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We propose the continuous wavelet transform for non-stationary vibration measurement by distributed vibration sensor based on phase optical time-domain reflectometry (OTDR). The continuous wavelet transform approach can give simultaneously the frequency and time information of the vibration event. Frequency evolution is obtained by the wavelet ridge detection method from the scalogram of the continuous wavelet transform. In addition, a novel signal processing algorithm based on the global wavelet spectrum is used to determine the location of vibration. Distributed vibration measurements of 500Hz and 500Hz to 1kHz sweep events over 20 cm fiber length are demonstrated using a single mode fiber.
©2012 Optical Society of America
1. Introduction
Fully distributed capability of optical fiber sensor makes it a powerful tool to measure the environment changes along the sensing fiber. For health monitoring on civil structures and fault diagnostics of mechanical structures, optical fiber vibration sensors can be used to evaluate structural condition and identify internal damages at early stage through measuring the intrinsic frequency. Different schemes have been proposed to fulfill the distributed vibration sensing, such as using the polarization optical time-domain reflectometry (OTDR) [1, 2], Sagnac or Mach-Zehnder interferometers [3–5]. In the aforementioned methods, vibration events are usually under the stationary conditions so that the vibration frequency analysis has generally relied on the spectral analysis via fast Fourier transform (FFT). An important limitation of the FFT is its inability to provide the time dependence of the signal spectrum. In practice, the interested realistic signals such as the cracks and fatigues inside the structures are usually transient and non-stationary which means that the conventional FFT approach is not able to reveal the inherent feature within the signal. Consequently, supplementary signal processing scheme of non-stationary signals is critical for health monitoring and fault diagnostics based on fiber vibration sensing.
Time-frequency signal analysis methods can interpret signals in both time and frequency domain simultaneously compared to the commonly-used FFT. A number of time-frequency analysis techniques are available to decompose complex signals in time–frequency domain, including short time Fourier transform (STFT), Wigner-Ville transform (WVT), and continuous wavelet transform (CWT). These methods map the signal under investigation into a two dimensional (2D) function of time and frequency and therefore can provide true time-frequency representations. However the time and frequency resolutions of the STFT cannot be improved simultaneously because of the fixed time-bandwidth product for a defined window function. The WVT will generate the interference terms and tend to mislead the signal analysis. In comparison, the CWT has developed as the most favored tool [6, 7] because of its ability to decompose the signal by employing a window of variable width compared to other methods. Another key advantage of the wavelet techniques is the variety of wavelet functions available to capture the subtle changes of the signal under investigation.
In this paper, we introduce the CWT method to the phase-sensitive OTDR vibration sensor with coherent detection to analyze the non-stationary vibration signals. Experimental results show a sweep vibration event with a frequency range of 500Hz to 1kHz can be detected along a single mode sensing fiber over 20cm disturbance length. The time dependent frequency information of the vibration event is determined from the CWT scalogram by wavelet ridge detection. The vibration position information is obtained by the wavelet global spectrum. This CWT method shows the potential of interrogating the complex non-stationary vibration signals for damage detection in civil and mechanical related applications with the phase sensitive OTDR.
2. Continuous wavelet transform
Wavelet transform decomposes a signal into a series of base functions of dilated and translated versions of the mother wavelet function. The form of the continuous wavelet transform of a signal is defined as [8]:
where is the original signal and and are the dilation and translation parameters, respectively. is the conjugate function of the mother wavelet. The CWT can measure the time evolution of frequency transients by using a complex analytic wavelet which can separate the amplitude and phase information of the signals. To be classified as a complex analytic wavelet, the wavelet function must meet the mathematical criteria that the Fourier transform of it should vanish for negative frequencies. Square of modulus of the CWT is known as the scalogram and is defined as:In Eq. (1), the spectral components of the signal are inversely proportional to the wavelet scale and the location represents the time information. Considering is the characteristic frequency of the mother wavelet function, the frequency corresponding to a certain wavelet at an arbitrary scale is defined as:The time-frequency resolution of the complex analytic wavelet depends on the time-frequency spread of the wavelet which corresponds to a Heisenberg box centered at in the time-frequency plane. The area of this box remains constant at all scales but the resolution in time and frequency depends on the scale. Hence the frequency accuracy will be affected when the CWT is used to analyze the vibration signal in phase-OTDR system. In general, the instantaneous frequency could be extracted by wavelet ridge detection in the time-frequency analysis using CWT method which could further alleviate the impact of noises on the signal interested. The wavelet ridge is defined by:
which corresponding to the local maxima at a specific scale .If the integral taken over all the time sections at a specific scale, a global wavelet power spectrum similar to the classical power spectrum via FFT method could be obtained which is shown in Eq. (5):
The peaks in this global power spectrum represent the dominant frequency components in the signal under investigation.3. Experimental setup and signal processing
Compared with the conventional OTDR, the phase-sensitive OTDR [9–11] utilizes a laser source with narrow linewidth and low frequency shift to form the interference of the backscattered light waves. The Rayleigh backscattering trace returned from the test fiber is modulated in the form of speckle-like profile [12] because of the coherent interaction of multiple scattering centers within the injected pulse duration. If there is no vibration along the sensing fiber, the amplitude of all the Rayleigh backscattering traces will be constant. When the vibration is applied on the certain section of the fiber, the amplitude of the Rayleigh backscattering light will be modulated and change in different time traces.
The experimental setup of the phase-sensitive OTDR with coherent detection is shown in Fig. 1 . The light source is a narrow linewidth laser (TeraXion) with maximum output power of 30mW and linewidth of 535kHz. The continuous-wave (CW) light from the laser is split by a 3dB coupler. One output arm of the CW light from the coupler is injected into an electro-optic modulator (EOM) to generate the pulses. An Erbium-doped fiber amplifier (EDFA) is used to amplify the pulses and the ASE noise is filtered by an optical fiber Bragg grating. The amplified pulses are launched into a single mode sensing fiber by a circulator. Another arm of the coupler is used as a local oscillator with a 200MHz frequency shift induced by an acoustic-optic modulator (AOM). Then the light of the local oscillator is combined with the backscattered Rayleigh signals returned from the fiber under test by a 3dB coupler. A balance detection scheme is used to eliminate the DC and common components and obtain 3dB signal-to-noise-ratio (SNR) improvement. The final signals are acquired by a high-speed oscilloscope and then the signal processing scheme is accomplished by a software program.
Fig. 1 Experimental setup for coherent phase-sensitive OTDR, NLL: narrow linewidth laser; AOM: acoustic-optic modulator; EOM: electro-optic modulator; EDFA: Erbium-doped fiber amplifier; PC: polarization controller, Filter: optical fiber grating filter.
In our former experimental setup [9], an electrical mixer is used to convert the intermediate frequency (IF) signal to the baseband. However this synchronous demodulation scheme is sensitive to the variation of the intermediate frequency, and typically a frequency locked loop is needed. The frequency drift of the AOM during measurement makes the SNR of the detection system fluctuate. Here we simplify our setup by removing the demodulation scheme. Due to the interference effect of the backscattering Rayleigh light wave within the pulse width, both the amplitude and phase of the signals have been encoded by the external vibration information before being combined with the local oscillator light so that the vibration event can be determined without recovering the baseband signals.
In order to reduce the amplitude fluctuation in the Rayleigh signal traces due to phase noise of the laser, partial interferometric problem (random polarization), and electrical noises such as thermal noise and shot noise, the wavelet denoising method [13] is introduced to enhance the interested signals. This signal denoising method is based on the idea of threshold wavelet coefficients of the noisy signal using the discrete wavelet transform to remove the noise. Hundreds of the Rayleigh backscattering traces from the sensing fiber are collected and then pass a digital low pass filter to remove firstly the high frequency noise. The points on each curve represent the positions along the fiber. The signals at a certain position of all the traces will be extracted and plotted versus time to form a new time domain sequence. After wavelet denoising on these new sequences, the CWT method will be applied to find the time-evolved frequency information. In order to determine the position of the vibration event, the global wavelet power spectrum at each position is calculated and a 2-D graph with fiber length and frequency information can be drawn to show the location of the vibration.
4. Experimental results and discussions
A PZT cylinder with 0.2m single mode fiber wound is put at 72.3m over 200m sensing fiber in our system as a vibration source. The PZT is driven by a function generator and the frequency can be adjusted from several Hz up to kHz. The fiber is fixed on the surface of the PZT cylinder by instant glue and the vibration of the PZT can be transmitted to the fiber. In our experiment, the pulse width is 10ns and 1000 traces are recorded by a high-speed oscilloscope with 1GHz sampling rate. The pulse repetition rate is fixed at 10 kHz and total data acquisition time is 0.1s. The maximum sensing length of the phase OTDR system is related to the repetition rate of the pulse. The time interval among the pulses should be larger than the round trip time that the pulses travel in the sensing fiber to keep only one pulse inside the sensing fiber. For the 10kHz repetition rate, the detection range is around 10km which is determined by, where L is the sensing fiber length, c the speed of light, n the group index of the fiber and f the repetition rate of the pulse. The sensing length can be extended by launching pulses with a higher peak power. A stationary vibration event with the frequency 500Hz is tested firstly. The position is determined by the amplitude difference among the global wavelet spectrum of different time sequences. Compared to the small amplitudes of the global wavelet spectrum at the fiber sections without vibration, there will be some peaks corresponding to the frequency of vibration at the location under the disturbance. Figure 2 shows the contour plot of the global wavelet power spectrum along the fiber length for the stationary vibration detection of 500Hz using the digital low pass filter with different cutoff frequency.
Fig. 2 Contour plot of the global wavelet power spectrum along the fiber length for the stationary vibration detection of 500Hz (a) 200MHz low pass filter (b) 350MHz low pass filter.
Since the signal with the vibration information is in the IF range around 200MHz and the bandwidth of the detector is 350MHz the high frequency electrical noise should be removed in order to obtain a better SNR. As shown in Fig. 2, two low pass filters with cutoff frequency 200MHz and 350MHz are tested separately. We find that the SNR can be optimized by a digital low pass filter with 200MHz cutoff frequency compared to 350MHz which can remove the high frequency electrical noise above the IF frequency and retain the useful information simultaneously. Here the SNR of the location information is defined as an amplitude ratio between the signal peak amplitude and the background noise level SNR = 10*log (Asignal/Anoise). According to this definition, the SNR of the 200MHz cutoff frequency filter is 20.7dB compared to the the 350MHz case which is 18.1dB. The global wavelet transform spectrum at each point along the sensing fiber is obtained to determine the position by integral over the whole data acquisition time which can amplify the signal section and suppress the noise level.
The position profile of the 500Hz vibration event with 10ns pulse width using the 200MHz low pass filter is shown in Fig. 3 . Here the spatial resolution is defined by the average of the rise and fall time equivalent fiber length for the small section under the disturbance [14],then spatial resolution around 0.172m is achieved. The 20cm fiber section under vibration is also clearly identified by the full width half maximum (FWHM) in Fig. 3.
Fig. 3 Position profile of the 500Hz vibration event with 10ns pulse width using the 200MHz low pass filter.
Figure 4(a) is the scalogram of the 500Hz stationary vibration event at the vibration position. It shows the frequency evolution with time of the vibration event which keeps constant in this case during the data acquisition process. The scale value is 256 which mean there are 256 points corresponding to the half sample frequency after CWT. The frequency accuracy is about 5000Hz/256 = 19.5Hz. The amplitude of the scalogram at around 500Hz changes along the time because the amplitude fluctuation of the time-domain traces induced by the noises. From Fig. 4(a), we can also find a frequency spreading which is determined by the time-frequency resolution of the CWT method. The frequency spreading will cause the accuracy of frequency measurement to decrease. This problem could be solved by the wavelet ridge detection method which can determine the instantaneous frequency. The instantaneous frequency in Fig. 4(b) is extracted from the scalogram by wavelet ridge detection method. It provides more accurate frequency information for the further analysis in real applications. The inset graph in Fig. 4(b) shows the sinusoid electrical signal applied on the PZT cylinder generated by the function generator.
Fig. 4 (a) Scalogram of the 500Hz stationary vibration event at the vibration position (b) instantaneous frequency obtained by wavelet ridge detection.
For non-stationary vibration measurement, a sinusoid signal with frequency from 500Hz to 1 kHz and one sweep time 0.05s generated by the function generator is continuously applied on the PZT cylinder. Figure 5 is the position information of the sweep non-stationary vibration signal determined by the global wavelet power spectrum using the digital low pass filter with different cutoff frequency. As similar as the stationary vibration case, the SNR of the 200MHz and 350MHz filters are 19.8dB and 15.9dB respectively which means 4dB SNR improvement by using the 200MHz filter. We can also find the frequency range of the vibration event at the vibration position but the time-dependent information has already been missed in the global wavelet power spectrum.
Fig. 5 Contour plot of the global wavelet power spectrum along the fiber length for the non-stationary vibration detection of 500Hz-1000Hz sweep signal (a) 200MHz low pass filter (b) 350MHz low pass filter.
The Fig. 6 presents the frequency evolution with time increasing from 500Hz to 1000Hz linearly in a time section 0.05s. Figure 6(a) is the scalogram of the sweep signal corresponding to two sweep periods. Similar to the stationary case, the instantaneous frequency of the sweep signal is shown in Fig. 6(b) and the inset graph in Fig. 6(b) shows the sweep electrical signal driving the PZT cylinder.
Fig. 6 (a) Scalogram of the 500Hz-1000Hz sweep non-stationary vibration event at the vibration position (b) instantaneous frequency obtained by wavelet ridge detection.
5. Conclusion
The application of continuous wavelet transform to analyze the non-stationary signal in the distributed vibration sensing based on phase-sensitive OTDR system is demonstrated in this paper. The frequency evolution with time of a sweep sinusoid vibration signal is found from the scalogram through the CWT method. This reported time-frequency analysis scheme is proved to be a powerful tool to evaluate both the stationary and non-stationary vibration events using the phase OTDR system for damage detection of civil or mechanical structures.
Acknowledgments
The authors would like to thank the Canadian Institute for Photonic Innovations (CIPI) for the financial support. The authors would like to thank the support of TeraXion. Z.G.Qin would like to acknowledge the support from the China Scholar Council (CSC).
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