## A multi-frequency signal processing method for fiber-optic gyroscopes with square wave modulation |

Optics Express, Vol. 22, Issue 2, pp. 1608-1618 (2014)

http://dx.doi.org/10.1364/OE.22.001608

Acrobat PDF (1052 KB)

### 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

1. E. J. Post, “Sagnac effect,” Rev. Mod. Phys. **39**, 475–493 (1967). [CrossRef]

2. V. Vali and R. W. Shorthill, “Fiber ring interferometer,” Appl. Opt. **15**, 1099–1100 (1976). [CrossRef] [PubMed]

3. H. C. Lefèvre, P. Martin, J. Morisse, P. Simonpiètri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital processing,” Proc. SPIE **1367**, 72–80 (1990). [CrossRef]

4. G. A. Pavlath, “Closed-loop fiber optic gyros,” Proc. SPIE **2837**, 46–60 (1996). [CrossRef]

5. R. C. Rabelo, R. T. de Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. **18**, 2146–2150 (2000). [CrossRef]

8. D. W. Heckman and M. Baretela, “Interferometric fiber optic gyro technology (IFOG),” IEEE Aerosp. Electron. Syst. Mag. **15**, 23–28 (2000). [CrossRef]

9. Z. Wang, Y. Yang, Y. Li, X. Yu, Z. Zhang, and Z. Li, “Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes,” Opt. Express **20**, 25421–25431 (2012). [CrossRef] [PubMed]

## 2. Theoretical analysis

*ϕ*(

_{m}*t*) = ±

*ϕ*

_{0}and combined with the compensation signal

*ϕ*= −

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

_{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.

9. Z. Wang, Y. Yang, Y. Li, X. Yu, Z. Zhang, and Z. Li, “Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes,” Opt. Express **20**, 25421–25431 (2012). [CrossRef] [PubMed]

9. Z. Wang, Y. Yang, Y. Li, X. Yu, Z. Zhang, and Z. Li, “Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes,” Opt. Express **20**, 25421–25431 (2012). [CrossRef] [PubMed]

*n*refers to the odd harmonics. The constant coefficient

*and Ω*

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

_{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

13. Y. Gronau and M. Tur, “Digital signal processing for an open-loop fiber-optic gyroscope,” Appl. Opt. **34**, 5849–5853 (1995). [CrossRef] [PubMed]

5. R. C. Rabelo, R. T. de Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. **18**, 2146–2150 (2000). [CrossRef]

*dBm*. In a 1991 study, a subtraction circuit that utilized the unused port of a source coupler as a noise reference was employed [14

14. R. P. Moller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett. **16**, 1902–1904 (1991). [CrossRef]

*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

15. J. Blake and I. S. Kim, “Distribution of relative intensity noise in the signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. **19**, 1648–1650 (1994). [CrossRef] [PubMed]

*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}t*= Ω +*

_{n}*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: where vector

**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: where

*σ*respectively. In our experiment we set

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

*and Ω*

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

_{Q}*N*(

_{I}*t*),

*N*(

_{Q}*t*) are independent random noise parts [9

**20**, 25421–25431 (2012). [CrossRef] [PubMed]

*and Ω*

_{I}*is a bivariate Gaussian distribution where*

_{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}*, Ω*

_{I}*)′s CRB because of the likelihood that their function is a joint Gaussian distribution, where Ω̂ is the estimation of Ω and*

_{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.

*, however, the variance is and since in our experiment these two channels have balanced demodulation therefore*

_{div}*σ*=

_{I}*σ*=

_{Q}*σ*in theory.

*ρ*between Ω

*and Ω*

_{I}*[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}**Ω′**as where

*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}**Σ**is

*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 which is simply the CRB of the quadrature demodulation. The common mode variance of Ω

*introduced by relation coefficient*

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

## 3. Experimental results

*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*.

*having been obtained from the sinusoidal modulation and Ω*

_{t}*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}*τ*as shown in Fig. 3 [16

16. F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE **74**, 162–168 (1986). [CrossRef]

*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

*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

*, Ω*

_{I}*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.

*represents the data direct difference without PCA processing while Ω*

_{dvi}*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

## 4. Conclusions

## Acknowledgments

## References and links

1. | E. J. Post, “Sagnac effect,” Rev. Mod. Phys. |

2. | V. Vali and R. W. Shorthill, “Fiber ring interferometer,” Appl. Opt. |

3. | H. C. Lefèvre, P. Martin, J. Morisse, P. Simonpiètri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital processing,” Proc. SPIE |

4. | G. A. Pavlath, “Closed-loop fiber optic gyros,” Proc. SPIE |

5. | R. C. Rabelo, R. T. de Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. |

6. | J. Blake and B. Szafraniec, “Rodom noise in PM and depolarized fiber gyros,” in Conference on Optical Fiber Sensors, Technicol Digest (CD) (Optical Society of America, 1997), paper OWB2. |

7. | R. B. Morrow Jr. and D. W. Heckman, “High precision IFOG insertion nto the strategic submarine navigation system,” in |

8. | D. W. Heckman and M. Baretela, “Interferometric fiber optic gyro technology (IFOG),” IEEE Aerosp. Electron. Syst. Mag. |

9. | Z. Wang, Y. Yang, Y. Li, X. Yu, Z. Zhang, and Z. Li, “Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes,” Opt. Express |

10. | S. J. Sanders, L. K. Strandjord, and D. Mead, “Fiber optic gyro technology trends-a Honeywell perspective,” in Proceedings of Optical Fiber Sensors Conference Technical Digest (Academic, 2002), pp. 5–8. |

11. | I. T. Jolliffe, |

12. | S. V. Vaseghi, |

13. | Y. Gronau and M. Tur, “Digital signal processing for an open-loop fiber-optic gyroscope,” Appl. Opt. |

14. | R. P. Moller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett. |

15. | J. Blake and I. S. Kim, “Distribution of relative intensity noise in the signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. |

16. | F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE |

17. | H. C. Lefevre, “Sagnac effect centenary: a special occasion to share the “serendipity” of the fibre-optic gyroscope,” in |

**OCIS Codes**

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(060.2800) Fiber optics and optical communications : Gyroscopes

**ToC Category:**

Sensors

**History**

Original Manuscript: September 19, 2013

Revised Manuscript: January 4, 2014

Manuscript Accepted: January 6, 2014

Published: January 16, 2014

**Citation**

Yongxiao Li, Zinan Wang, Yi Yang, Chao Peng, Zhenrong Zhang, and Zhengbin Li, "A multi-frequency signal processing method for fiber-optic gyroscopes with square wave modulation," Opt. Express **22**, 1608-1618 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1608

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### References

- E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–493 (1967). [CrossRef]
- V. Vali, R. W. Shorthill, “Fiber ring interferometer,” Appl. Opt. 15, 1099–1100 (1976). [CrossRef] [PubMed]
- H. C. Lefèvre, P. Martin, J. Morisse, P. Simonpiètri, P. Vivenot, H. J. Arditty, “High dynamic range fiber gyro with all-digital processing,” Proc. SPIE 1367, 72–80 (1990). [CrossRef]
- G. A. Pavlath, “Closed-loop fiber optic gyros,” Proc. SPIE 2837, 46–60 (1996). [CrossRef]
- R. C. Rabelo, R. T. de Carvalho, J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]
- J. Blake, B. Szafraniec, “Rodom noise in PM and depolarized fiber gyros,” in Conference on Optical Fiber Sensors, Technicol Digest (CD) (Optical Society of America, 1997), paper OWB2.
- R. B. Morrow, D. W. Heckman, “High precision IFOG insertion nto the strategic submarine navigation system,” in Proceedings of IEEE Positions Location and Navigation Symposium (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 332–338.
- D. W. Heckman, M. Baretela, “Interferometric fiber optic gyro technology (IFOG),” IEEE Aerosp. Electron. Syst. Mag. 15, 23–28 (2000). [CrossRef]
- Z. Wang, Y. Yang, Y. Li, X. Yu, Z. Zhang, Z. Li, “Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes,” Opt. Express 20, 25421–25431 (2012). [CrossRef] [PubMed]
- S. J. Sanders, L. K. Strandjord, D. Mead, “Fiber optic gyro technology trends-a Honeywell perspective,” in Proceedings of Optical Fiber Sensors Conference Technical Digest (Academic, 2002), pp. 5–8.
- I. T. Jolliffe, Principal Component Analysis (Springer, 2002), pp. 150–166.
- S. V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction (Wiley, 2008), pp. 107–134.
- Y. Gronau, M. Tur, “Digital signal processing for an open-loop fiber-optic gyroscope,” Appl. Opt. 34, 5849–5853 (1995). [CrossRef] [PubMed]
- R. P. Moller, W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett. 16, 1902–1904 (1991). [CrossRef]
- J. Blake, I. S. Kim, “Distribution of relative intensity noise in the signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. 19, 1648–1650 (1994). [CrossRef] [PubMed]
- F. L. Walls, D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986). [CrossRef]
- H. C. Lefevre, “Sagnac effect centenary: a special occasion to share the “serendipity” of the fibre-optic gyroscope,” in Proceedings of European Workshop on Fibre Sensors, (Academic, 2013), p. 25.

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