Timefrequency techniques for modal parameters identification of civil structures from acquired dynamic signals
Carlos Andres PerezRamirez^{1} , Juan Pablo AmezquitaSanchez^{2} , Hojjat Adeli^{3} , Martin ValtierraRodriguez^{4} , Rene de Jesus RomeroTroncoso^{5} , Aurelio DominguezGonzalez^{6} , Roque Alfredo OsornioRios^{7}
^{1, 2, 4, 6, 7}Faculty of Engineering, Autonomous University of Queretaro, Campus San Juan del Rio, Rio Moctezuma 249, Col. San Cayetano, 76807 San Juan del Rio, Queretaro, Mexico
^{3}Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, 470 Hitchcock Hall, 2070 Neil Avenue, Columbus, OH 43220, USA
^{5}CA Telemática, DICIS, University of Guanajuato, Carr. SalamancaValle de Santiago Km. 3.5 + 1.8 Km., 36885, Salamanca, Mexico
^{3}Corresponding author
Journal of Vibroengineering, Vol. 18, Issue 5, 2016, p. 31643185.
https://doi.org/10.21595/jve.2016.17220
Received 18 February 2016; received in revised form 10 May 2016; accepted 30 May 2016; published 15 August 2016
A major trust of modal parameters identification (MPI) research in recent years has been based on using artificial and natural vibrations sources because vibration measurements can reflect the true dynamic behavior of a structure while analytical prediction methods, such as finite element models, are less accurate due to the numerous structural idealizations and uncertainties involved in the simulations. This paper presents a stateoftheart review of the timefrequency techniques for modal parameters identification of civil structures from acquired dynamic signals as well as the factors that affect the estimation accuracy. Further, the latest signal processing techniques proposed since 2012 are also reviewed. These algorithms are worth being researched for MPI of large reallife structures because they provide good timefrequency resolution and noiseimmunity.
Keywords: modal parameters identification, timefrequency algorithms, wavelet transform, synchrosqueezing transform, civil structures, dynamic excitation sources.
1. Introduction
In the past two decades, modal parameters identification (MPI) has become a significant and growing research discipline in several areas such as aeronautics, mechanical engineering, and civil engineering because it can be used to assess the health of the structure and control its vibrations during dynamic events such as an earthquake. For civil structures, it can also be used to satisfy the seismic demands during ground motions in the form of response spectrum analysis. In general, MPI consists of three main steps: excitation and acquisition, signal processing, and modal parameters estimation, such as natural frequencies, damping ratios, and mode shapes (As seen in Fig. 1).
A major trust of MPI research in recent years has been based on using artificial and natural vibrations sources because vibration measurements can reflect the true dynamic behavior of a structure while analytical prediction methods, such as finite element models, are less accurate due to the numerous structural idealizations and uncertainties involved in the simulations. MPI based on vibration data represents a challenge because measured data is nonstationary and is embedded in highlevel noise. Furthermore, closelyspaced modal parameters found in a structure due to symmetric geometries or similar physical properties in different directions represent an additional challenge to MPI schemes [1]. For these reasons, it is of paramount importance to have an accurate signal processing algorithm capable of estimating the modal parameters of a civil structure using signals that are nonstationary with a highlevel of noise and possibly closelyspaced modes.
Fig. 1. Main steps in MPI
Sirca Jr. and Adeli [2] present a review of the strategies used to perform structural system identification. AmezquitaSanchez and Adeli [1] present an extensive review of signal processing techniques to extract features from the acquired vibrations signals for the purpose of structural health monitoring. This paper presents a stateoftheart review of journal articles on timefrequency techniques for vibrationbased MPI. The focus is on civil structures including highrise buildings and bridges. The rest of the paper is organized as follows. First, the excitation source and the effect of temperature on MPI accuracy is discussed. Next, vibrationbased timefrequency techniques used for MPI as well as their advantages and disadvantages are presented. Then, novel signal processing techniques that are potential candidates to be utilized in MPI schemes are presented. The article ends with some final remarks.
2. Influence of the excitation source and the temperature effect on the MPI accuracy
The excitation source and environmental temperature influence the MPI accuracy significantly. For instance, if a small excitation force is applied to the structure, the measured response might not reflect the full behavior of the structure. Furthermore, the acquired dynamic signals could contain mostly noise or the useful information might be embedded in a highlevel of noise. In contrast, a large excitation force can produce damage in the structure. On the other hand, the environmental temperature can modify the mechanical properties of the materials used in the structure thus producing changes in the identified modal parameters.
Different kinds of excitation sources have been used in vibrationbased MPI research. The most common ones are reviewed and their advantages and disadvantages are pointed out in this section. Further, the main effects produced by environmental temperature on the MPI accuracy as well as possible solutions to overcome them are addressed.
2.1. Excitation source
The excitation source is an important step in the MPI of a civil structure as it induces energy into the structure in form of vibrations to be monitored to observe its dynamic behavior. To perform this task, artificial and natural or ambient forces have been used to excite civil structures. Artificial excitations are generated by using a manmade mechanical equipment such as drop weights, hammers, and shakers, among others; however, an easy access to the structure and its temporary closing during the test are required [3] which may not be possible for certain structures such as a residential condominium building. Consequently, a significant part of MPI research has focused on natural excitations or ambient vibrations such as wind, microearthquakes, and traffic loadings since this type of excitation requires neither excitation equipment nor the interruption of the structure’s normal operation.
In general, the use of artificial or natural excitations depends on the nature of the structure under study and the available budget for testing. For example, if the structure has a relatively small size, the cost and complexity of mechanical equipments to generate the artificial vibrations are lower than those required for large structures. In the latter case ambient vibrations are preferred since they are capable of exciting a large structure without requiring sophisticated mechanical equipments. In addition, they allow the monitoring of the structure under real operating conditions. Moreover, they are useful in providing information in the lowfrequency band (below 1 Hz) because mechanical and electrical components used in the artificial exciters cannot reproduce them easily [4]. On the other hand, if highfrequency information is required, artificial exciters are recommended since the frequency range can be adjusted by controlling some features of the exciter [4]. In both cases, it is worth noting that if the energy injected in the structure is small, the structure will not be excited correctly and the measured dynamic signals will only contain noise; on the contrary, if the excitation is too high, the structure can suffer damage [5]. Table 1 presents a comparison of artificial and natural sources for exciting civil structures.
Table 1. Comparison of artificial and natural sources for exciting civil structures
Excitation source

Cost

Frequency Content

Availability

Ease of use

Realtime assessment

Artificial

Depends on the structure size

Broadband

Depends on the mechanical equipment

Depends on the mechanical equipment

No

Natural

Free

Low Frequency

Always

Easy

Yes

2.2. Environmental temperature effects on the MPI accuracy
Temperature can produce changes in the physical properties of the materials used in the structure. For instance, an increase in the temperature results in an increase in the material length thus producing a decrease in the structure stiffness. Furthermore, temperature variations may cause changes in the boundary conditions thus affecting the modal parameters values [6].
In recent years, researchers have proposed various schemes to quantify and compensate for the effects of temperature in the modal parameter values. Xia et al. [7] show that modal frequencies, in particular the first modal frequency, are affected greatly by the change of temperature, but not the damping ratios and modal shapes. They examine and compare three methods: linear regression (RM) models, blind source separation (BSS) models [8], and autoregressive (AR) models to compensate the temperature effects in a steel truss girder suspension bridge and a reinforced concrete (RC) highrise building and conclude that the RM algorithm allows compensating for the temperature effects on the modal frequencies when the structure behaves linearly, but when the relationship is slightly nonlinear, the AR and BSS algorithms have an edge for compensating the temperature effects on the modal frequencies. The AR, however, cannot model complex problems such as complicated nonlinear behavior of civil structures and noisecontaminated measured structural response which is usually the case [1].
To model the nonlinear relation between the temperature and modal parameters, Nandan and Singh [9] propose a subspace system identificationbased approach, combined with input data filtering to model the temperature–frequency relationship of two realistic bridge configurations: a simplysupported concrete boxgirder and a Tbeam girder superstructure, located in North Carolina, USA. They use the acquired data from these reallife bridges and note that the data must be filtered before utilization because of the seasonal variation in the data. Laory et al. [10] examine and compare multiple linear regression (MLR), artificial neural networks (ANN) [11, 12], support vector regression (SVR) [13, 14], regression trees (RT) [15], and random forest (RF) [16] algorithms to estimate the natural frequencies of a steel truss bridge under different environmental conditions and conclude that SVR and RF algorithms can be used to model the nonlinear relationship between the environmental conditions and modal parameters. Despite some promising results to compensate for the effects of the environmental conditions in the MPI, the problem has not been completely resolved because of high nonlinearities in the acquired signal. For this reason, it is desirable to investigate new algorithms to model the environmental conditionsmodal parameters relationship more appropriately.
In recent years, Adeli and associates have proposed new hybrid approaches for modeling or predicting highly nonlinear systems. Adeli and Jiang [17] present a nonlinear autoregressive moving average with exogenous inputsbased timedelay fuzzy wavelet neural network (WNN) algorithm for system identification of multistory building structures. They use concepts from the chaos theory, wavelets [18], and soft computing techniques, neural networks and fuzzy logic [19, 20], to model the highly nonlinear earthquakestructure system and to incorporate the inherent imprecision of the acquired data. Wang and Adeli [21] present a novel selfconstructing WNN for vibration control of nonlinear structures. They integrate a selfconstructing WNN with an adaptive fuzzy sliding mode control approach to construct a model that does not need apriori knowledge of the structure’s dynamics. They test the methodology using a benchmark problem, the finite element model (FEM) model of a continuous castinplace prestressed concrete boxgirder bridge [22]. To the best of the authors’ knowledge, these new approaches have not been reported in the literature to model the environmental conditionsmodal parameters relationship. The authors believe that these advanced models have great potentials and their application should be explored for compensating the environmental effects in the modal parameters, especially the modal frequencies.
3. Signal processing techniques for MPI
In reallife structures, the acquired dynamic signals are embedded in different levels of noise. The noise in signals generated using artificial excitations is in general lower than the noise in signals acquired for civil structures excited with ambient sources. This difference requires the utilization of more robust algorithms. Several signal processing techniques have been proposed and used to deal with this type of signals. In this section, the most common ones are reviewed in a chronological order and their advantages and shortcomings are pointed out.
3.1. Timedomain methods
Timedomain methods belong to a class of algorithms that do not require a space transformation (e.g. the frequency domain) to estimate the modal parameters of civil structures.
3.1.1. Statistical time series
Statistical time series models (STSM) use the measured response to establish an approximate mathematical model to represent the acquired dynamic behavior. They are known to be efficient for modeling timeinvariant linear systems [1]. They include autoregressive (AR), movingaverage (MA), autoregressive with exogenous inputs (ARX), autoregressivemovingaverage (ARMA), autoregressivemovingaverage with exogenous inputs (ARMAX), and BoxJenkins (BJ) models.
Takewaki and Nakamura [23] present an experimental analysis to estimate the natural frequencies and damping ratios of a baseisolated threestorey reinforced concrete (RC) building subjected to the 2004 Tokaidooki, Japan, earthquake using a batch processing leastsquares estimation method combined with an ARX model. They show that using the poles (the roots of the denominator) of the estimated ARX model, the modal parameters as well as their variations during the earthquake can be estimated. Gomez et al. [24] use an ARX model to identify the natural frequencies of a threespan curved RC bridge located in California subjected to trafficinduced vibrations and report that higher modes are not detected because the earthquakes excite only the first four modes.
Civil structures are known for having nonlinear and timevariant behavior [25] which makes the linear STSM ineffective for modeling their dynamical behavior because they can model only timeinvariant systems thus limiting an accurate estimation of the modal parameters. To lessen this drawback, Maosheng et al. [26] combine a recursive adaptive forgetting factor with an ARX model for MPI of the scaled model of 12story RC frame structure subjected to the 1940 El Centro earthquake. The model detects the response of the structure when the response is timeinvariant where the modal parameters are estimated from the entire acquired response using the poles of the ARX model. In contrast, when the response is timevarying, the model identifies the instance when the signal changes abruptly in amplitude. Once the change is detected, the previous samples are used to estimate the modal parameters using the poles of the ARX model. Su et al. [27] present a timevarying ARX (TVARX) model to estimate the instantaneous modal parameters through integration with wavelets and apply it to two 2D structures, a fivestory and an eightstory steel frame subjected to the 1999 ChiChi, Taiwan earthquake. They report an identiﬁcation error of about 2 % and 20 % for the natural frequencies and damping ratios, respectively. Other applications of statistical time series have been reported by Saito and Beck [28] and Niu et al. [29].
In statistical time series method, in general, the model order selection is a key to the accurate estimation of modal parameters. This selection is made either by a trialanderror or an adaptive approach. The trialanderror approach is timeconsuming and impractical. Consequently, other alternatives should be explored. For instance, Saito and Beck [28] use a Bayesian approach for identifying the best order for the ARX model used for estimating the MPI of a highrise building.
3.1.2. Free decaybased methods
When a civil structure is excited by a hammer or a dropweight, the freedecay response of the structure is measured. The term free is used because the excitation forces used can be treated as impulses. However, in reallife, the impulse excitation is seldom utilized because it cannot excite large civil structures adequately. Therefore, techniques such as the random decrement technique (RDT) [30] or the natural excitation technique (NExT) [31] are proposed to transform the response acquired by another type of excitation such as ambient or maninduced vibrations into a freedecay response.
RDT aims to convert the acquired signal into a freedecay response by averaging segments of the acquired response with a common initial condition using a given threshold [32]. A graphical illustration of the method is shown in Fig. 2. A threshold value ($a$) is used for obtaining the samples that fulfill the aforementioned condition (Fig. 2(a)). This value is obtained using the level crossing condition [33]:
where ${\sigma}_{x}$ is the signal standard deviation. Then, a segment of equal sample length (the curves labeled as s1, s2, s3, and s4) is extracted for every sample that satisfies the aforementioned condition (Fig. 2b). The resulting signal ($\delta $) is obtained as [34]:
where $N$ is the number of obtained segments, and can be treated as the structure’s freedecay response. He et al. [35] use the RDT algorithm to calculate the MPI of a threespan continuous steel truss bridge subjected to ambient vibrations produced by a moving train. The modal parameters are obtained iteratively solving a set of nonlinear equations formed using the freeresponse equation. The results obtained are compared with the peakpicking (PP) method. They conclude that RDT estimates the first four modes whereas PP identifies only the first three. Other applications of RDT for MPI are presented by Cury et al. [36] and Wang and Chen [37].
Despite good results reported in the aforementioned works, some unresolved difficulties remain using the RDT technique. RDT should be used with stationary or quasistationary signals as any slight variation can cause a change of the freedecay signal amplitude, leading to miscalculation of the damping ratio since a higher damped signal is estimated by the algorithm [38]; but, the measured signals obtained for reallife structures from dynamic excitations exhibit nonlinear and nonstationary properties. In order to overcome this limitation, Lin and Chiang [39] present a modified RDT technique combined with the Ibrahim time domain (ITD) method [40] to estimate the modal parameters of a 2D 3bay 17member cantilever space truss subjected to bandlimited noise as excitation. They report identiﬁcation error of less than about 1 % and 20 % for the natural frequencies and damping ratios, respectively. Lin and Tseng [41] use RDT with a timevarying threshold value on a 6degreeoffreedom model with viscous damping and an added 10 % noise. They report maximum errors in the natural frequencies and damping ratios of 1 % and 18 %, respectively.
Fig. 2. RDT estimation
Unlike RDT technique, the NExT technique obtains the freedecay response of a signal either by applying directly the cross correlation function to the signal or using the FFT (FFT is used only to obtain the crosscorrelation function) and its inverse to calculate the crossspectral density function [42]. Several applications of the NExT technique combined with the eigensystem realization algorithm (ERA) [43] have been reported for finding the MPI of a 1/40 scale cablestayed bridge [44], suspension bridges [45], a twospan continuous steel frame footbridge [46], a 16bay truss structure [47], Masonry Monuments [48, 49], and a Langertype arch steel bridge [50].
A major drawback of the NExT method is its increased computational burden in comparison with RDT. In order to lessen this problem, Chang and Pakzad [51] propose an enhanced NExT algorithm to estimate the modal parameters of the Golden Gate Bridge located at San Francisco, California, USA. A wireless sensor network is used to measure its response when it is subjected to ambient vibrations. They report maximum errors of about 1 % and 23 %, for the estimation of the natural frequency and damping ratio, respectively.
3.1.3. Subspace methods
Stochastic subspace identification (SSI) algorithms are another class of timedomain methods for estimating the modal parameters of civil structures subjected to ambient dynamic vibrations [52]. Different types of SSI methods have been proposed such as covariancedriven SSI (SSICOV) [53], covariancevariate SSI (SSICV) [54], and datadriven SSI (SSIDATA) [55]. Van Overschee and De Moor [52] show that all SSI methods can be generalized into a unified theory depending on the weighting matrix selection before the numerical decomposition. A review of the aforementioned methods is presented by Peeters and De Roeck [56].
SSI has been used to estimate the modal parameters of steel footbridges [57, 58], cablestayed footbridges [59], a stone arch bridge [60], a stressribbon footbridge [61], suspension bridge [62], a highway RC bridge [63], RC buildings [64], a super highrise tower [65], and masonry buildings [66].
SSI algorithms in general present high immunity to noise in the signals [65], but are computationally intensive, requiring significant computational resources especially for large structures [67, 68]. Furthermore, they cannot estimate closelyspaced modes accurately because they require setting the order or number of modes which can produce an underdetermined or overdetermined system. If the former model is set, some modes would not be detected, whereas the latter could produce spurious modes [57, 68, 69]. In order to lessen some of the problems found in the SSI methods, Reynders et al. [70] propose a modification to the SSI algorithm based on using both the measured responses from the ambient vibrations and the ones obtained with an artificial exciter to perform the MPI of a steel footbridge subjected to ambiental vibrations and artificial excitations. They show an improvement in the quality of the estimation of the modal parameters. Zhang et al. [68] introduce a modification to the SSIDATA algorithm in order to reduce its computational burden and apply it to a suspension bridge subjected to ambient excitations. The original SSIDATA method uses QRfactorization, the singular value decomposition (SVD), and least squares to determine the modal parameters which is computational intensive. In contrast, the modified method avoids the use of the QRfactorization by employing an eigenvalue decomposition. Further, they propose a similarity index to eliminate the fictitious modes. Li and Chang [67] introduce an optimization scheme for the online operation of the SSICOV method and test it using the ASCE benchmark frame structure. In this algorithm, QRfactorization is substituted with the Householder biiteration subspace tracker, which along the utilization of a statespace constructed subspace, minimizes the required time to compute the modal parameters since only part of the measured signal needs to be used. Their approach can deal with colored noise (noise whose frequency content is not uniform). Hong et al. [69] present an enhanced version of the canonical correlation analysis (CCA) to allow a better detection of the closelyspaced modes and test it using the actual measurements obtained from a suspension bridge subjected to wind excitation.
Another type of subspace method is the proper orthogonal decomposition (POD) (or KarhunenLoève decomposition). POD allows analyzing multidimensional data. A detailed explanation of this algorithm can be found in [71]. Kallinikidou et al. [72] use POD for MPI of a cablesuspended bridge subjected to trafficinduced vibrations as excitation. POD calculates the modal parameters using the covariance matrices of the bridge measurements at selected locations. The authors conclude that the POD method can deal with a huge amount of data, as only the most relevant sensors are used, allowing its utilization in realtime structural health monitoring (SHM) schemes. Wang and Cheng [73] propose a modification for calculating nonproportional damping ratios using POD. They test the methodology employing a steel cantilever beam subjected to a bandlimited excitation. They identify the natural frequencies with high accuracy, as a maximum deviation of 2 % is achieved; but, the error for the estimation of the damping ratio is over 50 % compared with the theoretical values.
3.2. Frequencydomain methods
Unlike timedomain methods, frequencydomain approaches must use a space transform in order to identify the existent components of the measured signal and estimate the modal parameters. This class of algorithms is known for their easeofuse. In some cases, the required computational burden is lower than that of the timedomain methods. The most widelyused algorithms are discussed in this subsection.
3.2.1. Fourier Transformbased methods
Fourier Transform (FT) is one of the wellknown and most popular algorithms used to identify relevant information in the frequency domain. In MPI applications, the optimized discrete version, known as the Fast Fourier Transform (FFT) is the one chosen for implementation on personal computers (PC) and portable platforms such as digital signal processors (DSP), microcontrollers, and field programmable gate arrays (FPGA). FFT has been used to estimate the natural frequencies of 12, 19, and 23story RC building structures [74], and a prestressed RC bridge [75]. Valla et al. [76] calculate the natural frequencies of two 30storey RC buildings subjected to ambiental vibrations employing the FFT.
In the last two decades, different algorithms based on the FFT such as peakpicking (PP) method, frequency domain decomposition (FDD) [77], and frequency response function (FRF) have been proposed to estimate the modal parameters of civil structures subjected to dynamic vibrations. Proposed by Bendat and Piersol [78], the PP method uses FFT to obtain the signal amplitude spectrum where the natural frequencies are the highest peaks of the spectrum, and the damping ratios are calculated using either the halfpower bandwidth or the logarithmic decrement methods while the mode shapes are estimated using SVD. The FDD method is based on the decomposition of the power spectral density matrix using the SVD method to estimate the modal parameters of the structure [77], whereas the FRF method uses the Fourier transform of the input excitation and the measured response to estimate a transfer function which is used for calculating the modal parameters of a civil structure [32].
Several examples of PP, FDD, and FRF methods for MPI of civil structures have been reported. Li et al. [35] compare the PP, FDD and RDT methods to estimate the modal parameters of a 101storey office highrise building with five basement levels subjected to the 2008 Shichuan, China earthquake. They use FDD and PP to estimate the natural frequencies, whereas RDT and FDD are employed to compute the damping ratios. GarcíaPalencia and SantiniBell [79] use the FRF method to estimate stiffness, mass, and viscous damping matrices of a linear elastic damped structure. They conclude that the FRF method does not work well under severe ambient conditions. Erdogan and Gülal [80] apply the PP method to calculate the modal parameters of a suspension bridge subjected to ambient dynamic vibrations and conclude that only lower modes can be identified.
Although FDD improves the resolution of PP by using the SVD method, the selection of the frequencies still remains as a manual task which can be a timeconsuming procedure [81]. In order to lessen this problem, Gade et al. [82] propose a modified version of FDD method known as enhanced FDD (EFDD) method which automates the process of identifying the natural frequencies by using a zerocrossing criteria. Furthermore, the inverse discrete Fourier transform is used to extract the individual mode and to estimate the modal parameters. Altunisik et al. [83] compare the SSI and EFDD methods for estimating the modal parameters (natural frequencies and damping ratios) of a box girder bridge subjected to traffic as excitation source. They report both algorithms estimate similar values. Other similar works have been presented by Magalhães et al. [84], Soyoz et al. [85], and Cismaşiu et al. [86], among others. Another modification of the original FDD method is the frequency spatial domain decomposition (FSDD) [87] which uses spatial filtering to improve the estimation of modal frequencies and damping ratios.
FFT and its derived algorithms present significant limitations. They cannot be used for estimating the modal parameters of structures subjected to ambient dynamic excitations because the monitored signals in a structure exhibit nonlinear and nonstationary properties which cannot be modelled by FFT adequately [1]. In order to lessen the limitation of FFT, Agneni et al. [88] propose a Hilbert transform (HT)based algorithm for obtaining the FRFs.
3.2.2. Highresolution methods
Highresolution (HR) methods are known for detecting frequencies, especially closelyspaced ones, in signals with a low signaltonoise ratio (highlevel noise) [89]. Among the HR methods, the multiple signal classification (MUSIC) [90] has been used to estimate the natural frequencies of civil structures. Jiang and Adeli [89] present a MUSIC method to estimate the pseudospectrum of highrise buildings subjected to artificial vibrations created by a shake table. The estimated frequencies are used to construct a damage indicator. The authors were the first to use the MUSIC approach in the field of structural engineering. OsornioRios et al. [91] employ MUSIC for calculating the natural frequencies of a fivebay trusstype structure subjected to artificial vibrations. The natural frequencies are used as inputs to a classifier to detect and quantify the damage severity. AmezquitaSanchez et al. [92] compute the natural frequencies of a fivebay truss structure with 70 members using the MUSIC method. They show that closelyspaced modes can be estimated. The pseudospectrum calculated by MUSIC cannot be used for estimating the damping ratio because its amplitudes have no physical significance, which is the opposite to the FFT spectrum. For this reason, other methods must be explored to obtain the damping ratio values [93].
None of the techniques discussed so far can determine the instantaneous changes or the evolution over time of the modal parameters. If those features are required, timefrequency methods must be used.
3.3. Timefrequency methods
These methods combine the properties of both time and frequency methods to provide an enhanced detectability of frequencies, especially closelyspaced ones. This subsection presents the main algorithms used in MPI schemes.
3.3.1. Wavelet transform
During the last two decades, wavelet transform (WT) has become the most widelyused timefrequency algorithm for signal processing in different engineering applications such as seismic engineering [94], filtering approaches [95], structural control [9698], structural reliability analysis [99, 100], pavement structural evaluation [101], damage detection [102], and biomedical signal processing [103, 104] because it provides a multiresolution timefrequency analysis, allowing the detection of sudden frequency changes, transients, and other features that are invisible in time domain [105]. The continuous wavelet transform (CWT), discrete wavelet transform (DWT), and wavelet packet transform (WPT) have been the most widelyused methods. The main difference between CWT and DWT is how the shifting and scaling is performed. DWT estimates the WT using dyadic blocks, whereas CWT does not, that is, the rates used in the aforementioned processes are chosen by the final user. Fig. 3 shows the decomposition tree for DWT and WPT where it seen that WPT decomposes both detail coefficients (dC) and approximation coefficients (aC) (Fig. 3(a)), whereas DWT decomposes only the aC (Fig. 3(b)). In this regard, it is seen that WPT is an improvement to DWT to allow the extraction of individual frequencies.
Fig. 3. Tree decompositions for a) WPT and b) DWT
a)
b)
In recent years, WT has risen as an important signal processing tool for MPI of different civil structures. ÜlkerKaustell and Karoumi [106] examine the Morlet CWT for MPI of a concretesteel composite railway bridge subjected to ambient dynamic vibrations caused by moving trains. The amplitude and phase of extracted signals are used to calculate the natural frequencies and damping ratios. They conclude that the values of the identified natural frequencies decrease and the damping ratio increases when the train increases its speed and vice versa. Le and Paultrec [107] apply CWT and the Cauchy wavelet to estimate the natural frequencies and damping ratios of a full scale twostorey RC building subjected to bandlimited noise as excitation. The authors report that the method can identify closelyspaced modes. Kim and Chen [108] examine WPT to estimate natural frequencies and damping ratios of a linear threedegreesoffreedom vibration system and report that the modal parameters of the simple system can be estimated with accuracy.
WTbased approaches have some drawbacks. For example, DWT method lacks translationinvariance which can affect the estimation of the modal parameters. In order to lessen this problem, Holschneider et al. [109] propose the stationary WT (SWT) which achieves translationinvariance by removing the downsamplers and upsamplers in the DWT and upsampling the filter coefficients by a factor of ${2}^{j1}$ in the $j$th level of the algorithm [110]. Sadhu et al. [111] examine SWT algorithm for computing the modal parameters of a 15story steelframe building subjected to ambient dynamic vibrations. SWT is used to decompose the time signal into several frequency bands, where the modes are calculated by using the parallel factor tool. Su et al. [112] apply the SWT algorithm to compute the modal parameters of an 8story steel frame and a steel box girder cablestayed bridge subjected to bandlimited noise and ambient dynamic vibrations, respectively. They report maximum accuracies of 2 % and 20 % for the natural frequencies and damping ratios, respectively.
The aforementioned WT algorithms have a fixed resolution which might require the utilization of other signal processing techniques such as EMDbased methods to fully extract the individual modes. In order to overcome these limitations, Daubechies et al. [113] proposed the Synchrosqueezed Wavelet Transform (SQWT) as a new adaptive WT capable of working with signals with highlevel of noise. To test the effectiveness of the SQWT, PerezRamirez et al. [93] apply the SQWT and the Gaussian wavelet to estimate the natural frequencies and damping ratios of a scaled 3D fourstory twobay by twobay steelbraced frame subjected to ambient dynamic vibrations. The results indicate the modal parameters of both structures are estimated with high accuracy, especially the closelyspaced modes.
3.3.2. HilbertHuang transform
Huang et al. [114] introduced the empirical mode decomposition (EMD) combined with Hilbert transform (HT) known as HilbertHuang transform or HHT as an adaptive signal processing method capable of analyzing stationary, nonlinear, nonstationary, and transient signals. The EMD method decomposes any time series data into a set of bandlimited quasistationary functions, called intrinsic mode functions (IMF). Next, the HT is applied to each IMF to obtain its amplitude and phase angle, which are used to estimate the natural frequencies ($\omega $) and damping ratios ($\zeta $) of a civil structure using the following equations:
where $\varphi $ and $A$ are the phase and amplitude of each IMF, respectively. Fig. 4 illustrates the application of HHT to calculate the natural frequencies and damping ratios of a synthetic signal generated using the following equation [115]:
where the signal is composed of two modes whose frequencies (${f}_{i}$) are 20 and 50 Hz, respectively, the damping ratios (${\zeta}_{i}$) are 1 % and 0.5 %, respectively, and the amplitudes (${A}_{i}$) are 1.0 for every mode (Fig. 4(a)). The test signal is not corrupted with noise [$n\left(t\right)=$0] and the phase (${\theta}_{i}$) for every mode is 0. The sampling frequency used is 1000 Hz, during 2 s, to obtain 2000 samples. Fig. 4(b) illustrates the individual components extracted using EMD, whereas Fig. 4(c) depicts the modal parameters estimated using HT showing their evolution over time. HHT method has been applied in different fields such as biomedical engineering [116], mechanical engineering [117], and earthquake engineering [118].
Fig. 4. MPI estimation using HHT: a) a sample test signal, b) individual modes, and c) the estimated natural frequencies and damping ratios.
a)
b)
c)
Shi et al. [119] compare the HHT and PP methods for estimating the modal frequencies and damping ratios of the 101story Shanghai World Financial Center subjected to ambient dynamic vibrations. The results show both methods estimate similar values for the natural frequencies but HHT is more accurate than PP for estimation of the damping ratios. Ditommaso et al. [120] compare the HHT, STFT, and the STransform (ST) for identifying the natural frequencies of a brickmasonry bearing wall structure subjected to the vibrations created from the explosion of a World War II bomb. They conclude that ST allows following the time evolution of the natural frequencies detected, especially for the higher ones. On the other hand, they report that HHT does not have the expected performance. They point out that some of the highfrequency modes extracted using EMD contain lowfrequency information, known as modemixing effect, this impeding its application to MPI.
A major problem of the HHT method is the modemixing effect encountered in the EMD process which means that waves with the same value of frequency are assigned to different IMFs in the sifting process [121]. Different strategies have been proposed in recent years in order to lessen this undesirable effect. Pai et al. [122] modify the EMD process by coupling it with the conjugatepair decomposition (CPD) method in order to detect if the IMF has two different frequencies or not. If they have, the CPD method is used to further decompose the IMF. They use the method to estimate the modal parameters of three damped modes of a horizontally cantilevered steel beam subjected to an initial tip displacement and report better performance than the EMD method when dealing with transient and noisy signals.
3.3.3. Blind source separation
Blind Source Separation (BSS) is a nonparametric algorithm capable of recovering the original sources from a mixture of signals [8, 123]. An explanation of the algorithm is presented in a recent article by AmezquitaSanchez and Adeli [1]. The most known BSS algorithms are the independent component analysis (ICA) [124], and the secondorder blind identification (SOBI) [125]. The former assumes that the measured data is a linear combination of statistically independent sources. On the other hand, the latter uses the sources temporal structure [126].
Poncelet et al. [126] evaluate the performance of ICA and SOBI methods to estimate the modal parameters of a 3DOF model and report that ICA can process signals from weakly excited systems, whereas SOBI can also handle signals obtained from moderately damped systems. A shortcoming of SOBI is its inability to identify closelyspaced modes [127]. Hazra et al. [127] propose a hybrid timefrequency approach based on SWT and EMD to alleviate the limitation of SOBI. The method also solves the underdetermined case (the number of acquired sensors is fewer than the number of modes to be identified). They carry out the MPI of a 10story steel tower with a composite steel deck subjected to ambiental vibrations. Yang et al. [128] modify the ICA algorithm to detect moderate damping modes and apply it to a cantilever beam excited with an impact hammer. They show the method can estimate modes whose damping ratio does not exceed 1 %.
3.4. Probabilistic methods
In recent years, probabilistic methods (PM) have become an important signal processing tool to estimate the modal parameters of civil structures using incomplete data. Cara et al. [129] use the ExpectationMaximization (EM) fused with the maximum likelihood estimation (MLE) methods and SSIDATA to compute the natural frequencies and damping ratios of a fourstory twobay by twobay braced steel frame subjected to ambient dynamic vibrations. Although the method can estimate the natural frequencies accurately, the damping ratio values show large variances to their theoretical counterparts.
Bayesian probability has been explored the most to identify the modal parameters of civil structures because it can deal with the unavoidable uncertainties of the measured data. Cheung and Beck [130] propose a Bayesianbased methodology to estimate the modal parameters using the numerical response of a tenstory shear frame with nonclassical damping. They point out that the computational burden required for Bayesianbased estimators is high. To overcome this limitation, Au [131] proposes a fast Bayesian FFTbased algorithm to estimate the modal parameters and apply it to the measurements of a 15story steelframe subjected to ambient vibrations. The method focuses on detecting wellspaced modes. Yan and Katafygiotis [132] presents a Bayesian approach for estimating the MPI using the statistical properties of the autospectral density sum and the statistical information of the spectral density matrix and apply it to a 3story building structure made of aluminum subjected to horizontal and torsional vibrations generated by a shake table.
3.5. Advantages and disadvantages of the signal processing techniques used in MPI.
The knowledge about the features (e.g. frequency resolution or noise immunity) of the different MPI schemes allows the selection of the appropriate algorithm. An incorrect selection of the algorithm may lead to unreliable results. Table 2 presents a summary of the main advantages and drawbacks of the signal processing techniques used for MPI.
4. Novel signal processing techniques
Recent advances in mathematics and signal processing fields have led to the development of novel algorithms for performing time, frequency, or timefrequency analyses. In this section, some of these algorithms are reviewed. To the best of the authors’ knowledge, these algorithms have not been used for MPI. They have features that make them potential candidates to be employed for the MPI.
Table 2. Advantages and disadvantages of the signal processing techniques used in MPI of civil structures
Method

Advantages

Disadvantages

STSM

Straightforward use.
Modal parameters can be directly obtained from the model.
Can deal with small noiselevel signals.

Linear and stationary model.
Accuracy depends on the level of noise. Model order selection can be a timeconsuming procedure.

Freedecay methods

Filtering properties.
Can process raw ambiental responses.
Simplicity.
Easy implementation.

Multistage schemes.
Increased computational burden.
Sensitive to noise.
Used mainly to process stationary responses.

SSI methods

Noise immunity.
Closelyspaced modes detection.
Modal parameters are directly obtained.
Can process slightly nonstationary signals.

Heavy computational burden.
Require calibration.
Generation of spurious modes.

FTbased
methods

Straightforward use.
Modal parameters are directly obtained.
Simplicity (PP).
Non apriori knowledge of the number of modes is required.
Filtering properties (FRF).

Fixed resolution.
Can deal with only stationary signals.
Sensitive to noise.
Closelyspaced modes are not detected.
The input excitation should be available for FRF estimation.
If the excitation source is a harmonic signal, the estimated natural frequency
can be biased (FDD).

Highresolution methods

Noise immunity.
Closelyspaced modes can be detected.

Computational burden.
Require calibration.

HHT and its variants

Adaptive method.
Straightforward use.
The individual modes are extracted.
No user interaction is required.

Modemixing.
EMD variants require calibration.
Its computational burden and accuracy depends on selected algorithm: EMD (lower), ensemble EMD (higher).

BSS

Good accuracy to separate frequency components.
Can identify modes with low energy.

A prefiltered stage is required for signal embedded in highlevel noise.
High damping ratios cannot be accurately estimated.

Probabilistic methods

Closelyspaced mode detection.
Apriori knowledge can be incorporated.
Can process noisecorrupted and incomplete data.

High computational burden.
Requires calibration.

4.1. Local characteristicscale decomposition
As mentioned earlier, the EMD algorithm suffers from the socalled modemixing effect which impedes the accurate identification and extraction of closelyspaced modes [1]. In order to lessen this limitation, recently Zheng et al. [133] proposes a new method called Local Characteristicscale Decomposition (LCD). It uses the concept of monocomponent intrinsic scale component (ISC), the equivalent of the IMF in EMD. The main difference between an ISC mode and an IMF mode is that the latter also uses a criterion in order to ensure the smoothness and symmetry of the potential mode. In addition, LCD uses a thresholdbased criterion to mitigate the end effect and the modemixing issues. The authors used LCD to detect faults in rolling bearings [133] where the modemixing and end effect issues have been reportedly alleviated.
4.2. Compact empirical mode decomposition
Proposed by Chu et al. [134], compact empirical mode decomposition (CEMD) is another timefrequency algorithm proposed to overcome the EMD limitation. It uses a criterion to identify false extrema points and a set of algebraic equations to set the upper and lower envelopes using Hermitian polynomials. These features reduce the modemixing effect and improve the IMF extraction quality, as the end effect (distortions in the beginning and the end of the component) is corrected.
4.3. MUSICEmpirical Wavelet Transform
Empirical Wavelet Transform (EWT) is an adaptive wavelet transform that uses FT to detect the frequency bands to contain the modes in order to construct an effective wavelet filter bank so that the modes can be retrieved accurately. Unfortunately, FT accuracy is compromised under noisy environments, affecting the EWT results. Recently, AmezquitaSanchez and Adeli [135] combined MUSIC algorithm with EWT to improve the accuracy of the original EWT. The proposal uses the MUSIC algorithm to estimate the contained frequencies in the signal and build the appropriate boundaries to create the wavelet filter bank. The authors demonstrate the new methodology is efficient for analyzing nonlinear and nonstationary signals embedded in highlevel noise thus allowing the detection of closelyspaced modes accurately.
4.4. Novel blind source separation algorithm
The performance/accuracy of BSSbased algorithms degrades in noisy environments. A solution is to use prefiltering stages in order to denoise the acquired dynamic signal. This solution, however, increases the computational burden of the proposal thus limiting its application for realtime strategies. In order to mitigate this shortcoming, Chui and Mhaskar [136] propose an adaptive harmonic model for performing the BSS of a multicomponent signal. The proposal removes any existent trend that might affect the component’s detection adversely. Then, signal’s components are extracted using signal separators based on a threshold value. The results show that the method can detect closelyspaced modes. Further, the performance in noisy environments is not degraded. These features make the algorithm a potential candidate for its utilization for the MPI.
5. Final remarks
MPI has become a very active area of research in structural engineering because of the evolution and recent advances in new sensors, signal processing algorithms, as well as the enhanced processing capabilities of the new computers. This paper presented an overview of the main signal processing techniques used in MPI.
The signal processing algorithms commonly used in MPI are time, frequency and timefrequency domain. Timefrequency techniques are the most preferred to estimate the modal parameters of civil structures because they allow observing the evolution of the modal parameters over time.
The newest signal processing techniques, LCD, CEMD and MUSICEWT, proposed since 2012 have not been used in MPI. These algorithms are worth being researched for MPI of large reallife structures because they provide good timefrequency resolution and noiseimmunity, among others.
In spite of the large amount of work presented on MPI, most deal with small and academic problems. New methodologies that use enhanced signal processing techniques capable of handling noisy data and nonlinear signals effectively, accurately, and reliably with computational efficiency for realtime applications should be explored for monitoring the modal parameters of large reallife structures for use in realtime SHM applications.
Acknowledgements
This work was partially supported by the Mexican Council of Science and Technology (CONACyT) by the scholarship 289377.
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