Exponential Window Research Articles

Improving the double-exponential windowing method to identify modal frequencies and damping ratios of dynamically large structures

Dynamically large structures can have many closely spaced modes. A particular example is an extruded aluminium panel used for building high-speed trains. Such a structure may be modelled deterministically with a large number of degrees of freedom (dofs), or statistically represented as a statistical energy analysis (SEA) system having particular SEA parameters. In both cases, experimental modal parameter identification may be required. Although there are several commercial packages which can be used for modal parameter identification, alternative methods with different strengths and/or convenience are still desirable. In this paper, the double-exponential windowing method developed by the second author is revisited and improved. Improvements are achieved by taking into account the effect of the finite time duration of Fourier transforms and making use of the real parts of driving point mobilities. The method is easy to implement and the improvements make the choice of the decay rate in the exponential window much flexible: for a very lightly damped structure, it allows to use a large positive decay rate to depress measurement noise, and for a highly damped structure, it allows to use a negative decay rate to make modes more evident or to separate overlapped modes. The usefulness of the improved method are demonstrated for a 30-bladed wheel model with known modal parameters, and for an extruded aluminium panel.

Detection of faulty accelerometer mounting from response measurements

Data integrity has become more critical with the growth in remote semi-automated machine health monitoring. The mounted resonance frequency of an accelerometer, if correctly mounted on a flat surface, is normally quite repeatable, for example at different locations on the same object. A method has recently been developed which largely separates forcing and transfer functions in response signals, by applying an exponential window in the cepstrum. This technique can show the resonance peaks in response spectra at all measurement points, and has been applied successfully in operational modal analysis. This paper proposes how this technique can be used to detect a faultily mounted accelerometer in industrial applications. Changes in the natural frequencies of the structure should show up equally in all measurements, but deterioration in the mounting condition of an individual accelerometer would only affect the mounted resonance frequency for that transducer. This paper demonstrates these principles using measurements on a gear test rig, operating with a range of different speed profiles, with two similar accelerometers, one having correct and then faulty mounting of two different severities. Measurements are also compared for accelerometers with a wider frequency range.

Modeling energy release parameters in stochastic simulation of ground motions generated by Vrancea intermediate-depth seismic source

Ground motion time-domain stochastic simulations are performed by generating a white or a Gaussian noise, shaping it with a window, loading its normalized spectrum with the spectrum of the motion at a specific site and transforming it back in time-domain (Boore in Pure Appl Geophys 160: 635–676, 2003). The outcome of the resulted simulations is compared to the target parameters, namely amplitude, frequency content and duration parameters. Using SMSIM (Boore in SMSIM: Fortran programs for simulating ground motions form earthquakes, 2005) and EXSIM (Motazedian and Atkinson in Bull Seismol Soc Am 95(3): 995–1010, 2005. https://doi.org/10.1785/0120030207), Coțovanu (J Mil Technol 1(2): 27–32, 2018. https://doi.org/10.32754/jmt.2018.2.05) and Coțovanu and Vacareanu (J Seismol 24(1): 229–241, 2020. https://doi.org/10.1007/s10950-019-09892-5) applied the stochastic simulation method for generating ground motions produced by intermediate-depth Vrancea (Romania) earthquakes and found inconsistent results for duration parameters (significant duration and energy distribution in time-domain). In simulations, the duration of the motion results from the source and path duration, but the significant duration (that contains 90% of the energy) depends on the way the energy is distributed in time, and this distribution is controlled through a noise shaping window. The objective of this paper is to provide an appropriate form of the shaping window that can describe the specific energy release of ground motions generated by Vrancea intermediate depth seismic source (the first 50–70% of energy being rapidly released, the rest of it having a very slow release). For this matter, 371 recorded motions were analyzed and 5 types of energy release were defined. The set with the above described pattern that contained almost half of the analyzed accelerograms was used to define a mean normalized cumulative energy that describes the Vrancea specificity. On this basis, the parameters for the SMSIM implemented exponential window were determined and, because the results were inconsistent, a more adequate window function was defined and implemented in SMSIM. For the newly defined two-interval window function an algorithm for calculating the parameters was described, and the parameters of the mean release of energy were determined. By using the two-interval function for describing the Vrancea specific release of energy, more appropriate ground motion simulations resulted.

A meshless singular boundary method for elastic wave propagation in 2D partially saturated poroelastic media

This paper presents a boundary meshless method, singular boundary method (SBM), in conjunction with the exponential window method (EWM) to simulate the dynamic response of the 2D partially saturated poroelastic media. The problem is firstly solved in the frequency-domain resulted from the time-domain governing equations via the Fourier transformation. Then the SBM approximates the solutions with a linear combination of fundamental solutions with respect to the source points on the physical boundary. To successfully apply the SBM, two issues are addressed. Firstly, the kernel functions, namely the fundamental solutions of frequency-domain governing equations, are derived based on the eigen-analysis. Secondly, the source singularities of the fundamental solutions are desingularized with analytical formulas. After obtaining the SBM formulation, the time-domain solutions can be obtained by the EWM, where the frequency domain windowing technique is introduced to stabilize the oscillations caused by damping parameters. Four numerical examples are carried out to show the validity and accuracy of the proposed method.

DFT-Based Identification of Oscillation Modes from PMU Data Using an Exponential Window Function

The characteristics of oscillation modes, such as interarea, regional, and subsynchronous modes, can vary during a power system fault, which can cause switching and control actions in the power system. Transient data of the modal response due to such a fault can be acquired through phasor measurement units (PMUs). When the transient data have a long duration, it is desirable to perform modal identification separately on each segment of the transient data, so as to reflect the varying characteristics of oscillation modes. A conventional discrete Fourier transform (DFT)-based method for parametric modal identification cannot be efficiently applied to such a segment dataset. In this paper, a DFT-based method with an exponential window function is proposed to identify oscillation modes from each segment of transient data in PMUs. This window function allows the application of the least squares method (LSM) for modal identification in the same manner as the conventional method. The accuracy of identification of the proposed method is compared with those of the conventional method and a Prony method through synthetic data of transient responses. Its feasibility is also verified by identifying real-world oscillation modes from transient data in PMUs.

Identification of modal parameters in a lightly damped system based on impact vibration testing: Application of exponential window and removal of its effect

The experimental environments for impact tests are simple to construct compared to other vibration tests in experimental modal analysis. However, given that sufficient sampling time is necessary for the response to attenuate, the following two problems arise: The first is that work efficiency is lowered. The second is that the measurement time increases, which makes it more susceptible to noise. To solve these problems, it is common to shorten the sampling time and apply an exponential window to reduce the leakage error of the frequency response function (FRF). Although studies on the influence on the FRF exist, identification accuracy for damping characteristic and the influence on the residue have not discussed. In this study, the influence on the modal properties obtained and the accuracy of the results are clarified using numerical examples based on a circle fit and method using simultaneous equations of the real and imaginary parts of the FRF. Finally, the identification results were compared and made sure that the method using simultaneous equations of the real and imaginary parts of FRF can identify modal properties with high accuracy. This method is effective for improving the working efficiency and reducing the influence of noise.

A frequency domain formulation of the singular boundary method for dynamic analysis of thin elastic plate

This paper presents a frequency domain formulation of the singular boundary method (SBM) for the analysis of thin elastic plates subjected to dynamic loads. In the SBM, the fundamental solutions for the dynamic thin plate problem in frequency domain are taken as basis functions. To isolate the singularities, we derive the origin intensity factors (OIFs) for the kernels of the SBM which allows the source points to coincide with the collocation points. For the transient analysis, the exponential window method (EWM) is applied in conjunction with the SBM to obtain the time-domain result. To demonstrate the efficiency and accuracy of the present method, several examples under harmonic and transient loads are carried out in comparison with the exact solutions and finite element solutions.

Kaiser ve Von-Hann Pencereleri Tabanlı Yeni bir Pencere Fonksiyonu ve Onun Sayısal Süzgeç tasarımındaki Performans Analizi

Digital filters, which are used in many signal processing applications, can be classified as recursive or nonrecursive. Since nonrecursive digital filters can provide inherent stability and exact linear phase characteristic, they have an important place in the literature. In this paper, a new hybrid function is proposed for the design of nonrecursive digital filters. This new window was obtained by combining two different window functions known as Kaiser and Von-Hann in the literature. First of all, the effects of two independent parameters of the proposed window function on the digital filter characteristic are analysed in terms of minimum stopband attenuation and transition bandwidth. Later, comparative examples for different filter lengths are given to compare the performance of the proposed window function in the design of digital filter with the well-known other two parameter window functions. Simulation results demonstrate that the proposed window function can provide a better filter design than Kaiser-Hamming, Saramaki, Kaiser, Dolph-Chebychev, Cosh, Exponential, and Gaussian window functions.

Enhanced window increment and adaptive recovery TCP for multi‐hop wireless networks

In this Letter, a combined algorithm known as receiver-assisted slow start (RASS) and feedback-assisted recovery (FAR) proposed to address the joint optimisation problems of transmission control protocol (TCP) such as non-optimal initial window (IW) size, slower window growth and spurious window deflation under multi-hop wireless (MHW) networks. The RASS fix the sender's optimum IW size from the receiver's advertised value and replaces the conventional exponential window growth with enhanced window increment mechanism that significantly accelerates the transmission rate. The FAR uses relay router's feedback information to regulate false transmission rate reduction during packet drops. The simulation result reveals that the proposed algorithms had a substantial improvement in throughput and reduction in packet latency over the existing standard New Jersey and non-congestion events algorithms under MHW environment.

Nonparametric identification of structural modifications in Laplace domain

This paper proposes and experimentally verifies a Laplace-domain method for identification of structural modifications, which (1) unlike time-domain formulations, allows the identification to be focused on these parts of the frequency spectrum that have a high signal-to-noise ratio, and (2) unlike frequency-domain formulations, decreases the influence of numerical artifacts related to the particular choice of the FFT exponential window decay. In comparison to the time-domain approach proposed earlier, advantages of the proposed method are smaller computational cost and higher accuracy, which leads to reliable performance in more difficult identification cases. Analytical formulas for the first- and second-order sensitivity analysis are derived. The approach is based on a reduced nonparametric model, which has the form of a set of selected structural impulse responses. Such a model can be collected purely experimentally, which obviates the need for design and laborious updating of a parametric model, such as a finite element model. The approach is verified experimentally using a 26-node lab 3D truss structure and 30 identification cases of a single mass modification or two concurrent mass modifications.

An experimental investigation on the effects of exponential window and impact force level on harmonic reduction in impact-synchronous modal analysis

A novel method called Impact-synchronous modal analysis (ISMA) was proposed previously which allows modal testing to be performed during operation. This technique focuses on signal processing of the upstream data to provide cleaner Frequency response function (FRF) estimation prior to modal extraction. Two important parameters, i.e., windowing function and impact force level were identified and their effect on the effectiveness of this technique were experimentally investigated. When performing modal testing during running condition, the cyclic loads signals are dominant in the measured response for the entire time history. Exponential window is effectively in minimizing leakage and attenuating signals of non-synchronous running speed, its harmonics and noises to zero at the end of each time record window block. Besides, with the information of the calculated cyclic force, suitable amount of impact force to be applied on the system could be decided prior to performing ISMA. Maximum allowable impact force could be determined from nonlinearity test using coherence function. By applying higher impact forces than the cyclic loads along with an ideal decay rate in ISMA, harmonic reduction is significantly achieved in FRF estimation. Subsequently, the dynamic characteristics of the system are successfully extracted from a cleaner FRF and the results obtained are comparable with Experimental modal analysis (EMA).

Method for finding optimal exponential decay coefficient in numerical Laplace transform for application to linear convolution

In this paper, a method based on the numerical Laplace transform is used for calculating the full linear convolution of real or complex signals. An algorithm for obtaining the last N values of the convolution is presented, along with a method for finding an optimal value for the decay coefficient of the transform. It is shown that the use of the numerical Laplace transform formulation allows the calculation of each half of the linear convolution independently, which has computational benefits. The numerical Laplace transform is expressed as the fast Fourier transform of signals that have been premultiplied by a decreasing exponential window characterized by decay coefficient c. The error of the resulting linear convolution depends on the value of the decay coefficient; undervalue results in the generation of wrap-around error whereas overvalue causes amplification of Gibbs phenomenon. In this paper, a formula that optimizes the value of the decay coefficient is developed. A trade-off value for c is obtained and error analysis shows that it outperforms other coefficients proposed in the literature when applied to the calculation of linear convolution. The relative errors obtained are of the order of 10−6% and 10−9% for single and double precisions.

An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher-order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency-domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap-around errors brought by the discrete Fourier transform. The frequency-domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.

A de-noising algorithm based on wavelet threshold-exponential adaptive window width-fitting for ground electrical source airborne transient electromagnetic signal

The ground electrical source airborne transient electromagnetic system (GREATEM) on an unmanned aircraft enjoys considerable prospecting depth, lateral resolution and detection efficiency, etc. In recent years it has become an important technical means of rapid resources exploration. However, GREATEM data are extremely vulnerable to stationary white noise and non-stationary electromagnetic noise (sferics noise, aircraft engine noise and other human electromagnetic noises). These noises will cause degradation of the imaging quality for data interpretation. Based on the characteristics of the GREATEM data and major noises, we propose a de-noising algorithm utilizing wavelet threshold method and exponential adaptive window width-fitting. Firstly, the white noise is filtered in the measured data using the wavelet threshold method. Then, the data are segmented using data window whose step length is even logarithmic intervals. The data polluted by electromagnetic noise are identified within each window based on the discriminating principle of energy detection, and the attenuation characteristics of the data slope are extracted. Eventually, an exponential fitting algorithm is adopted to fit the attenuation curve of each window, and the data polluted by non-stationary electromagnetic noise are replaced with their fitting results. Thus the non-stationary electromagnetic noise can be effectively removed. The proposed algorithm is verified by the synthetic and real GREATEM signals. The results show that in GREATEM signal, stationary white noise and non-stationary electromagnetic noise can be effectively filtered using the wavelet threshold-exponential adaptive window width-fitting algorithm, which enhances the imaging quality.

Noncoaxial Bessel-Gauss beams.

We proposed a new family of noncoaxial Gauss-truncated Bessel beams through multiplying conventional symmetrical Bessel beams by a noncoaxial Gauss function. These beams can also be regarded as the exponential-truncated version of Bessel-Gauss beams since they can be transformed into the product of Bessel-Gauss beams and an exponential window function along a certain Cartesian axis. The closed-form solutions of the angular spectra and paraxial propagation of these beams were derived. These beams have asymmetrical intensity distributions and carry the same orbit angular momentum per photon as the corresponding Bessel-Gauss beams. While propagating along the z axis, the mth (m≠0) noncoaxial Bessel-Gauss beams rotate their intensity distributions and the mth-order vortex at the beam center has a transverse shift along the direction perpendicular to the offset axis. Depending on the product of the transverse scalar factor of the Bessel beams and the offset between the Gaussian window function and the center of the Bessel beams, the noncoaxial Bessel-Gauss beams can produce unit vortices with opposite signs in pairs during propagation.

Simulation Based Design Analysis of an Adjustable Window Function

Window based Finite Impulse Response filters have the problem that in order to obtain better performance from these filters in terms of minimum stopband attenuation cost has to be paid for half main-lobe width and vice-versa. A solution of this contradictory behavior is to increase the length of the window which in turn requires more hardware hence increasing the cost of system. This paper proposes a novel window based on two shifted hyperbolic tangent functions. The proposed window contains an adjustable parameter, with the help of which desired time and frequency domain characteristics may be achieved for relatively shorter window length. The characteristics of the proposed window are compared with those of the two well-known adjustable windows namely Cosh window and Exponential window. MATLAB simulation results show that for the same value of window length, the proposed window provides improved output, and thus it makes a good compromise between minimum stopband attenuation and half main-lobe width compared to the windows mentioned previously.

Comparison of the convolution quadrature method and enhanced inverse FFT with application in elastodynamic boundary element method

Transient problems can often be solved with transformation methods, where the inverse transformation is usually performed numerically. Here, the discrete Fourier transform in combination with the exponential window method is compared with the convolution quadrature method formulated as inverse transformation. Both are inverse Laplace transforms, which are formally identical but use different complex frequencies. A numerical study is performed, first with simple convolution integrals and, second, with a boundary element method (BEM) for elastodynamics. Essentially, when combined with the BEM, the discrete Fourier transform needs less frequency calculations, but finer mesh compared to the convolution quadrature method to obtain the same level of accuracy. If further fast methods like the fast multipole method are used to accelerate the boundary element method the convolution quadrature method is better, because the iterative solver needs much less iterations to converge. This is caused by the larger real part of the complex frequencies necessary for the calculation, which improves the conditions of system matrix.

Vibrational Power Flow Analysis of a Cylindrical Shell Using a Four-Point Technique

The aim of studying and analysing vibrational characteristics of structures using power flow is to control the noise and vibration within the structure and prevent it from being transmitted into the environment. The power flow within a cylindrical shell is investigated because of its importance in designing spacecraft and marine structures. In this paper, a four-point power flow technique was used to examine the effects of flexural and shear forces on the total power flow in a cylindrical shell in free-free conditions. To obtain better results, the exponential window was used considering the use of a hammer for the excitation of the structure. The examining of the results and obtained diagrams determined that the effect of shear forces on the total power flow was more than the effect of flexural forces. Moreover, the precision of two-point and four-point techniques was compared.

Exponential Window Family

In this paper we propose a new class of 2-parameter adjustable windows, namely Exponential window, based on the exponential function [1,2]. The Exponential window is derived in the same way as Kaiser window was derived, but our proposed window is more computationally efficient because in its time domain function it has no power series expansion. First, the spectrum design equations for the Exponential window are established, and the spectral comparisons are performed with Cosh, Kaiser and ultraspherical windows. The proposed window is compared with Cosh and Kaiser windows, and the results show that for the same window length and mainlobe width the Exponential window provides better sidelobe roll-off ratio characteristic, which may be important for some applications, but worse ripple ratio. The second comparison is performed with ultraspherical window for the same window length, mainlobe width and sidelobe roll-off ratio and the results demonstrate that the Exponential window exhibits better ripple ratio for the narrower mainlobe width and larger sidelobe roll-off ratio, but worse ripple ratio for the wider mainlobe width and smaller sidelobe roll-off ratio.

THE EFFECT OF EXPONENTIAL/COSINE HYPERBOLIC/ POLYNOMIAL/ KAISER WINDOW PARAMETERS CG-ICG-ENBW-SL ON THE SNR IMPROVEMENT OF MST RADAR SIGNALS

In this paper, the effect of Exponential/Cosine Hyp erbolic/Polynomial/Kaiser window parameters Coheren t Gain (CG), Incoherent Power Gain (ICG), Equivalent Noise Band Width (ENBW ) and Scalloping Loss (SL) on the SNR values of MST Radar returns has been investigated. Six sets of multibeam observatio ns of the lower atmosphere made by the Indian Mesos phere-StratosphereTroposphere (MST) RADAR are used for the result ana lysis. Prior to the Fourier Transform, the in-phase and quadrature components of the Radar echo samples are weighted with Exponen tial/Cosine Hyperbolic/Polynomial/Kaiser window Fun ctions. It is observed that, with the increase in the parameter “ α” of Exponential and Cosine Hyperbolic Window and p“m” of Polynomial Window increases ENBW and SL, but decreases in CG a This observation can be used to establish in improving the Signal to Noise Ratio (SNR) of MST RADAR signals. From this s tudy it is reported that, the window e parameter “ α=6/m=4”can be suggested for optimum results of SNR improvement in MST RADAR signals. The optimum widow parameter“ α/m” in turn yields optimum Exponential window parameters as “CG=0.4768, ICG=0. 3495, ENBW=1.5371, SL=0.8536”, Cosine Hyperbolic wi ndow parameters as “CG=0.4769, ICG=0.3495, ENBW=1.5366, SL=0.8535”, Polynomial window parameters as “CG=0.5719, ICG=0. 4799, ENBW=1.4673, SL=0.8011”, and Kaiser window paramete rs as “CG=0.4991, ICG=0.3660, ENBW=1.4695, SL=0.841 0”.Which corresponds to optimum mail lobe width and side lob e attenuation to increase the SNR of MST Radar nois y data..