PR receiver with adaptive intensity transformation (AIT_PR), respectively. Meanwhile, the WD_AIT_PR receiver is robust to transmission impairments, including transmitter-side I/Q imbalance, receiver-side skew, the noise imbalance between two intensity measurements, laser phase noise, and laser wavelength drift. Finally, the enhanced performance of the WD_AIT_PR receiver is experimentally verified through recovering 25 GBaud 16QAM signals under the condition of 40 km SSMF transmission.

Fast Adaptive Beamforming for Weather Observations with Convolutional Neural Networks

Polarimetric phased array radar (PAR) can achieve high temporal resolutions for improved meteorological observations with digital beamforming (DBF). The Fourier method performs DBF deterministically, and produces antenna radiation patterns with fixed sidelobe levels and angular resolution by pre-computing the beamforming weights based on the geometry of receivers. In contrast, the Capon method performs DBF adaptively in response to the changing environment by computing the beamforming weights from the received signals at multiple channels. However, it becomes computationally expensive as the number of receivers grows. This paper presents computationally efficient adaptive beamforming with an application of convolutional neural networks, named ABCNN. ABCNN is trained with the phase and amplitude of complex-valued time-series IQ signals and the Capon beamforming weights as input and output. ABCNN is tested and evaluated using simulated time-series data from both point targets and weather scatterers for a planar of fully digital PAR architecture. The preliminary results show that ABCNN lowers computation time by a factor of three, compared to the Capon method, for a phased array antenna with 1024 elements, while mitigating the contamination from sidelobes by placing nulls at the location of the clutter. Furthermore, ABCNN produces antenna patterns similar to those from the Capon method, which shows that it has successfully learned the data.