## Abstract

A noise-resilient demodulation technique is proposed and demonstrated with an interferometric-noise-suppressing (INS-) Golay coded optical pulse source for a quasi-distributed sensor array constructed by identical ultra-weak fiber Bragg gratings (UWFBGs).In combination with a medium coherence light source, coding-based time-efficient noise reduction is facilitated for closely-multiplexed UWFBG arrays, for which conventional Golay coding is inapplicable. With 32-bit INS-Golay coded pulse trains, $5.6$-dB signal-to-noise ratio improvement is achieved as compared to the case where an uncoded pulse train is adopted. While the performance is almost as good as 32-time averaging, the time consumption for signal acquisition is only $\sim 1/8$. The proposed INS-Golay coding method exhibits linearity up to $0.9986$ in temperature sensing with a wavelength demodulation error of $\sim \pm 5$ pm. The time efficiency of the method increases with increased code length, providing a solution that alleviates the trade-off between the demodulation accuracy and speed in the UWFBG-based sensing systems with closely-multiplexed sensors.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fiber Bragg grating (FBG) sensor has found broad applications in mechanical equipment and structural health monitoring [1–4], due to its capability of temperature and strain sensing. Multiplexing of identical FBGs forms a sensor array, which facilitates quasi-distributed sensing [5–7]. To achieve long distance operation with high spatial resolution, the sensor array contains a large number of FBGs, resulting in signal crosstalk induced by multiple-reflection and spectral-shadowing effects [8, 9]. Such crosstalk can be reduced using ultra-weak (UW) FBGs with reflectivity as low as $-30$ dB [10]. UWFBGs can be serially inscribed on an optical fiber while it is drawn from the tower [11–13]. The technique guarantees rapid and reliable fabrication of identical gratings, making large-scale sensor multiplexing and long-range quasi-distributed sensing practical. However, the relatively weak sensing signals are easily tampered by noise. Averaging method is commonly adopted to improve the signal-to-noise ratio (SNR), at the expense of time-consuming signal acquisition and processing. The demodulation speed is thus limited when high accuracy is required. It is therefore necessary to find an alternative method that improves SNR with less time consumption.

During the past several years, noise reduction methods based on coded optical sources have been proposed for scattering-based fiber sensing systems [14, 15]. Using complementary sequences such as the Simplex code, the complementary correlated Prometheus orthogonal sequence, and the Golay code, a delta function can be generated through auto-correlation, which enhances the SNR and the dynamic range of the system [16, 17]. Since the overlap of the sensing pulses, resulting from the distributed reflections in the sensing fiber, is taken care of by the decoding algorithm, the techniques do not require the pulse period to be larger than the round-trip time in the sensing fiber. Consequently, the time consumed for signal acquisition can be significantly reduced as compared to the averaging-based noise reduction method, especially when a long sensing fiber is adopted. Among the complementary sequences, the Golay code is commonly employed because it is easy to implement. With the correlation property of the Golay code, the performance of an optical time-domain reflectometry (OTDR) sensing system, which explores the back-scattered optical signal, is improved while maintaining the spatial resolution [18]. In addition, coded optical pulse source is investigated numerically for two time-domain multiplexed FBGs with high reflectivity [19]. With the use of a 256-bit Golay sequence, 9-dB SNR improvement is achieved.

Even though ultra-weak, the online fabricated UWFBGs still present reflectivity controllable from $\sim 5\times {10}^{-5}$ to $\sim 3\times {10}^{-3}$, which is orders of magnitude higher than the scattering effects [11, 12]. With higher reflectivity, the SNR of the sensing signal is improved as compare to those in the scattering-based systems, which guarantees better demodulation performance [20, 21]. Nonetheless, to use the coded optical source with closely-multiplexed UWFBG sensor arrays for further noise reduction, there is actually one more concern. As compared to the back-scattered signals, signals reflected by UWFBGs highly resemble the coherence property of the light source. To implement conventional (C-) Golay coding, a coded optical pulse train, instead of an individual pulse, is used to generate the sensing signals. As a result, the multiple pulse trains reflected by multiple identical UWFBGs overlap and interfere with each other, generating strong interferometric noise, which cannot be suppressed by the decoding algorithm. Consequently, the system performance is severely degraded, making the C-Golay coding inapplicable for time-efficient noise reduction in fiber sensing systems based on closely-multiplexed FBGs. To overcome this issue, we propose an interferometric-noise suppressing (INS-) Golay coded optical source constructed using return-to-zero (RZ) formatted pulse trains. Because the pulse overlap due to the distributed reflections is strongly reduced, the interferometric noise is significantly suppressed. The proposed optical source facilitates the use of UWFBGs with identical center wavelengths in the sensor array, while achieving time-efficient noise reduction. With a 32-bit INS-Golay coded medium-coherence optical source, noise reduction is achieved with a $9.06$-km sensing fiber containing 3019 identical UWFBGs, exhibiting $5.6$-dB SNR improvement as compared to using an uncoded optical pulse train, which is almost as good as the performance of 32-time averaging. When adopted for temperature sensing, the method exhibits linearity up to $0.9986$ with a wavelength demodulation error of $\sim \pm 5$ pm. With the proposed methods, compromise between the demodulation accuracy and speed is strongly alleviated, as compared to the averaging method. Also, additional time consumption for signal acquisition, due to the use of INS-Golay coding, is almost negligible compared to the C-Golay coding method.

## 2. Principle of operation

Figure 1 shows the experimental setup of the proposed sensing system. A distributed-feedback (DFB) laser diode is used to construct the Golay coded optical source. To evaluate the influence of the interferometric noise, the linewidth of the laser is made variable via direct modulation using an electrical noise source. With the wavelength of emission tuned to *λ _{i}*, the continuous-wave output of the laser is directed to a semiconductor optical amplifier (SOA) for optical pulse generation. A bit pattern generator (BPG) produces coded electrical pulse sequences with 16-ns pulse width, which is suitable for the 3-m grating spacing adopted in the experiment. By driving the SOA with the electrical pulse sequence, the correspondingly coded optical source is obtained. By switching the electrical pulse sequence between the non-return-to-zero (NRZ) and the RZ formats, the optical source switches between C-Golay coded and the INS-Golay coded modes. In practice, the noise modulation and pulse generation can be integrated using an external circuit to directly modulate the laser diode, in order to further reduce the cost. After being amplified by an erbium-doped fiber amplifier (EDFA), the optical pulse train is directed to a UWFBG array using an optical circulator. The optical signals reflected by the sensor array are converted to electrical signals by a photodetector (PD) and collected by a data acquisition card (DAQ). The impulse response function of the UWFBG array at

*λ*is then recovered using the proposed decoding algorithm. By scanning the wavelength of the tunable laser from

_{i}*λ*

_{1}to

*λ*, the reflection spectra of the UWFBGs in the sensor array are reconstructed.

_{n}In the traditional sensing techniques where uncoded pulses are adopted as the optical sources, it is required that the pulse repetition rate ${f}_{\text{rep}}\le 1/{\tau}_{\text{SF}}=c/2{n}_{\text{eff}}{L}_{\text{SF}}$, where *c* is the speed of light in vacuum, ${\tau}_{\text{SF}}$, *n*_{eff}, and *L*_{SF} are the round-trip time, effective refractive index, and physical length of the sensing fiber, respectively. Through optical coding, such requirement is relaxed in the scattering-based sensing techniques [16–18], because the pulse overlap issue is taken care of by the decoding algorithm. Consequently, the time consumed for sensing signal acquisition is reduced significantly as compared to the averaging method, especially when a long sensing fiber is adopted.

Nevertheless, the situation is different for the FBG-based sensing systems, due to the occurrence of interference when pulses overlap. In the following, the optical coding, sensing, detection, and decoding processes are explained in detail. Based on C-Golay coding, we propose an INS-Golay coding method, which significantly reduces the generation of interferometric noise.

#### 2.1. Optical coding and sensing

For *N*-bit Golay coding, we start from two *N*-bit Golay complementary sequences. Taking the case with *N* = 8 as an example, the original Golay sequences are given by:

The codes are modulated on the optical carrier in a differential manner using the four sequences as follows:

*τ*

_{p}, the output electrical pulse trains present the NRZ format if the sequences are loaded directly into the BPG.

Assuming the modulation response of the SOA is linear and instantaneous, the envelopes of the optical signals present the same shape as the electrical pulse trains. Since the optical signals are detected and sampled for digital signal processing, they are described as discrete signals in the following. With the tunable laser emitting at wavelength *λ _{i}*, the complex electric field of the coded single-polarization optical pulse train is described by

*k*is the time index,

*τ*

_{s}is the sampling period of the DAQ, ${\varphi}_{{\lambda}_{i}}$ describes the phase of the tunable laser at

*λ*, and

_{i}*δ*is the Kronecker delta. $K={\tau}_{\mathrm{t}}/{\tau}_{\mathrm{s}}=N{\tau}_{\mathrm{p}}/{\tau}_{\mathrm{s}}$ is the length of the discrete signal ${E}_{{A}_{+}}$, with ${\tau}_{\mathrm{t}}=N{\tau}_{\mathrm{p}}$ being the duration of the pulse train. The subscripts in ${E}_{{A}_{+},{\lambda}_{i}}$ are used to specify the sequence and input wavelength, while sequence ${A}_{+}$ is taken as an example in the analysis. The time-index-matched sequence ${A}_{\mathrm{s}+}$ is obtained by replacing each bit in ${A}_{+}$ with ${\tau}_{\mathrm{p}}/{\tau}_{\mathrm{s}}$ bits of the same value.

The optical pulse train is then directed to the UWFBG sensor array, and the reflected signal is described by

*λ*, denoted by ${h}_{\text{SF},{\lambda}_{i}}$, is given by

_{i}*λ*and position ${p}_{\text{SF}}=m{\tau}_{\mathrm{s}}c/2{n}_{\text{eff}}$ of the sensing fiber. Along the sensing fiber, the UWFBGs produce relatively strong reflection when the wavelength

_{i}*λ*falls into their reflection spectra. It is worth noting that the reflection at the FBGs are considered instantaneous due to the negligibly short length. In the meantime, multiple-reflection and spectral-shadowing effects are ignored in the analysis due to the low FBG reflectivity. Substituting Eqs (3) and (5) into Eq (4), we obtain the discrete time series that describes the reflected sensing signal:

_{i}*k*and reflection at fiber position ${p}_{\text{SF}}=m{\tau}_{\mathrm{s}}c/2{n}_{\text{eff}}$. In Eq (6), different combinations of

*m*and

*k*can generate the same $l=m+k$ value, indicating a set of signals that arrive at the PD simultaneously. More specifically, Eq (6) can be rewritten into

Considering a specific *n* value, terms ${r}_{{\lambda}_{i}}\left[n-k\right]{A}_{\mathrm{s}+}\left[k\right]\text{exp}\left\{j{\varphi}_{{\lambda}_{i}}\left(k{\tau}_{\mathrm{s}}\right)\right\}$ with $max\left(0,1-M+n\right)\le k\le min\left(K-1,n\right)$ describe all the signals at time index *n*.

#### 2.2. Optical detection and decoding

Upon the optical-to-electrical conversion at the PD and sampling by the DAQ, the output time series is given by

*c*

_{oe}is the conversion coefficient concerning both the PD and the DAQ, which is assumed wavelength independent. As opposed to the scattering-based sensing techniques, the sensing signals reflected by the UWFBGs, even though ultra-weak, highly resemble the coherence property of the optical source. Interference can thus occur among the signals that arrive at the PD simultaneously. Consequently, the noise components in the output time series depends strongly on the coherence length of the optical source. In the following, we analyze how the coherence property of the light source affects the decoding process.

### 2.2.1. Incoherent optical source

Considering an incoherent optical source, the time series ${X}_{{A}_{+},{\lambda}_{i}}^{{}^{\text{'}}}$ is given by

*λ*. Eq (9) indicates that different signals arriving at the PD simultaneously do not interfere with each other, due to the use of the incoherent optical source. Let ${c}_{\text{oe}}=1$, the equation can be written as

_{i}When the input electrical sequence is switched to ${A}_{-}$, ${B}_{+}$, and ${B}_{-}$, the corresponding time series ${X}_{{A}_{-},{\lambda}_{i}}^{{}^{\text{'}}}$, ${X}_{{B}_{+},{\lambda}_{i}}^{{}^{\text{'}}}$, and ${X}_{{B}_{-},{\lambda}_{i}}^{{}^{\text{'}}}$ can be obtained.

The decoding process relies on the correlation of the four time series with the original Golay sequences. The two sets of sequences should have matched length so that decoding can be performed correctly. Therefore, sequences ${A}_{\mathrm{e}+}$, ${A}_{\mathrm{e}-}$, ${B}_{\mathrm{e}+}$, and ${B}_{\mathrm{e}-}$ are introduced by adding $\left({\tau}_{\mathrm{p}}/{\tau}_{\mathrm{s}}-1\right)$ “0”s after each bit in series ${A}_{+}$, ${A}_{-}$, ${B}_{+}$, and ${B}_{-}$. In the following, we let ${\tau}_{\mathrm{p}}/{\tau}_{\mathrm{s}}=4$, which is in accordance with that in the experimental demonstration. Let ${X}_{A,{\lambda}_{i}}^{{}^{\text{'}}}={X}_{{A}_{+},lambd{a}_{i}}^{{}^{\text{'}}}-{X}_{{A}_{-},{\lambda}_{i}}^{{}^{\text{'}}}$, ${X}_{B,{\lambda}_{i}}^{{}^{\text{'}}}={X}_{{B}_{+},{\lambda}_{i}}^{{}^{\text{'}}}-{X}_{{B}_{-},{\lambda}_{i}}^{{}^{\text{'}}}$, ${A}_{\mathrm{e}}={A}_{\mathrm{e}+}-{A}_{\mathrm{e}-}$, and ${B}_{\mathrm{e}}={B}_{\mathrm{e}+}-{B}_{\mathrm{e}-}$, the decoded result ${y}_{{\lambda}_{i}}$ is defined and derived as follows:

Substituting Eq (13) into Eq (12), we obtain

*λ*, the reflectivity along the sensing fiber over the wavelength range of interest can be obtained, so that the reflection spectra of the UWFBGs can be reconstructed.

_{i}### 2.2.2. Coherent optical source

Considering a coherent optical source with coherence time *τ*_{c}, the time series ${X}_{A,{\lambda}_{i}}^{{}^{\text{'}}}$ is given by

In practice, coherent optical sources such as DFB laser diodes are favored due to the faster wavelength scanning capability, as compared to the incoherent light sources such as those based on filtered amplified spontaneous emission noise. With the latter exhibiting several kilohertz wavelength tuning speed, at least 100-kHz wavelength scanning can be facilitated using a DFB laser diode via direct modulation [22]. Meanwhile, the linewidth of a filtered ASE source is limited by the bandwidth of the optical filter to usually no less than a few gigahertz. On the contrary, a DFB laser diode typically presents linewidth ranging from several to tens of megahertz, which leads to higher wavelength demodulation accuracy. The corresponding coherence length thus ranges from several to tens of meters. Consequently, signals reflected by several or even tens of gratings interfere with each other, resulting in severe interferometric noise. To guarantee the performance of the sensing system, it is desired that the interferometric noise terms be suppressed as much as possible. We therefore propose to implement the INS-Golay coding by inserting a number of “0”s after each bit of the Golay sequences.

To quantify the level of interferometric noise, the number of non-zero terms in the last two terms of Eq (15) at each time index is obtained, denoted by ${N}_{\text{NZ},{A}_{+}}\left[n\right]$. The number of interfering terms are thus given by

For the other three sequences, ${N}_{\text{IN},{A}_{-}}$, ${N}_{\text{IN},{B}_{+}}$, and ${N}_{\text{IN},{B}_{-}}$ are obtained, respectively. We then define the probability of occurrence as

Figure 2 compares the results of the C-Golay and the INS-Golay coding methods. The grating spacing of the adopted FBG array is 3 m, corresponding to $\sim 30$-ns round-trip time delay. C-Golay coding is implemented based on a set of 32-bit Golay sequences with 16-ns pulsewidth. For INS-Golay coding, 50%, 25%, and $12.5\%$ duty cycle values are investigated by inserting 1, 3, and 7 “0”s after each bit of the sequences, respectively. The corresponding results are shown in Fig. 2 as squares, up triangles, dots, and down triangles. With smaller linewidth, more FBGs fall into the coherence length, resulting in stronger interferometric noise for all four cases, as shown in Fig. 2. Due to the insertion of “0”s, interference is less likely to occur for INS-Golay coded sequences. Also, lower duty cycle leads to weaker interferometric noise, at the expense of lengthened sequences and more time-consuming post-processing.

Additionally, it should be noted that signals only become partially incoherent, instead of fully incoherent, when they have time delay $\mathrm{\Delta}t\sim {\tau}_{\mathrm{c}}$. The previous assumption is only made to simplify the derivation of Eq (15) and the subsequent equations. Therefore, the occurrence probability of interference shown in Fig. 2 should be considered as relative values, intended for the comparison of the C-Golay and the INS-Golay coding methods, instead of absolute values.

#### 2.3. Time consumption and multiplexing capacity

The time consumption for signal acquisition is affected by both the round-trip propagation time in the sensing fiber and the pulsewidth/sequence duration. Since *N*-time averaging and *N*-bit Golay coding exhibit similar performance [18], we compare the time consumption of *N*-time averaging, *N*-bit C-Golay coding, and *N*-bit INS-Golay coding with 25% duty cycle. As listed in Table 1, while the contribution by fiber propagation and pulsewidth both scale with *N* in the averaging methods, only the contribution by sequence duration scales with *N* in the coding methods.

In a long-distance sensing system, ${\tau}_{\text{SF}}\gg {\tau}_{\mathrm{p}}$. The time consumption of *N*-time averaging is therefore $\sim N/4$ times that of the coding methods. For example, considering a 10-km sensing fiber, 16-ns pulsewidth, and 32-bit Golay coding, the time consumptions for data acquisition in the three cases are $3.20$ ms, $0.40$ ms, and $0.41$ ms, respectively. We can see that the time consumption difference between the C-Golay and the INS-Golay coding methods is negligibly small. With larger *N*, the difference between the averaging and the Golay coding methods can be further increased. In practice, the cost on data storage and post-processing should also be taken into consideration in the choice of *N*.

The multiplexing capacity of an UWFBG array is calculated by multiplying the number of UWFBGs per unit length and the length of the sensing fiber. For a given laser source with fixed coherence length, a larger number of UWFBGs per unit length means that more UWFBGs involve in the generation of the interferometric noise, resulting in more significant SNR degradation. Therefore, the duty cycle of the INS-Golay coded pulse source needs to be reduced accordingly, in order to keep the interferometric noise under control. As for the length of the sensing fiber, it is mainly limited by the reflectivity of a single UWFBG. With lower UWFBG reflectivity, the allowed sensing fiber length is longer, because the insertion loss of a single sensor is lower, and the multiple-reflection and spectral-shadowing effects are less significant. The optical coding does not depend on the length of the sensing fiber. In fact, the reduced time consumption is even more pronounced when a long sensing fiber is adopted, as shown previously.

## 3. Experimental results and analysis

A $9.06$-km UWFBG array with 3-m grating spacing, 1-cm grating length, and $\sim 1551$-nm center wavelength at 25°C is fabricated for the experimental demonstration. To ensure high multiplexing capacity, UWFBGs with low reflectivity of about $-30$ dB are adopted. The DAQ provides 16-bit resolution and 250-MHz sampling rate, thus ${\tau}_{\mathrm{s}}=4$ ns. 32-bit Golay coding is adopted and 25% duty cycle is selected for the INS-Golay coding. The natural linewidth of the laser is $\mathrm{\Delta}\nu =10$ MHz, which can be broadened gradually to $\sim 80$ MHz by increasing the noise modulation index.

Figure 3 shows the recovered reflectivity ${R}_{{\lambda}_{i}}$ along the sensing fiber obtained with ${\lambda}_{i}=1550.9$ nm, when the uncoded, the C-Golay coded, and the INS-Golay coded pulse trains are adopted as the optical source, respectively. To display the detail, we only present the results between fiber positions 8700 m and 8900 m. Figures 3(a) and 3(b) are obtained at 10 MHz laser linewidth using uncoded pulses with 4 and 32 times of averaging, respectively. By keeping the pulse repetition rate ${f}_{\text{rep}}<c/2{n}_{\text{eff}}{L}_{\text{SF}}$, there is no pulse overlap in this case, and the results are not affected by the interferometric noise. In the following demonstration, the laser linewidth is fixed at 10 MHz for the averaging method. Due to the increased time of averaging, Fig. 3(b) exhibits a much less noisy result as compared to Fig. 3(a), as expected. Since 32-time averaging and 32-bit Golay coding should perform similarly, we use the results of the former as a reference. When the laser operates at its natural linewidth, the coding method fails to work, no matter the C-Golay or the INS-Golay coding is adopted, as shown in Figs. 3(c) and 3(d). The reason is that more than 40 FBGs involve in the interfering process at a laser linewidth of 10 MHz, and even INS-Golay coding cannot suppress the interferometric noise effectively enough. Through direct noise modulation, the laser linewidth is increased to 26MHz. The corresponding results of the C-Golay and the INS-Golay coding methods are shown in Figs. 3(e) and 3(f), respectively. While the INS-Golay coding starts to resemble the results in Fig. 3(b), the C-Golay coding shows no resemblance. We then further increase the laser linewidth to 80 MHz through strengthened noise modulation. While the C-Golay coding still fails to work, as shown in Fig. 3(g), results obtained using the INS-Golay coding displays a quite similar waveform to that shown in Fig. 3(b), as shown in Fig. 3(h). To quantify the performance, we compare the noise powers in three cases: measurement using an uncoded pulse train, uncoded pulse trains with 32-time averaging, and 32-bit INS-Golay coded pulse trains. The latter two show $6.5$-dB and $5.6$-dB SNR improvement over the former one. The slightly lower SNR improvement from the INS-Golay coding method is attributed to the increased relative intensity noise of the laser diode due to the noise modulation. Meanwhile, the remaining interferometric noise also affects the SNR of the decoded result.

We then compared the spectra reconstructed using 32-time averaging, the C-Golay coding, and the INS-Golay coding methods. 80-MHz laser linewidth is adopted for the coding methods. By tuning the wavelength from 1550 nm to 1552 nm with a 20-pm step, a series of ${R}_{{\lambda}_{i}}$ are obtained to recover the reflection spectra of the UWFBGs. Figure 4 shows the reconstructed reflection spectra of the four UWFBGs at the end of the $9.06$-km sensing fiber. From Eq (14), the reflectivity at each position in the sensing fiber is actually sampled four times. We simply collect the maximum value at each position. Figure 4(a) obtained from 32-time averaging and Fig. 4(c) obtained from 32-bit INS-Golay coding show similar spectral shapes with high SNR. As for the C-Golay coding, due to the strong interferometric noise, as shown in Fig. 3(g), the reflection spectra cannot be reconstructed correctly.

We then apply the uncoded pulse trains with 32 times of averaging and the 32-bit INS-Golay coded pulse trains for temperature sensing. FBG#3019 is heated by a temperature controller with the temperature varied from 25°C to 80°C at a step of 5°C. Figure 5 plots the change of center wavelength with the increase of temperature. The center wavelengths are obtained from the reconstructed spectra using centroid method. The closed squares correspond to the averaging method and the open circles the INS-Golay coding method. Both methods present similar accuracy. With 32-bit INS-Golay coding, the temperature sensing result exhibits linearity up to $0.9986$ with a wavelength demodulation error of $\sim \pm 5$ pm.

In the theoretical derivation, it is predicted that the occurrence probability of interference can be reduced to $<10\%$ using INS-Golay coding with 25% duty cycle, when the laser linewidth is increased to $\sim 33.3$ MHz. Experimentally, the interferometric noise gets under control at 80-MHz laser linewidth. The discrepancy is explained as follows. As mentioned previously, signals actually become partially coherent, instead of fully incoherent, when the time delay between them reaches the coherence time. Therefore, by assuming they are only coherent within the coherence length, we have underestimated the degree of interference. Second, in the quantification of the laser linewidth using the self-heterodyne measurement, the spectral line shape of the laser is assumed to be Lorentzian. In reality, noise modulation distorts the line shape, resulting in misestimation of the laser linewidth and thus the coherence length.

## 4. Conclusion

In conclusion, we demonstrate a time-saving noise-resilient demodulation technique based on the INS-Golay coding method. In combination with a medium coherent optical source, coding based noise reduction is facilitated for sensor arrays based on closely-multiplexed identical UWFBGs. Compared to the case using an uncoded pulse train, implementation of the 32-bit INS-Golay coded pulse train is able to increase the SNR by $5.6$ dB, which is almost as good as adopting 32-time averaging. The time consumption is only quadrupled, which is only $1/8$ the time consumption by 32-time averaging. The proposed INS-Golay coding method exhibits linearity up to $0.9986$ in temperature sensing with a wavelength demodulation error of $\sim \pm 5$ pm. The method becomes even more time-efficient with increased code length, providing a solution that alleviates the trade-off between the demodulation accuracy and speed in fiber sensing systems based on closely-multiplexed FBG sensors.

## Funding

National Natural Science Foundation of China (NSFC) (61735013, 61705169); Natural Science Foundation of Huber Province of China (2018CFA056).

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