Complex Analytic Signal Research Articles

An Innovative EEG-Based Pain Identification and Quantification: A Pilot Study.

The present pilot study aimed to propose an innovative scale-independent measure based on electroencephalographic (EEG) signals for the identification and quantification of the magnitude of chronic pain. EEG data were collected from three groups of participants at rest: seven healthy participants with pain, 15 healthy participants submitted to thermal pain, and 66 participants living with chronic pain. Every 30 s, the pain intensity score felt by the participant was also recorded. Electrodes positioned in the contralateral motor region were of interest. After EEG preprocessing, a complex analytical signal was obtained using Hilbert transform, and the upper envelope of the EEG signal was extracted. The average coefficient of variation of the upper envelope of the signal was then calculated for the beta (13-30 Hz) band and proposed as a new EEG-based indicator, namely Piqβ, to identify and quantify pain. The main results are as follows: (1) A Piqβ threshold at 10%, that is, Piqβ ≥ 10%, indicates the presence of pain, and (2) the higher the Piqβ (%), the higher the extent of pain. This finding indicates that Piqβ can objectively identify and quantify pain in a population living with chronic pain. This new EEG-based indicator can be used for objective pain assessment based on the neurophysiological body response to pain. Objective pain assessment is a valuable decision-making aid and an important contribution to pain management and monitoring.

Ultrasonic array imaging of highly attenuative materials with spatio-temporal singular value decomposition

With the increasing use of high density polyethylene (HDPE) pipes in nuclear industry which puts safety at the forefront, ultrasonic array imaging methods play a vital role in the structural integrity of HDPE pipe materials. However, the viscoelastic attenuation of HDPE pipe materials significantly decreases the level of signals, leading to a low signal-to-noise ratio caused by electronic noise. In this work, a domain-adapted spatio-temporal singular value decomposition (STSVD) processing algorithm combined with the total focusing method is proposed to improve the ultrasonic array image quality. First, the real-valued radio frequency (RF) data or A-scan signals are demodulated into the complex analytic signals containing in-phase and quadrature (I/Q) components. Then, the STSVD processing algorithm is used to filter the I/Q data, and the filtered I/Q data is converted into RF signals. Finally, the total focusing method is applied to the processed RF signals to produce the image of the region under detection as a stage of post-processing. Experiments are carried out with an ultrasonic linear phased array in contact with the HDPE pipe materials containing multiple side-drilled holes and through-wall notches. Results show that the proposed method can produce images with high quality to provide good inspection and characterization of defects in highly attenuative materials, especially the deeper defects.

Estimating phase error using a Hilbert transform‐based time‐domain technique

SummaryMeasuring phase noise in oscillators is crucial in communication systems, vibration analysis, and frequency synthesizers. Traditionally, this measurement is done in frequency domain by estimating the ratio of the power density at an offset frequency from the carrier to the power of the carrier signal. This approach is hardware intensive and dependent on the the offset frequency, for which there exists no standard. Here, we propose an alternative method to quantify phase noise but in the time domain in the form of a root‐mean‐square true phase angle deviation. This is done using a self‐created reference signal and by applying the Hilbert transform to generate a complex analytic signal from the real time‐domain data sequence. This enables the computation of instantaneous metrics such as phase noise. Simulations and experimental results are provided to validate the proposed technique. In addition, its relationship to the origin of phase noise is mathematically explained.

Identification of ${S}_{\text{1}}$ and ${S}_{\text{2}}$ Heart Sounds Using Spectral and Convex Hull Features

A new set of morphological characteristics of Phonocardiogram (PCG) signal is presented for recognition of first ( ${S}_{{1}}$ ) and second ( ${S}_{{2}}$ ) heart sounds (HSs). Initially, variational mode decomposition on PCG signal generates a set of amplitude and frequency modulated narrow band-limited components (NBCs) and Hilbert transformation of these NBCs comprehends its complex plane analytic signal representation (ASR). Instantaneous spectral attributes encompassing amplitude modulation bandwidths and convex hull area measure from the ASRs are concatenated to form the feature set. Experimental results on both publicly available and experimentally recorded HSs signals outperform the existing state-of-the-art. Also, the proposed technique does not require any timing information between ${S}_{{1}}$ and ${S}_{{2}}$ and electrocardiogram (ECG) signal reference and is highly robust to noisy real-world PCGs as shown by noise analysis.

Deconvolution of Gaussian peaks with mixed real and discrete‐integer optimization based on evolutionary computing

This study describes an alternative method for deconvolution of overlapping characteristic Gauss peaks with the help of optimization of a mixed variable genetic algorithm. Continuous and discrete variables and nonlinear discrete variables in optimization problems cause solution complexity. The processing and analysis of complex analytical signals is important not only in analytical chemistry but also in other fields of science. As the amount of data increases and linearity decreases, high‐performance computations are needed to solve analytical signals. It takes a long time to perform these calculations with traditional processor systems and algorithms. We have used NVIDIA graphical processing units (GPUs) to shorten the duration of these calculations. Solving such analytical signals with genetic algorithms is widely used in computational sciences. In this study, we present a new curve‐fitting method using a genetic algorithm based on Gauss functions used to deconvolve overlapping peaks and find the exact peak number in absorption spectroscopy. The deconvolution of individual bands in the UV‐VIS region is a complex task, because the absorption bands are broad and often strongly overlap. Useful information about the molecular structure and environment can only be obtained by appropriate and truthful separation of these peaks.

Multi‐scale transient modelling and simulation for distribution network‐interactive CHB multilevel converters

This study presents a multi-scale transient model of cascaded H-bridge (CHB) converter for distribution network-interactive studies, which is capable of simulating slow-changing dynamics up to high-frequency switching transients. The switching pulse behaviours of H-bridges are described using the waveform propagation theory. By taking individual H-bridge sub-modules into account, a semi-analytic function (SAF) model of the CHB multilevel converter is developed. It allows for accelerated simulation of switch-type electromagnetic transients (EMTs). During slow-changing dynamics, the developed SAF is converted to a shifted-frequency phasor (SFP) model using the Fourier series expansion. Due to the consistent modulation and complex analytical signal processes, smooth transitions between the SAF and SFP are achieved for different running stages. Moreover, the proposed model is readily integrated with external multi-scale models of the distribution network. The validity of the proposed multi-scale model is performed through case studies that include the CHB internal fault, distribution network faults, and also interactive dynamics between them. The accuracy and efficiency of the multi-scale model is verified through comparisons of the results with those obtained from the full EMT-type programmes.

OPTIMIZATION OF DIRECT DIGITAL METHOD OF CORRELATIVE-INTERFEROMETRIC DIRECTION FINDING WITH RECONSTRUCTION OF SPATIAL ANALYTICAL SIGNAL

Context. At present, in automated radio monitoring systems, direction-finding of radio-electronic means is carried out under conditions of a complex electromagnetic environment, a large apriori uncertainty about the parameters of radio emissions, as well as real-time implementation. A promising direction for these conditions is the use of broadband correlation-interferometric radiodirection finders using digital processing of the complex spectrum of the received mixture of radio emissions.Objective. The aim of the article is to study and optimize the direct digital method of correlation-interferometric direction finding with reconstruction of the spatial analytical signal.Method. The paper analyzes the features of the implementation and accuracy of the direction-finding method studied, as well as analytical optimization of the non-exploratory digital method of correlation-interferometric direction finding with reconstruction of the spatial analytical signal.Results. The parameters that are included in the dispersion equation for the error in estimating the direction to the radio source for the non-exploratory digital method of correlation-interferometric direction finding with reconstruction of the spatial analytical signal that are to be optimized are determined. It is shown that the main parameters that it is advisable to optimize are the separation distance between the selected elements of the antenna array, for the spatial positions of which the complex analytical signal is reconstructed, and their numbers. A theoretical optimization of the parameters of the studied method, as well as a comparative analysis of analytical calculations and simulation results were carried out. As a result of modeling, the dependence of the method error of the bearing estimate and of the average square estimate of the bearing under the action of normal Gaussian noise on thevalues of the optimized parameters was obtained.Conclusions. A theoretical optimization of the parameters of the studied method has been carried out and it has been determined that it is advisable to use the symmetric separation of the 64-element antenna array by 28 steps to minimize the variance of the error in estimating the direction to the radiation sourse and to ensure maximum accuracy of direction finding.

On ‘Orbital’ and ‘Spin’ Angular Momentum of Light in Classical and Quantum Theories – A General Framework

AbstractWe develop a general framework to analyze two much discussed questions concerning (a) ‘orbital’ and ‘spin’ angular momentum carried by light and (b) the paraxial approximation of the free Maxwell system both in the classical as well as quantum domains. After formulating the classical free Maxwell system in the transverse gauge in terms of complex analytical signals we derive expressions for the constants of motion (COM) associated with its Poincaré symmetry. In particular, we show that the COM corresponding to the total angular momentum J naturally splits into an ‘orbital’ part L and a ‘spin’ part S each of which is a COM in its own right. We then discuss quantization of the free Maxwell system and construct the operators generating the Poincaré group and analyze their algebraic properties and find that while the quantum counterparts and of L and S go over into bona fide observables, they fail to satisfy the angular momentum algebra making their interpretation as ‘orbital’ and ‘spin’ operators untenable at the quantum level. On the other hand does satisfy the angular momentum algebra and together with generates the group E(3). We then present an analysis of single photon states, paraxial quantization both in the scalar as well as vector cases, and single photon states in the paraxial regime. All along a close connection is maintained with the Hilbert space that naturally arises in the classical context.

Position estimation of active magnetic bearing shaft using auxiliary coils

PurposeThe purpose of this paper is to develop a system for estimating the position of the active magnetic bearing (AMB) shaft. A new approach using the static and dynamic inductances and complex analytic signal to simplify the estimation procedure. Finite element (FE) simulations are introduced as a part of the system synthesis.Design/methodology/approachThe paper presents an AMB displacement estimation system. The system is created with three inductive sensors. The position is computed from refined static and dynamic inductance obtained from complex analytic signals of flux and current. FE simulation is used to relate refined inductances to the displacement and to verify the model.FindingsThis paper shows the applicability of complex analytic signal transformation on estimation systems. The use of new refined inductance is presented in contrast to the classical approach of static and dynamic inductances. The paper shows that classical approach of static and dynamic inductance is not usable for the presented estimation system.Practical implicationsFor the practical implementation of the presented system, it is necessary to know the exact dimensions of the AMB stator and the voltage and frequency used to supply the inductance estimation system.Originality/valueThe paper presents a system for estimating the displacement of AMB. The paper introduces the application of complex analytic signal to the estimation of AMB displacement. The mentioned signal is used to compute the new refined inductances. The comparison to the classical approach of static and dynamic inductances is given in this paper. The paper introduces FE simulations to the estimation system synthesis.

Multisensor Systems by Electrochemical Nanowire Assembly for the Analysis of Aqueous Solutions.

The development of electrochemical multisensor systems is driven by the need for fast, miniature, inexpensive, analytical devices, and advanced interdisciplinary based on both chemometric and (nano)material approaches. A multicomponent analysis of complex mixtures in environmental and technological monitoring, biological samples, and cell culture requires chip-based multisensor systems with high-stability sensors. In this paper, we describe the development, characterization, and applications of chip-based nanoelectrochemical sensor arrays prepared by the directed electrochemical nanowire assembly (DENA) of noble metals and metal alloys to analyze aqueous solutions. A synergic action of the electrode transducer function and electrocatalytic activity of the nanostructured surface toward analytes is achieved in the assembled metal nanowire (NW) sensors. Various sensor nanomaterials (Pd, Ni, Au, and their multicomponent compositions) can be electrochemically assembled on a single chip without employing multiple cycles of photolithography process to realize multi-analyte sensing applications as well as spatial resolution of sensor analysis by this single-chip multisensor system. For multi-analyte electrochemical sensing, individual amperometric signals of two or more nanowires can be acquired, making use of the specific electrocatalytic surface properties of the individual nanowire sensors of the array toward analytes. To demonstrate the application of a new electrochemical multisensor platform, Pd-Au, Pd-Ni, Pd, and Au NW electrode arrays on a single chip were employed for the non-enzymatic analysis of hydrogen peroxide, glucose, and ethanol. The analytes are determined at low absolute values of the detection potentials with linear concentration ranges of 1.0 × 10−6 − 1.0 × 10−3 M (H2O2), 1.5 × 10−7 − 2.0 × 10−3 M (glucose), and 0.7 × 10−3 − 3.0 × 10−2 M (ethanol), detection limits of 2 × 10−7 M (H2O2), 4 × 10−8 M (glucose), and 5.2 × 10−4 M (ethanol), and sensitivities of 18 μA M−1 (H2O2), 178 μA M−1 (glucose), and 28 μA M−1 (ethanol), respectively. The sensors demonstrate a high level of stability due to the non-enzymatic detection mode. Based on the DENA-assembled nanowire electrodes of a compositional diversity, we propose a novel single-chip electrochemical multisensor platform, which is promising for acquiring complex analytical signals for advanced data processing with chemometric techniques aimed at the development of electronic tongue-type multisensor systems for flexible multi-analyte monitoring and healthcare applications.

On using the Hilbert transform for blind identification of complex modes: A practical approach

The modal identification of dynamical systems under operational conditions, when subjected to wide-band unmeasured excitations, is today a viable alternative to more traditional modal identification approaches based on processing sets of measured FRFs or impulse responses. Among current techniques for performing operational modal identification, the so-called blind identification methods are the subject of considerable investigation. In particular, the SOBI (Second-Order Blind Identification) method was found to be quite efficient. SOBI was originally developed for systems with normal modes. To address systems with complex modes, various extension approaches have been proposed, in particular: (a) Using a first-order state-space formulation for the system dynamics; (b) Building complex analytic signals from the measured responses using the Hilbert transform. In this paper we further explore the latter option, which is conceptually interesting while preserving the model order and size. Focus is on applicability of the SOBI technique for extracting the modal responses from analytic signals built from a set of vibratory responses. The novelty of this work is to propose a straightforward computational procedure for obtaining the complex cross-correlation response matrix to be used for the modal identification procedure. After clarifying subtle aspects of the general theoretical framework, we demonstrate that the correlation matrix of the analytic responses can be computed through a Hilbert transform of the real correlation matrix, so that the actual time-domain responses are no longer required for modal identification purposes. The numerical validation of the proposed technique is presented based on time-domain simulations of a conceptual physical multi-modal system, designed to display modes ranging from normal to highly complex, while keeping modal damping low and nearly independent of the modal complexity, and which can prove very interesting in test bench applications. Numerical results for complex modal identifications are presented, and the quality of the identified modal matrix and modal responses, extracted using the complex SOBI technique and implementing the proposed formulation, is assessed.

Imaginary part-based correlation mapping optical coherence tomography for imaging of blood vessels in vivo.

We present an imaginary part-based correlation mapping optical coherence tomography (IMcmOCT) technique for in vivo blood vessels imaging. In the conventional correlation mapping optical coherence tomography (cmOCT) method, two adjacent frames of intensity-based structural images are correlated to extract blood flow information and the size of correlation window has to be increased to improve the signal-to-noise ratio of microcirculation maps, which may cause image blur and miss the small blood vessels. In the IMcmOCT method, the imaginary part of a depth-resolved complex analytic signal in two adjacent B-scans is correlated to reconstruct microcirculation maps. Both phantom and in vivo experiments were implemented to demonstrate that the proposed method can provide improved sensitivity for extracting blood flow information in small vessels.

Broadband performance of superdirective delay-and-sum beamformers steered to end-fire.

Linear arrays steered to end-fire provide superdirective robust performance if a constraint is imposed on the white-noise gain. Filter-and-sum beamformers achieve the maximum constrained directivity by tuning their complex weights over the frequency. Delay-and-sum beamformers have simpler structures, but their weights are fixed and optimized at a given frequency. This letter investigates the constrained directivity provided over a broad band by different delay-and-sum techniques. Complex weights and analytic signals attain near-optimal broadband performance over four octaves. Oversteered arrays using real weights and signals were found to attain superdirective performance over approximately two octaves. Hearing aids and directional hydrophones are potential applications for the considered arrays.

Correlation properties of the vector signal representation for speckle pattern.

In one-dimensional (1D) signal analysis, the complex analytic signal built from a real-valued signal and its Hilbert transform is an important tool providing a mathematical foundation for 1D statistical analysis. For a natural extension beyond 1D signal, Riesz transform has been applied to high-dimensional signal processing as a generalized Hilbert transform to construct a vector signal representation and therefore, to enlarge the traditional analytic signal concept. In this paper, we introduce the vector correlations as new mathematical tools for vector calculus for statistical speckle analysis. Based on vector correlations of a real-valued speckle pattern, we present the associated correlation properties, which can be regarded as mathematical foundation for the vector analysis in speckle metrology.

Spectral characteristic of a pulsed hollow Gaussian beam passing through a circular aperture

The spectral characteristics of a pulsed hollow Gaussian beam passing through a circular aperture are studied. Based on the vectorial Rayleigh diffraction integrals, the analytical expressions of the spectra for the complex analytic signal representation and for the complex amplitude envelope representation are derived, respectively, and the comparison between them is made. The influences of the truncation parameter and the beam order on the spectral shifts and on the spectral switches are illustrated. It is shown that the spectrum for the complex analytic signal representation and that for the complex amplitude envelope representation are obvious different as the pulse shorter than an optical period.

Dual tree fractional quaternion wavelet transform for disparity estimation

This paper proposes a novel phase based approach for computing disparity as the optical flow from the given pair of consecutive images. A new dual tree fractional quaternion wavelet transform (FrQWT) is proposed by defining the 2D Fourier spectrum upto a single quadrant. In the proposed FrQWT, each quaternion wavelet consists of a real part (a real DWT wavelet) and three imaginary parts that are organized according to the quaternion algebra. First two FrQWT phases encode the shifts of image features in the absolute horizontal and vertical coordinate system, while the third phase has the texture information. The FrQWT allowed a multi-scale framework for calculating and adjusting local disparities and executing phase unwrapping from coarse to fine scales with linear computational efficiency.

Quasi-analytic multidimensional signals

Abstract In a recent paper, the authors have presented the unified theory of n-dimensional (n-D) complex and hypercomplex analytic signals with single-orthant spectra. This paper describes a specific form of these signals called quasi-analytic. A quasi-analytic signal is a product of a n-D low-pass (base-band) real (in general non-separable) signal and a n-D complex or hypercomplex carrier. By a suitable choice of the carrier frequency, the spectrum of a low-pass signal is shifted into a single orthant of the Fourier frequency space with a negligible leakage into other orthants. A measure of this leakage is defined. Properties of quasi-analytic signals are studied. Problems of polar representation of quasi-analytic signals and of its lower rank representation are discussed.

Deterministic signal associated with a random field

Stochastic fields do not generally possess a Fourier transform. This makes the second-order statistics calculation very difficult, as it requires solving a fourth-order stochastic wave equation. This problem was alleviated by Wolf who introduced the coherent mode decomposition and, as a result, space-frequency statistics propagation of wide-sense stationary fields. In this paper we show that if, in addition to wide-sense stationarity, the fields are also wide-sense statistically homogeneous, then monochromatic plane waves can be used as an eigenfunction basis for the cross spectral density. Furthermore, the eigenvalue associated with a plane wave, exp[i(k · r-ωt)], is given by the spatiotemporal power spectrum evaluated at the frequency (k, ω). We show that the second-order statistics of these fields is fully described by the spatiotemporal power spectrum, a real, positive function. Thus, the second-order statistics can be efficiently propagated in the wavevector-frequency representation using a new framework of deterministic signals associated with random fields. Analogous to the complex analytic signal representation of a field, the deterministic signal is a mathematical construct meant to simplify calculations. Specifically, the deterministic signal associated with a random field is defined such that it has the identical autocorrelation as the actual random field. Calculations for propagating spatial and temporal correlations are simplified greatly because one only needs to solve a deterministic wave equation of second order. We illustrate the power of the wavevector-frequency representation with calculations of spatial coherence in the far zone of an incoherent source, as well as coherence effects induced by biological tissues.

Diffraction of broadband pulsed light beams

Intensity of broadband light pulses propagating in the free space has been calculated in the approximation of the narrowband signal envelope and based on the complex analytic signal. The approximation of the narrowband signal envelope is shown to be applicable for calculating the diffraction of pulsed light beams regardless of the duration and coherence of the pulse in space and time.

IMAGE DISPARITY ESTIMATION BASED ON FRACTIONAL DUAL-TREE COMPLEX WAVELET TRANSFORM: A MULTI-SCALE APPROACH

In this paper, an efficient multi-scale image disparity estimation algorithm is proposed which estimates the local displacements needed to align different regions between a pair of images. This phase-based approach is based on fractional dual-tree complex wavelet transform (FrDTCWT). In the proposed FrDTCWT, initially we obtained the fractional Fourier transform of the image. We decomposed the fractional transformed image by dual-tree complex wavelet transform into real and imaginary parts. The complex analytic signal is obtained from the real-valued function which is obtained from this FrDTCWT. The FRDTCWT inherits the excellent mathematical properties of DTCWT and FrFT. Phase and amplitude of the image are computed from the complex analytic signal. The disparity is estimated as the optical flow using the phase difference method. The efficiency of the proposed algorithm is carried out by different experiments on synthetic as well as on realistic image sequences.