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Data compression and SNR enhancement with compressive sensing method in phase-sensitive OTDR
Accepted Manuscript Data compression and SNR enhancement with compressive sensing method in phase-sensitive OTDR Shuai Qu, Jun Chang, Zhenhua Cong, Hui Chen, Zengguang Qin
PII: DOI: Reference:
S0030-4018(18)30853-8 https://doi.org/10.1016/j.optcom.2018.09.064 OPTICS 23503
To appear in:
Optics Communications
Received date : 8 August 2018 Revised date : 21 September 2018 Accepted date : 26 September 2018 Please cite this article as: S. Qu, et al., Data compression and SNR enhancement with compressive sensing method in phase-sensitive OTDR, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.09.064 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
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Data Compression and SNR Enhancement with Compressive Sensing Method in Phase-sensitive OTDR Shuai Qua, Jun Changa, Zhenhua Conga, Hui Chena and Zengguang Qina,* a
School of Information Science and Engineering and Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University, Jinan 250100, China *Corresponding author: [email protected]
Abstract—An efficient signal processing method based on compressive sensing (CS) is presented for vibration sensing in phase sensitive optical time domain reflectometry system (φ-OTDR), which allows for compressing and denoising the raw data simultaneously. The main information of Rayleigh backscattering curves can be identified by sparse transformation. Then the data is compressed by observation matrix while the unwanted information is discarded. Orthogonal matching pursuit (OMP) algorithm is used to recovery the signal under the sparsity determined by a threshold rule. Experiment results show that the SNR is increased to 34.39 dB with the compression ratio of 18.9 for the vibration event of 100 Hz along 3 km sensing fiber which indicates the prominent performance of the novel method for vibration detection.
Keywords: φ-OTDR, signal-to-noise ratio, compressive sensing, compression ratio, orthogonal matching pursuit
1. Introduction In recent years, distributed fiber optical vibration sensors have attracted more and more research attention and been widely applied in many fields such as perimeter security, oil pipeline monitoring and damage detection of bridges to determine the locations of anomalous points [1-4]. As one of the most valuable technologies for distributed fiber optical vibration sensing, φ-OTDR has been studied intensively during the last few years due to its high detection sensitivity, relatively low cost, resistance to corrosion and immunity to electromagnetic interference. Compared with the traditional OTDR, φ-OTDR uses a highly coherent light source with narrow linewidth and low frequency drift. Rayleigh backscattering traces returned from sensing fiber are jagged because of the coherent interferences of the backscattered lights from different scattering centers within the optical pulse. The vibration induced dynamic strain would change the refractive index of the sensing fiber around the vibration location and the phase of the light wave after the vibration location. Therefore, the intensity of the Rayleigh backscattering light varies with time at the vibration location and the vibration can be detected. In φ-OTDR system, the sensing performance is characterized by three key parameters including frequency response, spatial resolution and detection distance. A mass of work has been done to enhance the performance of the φ-OTDR system. The maximum detectable frequency has been increased to 3 MHz by combining φ-OTDR
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with Mach-Zehnder interferometer with a spatial resolution of 5 m [5]. By using an ultra-long Raman fiber laser cavity, sensing distance over 125 km with a resolution of 10 m is realized [6]. Sensing range of 175 can be achieved with 25 m spatial resolution by mixing use of various amplification techniques [7]. With the improvement spatial resolution and detection distance, volumes of data collected by the system become prodigious which influences the data transmission, storage and retrieval. Generally, the single mode fiber is susceptible to external environmental disturbances, which would induce random noises on the light signal transmitting along the fiber and deteriorate the signal-to-noise ratio. Meanwhile, the attenuation of position-dependent signal caused by electrical noise and the phase noise of laser brings a lot of useless information. These increase the data amount and lead to low SNR which compromises the performance and efficiency of the system. Moving averaging and moving differential method has been proposed to remove the random noise which can realize spatial resolution of 5 m and frequency response of 1 kHz [8]. However, the SNR is only 6.5 dB that is not high enough to locate vibration accurately. Detection for 20 Hz and 8 kHz vibration events has also been achieved by wavelet denoising method with spatial resolution of 0.5 m [9]. But the approach does not reduce the data amount and requires large amount of time for data transmission and storage, which thus cannot guarantee real-time monitoring. Therefore, it is necessary to find a new method that can compress data and enhance the SNR simultaneously. In this paper, we introduce a novel method called sparse denoising method based on compressive sensing to reduce the volumes of the raw data and remove background noise for distributed optical fiber vibration sensor based on φ-OTDR. The data consists of many Rayleigh backscattering curves. When processed by discrete Fourier transform (DFT) matrix and Gaussian measurement matrix, noisy signal becomes sparse and compressed. Hence the data volume is reduced while the main components of signal are retained. Signal reconstruction is realized by orthogonal matching pursuit (OMP) algorithm. We experimentally demonstrate that the optimal compression ratio is 18.9 and the SNR is increased to 34.39 dB for vibration events of 100 Hz with fiber length of 3 km.
2. Principle of compressive sensing The theory of compressive sensing provides the solution for under-sampled signal reconstruction, which breaks the limit of the Shannon-Nyquist sampling theorem [10,11]. In a specific transform domain, the signal is sparse which means that only a few components are non-zero. A given signal X can be expressed as an N×1 column vector in time domain. After the sparse transformation, the signal X can be described as follows: (1) Where Ψ is an N×N matrix and S is an N×1 column vector. Each column of the matrix is the orthonormal basis of the Ψ domain, so the matrix is called sparse matrix. Discrete Fourier transform (DFT), discrete cousin transform (DCT) and discrete wavelet transform (DWT) are common sparse matrix. S is the representation of the
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same signal in Ψ domain. When S has only K non-zero elements, the signal X is regarded as K-sparse in the domain. In other words, the signal can be compressed while N>>K. An M×N observation matrix (M
Where Y is an M×1 column vector called observation vector and represents the compressed signal. A is an M×N matrix called information operator. Hence the signal is compressed and we can use Y instead of X for data transmission and storage. Then the Orthogonal matching pursuit (OMP) [12] algorithm is introduced to reconstruct the original signal and the input and output of the algorithm is shown in Fig. 1.
Fig. 1. Schematic diagram of OMP reconstruction algorithm.
In φ-OTDR system, there are different amplitude DC components along the fiber due to the coherent effect of Rayleigh scattering light. The data is composed of DC component and random noise when there is no vibration and we set a simulation signal for this situation shown in Fig. 2.
Fig. 2. (a) Raw data consisted of DC component and random noise. (b) Frequency spectrum for the raw data.
Fig. 2(a) shows the time domain information of the raw data that consists of a DC component with the amplitude of 1V and Gaussian white noise with the amplitude of 0.3V. The frequency domain information of the data is shown in Fig. 2(b) and no distinct peaks are found. It means that random noise is not sparse so we will focus on the situation when there is vibration. Fig. 3 shows the time domain and frequency domain information of two kinds of vibration events. In Fig. 3(a), the sinusoidal signal with amplitude of 1 V and frequency of 200 Hz is mixed with the same white noise in Fig. 2. Meanwhile, superposition of four sinusoidal signals with the frequency of 50 Hz, 100 Hz, 200 Hz and 400 Hz is shown in Fig. 3(c). Fig. 3(b) and (d) give the frequency spectrum for both cases. Obvious peaks at corresponding frequency are obtained because the signals are sparse.
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Fig. 3. Simulation signals of different vibration events. (a) sinusoidal signal mixed with DC component and Gaussian white noise. (b) frequency spectrum for the first simulation signal. (c) multiple sinusoidal signal mixed with DC component and Gaussian white noise. (d) frequency spectrum for the second simulation signal.
In our experiment, the observation matrix and sparse matrix are the Gaussian measurement matrix and DFT matrix respectively. Here, the Pearson correlation coefficient (PCC) [13] is used to describe the similarity between the original signal and the reconstructed signal. PCC is a common parameter that quantifies the correlation between two variables. The PCC is defined as: (3) Where and are the two variables, respectively. and are the arithmetic mean of the two variables, respectively. is the length of the variables. Fig. 4 shows the PCC value between original signal and reconstructed signal under different values of K and different sizes of observation matrix which reflects the reconstruction effect. In general, the K value of signal is relatively small. Here the K value is chosen as 4, 10 and 16 to compare the effect of reconstruction. We think that the minimum M value represents the minimum length that can be compressed when PCC value reaches maximum value. Compression ratio is introduced to quantify the reduction in data-representation size produced by compressed sensing algorithm. It is defined as , where N is the array size of the original signal and M is the array size of the observation vector. When K is equal to 4, the minimum M value is 39, so the compression ratio is 25.6. When K is 10 and 16, the compression ratio is 9.5 and 5.4 respectively. It shows that the smaller the K, the greater the compression of the data. Fig. 5 shows the signal reconstruction with the compression ratio mentioned above. Compared to Fig. 5(b) and (c), the result shown in Fig. 5(a) is closest to the uncontaminated signal. It proves that the data is compressed and recovery while the noise is suppressed.
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Fig. 4. PCC between original signal and reconstructed signal with different sparsity and different size of observation matrix.
Fig. 5. Signal recovery under maximum compression ratio. (a)K=4 (b) K=10 (c) K=16
In sparse denoising theory, all the signals can be compressed and sparse in a certain domain. But there is no transform domain for random noise to be sparse [14]. In applications, noise is unavoidable and equation (2) can be revised as: (4) Where n represents the noise and Yn represents the observation vector corresponding to the noise. Through Fig. 2 and Fig. 3, it is obvious that signal is sparse while noise is not. When applying compressive sensing, most of the useful information of signal is preserved in Y. But noise information is lost when n is compressed as Yn. Based on Fig. 5, we can find that the denoising performance is directly affected by the K value. Thus a threshold is required to determine the selection of K value [12, 15] and it can be set by using the expression as given below: (5) Where σ is the standard deviation of the signal, N is the length of the signal. The K value can be determined as the number of the components that are larger than T after DFT is applied on X. But it is possible that partial information could be lost in this way. In fact, selection of K value in the method is defined as K*=K+1. Furthermore, shrinkage thresholding is more appropriate to be employed for retaining the signal as much as possible. It is defined as: (6) Where η is the shrinkage coefficient and is chosen with 0.05, recovered by OMP.
S
is the coefficient of S
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Fig. 6. Schematic diagram of denoising method based on compressive sensing
The sparse denoising method based on the compressive sensing is applied in the φ-OTDR system. The schematic diagram is shown in Fig. 6 and the procedure contains the following steps: 1) calculate the threshold T and apply DFT on raw Rayleigh backscattering curves to obtain original S vector. 2) compute the value of K and K* which is the number of the elements larger than T in S vector. 3) adopt the OMP algorithm and shrinkage thresholding. 4) combine the estimation of S vector from both cases and acquire the denoising signal based on equation (1). 5) repeat step 1 to 4 when processing the signal of another location in sensing fiber.
3. Experimental setup and results The setup of φ-OTDR system is shown in Fig. 7. The laser source is an external cavity laser (ECL) with a central wavelength of 1550 nm and a narrow linewidth of 3 kHz. The maximum output power of the laser through the isolator is about 10mW. An acoustic optical modulator (AOM) is used to modulate the laser source and generate optical pulses. A function generator is used to drive the AOM with the pulse width and repetition rate of 50 ns and 10 kHz respectively. Optical pulse is amplified by an erbium-dope fiber amplifier (EDFA) and the spontaneous emission noise is removed by a tunable filter. The filtered pulse is launched into the sensing fiber with a length of 3 km through a circulator. The sensing fiber is composed of two bundles of single mode fiber with 2 km and 1km length respectively. The vibration source is a cylinder PZT which is put at the location of 2 km. About 1 m fiber is wounded on the PZT to vibrate together. The vibration is generated by a function generator and the frequency can be set from Hz to kHz. The Rayleigh backscattering light is amplified by another EDFA and filtered by a narrow bandwidth filter. The photodetector collects the Rayleigh signals and converts light signals into electrical signals gathered by a high-speed oscilloscope with a sampling rate of 100 MHz.
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Fig. 7. The experimental setup of φ-OTDR system.
Fig. 8 shows 1000 consecutive raw Rayleigh backscattering traces for vibration events of 100 Hz. It can be seen that the raw traces present jagged appearance which is resulted from the coherent interaction of multiple scattering centers within the duration of the optical pulse. Because of the raw Rayleigh backscattering traces mixed with strong random noise induced by environmental fluctuation or other factors, it is difficult to locate the vibration simply from the original data.
Fig. 8. 1000 consecutive raw Rayleigh backscattering traces for vibration event of 100Hz.
Fig. 9(a) shows the sparsity distribution along the sensing fiber calculated by threshold rule. There is an obvious peak at 2115 m indicating that the value of K at vibration location is different from other locations. We can obtain the sparsity of 3 at vibration location and explore the optimal compression ratio under the sparsity. In Fig. 9(b), the maximum PCC value is around 0.8 and the minimum M value is 53. So the compression ratio can be reached 18.9.
Fig. 9. (a) Sparsity distribution along the sensing fiber based on threshold rule (b) PCC between original signal and recovery signal under different size of observation matrix
The denoising performance of the compressive sensing method is compared with other signal processing algorithms including moving average and moving differential method (MAMD)[8], wavelet denoising method (WD) [9] and nonlocal mean method (NLM)[16,17]. The position information obtained by the moving average and moving differential method with the average number of 20 and differential number of 20 is shown in Fig. 10(a). The vibration location can be identified in view of the peak at 2115 m. However, the background noise is not removed sufficiently which may result
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in wrong alarm. Fig. 10(b) gives the location information obtained by wavelet denoising method and the noise is restrained to some extent. Fig. 10(c) shows the location of vibration using nonlocal mean method which the full width of search window and similar window are 11 and 10, respectively. Fig. 10(d) gives the result under the sparse denoising method based on compressive sensing method. It is obvious that the background noise is the lowest which confirms the effectiveness of the method in denoising. The denoising performance could be quantified by SNR, which is defined as follows: SNR=20*log10(Vsignal/RMS(Vnoise)), where Vsignal is the amplitude of the signal and RMS(Vnoise) is the root-mean-square of the background noise. In Fig. 10(a), the SNR of location information is 18.17dB for the moving average and moving differential method. In Fig.10 (b), the SNR of location information can increase to 25.61dB for the wavelet denoising. In Fig. 10(c), the SNR of location information is 21.25dB when the nonlocal mean method is adopted. In Fig. 10 (d), the SNR of location information based on the compressed sensing method increases to as high as 34.39dB.
Fig. 10. Vibration detection of 100 Hz processed by different denoising methods.(a) moving average and moving differential method. (b) wavelet denoising method. (c) nonlocal mean method. (d) compressive sensing method.
Fig. 11 illustrates the spatial resolution obtained by the four different signal processing methods. The spatial resolution is 5m, 4.8m and 5m respectively for the reported MAMD, WD and NLM methods, shown in Fig. 11(a)-(c). The spatial resolution achieved by the proposed compressive sensing method is 4m which is shown in Fig.11 (d).
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Fig.11. Spatial resolution of the four methods. (a) moving average and moving differential method. (b) wavelet denoising method. (c) nonlocal mean method.(d) compressive sensing method.
For more clearly contrast the performance of the four methods, several main parameters are listed in Table 1 including SNR, processing time, spatial resolution and the size of data. When considering all the parameters simultaneously, the proposed method has the most potential for real-time measurements by using FPGA and DSP in practice. Note that the same original data and computer is used for the performance comparison of the four methods in Table 1. Table 1. Performance comparison of different algorithms
Method
SNR
Processing time
Spatial resolution
The size of data
MAMD
18.17dB
2.33s
5.0m
1000×3000
WD
25.61dB
18.66s
4.8m
1000×3000
NLM
21.25dB
56.63s
5.0m
1000×3000
CS
34.39dB
17.29s
4.0m
53×3000
Finally, we change the size of observation matrix and the processing result is shown in Fig. 12. Fig. 12(a)-(c) shows the location information when observation matrix is 200,400 and 600. The SNR is 37.20 dB, 39.35 dB and 42.09 dB respectively. It is concluded that the SNR is increased with the size of the observation matrix. So if the SNR is not satisfactory under the optimal compression ratio, we can reduce the compression ratio in order to achieve a higher SNR.
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Fig. 12. Location information obtained by different size of observation matrix. (a) 200 (b) 400 (c) 600
4. Conclusion In this paper, a new sparse denoising method based on compressive sensing for compressing data and removing random noise in φ-OTDR system is proposed. The processing object is the one-dimensional time domain noisy signal obtained from the Rayleigh backscattering curves at the same location. Gaussian measurement matrix and DFT matrix are used to obtain the measure vector which contains the important components of signal. A threshold rule is introduced to determine the sparsity and the signal reconstruction is realized by OMP algorithm. The experiment results demonstrate that the SNR for 100 Hz vibration events is enhanced to 34.39 dB with the compression ratio of 18.9. The new method can be regarded as an effective instrument for performance improvement of φ-OTDR system.
5. Acknowledgements This work was supported by National Natural Science Foundation of China (61405105&61475085), the Shandong Provincial Natural Science Foundation, China (ZR2014FQ015) and Science and technology development project of Shandong province (2014GGX101007), and the Fundamental Research Funds of Shandong University (2017JC023&2014YQ011).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
J.P. Dakin, D.A.J. Pearce, A.P. Strong, C.A. Wade, Proc. SPIE 838 (1988) 325. J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, J. Lightw. Technol. 23 (2005) 2081. A. J. Rogers, Appl. Opt. 20, (1981) 1060. Y. K. Dong, D. X. Ba, T. F. Jiang, D. W. Zhou, H. Y, Zhang, C. Y. Zhu, Z. W, Lu, H. Li, L. Chen, X. Y. Bao, IEEE Photonics J. 5, (2013) 2600407. T. Zhu, Q. He, X. H. Xiao, X. Y Bao, Opt. Express. 21, (2013) 2953. H. F. Martins, S. Matin-Lopez, P. Corredera, J. D. Ania-Castanon, O. Frazao, M. Gonzalez-Herraez, J. Lightw. Technol. 33, (2015) 2628. Z. N. Wang, J. J. Zeng, M. Q. Fan, H. Wu, F. Peng, L. Zhang, Y. Zhou, Y. J. Rao, Opt. Lett. 39, (2014) 5866. Y. L. Lu, T. Zhu, L. Chen, X. Y. Bao, J. Lightw. Technol. 28, (2010) 3243. Z. G. Qin, L. Chen, X. Y. Bao, IEEE. Photon. Technol. Lett. 24, (2012) 542. D. Donoho, IEEE Trans. Inf. Theory. 52, (2006) 1289. R. G. Baraniuk, IEEE Signal Process. Mag. 24, (2007) 118. J. A. Tropp, IEEE Trans. Inf. Theory. 53, (2007) 4655. M. M. Mukaka, Malawi Med. J. 24, (2012) 69. L. Zhu, Y. L. Zhu, H. Mao, M. H. Gu, KAM'09. Second International Symposium on. IEEE (2009) 35. S. Stankovic, I. Orovic, L. Stankovic, Signal Process. 104, (2014) 43. M. A. Soto, J. A. Ramírez, and L. Thévenaz, Nat. Commun. 7,(2016) 10870. J. Wang, Y. Guo, Y. Ying, Y. Liu, and Q. Peng, Proc. IEEE Int. Conf. Image Process. (2006) 1429.
PII: DOI: Reference:
S0030-4018(18)30853-8 https://doi.org/10.1016/j.optcom.2018.09.064 OPTICS 23503
To appear in:
Optics Communications
Received date : 8 August 2018 Revised date : 21 September 2018 Accepted date : 26 September 2018 Please cite this article as: S. Qu, et al., Data compression and SNR enhancement with compressive sensing method in phase-sensitive OTDR, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.09.064 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
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Data Compression and SNR Enhancement with Compressive Sensing Method in Phase-sensitive OTDR Shuai Qua, Jun Changa, Zhenhua Conga, Hui Chena and Zengguang Qina,* a
School of Information Science and Engineering and Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University, Jinan 250100, China *Corresponding author: [email protected]
Abstract—An efficient signal processing method based on compressive sensing (CS) is presented for vibration sensing in phase sensitive optical time domain reflectometry system (φ-OTDR), which allows for compressing and denoising the raw data simultaneously. The main information of Rayleigh backscattering curves can be identified by sparse transformation. Then the data is compressed by observation matrix while the unwanted information is discarded. Orthogonal matching pursuit (OMP) algorithm is used to recovery the signal under the sparsity determined by a threshold rule. Experiment results show that the SNR is increased to 34.39 dB with the compression ratio of 18.9 for the vibration event of 100 Hz along 3 km sensing fiber which indicates the prominent performance of the novel method for vibration detection.
Keywords: φ-OTDR, signal-to-noise ratio, compressive sensing, compression ratio, orthogonal matching pursuit
1. Introduction In recent years, distributed fiber optical vibration sensors have attracted more and more research attention and been widely applied in many fields such as perimeter security, oil pipeline monitoring and damage detection of bridges to determine the locations of anomalous points [1-4]. As one of the most valuable technologies for distributed fiber optical vibration sensing, φ-OTDR has been studied intensively during the last few years due to its high detection sensitivity, relatively low cost, resistance to corrosion and immunity to electromagnetic interference. Compared with the traditional OTDR, φ-OTDR uses a highly coherent light source with narrow linewidth and low frequency drift. Rayleigh backscattering traces returned from sensing fiber are jagged because of the coherent interferences of the backscattered lights from different scattering centers within the optical pulse. The vibration induced dynamic strain would change the refractive index of the sensing fiber around the vibration location and the phase of the light wave after the vibration location. Therefore, the intensity of the Rayleigh backscattering light varies with time at the vibration location and the vibration can be detected. In φ-OTDR system, the sensing performance is characterized by three key parameters including frequency response, spatial resolution and detection distance. A mass of work has been done to enhance the performance of the φ-OTDR system. The maximum detectable frequency has been increased to 3 MHz by combining φ-OTDR
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with Mach-Zehnder interferometer with a spatial resolution of 5 m [5]. By using an ultra-long Raman fiber laser cavity, sensing distance over 125 km with a resolution of 10 m is realized [6]. Sensing range of 175 can be achieved with 25 m spatial resolution by mixing use of various amplification techniques [7]. With the improvement spatial resolution and detection distance, volumes of data collected by the system become prodigious which influences the data transmission, storage and retrieval. Generally, the single mode fiber is susceptible to external environmental disturbances, which would induce random noises on the light signal transmitting along the fiber and deteriorate the signal-to-noise ratio. Meanwhile, the attenuation of position-dependent signal caused by electrical noise and the phase noise of laser brings a lot of useless information. These increase the data amount and lead to low SNR which compromises the performance and efficiency of the system. Moving averaging and moving differential method has been proposed to remove the random noise which can realize spatial resolution of 5 m and frequency response of 1 kHz [8]. However, the SNR is only 6.5 dB that is not high enough to locate vibration accurately. Detection for 20 Hz and 8 kHz vibration events has also been achieved by wavelet denoising method with spatial resolution of 0.5 m [9]. But the approach does not reduce the data amount and requires large amount of time for data transmission and storage, which thus cannot guarantee real-time monitoring. Therefore, it is necessary to find a new method that can compress data and enhance the SNR simultaneously. In this paper, we introduce a novel method called sparse denoising method based on compressive sensing to reduce the volumes of the raw data and remove background noise for distributed optical fiber vibration sensor based on φ-OTDR. The data consists of many Rayleigh backscattering curves. When processed by discrete Fourier transform (DFT) matrix and Gaussian measurement matrix, noisy signal becomes sparse and compressed. Hence the data volume is reduced while the main components of signal are retained. Signal reconstruction is realized by orthogonal matching pursuit (OMP) algorithm. We experimentally demonstrate that the optimal compression ratio is 18.9 and the SNR is increased to 34.39 dB for vibration events of 100 Hz with fiber length of 3 km.
2. Principle of compressive sensing The theory of compressive sensing provides the solution for under-sampled signal reconstruction, which breaks the limit of the Shannon-Nyquist sampling theorem [10,11]. In a specific transform domain, the signal is sparse which means that only a few components are non-zero. A given signal X can be expressed as an N×1 column vector in time domain. After the sparse transformation, the signal X can be described as follows: (1) Where Ψ is an N×N matrix and S is an N×1 column vector. Each column of the matrix is the orthonormal basis of the Ψ domain, so the matrix is called sparse matrix. Discrete Fourier transform (DFT), discrete cousin transform (DCT) and discrete wavelet transform (DWT) are common sparse matrix. S is the representation of the
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same signal in Ψ domain. When S has only K non-zero elements, the signal X is regarded as K-sparse in the domain. In other words, the signal can be compressed while N>>K. An M×N observation matrix (M
Where Y is an M×1 column vector called observation vector and represents the compressed signal. A is an M×N matrix called information operator. Hence the signal is compressed and we can use Y instead of X for data transmission and storage. Then the Orthogonal matching pursuit (OMP) [12] algorithm is introduced to reconstruct the original signal and the input and output of the algorithm is shown in Fig. 1.
Fig. 1. Schematic diagram of OMP reconstruction algorithm.
In φ-OTDR system, there are different amplitude DC components along the fiber due to the coherent effect of Rayleigh scattering light. The data is composed of DC component and random noise when there is no vibration and we set a simulation signal for this situation shown in Fig. 2.
Fig. 2. (a) Raw data consisted of DC component and random noise. (b) Frequency spectrum for the raw data.
Fig. 2(a) shows the time domain information of the raw data that consists of a DC component with the amplitude of 1V and Gaussian white noise with the amplitude of 0.3V. The frequency domain information of the data is shown in Fig. 2(b) and no distinct peaks are found. It means that random noise is not sparse so we will focus on the situation when there is vibration. Fig. 3 shows the time domain and frequency domain information of two kinds of vibration events. In Fig. 3(a), the sinusoidal signal with amplitude of 1 V and frequency of 200 Hz is mixed with the same white noise in Fig. 2. Meanwhile, superposition of four sinusoidal signals with the frequency of 50 Hz, 100 Hz, 200 Hz and 400 Hz is shown in Fig. 3(c). Fig. 3(b) and (d) give the frequency spectrum for both cases. Obvious peaks at corresponding frequency are obtained because the signals are sparse.
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Fig. 3. Simulation signals of different vibration events. (a) sinusoidal signal mixed with DC component and Gaussian white noise. (b) frequency spectrum for the first simulation signal. (c) multiple sinusoidal signal mixed with DC component and Gaussian white noise. (d) frequency spectrum for the second simulation signal.
In our experiment, the observation matrix and sparse matrix are the Gaussian measurement matrix and DFT matrix respectively. Here, the Pearson correlation coefficient (PCC) [13] is used to describe the similarity between the original signal and the reconstructed signal. PCC is a common parameter that quantifies the correlation between two variables. The PCC is defined as: (3) Where and are the two variables, respectively. and are the arithmetic mean of the two variables, respectively. is the length of the variables. Fig. 4 shows the PCC value between original signal and reconstructed signal under different values of K and different sizes of observation matrix which reflects the reconstruction effect. In general, the K value of signal is relatively small. Here the K value is chosen as 4, 10 and 16 to compare the effect of reconstruction. We think that the minimum M value represents the minimum length that can be compressed when PCC value reaches maximum value. Compression ratio is introduced to quantify the reduction in data-representation size produced by compressed sensing algorithm. It is defined as , where N is the array size of the original signal and M is the array size of the observation vector. When K is equal to 4, the minimum M value is 39, so the compression ratio is 25.6. When K is 10 and 16, the compression ratio is 9.5 and 5.4 respectively. It shows that the smaller the K, the greater the compression of the data. Fig. 5 shows the signal reconstruction with the compression ratio mentioned above. Compared to Fig. 5(b) and (c), the result shown in Fig. 5(a) is closest to the uncontaminated signal. It proves that the data is compressed and recovery while the noise is suppressed.
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Fig. 4. PCC between original signal and reconstructed signal with different sparsity and different size of observation matrix.
Fig. 5. Signal recovery under maximum compression ratio. (a)K=4 (b) K=10 (c) K=16
In sparse denoising theory, all the signals can be compressed and sparse in a certain domain. But there is no transform domain for random noise to be sparse [14]. In applications, noise is unavoidable and equation (2) can be revised as: (4) Where n represents the noise and Yn represents the observation vector corresponding to the noise. Through Fig. 2 and Fig. 3, it is obvious that signal is sparse while noise is not. When applying compressive sensing, most of the useful information of signal is preserved in Y. But noise information is lost when n is compressed as Yn. Based on Fig. 5, we can find that the denoising performance is directly affected by the K value. Thus a threshold is required to determine the selection of K value [12, 15] and it can be set by using the expression as given below: (5) Where σ is the standard deviation of the signal, N is the length of the signal. The K value can be determined as the number of the components that are larger than T after DFT is applied on X. But it is possible that partial information could be lost in this way. In fact, selection of K value in the method is defined as K*=K+1. Furthermore, shrinkage thresholding is more appropriate to be employed for retaining the signal as much as possible. It is defined as: (6) Where η is the shrinkage coefficient and is chosen with 0.05, recovered by OMP.
S
is the coefficient of S
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Fig. 6. Schematic diagram of denoising method based on compressive sensing
The sparse denoising method based on the compressive sensing is applied in the φ-OTDR system. The schematic diagram is shown in Fig. 6 and the procedure contains the following steps: 1) calculate the threshold T and apply DFT on raw Rayleigh backscattering curves to obtain original S vector. 2) compute the value of K and K* which is the number of the elements larger than T in S vector. 3) adopt the OMP algorithm and shrinkage thresholding. 4) combine the estimation of S vector from both cases and acquire the denoising signal based on equation (1). 5) repeat step 1 to 4 when processing the signal of another location in sensing fiber.
3. Experimental setup and results The setup of φ-OTDR system is shown in Fig. 7. The laser source is an external cavity laser (ECL) with a central wavelength of 1550 nm and a narrow linewidth of 3 kHz. The maximum output power of the laser through the isolator is about 10mW. An acoustic optical modulator (AOM) is used to modulate the laser source and generate optical pulses. A function generator is used to drive the AOM with the pulse width and repetition rate of 50 ns and 10 kHz respectively. Optical pulse is amplified by an erbium-dope fiber amplifier (EDFA) and the spontaneous emission noise is removed by a tunable filter. The filtered pulse is launched into the sensing fiber with a length of 3 km through a circulator. The sensing fiber is composed of two bundles of single mode fiber with 2 km and 1km length respectively. The vibration source is a cylinder PZT which is put at the location of 2 km. About 1 m fiber is wounded on the PZT to vibrate together. The vibration is generated by a function generator and the frequency can be set from Hz to kHz. The Rayleigh backscattering light is amplified by another EDFA and filtered by a narrow bandwidth filter. The photodetector collects the Rayleigh signals and converts light signals into electrical signals gathered by a high-speed oscilloscope with a sampling rate of 100 MHz.
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Fig. 7. The experimental setup of φ-OTDR system.
Fig. 8 shows 1000 consecutive raw Rayleigh backscattering traces for vibration events of 100 Hz. It can be seen that the raw traces present jagged appearance which is resulted from the coherent interaction of multiple scattering centers within the duration of the optical pulse. Because of the raw Rayleigh backscattering traces mixed with strong random noise induced by environmental fluctuation or other factors, it is difficult to locate the vibration simply from the original data.
Fig. 8. 1000 consecutive raw Rayleigh backscattering traces for vibration event of 100Hz.
Fig. 9(a) shows the sparsity distribution along the sensing fiber calculated by threshold rule. There is an obvious peak at 2115 m indicating that the value of K at vibration location is different from other locations. We can obtain the sparsity of 3 at vibration location and explore the optimal compression ratio under the sparsity. In Fig. 9(b), the maximum PCC value is around 0.8 and the minimum M value is 53. So the compression ratio can be reached 18.9.
Fig. 9. (a) Sparsity distribution along the sensing fiber based on threshold rule (b) PCC between original signal and recovery signal under different size of observation matrix
The denoising performance of the compressive sensing method is compared with other signal processing algorithms including moving average and moving differential method (MAMD)[8], wavelet denoising method (WD) [9] and nonlocal mean method (NLM)[16,17]. The position information obtained by the moving average and moving differential method with the average number of 20 and differential number of 20 is shown in Fig. 10(a). The vibration location can be identified in view of the peak at 2115 m. However, the background noise is not removed sufficiently which may result
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in wrong alarm. Fig. 10(b) gives the location information obtained by wavelet denoising method and the noise is restrained to some extent. Fig. 10(c) shows the location of vibration using nonlocal mean method which the full width of search window and similar window are 11 and 10, respectively. Fig. 10(d) gives the result under the sparse denoising method based on compressive sensing method. It is obvious that the background noise is the lowest which confirms the effectiveness of the method in denoising. The denoising performance could be quantified by SNR, which is defined as follows: SNR=20*log10(Vsignal/RMS(Vnoise)), where Vsignal is the amplitude of the signal and RMS(Vnoise) is the root-mean-square of the background noise. In Fig. 10(a), the SNR of location information is 18.17dB for the moving average and moving differential method. In Fig.10 (b), the SNR of location information can increase to 25.61dB for the wavelet denoising. In Fig. 10(c), the SNR of location information is 21.25dB when the nonlocal mean method is adopted. In Fig. 10 (d), the SNR of location information based on the compressed sensing method increases to as high as 34.39dB.
Fig. 10. Vibration detection of 100 Hz processed by different denoising methods.(a) moving average and moving differential method. (b) wavelet denoising method. (c) nonlocal mean method. (d) compressive sensing method.
Fig. 11 illustrates the spatial resolution obtained by the four different signal processing methods. The spatial resolution is 5m, 4.8m and 5m respectively for the reported MAMD, WD and NLM methods, shown in Fig. 11(a)-(c). The spatial resolution achieved by the proposed compressive sensing method is 4m which is shown in Fig.11 (d).
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Fig.11. Spatial resolution of the four methods. (a) moving average and moving differential method. (b) wavelet denoising method. (c) nonlocal mean method.(d) compressive sensing method.
For more clearly contrast the performance of the four methods, several main parameters are listed in Table 1 including SNR, processing time, spatial resolution and the size of data. When considering all the parameters simultaneously, the proposed method has the most potential for real-time measurements by using FPGA and DSP in practice. Note that the same original data and computer is used for the performance comparison of the four methods in Table 1. Table 1. Performance comparison of different algorithms
Method
SNR
Processing time
Spatial resolution
The size of data
MAMD
18.17dB
2.33s
5.0m
1000×3000
WD
25.61dB
18.66s
4.8m
1000×3000
NLM
21.25dB
56.63s
5.0m
1000×3000
CS
34.39dB
17.29s
4.0m
53×3000
Finally, we change the size of observation matrix and the processing result is shown in Fig. 12. Fig. 12(a)-(c) shows the location information when observation matrix is 200,400 and 600. The SNR is 37.20 dB, 39.35 dB and 42.09 dB respectively. It is concluded that the SNR is increased with the size of the observation matrix. So if the SNR is not satisfactory under the optimal compression ratio, we can reduce the compression ratio in order to achieve a higher SNR.
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Fig. 12. Location information obtained by different size of observation matrix. (a) 200 (b) 400 (c) 600
4. Conclusion In this paper, a new sparse denoising method based on compressive sensing for compressing data and removing random noise in φ-OTDR system is proposed. The processing object is the one-dimensional time domain noisy signal obtained from the Rayleigh backscattering curves at the same location. Gaussian measurement matrix and DFT matrix are used to obtain the measure vector which contains the important components of signal. A threshold rule is introduced to determine the sparsity and the signal reconstruction is realized by OMP algorithm. The experiment results demonstrate that the SNR for 100 Hz vibration events is enhanced to 34.39 dB with the compression ratio of 18.9. The new method can be regarded as an effective instrument for performance improvement of φ-OTDR system.
5. Acknowledgements This work was supported by National Natural Science Foundation of China (61405105&61475085), the Shandong Provincial Natural Science Foundation, China (ZR2014FQ015) and Science and technology development project of Shandong province (2014GGX101007), and the Fundamental Research Funds of Shandong University (2017JC023&2014YQ011).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
J.P. Dakin, D.A.J. Pearce, A.P. Strong, C.A. Wade, Proc. SPIE 838 (1988) 325. J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, J. Lightw. Technol. 23 (2005) 2081. A. J. Rogers, Appl. Opt. 20, (1981) 1060. Y. K. Dong, D. X. Ba, T. F. Jiang, D. W. Zhou, H. Y, Zhang, C. Y. Zhu, Z. W, Lu, H. Li, L. Chen, X. Y. Bao, IEEE Photonics J. 5, (2013) 2600407. T. Zhu, Q. He, X. H. Xiao, X. Y Bao, Opt. Express. 21, (2013) 2953. H. F. Martins, S. Matin-Lopez, P. Corredera, J. D. Ania-Castanon, O. Frazao, M. Gonzalez-Herraez, J. Lightw. Technol. 33, (2015) 2628. Z. N. Wang, J. J. Zeng, M. Q. Fan, H. Wu, F. Peng, L. Zhang, Y. Zhou, Y. J. Rao, Opt. Lett. 39, (2014) 5866. Y. L. Lu, T. Zhu, L. Chen, X. Y. Bao, J. Lightw. Technol. 28, (2010) 3243. Z. G. Qin, L. Chen, X. Y. Bao, IEEE. Photon. Technol. Lett. 24, (2012) 542. D. Donoho, IEEE Trans. Inf. Theory. 52, (2006) 1289. R. G. Baraniuk, IEEE Signal Process. Mag. 24, (2007) 118. J. A. Tropp, IEEE Trans. Inf. Theory. 53, (2007) 4655. M. M. Mukaka, Malawi Med. J. 24, (2012) 69. L. Zhu, Y. L. Zhu, H. Mao, M. H. Gu, KAM'09. Second International Symposium on. IEEE (2009) 35. S. Stankovic, I. Orovic, L. Stankovic, Signal Process. 104, (2014) 43. M. A. Soto, J. A. Ramírez, and L. Thévenaz, Nat. Commun. 7,(2016) 10870. J. Wang, Y. Guo, Y. Ying, Y. Liu, and Q. Peng, Proc. IEEE Int. Conf. Image Process. (2006) 1429.