Signal-to-noise ratio analysis of computational distributed fiber-optic sensing

Dayong Shu; Da-Peng Zhou; Da-Peng Zhou; Xinlei Zhou; Wei Peng; Liang Chen; Xiaoyi Bao

doi:10.1364/OE.390324

1. Introduction

Distributed fiber-optic sensors (DFOSs) measure a spatially distributed profile of environmental quantities such as temperature, strain, pressure, etc., by detecting and analyzing specific optical effects along optical fibers [1]. This distributed-measurement capability offers unique advantage compared to conventional discrete sensing techniques, especially for the fact that DFOSs are able to achieve long-distance measurement range using a single unaltered optical fiber as the sensing element [2]. DOFSs generally utilize natural scattering processes [3,4], such as Rayleigh scattering, spontaneous Raman scattering and spontaneous or stimulated Brillouin scattering, in optical fibers along with the interrogating methods which are primarily based on time-domain techniques [5] or frequency-domain techniques [6]. For certain applications requiring the ability of continuous long range measurement (over many tens of kilometers) with moderate spatial resolution (on the order of meters) which is also one of the superior advantages of DFOSs, the configurations based on time-domain technique are usually adopted owing to their considerably reduced complexity and greater system bandwidth [1,7,8]. This technique relies on sending an optical pulse into an optical fiber, and collecting backscattered light by a photodetector; the resultant time-domain electrical signal needs to be acquired by a fast-speed digitizer whose sampling rate is required to meet Nyquist-Shannon sampling theorem [9], so that the sampling time interval must be shorter than half of the pulse duration which determines the spatial resolution of the sensors [1]. Therefore, DFOSs require data acquisition (DAQ) modules to have high sampling rate and fine resolution to detect fast and weak signals, which makes DFOSs at a high cost, and this may be one of the factors preventing DFOSs from widespread applications in practice.

Inspired by recent advances in single-pixel imaging [10,11], and ghost imaging technique in both the spatial [12,13] and temporal domain [14–19], we demonstrated a computational distributed fiber-optic sensing technique [20] focusing on stationary measurement recently which can reduce the sampling rate requirement significantly compared to the conventional time-domain technique. The sensing paradigm is to send optical pulse sequences with pre-known binary patterns to illuminate the spatial scattering object sequentially in the time domain, and in the meantime, the sensing fiber itself “integrates” backscattered light continuously and the resultant integrated optical signal only needs to be collected at discrete specific moments with an acquisition rate inversely proportional to the total time duration of the light sequence [20]. Correlation calculations between the collected data and the pre-known binary patterns are performed afterwards to retrieve the temporal “image”; therefore, the sampling rate can be greatly reduced compared to the conventional method where the temporal “image” containing the distributed scattering information needs to be acquired with high sampling rate. It is worth mentioning that the computational technique requires acquisition of the integrated backscattered signals associated to a number of different pulse patterns, and for each pattern, a number of averages are also needed due to the weak scattering; while, for the conventional method, the time-domain trace only needs to be obtained after averaging to achieve sufficient signal-to-noise ratio (SNR). Hence, it is necessary to investigate acquisition time difference between the computational method and the conventional method, especially to find out the total number of acquisitions required for the computational method to achieve the same SNR level as the conventional method. This analysis is particularly interesting from practical point of view. Even though the sampling rate can be reduced greatly by using the computational method, one still expect that the approach should have similar performance as the conventional method in terms of achieving the same measurement accuracy with reasonable measurement time.

Most DFOSs based on time-domain techniques rely on measuring the distributed backscattered signals, the variations of which determine the external physical quantities; therefore, the higher the SNR of the measured backscattered signal is, the better accuracy the measurements can achieve. In this work, we investigate the SNR of the computational distributed fiber-optic sensing technique in detail. As the technique is developed for stationary measurement, i.e, the external physical quantity which needs to be measured varies so slow that it can be assumed to be constant during one measurement, we only consider the influence of the white Gaussian noise (WGN) from the detector. We have already shown in [20] that using Walsh-Hadamard sequence instead of using random sequence, the required number of iterations are greatly reduced, because ghost imaging technique using random illumination exists a noise source associated to the random variable which needs to be averaged. Hence, even without detection noise, using random sequences normally requires large amount of averages to recover the image. For the computational sensing technique using Walsh-Hadamard sequence, because of the orthogonality of the sequence, the image recovery needs much less number of averages, especially when the number of temporal “pixels” is not large [20]. Therefore, we choose to analyze the SNR of the case using Walsh-Hadamard sequence, and compare it with the conventional time-domain method. Both the theoretical and experimental results show that the computational distributed fiber-optic sensing technique requires twice more averages than that needed by the traditional method to achieve the same level of SNR. These results make the computational method a very promising technique in practical applications since one only needs to double the number of averages, but the acquisition sampling rate can be significantly reduced.

2. Theoretical analysis

Ghost imaging normally uses random illumination patterns for reconstructing an image. The SNR of the recovered image is proportional to the number of iterations or averages [21,22], even without considering the detection noise. The reason for non-perfect SNR is due to the random illumination pattern used so that there is noise associated to the random variable which needs to be averaged during the correlation calculations. Since Walsh-Hadamard matrix is orthogonal, the SNR of recovered image based on the Walsh-Hadamard illumination patterns should be in principle higher than that based on random illuminations, provided that the same number of iterations or averages are performed. Therefore, we will only investigate the case of temporal image recovery based on Walsh-Hadamard patterns, analyzing the system’s SNR when there are measurement noises. There are many noise sources in a DFOS system which is hard to eliminate, such as laser intensity noise, detector noises, etc. In the following, only detection noises are considered and WGN is assumed. We compare the performance of our computational distributed fiber-optic sensing technique with respect to that of the conventional time-domain methods in terms of averaging number required by each approach to achieve the same level of SNR.

2.1 Analytical analysis

For the computational distributed fiber-optic sensing technique, we have already shown in our previous work [20] that differential ghost imaging (DGI) protocol using Walsh-Hadamard sequence is equivalent to performing an inverse Walsh-Hadamard transformation; therefore if no noises are present in the system, the detected signal with a dimension of ${2^k}$ ($k$ is a positive integer) can be expressed as [20],

(1)$${\boldsymbol D = }{{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}}{\boldsymbol S},$$

where ${\boldsymbol D} = \textrm{[}{D_1} - {\tilde{D}_1},{D_2} - {\tilde{D}_2}, \cdots ,{D_{{2^k}}} - {\tilde{D}_{{2^k}}}]$ is the detected backscattering signal vector, ${\boldsymbol S} = [S(1),S(2), \cdots ,S({2^k})]$ is the temporal image vector that needs to be recovered, and does not vary over the duration of one measurement, and ${{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}}$ is the natural-order Walsh-Hadamard matrix. The detected signals ${D_i}$ is associated to Walsh-Hadamard pattern and ${\tilde{D}_i}$ is associated to the inverse pattern:

(2)$${D_i} = \sum\limits_{j = 1}^{{2^k}} {{I_i}(j)S(j)} ,$$

(3)$${\tilde{D}_i} = \sum\limits_{j = 1}^{{2^k}} {{{\tilde{I}}_i}(j)S(j)} ,$$

where ${I_i}(j)$ is the Walsh-Hadamard pattern from the ${i^{th}}$ row of the matrix with $i = 1,2, \cdots ,{2^k}.$ The values of ${I_i}(j)$ is binary, either 1 for the value + 1 in ${{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}}$ or 0 for the value −1 in ${{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}},$ and ${\tilde{I}_i}(j) = 1 - {I_i}(j).$ Thus, ${I_i}(j) - {\tilde{I}_i}(j)$ represents matrix elements $(i,j)$ in ${{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}}.$ Note that the total duration of a sequence is $T = {2^k}\Delta t$ with $\Delta t$ being a bit period, which determines the image sampling time resolution (time between samples j and $j\textrm{ + 1}$). However, in practice, there are noises from the detector, so that we can express the actual detected values as,

(4)$${M_i} = {D_i} + {e_i},$$

(5)$${\tilde{M}_i} = {\tilde{D}_i} + {\tilde{e}_i},$$

where ${e_i}$ and ${\tilde{e}_i}$ are the noises which are assumed to be WGN. From Eqs. (2)–(5), the average of the signals after measuring n times under the same sequence can be expressed as:

(6)$$\langle M_i^n\rangle = \langle D_i^n + e_i^n\rangle = {D_i} + \langle e_i^n\rangle ,$$

(7)$$\langle \tilde{M}_i^n\rangle = \langle \tilde{D}_i^n + \tilde{e}_i^n\rangle = {\tilde{D}_i} + \langle \tilde{e}_i^n\rangle ,$$

where $M_i^n(n = 1,2,3, \cdots )$ is the n^th measurement of ${M_i}.$ $\langle \cdots \rangle $ denotes an average. Because ${D_i}$ and ${\tilde{D}_i}$ are the ideal values, one can obtain $\langle D_i^n\rangle = {D_i},\;\;\langle \tilde{D}_i^n\rangle = {\tilde{D}_i}.$ $\langle e_i^n\rangle $ and $\langle \tilde{e}_i^n\rangle $ are the average of the detection noises. The reconstructed image vector $\hat{{\boldsymbol S}}$ can be expressed as follows:

(8)$$\begin{aligned}\hat{{\boldsymbol S}} &= \frac{\textrm{1}}{{{\textrm{2}^k}}}{{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}}\langle {{\boldsymbol M}^n}\rangle\\ & = \frac{\textrm{1}}{{{\textrm{2}^k}}}\left[ {\begin{array}{cccc} {{I_1}(1) - {{\tilde{I}}_1}(1)}&{{I_1}(2) - {{\tilde{I}}_1}(2)}& \cdots &{{I_1}({2^k}) - {{\tilde{I}}_1}({2^k})}\\ {{I_2}(1) - {{\tilde{I}}_2}(1)}&{{I_2}(2) - {{\tilde{I}}_2}(2)}& \cdots &{{I_2}({2^k}) - {{\tilde{I}}_2}({2^k})}\\ \vdots & \vdots & \cdots & \vdots \\ {{I_{{2^k}}}(1) - {{\tilde{I}}_{{2^k}}}(1)}&{{I_{{2^k}}}(2) - {{\tilde{I}}_{{2^k}}}(2)}& \cdots &{{I_{{2^k}}}({2^k}) - {{\tilde{I}}_{{2^k}}}({2^k})} \end{array}} \right]\left[ {\begin{array}{c} {\langle M_1^n\rangle - \langle \tilde{M}_1^n\rangle }\\ {\langle M_2^n\rangle - \langle \tilde{M}_2^n\rangle }\\ \vdots \\ {\langle M_{{2^k}}^n\rangle - \langle \tilde{M}_{{2^k}}^n\rangle } \end{array}} \right],\end{aligned}$$

where $\langle {{\boldsymbol M}^n}\rangle = [\langle M_1^n\rangle - \langle \tilde{M}_1^n\rangle ,\langle M_2^n\rangle - \langle \tilde{M}_2^n\rangle , \cdots ,\langle M_{{2^k}}^n\rangle - \langle \tilde{M}_{{2^k}}^n\rangle ]$ is the average of the detected signal vector, and $\hat{{\boldsymbol S}} = [\hat{S}(1),\hat{S}(2), \cdots ,\hat{S}({2^k})]$ is the reconstructed image vector. Please note that ${\boldsymbol H}_{{\textrm{2}^k}}^{ - 1}\textrm{ = 1/}{\textrm{2}^k}{{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}}$ has been used in Eq. (8). The noises associated to the reconstructed image are the difference between the reconstructed image and the original image. Through the above equations, we can obtain the noise vector:

(9)$${\boldsymbol N}\textrm{ = }{\hat {\boldsymbol{S}} - \boldsymbol{S}}\textrm{ = }\frac{1}{{{2^k}}}{{\boldsymbol H}_{{{\boldsymbol 2}^{\boldsymbol k}}}}\left[ {\begin{array}{c} {\langle e_1^n\rangle - \langle \tilde{e}_1^n\rangle }\\ {\langle e_2^n\rangle - \langle \tilde{e}_2^n\rangle }\\ \vdots \\ {\langle e_{{2^k}}^n\rangle - \langle \tilde{e}_{{2^k}}^n\rangle } \end{array}} \right],$$

where ${\boldsymbol N} = [N(1),N(2), \cdots ,N({2^k})] = [\hat{S}(1) - S(1),\hat{S}(2) - S(2), \cdots ,\hat{S}({2^k}) - S({2^k})]$ represents the noise vector in an actual measurement. For $N(j) = \hat{S}(j) - S(j),$ the mean square error (MSE) can be calculated as follows [23]:

(10)$$MSE = E\{{{{[{N(j)} ]}^2}} \}= E\left\{ {{{\left[ {\frac{{(\langle e_1^n\rangle - \langle \tilde{e}_1^n\rangle ) + (\langle e_2^n\rangle - \langle \tilde{e}_2^n\rangle ) + \cdots (\langle e_{{2^k}}^n\rangle - \langle \tilde{e}_{{2^k}}^n\rangle )}}{{{2^k}}}} \right]}^2}} \right\} = \frac{{2{\sigma ^2}}}{{{2^k}n}},$$

where $E \{ \cdots \} $ is the expectation value. For simplicity, we assume the noise $e_i^n$ and $\tilde{e}_i^n$ are uncorrelated, zero-mean random processes with variance ${\sigma ^\textrm{2}}$, so that $E\{{e_i^n} \}= E\{{\tilde{e}_i^n} \}= \textrm{0}$, $E\{{{{(e_i^n)}^2}} \}= E\{{{{(\tilde{e}_i^n)}^2}} \}= {\sigma ^2}$, and $E\{{e_i^n\tilde{e}_i^n} \}\textrm{ = }E\{{e_i^ne_j^n} \}= E\{{\tilde{e}_i^n\tilde{e}_j^n} \}= 0(i \ne j)$. Therefore, if the power of the temporal image signal is ${\bar{S}^2},$ the SNR can be express as:

(11)$$SNR = 10{\log _{10}}\frac{{{{\bar{S}}^2}}}{{MSE}} = 10{\log _{10}}\frac{{{2^k}n{{\bar{S}}^2}}}{{\textrm{2}{\sigma ^\textrm{2}}}}.$$

The above equation represents the SNR of the computational method with a total number of measurements of ${2^k}\ast n.$ For the conventional method with the same amount of measurements, the MSE is ${\sigma ^2}/{2^k}n,$ so that the SNR is 3 dB higher than the computational method. In other words, the computational method using Walsh-Hadamard sequence needs twice more number of measurements in order to achieve the same SNR level as the conventional method. Note that even though we modulate optical inputs adopting Walsh-Hadamard sequence, our method is fundamentally different from the conventional coding technique which still needs to acquire the entire time-domain traces with high sampling rate and processes them afterwards based on known mathematical properties of the selected code to enhance the system’s SNR due to the coding gain that depends on the length of the code [20]. Our method is not proposed for the SNR enhancement and there is no gain related to the code length, but instead, is developed for reducing the sampling rate requirement. Therefore, we only compare the SNR with respect to the conventional case using single pulse.

2.2 Simulation results

Before conducting experiments, we perform a series of simulations to confirm the above theoretical analysis. We first take 256 data points to simulate a time-domain trace for a conventional DFOS with the maximum value normalized to 1. Using computational method as proposed in [20], the temporal image can be clearly reconstructed as shown in Figs. 1(a) and 1(b) using 256 order Walsh-Hadamard matrix. According to Eq. (11), if there is WGN in the detection, the reconstruction process is equivalent to performing 128 averages using conventional method when $n = 1.$ We, therefore, add WGN of 10 dB and 15 dB to the original time-domain trace, respectively, and proceed with averaging for 128 times. Then, using the computational method, we assume that the same detector is used, and the reconstructed temporal images are shown in Figs. 1(a) and 1(b), respectively. The computational and conventional methods give similar SNR confirming our above analytical analysis.

Fig. 1. Comparison of the conventional method and computational method after averaging for the cases with SNR of (a) 10 dB and (b) 15 dB. The number of measurements for the computational method is twice more than that of the conventional method.

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Next, we investigate the SNR variation as a function of average number and the Walsh-Hadamard order of the computational method. The same original time-domain trace in Fig. 1 is used for the simulation, so that we are able to obtain the SNR value after the average. From Eq. (11) and the discussion in the last section, if only WGN in the detection is considered, the reconstructed temporal image in the computational method requires twice more number of measurements compared to that required by the conventional method. For the conventional method, we add WGN of 10 dB, 15 dB and 20 dB on the original time-domain trace, and calculate the SNR values for different average numbers, so that the corresponding MSE could also be obtained. The results are shown in Figs. 2(a) and 2(b). For the computational method, we consider the case using the same detector and 256-order Walsh-Hadamard matrix to generate signal sequences. The SNR and MSE with respect to the average number are also shown in Figs. 2(a) and 2(b). Both the methods have the same SNR values, but the actual total number of measurements using the computational method to complete the image reconstruction is twice more than the average number shown in both the figures. This is consistent with our theoretical analysis in the last section.

Fig. 2. (a) SNR and (b) MSE as a function of the number of averages for different WGN levels. The horizontal axis represents the number of average for the conventional case, which is half of the value for the computational method.

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3. Experimental results

We setup a BOTDA system to confirm the analysis. The experiment setup is shown in Fig. 3. A laser beam from the semiconductor laser with a wavelength of 1554 nm is split by an optical coupler. One path is frequency shifted via a carrier-suppressed single sideband modulator (SSBM) which is controlled by a synthesizer. The resultant beam is enhanced by an erbium-doped fiber amplifier (EDFA), and then passes through an isolator to the fiber under test (FUT). The other path beam is boosted by the other EDFA and then is modulated into optical pulses by an electro-optic modulator (EOM) driven by a function generator. The pulse is delivered into the FUT through an optical circulator. The signal out of the photodetector (PD) is acquired by a digitizer, and the results are analyzed on a computer. Note that the polarization scrambler is removed from the system, as we are only considering WGN noise in this work. The polarization impact on the computational BOTDA system will be discussed at the end of the section.

Fig. 3. Schematic diagram of computational BOTDA system: SSBM: single sideband modulator; EOM: electro-optic modulator; EDFA: erbium-doped fiber amplifier; PD: photodetector.

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We choose to use 25 ns optical pulse corresponding to a 2.5 m spatial resolution and a FUT of 1 km for the validation purpose. The FUT has a Brillouin peak frequency around 10850 MHz, and we display the corresponding time-domain trace in Fig. 4, where we choose to use ${2^k} = \textrm{ }256$ Walsh-Hadamard matrix and average 128 times for each measurement. The detailed reconstruction procedure has been described in detail in [20]. For the conventional method using single pulse, the time-domain trace is also shown in Fig. 4 for comparison with the average number of 2¹⁴. The time-domain trace shows large variations along the sensing fiber due to the polarization state difference between the two counter-propagating beams, as there is no polarization scrambler used in the present setup. Since we are only considering the WGN in the system, longer FUT is not necessary for testing as we only need to focus on the MSE values for the current analysis. We next investigate the MSE as a function of the average times, and compare the result of the computational method to that of the conventional method.

Fig. 4. Measured time-domain traces of the 1 km FUT with 2.5 m spatial resolution at fixed frequency difference of 10850 MHz between the two laser beams using both the computational and conventional methods.

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We still fix the frequency difference between the two laser beams of 10850 MHz to obtain the time-domain traces using the conventional method. The MSE with respect to different average numbers are shown in Fig. 5. Then, we proceed with the computational method to reconstruct the time-domain traces. Similar to Fig. 3, total number of averages are performed twice more than that of the conventional method, and the resultant MSE value are shown in Fig. 5 for the comparison. We can see that both the results agree well with each other, and they are consistent with our analytical and simulation analysis in the last section. However, the MSE in our computational method is slightly larger than that of the conventional method as shown in Fig. 5. We believe that this is because the peak power of each optical pulse in the sequence are not the same; therefore, this introduces an extra source of noise as indicated in Eqs. (2) and (3) in measuring ${D_i}$ and ${\tilde{D}_i}$ when peak values of ${I_i}(j)$ and ${\tilde{I}_i}(j)$ are not the same. Even though we use the computational BOTDA system for the illustration, the above analysis can be applied to other kinds of computational DFOSs based on Rayleigh scattering, Raman scattering, and spontaneous Brillouin scattering where the temporal image is not sensitive to the polarization state of the pulsed light.

Fig. 5. MSE as a function of the number of averages for the time-domain traces obtained when the frequency difference between the two laser beams is 10850 MHz. The horizontal axis represents the number of average for the conventional case, which is half of the value for the computational method.

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For BOTDA technique, different polarization state between two counter-propagating beams results in different temporal images. In the experiment, we did not use a polarization scrambler as the conventional BOTDA system does, because the present work is focusing on analyzing the SNR of the computational method when only WGN presents in the detector. However, in a BOTDA system, the temporal image, which needs to be recovered is proportional to the interaction strength of the counter-propagating laser beams through stimulated Brillouin scattering, so that it strongly depends on their mutual polarization states between the two laser beams. If a polarization scrambler is inserted in the system, one needs to obtain the averaged Brillouin gain spectrum for the sensing purpose to reduce the polarization impact. For this case, the temporal image becomes a polarization-averaged time-domain trace which needs to be recovered, and from Eqs. (2)-(5), one can conclude that there will be an extra noise source due to polarization scrambling so that the computational method requires much more averages to achieve similar SNR level compared to the conventional method. However, for a BOTDA system, the polarization scrambler is not needed if balanced detection and orthogonal probe sidebands are used in the system as explained in [24,25]. Alternatively, polarization-maintaining fiber may be used as the sensing fiber to avoid using the polarization scrambler. Therefore, in principle, the computational distributed fiber-optic sensing technique can achieve similar SNR level compared to the conventional time-domain method by performing twice more averages even for BOTDA technique.

4. Conclusion

In conclusion, we investigate the SNR of the computational distributed fiber-optic sensing technique focusing on stationary measurement when only the WGN is considered in the detector of the sensing system. Both the theoretical and the experimental results confirm that the computational method requires twice more number of averages than the conventional time-domain method to achieve the same SNR level. Since the computational method is proposed for stationary measurement, doubling the acquisition time is normally acceptable in practice. However, the approach can offer a great design simplification for the distributed sensing systems based on time-domain technique because the sampling rate requirement is significantly reduced.

Funding

National Natural Science Foundation of China (61520106013, 61727816); Fundamental Research Funds for the Central Universities (DUT17RC(3)074).

Acknowledgments

D. –P. Zhou would also like to thank the financial support of the Academic Program Development Funds from Dalian University of Technology, and the Research Fund for International Cooperation (ICR1907).

Disclosures

The authors declare no conflicts of interest.

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Signal-to-noise ratio analysis of computational distributed fiber-optic sensing

Abstract

1. Introduction

2. Theoretical analysis

2.1 Analytical analysis

2.2 Simulation results

3. Experimental results

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (5)

Equations (11)

Optics Express