Real-time self-calibration PGC-Arctan demodulation algorithm in fiber-optic interferometric sensors

Zhiyu Qu; Shuai Guo; Changbo Hou; Changbo Hou; Jun Yang; Jun Yang; Libo Yuan; Libo Yuan

doi:10.1364/OE.27.023593

1. Introduction

Fiber-optic interferometric sensors (FOISs) such as Mach-Zehnder, Michelson, Fabry-Perot and Sagnac interferometers are applied to both military and civilian fields due to their high sensitivity, large dynamic range, immunity to electromagnetic interference, etc [1–5]. The demodulation algorithms in FOISs directly affect the performance of the demodulation system. The typical demodulation algorithms mainly include active homodyne, passive homodyne using a 3 × 3 optical fiber coupler, general heterodyne method and phase-generated carrier (PGC) [6, 7]. PGC has many advantages such as wide dynamic range, good linearity, high sensitivity, high phase measurement accuracy and sensor multiplexing [8–10]. So PGC has become the most widely used demodulation method in FOISs [11, 12].

The differential-and-cross-multiplying approach of phase-generated carrier (PGC-DCM) demodulation algorithm and the arctangent approach of phase-generated carrier (PGC-Arctan) demodulation algorithm are the conventional PGC algorithms [13, 14]. PGC-DCM demodulation algorithm is influenced by the light intensity disturbance (LID), the carrier phase delay, nonideal performance of the low-pass filter and the drift of phase modulation depth C [15–18]. These factors will cause the nonlinear distortion of the demodulation results. The PGC-Arctan algorithm proposed later can eliminate the influence of LID [15]. However, the other factors such as carrier phase delay and the drift of phase modulation depth C still lead to nonlinear distortion. Especially when C value deviates from 2.63 rad, the PGC-Arctan algorithm will have serious harmonic distortion [13, 19]. C value is determined by the amplitude of the sinusoidally carrier and the parameters of phase modulator [20, 21]. These parameters will vary with laser wavelength, temperature and humidity [22, 23]. So it is necessary to estimate and calibrate fluctuating C value in real time in practice.

In order to solve the problem of nonlinear distortion in PGC demodulation algorithm, the scholars have proposed several methods. Some algorithms only focused on C, for example, J. He et al. presented a PGC demodulation algorithm based on arctangent function and DSM [24]. The algorithm has low harmonic distortion with a range of phase modulation depth from 1.5 to 3.5 rad. Nevertheless, the method can not obtain C value and calibrate it in real time. Y. W. Tong et al. proposed a PGC demodulation algorithm which can eliminate the influence of LID and the drift of C value [25]. The algorithm introduces a third harmonic carrier and a high-pass filter which make the calculation too complex to implement on the hardware. A. V. Volkov et al. presented a C evaluation and correction scheme [26]. The scheme uses the third and the fourth harmonic carrier to calculate C value and correct it based on the integral control feedback. But the algorithm can not eliminate the influence of the carrier phase delay and some other factors. The system requires a separate module to evaluate and correct C value which will increase the hardware complexity. Some other algorithms comprehensively considered all the factors such as LID and carrier phase delay et al. to reduce the nonlinear distortion. C. Ni et al. proposed an ellipse fitting algorithm (EFA) to compensate the nonlinear distortion caused by LID, carrier phase delay and the drift of C [27]. While the variation range of measured phase signal should be more than $π / 2$ rad. Otherwise, the demodulation results will be inaccurate or even incorrect. To solve this problem, L.P. Yan et al. used a triangular signal as an additional modulation signal [28]. The algorithm can still work when the variation range of measured phase signal less than $π / 2$ rad. These two algorithms have a common problem that C can not be calibrated to the optimal value to suppress nonlinear distortion further. In practical applications, we hope that the demodulation algorithm in high-precision optical fiber sensor systems can not only use EFA for nonlinear distortion suppression, but also estimate the C value and calibrate it to the optimal value, thus further eliminating the nonlinear distortion impact on the system. And we also hope that the demodulation algorithm has low computational complexity and it is easy to implement on hardware.

PGC-Arctan-SC demodulation algorithm is proposed in this paper. First, the algorithm uses EFA to suppress the nonlinear distortion caused by LID, the carrier phase delay, nonideal performance of the low-pass filter and the drift of C. Then, the accurateC value can be estimated jointly by the ellipse fitting parameters and C-related components. Last, PID module can realize closed-loop feedback control to calibrate C to the optimal value. So that the nonlinear distortion of the demodulation results can be suppressed further. The algorithm is easier to implement on the hardware and has high real-time performance because of its low computational complexity.

In this work, PGC-Arctan-SC demodulation algorithm is proposed to estimate and calibrate C while suppressing nonlinear distortion. In section 2, the principle of the conventional PGC-Arctan demodulation algorithm is introduced. In section 3, the novel algorithm which uses the elliptical parameters to estimate and calibrate C to suppress the nonlinear distortion is discussed. In section 4, the effectiveness of the proposed algorithm is validated by simulation. In section 5, the all-digital PGC demodulation system is implemented on embedded SoC platform (FPGA+ARM). The experiments and results are given.

2. The principle of PGC demodulation algorithm

2.1. PGC demodulation system description

The scheme for PGC demodulation system is shown in Fig. 1. The light beam emitted from a RIO laser source which is divided into two beams by Y waveguide. And the two beams enter the reference arm and the sensing arm respectively. The sensing arm is used to detect the vibration of the environment. The reference arm is not affected by the sensing signal. So that the optical path difference of transmitted light in the two arms will change when the external signal varies. The two light beams output from the two arms will interfere in the coupler. Then the interference signal is detected by a differential photodetector (PD). This interference signal is sampled by an analog to digital converter (ADC) controlled by a FPGA for further phase demodulation. A carrier signal is generated by FPGA and then converted to analog signal by a digital to analog converter (DAC). The output signal of DAC is applied to the Y waveguide to modulate the interference signal. FPGA transmits the system parameters to an ARM. ARM uploads the parameters to a PC and PC sends instructions to ARM to control the system.

Fig. 1 Scheme for PGC demodulation system.

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The interference signal [29–32] is expressed by

I (t) = A + B cos [C cos ω_{0} t + φ (t)],

where A and B are the DC offset and the AC amplitude, respectively. C is the phase modulation depth. ω₀ is the carrier frequency.

φ (t) = D \cos (ω_{s} t) + ψ (t)

is the sum of the measured phase signal and the environmental noise. D is the amplitude of the measured phase signal. ω_s is the frequency of the measured phase signal.

Fig. 2 Principle of PGC-Arctan demodulation algorithm. DDS: direct digital frequency synthesizer. MUL: multiplier. LPF: low-pass filter. DIV: division. ATAN: arctangent operation.

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2.2. PGC demodulation algorithm description

The principle of PGC demodulation algorithm is shown in Fig. 2. The interference signal is separately multiplied with a fundamental carrier and a second-harmonic carrier. After the high-frequency components are filtered out by two low-pass filters, a pair of non-strict quadrature components $I_{x} (t)$ and $I_{y} (t)$ are obtained

I_{x} (t) = - B J_{1} (C) sin φ (t),

I_{y} (t) = - B J_{2} (C) cos φ (t),

where

J_{1} (C)

and

J_{2} (C)

are the first-order and second-order Bessel functions with C respectively. Dividing

I_{x} (t)

and

I_{y} (t)

yields

I_{d i v} (t) = \frac{J_{1} (C)}{J_{2} (C)} tan φ (t) .

Then the arctangent operation is performed to derive the measured phase signal.

φ^{'} (t) = arctan [\frac{J_{1} (C)}{J_{2} (C)} tan φ (t)] .

In order to minimize the impact of Bessel function on the demodulation results, it is necessary to find an appropriate C value to make $J_{1} (C) = J_{2} (C)$ .

2.3. Selection of the optimal C

Figure 3(a) illustrates the dependence of $J_{1} (C)$ and $J_{2} (C)$ on C. There are three optimal values of C in the range from 0 to $2 π$ rad. When $C = 2.63$ rad, $C = 4.478$ rad and $C = 6.087$ rad respectively, the absolute value of first-order Bessel function and the second-order Bessel function are approximately equal.

Fig. 3 (a) The dependence of J₁ (C) and J₂ (C) on C. (b) The dependence of J₁ (C)/J₂ (C) on C.

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Figure 3(b) illustrates the dependence of $J_{1} (C) / J_{2} (C)$ on C. The relationship between $J_{1} (C)$ and $J_{2} (C)$ can be seen more clearly. Considering that the hardware system requires towork continuously for a long time, it is necessary to select a C value to make the demodulation system work steadily.

Fig. 4 The dependence of d(J₁ (C)/J₂ (C))/d(C) on C.

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The dependence of $d (J_{1} (C) / J_{2} (C)) / d (C)$ on C is shown in Fig. 4. When $C = 2.63$ rad, C has a minimum impact on the demodulation system because of the minimum change rate of $J_{1} (C) / J_{2} (C)$ . For this reason, 2.63 rad is usually chosen as the optimal operating value of C in hardware systems.

Fig. 5 PGC-Arctan-SC algorithm for suppressing nonlinear distortion and calibrating C value. EF: ellipse fitting. EC: ellipse correction. EFA: ellipse fitting algorithm. E, F, G, H, M: the parameters of ellipse fitting. h, k, a, b, δ: the parameters of ellipse correction. C₀: predefined optimal value of phase modulation depth. P: proportion. I: integral. D: differential. DAC: digital to analog converter.

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3. The principle of PGC-Arctan-SC demodulation algorithm

The principle of PGC-Arctan-SC demodulation algorithm is shown in Fig. 5. The EFA module and the PID module are introduced in PGC-Arctan demodulation algorithm. EFA is used to suppress the nonlinear distortion. PID uses the elliptical parameters which generated by EFA to estimate and calibrate the C value. The C value is mainly determined by the amplitude of the sinusoidally carrier and parameters of phase modulator. Y waveguide is used as phase modulator in this paper. The main parameter related to C

value of Y waveguide is half-wave voltage. The half-wave voltage will vary with laser wavelength, temperature and humidity, which leads to the drift of C value. So that it is necessary to estimate and calibrate C value in real time.

3.1. EFA description

In order to suppress the nonlinear distortion, EFA is introduced in PGC-Arctan demodulation algorithm as shown in Fig. 5.

The general expression of the nonlinear distortion [33–35] can be expressed as

I_{x} (t) = h + a cos ϕ (t),

I_{y} (t) = k + b cos [ϕ (t) - δ],

where h and k are the DC offsets, a and b are the AC amplitudes,

ϕ (t)

is the measured phase signal, δ is the phase difference between

I_{x} (t)

and

I_{y} (t)

. Equations (6) and (7) can be rewritten as following because of

\sin^{2} ϕ (t) + \cos^{2} ϕ (t) = 1

\begin{array}{l} I_{x}^{2} (t) - \frac{2 a cos δ}{b} I_{x} (t) I_{y} (t) + \frac{a^{2}}{b^{2}} I_{y}^{2} (t) + (\frac{2 a k cos δ}{b} - 2 h) I_{x} (t), \\ \begin{array}{l} + (\frac{2 a h cos δ}{b} - \frac{2 a^{2} k}{b^{2}}) I_{y} (t) + (h^{2} + \frac{a^{2} k^{2}}{b^{2}} - \frac{2 h k a cos δ}{b} - a^{2} {sin}^{2} δ) = 0. \end{array} \end{array}

The general form of the elliptic equation can be expressed as

x^{2} + E x y + F y^{2} + G x + H y + M = 0.

Comparing Eqs. (8) and (9) can obtain

{\begin{array}{l} E = - (2 a cos δ) / b \\ F = a^{2} / b^{2} \\ G = (2 a k cos δ) / b - 2 h \\ H = (2 a h cos δ) / b - (2 a^{2} k) / b^{2} \\ M = h^{2} + (a^{2} k^{2}) / b^{2} - (2 h k a cos δ) / b - a^{2} \overset{2}{\sin} δ \end{array}

The parameters $h, k, a, b and δ$ can be calculated by Eq. (10).

{\begin{array}{l} h = (2 F G - E H) / (E^{2} - 4 F) \\ k = (2 H - E G) / (E^{2} - 4 F) \\ a = {[(h^{2} + k^{2} F + h k E - M) / (1 - E^{2} / 4 F)]}^{\frac{1}{2}} \\ b = {(a^{2} / F)}^{\frac{1}{2}} \\ δ = \overset{- 1}{\cos} [- E / 2 \sqrt{F}] \end{array}

Therefore, the ellipse correction parameters $h, k, a, b$ and δ can be calculated by the ellipse fitting parameters E, F, G, H and M. The nonlinear distortion of I_x and I_y obey the normal distribution because they are generated independently and randomly [28]. The ellipse fitting parameters can be obtained by processing N groups of I_x and I_y data with the least-squares method. Substituting N groups of data into Eq. (9) yields the residuals as following

r_{i} = I_{x i}^{2} + E I_{x i} I_{y i} + F I_{y i}^{2} + G I_{x i} + H I_{y i} + M,

where

(I_{x i}, I_{y i}) (i = 1, 2, 3, ..., N)

are sampled data. Adding the squares of the residuals yields

P = \sum_{i = 1}^{N} r_{i}^{2} = \sum_{i = 1}^{N} {(I_{x i}^{2} + E I_{x i} I_{y i} + F I_{y i}^{2} + G I_{x i} + H I_{y i} + M)}^{2} .

The extremum of P satisfies the partial derivative equations as following

\frac{\partial P}{\partial E} = \frac{\partial P}{\partial F} = \frac{\partial P}{\partial G} = \frac{\partial P}{\partial H} = \frac{\partial P}{\partial M} = 0.

A matrix can be obtained from Eq. (14) as following

[\begin{array}{l} E \\ F \\ G \\ H \\ M \end{array}] = {[\begin{array}{l} \sum_{i = 1}^{N} I_{x i}^{2} I_{y i}^{2} & \sum_{i = 1}^{N} I_{x i} I_{y i}^{3} & \sum_{i = 1}^{N} I_{x i}^{2} I_{y i} & \sum_{i = 1}^{N} I_{x i} I_{y i}^{2} & \sum_{i = 1}^{N} I_{x i} I_{y i} \\ \sum_{i = 1}^{N} I_{x i} I_{y i}^{3} & \sum_{i = 1}^{N} I_{y i}^{4} & \sum_{i = 1}^{N} I_{x i} I_{y i}^{2} & \sum_{i = 1}^{N} I_{y i}^{3} & \sum_{i = 1}^{N} I_{y i}^{2} \\ \sum_{i = 1}^{N} I_{x i}^{2} I_{y i} & \sum_{i = 1}^{N} I_{x i} I_{y i}^{2} & \sum_{i = 1}^{N} I_{x i}^{2} & \sum_{i = 1}^{N} I_{x i} I_{y i} & \sum_{i = 1}^{N} I_{x i} \\ {\sum_{i = 1}^{N} I}_{x i} I_{y i}^{2} & \sum_{i = 1}^{N} I_{y i}^{3} & \sum_{i = 1}^{N} I_{x i} I_{y i} & \sum_{i = 1}^{N} I_{y i}^{2} & \sum_{i = 1}^{N} I_{y i} \\ \sum_{i = 1}^{N} I_{x i} I_{y i} & \sum_{i = 1}^{N} I_{y i}^{2} & \sum_{i = 1}^{N} I_{x i} & \sum_{i = 1}^{N} I_{y i} & \sum_{i = 1}^{N} 1 \end{array}]}^{- 1} \cdot (- [\begin{array}{l} \sum_{i = 1}^{N} I_{x i}^{3} I_{y i} \\ \sum_{i = 1}^{N} I_{x i}^{2} I_{y i}^{2} \\ \sum_{i = 1}^{N} I_{x i}^{3} \\ \sum_{i = 1}^{N} I_{x i}^{2} I_{y i} \\ \sum_{i = 1}^{N} I_{x i}^{2} \end{array}]) .

The ellipse fitting parameters E, F, G, H and M can be obtained from Eq. (15), then the ellipse correcting parameters $h, k, a, b and δ$ can be obtained from Eq. (11). Finally, the corrected signal can be obtained from Eqs. (6) and (7) as following

I_{x}^{'} (t) = cos [ϕ (t)] = [I_{x} (t) - h] / a,

I_{y}^{'} (t) = sin [ϕ (t)] = [\frac{I_{y} (t) - k}{b} - cos ϕ (t) cos δ] / sin δ,

where

I_{x}^{'} (t)

and

I_{y}^{'} (t)

are a pair of strictly quadrature signals. So that EFA can eliminate the nonlinear distortion effectively.

3.2. PGC-Arctan-SC demodulation algorithm description

PGC-Arctan-SC demodulation algorithm is proposed, which can use the elliptical parameters to calculate C value. Figure 5 shows the principle of estimating and calibrating C value. First, a, b and δ arrive at the C estimation block. Equations (2) and (3) can be rewritten as

sin φ (t) = I_{x} (t) / [- B J_{1} (C)],

cos φ (t) = I_{y} (t) / [- B J_{2} (C)] .

Equations (6) and (7) can be rewritten as

cos ϕ (t) = [I_{x} (t) - h] / a,

sin ϕ (t) = [\frac{I_{y} (t) - k}{b} - \frac{I_{x} (t) - h}{a} cos δ] / sin δ .

Comparing Eqs. (18) and (19) with Eqs. (20) and (21) respectively can obtain the identities as

sin φ (t) \equiv cos ϕ (t),

cos φ (t) \equiv sin ϕ (t) .

so that Eqs. (24) and (25) can be obtained from Eqs. (22) and (23) as

I_{x} (t) / [- B J_{1} (C)] \equiv [I_{x} (t) - h] / a,

I_{y} (t) / [- B J_{2} (C)] \equiv [\frac{I_{y} (t) - k}{b} - \frac{I_{x} (t) - h}{a} cos δ] / sin δ .

Equations (24) and (25) can be rewritten as

[a + J_{1} (C) B] I_{x} - J_{1} (C) B h \equiv 0,

\frac{cos δ}{a} I_{x} - [\frac{sin δ}{B J_{2} (C)} + \frac{1}{b}] I_{y} + \frac{k}{b} - \frac{h cos δ}{a} \equiv 0.

According to the undetermined coefficient method, Eq. (28) can be obtained from Eq. (26) as

{\begin{array}{l} a + J_{1} (C) B = 0 \\ - J_{1} (C) B h = 0 \end{array}

so that a and h can be expressed as

{\begin{array}{l} a = - J_{1} (C) B \\ h = 0 \end{array}

Equation (30) can be obtained from Eq. (27) as

{\begin{matrix} \frac{cos δ}{a} = 0 \\ - [\frac{sin δ}{B J_{2} (C)} + \frac{1}{b}] = 0 \\ \frac{k}{b} - \frac{h cos δ}{a} = 0 \end{matrix},

so that b, k and δ can be expressed as

{\begin{array}{l} b = \frac{- B J_{2} (C)}{sin δ} \\ k = 0 \\ δ = \frac{π}{2} + n π n = 0, 1, 2, 3... \end{array}

Equation (32) can be obtained from Eqs. (29) and (31) as

\frac{J_{1} (C)}{J_{2} (C)} = \frac{a}{b sin δ},

so that C value can be estimated in real time by a, b and δ. After estimating C value, its calculated value arrives at the error estimation block. Then the error signal

ε (t)

is obtained by subtracting the predefined optimal value C₀ from the current value.

ε (t) = C_{0} - C^{'} (t) .

Next, the output signal $U (t)$ of PID can be obtained using proportional calculation, integral calculation and differential calculation.

U (t) = k_{p} * ε (t) + k_{i} * \int ε (t) d t + k_{d} * \frac{d ε (t)}{d t},

where k_p, k_i, k_d are proportional coefficient, integral coefficient, differential coefficient, respectively.

U (t)

will return to the input unit of PID to adjust the input signal which realizes closed-loop automatic control. Afterwards, the input signal

C (t)

of PID is updated to

C^{'} (t)

.

C^{'} (t)

can be obtained as

C^{'} (t) = C (t) + U (t) .

Last, $C^{'} (t)$ is multiplied by the carrier signal generated by DDS. The multiplication result is converted to analog signal by DAC then applied to Y waveguide to modulate the interference signal.

Fig. 6 (a) Fixed step method for calibrating C value. (b) PID method for calibrating C value.

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Fixed step method is shown in Fig. 6(a). The C value is calibrated to 2.63 rad from 1 rad step by 0.01 rad. It has advantages of simple structure and stable operation. However, the working speed and accuracy of the fixed step method are related to step length. PID method is shown in Fig. 6(b). The accuracy of PID method is 0.01 rad. PID method can also be used for automatically calibrating C

to its optimal value. And PID method merely requires 4 seconds while fixed step method requires 163 seconds in the same accuracy condition. So PID method has higher speed than the fixed step method in the same accuracy condition.

According to the above analysis, PGC-Arctan-SC demodulation algorithm can use the elliptical parameters to suppress the nonlinear distortion and estimate C simultaneously. Then PID module can calibrate C to the optimal value rapidly to suppress the nonlinear distortion further.

4. Simulation

4.1. Effectiveness verification for EFA

EFA includes ellipse fitting part and ellipse correction part. In order to verify the effectiveness of EFA in suppressing nonlinear distortion, a comparison experiment was carried out. The demodulation system with EFA and without EFA were tested respectively. In the system of this paper, the sampling frequency f_s of the interference signal was 10 MHz, the carrier frequency f₀ was 1 MHz, the measured phase signal frequency f was 1 kHz. SINAD is used to evaluate the quality of the demodulation results [36]. C value took a step length of 0.01 rad, increasing from 0.01 to 5.00 rad.

As shown in Fig. 7(a), because of the existence of nonlinear distortion, the Lissajous figure of the components $I_{x} (t)$ and $I_{y} (t)$ is apparently an ellipse when C = 1 rad before using EFA, while it changes into a unit circle after using EFA. The elliptic parameters vary with C value which are listed in Table 1. h, k, a, b and δ are the ellipse fitting parameters. h^′, k^′, a^′, b^′ and $δ^{'}$ are the ellipse correction parameters. e and e^′ are the elliptical eccentricity of the Lissajous figure. Because the smaller the elliptical eccentricity is, the closer the Lissajous figure is to the circle. The elliptical eccentricities e^′ are all equal to 0. It is obviously that a pair of strict quadrature signals are obtained after EFA.

As shown in Fig. 7(b), when the system does not include EFA, SINAD achieves the maximum value 86.98 dB when $C = 2.63$ rad. When the system includes EFA, SINAD achieves the maximum value 83.41 dB which is less than 86.98 dB. That is because the correction operation causes the loss of system performance. But SINAD is varies slightly with the variation of the C value. According to Fig. 7(b), after ellipse fitting and correcting, SINAD is much higher and more stable than the system without EFA when the C value is in the range of 0 to 5 rad. The results can validate the effectiveness of EFA in suppressing nonlinear distortion.

Fig. 7 (a) Lissajous figure of I_x (t) and I_y (t). (b) The dependence of SINAD and C.

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Table 1. Effectiveness verification of EFA

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4.2. Evaluation of the optimal C

In actual working environment, C value will not only deviate from the optimal value, but also fluctuate randomly because of the random drift of the half-wave voltage. The dependence of SINAD on C with different random fluctuations is shown in Fig. 8(a). The random fluctuations are 0.0001 rad, 0.001 rad, 0.01 rad and 0.1 rad respectively. The maximum value of SINAD is stable nearby 2.63 rad at different random fluctuations. And the stable range of SINAD decreases with the increase of random fluctuation. The dependence of d(SINAD)/dC

on C with different random fluctuation is shown in Fig. 8(b). When random fluctuation is 0.1 rad and the C value is in the range from 1.65 to 2.96 rad, d(SINAD)/dC is approximately equal to 0. Therefore, the demodulation system with EFA has the optimal C working interval.

Fig. 8 (a) The dependence of SINAD on C with different random fluctuations. (b) The dependence of d(SINAD)/dC on C with different random fluctuations.

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In order to show the influence of different C values and the random fluctuation of C on demodulation performance, C value increases from 0.1 to 5 rad step by 0.01 rad, the random fluctuation amplitude increases from 0 to 0.01 rad step by 0.001 rad. The results are shown in Fig. 9. SINAD is relatively stable when C value is nearby 2.63 rad. At the same C value, SINAD deteriorates with the increase of random fluctuation amplitude. Therefore, though the demodulation system with EFA has the stable results in a range of C value, the random fluctuation of C value also has a great impact on the performance. In order to avoid the demodulation performance deteriorating caused by the excessive drift of C value and random fluctuation of C value, it is necessary to stabilize C at the optimal value in real time.

Fig. 9 The dependence of SINAD on C and different random fluctuation amplitudes of C.

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4.3. Verification for real-time estimation of C

Under the conditions set in the section 4.1, the results of $J_{1} (C) / J_{2} (C)$ and $a / (b \sin δ)$ vary with C value are listed in Table 2. a, b and δ are the elliptical parameters which can be obtained from EFA. $J_{1} (C) / J_{2} (C)$ is the ratio of the first-order and second-order Bessel functions with C. $a / (b \sin δ)$ is the estimated value of $J_{1} (C) / J_{2} (C)$ . The errors between the theoretical value and the estimated value are tiny according to Table 2. So the high precision C values can be obtained from elliptical parameters.

Table 2. Effectiveness evaluation of monitoring C

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5. Experimental results and discussion

In order to verify the effectiveness of PGC-Arctan-SC demodulation algorithm, an experimental device is constructed as shown in the Fig. 10. In this paper, Zynq-7100 SoC (XC7Z100, Xilinx) is used as the hardware development platform of the digital PGC demodulation system. XC7Z100 is consisted of Processing System (PS) part and Programmable Logic (PL) part. PS part integrates the software programmability of a dual-core ARM Cortex-A9 MP Core processor, with processing speed up to 1 GHz. This part sends instructions to FPGA and upload signal data to PC. PL part integrates the hardware programmability of a 28-nm FPGA Kintex-7, including 444K Logic Cells, 26.5Mb Block RAM, 2020 DSP Slices, 400 Maximum I/O Pins and 16 Maximum Transceiver Count, which can realize important analysis and hardware acceleration. This part fulfills the demodulation algorithm including multiplication module, low-pass filter module, ellipse fitting module and PID module, etc.

Fig. 10 Experimental setup.

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A four-channel PGC demodulation system is implemented on the hardware development platform. The system is consisted of the PGC modulation part and the digital PGC demodulation part. In the PGC modulation part, the carrier signal generated by DAC modulates the interference signal through Y waveguide modulator. The carrier signal amplitude can be controlled by ARM through network instruction. Then the interferometric signal emitted by Mach-Zehnder interferometer is converted to an electrical signal by a differential PD. And the electrical signal is sampled by ADC for demodulation. In the digital PGC demodulation part, the sampled digital signal is separately multiplied with a fundamental carrier and a second-harmonic carrier generated by DDS. Then the two signals pass through a low-pass filter for EFA. And the measured phase signal can be obtained by the arctangent operation.

5.1. Experimental verification for C value drift

In order to verify the necessity of real-time estimation and calibration of C, an experiment is designed to observe the drift of C value. As shown in Fig. 11, C value drifts with time in the actual working environment. If C value deviates from 2.63 rad too much in the PGC-Arctan demodulation system, it will cause serious harmonic distortion and even demodulation failure. Therefore, it is necessary to estimate C value and calibrate it to the optimal operation value in real time.

Fig. 11 The drift of C value with time.

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5.2. Experimental verification for PGC-Arctan-SC

In the hardware system, the measured phase signal frequency is 1 kHz, the sampling frequency is 10 MHz, the carrier signal frequency of the hardware system is 1 MHz. C is set to different values by ARM instruction. The output data of the low-pass filter and the data of the ellipse fitting are uploaded to PC by ARM. And The corresponding normalized Lissajous figures are shown in Fig. 12(a). As shown in Fig. 12(b), the ellipse is corrected into the circle with EFA in real time. The non-strict quadrature components $I_{x} (t)$ and $I_{y} (t)$ are corrected into a pair of strict quadrature components. The ellipse fitting parameters, the ellipse correcting parameters and the elliptical eccentricity are listed in Table 3. The elliptical eccentricity e drops first and rises later with the increasing of C value from 1.00 to 3.00 rad. When $C = 2.63$ rad, the elliptical eccentricity e is approximately equal to 0. After EFA, the elliptical eccentricity e^′ are all equal to 0 or approximately equal to 0. Therefore, EFA can improve the performance of the demodulation system when C deviates the optimal operating value.

Fig. 12 (a) Lissajous figure of I_x (t) and I_y (t) before real-time correction. (b) Lissajous figure of I_x (t) and I_y (t) after real-time correction.

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Table 3. Verification of hardware-implemented EFA

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SINAD of the hardware demodulation system without EFA and with EFA are compared and listed in Table 4. In both cases, the average SINAD rises first and drops later with the increase of the C value from 1.00 to 3.00 rad. And SINAD reaches to the maximum value when $C = 2.63$ rad. In general, the performance of the system with EFA is better than the system without EFA except $C = 2.63$ rad. When the C value deviates from 2.63 rad, PGC-Arctan-SC will calibrate it back. The C value will remain near 2.63rad, so PGC-Arctan-SC always maintains the optimal system performance.

Table 4. Results of the system without EFA and with EFA

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As shown in Fig. 13, when $C = 2.63$ rad, SINAD of one measurement reached 61.33 dB.

Fig. 13 SINAD analysis of the demodulated phase signal after real-time calibration.

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6. Conclusion

In this paper, a PGC demodulation algorithm called PGC-Arctan-SC is proposed. The effectiveness of the algorithm is verified in theory and simulation. The algorithm is accomplished on SoC (FPGA + ARM). Several experiments were carried out for the effectiveness verification of PGC-Arctan-SC. The experimental results show that PGC-Arctan-SC uses EFA to correct the non-strict quadrature components into a pair of strict quadrature components to suppress the nonlinear distortion caused by LID, the carrier phase delay et. al. Simultaneously, PGC-Arctan-SC estimates C value and calibrates it to the optimal value in real time to decrease the nonlinear distortion further. And it is easier to implement on the hardware because of its low computational complexity.

Funding

National Natural Science Foundation of China (61801143); Natural Science Foundation of Heilongjiang Province (LH2019F005); Fundamental Research Funds for the Central Universities (3072019CF0801 and 3072019CFM0802).

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	C(rad)	1.00	1.50	2.00	2.50	2.63	3.00
Before	h	-5.1677e-9	-6.5521e-9	-6.7728e-9	-5.8378e-9	-5.4299e-9	-3.9827e-9
	k	7.5133e-5	1.5176e-4	2.3070e-4	2.9159e-4	3.0224e-4	3.1752e-4
	a	0.8911	1.1298	1.1678	1.0066	0.9363	0.6867
	b	0.2327	0.4700	0.7146	0.9032	0.9361	0.9835
	δ	1.5708	1.5708	1.5708	1.5708	1.5708	1.5708

	e	0.9653	0.9094	0.7909	0.4415	0.0207	0.7159

After	$h^{'}$	-5.9059e-16	-5.0817e-16	-4.7274e-16	-4.0615e-16	-4.5679e-16	-5.8618e-16
	$k^{'}$	1.4731e-14	1.1440e-14	1.7821e-15	1.4869e-15	6.8254e-15	6.3698e-15
	$a^{'}$	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	$b^{'}$	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	$δ^{'}$	1.5708	1.5708	1.5708	1.5708	1.5708	1.5708

	$e^{'}$	0	0	0	0	0	0

C(rad)	$J_{1} (C) / J_{2} (C)$	$a / (b \sin δ)$	ERROR
1.00	3.8297	3.8290	0.0183 $%$
1.50	2.4040	2.4035	0.0208 $%$
2.00	1.6345	1.6343	0.0122 $%$
2.50	1.1144	1.1145	0.0233 $%$
2.63	0.9999	1.0001	0.0090 $%$
3.00	0.6975	0.6983	0.1146 $%$

	C(rad)	1.00	1.50	2.00	2.50	2.63	3.00
Before	h	-1.7936e15	-1.8006e14	2.0187e13	1.3329e12	-3.2932e12	-1.1481e14
	k	-4.3984e14	-2.0999e13	-6.8794e12	4.8111e12	7.1290e12	3.4350e14
	a	7.0037e15	1.1032e16	1.1203e16	9.3620e15	8.6582e15	6.0740e15
	b	1.8302e15	4.5901e15	6.8521e15	8.3957e15	8.6501e15	8.6993e15
	δ	1.7669	1.5774	1.5735	1.5709	1.5710	1.4781

	e	0.9653	0.9093	0.7911	0.4425	0.0432	0.7159

After	$h^{'}$	1.5959e-5	6.3705e-7	-3.3272e-5	2.0575e-5	-6.3011e-5	-8.9389e-5
	$k^{'}$	-2.8113e-5	1.0012e-6	9.6775e-6	-2.0441e-5	-4.7651e-5	1.2411e-5
	$a^{'}$	0.9998	0.9997	0.9998	0.9996	0.9998	0.9999
	$b^{'}$	0.9998	0.9999	0.9998	0.9995	0.9999	0.9999
	$δ^{'}$	1.5709	1.5708	1.5708	1.5708	1.5707	1.5707

	e^′	0	0.0200	0	0.0141	0.0141	0

C(rad)	SINAD(dB)
C(rad)	without EFA	with EFA
1.00	8.8984	46.4538
1.50	20.4144	53.5524
2.00	23.3920	53.8652
2.50	37.3319	57.7880
2.63	63.1739	61.5691
3.00	27.6188	37.2396

	C(rad)	1.00	1.50	2.00	2.50	2.63	3.00
Before	h	-5.1677e-9	-6.5521e-9	-6.7728e-9	-5.8378e-9	-5.4299e-9	-3.9827e-9
	k	7.5133e-5	1.5176e-4	2.3070e-4	2.9159e-4	3.0224e-4	3.1752e-4
	a	0.8911	1.1298	1.1678	1.0066	0.9363	0.6867
	b	0.2327	0.4700	0.7146	0.9032	0.9361	0.9835
	δ	1.5708	1.5708	1.5708	1.5708	1.5708	1.5708

	e	0.9653	0.9094	0.7909	0.4415	0.0207	0.7159

After	$h^{'}$	-5.9059e-16	-5.0817e-16	-4.7274e-16	-4.0615e-16	-4.5679e-16	-5.8618e-16
	$k^{'}$	1.4731e-14	1.1440e-14	1.7821e-15	1.4869e-15	6.8254e-15	6.3698e-15
	$a^{'}$	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	$b^{'}$	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	$δ^{'}$	1.5708	1.5708	1.5708	1.5708	1.5708	1.5708

	$e^{'}$	0	0	0	0	0	0

Real-time self-calibration PGC-Arctan demodulation algorithm in fiber-optic interferometric sensors

Abstract

1. Introduction

2. The principle of PGC demodulation algorithm

2.1. PGC demodulation system description

2.2. PGC demodulation algorithm description

2.3. Selection of the optimal C

3. The principle of PGC-Arctan-SC demodulation algorithm

3.1. EFA description

3.2. PGC-Arctan-SC demodulation algorithm description

4. Simulation

4.1. Effectiveness verification for EFA

4.2. Evaluation of the optimal C

4.3. Verification for real-time estimation of C

5. Experimental results and discussion

5.1. Experimental verification for C value drift

5.2. Experimental verification for PGC-Arctan-SC

6. Conclusion

Funding

References

Cited By

Figures (13)

Tables (4)

Equations (35)

Optics Express