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Abstract
In a multi-channel interferometric fiber-optic sensor system using space-division multiplexing (SDM) and phase-generated-carrier (PGC) demodulation, the phase delay and phase modulation depth fluctuation of each channel will affect the amplitude consistency and harmonic distortion of the demodulation results. In this paper, an improved demodulation scheme based on the Goertzel algorithm is proposed to calculate the multi-channel phase delay and phase modulation depth and to compensate for their fluctuations simultaneously. First, the carrier's 1st to 6th harmonic amplitudes in the interference fringe are extracted using the Goertzel algorithm. Then, the phase delay is calculated using the real and imaginary components of the 1st harmonic amplitude. The phase modulation depth is calculated with a combinatorial operation of the 1st to 6th harmonic amplitudes. In addition, a reference channel is introduced to implement phase delay and modulation depth feedback control. The experimental results demonstrate that the improved scheme can effectively suppress the harmonic distortion and improve the amplitude consistency of multi-channel interferometric fiber-optic sensors with low resource consumption.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, multi-channel interferometric fiber-optic sensors have been widely used in sonar systems [1], structural health monitoring [2], microseismic monitoring [3,4], and many other fields. Several multiplexing technologies have been proposed, such as time, frequency, coherence, space, and wavelength-division multiplexing [5–7]. Among them, the space-division multiplexing technology (SDM) has the advantages of high array optical efficiency and low sensor-to-sensor crosstalk [8]. Phase-generated-carrier (PGC) is one of the most widely used demodulation schemes for SDM-based systems due to its high resolution, large dynamic range, and good linearity [9–11]. For this type of demodulation scheme, carrier phase delay (CPD) inevitably occurs due to the leading fiber, sensing fiber, photoelectric conversion, and circuit transmission [9]. The phase modulation depth (PMD) of each channel also deviates from the ideal value due to the frequency shift of the laser source, ambient temperature fluctuation, and manufacturing error. The imperfect CPD and PMD will lead to amplitude fading and harmonic distortion of the demodulation results [12–14] and cause poor multi-channel amplitude consistency of the SDM-based multi-channel sensor system [9].
Several methods to eliminate the influences of the imperfect CPD or PMD separately in the PGC scheme have been proposed yet [15–20]. However, these imperfections always coexist and need to be solved simultaneously [21,22]. Many methods have been proposed, for example, the orthogonal detection and AC-DC component extraction scheme [22], differential-cross-dividing-and-multiplying (DCDM) scheme [23], and machine learning [24]. In addition, the ellipse fitting algorithm [14,25–29] is a classical scheme, but it depends on historical data and requires a large amount of data to implement matrix operations [21,30]. L.S. Zhang [31,32] proposed an improved scheme based on amplitude normalization of orthogonal signal pairs. It is unsuitable for large-scale fiber-optic sensor systems because the Fast Fourier Transform (FFT) and Hilbert transform in the scheme requires many computational resources. J. Xie [33] proposed a quadrature multi-harmonic frequency down-conversion scheme. This scheme is complicated and resource-consuming due to calculating four pairs of quadrature harmonic components [21]. L.P. Yan [21] proposed another scheme by adopting the fundamental in-phase and quadrature harmonic components and their differential components. This scheme effectively reduces resource consumption, but five low-pass filters and three carrier harmonics are needed to achieve single-channel CPD and modulation phase depth compensation. From the above, although some progress has been made to eliminate the influences of CPD and PMD fluctuations, for the large-scale fiber-optic sensor systems, it would be much better if the complexity and resource consumption could be reduced.
This paper presents an improved phase demodulation scheme based on the Goertzel algorithm for SDM-based multi-channel interferometric fiber-optic sensor system. We estimate and compensate the CPD and PMD simultaneously using the Goertzel algorithm, which has the characteristics of high parameter calculation accuracy, effectively suppressing the influences of CPD and PMD. And this scheme has the advantage of low resource consumption, so it is especially suitable for SDM-based systems. The principle of the proposed scheme is introduced in Section 2. The experiments and analysis are presented in Section 3. The experimental results demonstrate that the improved scheme can effectively suppress the harmonic distortion and improve the multi-channel amplitude consistency of the interferometric fiber-optic sensor system.
2. Principle
2.1 Effects of CPD and PMD on the PGC scheme
In an SDM-based multi-channel interferometric fiber-optic sensor system using PGC demodulation, a frequency modulation signal is introduced to the narrow linewidth laser to effectively obtain the interferometric fiber-optic sensor's disturbance. The interference signal [15] of a single-channel interferometric fiber-optic sensor is expressed by:
In the traditional PGC-ACTAN scheme [10], the interference signal Vi(t) is multiplied by the first sine and second cosine harmonic components of the carrier signal. Then, a pair of in-phase and quadrature components are obtained through low-pass filtering, which can be expressed as:
Then, the demodulated result is obtained by:
It is directly related to the θi and Ci. For optimum harmonic distortion and multi-channel amplitude consistency in the sensing system, Ci needed to be maintained to 2.63 rad, and θi needed to be kept to kπ (k = 0, ±1, …).
In practical applications, the CPD is caused by the time delay of the optical path [12], circuit transmission, and photoelectric conversion. For an SDM-based system, the transmission fiber lengths of each channel are very close, and each channel uses the same photoelectric conversion circuit, analog-to-digital conversion circuit, and signal processing flow. Therefore, the CPD of each channel can be regarded as being the same.
The Ci can be expressed as:
where n is the fiber's refractive index, Δl is the arm length difference of the fiber-optic interference sensor, c is the velocity of light, and Δν represents the frequency shift amplitude of the laser. Ci is greatly affected by the ambient temperature fluctuation and the amplitude of the modulation voltage [9], so it is difficult to stabilize at the ideal values. For the SDM-based sensor array, n of each channel is different due to the different installation positions, and Δl of each channel is different due to the manufacturing process. Therefore, the PMD of each channel needs to be estimated and compensated independently.2.2 Improved phase demodulation scheme based on the Goertzel algorithm
This paper proposes a CPD and PMD estimation and compensation scheme for multi-channel fiber-optic sensor demodulation based on the Goertzel algorithm, called PGC-GOERTZEL, shown in Fig. 1.
Fig. 1. Schematic of PGC-GOERTZEL scheme. LPF, low-pass filter; DAC, digital-to-analog converter; PID, proportion integration differentiation; LUT, lookup table; VREF, voltage reference.
Expanding Eq. (1) in terms of Bessel functions gives:
According to the Bessel recurrence relation, the PMD Ci can be computed directly with the following equation [34]:
According to Eq. (6), only the carrier's 1st to 6th harmonic amplitudes are required to calculate the PMD, not the entire amplitude spectrum. In such a case, a direct computation of the desired values is more efficient [38]. The Goertzel algorithm computes the spectrum evenly across the signal's bandwidth and can be efficiently tuned to analyze only specific frequencies [39]. The FFT algorithm requires (N/2)log2N multiplications and Nlog2N additions, while the Goertzel algorithm only requires 2N+6 real multiplications and 4N+8 real additions [40,41], where N is the length of the sampled data. In addition, the recursive feature of the Goertzel algorithm enables it to start calculation in the sampling process without waiting for data blocks to be completely stored, which can save a lot of resources in combination with the resource-sharing approach [42]. These characteristics of the Goertzel algorithm make it especially suitable for calculating multi-channel CPDs and PMDs.
The difference equations describe the realization of the Goertzel algorithm:
In addition, we can see that the 1st harmonic of the carrier from Eq. (5) is:
Due to the REFI being placed on a vibration isolation platform, sin[φ0(t)+φ00] can be regarded as a constant. θ0 can be obtained by the following expression, according to the properties of the Goertzel algorithm [38]:
Since the transmission fiber length of each channel are very close, and the same circuit is used, it can be considered that θi =θ0. Thus far, the CPD compensation coefficients η1 = cos(θ0) and η2 = cos(2θ0) of each channel can be obtained. Then, CPD is controlled to kπ (k = 0, ±1, …) by changing the carrier phase.
Finally, the demodulation result of the improved phase demodulation scheme can be expressed as Eq. (10). Then, φi(t) can be obtained by phase unwrapping and high-pass filtering.
From the above analysis, the Goertzel algorithm extracts the 1st to 6th harmonics of the carrier in the interference fringe, which boasts low resource consumption. The proposed demodulation scheme can realize the synchronous calculation and compensation of multi-channel CPDs and PMDs.
3. Simulation, experiment results, and discussion
3.1 Simulation
In this subsection, the simulated results of the CPD and PMD calculation errors are presented to show the accuracy of the Goertzel-based method. A simulated interference fringe Vs was performed using the following settings, and Gaussian white noise was added to it to simulate A/D quantization noise. The sampling frequency fs was set to 200 kHz. The carrier frequency f0 was set to 5 kHz, consistent with the experiment in the following subsection. The DC component A and the AC component B of Vs were set to 1 V and 2 V, respectively. The phase signal to be measured φ(t) was set to 1 × cos(2π×100 × t). The initial phase φ0 of the interferometer was set to π/2. The length N of the sampled data was set to 1024.
A 1024-point FFT algorithm was compared with the Goertzel-based algorithm in the simulation. First, PMD was set to 2.63 rad, and CPD was changed within the range of (0, 2π) rad with a step of 0.01 rad. The simulation result is shown in Fig. 2(a). The errors between the set CPDs and the calculated CPDs are both lower than ±0.0005 rad for the FFT algorithm and Goertzel-based algorithm. Then, CPD was set to 0.2 rad, and PMD was changed within the range of (2.4, 2.8) rad with a step of 0.01 rad. As shown in Fig. 2(b), the errors between the set CPDs and the calculated CPDs are bose lower than ±0.004 rad for the two algorithms we used. Based on the simulation results, the proposed Goertzel-based algorithm can accurately calculate CPD and PMD, with an accuracy close to the FFT algorithm. In subsection 3.3, we will demonstrate that the Goertzel-based algorithm has the advantage of low resource consumption by implementing these algorithms in FPGA.
3.2 Influence of accompanied optical intensity modulation
The theoretical analysis and simulation are based on the ideal light source, however, accompanied optical intensity modulation (AOIM) is inevitably due to the direct frequency modulation [22]. Therefore, the interference signal Vi(t) in Eq. (1) can be modified to [43]:
According to Eq. (11) and Eq. (12), AOIM may make PGC-GOERTZEL invalid or inaccurate when m is sufficiently large. In our experiment, a low-noise single-frequency fiber laser (Koheras BASIK X15, NKT Photonics, Birkerød, Denmark) was used as the light source. Its center wavelength is 1550.12 nm, Lorentzian linewidth is less than 0.1 kHz, fast wavelength modulation speed up to 20 kHz, max phase noise is better than 0.3 µrad/√Hz/m @ 100 Hz, and RIN level is better than −135 dBc/Hz @ 10 MHz. The laser includes the capability to use an external electrical signal to modulate the emission wavelength using a Piezo and can operate in current mode or power mode. The power mode is a better choice to reduce AOIM, which keeps the output power of the fiber laser at a fixed level. In this mode, m was tested to evaluate the effect of AOIM. The system configuration is shown in Fig. 3(a).
Fig. 3. (a) Experimental setups for AOIM testing. (b) Modulation signal Vm, the output intensity of laser V0, interference fringe V1, and bandpass filtered signal of V0. (c) Calculation error of CPD. (d) Calculation error of PMD.
The arm length difference of the Michelson interferometer was set to 1 m (same as FOA in subsection 3.3), and the modulation frequency was set to 5 kHz. V0 was used to record the output intensity of the laser, and V1 was used to record the interference fringes to ensure that the PMD of the Michelson interferometer was stable at 2.63 rad. The results are shown in Fig. 3(b). We can see that the amplitude of the modulation signal Vm is 0.77 V. Under this condition, the average value of V0 is 3.185 V. The first-order AOIM signal BPF(V0) was extracted from V0 through a bandpass filter with cutoff frequencies of [4 kHz, 6 kHz]. Its peak-to-peak value is 0.0012 V, so the m is about 0.0002 using the m calculation method [22,44]. Therefore, the AOIM of Koheras BASIK X15 laser in power mode is very insignificant.
Based on the m test results, we have analyzed the effect of the AOIM on the calculation error of CPD and PMD, using the same simulation strategy as in subsection 3.1, as shown in Fig. 3(c) and Fig. 3(d). For the Koheras BASIK X15 (m = 0.0002), the calculation error of CPD is less than ±0.0005 rad in the range of (0, 2π) rad. This result is consistent with the ideal laser (m = 0). The PMD calculated value is still high-precision. The calculation error of PMD is in the range of (0.001, 0.004) rad and increases with PMD, slightly increased compared to the ideal laser. This phenomenon is because the AOIM and modulation carrier have the same frequency, leading to an error in V i n. Based on the characteristics of the Bessel function and Fourier Transform, multiplication at the same frequency will increase the result of Eq. (6).
Based on the simulation results and the theoretical analysis in Refs. [43,44], the AOIM of Koheras BASIK X15 laser hardly affects the performance of the PGC-GOERTZEL scheme. In future research, we will improve the ability of PGC-GOERTZEL to eliminate the effects of AOIM, making PGC-GOERTZEL can be applied to more lasers.
3.3 Experimental setup
An SDM-based multi-channel interferometric fiber-optic sensor system was constructed to verify the performance of the PGC-GOERTZEL scheme, as illustrated in Fig. 4. Three identical interferometric fiber-optic accelerometers (FOAs) were glued on the surface of the aluminum plate using marble adhesive (Hercules, Keda Marble Protective Materials Co., Ltd., Wuhan, China. E-modulus: ≥3000 MPa). They are located at the three vertices of an equilateral triangle with a side length of 200 mm. The FOA is based on an unbalanced Michelson interferometer with an arm length difference of 1 m. Inside the FOA, two clamped metal diaphragms with the supported mass and movable lid constitute the sensing element of the accelerometer [45]. The sensing arm of the interferometer is wrapped around the surface of the upper moving lid and the lower fixed lid with a certain prestress. When the sensor is accelerated, the supported mass will have a displacement relative to the base and make the metal diaphragm deform, thus causing compression or extension of optical fibers wrapped around the surface of the upper and lower lids. This effect produces a change in the length of optical fibers, leading to a phase shift in the optical fiber interferometer. The aluminum plate was fixed through bolts on the vibration exciter (4808, Brüel & Kjaer Sound & Vibration Measurement, Virum, Denmark). A signal generator and a power amplifier were used to drive the vibration exciter. A REFI was placed on a vibration isolation platform, and its fiber length and arm length difference were the same as those of the FOA. Koheras BASIK X15 laser was used as the light source. D/A conversion was performed with 14-bit resolution and 125 MSPS update rate. A/D conversion was performed with 16 channels simultaneously sample inputs, 16-bit resolution, and 200 kSPS.
Fig. 4. Experimental configuration of the multi-channel interferometric fiber-optic sensor system. ISO, fiber laser isolator; PD, photodetector; FC, fiber coupler; FOA, fiber-optic accelerometer; FPGA, field programmable gate array; A/D, analog-to-digital; D/A, digital-to-analog; SignalGen, signal generator; PA, power amplifier; VE, vibration exciter.
The Goertzel-based algorithm was carried out in a Xilinx FPGA evaluation board (ZCU102, Xilinx, CA, USA) by using the resource-sharing approach [42], N in Eq. (8) was set to 1000, so the CPD and PMD can be updated in 5 ms, k in Eq. (8) was set to 25, 50, 75, 100, 125, and 150 to calculate Vn. In the LUT of compensation coefficients γ1 and γ2, J1(2.4∼2.8) and J2(2.4∼2.8) are stored in two ROMs with a depth of 256, so the accuracies of γ1 and γ2 can reach 4.3 × 10−4 and 1.8 × 10−4, respectively. In addition, 1024-point FFT algorithm using Xilinx LogiCORE IP and the time-division locking (TDL) algorithm proposed in Ref. [9] were implemented in the same hardware for comparison. Details about the hardware resources utilized are reported in Table 1, which indicates the low resource consumption advantage of the proposed Goertzel-based algorithm. FFT algorithm needs to wait for the data block to be completely stored before starting the calculation, and it requires a lot of multiplication, so more DSP slices and BRAM are required. TDL algorithm contains four additional FIR filters, thus requiring more BRAM to store filter coefficients and more DSP for convolution. Therefore, the Goertzel-based algorithm is advantageous for large-scale SDM-based cost fiber-optic sensor systems.
Table 1. Hardware Resources Utilized (Single-Channel Demodulation Unit)
3.4 Multi-channel CPD and PMD stability
In this subsection, we choose PGC-ARCTAN as the comparison scheme, it is the prototype of the PGC-GOERTZEL scheme, and many of its optimization schemes have been proposed [9,14,21,26,33]. We used a 100 Hz sinusoidal signal to drive the vibration exciter to emit vibrations and adjusted its amplitude until the peak-to-peak value of the sinusoidal signals detected by FOAs was close to 2 rad. The REFI were placed in a quiet, vibration-isolated environment. The demodulation results were recorded continuously for more than 720 minutes.
Figure 5(a) shows the PMDs of three FOAs using the PGC-ARCTAN scheme. We can see that PMDs can’t be stabilized at the ideal value of 2.63 rad during the test period. As the test time increases, the difference in PMDs of the three FOAs tends to increase. The standard deviation of PMDs reaches 0.0291 rad, 0.0157 rad, and 0.0161 rad, respectively, showing significant instability. Figure 5(c) shows the CPD of the REFI, which fluctuates around 0.603 rad (34.5°) with an amplitude of 0.01 rad (0.57°). These results indicate that it is necessary to stabilize PMDs and CPDs. Then, we adopted the PGC-GOERTZEL scheme for testing. We set PMD and CPD of the REFI to 2.63 rad and π rad, respectively. Figure 5(b) shows the test results of PMD. The mean value of PMD for REFI is 2.6296 rad, which is very close to the set value. Its standard deviation is 0.0005 rad, reaching the level of the previously reported single-channel PMD stabilization scheme [18,21]. The mean values of PMD for three FOAs are 2.6310 rad, 2.6273 rad, and 2.6331 rad, respectively. Their deviation from the REFI is due to the interferometer's 1 mm manufacturing error of FOAs. The standard deviations of PMD for three FOAs reach 5.5 × 10−4 rad, 5.2 × 10−4 rad, and 5.1 × 10−4 rad, respectively. Figure 5(d) shows the CPD of the REFI. It fluctuates near 3.1416 rad with an amplitude of 0.0017 rad (0.09°), which is better than the value of 0.003 rad in our previous research [9].
Fig. 5. (a) PMDs measured by PGC-ARCTAN. (b) PMDs measured by PGC-GOERTZEL. (c) CPD measured by PGC-ARCTAN scheme. (d) CPD measured by PGC- GOERTZEL.
The above results indicate that the PGC-GOERTZEL scheme can simultaneously measure and stable the CPDs and PMDs of multi-channel interferometric fiber-optic sensors with high accuracy.
3.5 Multi-channel amplitude consistency, THD, and SINAD
First, the multi-channel amplitude consistency, THD, and SINAD were tested and compared. Figure 6(a) and Fig. 6(b) show the amplitude peak-to-peak (PK-PK) of three FOAs using the PGC-ARCTAN scheme and PGC-GOERTZEL scheme, respectively. For the PGC-GOERTZEL scheme, the PK-PK mean values of three FOAs are 2.1125 rad, 2.1115 rad, and 2.1105 rad, and standard deviations are 0.0001 rad, 0.0001 rad, and 0.0002 rad, respectively. The PK-PK maximum deviation of three FOAs is 0.0024 rad. Compared with the results of the PGC-ARCTAN scheme, the amplitude consistency and stability of the multi-channel sensors are greatly improved. Figures 6(c)∼(e) and Figs. 6(f)∼(h) show the power spectral densities (PSDs) of the signal detected by the three FOAs using two different demodulation schemes, respectively. For the PGC-GOERTZEL scheme, the THDs are 0.05%, 0.07%, and 0.08%, respectively, and each is significantly lower than the results of the PGC-ARCTAN scheme. The SINADs are 66.02 dB, 63.10 dB, and 61.94 dB, respectively, about 30 dB higher than the results of the PGC-ARCTAN scheme.
Fig. 6. (a) PK-PK fluctuations, PGC-ARCTAN. (b) PK-PK fluctuations, PGC-GOERTZEL. (c)∼(e) PSD, PGC-ARCTAN. (f)∼(h) PSD, PGC-GOERTZEL.
The experimental results of multi-channel amplitude consistency, THD, and SINAD using PGC-ARCTAN and PGC-GOERTZEL are listed in Table 2 to make the comparison more obvious. This comparison indicates that the proposed PGC-GOERTZEL scheme can effectively suppress the influences of CPD and PMD, improve the multi-channel amplitude consistency, suppress harmonic distortion, and improve SINAD.
Table 2. Multi-channel Amplitude Consistency, THD, and SINAD
Next, similar experiments were repeated, mean SINAD of three FOAs under different PMDs and CPDs were estimated and compared. The PGC time-division locking (PGC-TDL) scheme [9], which based on the PMD and CPD calculation methods proposed in Refs. [15,18], was reproduced and compared with the PGC-GOERTZEL and PGC-ARCTAN. First, CPD was set to π rad, and PMD was changed within the range of (2.42, 2.78) rad with a step of 0.03 rad. Then, PMD was set to 2.63 rad, and CPD was changed within the range of (0, 2π) rad with a step of π/6 rad. As shown in Fig. 7, for the PGC-ARCTAN method, SINAD will degrade significantly with PMD deviating from 2.63 rad and CPD deviating from kπ (k = 0, 1, 2) rad. The proposed PGC scheme maintained the SINAD larger than 63 dB at all tested PMD and CPD, and this performance is close to PGC-TDL. These results indicate that the PGC-GOERTZEL scheme can effectively suppress the influences of CPD and PMD.
Fig. 7. Estimated SINAD for three methods under different phase delays and phase modulation depths. (a) Phase modulation depth. (b) Phase delay.
3.6 Resolution and dynamic range
The noise floor level determines a minimum detectable signal, which is a crucial index to evaluate the resolution of the system [26,46]. The demodulation system was placed in a quiet and vibration-isolated environment. The result is shown in Fig. 8(a). We can find that the noise floor level of the selected channel is about −114 dB (re 1 rad/√Hz) @ 100 Hz, which is close to the results of PGC-TDL [9] and PGC-ARCTAN-EKF [26].
The carrier frequency fm for the PGC technique is usually centered at one-quarter Nyquist, i.e. (fs/2)/4, the maximum signal frequency observed before overloading occurs is fmax ≤ fc/(2φ0) [4], where φ0 is the amplitude of the measured phase. In our experiment, fm was set to 5 kHz, which satisfies the above theory, so the maximum signal frequency fmax is (2500/φ0) Hz. A further experiment was carried out to measure the system's dynamic range in the frequency of (5, 500) Hz. Figure 8(b) shows the dynamic range in a 1 Hz bandwidth at 4% THD. The dynamic range can reach 125 dB @ 100 Hz, close to our previous research [10], and decreases as the signal frequency increases. It can be explained by increasing the useful harmonics of the input signal, which are arranged in fixed low-pass filter passbands [47].
The above results indicate that the PGC-GOERTZEL scheme has high resolution and large dynamic range.
4. Conclusion
In this paper, an improved multi-channel interferometric fiber-optic sensor demodulation scheme based on the Goertzel algorithm is proposed, which has the advantage of requiring low resource consumption. This scheme can realize the synchronous calculation and compensation of multi-channel CPDs and PMDs. The experimental results demonstrate that the improved scheme can effectively improve the multi-channel amplitude consistency and suppress harmonic distortion. Therefore, we believe that the proposed SDM-based multi-channel interference fiber-optic sensor, which demodulates based on the Goertzel algorithm, has excellent potential in the field of multipoint parameter monitoring, such as structural health monitoring and microseismic monitoring. The result also implies that the proposed PGC-GOERTZEL is promising in the many types of fiber-optic sensors based on interferometric configurations, such as fiber-optic ultrasound and ultra-precision displacement sensors, etc. In future research, we will improve the ability of PGC-GOERTZEL to eliminate the effects of AOIM.
Funding
National Natural Science Foundation of China (61875185, U1939207); Scientific Instrument Developing Project of the Chinese Academy of Sciences (YJKYYQ20210036); Shenzhen Science and Technology Innovation Program (JCYJ20190814110601663).
Acknowledgments
The authors thank YingBo Luo, Bing Lv, and Kun Cheng for their help with the sensor package and experiments.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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