IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 26, NO. 16, AUGUST 15, 2014
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A Cross-Correlation Method in Wavelet Domain for Demodulation of FBG-FP Static-Strain Sensors Wenzhu Huang, Wentao Zhang, Tengkun Zhen, Fusheng Zhang, and Fang Li
Abstract— Static-strain can be detected by measuring a crosscorrelation of two FBG-based Fabry–P´erot interferometers (FBG-FPs). This letter addresses applying the wavelet transform to cross-correlation processing in noise-contaminated FBG-FP reflection spectrums for crust deformation measurement. Since the wavelet transform has a unique feature of varying timefrequency resolution, cross-correlation processing in the wavelet domain is more robust with correlated noise and window size and is less sensitive to nonstationary data. We discuss the experimental results in respect of the factors that could influence the sensor resolution. In addition, a staticstrain resolution of 1.3 nε, higher than conventional cross-correlation method is obtained in the laboratory test by using this technique. Index Terms— Cross-correlation, wavelet transform, FBG-FP, static-strain, wavelength demodulation.
Fig. 1. The schematic configuration of the demodulation system. ISO, isolator; CP, coupler; CIR, circulator; PC, polarization controller; PD, photodiode; A/D, analog-to-digital converter; FG, function generator; PCA, piezoelectric ceramic amplifier.
I. I NTRODUCTION
F
IBER Bragg gratings (FBGs) and FBG based Fabry–P´erot interferometers (FBG-FPs) have found several applications in different fields owing to their capability of real-time, in situ, sensitive strain measurement, with the advantage of low cost, small size, fast response and immunity to electromagnetic interference [1], [2]. For this purpose, different interrogation systems have been developed so far, mostly based on broademission radiation sources combined to optical spectrum analyzers and filters. More recently, a scheme based on a narrow linewidth tunable laser and highprecision FBG-FPs is devised, achieving a higher sensitivity level for static-strain measurement [3]–[5] which is required in seismic and crust deformation measurement [6]. Unlike dynamicstrain detection, an extra reference is required for statics-train measurement, which is usually an additional sensor head identical to the strain sensor but free of strain [5]. Then the static-strain can be obtained by calculating the Bragg wavelength difference between the
Manuscript received February 8, 2014; revised March 24, 2014; accepted April 20, 2014. Date of publication June 5, 2014; date of current version July 24, 2014. This work was supported in part by the National Science Foundation of China under Grant 41074128, in part by the 863 Program of China under Grant 2013AA09A413 and Grant 2014AA093406, and in part by the Key Instrument Developing Project, Chinese Academy of Sciences, Beijing, China, under Grant ZDYZ2012-1-08-03. W. Huang, W. Zhang, and F. Li are with the Optoelectronic System Laboratory, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China (e-mail:
[email protected];
[email protected];
[email protected]). T. Zhen and F. Zhang are with the Institute of Electrical and Electronic Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2014.2327969
two FBGs or FBG-FPs from their spectra using a sort of algorithm, such as the centroid detection algorithm [7], the autocorrelation algorithm [8] and the cross-correlation (CC) algorithm [9]. Among them, the cross-correlation algorithm can determine the Bragg wavelength difference directly and exhibits good ability of suppressing random uncertainty [3]. Actually, cross-correlation processing is a traditional method for time delay estimation, whose principle is the same with the cross-correlation calculation for the Bragg wavelength difference. However, the performance of crosscorrelation method is sensitive to window size and correlated noise [10]. However, the environmental disturbance make the sensing signal of FBG based sensor contain much noise in actual application, which seriously influences the precision of demodulation. So an advanced demodulation scheme should consist of a good denoising method and a high-resolution wavelength demodulation algorithm. The cross-correlation algorithm is not suit to deal with non-stationary signal or noise. Recently, wavelet transform, which is proved to be a good time-frequency analysis method [11], have been developed to get a better denoising effect for FBG sensors. But there are no such reports where the wavelet transform is used to calculate the wavelength difference of FBG. This letter presents a cross-correlation method in wavelet domain for demodulation of FBG-FP based static-strain sensors, which is used for crust deformation measurement. The schematic configuration of the demodulation system is shown in figure 1. The beam from the tunable fiber laser (NKT, linewidth 100 Hz) is spilt into a pair of FBG-FPs, one is for the sensing of strain, and the
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 26, NO. 16, AUGUST 15, 2014
Fig. 3.
Fig. 2. The reflectance spectrums of the referencing FBG-FP (FBG-FP 1) and the sensing FBG-FP (FBG-FP 2).
other is strain-free working as a reference for compensating the error due to temperature disturbance and light source drift. The polarization controller is used to eliminate the influence of polarization instability and the two reflectance spectrums of FBG-FPs are acquired by photodiode. The strain of the sensing FBG-FP is demodulated by a demodulation algorithm. This is a common method for static or low frequency strain sensing by distinguishing the resonance wavelength shift. In the demodulation algorithm, we apply the wavelet transform to cross-correlation processing in noisecontaminated FBG-FP reflection spectrums for static-strain measurement. Since the wavelet transform has a unique of feature of varying time-frequency resolution, cross-correlation processing in the wavelet domain has the following advantage compared with conventional demodulation methods: 1) more robust with correlated noise and window size; 2) less sensitive to non-stationary data; 3) higher static-strain demodulation resolution. II. T HEORY AND A NALYSIS Assuming that the reflectance spectrums of the referencing FBG-FP and the sensing FBG-FP are x(t) and y(t) respectively, which have correlated noise and a definite window size shown in figure 2. The part in bandwidth of the reflection peck of x(t) and y(t) is s(t). The other part and noise is z 1 (t), z 2 (t) respectively. And the delay inequality is w. The expressions of x(t) and y(t) are shown as below: x(t) = s(t) + z 1 (t) (1) y(t) = s(t − w) + z 2 (t) The cross-correlation technique estimates the time delay (the Bragg wavelength difference) w by maximizing the crosscorrelation function of the two spectrums over τ : R1 (τ ) = x(t)y(t + τ )dt (2) In fact, for the measurement of the Bragg wavelength difference, the measured reflectance spectrums contain signals at different moments. The two spectrums can be expressed as: x(t) = s(t) + z 1 (t) (3) y(t) = s(t − w(t)) + z 2 (t)
The scaling function and wavelet function of daubechies 4 wavelet.
where w(t) implies that the time delay is a function of t. To estimate the wavelength differential of two specific spectrums, a windowed cross-correlation technique must be used. To estimate the time delay w(t) at t = t0 , this technique first computes the windowed cross-correlation given by: t 0 +T
R2 (t0 , τ ) =
x(t)y(t + τ )dt
(4)
t0
where T denotes the window size, and then provides the estimate of w(t) by maximizing R2 (t0 , τ ) over τ . Clearly, the performance of windowed cross-correlation depends on the window size employed in computing the crosscorrelation. In other words, the performance depends on the time-frequency resolution of the window. Since it is well known that the wavelet transform has features of varying timefrequency resolution and high localization in time domain and frequency domain, it will be advantageous to employ it in the cross-correlation processing. Assuming ψ is an admissible function, the wavelet transform of function f (t) with respect to ψ is defined as: t −b )dt (5) f (t)ψ( Wψ f (a, b) = |a|−1/2 a In (5) ψ is called the mother wavelet, a (the scaling value) represents dilation and b (the delay) represents translation. A discrete wavelet is defined when only discrete values of scaling and translation are used, e.g. j/2
j
ψ j,k (t) = a0 ψ(a0 t − kb0 )
(6)
for certain a0 and b0 . If a is chosen to be 2 and b0 is chosen to be 1, and furthermore, {ψ j , k} are mutually orthogonal. Then ψ is called an orthogonal wavelet. Here, we chose daubechies four wavelet as ψ, whose scaling function and wavelet function are shown in figure 3. The daubechies wavelets, based on the work of ingrid daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support [12]. With each wavelet type of this class, there is a scaling function (called the father wavelet) which generates an orthogonal multi-resolution analysis. Daubechies wavelets are widely used in solving a broad range of problems, e.g. self-similarity properties of a signal or fractal problems, signal discontinuities, etc.
HUANG et al.: CROSS-CORRELATION METHOD IN WAVELET DOMAIN
Fig. 4.
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The results of cross-correlation and wavelet cross-correlation.
So, it is obvious that (2) can be interpreted as a wavelet transform: (7) R(τ ) = x(t)y(t + τ )dt = W y x(1, −τ ) where W y x(1, −τ ) denotes the wavelet transform of x with respect to y at scale 1 and translate –τ . Using the crosscorrelation transform, (7) can be expressed as dadb 1 (8) R(τ ) = Wψ x(a, b)Wψ y(a, b + τ ) 2 cψ a where ψ is assumed to be a mother wavelet and cψ is a normalization constant depending on ψ. (8) implies that the cross-correlation between x and y can be computed from their wavelet transform with respect to ψ. If an orthogonal wavelet is employed, then (8) becomes Wψ x(m, n)Wψ y(m, n + τ ) (9) R(τ ) = m
n
Looking at the right hand of (9), for each fixed scaling level (i.e., fixed m) the summation over n is a regular discrete crosscorrelation. When (9) are applied to (3) for measurement of wavelength differential, the computed R(τ ) is location-dependent and window-size-dependent. In the wavelet domain crosscorrelation, noise exist only when the time domain and frequency domain between noise and effective signal overlap. But noise often differs from signal. So we can extract the accurate delay inequality and get a good noise suppression effect by using wavelet domain cross-correlation (WCC), which is suit to calculate the wavelength difference between the two FBG-FPs. Figure 4 shows the results of cross-correlation and wavelet cross-correlation from figure 2. Here, the measurement period is 10 seconds and the sampling frequency is 10 kHz. So the window size is 100000 sampling point, which has a great influence to the traditional cross-correlation. We can see that the peak value of CC is about 0.4 close to CC noise level and the one of WCC is about 0.8. So we can conclusion that the cross-correlation processing in the wavelet domain is more robust with correlated noise and window size and less sensitive to non-stationary data which implies that WCC is more suitable for calculating the wavelength difference.
Fig. 5. The demodulation algorithm based on cross-correlation in wavelet domain.
And a more accurate demodulation of the Bragg wavelength difference can be received by wavelet cross-correlation method ultimately. III. E XPERIMENT AND D ISCUSSION In the laboratory test each FBG-FPs is formed by writing two identical FBGs 20-cm distant in a single mode fiber. The parameters of the FBGs are: nominal center wavelength 1549.720 nm, bandwidth 0.22 nm and peak reflectivity 99.5%. The free spectral range (FSR) and the bandwidth of the FBG-FPs are 4.1 pm and 0.9 MHz respectively. Here, the measurement period is 1 seconds and the sampling frequency is 10 kHz. So the window size is 100000 sampling point. The demodulation algorithm based on cross-correlation in wavelet domain is shown in Figure 5. Firstly, Digital filtering is used to suppress the noise in noise-contaminated FBG-FP reflection spectrum. Then cross-correlation in wavelet domain is used to calculate the wavelength difference between the spectrums and suppression noise again. Gauss curve fitting is used to dispose the result of the cross-correlation in wavelet domain between the FBG-FPs. Finally we can get the wavelength shift by using peak detection method. For high-resolution static-strain sensing, an additional reference source is necessary and important. The consistency of sensing parameters and the external environment disturbance between sensing FBG-FP and referencing FBG-FP directly determines the veracity of final demodulation result. Here, the same FBG-FPs are packaged in a seal box and the seal box is hanging on the spring to suppress the low-frequency vibration. A 10-cm thick stainless steel tank is used to suppress environmental noise interference and maintain a relatively constant temperature, which is shown in Figure 6. In the laboratory test, we put the stainless steel tank in the basement, where the environment is quiet and the temperature is relatively constant. The demodulation system is put in the monitoring laboratory. And the FBG-FPs are connected to the demodulation system by a 100 m long armored optical cable.
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TABLE I T HE S TATIC -S TRAIN R ESOLUTION OF D IFFERENT M ETHODS
the wavelength difference directly has a 5.1 nε (equal to 0.0061 pm standard deviation) static strain resolution, which is shown in table I. IV. C ONCLUSION
Fig. 6.
The scheme of laboratory test.
This letter presents a cross-correlation method in wavelet domain for demodulation of highprecision FBG-FP based static-strain sensors. From the analysis of the principle and experiment results, we can draw a conclusion that crosscorrelation processing in the wavelet domain is more robust with correlated noise and window size and less sensitive to non-stationary data than traditional cross-correlation algorithm. And a higher static-strain resolution, 1.3 nε, is obtained in the laboratory test by using this technique, which implies that the proposed wavelet cross-correlation method has a good application prospect for geophysics applications. R EFERENCES
Fig. 7. The demodulation results of wavelet cross-correlation algorithm, traditional cross-correlation algorithm, and gauss curve fitting algorithm.
The demodulation results are compared with traditional cross-correlation algorithm, which is shown in Figure 7. We can see that the results by different methods are with the same change trend. But the demodulation algorithm based on cross-correlation method in wavelet domain has a higher accuracy of static-strain demodulation than traditional crosscorrelation algorithm and gauss curve fitting algorithm, which implies that cross-correlation processing in the wavelet domain is more robust with correlated noise and window size and less sensitive to non-stationary data because of its unique of feature of varying time-frequency resolution. The proposed demodulation algorithm has a 1.3 nε (equal to 0.0015 pm standard deviation) static-strain resolution, while the traditional cross-correlation algorithm has a 3.3 nε (equal to 0.0041 pm standard deviation) static strain resolution and the gauss curve fitting algorithm which is used to calculate
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