Three-stage Lamb-wave-based damage localization algorithm in plate-like structures for structural health monitoring applications

Signal Processing 168 (2020) 107360

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Three-stage Lamb-wave-based damage localization algorithm in plate-like structures for structural health monitoring applications Mohammad Mahdi Bahador a, Amir Zaimbashi b,∗, Reza Rahgozar a a b

Smart Materials and Structural Health Monitoring Laboratory, Department of Civil Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Optical and RF Communication Systems (ORCS) Laboratory, Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

a r t i c l e

i n f o

Article history: Received 5 March 2019 Revised 30 September 2019 Accepted 23 October 2019 Available online 7 November 2019 Keywords: Lamb waves Structural health monitoring Generalized likelihood ratio test Direction of arrival Amplitude and phase estimation Damage localization

a b s t r a c t In this paper, a new two-dimensional localization algorithm based on the combination of detection theory and array processing frameworks is presented. Our proposed method is capable to extract locations of multi-damage in a plate-like structure for structural health monitoring purpose. The proposed technique is a three-stage localization algorithm. In the first stage, a signal activity detection (SAD) algorithm is applied to extract curved delays of interested damages, which are in the near-field of the array. Then, a new high-resolution technique, named compensated curved-time delay version of amplitude and phase estimation (CTD-APES), is proposed to estimate damage direction of arrivals. Finally, at the third stage of our proposed algorithm, the location of detected damages are found based on the obtained results of the SAD and CTD-APES algorithms. Unlike most existing works, the proposed method is developed based on the spatial-wideband effect as well as near-field nature across the receiving antenna array; thus the proposed method along with the problem formulation can be considered as a new analytical framework in this topic. Extensive Monte Carlo simulation results are provided to demonstrate the superiority of the proposed technique as well as to validate the theoretical calculations. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Structural health monitoring (SHM) is the implementing process to obtain structural on-demand information about the design feedback, performance, and state of health. The goal of SHM research is detecting and interpreting adverse changes to the material and/or geometric properties of a structural system to improve reliability and reduce life-cycle costs. The SHM plays an important role in many fields of application. For example, many of civil structures and constructed facilities, including bridges, buildings, offshore platforms, tunnels, pipelines, towers, dams, and many others, that have a vital role in society, are in a condition of serious deterioration. In such cases, the SHM can help the owners, builders, and designers of structures to make logical decisions. Major benefits include enhancing the safety of structures with early identifying the possible damages and failures, prompting more efficient use of maintenance resources, and providing information that leads to improving designs. In the aerospace industry, the guarantee of lifesafety is always a strong motivation, the benefits of SHM in this field can appear as increased up-time usage rates for aerospace systems by defect/damage or load monitoring and improved ∗

Corresponding author. E-mail address: [email protected] (A. Zaimbashi).

https://doi.org/10.1016/j.sigpro.2019.107360 0165-1684/© 2019 Elsevier B.V. All rights reserved.

designs [1]. The high-performance structures, for example, unmanned aerial vehicles, may not be able to operate effectively without the SHM. The SHM based on ultrasonic guided waves (UGWs) has created much interest this past decade in many fields of application [2–6]. The UGWs can be defined as stress waves enforced to follow a path defined by the geometric boundaries of the structure, and they are usually generated by some high-frequency pulse signals [7]. These waves can also propagate in many different kinds of waveguides including thin plates, rods, tubes, and multilayered structures. The nature of UGWs, which can propagate over long distances with little loss in energy, yields tremendous potential causing cost-effective inspection. The UGWs based inspection techniques allow the inspection of hidden structures, coated structures, structures underwater or soil and structures encapsulated in insulation and concrete such as a wide variety of structures from railway tracks to pipelines or even aircraft skins. One of UGWs based promising approach for SHM in thin-walled structures is also referred to as Lamb waves (LWs). The LWs consist of a multimodal nature, where the velocity of each mode is dependent on the frequency content of the excitation signal and thickness of the plate. In other words, LWs can be divided into two kinds of mode; named symmetrical (S0 , S1 , . . . ) and anti-symmetrical (A0 , A1 , . . . ) modes. The LWs can propagate long distances with low attenuation

2

M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

through the thin solid plate or shell structures with free surfaces. However, the main challenge in the use of LWs is their dispersive nature, such that the wave velocities are functions of frequency. In such cases, different frequency contents of excitation signal produce several LW modes traveling at different and changing speeds, disturbing the damage imaging results. Thus, we should work with the specific modes to obtain an appropriate sensitivity. To do this, we can resort to the Lamb wave-mode tuning method to select an appropriate excitation frequency at which a single mode LW is excited [8]. To generate Lamb waves, we can use a Piezoelectric Wafer Active Sensor (PWAS). The guided Lamb wave generation characteristics by using PWAS have been studied in [9] and [10,11]. It has been shown that the A0 mode is used for the detection of surface defects and disbonds/delaminations, while S0 mode is used for through-the thickness damage detection [12]. In Lamb wave SHM applications, when there is a structural defect in the shell or plate-like structure, these waves are then reflected from some defects. The aim is to find the locations of structural defects. In this case, various LWs based signal processing algorithms have been proposed [13]. For instance, a pulse-echo method for crack detection in aircraft aluminum panel was studied in [14]. The time-reversal process was applied to improve the conventional pulse-echo and pitch-catch method to increase the signal to noise ratio (SNR) and suppress boundary effects [15]. These methods required undamaged baseline data [16,17]. To overcome this restriction, a damage identification method based on combining of consecutive outliers analysis and the time-reversal process has been proposed in [18]. Array signal processing methods are other powerful methods can be used to localize damages in a waveguide without any baseline data requirement. To do this, in general, there are two configurations presented in the literature. One class is configured with its antennas colocated for coherent transmission and detection, whereas the other is with receive antennas separated far away from each other, sometimes referred to as distributed configuration [19–23]. For colocated topology, a multiple antennas system has been shown to offer higher resolution and direct applicability of adaptive array techniques. The phased array processing methods, that are based on colocated configuration, exploit sensors as transmitter and receiver in various configurations to steer and capture the outputs of all sensors in a desired direction [24]. In [25] and [26], Delay-And-Sum (DAS) beamforming algorithm has been exploited to localize structural defect in aluminum and composite plates. The obtained results in [25] have shown that this method suffers from low resolution and high sidelobe levels. To circumvent this, recently, many researchers have developed high-resolution algorithms such as Multiple Signal Classification (MUSIC) algorithm, providing superior performance over the DAS approach [27–31]. It should be noted that these methods provide super-resolution when the sources are uncorrelated and the number of snapshots is high. However, none of these methods can cope with very low snapshot numbers, coherent or highly correlated sources, or severe noise. Besides, most of the above works have used the narrowband assumption in array signal processing, which is not valid in our problem. In this paper, our focus is on an UGWs propagation-based damage localization problem to localize the damages in plate-like structures for SHM applications. To do this, we consider a singleinput-multiple-output configuration and model the received signals from damages being in the near-field or far-field of the array. The another focus is on the colocated receive antenna array to estimate the range and angle of the damages due to near-field nature of our damage-detection problem. Then, we propose a three-stage localization algorithm based on the detection theory and a new array processing frameworks. In the first stage of the proposed algorithm, we model the signal activity detection problem as a composite hypothesis testing problem and develop a detector to

estimate curved delays of interested near-field damages. In the second stage, we introduce a compensated curved-time delay version of amplitude and phase estimation (CTD-APES) to estimate damage direction of arrival angles (DoAs). Finally, in the third stage, a localization algorithm is used to localize detected damages in the previous stages. We devise extensive multi-damage scenarios to validate our proposed algorithm and analytical framework. Our simulation results show the high resolution and multi-damage detection capabilities of our proposed algorithm even under few snapshots. The rest of this paper is organized as follows. In Section 4.2, the excitation signal model is described. The received signal model is derived in Section 3. In Section 4, a new three-stage localization algorithm is presented. Extensive simulation results are provided in Section 5. Finally, Section 6 includes the conclusion of this paper. Notation: In this paper, scalars are denoted by nonboldface lowercase letter, e.g., a. Vectors are denoted by boldface lowercase letters, e.g., a, and matrices by boldface uppercase letters, e.g., A. Superscripts (.)T , (.)∗ and (.)H denote transpose, complex conjugate, and complex conjugate transpose, respectively. The Euclidean norm of vector x is denoted by x, |x| represents the modulus of x, and the mth element of x is denoted by [x]m . The operator . denotes the smallest integer equal to or smaller than the argument and Pr( ) is an operator that represents the probability of its argument. Re{z(t)} stands for the real part of z(t). 1M represents an M-dimensional all-one vector. 2. Excitation waveform The usual excitation signal used in Lamb wave-based inspection (LWI) is windowed tone burst (TB) signal [32]. The TB is a modulated sinusoidal signal by the Hanning or Gaussian window in the time domain to reduce the effect of dispersion and maintain mode purity [33,34]. This signal can be described as

st (t ) = w(t ) sin(2π fc t )

0 ≤ t ≤ Td

(1)

where fc represents carrier frequency, and Td is called time duration of the signal, which is equal to the number of cycles divided by fc ; i.e., Td = N/ fc . Here, w(t) is window signal applied on carrier signal sin(2π fc t). The commonly used window in the LWI is the Hanning window, given by Giurgiutiu [32]



w(t ) = 0.5 1 − cos(2π

t ) Td



(2)

It should be noted that st (t) is a band-pass signal, represented by

st (t ) = Re{s(t ) e j2π fc t }

(3)

where s(t) is called the equivalent complex baseband signal, given by π

s(t ) = w(t ) e− j 2

(4)

The elements of the TB signal including the carrier, window and the resulted TB signal are shown in Fig. 1. There are some comments on the selection of parameters Td and fc . By selecting proper carrier frequency fc , the dispersion behavior of Lamb waves could be suppressed. As discussed in [12], the S0 mode of Lamb

Fig. 1. Elements of tone burst signal: a) Carrier signal, b) Hanning window, c) Tone burst signal.

M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

3

waves gets a much stronger echo from a through-the thickness crack than A0 mode. It was shown in [12] that the tone burst wave packets maintain their shape even after multiple reflections, thus indicating that the S0 waves have very little dispersion at the frequency of tuned S0 . As the effect of damage in the waveguide (e.g., aluminum plate) on transmitted signal must be distinguished, the excitation signal must be a short pulse. As the number of cycles would decrease, the bandwidth of signal would become wider and causes more dispersion and vice versa. The dispersive behavior of Lamb waves depends on the excitation frequency. As Lamb wave modes have dispersive behavior depending on propagation frequency, the propagation characteristics vary as the wave frequency content changes. For decreasing this, it could offer a trade-off between time duration and frequency content. Among selecting fc and Td , selecting fc is more effective than other. In this case, some researchers have tried to present methods to select the best center frequency. For instance, frequency tuning of the Lamb waves was shown to be a powerful method to reach the best guess of fc [9]. 3. Received signal model Consider a sensing system with a single transmit antenna element and a receiver array equipped with M colocated antenna, as illustrated in Fig. 2. In the receiver side, a uniform linear array (ULA) with M elements along the x-axis is considered. By definition ξ = M+1 2 , the x-y position of the mth element in the receiving array is denoted by ((m − ξ )d, 0 ) for m = 1, . . . , M. In the following, without loss of generality, we assume that M is even. Here, we assume a same inter-element spacing of d between receiving elements with d ≤ λ/2, where λ is the wavelength of the received signal. For simplicity, we consider the first element as the reference element. In the considered scenario, the transmitter antenna is assumed to be located at the coordinate of (0,0) in order to emitting signal st (t). Also, it is assumed that there are L damages1 in the polar coordinates of (rl , θ l ) for l = 1, . . . , L, where θ l is the broadside angle of the array and rl is the distance (slant range) from the middle of ULA (Tx. position) to the lth damage, as shown in Fig. 2. Now, the signal received by the mth element due to the lth damage can be written as (l ) xm (t ) = αm(l ) st (t − τm(l ) )

(5)

(l ) where αm is the complex-valued reflection coefficient of the lth damage. Here, τm(l ) is the traveling time from the transmitter to the lth damage and again to the mth receiver element, given by (l )

τm

(l )

r + rm = l c

(6)

where c is the group velocity in the waveguide. As shown in Fig. 2, rl is the distance between the lth damage and the transmitter and (l ) rm is the distance between the lth damage and the mth element in the receiving ULA, expressed as (l ) rm =



rl2 + ((m − ξ )d )2 − 2rl (m − ξ )dsin(θl )

(7)

(l ) To proceed, we use the second order Taylor expansion of rm , i.e., (l ) rm = rl − (m − ξ ) dsin(θl ) + (m − ξ )2

d2 cos2 (θl ) d3 + O ( 2 ) (8) 2rl rl

By substituting Eq. (8) into Eq. (6), we obtain (l )

τm 1

2r d3 = l − ω l ( m − ξ ) + φl ( m − ξ ) 2 + O ( 2 ) , c rl

It should be noted that L is an unknown parameter.

(9)

Fig. 2. Damage localization problem using an uniform linear array (ULA).

where

ωl =

dsin(θl ) , c

(10)

d2 cos2 (θl ) . 2rl c

(11)

and

φl =

3

In Eq. (9), the term O ( d2 ) corresponds to the term of order greater r

or equal to

d3 , r2

l

neglected here. By making use of Eq. (3), Eq. (5) can

l

be rewritten as (l ) (l ) xm (t ) = Re{αm(l ) s(t − τm(l ) ) e j2π fc (t−τm ) }

(12)

By substituting Eq. (9) into Eq. (12), we find

  2r αm(l ) s t − l + ωl (m − ξ ) c   2rl 2 − φl (m − ξ )2 e j2π fc (t− c +ωl (m−ξ )−φl (m−ξ ) )

(l ) xm (t ) = Re



By introducing t  t −

2rl c

(13)

and replacing it in Eq. (13), we get

  (l ) xm (t ) = Re αm(l ) s t + ωl (m − ξ )    2 − φl (m − ξ )2 e j2π fc (t +ωl (m−ξ )−φl (m−ξ ) )

(14)

By selecting the time of signal arriving at the first as origin, Eq. (14) can be expressed as (l ) xm (t ) = Re{αm(l ) s(t + ωl (m − ξ ) 2 − φl (m − ξ )2 ) e j2π fc (t+ωl (m−ξ )−φl (m−ξ ) ) }

(15)

In our problem, we cannot use spatial-narrowband assumption in order to approximate the term s(t − τm(l ) ) with s(t), which is called the spatial-wideband effect. In other words, the spatialnarrowband assumption said that the max τm(l )  1B , where B is

the bandwidth of the received signal, so all τm(l ) s can be replaced with a constant delay such as zero for simplicity to conclude s(t − τm(l ) ) s(t ). This inspires us to use the SAD algorithm to find different delays of interested damage across receive antenna elements.

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M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

Similarly, in the presence of L damages, the output of the mth antenna element can be written as

xm (t ) =

L (l ) xm (t )

(16)

l=1

Now, the equivalent baseband of Eq. (16), denoted as xm,b (t), can be expressed as

xm,b (t ) =

L αm(l ) s(t + ωl (m − ξ ) l=1

− φl (m − ξ )2 )e j2π fc ωl (m−ξ ) e− j2π fc φl (m−ξ )

2

(17)

If the received signal in each element is sampled with the sampling frequency fs , the nth sample of the received signal at the mth antenna element can be expressed as L 2 xm,b [n] = αm(l ) s[n + nm(l ) − p(ml ) ]e j2π fc ωl (m−ξ ) e− j2π fc φl (m−ξ )

(18)

l=1

(l ) where, nm and p(ml ) are the number of samples corresponding to (l ) the delay of ωl (m − ξ ) and φl (m − ξ )2 , i.e., nm = ωl ( m − ξ ) f s  (l ) 2 and pm = φl (m − ξ ) fs . In the presence of the system noise the received signal Eq. (18) can be rewritten as

x[n, m] =

l=1

(19)

In Eq. (19), w[n, m] is the nth sample of the received noise at the mth element and s[n] is the nth sample of baseband equivalent transmitted signal s(t). Remark 1. It is worth to note that two terms p(ml ) and 2 e− j2π fc φl (m−ξ ) in Eq. (19) are produced due to the near-field feature of our problem. With the spatial-narrowband and when damages are in the far-field of the array, Eq. (19) can be simplified as

x[n, m] = s[n]

L

α (l ) e j2π fc ωl (m−ξ ) + w[n, m]

(20)

l=1

Now, the M × 1 array output vector of an M element array at the nth snapshot can be described by the following simple model

xn = A(θ )sα [n] + wn

(21)

where sα [n] = s[n][α (1 ) , . . . , α (L ) ]T , θ = [θ1 , . . . , θL ]T , A(θ ) is the M × L steering matrix defined as A(θ ) = [a(θ1 ), . . . , a(θL )] with a(θl ) given by

a(θl ) = [e j2π fc ωl (1−ξ ) , e j2π fc ωl (2−ξ ) , . . . , e j2π fc ωl (M−ξ ) ]T

(22)

Note that for near-field damages, the steering vector should be modified as

a(θl ) = [e j2π fc ωl (1−ξ ) e− j2π fc φl (1−ξ ) , . . . , e j2π fc ωl (M−ξ ) e− j2π fc φl (M−ξ ) ]T 2

4.1. Proposed SAD algorithm As has been mentioned, our system measures the spatial distribution of damage reflectivity in the two-dimensional polar coordinate system of range and azimuth angle. Indeed, our system transmits a pulse with the width of Td and, then, collects the baseband received signal from each ULA element, which is sampled with the sampling frequency of fs in the range of hundreds of kilohertz. The n dimension of x[n, m] is called fast time or range dimension while the m dimension represents spatial sampling. The sampling interval, Ts = f1 , determines the spacing of the ULA range samples, s

L 2 αm(l ) e j2π fc ωl (m−ξ ) e− j2π fc φl (m−ξ ) s[n + nm(l ) − p(ml ) ]

+ w[n, m]

and Iterative Adaptive Approach for Amplitude and Phase EStimation (IAA-APES) to estimate damages’ DoA since it is not possible to assume spatial-narrowband and far-field assumptions. In this section, we introduce a new three-stage algorithm to localize the damage positions. In the first stage of our proposed algorithm, we use a signal activity detection (SAD) algorithm to estimate curved delays of considered near-field damages. Then, we use a compensated curved-time delay version of IAA-APES, abbreviated as CTDAPES, to estimate the damages’ DoA. Finally, a localization algorithm is used to localize damages in the x-y plane based on the results of the SAD and CTD-APES algorithms. In the following, each stage of the proposed algorithm will be described in detail.

2

(23) In our case, both spatial-narrowband and far-field assumptions are violated and the general signal model presented in Eq. (19) should be considered. In the following, we consider the above facts and propose a new three-stage localization algorithm accordingly. 4. Proposed three-stage localization algorithm In our case, we cannot directly apply the conventional and robust DoA algorithms such as MUltiple SIgnal Classifiction (MUSIC)

referred to as range cells, with the range cells spacing of cT2s . In spatial dimension, the array elements sample the wavefront that impinges on the array face. Thus the spacing of the array elements is yet another spatial sampling interval. The most fundamental task of our system is SAD problem performed in a fast time (range) dimension independently on each receiver element and done after receiving each transmitted pulse. The basic method of detection is to obtain the power of the received time signal and compare it to a threshold η. The threshold η is chosen to be high enough so that it is very unlikely that noise will produce a power large enough to cross the threshold η. As a result, we can assume that if the power of the received signal does cross the threshold at some range cell of i, then those high-amplitude echoes are caused by a target located at the corresponding range of i cT2s . To perform this important task perfectly, let xi [m] denote the sampled received vector signal from the ith sample to i + Nd at the mth element, i.e., xi [m] = [x[i, m], . . . , x[i + Nd , m]]T where Nd = Td fs . In the presence of a damage in the ith range cell, we have xi [m] = γm s + wi [m] 2 (l ) where γm = αm e j2π fc ωl (m−ξ ) e− j2π fc φl (m−ξ ) represents the complex-valued amplitude of the damage in the ith range cell, π s = [s[0], . . . , s[Nd − 1]]T with s[n] = 0.5(1 − cos(2π Nn )) e− j 2 and d

wi [m] = [w[i, m], . . . , w[i + Nd , m]]T . Thus, we can model the SAD as a binary hypothesis testing problem for a given range cell i as follows



H0 : xi [m] = wi [m] H1 : xi [m] = γm s + wi [m]

(24)

where hypothesis H0 presents the absence of a damage at range cell i, while the hypothesis H1 presents the presence of a damage at range cell i. In our detection problem, we assumed that the noise vector wi [m] is zero-mean complex white Gaussian noise H with the unknown covariance matrix of E {wi [m]wi [m] } = σm2 IN . Also, the received amplitude γ m is considered to be an unknown complex-valued parameter. According to the Neyman-Pearson (NP) criterion, the optimum solution to the hypothesis testing problem Eq. (24) is the likelihood ratio test (LRT), given by

l i ( xm ) =

H

L(xi [m]; γm , σm2 , H1 ) 1  ≷η L(xi [m]; σm2 , H0 ) H 0

(25)

M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

where L(xi [m]; γm , σm2 , H j ) is the likelihood function (LF) in our problem under hypothesis H j for j = 0, 1, given by

 L(xi [m]; γm , σm2 , H j ) =

 exp −

1

N2d

π σm2

(xi [m] − μγm s )H (xi [m] − μγm s ) σm2

(26)

where μ is zero for H0 and it is one under H1 . For the problem at hand, the NP detector cannot be implemented since it requires the knowledge of the parameters γ m and σm2 , which are unknown in practical situations. To circumvent this, we resort to generalized likelihood ratio test (GLRT) which is equivalent to replacing the unknown parameters with their maximum likelihood estimates (MLEs) in the LRT [35–40]. It can be shown that the MLEs of the unknown parameters under each hypothesis can be obtained as [38]

 xi [ m ]  2 2 H0 : σˆ m, 0 =

(27)

Nd

and

H1 :

⎧ sH xi [m] ⎪ ⎪ ⎨γˆm = H s s

(28)

⎪ xi [m]2 − Nd |γˆm |2 ⎪ 2 ⎩σˆ m, 1 = Nd

2 Here, γˆm is the MLE of γ m under H1 , σˆ m, is the MLE of 0

the σm2 assuming H0 is true (maximizes L(xi [m]; σm2 , H0 )), and 2 2 σˆ m, 1 is the MLE of the σm assuming H1 is true (maximizes L(xi [m]; γˆm , σ 2 , H1 )). By replacing these MLEs to the LRT, we obtain the generalized likelihood ratio test (GLRT) as

l i ( xi [ m ] ) =

H1 2 L(xi [m]; γˆm , σˆ m, 1 , H1 ) ≷ η 2 L(xi [m]; σˆ m,0 ), H0 ) H

(29)

0

or equivalently H1

| g( x i [ m ] ) | 2 ≷ η H0

where η is set subject to a constraint on the probability of false  alarm Pf a = P r |g(xi [m] )|2 > η|H0 . Consequently, if |g(xi [m] )|2 is smaller than a preassigned decision threshold η, the detector declares H0 , otherwise, H1 . The false alarm probability of the proposed detector can be obtained as [38]

Pf a = QF1,N−(P+1) (η ),

(32)

where QF1,N−(P+1) is the right-tail probability of central complex F distribution. As can be seen from (32), the threshold setting is feasible with no prior knowledge of any unknown parameters. Thus, the obtained detector ensures the constant false alarm rate (CFAR) property against any uncertainty in the noise power. This means that the false alarm probability of the designed system remains constant by changing the noise power. Some interpretation of the proposed detector is in order. The term sH xi [m] is the correlation between the sampled transmitted pulse vector s and the received vector xi [m]. Since s is identical to the signal to be detected under hypothesis H1 , namely the presence of the s in the vector under test (VUT) xi [m], this processing is called matched filter. The term xi [m]H xi [m] is the estimation of the noise variance under hypothesis H0 and the term sH s is the energy of the transmitted signal. Therefore, the GLR test calls for taking the square-law of the matched filter output sH xi [m], normalizing it by the transmitted signal energy and the estimation of noise variance, and comparing the result to a threshold η. The proposed detector consider only a one-dimensional vector of range cells in making a decision. For the presence or absence of damages across all range cells of interest, they should be tested one after another in a sliding manner. If the test statistic associated with a range cell exceeds the threshold, the detection algorithm declares that damage present at that range cell. This process is carried out for all range cells of interest. Fig. 3(a) shows the spatial-temporal signal received in the input of the SAD algorithm for a single-damage scenario, while their corresponding outputs of the proposed SAD algorithm are shown in Fig. 3(b). In Fig. 3(b), we compute the values of |g(xi [m] )|2 for all interested range cells, then for values greater than the fixed threshold η we use |g(xi [m] )|2 , otherwise they are replaced with zero value, i.e.,



| g( x i [ m ] ) | 2 , 0

| g( xi [ m ] ) | 2 > η

(30)

SAD [i, m] =

(31)

From Fig. 3, there are some points in order. It is seen that the received echoes over different antenna elements don’t have the same delay due to wideband-spatial effect, where they

where

sH xi [m] g( x i [ m ] ) = sxi [m]

5

otherwise

(33)

Fig. 3. Spatial-temporal representation of the input and output of the SAD algorithm for a high signal-to-noise scenario: a) Input of the SAD algorithm, b) SAD algorithm after thresholding, c) SAD results after finding the local maximums across different antenna elements.

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M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

approximately follow a quadratic structure, called curved delays, due to the near-field nature of our case. Then, denoted by τˆm ’s for m = 1, . . . M the estimations of these curved delays, where they are obtained as delay values for which the output of the SAD algorithm after thresholding reach local maximums in each antenna element, i.e.,

τˆm = argmaxi { SAD [i, m]}Ts

(34)

or, equvalentely,

nˆ m = argmaxi { SAD [i, m]}

(35)

where nˆ 1 , nˆ 2 , . . . , nˆ M are the range cell numbers for which the SAD algorithm declare the presence of damage across different antenna elements. In the sequel, the estimated delay times of (34) are called times of flight (ToF), where it is denoted by τˆm(l ) for the lth (l ) damage in a multi-damage scenario. Similarly, denoted by nˆ m for m = 1, . . . , M the range cell numbers for which the SAD algorithm declare the presence of the lth damage across different antenna elements. In Fig. 3(c), the range cells corresponding to the local maximums over different antenna elements for the considered singletarget scenario are shown. To the best of authors’ knowledge, this is the first work applying a SAD algorithm to estimate the curved delays of the considered near-field damages. 4.2. Proposed CTD-APES algorithm The main goal of array processing is to recognize direction and waveforms of the desired source by combining the received signals by aperture elements, therefore, the desired signal would be enhanced and unwanted signal such as interference and noise would be suppressed. The basic approach to sensor array signal processing is the delay and sum beamformer (DAS) that have many disadvantages such as low resolution and high sidelobe levels resulting in some limitations [41]. To cure this, high-resolution approaches have been proposed to provide better performance than the DAS. The standard Capon beamformer (SCB) [42] and multiple signal classification (MUSIC) [43,44] are well-known algorithms providing high-resolution DoA processing when the sources are uncorrelated and the number of snapshots is high. But none of them can work well in case of low numbers of snapshots, coherent or correlated sources, or severe noise level. The least squares-based iterative adaptive approach for amplitude and phase estimation (IAAAPES) was proposed to work with few snapshots (even one), uncorrelated, partially correlated, or coherent sources and arbitrary array geometries [45]. However, this algorithm has been proposed under spatial-narrowband and far-field assumption, which is violated in our case. In this work, for the first time, we introduce a modified version of IAA-APES called compensated curved-time delay amplitude and phase estimation (CTD-APES) to find damages’DoA. Consider a damage located at angle θ l which reflects the transmitter signal into the array. Suppose that nˆ 1 , nˆ 2 , . . . , nˆ M are the range cell numbers for which the SAD algorithm detected the damage across different antenna elements. In our proposed IAA-APES based algorithm, we feed the modified IAA-APES with the sampled snapshot constructed as

g [ n l ] = [ g ( [ n l ] 1 ) , . . . , g ( [ n l ] M )] T

(36)

where g[nl ] is an M × 1-dimensional vector with nl = (l ) T (l ) [n1(l ) , . . . , nM ] and g(.) is defined in Eq. (31), i.e., [nl ]m = nm for m = 1, . . . , M. Similar to Eq. (21), the spatial vector g[nl ] can be written as

g [ n l ] = γ [ n l ]  a ( θl ) + w [ n l ]

(37)

where (l ) T γ [nl ] = [γ [[nl ]1 ], . . . , γ [[nM ]1 ]]T = [γ [n1(l ) ], . . . , γ [nM ]]

(38)

and a(θl ) is an M × 1-dimensional vector, called pseudo-steering vector, given by

a(θl ) = [e j2π fc ωl (1−ξ ) e− j2π fc φl (1−ξ ) , . . . , e j2π fc ωl (M−ξ ) e− j2π fc φl (M−ξ ) ]T 2

2

(39) (l )

In (38), γ [nm ] represents complex-valued amplitude of the lth (l ) damage signal detected in the range cell nm at the mth element. Here, vector w[nl ] is the spatial noise at the range cells associated to the lth detected damage. To enhance the performance of our proposed DoA estimation, we should use several snapshots around the above detected range cells such as

g [ n l + i 1 M ] = γ [ n l + i 1 M ]  a ( θl ) + w [ n l + i 1 M ]

(40)

for i = −N/2, . . . , N/2 with N being the number of snapshots. It should be noted that in the original IAA-APES it was assumed (l ) that nˆ 1(l ) = nˆ 2(l ) = . . . = nˆ M , while in our problem they are different. Based on this, we call the new IAA-APES based algorithm as CTDAPES. Now, to determine θ l we can exploit a modified IAA-APES based algorithm. To do this, let us define

pˆ = [ pˆ (θ1 ), . . . , pˆ (θK )]T

(41)

where θk = −2π + (k − 1 ) πK for k = 1, . . . , K, in which |θk+1 − θk | = π defines the angle accuracy to search damage angles. In Eq. (41), K pˆ (θk ) can be estimated as

pˆ (θk ) =

|aH (θk )R−1 g[nl + i1M ]|2 l |aH (θk )R−1 a ( θk ) | 2 l i=−N/2

N/2 1 N

(42)

where the spatial covariance matrix Rl is defined as

Rl = a(θl )PaH (θl )

(43)

where

ˆ) P = diag(p

(44)

It is noticeable that, the matrix Rl is depended on unknown matrix P, therefore, the algorithm behaves recursively to estimate the required matrices. The proposed CTD-APES algorithm is summarized in Algorithm 1 for each detected damage of the SAD algorithm, where K is the number of searching points in the angle-domain, and l = 1, . . . , L with L being the number of damages detected by the SAD algorithm. By considering K > L, only a few components of vector pˆ will be non-zero. Now, the location of peak obtained from ˆ determine the angle of damage (or damages) detected by vector p the SAD algorithm. Our simulation results show that the CTD-APES does not provide significant improvements in performance after about 10 iterations. Algorithm 1 Proposed CTD-APES in the presence of L damages, i.e., l = 1, . . . , L and K > L. 1: 2:

Construct pˆ = [ pˆ (θ1 ), . . ., pˆ (θK )]T with N/2 pˆ (θk ) = N (aH (θ 1)a(θ ))2 |aH (θk )g[nl + i1M ]|2 , i=−N/2 k

k

repeat ˆ) R = A(θ )PAH (θ ), A(θ ) = [a(θ1 ), . . ., a(θK )], P = diag(p 4: for k = 1 to K do 5:  2 |aH (θk )R−1 g[nl +i1M ]|2 6: pˆ (θk ) = N1 N/ −1 H 2 i=−N/2

3:

end for 8: until convergence

| a ( θk ) R

a ( θk ) |

7:

If several damages are present in the considered scenario with different range cells, the curved time returns of one target could coincide with the curved time returns of other targets in different elements. For example, the curved time returns (CTRs) in the range dimension at the output of the proposed SAD algorithm resulted from two damages in four elements are shown in Fig. 4.

M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

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Fig. 4. Spatial-temporal representation of the input and output of the SAD algorithm for a high signal-to-noise regime and in multi-damage scenario: a) Input of the SAD algorithm, b) SAD algorithm after thresholding.

The damages are shaded differently so that the reader might easily distinguish them. In reality, we need a target association algorithm to combine different possible CTRs. For example, if there are NT(m ) detected targets by the SAD algorithm when applied for the mth  (m ) element, the number of virtual CTRs possibilities is M for m=1 NT DoA estimations. This is out of the scope of the present work and will be discussed in our future work. Thus, in this paper we assume that there are L damages far from each other such that they can be easily distinguished based on the output of SAD algorithm; i.e., they are different in the range dimension as shown in Fig. 4b. 4.3. Proposed localization algorithm The location of the damages may be estimated by using a multistatic structure with more than two mono-static or bistatic geometries, depending on the position of transmitter and receiver relatively to each other. In bistatic case, the line of the position of the damage can be represented by an ellipse whose foci are the locations of the transmitter and the receiver. Here, the transmittertarget-receiver distance is called the bistatic range. The damage location is determined by the intersection of the multiple ellipses. This needs to detect the damage echo signals emitted from the multiple transmitters. However, the detection of the damage echo signals may not be guaranteed even when the targets are near the receiver. To increase the detectability of the damage signals, DoA estimates can be used for finding the exact locations of the interested damages. As the DoA estimate of the damage echo signal indicates the direction of the damage from the receiver, the damage location can be determined by finding the crossing point between the DoA and the ellipse corresponding the detected damage. In contrast, in the case of the mono-static case, finding the crossing point between the DoA and the circle results in the damage position. Based on these facts, the localization stage of the proposed algorithm is devised. In other words, the detection results of the proposed SAD algorithm together with the DoA estimations of CTD-APES Algorithm are used to localize the interesting damages in the x-y plane. To do so, as shown in Fig. 5, we need to determine the radius of the circle with the center of transmitter and then find the crossing point between the obtained DoA and circle to localize detected damage. Suppose that the estimated time of flight (ToF) of the lth damage over different antenna

Fig. 5. Illustration of the third stage of the proposed algorithm.

elements are denoted by τˆm(l ) for m = 1, . . . , M. We compute the cτˆ

(l )

radius of circle corresponding to the lth damage as rˆl = 2av where  (l ) τˆa(vl ) = M1 M with 0 ≤ ζ m ≤ 1. Our extensive simulation m=1 ζm τˆm results indicate that for low-size receive array and when the transmitter antenna is placed in the middle of the receive array, the proper choice of ζ m ’s is ζm = 1 for m = 1, . . . , M. In the simulation section, the effectiveness of this stage of the proposed method will be examined for some multi-damage scenarios. 5. Simulation results and discussion In this section, we aim to examine the performance of the proposed algorithm, schematically depicted in Fig. 6. To do this, we first investigate the false alarm rate and detection performance of proposed SAD detector. Second, we study the performance of the proposed CTD-APES algorithms. Third, the performance of the localization algorithms is investigated. To do so, the carrier frequency fc is set equal to 391 kHz based on the frequency tuning method

8

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(l ) Fig. 6. Schematic of problem layout, where nl = [n1(l ) , . . . , nM ] and τˆ l = [τˆ1(l ) , . . . , τˆM(l ) ] with l = 1, . . . , L.

presented in [9] for a 6061 aluminum alloy square with dimensions of 500 × 500 × 5 mm, the sampling frequency fs is selected equal to 5 MHz. As a result, the number of samples of TB signal would be equal to 58, i.e., Nd = 58. It should be noted that fs plays important role to achieve more precision of the SAD and the CTD-APES algorithms. The configuration of ULA, as shown in Fig. 2, consists of a transmitter placed at the center of ULA and four receive antenna elements placed on the two sides of transmitter along the x-axis with inter-element spacing of 8mm, i.e., M = 4 and d = 8mm. To examine the effectiveness of the proposed algorithm, extensive scenarios are devised with characteristics listed in Table 1. They include three single-damage scenarios (SDS), three twodamage scenarios (2DS) and two three-damage scenarios (3DS). Here, the SNR of lth damage reflected signal is defined as

SNR

(l )

|γ ( l ) |2 = 2 σ Nd

(45)

where γ (l) is the complex amplitude of damage under test and σ 2 is the received noise variance. 5.1. Performance evaluation of SAD algorithm

Table 1 Different scenarios for performance evaluation of the proposed algorithm. Damage Position

θl

Scenario

◦

rl (cm)

SNR (dB)

I

SDS1 SDS2 SDS3

+27 0◦ −47◦

13.0 10.0 17.0

5 5 5

II

2DS1 2DS2 2DS3

+27◦ , 0◦ −47◦ , 0◦ −47◦ , +27◦

13.0,10.0 17.0,10.0 17.0,13.0

5 5 5

III

3DS1 3DS2

+27◦ , 0◦ , −47◦ +27◦ , 0◦ , −47◦

13.0,10.0,17.0 11.2,10.0,14.7

10 20

value η while no target is present. A good detector is one that maximizes the detection probability for a given false alarm probability. In the detection theory framework, the detection threshold is set according to a predetermined false alarm probability. In Fig. 7, the false alarm probability as a function of detection threshold η is depicted. In general, when it comes to the reliability of Monte Carlo evaluations, it is commonly agreed to adopt at least 100 Monte Carlo (MC) runs to obtain the detection threshold relip fa

As mentioned before, the probability of false alarm Pfa is the probability that an instantaneous sample be above the threshold

ably. Thus for the minimum value of p f a = 0.001, this means 105 runs. In our work, we use 106 MC runs to be clearly on the safe

Fig. 7. Probability of false alarm as functions of detection threshold η.

M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

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Fig. 8. Probability of detection as functions of input signal-to-noise ratio for different values of false alarm probability.

Fig. 9. Evaluation of the first and second stage of the proposed algorithm for the SDS1 scenario: a) SAD outputs before thresholding versus range cell across different antenna elements, b) CTD-APES output as functions of angle.

side and to make sure that results are accurate. It is seen as Pfa decreases, the threshold increases. In our case, we set the detection threshold for the false alarm probability of 10−3 equal to 0.334. From Fig. 7, it is also seen that the obtained closed-form expression for false alarm probability (Eq. (32)) is very accurate in terms of fitting the Monte Carlo simulated false alarm probability. Here, we use ’analytical’ to stand for the derived analytical false alarm probability and ’MC Simulation’ are representative of the empirical false alarm probability. In general, the detection probability of a detector is a function of SNR values and false alarm probability by which the detection threshold is set. In Fig. 8, the detection probability versus SNR for three values of false alarm 10−3 , 10−2 and 10−1 is plotted. Here, we assume a single damage and use SNR(l ) = SNR. It is seen that by decreasing the false alarm, the detection probability is decreased for a given SNR. As expected, when received SNR is increased, the detection probability is also increased. From Fig. 8,

it is observed that the required SNR is equal to -6.5dB for Pd = 0.9 and Pf a = 10−3 . 5.2. Performance evaluation of CTD-APES algorithm In this section, the performance of the proposed CTD-APES algorithm is evaluated through three general scenarios of Table 1, which are similar to previous works [9,12,14,46]. In these scenarios, we consider the scenario I as three single-damage scenarios denoted by SDS1, SDS2, and SDS3. In the single-damage scenarios, the considered damage is created at θ = 0, +27◦ and −47◦ with the distance of 10, 13 and 17 cm, respectively. By these scenarios, the effect of different time delays on the performance of the proposed algorithm can be studied. The results corresponding to scenario SDS1 and that of SDS2 and SDS3 are reported in Figs. 9 and 10, respectively. For example, the outputs of the SAD algorithm before thresholding across different antenna elements are shown in

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Fig. 10. Evaluation of the first and second stage of the proposed algorithm for two single-damage scenarios denoted as SDS2 and SDS3: a,c) SAD output versus range cell in the first antenna element, b,d) CTD-APES output as functions of angle.

Fig. 9(a). From this figure, the effect of the wideband-spatial effect and the near-filed nature of our problem can be seen. In other words, in the range dimension, it is observed that the range cells associated to the peaks in the SAD algorithm across different antenna elements are equal to 474, 481, 484 and 490. This inspires us to propose a SAD algorithm to estimate the curved delays of damage across different elements. From Fig. 9(b), it is also observed that the proposed CTD-APES algorithm determine the angle of the considered damage properly, where the vertical dotted lines represent the true source locations. Similarly, for the SDS2 and SDS3 scenarios, but for the first antenna element, the results of the SAD algorithm are presented in the Fig. 10(a) and (c). In this case, the results of the CTD-APES algorithm are also depicted in Fig. 10(b) and (d). From these figures, it is seen that reflected signal from

damage is correctly positioned in the range and spatial dimensions. In our problem, it should be noted that the CTD-APES algorithm is fed with the results of the SAD algorithm, providing damages DoA with low sidelobes and peak at the true damage angle. In the general scenario II, two damages with different angles and ranges are considered to devise three cases named as 2DS1, 2DS2 and 2DS3 with characteristics listed in Table 1. In these cases, the results of the SAD algorithm obtained in the first antenna element are shown in Fig. 11(a), (c) and (e). It is seen that the SAD algorithm provides accurate damages’ range cell estimates. For the spatial dimension, the results are shown in Fig. 11(b), (d) and (f). It is observed that the proposed CTD-APES algorithm is also able to resolve the sources successfully and provides accurate angle estimates.

M.M. Bahador, A. Zaimbashi and R. Rahgozar / Signal Processing 168 (2020) 107360

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Fig. 11. Evaluation of the first and second stage of the proposed algorithm for three double-damage scenarios denoted as 2DS1,2DS2 and 2DS3: a,c,e) SAD output versus range cell for the first antenna element, b,d,e) CTD-APES output as functions of angle.

For the more general case of scenario III, the above simulations are repeated and their results are shown in Fig. 12. Here, the scenario 3DS1 corresponds to the case considered in [47]. The results of Yu [47] have shown that the one damage in the triple-damage scenario can not be detected, while our results shown in Fig. 12 illustrate the superiority of the proposed high-resolution CTD-APES algorithm. These performance improvements are provided due to the matched filter gain of the proposed SAD algorithm as well as the consideration of the near-field nature to devise the proposed high-resolution CTD-APES algorithm. In Table 2, the results of the first and second stages of the proposed algorithm are summarized. By comparing the obtained re-

sults of Table 2 with that of Table 1, the superior performance of the proposed algorithm in both range and spatial dimensions can be observed. 5.3. Performance evaluation of localization algorithm In this section, the performance of the proposed localization algorithm is examined. To see the effectiveness of the third stage of the proposed three-stage localization algorithm, the performances of the proposed algorithm are investigated for damage scenarios SDS1, 2DS3, 3DS1 and 3DS2, where their corresponding results are depicted in Figs. 13–16, respectively. In these figures, SIMO x-y

12

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Fig. 12. Evaluation of the first and second stage of the proposed algorithm for two triple-damage scenarios denoted as 3DS1 and 3DS2: a,c) SAD output versus range cell for the first antenna element, b,d) CTD-APES output as a function of angle.

plane results of the proposed three-stage localization algorithm are depicted. As can be seen, the proposed three-stage localization algorithm can estimate the positions of the considered defects accurately. In these figures, the CTD-APES’ performance is also compared with that of the classical delay-and-sum (DAS) method [46]. It is observed that the DAS method suffers from low resolution to find damages’DoA in multi-damage scenarios, while the proposed method is a high-resolution method working well with few snapshots. These results of the Figs. 13–16 clearly show that all stages of the proposed three-stage localization algorithm perform well to localize the position of the damages with superior performance.

6. Conclusion In this paper, a novel three-stage localization algorithm for plate-like structural health monitoring purpose has been proposed. In our problem, it was shown that the spatial-narrowband and farfield assumptions were violated. Based on these facts, a new signal activity detection (SAD) algorithm has been proposed to estimate different time delays experienced by damage across different receiver antenna elements. It has been shown that the proposed SAD algorithm provides some processing gains resulting in significant damages localization improvements. In addition to this, it is

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Table 2 Results of the proposed three stage localization algorithm. Estimated Position

θˆl

Scenario

rˆl (cm) ◦

I

SDS1 SDS2 SDS3

+27 0◦ −47◦

13.0 10.1 17.1

II

2DS1 2DS2 2DS3

+27◦ , 0◦ −48◦ , 1◦ −46◦ , +27◦

13.0,10.1 15.4,10.1 17.1,13.0

III

3DS1 3DS2

+26◦ , 1◦ , −46◦ +27◦ , 0◦ , −47◦

13.1,10.1,17.0 11.3,10.2,14.8

Actual Position Estimated Position DAS Method

Fig. 15. Comparison of SIMO x-y plane results of the proposed three-stage algorithm and that of DAS method for 3DS1.

Fig. 13. Comparison of SIMO x-y plane results of the proposed three-stage algorithm and that of DAS method for SDS1.

Actual Position Estimated Position DAS Method

Fig. 16. Comparison of SIMO x-y plane results of the proposed three-stage algorithm and that of DAS method for 3DS2.

shown that the first stage of the proposed algorithm, namely the SAD algorithm, isolates the received damages echoes in the range dimension to reduce the interference due to other damages. The second stage of the proposed algorithm, CTD-APES algorithm, estimates the angles of incident wavefronts reflected by damages precisely. Finally, the third stage of the proposed algorithm localizes the interested damages in the x-y plane based on the results of the previous stages. Extensive simulation results have been provided to show the superior performance of the proposed three-stage localization algorithm to localize the interesting damages in the x-y plane. In our future work, we will experimentally verify the performance of the proposed algorithm. References Fig. 14. Comparison of SIMO x-y plane results of the proposed three-stage algorithm and that of DAS method for 2DS3.

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