## Abstract

The high spatial resolution, high accuracy, as well as real-time capability of distributed fiber-optic sensors are important for real-time structural health monitoring. As one of the promising technologies, the optical frequency domain reflectometry (OFDR) based sensors, have attracted lots of attention. Currently, for the demodulation, the conventional method based on short time Fourier transform requires long computational time, while a recent method based on group delay can remarkably improve the calculation speed but has low noise tolerance. In this study, we propose a fast demodulation method which employs the cross correlation, weighted sliding windowed Fourier transform and logistic activation function based thresholding process. This approach keeps good balance among the high spatial resolution, high accuracy, and the real-time capability that is expected to further improve the applicability of OFDR based sensors.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Distributed fiber-optic strain/temperature sensors show large potential in the structural health monitoring (SHM) applications, for their capability of carrying dense information along a single fiber and being immune to electromagnetic interference [1–3]. In SHMs, including the ones for airplane, high spatial resolution, accuracy, as well as sampling frequency (real-time capability) are required for the detection of deformation, damage, etc., of the host structures [4–6]. Thus, among various interrogating technologies, the optical frequency domain reflectometry (OFDR) is attracting lots of attentions for its sub-millimeter order spatial resolution and the sampling rate over 100 sampling per second (sps) [4,7]. Although the OFDR was essentially used to interrogate information carried wavelength shifts of Rayleigh scattering [8–11], lots of achievements have also been made by the combination of OFDR and fiber Bragg grating (FBG) [12–15, 18]. In general, the principles of both techniques are demodulating the wavelength shifts caused by external environmental variations (strain, temperature, etc.). Typically, there are two methods for the demodulation of OFDR system. In one of them, the fast Fourier transform (FFT) was firstly applied to the interference signal from the OFDR for obtaining the amplitude distribution at position domain. After sliding a bandpass filter along the positions, the inverse fast Fourier transform (iFFT) is used to reconstruct the frequency (or wavelength) information at corresponding positions. Then the wavelength shifts are calculated by using cross correlation of reference and test signals, or in the case of FBG, detecting the deferences of central wavelengths [9,12,19]. In the other method, a sliding window is directly applied to the interference signal at frequency (or wavelength) domain. Then the FFT is applied to each windowed segment of signals to acquire the spectrogram in spatial frequency domain. This process refers to a short time Fourier transform (STFT). In the end, the wavelength shift at each position is retrieved by the same technique as in the first method [13,14]. In general, these two methods show similar performance. The first one is more efficient for quasi-distributed measurement, because the non-sensor part can be filtered in the first step of demodulation. However, in the case of fully distributed measurement, the second method is more direct and a bit faster since extra time in the FFT-iFFT process is not required.

However, at high spatial resolution and sampling rate, both methods require large amount of computational time and memory, which limit the applicability of such systems in dynamic measurements. In recent years, some approaches on fast demodulation have been reported. For example, the time-resolved OFDR for Rayleigh scattering [20], and OFDR with fast processing algorithm for distributed acoustic sensing [21]. Another recent study introduced a fast demodulation algorithm by calculating the windowed group delay [22]. This method can remarkably reduce the computational time. However, the approach did not consider the noisy signal which is unavoidable in practical applications. In this study, we propose a fast demodulation process which is described in details in Section 2.2. This method mainly employs the cross correlation, FFT, and more importantly, a newly designed thresholding function based on logistic activation function. As a result, this approach shows good measurement accuracy at noisy condition without sacrificing the spatial resolution and computing speed, which may further improve the applicably of this technology in real-time dynamic SHM.

## 2. Principles

#### 2.1. Principle of OFDR based FBG sensing system

The principle of the OFDR used in this study has been well presented in [12–15]. Briefly, the sensing system consists of a tunable laser source (TLS), two interferometers, a sensing part and a computer. Figure 1 illustrates the configuration of the OFDR system. In this study, we employ a long length FBG as sensing component. The Interferometer 1 generates a clock signal to trigger the data acquisition of Detector 2, which is expressed as

where*n*is the effective refractive index of the fiber core,

_{eff}*L*is the path difference between Reflector 1 and 2,

_{R}*k*= 2

*π*/

*λ*is the wavenumber, respectively. Thus, the Detector 2 is triggered at a constant interval of wavenumber, Δ

*k*=

*π*/(

*n*), for the acquisition of the interference signal generated by the FBG and Reflector 3. If we divide the long length FBG into

_{eff}L_{R}*M*segments, the signal of Detector 2 can be expressed as where

*R*is the Bragg spectrum of the

_{i}*i*segment,

^{th}*z*is its position in Fig. 1, respectively. Then the signals are demodulated by using various signal processing algorithms to reconstruct the information carried by the FBG. In Eqs. (1) and (2), the phase terms which may cause noises [16, 17] in practical measurement are not shown for the simple description on the measurement mechanism at ideal condition.

_{i}#### 2.2. Principle of the fast demodulation process

In the proposed fast demodulation method, the interference signals obtained by Detector 2 of the FBG at initial state (initial signal) and under the test (test signal) are defined as *I _{a}*(

*k*) and

*I*(

_{b}*k*), respectively. Rather than applying cross correlation in the last step for the calculation of wavelength shift [9,19,20], we directly calculate the cross correlation of the reference and test signals using a fast approach as

*FFT*and

*iFFT*denote fast Fourier transform and its inverse, respectively [23]. The bar denotes the complex conjunction. With the sampling number of

*N*in both

_{s}*I*and

_{a}*I*, since the complexity of FFT is

_{b}*O*(

*N*), it can be much more efficient than direct computation of the linear correlation, whose complexity goes as $O\left({N}_{s}^{2}\right)$. Then the distribution of wavenumber shifts can be acquired by using the weighted sliding windowed FFT (W-WFFT), which is expressed as

_{s}logN_{s}*ĥ*(

*w*) is the window function with the length of

*w*,

*Ĉ*

_{kn}is the segment of

*C*with the same length,

*k*(regarded as the weight) is the central wavenumber of the

_{n}*n*window,

^{th}*N*is the total number of windows, respectively. Figure 2 schematically represents the W-WFFT process. When applying the sliding window to the cross correlated interference signal,

*C*(

*k*), the corresponding intensity distribution of reflected light at position domain is calculated by |

*FFT*{

*ĥ*(

*w*)

*Ĉ*

_{kn}}|

^{2}at the same time. Then, the weighted average process is equivalent to the calculation of centroids of spectra at wavenumber domain, but without acquiring the whole spectrograms of initial and test signals as in the STFT method. Thus, it is more efficient in the computational speed and memory use.

However, in the W-WFFT process by only using Eq. (4), the noisy signal which is unavoidable in practice is not considered. For the improvement, a novel fast thresholding process has been designed. Classically, a soft thresholding function is expressed as

*s*,

*R*and

*τ*are the signal, response and threshold, respectively. However, since the equation is not differentiable at

*τ*, large amount of “if functions” are necessary in the algorithm. Thus, the thresholding process will enormously increase the computational time. To address this issue, we propose a thresholding process based on the logistic activation function widely used in artificial neural networks [24], which is expressed as where the steepness coefficient,

*η*, should be much larger than

*τ*(> 0) to make the function curve close to the one of Eq. (5). In this study,

*η*is set to be 10

^{10}

*τ*. As shown in Fig. 3, in both thresholding process, the non-zero response can only be obtained when the input signal is larger than the threshold,

*τ*. By substituting

*s*=

*g*= |

_{n}*FFT*{

*ĥ*(

*w*)

*Ĉ*

_{kn}}| in to Eq. (6), and introducing the thresholding coefficient,

*β*, which is expressed as the thresholding process can be implemented into Eq. (4), and be derived to

_{n}*τ*, which is the equation at position domain, is calculated as

*θ*is the threshold ratio whose value is within (0, 100%). In this process, the

*g*will only be activated when its value is larger than

_{n}*τ*that results in the fast performance of thresholding. In this paper, the improved approach is named as activated W-WFFT (AW-WFFT) for short. The detailed diagram of the algorithm is shown in Fig. 4.

## 3. Evaluation of the fast demodulation by numerical simulation

In previous studies, our group and collaborators have demonstrated the distributed sensing using continuously inscribed long-length FBGs whose measurement length is up to 26.5 m at 1.6 mm spatial resolution [5]. In this study, a 100 mm uniform stress-free FBG (Bragg wavelength, *λ _{B}* = 1549 nm), which is efficient enough for the performance validation, was simulated to generate the initial signal,

*I*. And a step distributed stressed FBG of the same length was simulated to generate the test signal,

_{a}*I*. The distribution of corresponding Bragg wavelength is described as

_{b}*L*,

_{s}*L*,

_{e}*L*

_{1}and

*L*

_{2}are the position of starting point, ending point and two singularities, respectively.

*z*is the position on the FUT, as shown in Fig. 1. The detailed simulation model refers to [13,14,25]. The parameter values are given in Table 1. We firstly conducted the calculation in Eqs. (3) and (4) when no noise was added to

*I*and

_{a}*I*. Then the same process was repeated when Gaussian noise with the signal to noise ratio (SNR) of 50 dB was added for generating similar condition as in practical measurements where phase noise, electrical noise, etc. exist. The chosen noise level is large enough to show clear differences in the demodulation performances. To give more intuitive results, compared with the applied situation (Eq. (10)), the wavenumber shifts were converted to Bragg wavelength shifts by using where

_{b}*λ*

_{1}and

*λ*

_{2}are the start and the end of the sweep range of TLS, respectively. In the study, the maximum relative error caused by this approximation was only 0.3% that can be ignored. The converted results are shown in Figs. 5(a) and (b). In the demodulation, the window length and step length are 2 nm and 0.5 nm, respectively. We can see that, although the W-WFFT process is already able to demodulate the distribution, it does not have good tolerance to the noise of measured signals. By contrast, as shown in Fig. 5(c), the employment of the AW-WFFT process, from Eqs. (6)–(9), has effectively reduced the influence of noise.

## 4. Comparison with STFT and group delay based approaches

#### 4.1. Numerical simulation

To give an objective evaluation of the proposed method, we performed horizontal
comparisons of it with the methods based on STFT, and group delay. In
all the methods, we selected hamming window, and set the window
length, step length and SNR to be 2 nm, 0.5 nm and 50 dB,
respectively. The threshold in the peak detection process of STFT
based demodulation is set to be the same as in proposed method
(*θ* = 35%). And the rest
parameters are the same as given Table 1. The demodulated distribution of all three methods are
shown in Fig. 6. It is clear
that in the STFT based method and the proposed method, the accuracies
are better than group delay based one. To make a quantitative
comparison, we define the accuracy as the root mean square deviation
(RMSD) between the applied and measurement values within [410,
430] mm, [440, 460] mm, and [470,
490] mm. Meanwhile, the computational time is compared by
calculating the relative consumed time as

*t*is the total time used by STFT method, and

_{st}*t*is the total time used by the other methods. In this work, the total times are the time consumed by 100 times demodulation of each method. According to the results shown in Table 2, the conventional STFT method shows the best measurement accuracy, however, takes long time. The group delay based method takes much less time but shows the lowest accuracy (

_{x}*δ*). And the proposed fast demodulation method in this study keeps good balance in high speed and high accuracy. In all the methods, the spatial resolutions (

*σ*), defined as the 10% to 90% response distances of the pure step distributions [3,14], were achieved to be ∼ 0.3 mm. As pure step distribution is simulated, the spatial resolution and response distance are identical. In this study, the comparison was conducted by Matlab 2018a on the same computer (Intel Core (TM) i7-3770 @3.4 GHz).

#### 4.2. Experiment

Experimentally, the three demodulation methods were also compared. An FBG with the length of about 100 mm was inscribed by using phase mask and UV light. The Bragg wavelength is ∼ 1551 nm at stress-free condition. As shown in Fig. 7(a), the FBG in the middle (green circle) was bonded to a steel wire which is fixed on a stationary stage. The diameter of the steel is ∼ 0.5 mm. Then we stretched one half (right hand side) of the grating using a translation stage while the other half was left stress-free. The displacement of the translation stage was monitored by a laser displacement sensor whose accuracy is ±0.2 *μ*m. Considering the total length of stress-applied fiber, *L _{sap}*, is 1 m, the corresponding strain accuracy of the setup is ±0.2

*με*. Figure 7(b) shows the picture of FBG fixed area. The spectrogram of the FBG at that condition is shown in Fig. 7(c). The Bragg wavelength of stretched part is ∼ 1552.5 nm. Respectively, the STFT, group delay, W-WFFT, and AW-WFFT based methods were applied for the demodulation. The parameter settings are the same as in the simulation without adding extra noises. Because of the size of the adhesive spot (about 0.7 ∼ 1.0 mm), the applied distribution is not a pure step. Thus, the response distances are larger than the spatial resolutions in the experimental results. From the demodulated results shown in Fig. 8 and Table 3, the STFT and AW-WFFT based methods show similar performance regarding the accuracy, while the group delay and W-WFFT based method shows the lowest accuracy. The experimental measurement accuracy was calculated as the RMSD of wavelength shifts within [200, 230] mm and [250, 280] mm where uniform distribution is applied, as shown in Fig. 8. In general, the experiment and simulation have shown good agreement in the results.

## 5. Discussions

#### 5.1. Optimization of threshold ratio

In the both STFT and AW-WFFT methods, thresholding is a crucial process for the reduction of noise in the measurement. As shown in Section 4, in this study, all the threshold ratios were set to be 35% for controlling variable. However, in practical measurements, the threshold ratio need to be optimized. Figure 9(a) shows the acquired spectrum of wavelength shift at 238 mm in the experiment. Due to the steeply changed strain, the multi-peak feature as well as the background noise can be observed. From the relation of threshold ratio and accuracy shown in Figs. 9(b) and (c), we can see that when the weak-threshold (Wk.T.) which is below the noise level is applied, the increase of threshold ratio can always result in better accuracy. Then, when the threshold reaches the waist of the spectrum, the obtained accuracy becomes relatively stable. This is the optimal-threshold (Op.T.) region. However, when it keeps increasing to the over-threshold region where multi-peak pattern occurs, there will be a sudden accuracy degradation. Although, after that, the accuracy appears to improve again, the measurement has already lost the actual profile of the steeply changed strain. Thus, practically, to achieve accurate and reliable measurement, the threshold ratio should be in the Op.T. region. In this study, the chosen threshold ratio (35%) is optimal for both STFT and AW-WFFT methods in the experiment.

#### 5.2. Overall comparison of different demodulation methods

To make the comparison more clear, the results in Tables 2 and 3 are plotted in two spider charts, as shown in Fig. 10. Regarding the application in real-time SHMs, the sensing systems are expected to perform with higher spatial resolution (smaller value), higher accuracy (smaller value) and shorter computational time, which refers to a smaller area in the spider charts. In both simulation and experiment, the smallest area was achieved by the proposed AW-WFFT method in this study. In addition, once the requirements for one specific application is given, the spider charts may provide an intuitive method for selecting the most efficient demodulation technique.

## 6. Conclusions

Regarding the high spatial resolution, high sampling rate and high accuracy requirements of distributed fiber optic sensors in dynamic SHM applications, we developed a fast demodulation process for OFDR based sensing system. This method employs weighted sliding windowed FFT to reconstruct the distribution of Bragg wavelength shifts from the direct cross correlation of the reference and test signals. Specially, the employment of a novel thresholding process based on logistic activation function leads to good measurement accuracy at noisy condition without much scarification in computing speed. In the future, this method is expected to be one of the promising demodulation solutions in real-time SHM applications.

## Acknowledgments

We would like to thank Mr. Makoto Kanai for his help in the preparation of the experiment.

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