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Abstract
An improved phase generated carrier arctangent demodulation algorithm based on harmonic mixing and phase quadrature technology (PGC-Arctan-HP) is proposed in this paper, which can eliminate the effects of modulation depth shift, carrier phase delay, and light intensity disturbance (LID) on the demodulation results. The simulation results are consistent with theoretical analysis, and indicate that the PGC-Arctan-HP algorithm can achieve optimal demodulation compared with other demodulation algorithms. The results of interferometric experiments show that the demodulated waveforms of the improved algorithm are relatively stable with an amplitude error of 0.0294%. The total-harmonic-distortion (THD) and the signal-to-noise-and-distortion (SINAD) can reach -60.0286 dB and 59.5388 dB.
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1. Introduction
Fiber optic interferometric sensors (FOISs) have become the sensing part of precision measurement systems such as geophones, hydrophones, and fiber optic gyroscopes due to their high sensitivity, strong anti-interference ability, and large measurement range. They are extensively employed in fields such as pipeline monitoring, oil and gas exploration, and hydroacoustic detection [1–3]. In order to retrieve the measured signal, FOISs can transform external physical variables into phase changes of optical signals, turn those optical signals into electrical signals via photodetectors (PD), and then recover the phase changes via demodulation technology. Common phase demodulation techniques are mainly divided into two categories: heterodyne demodulation and homodyne demodulation. Heterodyne demodulation is more expensive and has a more sophisticated system as a result of the usage of frequency shifters than homodyne demodulation. The common methods of homodyne demodulation are coupler demodulation and phase generated carrier (PGC) demodulation. Between them, the PGC method is widely used due to the advantages of large measurement dynamic range, high resolution, and good linearity [4].
When using the PGC demodulation method, a high-frequency carrier modulation signal is introduced into the interferometer, and then the modulated interference signal is demodulated using a demodulation algorithm. The PGC differential-and-cross-multiplying (PGC-DCM) algorithm [5] and the PGC arctangent (PGC-Arctan) algorithm [6] are the most classical PGC demodulation algorithms. They mix the interference signal with the first and second harmonic signals of the carrier signal, and filter out the high-frequency components through low-pass filters. The PGC-DCM algorithm obtains the phase shift signal to be measured by performing differential cross multiplication on the filtered two signals. The demodulation result of this algorithm contains AC component of light intensity and modulation depth, which is easily affected by light intensity disturbance (LID) and modulation depth shift in practice. The PGC-Arctan algorithm directly divides the two filtered components, and then performs an arctangent operation to obtain the phase shift signal to be measured, which avoids the effect of LID. However, when the modulation depth deviates from 2.63 rad, the demodulation result has serious harmonic distortion. In addition, the carrier phase delay caused by signal transmission delay can also affect the demodulation effect of both algorithms in practical applications.
In order to suppress the effects of three factors on the demodulation results of the PGC algorithms, numerous improved PGC demodulation algorithms have been proposed. They are mainly divided into three categories: (a) Demodulation algorithms based on real-time monitoring and calibration, including algorithms that compute the modulation depth in real-time and calibrate through integral control feedback [7], and demodulation algorithms that correct nonlinear errors through filtering and ellipse fitting algorithms [8–12]. However, the calibration delays of the above algorithms introduce nonlinear errors and are too complex in hardware implementation. (b) Direct computational demodulation algorithms [13–18], which refer to the use of two orthogonal signals after filtering to do the relevant mathematical operations to eliminate the effects of interfering factors. Examples include the PGC-DSM-Arctan algorithm based on arctangent function and differential self-multiplication to eliminate the influence of three factors [13], the PGC-SDD algorithm that utilizes single differential to eliminate the LID [14], the SR-PGC algorithm that removes the effects of interfering factors by dividing the two filtered components by their differential signals [16], and the PGC-SDD-DSM algorithm based on single differential division and differential self-multiplication [18]. The differentiation and division parts of these algorithms introduce some distortion points that cause nonlinear distortions, and the integration part also generates nonlinear errors due to non-zero initial values. In addition, these algorithms only consider the case where the LID is a slow variable. (c) Demodulation algorithms based on constructing reference signals [19–22], which eliminate the effects of interfering factors by mixing the interference signal with other signals, such as A. N. Nikitenko et al. [19] mixed the interference signal with the sine and cosine components of carrier first and second harmonic signals to eliminate the influence of carrier phase delay. Y. Li et al. [20] obtained a function related to the modulation depth by dividing the result of mixing and filtering the interference signal with the carrier second harmonic signal by the result of low-pass filtering of the interference signal, and then calculated the value of the modulation depth. After operation, the influence of modulation depth was eliminated. Such algorithms have high reliability, but most of them only consider a single factor. All the above three types of algorithms have obvious defects, so it is necessary to propose a new demodulation algorithm to eliminate the influence of three factors and improve the accuracy of the demodulation system, as well as to meet the demodulation of high stability and practicality.
In this paper, an improved phase generated carrier arctangent demodulation algorithm based on harmonic mixing and phase quadrature technology (PGC-Arctan-HP) is proposed, which utilizes the phase quadrature property of carrier sine and cosine components to eliminate the carrier phase delay, utilizes the carrier first harmonic, second harmonic, third harmonic and fourth harmonic signals to mix with the interference signal to eliminate modulation depth and LID, and ultimately removes the effects of three factors on the demodulation results. The algorithm has high demodulation accuracy because its calculation process has no calculus operation and feedback link, which avoids the introduction of nonlinear errors. And the demodulation system has high stability when the interfering factors change. This paper is organized as follows. Section 2 describes the principle of PGC-Arctan-HP algorithm. The simulation analysis and comparison of the improved demodulation algorithm with the classical demodulation algorithm and other improved algorithms are carried out in Section 3 while taking various parameters into consideration. The demodulation performance of the PGC-Arctan-HP algorithm is experimentally investigated in Section 4 using an interferometric system. Section 5 offers a summary as a conclusion.
2. Principle
The schematic diagram of PGC-Arctan-HP demodulation algorithm is shown in Fig. 1. After introducing a high-frequency cosine carrier modulation signal, the interference signal detected by the photodetector can be expressed as
Equation (1) can be expanded according to the Bessel functions as
Multiply the interference signal with cos(ωct), sin(ωct), cos(2ωct), sin(2ωct), cos(3ωct), sin(3ωct), cos(4ωct), and sin(4ωct) to obtain eight mixing signals, and then filter out ωc and its higher-order harmonic components through low-pass filters to obtain P1(t) to P8(t):
According to the phase quadrature property of sine and cosine signals, add the square of P1(t) and P2(t), then extract the square root to eliminate carrier phase delay and obtain Q1(t). Repeat the above operations for P3(t) and P4(t), P5(t) and P6(t), and P7(t) and P8(t) to obtain Q2(t) to Q4(t):
Add Q1(t) and Q3(t), and add Q2(t) and Q4(t) to obtain
By the recursive property of the Bessel functions:
Equation (5) and Eq. (6) can be transformed into
Then divide R1(t) by R2(t) to get
Divide Q3(t) by Q2(t) to obtain:
At this point, W1(t) and W2(t) do not contain the light intensity AC component B and its disturbance 1 + mcos(ωnt), eliminating the influence of the LID on the demodulation result. Then, W1(t) and W2(t) are directly multiplied to eliminate the second-order and third-order Bessel functions, and the result is multiplied by 1.5 to obtain
Then, through square root operation, arctangent operation, phase unwrapping, and high pass filtering, the phase shift signal φ(t) to be measured can be obtained.
Based on the above principles, it can be seen that the demodulation result of the improved demodulation algorithm only contains the phase shift signal to be measured, which is no longer a function of interference DC/AC component, LID, carrier phase delay, and modulation depth. This avoids demodulation distortion caused by the LID, carrier phase delay, and modulation depth shift, and good demodulation can be expected from the PGC-Arctan-HP algorithm.
3. Simulation and analysis
In this section, in order to verify the effectiveness of the PGC-Arctan-HP algorithm, we simulated the PGC-Arctan-HP algorithm, two classical PGC algorithms (PGC-DCM [5], PGC-Arctan [6]), and some improved algorithms (PGC-DSM-Arctan [13], SR-PGC [16], PGC-SDD-DSM [18]) on the Matlab platform, and analyzed the stability of different demodulation algorithms in the presence of the modulation depth shift, the carrier phase delay, and the LID. Code examples of the PGC-Arctan-HP algorithm are given in the supplementary material Code 1 (Ref. [23]).
The basic simulation parameters are shown in Table 1. The DC/AC component of the interference signal is 1 V, the sampling rate is 16 Ms/s, the phase shift signal to be measured is a cosine signal with an amplitude of 10 rad and a frequency of 2000Hz, the carrier modulation signal frequency is 1 MHz, the low-pass filter is a FIR equal-ripple filter, the passband cutoff frequency is 0.35 MHz, the stopband cutoff frequency is 0.65 MHz, the passband ripple is 0.0001, and the stopband ripple is 120.
Table 1. Basic Parameters of Algorithm Simulation
3.1 Performance of demodulation algorithms with modulation depth shift
In theory, when the modulation depth C reaches 2.37 rad (max [J1 (C) J2 (C)]), the PGC-DCM algorithm has the best demodulation performance, and the optimal modulation depth for the PGC-Arctan algorithm is 2.63 rad (J1 (C) = J2 (C)). In practice, the modulator is affected by the environment and other factors, the modulation depth is difficult to stabilize at the optimal value, which will produce a shift within 1.5 rad, leading to demodulation errors. Therefore, we set variables with modulation depth C ranging from 1 to 3.5 rad and intervals of 0.5 rad, and chose two typical values (2.37 and 2.63) to simulate and compare the PGC-Arctan-HP algorithm with other demodulation algorithms.
The time-domain waveforms of some demodulation results are shown in Fig. 2. It can be seen that the demodulation amplitude of the PGC-DCM algorithm is smaller than the original phase shift signal, which is due to the J1 (C) J2 (C) contained in the demodulation result of the algorithm is less than 1. When C deviates from the optimal modulation depth, J1 (C) J2 (C) changes, causing amplitude changes in the demodulation result of the PGC-DCM algorithm, resulting in linear distortion in the demodulation result. Under the condition that J1 (C) is not equal to J2 (C), the demodulation waveform of the PGC-Arctan algorithm undergoes significant nonlinear distortion, while other improved algorithms have no obvious distortion.
The performance of these algorithms is usually evaluated in terms of amplitude relative error (Rerror), total-harmonic-distortion (THD), and signal-to-noise-and-distortion (SINAD). Where Rerror is defined as the ratio of the difference between the demodulation result and the amplitude of the original phase shift signal to the original signal. THD is defined as the ratio of the equivalent root-mean-square (RMS) amplitude of all harmonics to the fundamental frequency amplitude. SINAD is expressed as the ratio of the fundamental frequency power to the sum of the power of all noises and harmonics. Figure 3 shows the performance of different algorithms when the modulation depth changes, and it can be seen that the PGC-DCM algorithm cannot demodulate the original phase shift signal amplitude, but has lower THD with higher SINAD. The demodulation performance of the PGC-Arctan algorithm depends on the stability of C. When C is stabilized at the optimal value, the PGC-Arctan algorithm can achieve lower amplitude error, THD, and higher SINAD. The demodulation results of other improved demodulation algorithms can maintain relative stability when the modulation depth C changes, but the Rerror and THD are higher and the SINAD is lower. Under the same conditions, the maximum Rerror of the PGC-Arctan-HP algorithm is 1.15 × 10−7%, which is 1000 times less than the Rerror of the PGC-SDD-DSM algorithm. The THD is lower than -145 dB, which is a reduction of 30 dB compared to the PGC-DCM algorithm. The SINAD is as low as 118 dB, which is much better than that of other demodulation algorithms. In terms of the overall performance of demodulation, PGC-Arctan-HP has better demodulation performance than other demodulation algorithms, has high demodulation accuracy, and can maintain relatively stable demodulation when the modulation depth is shifted.
Fig. 3. (a) Rerror of demodulation results of various algorithms under different modulation depths. (b) THD of demodulation results of various algorithms under different modulation depths. (c) SINAD of demodulation results of various algorithms under different modulation depths.
3.2 Performance of demodulation algorithms with carrier phase delay
When the fiber optic interference system works in practice, there are delays in the processes of optical path propagation, photoelectric detection, and signal transmission, resulting in carrier phase delay between the interference signal and the mixing signal, which is superimposed on the demodulation signal and causes demodulation distortion. We set the modulation depth C to 2.63, and set the carrier phase delay to a variable from 0 to 2π to simulate each algorithm.
The demodulation waveforms at partial carrier phase delay values are shown in Fig. 4. When the carrier phase delay is 2π/5, 4π/5, and π, there is a phase difference of π/2 between the demodulated signal and the original signal. This is due to the fact that when the carrier phase delay is within the range of π/4 to π/2, 3π/4 to 5π/4, and 3π/2 to 7π/4, the results of mixing and filtering the carrier first and second harmonic signals with interference signal are opposite to the sign of the original phase shift signal, resulting in a phase shift of π/2 in the demodulation result. For fiber optic interferometric sensors, only the amplitude and frequency of the signal are generally considered, without the phase of the signal to be measured. When the carrier phase delay is not equal to 0, π, 2π, the PGC-DCM algorithm exhibits significant linear distortion, the PGC-Arctan algorithm exhibits nonlinear distortion, and the demodulation waveform is no longer sinusoidal, while the other improved demodulation algorithms do not have significant distortion in the waveform.
Figure 5 shows the performance of different demodulation algorithms when carrier phase delay exists. The results show that when the carrier phase delay exists, compared with other demodulation algorithms, the Rerror of the PGC-Arctan-HP algorithm can still maintain at the 10−7 level, with higher demodulation accuracy. Its THD is reduced by 30 dB and its SINAD is increased by 20 dB compared with the PGC-DCM algorithm. Moreover, when the carrier phase delay is changed, all the indexes of PGC-Arctan-HP algorithm can maintain high stability, and the demodulation effect of PGC-Arctan-HP is better than other demodulation algorithms.
Fig. 5. (a) Rerror of demodulation results of various algorithms under different carrier phase delays. (b) THD of demodulation results of various algorithms under different carrier phase delays. (c) SINAD of demodulation results of various algorithms under different carrier phase delays.
3.3 Performance of demodulation algorithms with light intensity disturbance
Light intensity disturbance (LID) is the third factor that affects the performance of demodulation algorithms. In practice, both optical power fluctuations and light source intensity noise are responsible for the generation of LID. The LID can be decomposed into a combination of a series of cosine signals. Therefore, in order to test the performance of the demodulation algorithms under different frequency LID, we set the modulation depth C to 2.63, the carrier phase delay to 0, the amplitude of LID m to 0.5, and the LID frequency f (f = ωn/ 2π) to 0, 10, 100, 1000, 10000, and 100000 Hz, and then performed the simulation.
The waveforms of the demodulation results are shown in Fig. 6. It can be seen that when the LID frequency is low, the demodulation algorithm is less affected. When the LID frequency is greater than 1000 Hz, the demodulation result of the PGC-DCM algorithm shows a significant amplitude distortion. When the LID frequency is at 10000 Hz and 100000 Hz, the demodulation waveforms of the SR-PGC algorithm and PGC-SDD-DSM algorithm will generate glitches, and even cause significant demodulation distortion. The PGC-DSM-Arctan algorithm also generates glitches at the LID frequency of 100000 Hz. This is because the differentiation and division stages of the above improved algorithms generate nonlinear errors when the frequency of the LID is large. The PGC-Arctan algorithm and the PGC-Arctan-HP algorithm proposed can maintain better demodulation stability under the effect of the LID.
Fig. 6. Demodulation results of various algorithms under different frequency light intensity disturbances.
We compared the performance of various algorithms under different frequencies of LID, and the results are shown in Fig. 7. It can be seen that compared with other demodulation algorithms, the PGC-Arctan-HP algorithm has smaller Rerror, lower THD, and higher SINAD, and the demodulation performance is more stable when the frequency of the LID changes.
Fig. 7. (a) Rerror of demodulation results of various algorithms under different frequency LID. (b) THD of demodulation results of various algorithms under different frequency LID. (c) SINAD of demodulation results of various algorithms under different frequency LID.
3.4 Stability of PGC-Arctan-HP algorithm with combined effect of three factors
To investigate the stability of the PGC-Arctan-HP algorithm under the combined effect of three factors, we first set the carrier phase delay to 2π/5, the LID frequency to 100000 Hz, and the modulation depth in the range of 1 to 10 rad with a variable step size of 0.02. Figure 8 displays the simulation results. The PGC-Arctan-HP algorithm maintains good demodulation stability when the modulation depth ranges from 1 to 4 rad, while nonlinear distortion occurs when it exceeds the range.
Secondly, the modulation depth was set to 3.5 rad, the LID frequency was set to 100000 Hz, and the carrier phase delay was set to a variable with a step size of π/100 within 0 to 2π. The simulation result (Fig. 9) shows that when the carrier phase delay is in the range of π/4 to π/2, 3π/4 to 5π/4, and 3π/2 to 7π/4, the demodulation result has a phase difference of π/2 to the original phase shift signal, but there is no waveform distortion.
Finally, we set the modulation depth to 3.5 rad, the carrier phase delay to 2π/5, and the LID frequency as a function of continuous variation f = 10i(i ranging from 0 to 5, with a step size of 0.01). The simulation results are shown in Fig. 10, and it can be clearly seen that the demodulation waveform of the PGC-Arctan-HP algorithm has no linear and nonlinear distortions and can maintain good stability when the LID frequency changes.
3.5 Dynamic range analysis of PGC-Arctan-HP algorithm
The dynamic range is defined as the logarithmic value of the ratio of the maximum detectable signal amplitude to the minimum detectable signal amplitude without significant distortion (Dh = 20log10(φmax/φmin) [dB]) [20]. The minimum detectable signal amplitude is usually described by background noise. The maximum detectable signal amplitude is limited by the carrier modulation signal angular frequency ωc and low-pass filter bandwidth ωb(ωb is the value of stopband cutoff frequency minus passband cutoff frequency). Suppose the phase shift signal to be measured is φ(t)=φ0cos(ωst), in which φ0and ωs are the amplitude and the angular frequency of the signal. To avoid the frequency aliasing, it should be satisfied (φ0 + 1)ωs < (ωc-ωb)/2. φmax = (ωc-ωb)/(2ωs)-1) decreases with increasing angular frequency ωs of the phase shift signal. Therefore, when ωc and ωb of the PGC demodulation system are determined, the dynamic range decreases with the increase of phase shift signal frequency. Under simulation conditions, there is no other noise except for established disturbances, so the minimum detectable signal is -230 dB. We set the modulation depth C to 2.63, without carrier phase delay and LID, and then obtained the dynamic range of the PGC-Arctan-HP algorithm at different frequencies, as shown in Fig. 11. It can be seen that as the frequency of the phase shift signal to be measured increases, the dynamic range of the improved demodulation algorithm gradually decreases, and this change tends to be gentle. When the frequency of the signal to be measured is 100 Hz, the dynamic range is 303.91 dB, and the maximum detectable signal amplitude is 73.91 dB (also 4960 rad). When the frequency of the signal to be measured is 4000 Hz, the dynamic range of the demodulation algorithm is 271.58 dB, and the maximum detectable signal amplitude is 41.58 dB (also 120 rad). However, the maximum detectable signal amplitude of existing typical demodulation schemes is about 55dB@100 Hz [13–18], in comparison, our improved algorithm obtains a gain of 18.91 dB with a larger dynamic range.
According to the simulation results, the PGC-Arctan-HP algorithm has high demodulation accuracy and stability. The PGC-DCM algorithm produces linear distortion due to the modulation depth, carrier phase delay, and LID. The PGC-Arctan algorithm exhibits nonlinear distortion due to modulation depth changes and carrier phase delay. Other improved algorithms exhibit significant waveform distortion when the frequency of LID is high, and the performance indexes such as Rerror, THD, and SINAD fluctuate significantly when the disturbance factors change. The PGC-Arctan-HP algorithm is able to maintain good demodulation stability in the range of modulation depth from 1 to 4 rad. There is no distortion in the demodulation waveform when carrier phase delay and LID are applied, and its performance indicators are relatively stable. In addition, compared with other demodulation algorithms, the PGC-Arctan-HP algorithm has the smallest demodulation amplitude error, the lowest THD, and the highest SINAD, with high demodulation accuracy, and it can maintain a large demodulation range in the high-frequency bands.
4. Experiments and results
To verify the performance of the PGC-Arctan-HP algorithm in practical applications, we built a demodulation system based on the Mach-Zehnder interferometer. According to the schematic of the experimental setup illustrated in Fig. 12(a) and the physical diagram of the system illustrated in Fig. 12(b), the laser emitted by a narrow linewidth semiconductor laser with a central wavelength of 1550 nm (CoSF-D-ER-M, Connet) passes through the isolator and depolarizer to reach the Y-waveguide phase modulator (KG-MIOC-15-300 M, CONQUER), and is divided into two channels. The two lasers pass through unequal-length interference arms to interfere with the optical coupler (OC). They are then photoelectrically converted into a voltage signal in a photodetector (PD, IAM-1003, Zi Guan), converted into a digital signal through a data acquisition (DAQ) card (PCIe-7820, BXIT), and finally sent to the PC for demodulation. Among them, the signal generator (AFG31051, Tektronix) sends a cosine signal with a frequency of 2000Hz to the semiconductor laser, and the modulated light passes through the interference arms with a difference of 1 m in arm length, generating a phase shift with a frequency of 2000Hz and an amplitude of 10 rad, which is used as the phase shift signal to be measured. The Y-waveguide phase modulator generates a high-frequency carrier modulated signal with an amplitude of 2.63 rad and a frequency of 1 MHz. The DAQ card acquisition rate is 31.25 Ms/s, and the signal is transmitted to a PC and downsampled to 16 Mbps using Matlab for demodulation processing.
Fig. 12. Mach-Zehnder interferometer experimental system. (a) Schematic of experimental setup. (b) Physical diagram of the experimental setup.
In the experiment, we set the modulation depth to 2.63 rad. But due to the unstable performance of the Y-waveguide modulator, the actual output amplitude of the carrier modulation signal fluctuated slightly around the set value of 2.63 rad. We set the output power of the laser to 13 mW, but the actual output optical power fluctuated slightly due to the light source noise. Meanwhile, some delays in the carrier modulation signal generation, optical transmission and DAQ acquisition process led to phase carrier delay. Therefore, the interference signals acquired included the comprehensive effects of modulation depth shift, LID, and carrier phase delay. Different demodulation algorithms were used to demodulate the interference signals, and the demodulated time-domain waveforms are shown in Fig. 13. The demodulated waveform amplitude of the PGC-DCM algorithm has significant errors with the phase shift signal to be measured. The demodulation results of PGC-Arctan algorithm and PGC-DSM-Arctan algorithm are affected by disturbance factors, which cause nonlinear distortion. The SR-PGC algorithm and PGC-SDD-DSM algorithm introduce nonlinear errors in the differentiation, division, and integration stages, which result in the unstable amplitude of the demodulated waveforms with breakpoints and glitches. In contrast, the demodulation result of the PGC-Arctan-HP algorithm proposed in this paper can effectively restore the phase shift signal to be measured.
The power spectral density (PSD) of different demodulation algorithms was calculated based on the experimental results, as shown in Fig. 14. The noise of the PGC demodulation system includes system background noise and demodulation noise. The system background noise includes the quantization noise of the DAQ card, photoelectric detection noise, and optical path noise. The demodulation noise is the noise caused by harmonic noise and other errors in the demodulation process. The SR-PGC algorithm and the PGC-SDD-DSM algorithm introduce nonlinear errors and harmonic components in the computation process, resulting in high system noise of -47 dB and -40 dB. The demodulation results of other algorithms contain a significant amount of harmonic noise. Among them, the PGC-Arctan-HP algorithm has the lowest noise level, and the PSD of the second harmonic noise is -66.28 dB. Based on the PSD, Table 2 shows the performance of different demodulation algorithms. It can be seen that the PGC-Arctan-HP algorithm has the smallest demodulation error (0.0294%), the lowest THD (-60.0286 dB), and the highest SINAD (59.5388 dB). Compared with the PGC-DCM, PGC-Arctan, PGC-DSM-Arctan, SR-PGC, and PGC-SDD-DSM algorithms, obtaining a SINAD gain of 30.7616 dB, 21.6386 dB, 20.1913dB, 28.0276 dB, and 36.4926 dB, respectively.
Table 2. Performance of Different Demodulation Algorithms
5. Conclusion
In this paper, we propose an improved demodulation algorithm (PGC-Arctan-HP) based on the research of classic PGC algorithms. PGC-Arctan-HP demodulation algorithm mixes the modulated interference signal with the sine and cosine components of the carrier to eliminate the carrier phase delay. The influence of modulation depth and LID are eliminated by utilizing the third and fourth harmonics of the carrier and the recursive property of the Bessel functions. The algorithm improves the accuracy and stability of the demodulation system. Different from current algorithms, the proposed algorithm can eliminate the effects of three factors simultaneously without performing calculus operations or adding additional phase compensation, and does not introduce other demodulation errors. It is simple, feasible, and conducive to real-time implementation.
The demodulation performance of the improved demodulation algorithm and current algorithms is compared through simulation. The results show that the PGC-Arctan-HP algorithm has optimal demodulation performance under the three kinds of disturbances, respectively. It is verified that the algorithm has good demodulation stability under the combined effect of the three factors, and the simulation results also show that the PGC-Arctan-HP algorithm can maintain a large dynamic range in the high-frequency bands. Finally, we apply the demodulation algorithms to the fiber optic interferometric system. The experimental results show that the PGC-Arctan-HP algorithm can accurately demodulate the phase shift signals to be measured with high demodulation accuracy, high stability in the time domain, and better performances compared with other typical algorithms. Therefore, the demodulation system based on this algorithm can significantly improve the performance of fiber optic sensors. Our next work will try to apply the PGC-Arctan-HP algorithm to fiber optic geophone systems and distributed fiber optic sensing systems.
Funding
National Natural Science Foundation of China (41704173); Science Foundation of China University of Petroleum, Beijing (2462020YXZZ025); Industry-university-research Innovation Fund for Chinese Universities (2020HYA08001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Matlab files (.m files) related to the algorithm simulation are provided in Code 1, [23].
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