Fourier Series Coefficients Research Articles

Signal Decomposition for Monitoring Systems of Processes

This article is devoted to the problem of signal decomposition into periodic and aperiodic components. According to the proposed approach, there is no need to evaluate the aperiodic component as a difference between the total signal of its periodic components. This research aims to create a general analytical approach that combines the Fourier and Maclaurin series methodologies into a single comprehensive series. As a result, analytical expressions for determining deposition coefficients were established for an aperiodic signal with a monoharmonic overlay. Recurrence relations were established to determine the coefficients of this series. These relations allow direct integrations of the obtained values of integrals to be avoided. The evaluated numerical values of the coefficients are also presented graphically and tabulated. It was proven that the values of these coefficients are universal numbers since they do not depend on the period/frequency of oscillations. The reliability of the proposed approach was confirmed by the fact that the evaluated coefficients are equal to the Fourier series coefficients in the case of a periodic signal. Also, for an aperiodic signal, these coefficients were reduced to the coefficients of the Maclaurin series. The usability of the proposed generalized analytical approach for signal decomposition is for control and monitoring systems of processes.

Towards a Unified Approach for Continuously-Variable Impedance Control of Powered Prosthetic Legs over Walking Speeds and Inclines.

Research in powered prosthesis control has explored the use of impedance-based control algorithms due to their biomimetic capabilities and intuitive structure. Modern impedance controllers feature parameters that smoothly vary over gait phase and task according to a data-driven model. However, these recent efforts only use continuous impedance control during stance and instead utilize discrete transition logic to switch to kinematic control during swing, necessitating two separate models for the different parts of the stride. In contrast, this paper presents a controller that uses smooth impedance parameter trajectories throughout the gait, unifying the stance and swing periods under a single, continuous model. Furthermore, this paper proposes a basis model to represent intertask relationships in the impedance parameters-a strategy that has previously been shown to improve model accuracy over classic linear interpolation methods. In the proposed controller, a weighted sum of Fourier series is used to model the impedance parameters of each joint as continuous functions of gait cycle progression and task. Fourier series coefficients are determined via convex optimization such that the controller best reproduces the joint torques and kinematics in a reference able-bodied dataset. Experiments with a powered knee-ankle prosthesis show that this simpler, unified model produces competitive results when compared to a more complex hybrid impedance-kinematic model over varying walking speeds and inclines.

Interpolation problems for random fields on Sierpinski’s carpet

The prediction of stochastic processes and the estimation of random fields of different natures is becoming an increasingly common field of research among scientists of various specialties. However, an analysis of papers across different estimating problems shows that a dynamic approach over an iterative and recursive interpolation of random fields on fractal is still an open area of investigation. There are many papers related to the interpolation problems of stationary sequences, estimation of random fields, even on the perforated planes, but all of this still provides a place for an investigation of a more complicated structure like a fractal, which might be more beneficial in appliances of certain industry fields. For example, there has been a development of mobile phone and WiFi fractal antennas based on a first few iterations of the Sierpinski carpet. In this paper, we introduce an estimation for random fields on the Sierpinski carpet, based on the usage of the known spectral density, and calculation of the spectral characteristic that allows an estimation of the optimal linear functional of the omitted points in the field. We give coverage of an idea of stationary sequence estimating that is necessary to provide a basic understanding of the approach of the interpolation of one or a set of omitted values. After that, the expansion to random fields allows us to deduce a dynamic approach on the iteration steps of the Sierpinski carpet. We describe the numerical results of the initial iteration steps and demonstrate a recurring pattern in both the matrix of Fourier series coefficients of the spectral density and the result of the optimal linear functional estimation. So that it provides a dependency between formulas of the different initial sizes of the field as well as a possible generalizing of the solution for N-steps in the Sierpinski carpet. We expect that further evaluation of the mean squared error of this estimation can be used to identify the possible iteration step when further estimation becomes irrelevant, hence allowing us to reduce the cost of calculations and make the process viable.

A novel algorithm for open switch fault detection and fault tolerant control of interleaved DC‐DC boost converters

AbstractThis paper presents a novel algorithm for fault diagnosis and fault tolerant control of interleaved boost converters (IBCs) in the presence of single or double simultaneous open‐circuit faults (OCFs) in power switches. An innovative diagnosis signal (recorded by a cheap current sensor) will be introduced whose waveform mainly depends on the healthy or faulty condition of the converter. Since the diagnosis signal is periodic in the steady‐state operation of the converters, its Fourier Series coefficients, together with the duty cycle of the converter are used as distinguishing features for fault diagnosis of the OCFs in the converter. The well‐known Kalman filter is utilized for robust estimation of the Fourier‐based features. Finally, the modulation of the remaining healthy phases is rearranged in a way that the pre‐fault performance recovers. The proposed algorithm is verified in a laboratory four‐phase IBC setup in which the experimental results show its satisfactory performance. Also, the structure of the proposed method is straightforward and can be implemented in the same microcontroller which is used for voltage and current regulation of the converter.

Optimal with Respect to Accuracy Recovery of Some Classes Functions by Fourier Series

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained. Keywords: function approximation, Fourier series, Fourier series coefficients, approximation error, computational complexity.

Fourier Analysis of the Vertical Ground Reaction Force During Walking: Applications for Quantifying Differences in Gait Strategies.

Time series biomechanical data inform our understanding of normal gait mechanics and pathomechanics. This study examines the utility of different quantitative methods to distinguish vertical ground reaction forces (VGRFs) from experimentally distinct gait strategies. The goals of this study are to compare measures of VGRF data-using the shape factor method and a Fourier series-based analysis-to (1)describe how these methods reflect and distinguish gait patterns and (2)determine which Fourier series coefficients discriminate normal walking, with a relatively stiff-legged gait, from compliant walking, using deep knee flexion and limited vertical oscillation. This study includes a reanalysis of previously presented VGRF data. We applied the shape factor method and fit 3- to 8-term Fourier series to zero-padded VGRF data. We compared VGRF renderings using Euclidean L2 distances and correlations stratified by gait strategy. Euclidean L2 distances improved with additional harmonics, with limited improvement after the seventh term. Euclidean L2 distances were greater in shape factor versus Fourier series renderings. In the 8 harmonic model, amplitudes of 9 Fourier coefficients-which contribute to VGRF features including peak and local minimum amplitudes and limb loading rates-were different between normal and compliant walking. The results suggest that Fourier series-based methods distinguish between gait strategies.

Spectrum Analysis for Fully Connected Neural Networks.

This article studies the meaning of parameters of fully connected neural networks with single hidden layer from the perspective of spectrum. Under the constraints of numerical range, the corresponding relationship between parameters and the spectrum of network function can be established by the Fourier series coefficients of the activation function, which is truncated and periodically extended. This work is substantiated on the Mixed National Institute of Standards and Technology (MNIST) handwritten dataset and two illustrative examples with certain spectra. The simulations complete the conversion between spectrum and parameters with high precision and give the significance of hidden nodes to the spectrum of network function. Some algorithms derived from these properties, such as the parameter initialization method using spectrum and the pruning method by sorting amplification weights, are also presented to introduce how spectrum analysis affects neural network decision-making. Thus, spectrum analysis has great potential in network interpretation.

DAGGER: Data AuGmentation GEneRative Framework for Time-Series Data in Data-Driven Smart Manufacturing Systems

As industries transition into the Industry 4.0 paradigm, the relevance and interest in concepts like Digital Twin (DT) are at an all-time high. DTs offer direct avenues for industries to make more accurate predictions, rational decisions, and informed plans, ultimately reducing costs, increasing performance and productivity. Adequate operation of DTs in the context of smart manufacturing relies on an evolving data-set relating to the real-life object or process, and a means of dynamically updating the computational model to better conform to the data. This reliance on data is made more explicit when physics-based computational models are not available or difficult to obtain in practice, as it's the case in most modern manufacturing scenarios. For data-based model surrogates to "adequately" represent the underlying physics, the number of training data points must keep pace with the number of degrees of freedom in the model, which can be on the order of thousands. However, in niche industrial scenarios like the one in manufacturing applications, the availability of data is limited (on the order of a few hundred data points, at best), mainly because a manual measuring process typically must take place for a few of the relevant quantities, e.g., level of wear of a tool. In other words, notwithstanding the popular notion of big-data, there is still a stark shortage of ground-truth data when examining, for instance, a complex system's path to failure. In this work we present a framework to alleviate this problem via modern machine learning tools, where we show a robust, efficient and reliable pathway to augment the available data to train the data-based computational models.
 Small sample size data is a key limitation in performance in machine learning, in particular with very high dimensional data. Current efforts for synthetic data generation typically involve either Generative Adversarial Networks (GANs) or Variational AutoEncoders (VAEs). These, however, are are tightly related to image processing and synthesis, and are generally not suited for sensor data generation, which is the type of data that manufacturing applications produce. Additionally, GAN models are susceptible to mode collapse, training instability, and high computational costs when used for high dimensional data creation. Alternatively, the encoding of VAEs greatly reduces dimensional complexity of data and can effectively regularize the latent space, but often produces poor representational synthetic samples. Our proposed method thus incorporates the learned latent space from an AutoEncoder (AE) architecture into the training of the generation network in a GAN. The advantages of such scheme are twofold: \textbf{(\textit{i})} the latent space representation created by the AE reduces the complexity of the distribution the generator must learn, allowing for quicker discriminator convergence, and \textbf{(\textit{ii})} the structure in the sensor data is better captured in the transition from the original space to the latent space. Through time statistics (up to the fifth moment), ARIMA coefficients and Fourier series coefficients, we compare the synthetic data from our proposed AE+GAN model with the original sensor data. We also show that the performance of our proposed method is at least comparable with that of the Riemannian Hamiltonian VAE, which is a recently published data augmentation framework specifically designed to handle very small high dimensional data sets.

Time-domain response of supercapacitors using their impedance parameters and Fourier series decomposition of the excitation signal

Supercapacitors are mostly recognized for their high power density capabilities and fast response time when compared to secondary batteries. However, computing their power in response to a given excitation using the standard formulæof capacitors is misleading and erroneous because supercapacitors are actually non-ideal capacitive devices that cannot be characterized with a single constant capacitance. In this study we show how to estimate accurately the time-domain power and energy of supercapacitors in response to any excitation signal represented in terms of its Fourier series coefficients with the sole knowledge of the frequency-domain impedance parameters of the device. The presented theory is first verified and validated with Spice simulations conducted on an RC network emulating the behavior of a circuit consisting of a resistance (Rs) in series with a constant phase element (CPE) of fractional impedance ZCPE=1/(sCα). Then we do the same for a commercial supercapacitor modeled as an Rs-CPE circuit, and subjected to both a periodic triangular voltage waveform and a random voltage excitation. The results are conclusive and very promising for adopting the proposed procedure to estimate the power and energy performance of supercapacitors in response to real-world charging and discharging signals.

Application of stochastic filter to three-phase nonuniform transmission lines

ABSTRACT This paper implements the stochastic filters for state and parameter estimation of nonuniform transmission lines (NTL). In general, transmission line (TL) problem is a continuous time and space problem. By taking the line loading noise into account, the TL equations become a stochastic partial differential equation (PDE) rather than a simple set of coupled finite stochastic differential equations (SDE). By transforming the spatial variables into the Fourier domain, the stochastic PDE can be transformed into an infinite sequence of SDE. After truncation to a finite set of Fourier series coefficients, it becomes a finite set of coupled linear SDE, which is the required domain in which extended Kalman filter (EKF) and unscented Kalman filter (UKF) can be applied. For state space equation of EKF and UKF, the voltage and current of periodic NTL are expanded into an infinite set of spatial harmonics. In this way, the voltage and current measurement of NTL become an eigenvalue problem. The NTL is considered as cascade of small infinite NTL and the four distributed primary parameters of the periodic NTL are expressed using Fourier series expansion. Finally, the Kalman filter (KF)-based state estimation and the EKF- and UKF-based parameter estimation have been compared with recursive least squares (RLS) method. The simulation results present the superiority of the method.

Energy-independent complex single-P-wavesNNpotential from the Marchenko equation

We extend our previous results of solving the inverse problem of quantum scattering theory (Marchenko theory, fixed-$l$ inversion). In particular, we apply an isosceles triangular-pulse function set for the Marchenko equation input kernel expansion in a separable form. The separable form allows a reduction of the Marchenko equation to a system of linear equations for the output kernel expansion coefficients. We show that in the general case of a single partial wave, a linear expression of the input kernel is obtained in terms of the Fourier series coefficients of $q^{1-m}(1-S(q))$ functions in the finite range of the momentum $0\leq q\leq\pi/h$ [$S(q)$ is the scattering matrix, $l$ is the angular orbital momentum, $m=0,1,\dots,2l$]. Thus, we show that the partial $S$--matrix on the finite interval determines a potential function with $h$-step accuracy. The calculated partial potentials describe a partial $S$--matrix with the required accuracy. The partial $S$--matrix is unitary below the threshold of inelasticity and non--unitary (absorptive) above the threshold. We developed a procedure and applied it to partial-wave analysis (PWA) data of $NN$ elastic scattering up to 3 GeV. We show that energy-independent complex partial potentials describe these data for single $P$-waves.

Describing function method with pointwise balancing in two-dimensional regularized time domain for quasi-periodic responses

This paper proposes a new semi-analytical approach, called as describing function method with pointwise balancing (DFPB), to solve quasi-periodic (QP) responses of nonlinear dynamical systems especially those with non-smooth nonlinearities. This method is to describe the QP motion by generalized Fourier series with multiple regularized time scales. By discretizing the solution in the regularized time domain, the original equation is transformed into over-determined algebraic equations governing Fourier series coefficients via pointwise balancing. The coefficients are determined by solving the balance equations via the Newton–Raphson iteration algorithm. In addition, the Tikhonov regularization is utilized to enhance the convergence of the iteration scheme, so that the least squares solution of the over-determined equations can be obtained efficiently. On the one hand, the pointwise balancing avoids repeated transformations of non-smooth functions between time- to frequency-domain. On the other hand, the regularized time scales make it convenient to discretize the system equations regardless of unknown fundamental frequencies. With these procedures, the DFPB is well-designed to solve non-smooth systems for QP response with frequencies to be determined. In the numerical examples, the effectiveness and accuracy are elucidated in detail through both smooth and non-smooth models. The DFPB results are in excellent agreement with those attained by the Runge–Kutta method. In addition, it is worthy of pointing out the DFPB can track both stable and unstable QP responses.

PROGRAMACIÓN DE INTERFAZ GRÁFICA EN APP DESIGNER DEL MATLAB PARA REPRESENTAR LA SERIE DE FOURIER EN CURSO INTRODUCTORIO DE TELECOMUNICACIONES

This article describes the procedure for the implementation of a graphical user interface using the App Designer development environment of the Matlab software and its Toolbox Signal Processing and Symbolic Math, which allowed obtaining the Fourier Series coefficients for seven periodic signals followed by a graphical representation in the domain of time and frequency through pulses. Likewise, this 
 interface was implemented using various components such as Drop Down, Button, Knob, Axes, among others, which allowed the choice of the number of sinusoidal components used in the decomposition, as well as the choice of the type of signal, its amplitude, frequency, DC level and duration time of the same. Therefore, this implementation complements the teaching-learning process developed remotely for a topic in the Telecommunications I course of the 6th academic semester of the Electronic Engineering Degree at Ricardo Palma University. Finally, the graphical results are presented for the case of four choices of periodic signals, with different amplitudes, frequencies and DC levels; as well as the numerical results in terms of the mean square error, showing a lower value for the triangular signal despite using a reduced number of sinusoidal components

Impact on the beam of the final length of the moving mass

There have been considered solving direct problem of deforming of isotropic, elastic and hingedly supported beam of finite length. A roller moving at a constant speed along the axis of the beam acts on the beam. The roller has a cylindrical shape of a certain radius and a length that is greater than or equal to the width of the beam. The differential equations of beam motion are analyzed from the point of view of the influence of their components and especially the right-hand parts of the equations on the dynamic behavior of the beam in the case of using certain common materials of the beam and roller, and an option to reduce the equations to a more simplified form is proposed. The unknown functions included in the equations are sought in the form of Fourier series. This allows us to reduce the original equations to ordinary differential equations, which are solved using the Laplace transform. Expressions for coefficients in Fourier series are found using operational calculus and the residue theory. The results of the first numerical experiment on the study of the influence of the roller speed on beam deflections are presented in the form of curves in the figure. For a specific calculated mechanical system in the form of a steel roller, which moves along a steel beam at a constant speed under zero initial conditions, the research results are presented in the form of graphs of beam deflections for different speeds of the roller. The second numerical experiment was carried out to study the propagation of vibrational waves of the beam in the case of motion of the roller at a sufficiently high speed. For this, the figure shows the combined shapes of the beam and the position of the roller at different moments of action of the moving mass. The behavior of the beam at a high speed of movement of the roller was analyzed and a comparison of the deflections of the beam with the deflections of the static model of the beam was made. Further directions for the development of the problem in applied fields of technology and in inverse problems of identifying unknown parameters by indirect manifestations are outlined.

Avoiding the bit-reversed ordering in parallel multi-digit multiplication based on FFT

The paper for the parallel model of computation, a modification of the method of implementing the multiplication of multi-digit integers based on the fast Fourier transform (FFT) avoiding the bit-reversed ordering is proposed. The paper researches the calculation of FFT according to the “butterfly” scheme based on decimation-in- frequency and decimation-in-time methods, an input signal with elements in direct and bit-reversed order, with an increase and decrease in the number Fourier series coefficients at each step of the "butterfly", the use of a list of Fourier series coefficients in direct and bit-reversed order. The standard FFT-based multiplication algorithm uses the same “butterfly” operation to compute the forward and inverse Fourier transforms. The paper analyzes two combinations of the FFTFDN–FFTTBN and FFTFBN–FFTTDN “butterfly” calculation schemes for calculating forward and inverse discrete Fourier transforms (DFT) in the case of implementing the multi-digit operation in parallel computational model to exclude bit-reversed permutation. A scheme for distributing calculations among four processors is proposed, in which forward and inverse Fourier transform calculations are localized within one parallel processor. The proposed modification does not reduce the computational complexity in terms of the number of complex operations, but due to the exclusion of bit-reversed permutation, the number of synchronization commands between processors and data is reduced, which reduces the algorithm execution time. The scheme can be adapted to distribute the computations among a larger number of processors. Four algorithms for implementing FFT based on decimation-in-frequency and decimation-in-time methods, an input vector with elements in direct and bit-reversed orders are presented. To check the result of the calculation, the algorithm of multiplication avoiding the steps of bit-reversed ordering was implemented in the APL programming language. An example of calculation is given in the form of a table.

Pseudo high-frequency boosts the generalization of a convolutional neural network for cassava disease detection

Frequency is essential in signal transmission, especially in convolutional neural networks. It is vital to maintain the signal frequency in the neural network to maintain the performance of a convolutional neural network. Due to destructive signal transmission in convolutional neural network, signal frequency downconversion in channels results into incomplete spatial information. In communication theory, the number of Fourier series coefficients determines the integrity of the information transmitted in channels. Consequently, the number of Fourier series coefficients of the signals can be replenished to reduce the information transmission loss. To achieve this, the ArsenicNetPlus neural network was proposed for signal transmission modulation in detecting cassava diseases. First, multiattention was used to maintain the long-term dependency of the features of cassava diseases. Afterward, depthwise convolution was implemented to remove aliasing signals and downconvert before the sampling operation. Instance batch normalization algorithm was utilized to keep features in an appropriate form in the convolutional neural network channels. Finally, the ArsenicPlus block was implemented to generate pseudo high-frequency in the residual structure. The proposed method was tested on the Cassava Datasets and compared with the V2-ResNet-101, EfficientNet-B5, RepVGG-B3g4 and AlexNet. The results showed that the proposed method performed 95.93% in terms of accuracy, 1.2440 in terms of loss, and 95.94% in terms of the F1-score, outperforming the comparison algorithms.

Development of Broken Rotor Bar Fault Diagnosis Method with Sum of Weighted Fourier Series Coefficients Square

This study proposes a broken rotor bar (BRB) fault diagnosis method for an induction motor using the sum of the weighted Fourier series coefficients squares of a complex current as a diagnosis signal. First, the sum of the squares of the Fourier series coefficients confirms the very narrow band-pass filter characteristics to derive a specific frequency component. This assists us in obtaining a BRB fault diagnosis signal that exists in a limited frequency range. Second, the magnitude of the Fourier series coefficients of the BRB fault signal is proportional to the slip frequency and load condition. A weighting factor is proposed to render the BRB fault signal irrelevant to the slip frequency and load condition. Consequently, the proposed fault diagnosis can be conducted without the slip frequency information or searching for the maximum coefficient component. Finally, the proposed fault diagnosis method is validated through experiments using a 55 kW induction motor with and without a BRB fault. It is implemented with a DSP controller at time intervals of 20, 10, 5, and 4 s for the Fourier series. The proposed diagnosis method performs well under various load conditions and shows that the derived fault signal exhibits a large difference between healthy and BRB faulty induction motors.

State and Parameter Estimation of Non-Uniform Transmission Line Using Kronecker Product Based Modeling

The distributed parameters of a non-uniform transmission line (NTL) are assumed to depend on a finite set of unknown parameters, for example, their spatial Fourier series coefficients. The line equations are set up taking random voltage and current line loading into account with the random loading being assumed to be white Gaussian noise in the time and spatial variables. Bypassing over to the spatial Fourier series domain, these line stochastic differential equations (SDEs) are expressed as a truncated sequence of coupled stochastic differential equations for the spatial Fourier series components of the line voltage and current. The coefficients that appear in these stochastic differential equations are the spatial Fourier series coefficients of the distributed line parameters and these coefficients in turn are linear functions of the unknown parameters. By separating these complex stochastic differential equations into their real and imaginary part, the line equations become a finite set of SDEs for the extended state defined to be the real and imaginary components of the line voltage and current as well as the parameters on which the distributed parameters depend. The measurement model is assumed to be white Gaussian noise corrupted versions of the line voltage and current sampled at a discrete set of spatial points on the line and this measurement model can be expressed as a linear transformation of the state variables, i.e, of the spatial Fourier series components of the line voltage and current plus a white measurement noise vector. This work uses sparse matrix factorization using Kronecker product to implement the Fourier unitary transform. This work implements perturbation theory for non-uniform transmission line. The extended Kalman filter (EKF) equations are derived for this state and measurement model and our real timeline voltage and current and distributed parameter estimates show excellent agreement with the true values based on MATLAB simulations. The use of sparse matrix factorization using Kronecker product ease the mathematical complexity and provides compact representation.

Discrete Time Sequence Reconstruction of a Signal Based on Local Approximation Using a Fourier Series by an Orthogonal System of Trigonometric Functions

The article considers the development of mathematical and algorithmic support for the sample’s reconstruction in problem sections of a discrete sequence of a continuous signal. The work aimed to ensure the reconstruction of lost samples or sections of samples with a non-constant distorted time grid when sampling a signal with a uniform step and at the same time to reduce the computational complexity of digital reconstruction algorithms. The solution to the stated problem is obtained based on the local approximation method. The specific of this method application was the use of two subsequences of samples located symmetrically concerning the reconstructed section of the sequence. The approximating model is a Fourier series on an orthogonal system of trigonometric functions. The optimal solution to the approximation problem is based on the minimum square error criterion. Mathematical equations are obtained for this type of error. They allow us to estimate its value depending on the model order and the samples number in the subsequences used in the reconstruction process. The peculiarity of the mathematical equations obtained in this paper for signal reconstruction is that they do not require the preliminary calculation of the Fourier series coefficients. They provide a direct calculation of the values of reconstructed samples. At the same time, when the number of samples in the subsequences used for reconstruction will be even, it is not necessary to perform multiplication operations. All this made it possible to reduce the computational complexity of the developed algorithm for signal reconstruction. Experimental studies of the algorithm were carried out based on simulation modeling using a signal model that is an additive sum of harmonic components with a random initial phase. Numerical experiments have shown that the developed algorithm provides the reconstruction result of signal samples with a sufficiently low error. The algorithm is implemented as a software module. The operation of the module is carried out on the basis of asynchronous control of the sampling reconstruction process. It can be used as part of metrologically significant software for digital signal processing systems.

Using radial pulse wave as hemodynamic measurements to quantify effects of acupuncture therapy for patients with traumatic brain injury and ischemia stroke

Background and aimTraumatic Brain Injury (TBI) and stroke are major sources of death and disability worldwide. Acupuncture has been used as a supplemental therapy for patients with TBI and stroke. This study was aimed to evaluate the effects of acupuncture therapy for patients with TBI and stroke by radial pulse spectrum. Experimental procedure22 patients (6 TBI and 16 stroke) were enrolled and underwent radial pressure wave measurement before and after acupuncture treatment at Dubi (ST-35), Zusanli (ST-36) and Jiexi (ST-41). The harmonic analysis of the radial pressure wave was calculated and transformed into Fourier series coefficients Cn, Pn and the variation coefficient CnCV. ResultsAfter acupuncture, systolic blood pressure, heart rate, and Glasgow Coma Scale changed very slightly. The harmonic index C4, C7, C9, C10, C3CV and C5CV had significant increases. (P < 0.05) After 3-week course acupuncture treatment, systolic blood pressure, C7, C8, C9, C10 and P10 had significant increases. (P < 0.05) ConclusionHarmonic analysis of radial pulse waves may detect earlier circulatory system changes of acupuncture treatment before they were evident with other hemodynamic readings or scale.