## Abstract

A signal processing method based on group delay calculations is introduced for distributed measurements of long-length fiber Bragg gratings (FBGs) based on optical frequency domain reflectometry (OFDR). Bragg wavelength shifts in interfered signals of OFDR are regarded as group delay. By calculating group delay, the distribution of Bragg wavelength shifts is obtained with high computational efficiency. We introduce weighted averaging process for noise reduction. This method required only 3.5% of signal processing time which was necessary for conventional equivalent signal processing based on short-time Fourier transform. The method also showed high sensitivity to experimental signals where non-uniform strain distributions existed in a long-length FBG.

©2014 Optical Society of America

## 1. Introduction

Distributed sensing capability of optical fiber sensors have been attracting enormous attention in the fields of telecommunication network monitoring and structural monitoring. In the case of structural monitoring such as strain measurements for the initial damage detection in structural parts, high spatial resolution is required for the sensing technique. Higher spatial resolution increases sensitivity against structural defects and provides detail data based on which following damage analysis is conducted. Representative techniques for distributed sensing use backscattering lights such as Raman [1], Brillouin [2] and Rayleigh [3]. Typical Raman and Brillouin based distributed sensing techniques employ optical time domain reflectometry (OTDR) [4–6], which is not well suited for the structural monitoring purposes due to its limitation of the spatial resolution, whereas recent studies of Brillouin based techniques demonstrated cm and sub-cm spatial resolution using correlation domain reflectometry [7], correlation domain analysis [8] and time domain analysis [9–11]. These techniques require two propagating lights inserted from both ends of a fiber. As a technique with high spatial resolution and simple system configurations, optical frequency domain reflectometry (OFDR) is eagerly developed. A number of studies have demonstrated distributed strain or temperature measurements based on OFDR system using Rayleigh backscattering [12, 13]. OFDR is also employed for the distributed sensing using fiber Bragg gratings (FBGs) [14–18]. A long-length FBG with more than 10 cm length is used, and strain or temperature distributions within the FBG are monitored with sub-mm spatial resolution. Optical low-coherence reflectometry (OLCR) is another representative technique for distributed FBG measurements with high spatial resolution [19].

So far, a major part of the reports of the distributed sensing techniques focuses on static measurements. This is because of the difficulty of the dynamic measurements for distributed sensing system. Sensing systems such as OLCR have complex configurations including mechanical moving parts, which are not suitable for fast measurements with accuracy and stability. On the other hand, OFDR requires intensive signal processing such as short-time Fourier transform (STFT) [14–17], which is a bottleneck in software aspects. Dynamic distributed measurement capability, if achieved, highly contributes to widen the range of optical fiber sensor applications, however, only a few reports have been made due to such difficulties. A. K. Sang et al. and S. T. Kreger et al. showed experimental results of dynamic distributed measurements based on OFDR using Rayleigh backscattering [20, 21]. D. P. Zhou et al. introduced time-resolved OFDR for distributed vibration measurements which divides interference data of Rayleigh backscattering into several portions in order to increase the measurement rate [22].

In this paper, we introduce a highly efficient signal processing method based on group delay calculation for an OFDR distributed sensing system. Reflected lights from an FBG and a reference mirror provide an interference signal when a tunable laser source sweeps wavelength. The interference signal includes amplitude profiles of Bragg spectrum and accompanying waves whose frequency corresponds to the reflection locations in the FBG. Strain or temperature variations at a certain location of the FBG cause the shift of the amplitude profile, which is regarded as a group delay of the interference signal. Therefore, calculation results of the group delay as a function of the frequency provides a spatial distribution of Bragg wavelength shifts. In addition, we conduct weighted averaging process for noise reduction. We investigate the performance of this method using OFDR interference signals. The interference signals are prepared both by numerical simulations and physical tests. The computational efficiency of this method indicates promising applicability for real-time measurements.

## 2. System configuration and algorithm principle

The system configuration of the OFDR system is shown in Fig.
1. *L _{R}* indicates the length of the optical path difference between
the mirror M1 and M2. In the same manner, the optical path difference between the mirror M3 and
an arbitrary location at the long-length FBG is defined as

*L*<

_{a}*L*<

*L*. For following experiments, we used a wavelength tunable laser source (ANDO AQ4321) and an A/D converter (NI PCI 6115). The reflectivity and bandwidth of the long-length FBG were less than 10% and 0.02 nm, respectively.

_{b}The interferometer which consists of the photodetector D1, the optical coupler C2 and the mirrors M1 and M2 outputs clock signal to trigger data acquisition. The interference signal at the detector D2 is sampled by the constant interval of wavenumber, Δ*k*, which is determined by *L _{R}* as

*n*is the effective refractive index of the fiber core [23]. The wavenumber,

_{eff}*k*, is in relationship with the wavelength,

*λ*, as

*k*= 2π/

*λ*.

When the interference signals of an unstrained FBG and a strained FBG are defined as *x _{1}* (

*k*) and

*x*(

_{2}*k*), respectively, as a function of wavenumber

*k*, we can calculate the group delay of

*x*and

_{1}*x*as below. The impulse response,

_{2}*h*(

*k*), is calculated as

*H*, can be expressed by magnitude,

*A*, and phase,

*b*, aswhere

*ω*is frequency and

*T*is sampling period. The group delay,

*T*, is defined as the derivative of phase with respect to frequency, and can be defined directly from the frequency response while avoiding the intermediate calculation of the phase response as

_{g}*z*-domain aswhere Re denotes taking the real part and

*z*= e

*, as seen in the reference [24]. In addition to the transform pair,*

^{jωT}*h*(

*k*) ⇔

*H*(

*z*), when we define the transform pair aswe haveTherefore, Eq. (5) is expressed asIn this manner, we can calculate the group delay aswhere

*fft*denotes the fast Fourier transform. The coefficient

*T*is set to be Δ

*k*when shifts of Bragg wavenumber are to be obtained.

In order to reduce the processing noise, we calculate the group delay with additional weighted averaging process. We extract a certain section of the interference signals by using a window function, and calculate the group delay for the section. We slide the window and conduct weighted averaging.

When we express *h _{k}* (

*k*) =

*k h*(

*k*), we apply a window function to

*h*(

_{k}*k*) and

*h*(

*k*), and extract certain sections of them which are expressed as ${\widehat{h}}_{k}$ and $\widehat{h}$, respectively. Using these sectional functions, group delay, ${\widehat{T}}_{g}$, is calculated using complex conjugates as

*i*indicates the

*i*th section of the window application.

In concept, the conventional signal processing method based on STFT [14], which is described in detail in the following section, calculates distributions of Bragg spectra for unstrained and strained FBGs, and Bragg peak shifts are measured. On the other hand, this signal processing method based on the group delay directly calculates Bragg peak shift distributions. The STFT provides abundant information including the shape of Bragg spectra, whereas the group delay has an advantage in calculation efficiency.

## 3. Calculation results for simulated signal and discussion

We investigated the performance of the signal processing method based on group delay calculation.
Firstly, we prepared two interference signals of initial and shifted FBG,
*x _{1}* (

*k*) and

*x*(

_{2}*k*), respectively, using a numerical model of the OFDR system based on piece-wise uniform approach [23]. Optical and geometric parameters which were used for the numerical model are listed in Table 1. Symbols such as M3 and D2 correspond to those in Fig. 1.

For comparison, we conducted signal processing using STFT. In the process of STFT, a hamming
window extracts a certain section of the interference signals and fast Fourier transformation
(FFT) is applied to it. Thus, the power profile as a function of the frequency is obtained. By
sliding the window through the entire range of the interference signal, the power profiles cover
the whole range of the wavelength. Frequency is converted to position of reflection. Eventually
one can obtain the power profile on the map of wavelength and position. Figure 2 shows the spectrograms of the initial and shifted FBG which were
obtained by STFT applied to *x _{1}* (

*k*) and

*x*(

_{2}*k*), respectively. The distribution of Bragg spectra of 100 mm FBG is observed at 1550 nm for the initial FBG, whereas 1 nm and 2 nm shifts are observed for the shifted FBG. These results agree with the parameter input as shown in Table 1. Bragg wavelength was determined by the peak of the spectra, and the distribution of the Bragg wavelength shifts was calculated. Figure 3 shows the calculated result of the distribution of the Bragg wavelength shifts. The width of the window, FFT length and the length of the window slide were set to be 604 pm (7340 data points), 2

^{15}data points and 10 pm (122 data points), respectively. FFT was conducted 937 times. In this case, spatial interval which expresses the distance between two adjacent points in position-wise was 0.3 mm.

We conducted the signal processing based on the group delay calculation. Figure 4 (a) shows the calculated result of the distribution of the Bragg wavelength shifts which did not include noise reduction process. This group delay was calculated as Eq. (3). Figure 4 (b) is the result when the noise reduction process was included as Eq. (5). It is observed that the accuracy increased by the noise reduction process. For the noise reduction process, hann window was employed. The width of the window and the length of its slide were set to be 1.35 nm (32816 data points) and 0.7 nm (17016 data points), respectively, which resulted in 13 times of FFT calculation. The spatial interval was 0.3 mm. Window width and its length of slide were far larger than those in STFT, however, these were sufficient to produce the equivalent spatial interval and contributed to computational efficiency. When the signal processing time was compared between two methods based on STFT and group delay, the method based on group delay required only 3.5% of the processing time which was necessary for the one based on STFT. The comparison was conducted by MATLAB and the time for drawing graphs was excluded. This performance indicates the promising applicability for real-time measurements. The processing time of group delay calculations is further reduced by applying smaller length of the window width and the larger length of the window slide, whereas the spatial interval is finer when larger length of the window width is set. The effect of the noise reduction decrease when the window width is larger. These parameters are adjusted in accordance with desirable spatial resolution, accuracy and rate of measurements.

## 4. Calculation results for physical test and discussion

In order to investigate the performance of the signal processing method based on group delay
calculation when signals of physical tests are of interest, we conducted distributed strain
measurements using an aluminum plate with holes [17]. The
geometrical information of the aluminum plate with three holes is depicted in Fig. 5.The length, width and thickness of the aluminum plate were 250 mm, 30 mm and 2 mm,
respectively. The three holes with diameters of 2 mm, 2 mm and 4mm were located with 15 mm
interval between each and to touch the center line of width of the plate. The longitudinal
direction of the plate is set as *L*-axis, and 50 mm length of a long-length FBG
was bonded 0.5 mm away from the center line. A tensile load of 300 kgf was applied along the
longitudinal direction. For reference, the theoretical strain distribution at the location of
the FBG was calculated using finite element method (FEM). The Young’s modulus of the
aluminum plate was set as 70.3 GPa and Poisson’s ratio as 0.345 for FEM. For the system
configuration of OFDR, *L _{R}* was 15.47 m and the wavelength sweep range
was from 1552.0 nm to 1555.0 nm. Initial Bragg wavelengths without loadings were approximately
around 1552.5 nm at 0 <

*L*< 50 mm. We converted 1 nm Bragg wavelength shift into strain shift of 830 με.

Figure 6 shows results of estimated strain distribution by
FEM and OFDR. For OFDR results, STFT and the group delay calculation methods were used in signal
processing. In the process of STFT, window length, FFT length and window slide length were set
to be 400 pm (7502 data points), 2^{15} data points and 5 pm, respectively, and the
spatial interval was 0.47 mm. In the process of group delay calculation, window length and
window slide length were set to be 0.4 nm (15094 data points) and 0.1 nm (3774 data points),
respectively, and the spatial interval was 1.03 mm. The three peaks of strain distributions
represent the effect of the stress concentrations due to the three holes. Both results of the
signal processing method based on STFT and group delay calculations showed proper sensitivity to
the assumed strain distribution by FEM. It can be said that the weighted averaging process does
not cause excessive smoothing. It is rather seen that the result of group delay calculations
showed slightly higher responsiveness compared with that of STFT especially where sharp
variations of strain existed. These results demonstrated robust applicability of the signal
processing method based on group delay calculations even when experimental data of non-uniform
strain distributions was of interest.

## 5. Conclusion

We introduced a signal processing method based on group delay calculations for distributed FBG measurements using OFDR. Bragg wavelength shifts in interfered signals of OFDR were regarded as group delay. By calculating group delay between two interfered signals of unstrained and strained FBG, the distribution of Bragg wavelength shifts was obtained with high computational efficiency. In order for noise reduction we introduced weighted averaging process. This method required only 3.5% of signal processing time which was necessary for conventional signal processing method based on STFT. The method of group delay calculations showed high sensitivity to experimentally obtained OFDR signals in which non-uniform strain distributions existed. These results indicated promising applicability of this method for real-time measurements.

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