## Abstract

The bias stability and random walk coefficients (RWC) of interferometric fiber-optic gyroscopes (IFOGs) are directly affected by characteristic noises produced by optoelectronics interactions in optic sensors. This paper documents a novel demodulation method for square wave modulated IFOGs, a method capable of suppressing the white noise that results from optical intensity noises and circuit noises as well as shot noises. In addition, this paper provides a statistical analysis of IFOG signals. Through use of orthogonal harmonic demodulation followed by deployment of matched filters to detract the Sagnac phase from the IFOGs, these channels we then processed, using principle component analysis (PCA), to establish optimal independent synchronous quadrature signal channels. Finally a difference procedure was carried out for the outputs. Our results showed that an experimental sample of the proposed IFOG (1982 m coil under uncontrolled room temperature) achieved a real-time output variance improvement in detecting the Earth’s rotation rate, which is well matched with theoretical calculations of its Cramèr-Rao bound (CRB).

© 2014 Optical Society of America

## 1. Introduction

First demonstrated at Utah University during the 1970s, interferometric fiber-optic gyroscopes (IFOGs) are rotation sensors that use optic wave information to detect the phase shift induced by the Sagnac effect [1, 2]. Over the course of several decades, integrated optics devices and digital processors have followed a development path such that nowadays the signal processing method typically used by high resolution IFOGs relies on square modulated digital closed loop construction [3,4]. In this type of IFOG, measurement noise, which a statistical assessment can divide into short-term noise and long-term noise, is primarily caused by light intensity noises and photoelectric conversion as well as environmental temperature drift [5].

Long term noises, known as bias stability, and short-term noises, known as random walk coefficients (RWC), represent the most notable effects of rotation measurements, especially in high precision detection areas. Due to the characteristics of optoelectronic conversion used in IFOGs, the noise in fiber gyroscopes is expressed as short-term white noise, especially in laboratory environments where drift induced by temperature is quite small [6]. To reduce the remaining short-term white noises, both gyro-level and system-level strategies must be deployed [7, 8]. However, gyro-level hardware improvement is constrained by costs and other factors, while system-level improvement is likewise limited by the nature of the environment since Kalman Filter technologies are sensitive to environmental variance.

Given these limitations, it seems appropriate to find new strategies for addressing noise in fiber gyroscopes. Towards this end, we recently developed and tested quadrature demodulation as a new method, designed for open-loop sine modulated IFOG systems, through which we may obtain two simultaneous outputs [9]. This environmentally robust method notably suppresses short-term noises, especially the most common noises in IFOG systems such as relative intensity noises (RIN), at no additional hardware cost. Its applications, however, are limited owing to the fact that high precision IFOGs are square wave modulated with closed loop compensation [10].

To address this need, this paper proposes a novel demodulation method appropriate to square wave modulated IFOGs, a method that could directly obtain the absolute rotation rate and also compliant with traditional zero value detection with the step wave feedback. First, through eigenfrequency correlation with local oscillation signals, we orthogonally extract the harmonic signals of the output square signal. Then through use of a simultaneous matched filter, we obtain the quadrature signals with an optimal signal-to-noise ratio (SNR). Lastly we process these two quadrature channels by use of principal component analysis (PCA), based on the closed loop gyroscope’s characteristic zero rotation rate detection [11]. This method allows us to obtain a significant improvement in random walk coefficients (RWC) for our IFOGs with complete decorrelation of signal difference, and a better performance of bias stability is also observed.

Experimental results showed a RWC and bias stability of
$0.0006\hspace{0.17em}\text{deg}/\sqrt{h}$ and 0.0035 deg/*h* in our square wave modulated IFOG; a notable improvement over the same style of IFOGs which is modulated via sine waves previously. In addition, the PCA method completely decor-related orthogonal signals, allowing us to realize the optimized variance bound of the system (Cramèr-Rao bound), a bound related to the correlation coefficient *ρ*. These results are well matched with theoretical estimates derived from the calculations from Fisher information [12].

## 2. Theoretical analysis

Figure 1 shows the closed loop polarization maintaining IFOG used in our experiment along with key implementation details. The phase modulation signal is a square wave *ϕ _{m}*(

*t*) = ±

*ϕ*

_{0}and combined with the compensation signal

*ϕ*= −

_{f}*ϕ*. The response function of IFOGs is written as

_{s}*ϕ*is the Sagnac phase shift and Δ

_{s}*ϕ*(

_{m}*t*) =

*ϕ*(

_{m}*t*) −

*ϕ*(

_{m}*t*−

*τ*), in our experiment the modulation depth is

*π*/2. The digital step wave signal is used to compensate the Sagnac phase for the zero rotation rate detection.

Figure 1 also shows the digital signal processing section of our experimental configuration. It has a relative 90° phase shift in each pair of LO signals, which is the same as the quadrature demodulation method used in the open loop IFOGs [9]. The key difference between sine modulation and square modulation IFOGs is their harmonics amplitudes, which are Bessel coefficients and odd numbers respectively. Thus the orthogonal demodulation of square wave modulation IFOGs results in the change of Bessel coefficients of quadrature demodulation for open loop IFOGs [9] to an odd number series. Accordingly, the synchronous I channel and Q channels are:

*n*refers to the odd harmonics. The constant coefficient $\frac{\sqrt{2}}{2}\frac{2}{n\pi}$ is illuminated in the demodulation, and thus it has no influence on the results. Accordingly, the rotation rate Ω

*and Ω*

_{I}*could be extracted from the I and Q channels simply as*

_{Q}*L*and

*D*refer to the length and diameter of the fiber loop respectively. Consequently the change of light source intensity

*I*

_{0}directly influences the measurement of closed loop IFOGs. This is quite different from open loop IFOGs which are independent of optical power

*I*

_{0}and modulation depth because of the division calculation of trigonometric function [13].

Given this, it is clear that for the closed loop IFOGs the most important error is induced by the optical intensity noise. In addition, the shot noise induced by photon detector (PD) and the thermal noise in the demodulation circuit are both related to the short-term noises and both expressed as white noises [5, 6]. All the noises are easily observed in our measurements which take the random walk coefficient as an evaluation index for the IFOGs. By contrast, the drift of the polarization state and environmental temperature induce long-term noises, which has limited impact on short-term measurement since the laboratory environment is relatively stable. Therefore, the noise model in our experiment is written as [5, 9]

*I*,

*S*and

*T*stand for intensity, shot and temperature respectively, while

*k*is the Boltzmann constant,

*T*is absolute temperature,

*e*is the electron charge, Δ

*ν*is the source bandwidth,

*R*is the detection load resistance, <

_{L}*i*> is the mean electric current at the PD, and B is the detection bandwidth.

The following theoretical analysis accounts for the normal distribution of noises. Since the 1550nm wide band light source is higher than the milliwatt scale, noises induced by broadband light sources constitute the dominant component of the FOG tests [5]. In our experiment, the optical source power is 8 *dBm*. In a 1991 study, a subtraction circuit that utilized the unused port of a source coupler as a noise reference was employed [14]. This method, however, increased the hardware complexity of the IFOGs.

We can also describe the noise of IFOGs according to linear system theory, which represents the noise as a Gaussian distributed signal modulated by a square wave. In that case,

where*N*(

_{I}*t*) is the optical intensity noise. For the expression of

*g*(

*t*), the Sagnac phase has been set to zero since first, a step wave has compensated for it, and second, those applications that require a low random walk coefficient for the most part operate near a zero rotation rate [15]. This means that the variance of closed loop IFOGs is completely induced by these noises, especially the optical intensity noise which obeys to the normal distribution.

As shown in Fig. 1, we applied *V _{I}*(

*t*) = sin(

*ω*) and

_{m}t*V*(

_{Q}*t*) = cos(

*ω*) as LO signals in order to obtain the optimal demodulation results; by that means the multi-frequency quadrature demodulation signals with different SNR were obtained. For the square modulated IFOG, the single extraction of basic frequency signals by LO correlation was not stable enough for our purposes, since the SNR was low in comparison to the square window used in conventional method. Thus in order to solve this problem, we designed a matched filter for multi-frequency signals.

_{m}tSince IFOGs are dominated by white noise, we were able to get the lowest variance estimate through use of the Bayes estimation. First we can see that output Ω obeys to normal distribution, meaning that for both I and Q channels, Ω* _{n}* = Ω +

*N*(0,

_{n}*σ*), where subscript

_{n}*n*means demodulated with the

*n*times LO signal,

*N*(0,

_{n}*σ*) represents the white noises with 0 mean and

_{n}*σ*represents the variance for different harmonics. Then the Bayes Posteriori Estimation function

_{n}*f*

_{Ω}|

**(Ω|**

_{Ω}**Ω**) is:

**Ω**is made up by Ω

*and*

_{n}*f*

**|**

_{Ω}_{Ω}(

**Ω**|Ω) is the priori distribution function. We can get the optimal estimate for Ω by calculating the maximum a posteriori probability (MAP) estimate, a derivative of log

*f*

_{Ω}|

**(Ω|**

_{Ω}**Ω**) to Ω. The optimal Ω makes this equation equal to 0. Thus we obtain:

*σ*respectively. In our experiment we set

_{n}*n*= 5. This filter functions synchronously and thus does not have any memory effect.

A matched filter is responsible for obtaining the maxim SNR signal Ω* _{I}* and Ω

*for the step wave feedback. In practice, the detection results can be written from a statistical perspective as*

_{Q}*N*(

_{I}*t*),

*N*(

_{Q}*t*) are independent random noise parts [9].Because of the simultaneous differences, the suppression of common noises in the outputs can be easily expressed as:

The difference between synchronous outputs represents the process by which the differential mode noise in the IFOG system is suppressed. As discussed above, the noise of IFOGs are for the most part normally distributed. Therefore the distribution function of quadrature outputs Ω* _{I}* and Ω

*is a bivariate Gaussian distribution*

_{Q}*ρ*is the correlation coefficient between Ω

*and Ω*

_{I}*,*

_{Q}*σ*and

_{I}*σ*is the standard variance of Ω

_{Q}*and Ω*

_{I}*respectively.*

_{Q}In order to obtain the optimal estimate of these measurements, we calculated the Cramèr-Rao bound (CRB) of Ω based on its Fisher information [12]. In statistical theory, CRB is the lowest bound of a deterministic parameter variance. We can thus easily obtain Ω(Ω* _{I}*, Ω

*)′s CRB because of the likelihood that their function is a joint Gaussian distribution,*

_{Q}*Var*[Ω̂ (Ω

*, Ω*

_{I}*] is the variance of the estimation. There is no difference to the final result if*

_{Q}*ρ*< 0, since it would induce the inversion of the sign during the calculation processing too. The sign of

*ρ*is dependent on the settings of the initial phase degrees of LO signals. It is easily to see that if

*ρ*= 0 then the minimum variance as the optimal RWC of IFOGs is obtained.

For the Ω* _{div}*, however, the variance is

*σ*=

_{I}*σ*=

_{Q}*σ*in theory.

Principle component analysis (PCA) is a classic method used to eliminate the correlation coefficient *ρ* between Ω* _{I}* and Ω

*[11]. PCA requires that the data mean be zero, which is perfectly in accordance with closed-loop gyroscopes because of their zero detection feature. Thus for practical applications when there is a change of rotation rate which introduces a change of the Sagnac phase, this change could be compensated for with sufficient speed in closed loop IFOGs, thus maintaining the stability of our analysis. In addition, we can use the PCA method to directly distribute the output signals into two independent orthogonal signals.*

_{Q}Using their previous statistical information, we established the covariance matrix of demodulation outputs. We assume the device is stationary, and therefore that the covariance matrix is stable. Then we calculate the eigenvectors of the covariance matrix, which are the same as the calculation of quadratic form. From that we obtain the **Ω′** as

*c*is the unified element of the eigenvector

_{ij}**C**

*for the covariance matrix*

_{i}**Σ**. The unified eigenvector

**C**ensures that variances of Ω

*and Ω*

_{I}*are still on the same scale after PCA.*

_{Q}The eigenvalue of covariance matrix **Σ** is
$\frac{{\sigma}_{I}^{2}+{\sigma}_{Q}^{2}\pm \sqrt{{\left({\sigma}_{I}^{2}-{\sigma}_{Q}^{2}\right)}^{2}+4{\rho}^{2}{\sigma}_{I}^{2}{\sigma}_{Q}^{2}}}{2}$, and can be simplified as (1 ± *ρ*)*σ*^{2} when *σ _{I}* =

*σ*. Thus after the PCA processing, the variances of Ω

_{Q}*and Ω*

_{I}*are redistributed into two new channels: one a common mode channel whose variance is (1 +*

_{Q}*ρ*)

*σ*

^{2}; the other a differential mode channel whose variance is (1 −

*ρ*)

*σ*

^{2}. The variance of the common mode channel is bigger than that of the differential mode channel; however the sum of these two channels variances is consistent with the previous one that had not been processed through PCA, thus verifying that the signal scale remained unchanged after PCA. Through the difference processing, we get

*introduced by relation coefficient*

_{div}*ρ*is compensated for through PCA.

It is noteworthy that by refreshing the eigenvectors and using data registers to update the covariance matrix this processing can be kept stable over long time periods. From a statistical perspective, uncorrelated signals can be viewed simply as independent gyroscopes which detect the same rotation rate. Hence an *N* times improvement for the variance was obtained, with *N* being the number of gyroscopes (in our experiment, *N* = 2). Therefore this method showed a better performance with respect to the random walk coefficients than simple orthogonal demodulation which contained a notable correlation.More specifically, this algorithm has no specific requirement for noise features, since the statistical feature applied in our algorithm wholly suppresses correlated noise.

The other important characteristic of our method is that it allows for real-time processing, which in turn means that there is no delay structure therefore and thus no memory effect. Our method can thus maintain the fast response of gyroscopes, an advantage which is extremely significant for practical applications. Furthermore, the irregular component of outputs produced by the irregular signal modulation can be suppressed by the orthogonal harmonic demodulation. Long-term bias stability, however, is limited to the gyro-level hardware as well as subtle noise types other than that of white Gaussian noise in our presumption. In order to achieve a potential significant improvement for such long-term bias stability, a further optimization in the gyro-level hardware of IFOGs is required.

## 3. Experimental results

We carried out several long-term tests targeting the Earth’s rotation rate (9.666°/h in theory at the laboratory latitude 39.99° N) under room temperature. In consideration of the Sagnac effect on our gyroscopes, the detection phase was set at about 0.01°. Figure 1 shows the experiment setup of the polarization maintained IFOG. We used a programmable signal generator to finish the Sagnac phase feedback. Our light source was a spontaneous emission source with a center frequency of 1550 nm and a band width of 40 nm. The PMF was 1982 meter long and 0.14 m diameter.

The sampling rate and sampling length were 2 *MHz* and 200000 respectively, and the resolution of the digitizer was 22 *bits*. The light source power was 8 *dBm*; therefore the previous analysis of the noise as the normal distribution is reasonable. The test bandwidth of the gyroscope was 10 *Hz*. The test length was 50000 points and the test time was 1.53 *hours*.

A typical set of experimental results is shown in Fig. 2, Ω* _{t}* having been obtained from the sinusoidal modulation and Ω

*from the square modulation. Both of these results were taken from the steadiest part of the two modulation methods respectively and here the Ω*

_{s}*is before the PCA processing. This, in turn, means that this comparison merely contrasts these two modulation and demodulation methods. It is obvious that Ω*

_{s}*has the lowest noise amplitude and is therefore more robust against long term bias effects.*

_{s}For further evaluation of the rotation data, the Allan variance analysis method was applied. This method evaluates the standard variance of the Ω versus the averaging time *τ* as shown in Fig. 3 [16]. In addition, the more detailed indices deduced from the Allan variance are shown in Table 1, where *Q*, *N*, *Bs*, *K* and *R* refer to quantization noise, random walk coefficient, bias stability, rate random walk, and rate ramp respectively. The bias stabilities of sine wave modulation and square wave modulation outputs are 0.0073 deg/*h* and 0.0035 deg*/h*. In addition, the random walk coefficient was improved from
$0.0011\hspace{0.17em}\text{deg}/\sqrt{h}$ to
$0.0006\hspace{0.17em}\text{deg}/\sqrt{h}$.

Finally the other performances of the Allan indices all saw improvement. These results are due to the fact that the square modulation produced more useful information about the Sagnac phase than the sine wave modulation during the same modulation circle.

To evaluate the PCA processing, we chose a measurement of length of 50000 after a long-term test, since PCA performance is directly relevant to the *ρ* which is in turn affected by background noises. For a better comparison, we chose the section of data which showed a relevant high noise amplitude after a long-term test, since the stability performance of these devices weakened over time. The data which were projected on the Ω* _{I}*, Ω

*axis before and after PCA are shown in Fig. 4 respectively. The data has a relevant coefficient*

_{Q}*ρ*= 0.4416 before PCA. After the PCA processing, this coefficient was eliminated.

The comparison between simultaneous difference detection before PCA processing and after PCA is shown in Fig. 5. There Ω* _{dvi}* represents the data direct difference without PCA processing while Ω

*represents the data direct difference after PCA. Before PCA, the variances of Ω*

_{pca}*, Ω*

_{I}*and Ω*

_{Q}*are 0.2057, 0,2194 and 0.1532 (°/*

_{dvi}*h*)

^{2}. After PCA, we obtained variances for Ω′

*, Ω′*

_{I}*and Ω*

_{Q}*of 0.3066, 0.1185 and 0.1063 (°/*

_{pca}*h*)

^{2}respectively. The variance improved from 0.1532 to 0.1063 (°/

*h*)

^{2}which is just equal to the $\frac{1}{2}\overline{{\sigma}^{2}}$, and well matched with the previous theoretical estimates, demonstrating that this method achieved the IFOG’s CRB. A further evaluation obtained through Allan analysis is shown in Fig. 6. The indices are listed in Table 2. The improvement in the variances of outputs is directly relevant to the random walking coefficient.

We can see the output is from a stable and highly consistent environment, thus there is no specific environmental factor required for this method. This performance is repeatable, with no other requirement for the instrumentation of current square modulation closed-loop IFOGs. Consequently, this method is a novel method as an optimization of traditional demodulation method used in square wave modulated IFOGs without increasing the hardware complexity.

Since the proposed method is independent of the optical and electrical performance of IFOG’s hardware, it could be applied in various applications to meet the requirements of both high and low grade standards under different scenarios, from navigation system to motion tracking and stabilization. With highly optimized components, manufacturing and feedback controls, the long-term bias stability of the best IFOG is able to surpass 10-5 deg/h [17]. Accompany with other advantages, such as highly-integrated, stable and no-moving part, IFOGs thus become a strong competitor of ring laser gyroscope (RLG) as ultra-high-precision rotation sensors.

## 4. Conclusions

In conclusion, we have documented a novel method of quadrature demodulation for square wave modulation IFOGs, where we detracted the orthogonal multi-frequency signal from the IFOGs and then deploy matched filter to detract the Sagnac phase from the gyroscopes, in order to obtain two synchronous channels. These channels we then processed, using principle component analysis (PCA), to establish optimal independent synchronous quadrature signal channels. Finally we carried out a difference procedure for the outputs. Our results showed that an experimental sample of the proposed IFOG (1982 m coil under uncontrolled room tempeature) achieved a real-time output variance improvement in detecting the Earth’s rotation rate, which is well matched with theoretical calculations from Fisher information. Moreover both short-term noise and long-term instability were reduced by the multi-frequency signal processing synchronous difference procedure. Experimental outputs from IFOGs were consistent with theoretical estimates. Unlike traditional methods of demodulation, this method reduced both the RWC and bias drift for the same IFOG.

This method requires no previous information about the IFOG system except for the covariance matrix. The other important advantage of the method is that it leaves the physical structure of the gyroscopes unchanged, meaning that this method could be applied to most square wave modulated IFOGs to suppress the noise amplitude without increasing the cost or configuration complexity of the system. In addition, it is a method that simultaneously suppresses short term noise and long term instability, both of which place significant limitations on IFOGs. Therefore, this multi-frequency quadrature demodulation method shows great profit potential as a strategy for improving the performance of high precision close-loop IFOGs.

## Acknowledgments

This work was supported by 973 Program of China No. 2013CB329205, 973 Program of China No. 2010CB328203, and the National Natural Science Foundation of China (NSFC) under grant No. 61307089.

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