Abstract

Signal processing for low-finesse fiber-optic Fabry–Perot sensors based on white-light interferometry is investigated. The problem is demonstrated as analogous to the parameter estimation of a noisy, real, discrete harmonic of finite length. The Cramer–Rao bounds for the estimators are given, and three algorithms are evaluated and proven to approach the bounds. A long-standing problem with these types of sensors is the unpredictable jumps in the phase estimation. Emphasis is made on the property and mechanism of the “total phase” estimator in reducing the estimation error, and a varying phase term in the total phase is identified to be responsible for the unwanted demodulation jumps. The theories are verified by simulation and experiment. A solution to reducing the probability of jump is demonstrated.

© 2013 Optical Society of America

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2011 (3)

2010 (1)

2008 (4)

2006 (3)

2005 (3)

2004 (2)

2003 (2)

B. Yu, D. W. Kim, J. Deng, H. Xiao, and A. Wang, “Fiber Fabry–Perot sensors for detection of partial discharges in power transformers,” Appl. Opt. 42, 3241–3250, 2003.
[CrossRef]

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

2000 (1)

T. Liu and G. F. Fernando, “A frequency division multiplexed low-finesse fiber optic Fabry–Perot sensor system for strain and displacement measurements,” Rev. Sci. Instrum. 71, 1275–1278 (2000).
[CrossRef]

1999 (1)

H. C. So, Y. T. Chan, Q. Ma, and P. C. Ching, “Comparison of various periodograms for sinusoid detection and frequency estimation,” IEEE Trans. Aerosp. Electron. Syst. 35, 945–952 (1999).
[CrossRef]

1998 (2)

T. Liu, M. Wu, Y. Rao, D. A. Jackson, and G. F. Fernando, “A multiplexed optical fibre-based extrinsic Fabry–Perot sensor system for in-situ strain monitoring in composites,” Smart Mater. Struct. 7, 550–556 (1998).
[CrossRef]

R. Cortés, A. V. Khomenko, A. N. Starodumov, N. Arzate, and L. A. Zenteno, “Interferometric fiber-optic temperature sensor with spiral polarization couplers,” Opt. Commun. 154, 268–272 (1998).
[CrossRef]

1996 (2)

V. Bhatia, M. B. Sen, K. A. Murphy, and R. O. Claus, “Wavelength-tracked white light interferometry for highly sensitive strain and temperature measurements,” Electron. Lett. 32, 247–249 (1996).
[CrossRef]

T. Liu, D. Brooks, A. R. Martin, R. A. Badcock, and G. F. Fernando, “Design, fabrication, and evaluation of an optical fiber sensor for tensile and compressive strain measurements via the use of white light interferometry,” Proc. SPIE 2718, 408–416 (1996).
[CrossRef]

1995 (1)

V. Bhatia, K. A. Murphy, R. O. Claus, T. A. Tran, and J. A. Greene, “Recent developments in optical-fiber-based extrinsic Fabry–Perot interferometric strain sensing technology,” Smart Mater. Struct. 4, 246–251 (1995).
[CrossRef]

1994 (2)

G. Ghosh, M. Endo, and T. Iwasaki, “Temperature-dependent Sellmeier coefficients and chromatic dispersions for some optical fiber glasses,” J. Lightwave Technol. 12, 1338–1342 (1994).
[CrossRef]

J. Schwider and L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. 19, 995–997 (1994).
[CrossRef]

1993 (2)

S. Taplin, A. G. Podoleanu, D. J. Webb, and D. A. Jackson, “Displacement sensor using channelled spectrum dispersed on a linear CCD array,” Electron. Lett. 29, 896–897 (1993).
[CrossRef]

C. Belleville and G. Duplain, “White-light interferometric multimode fiber-optic strain sensor,” Opt. Lett. 18, 78–80 (1993).
[CrossRef]

1991 (1)

1985 (1)

1982 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1974 (1)

D. Rife and R. Boorstyn, “Single tone parameter estimation from discrete-time observations,” IEEE Trans. Inf. Theory 20, 591–598 (1974).
[CrossRef]

Arzate, N.

R. Cortés, A. V. Khomenko, A. N. Starodumov, N. Arzate, and L. A. Zenteno, “Interferometric fiber-optic temperature sensor with spiral polarization couplers,” Opt. Commun. 154, 268–272 (1998).
[CrossRef]

Badcock, R. A.

T. Liu, D. Brooks, A. R. Martin, R. A. Badcock, and G. F. Fernando, “Design, fabrication, and evaluation of an optical fiber sensor for tensile and compressive strain measurements via the use of white light interferometry,” Proc. SPIE 2718, 408–416 (1996).
[CrossRef]

Barnes, T. H.

Belleville, C.

Bhatia, V.

V. Bhatia, M. B. Sen, K. A. Murphy, and R. O. Claus, “Wavelength-tracked white light interferometry for highly sensitive strain and temperature measurements,” Electron. Lett. 32, 247–249 (1996).
[CrossRef]

V. Bhatia, K. A. Murphy, R. O. Claus, T. A. Tran, and J. A. Greene, “Recent developments in optical-fiber-based extrinsic Fabry–Perot interferometric strain sensing technology,” Smart Mater. Struct. 4, 246–251 (1995).
[CrossRef]

Boorstyn, R.

D. Rife and R. Boorstyn, “Single tone parameter estimation from discrete-time observations,” IEEE Trans. Inf. Theory 20, 591–598 (1974).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Boulet, C.

Brooks, D.

T. Liu, D. Brooks, A. R. Martin, R. A. Badcock, and G. F. Fernando, “Design, fabrication, and evaluation of an optical fiber sensor for tensile and compressive strain measurements via the use of white light interferometry,” Proc. SPIE 2718, 408–416 (1996).
[CrossRef]

Chan, Y. T.

H. C. So, C. Kit Wing, Y. T. Chan, and K. C. Ho, “Linear prediction approach for efficient frequency estimation of multiple real sinusoids: algorithms and analyses,” IEEE Trans. Signal Process. 53, 2290–2305 (2005).
[CrossRef]

H. C. So, Y. T. Chan, Q. Ma, and P. C. Ching, “Comparison of various periodograms for sinusoid detection and frequency estimation,” IEEE Trans. Aerosp. Electron. Syst. 35, 945–952 (1999).
[CrossRef]

Ching, P. C.

H. C. So, Y. T. Chan, Q. Ma, and P. C. Ching, “Comparison of various periodograms for sinusoid detection and frequency estimation,” IEEE Trans. Aerosp. Electron. Syst. 35, 945–952 (1999).
[CrossRef]

Claus, R. O.

V. Bhatia, M. B. Sen, K. A. Murphy, and R. O. Claus, “Wavelength-tracked white light interferometry for highly sensitive strain and temperature measurements,” Electron. Lett. 32, 247–249 (1996).
[CrossRef]

V. Bhatia, K. A. Murphy, R. O. Claus, T. A. Tran, and J. A. Greene, “Recent developments in optical-fiber-based extrinsic Fabry–Perot interferometric strain sensing technology,” Smart Mater. Struct. 4, 246–251 (1995).
[CrossRef]

K. A. Murphy, M. F. Gunther, A. M. Vengsarkar, and R. O. Claus, “Quadrature phase-shifted, extrinsic Fabry–Perot optical fiber sensors,” Opt. Lett. 16, 273–275 (1991).
[CrossRef]

Cortés, R.

R. Cortés, A. V. Khomenko, A. N. Starodumov, N. Arzate, and L. A. Zenteno, “Interferometric fiber-optic temperature sensor with spiral polarization couplers,” Opt. Commun. 154, 268–272 (1998).
[CrossRef]

Deng, J.

Dong, B.

Duan, Y.

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

Duplain, G.

Endo, M.

G. Ghosh, M. Endo, and T. Iwasaki, “Temperature-dependent Sellmeier coefficients and chromatic dispersions for some optical fiber glasses,” J. Lightwave Technol. 12, 1338–1342 (1994).
[CrossRef]

Fernando, G. F.

T. Liu and G. F. Fernando, “A frequency division multiplexed low-finesse fiber optic Fabry–Perot sensor system for strain and displacement measurements,” Rev. Sci. Instrum. 71, 1275–1278 (2000).
[CrossRef]

T. Liu, M. Wu, Y. Rao, D. A. Jackson, and G. F. Fernando, “A multiplexed optical fibre-based extrinsic Fabry–Perot sensor system for in-situ strain monitoring in composites,” Smart Mater. Struct. 7, 550–556 (1998).
[CrossRef]

T. Liu, D. Brooks, A. R. Martin, R. A. Badcock, and G. F. Fernando, “Design, fabrication, and evaluation of an optical fiber sensor for tensile and compressive strain measurements via the use of white light interferometry,” Proc. SPIE 2718, 408–416 (1996).
[CrossRef]

Ghosh, G.

G. Ghosh, M. Endo, and T. Iwasaki, “Temperature-dependent Sellmeier coefficients and chromatic dispersions for some optical fiber glasses,” J. Lightwave Technol. 12, 1338–1342 (1994).
[CrossRef]

Gong, J.

Greene, J. A.

V. Bhatia, K. A. Murphy, R. O. Claus, T. A. Tran, and J. A. Greene, “Recent developments in optical-fiber-based extrinsic Fabry–Perot interferometric strain sensing technology,” Smart Mater. Struct. 4, 246–251 (1995).
[CrossRef]

Gunther, M. F.

Han, M.

Han, Y.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hathaway, M.

Ho, K. C.

H. C. So, C. Kit Wing, Y. T. Chan, and K. C. Ho, “Linear prediction approach for efficient frequency estimation of multiple real sinusoids: algorithms and analyses,” IEEE Trans. Signal Process. 53, 2290–2305 (2005).
[CrossRef]

Huang, H.

Huang, Z.

Y. Zhu, Z. Huang, F. Shen, and A. Wang, “Sapphire-fiber-based white-light interferometric sensor for high-temperature measurements,” Opt. Lett. 30, 711–713 (2005).
[CrossRef]

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

Huo, W.

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

Ina, H.

Iwasaki, T.

G. Ghosh, M. Endo, and T. Iwasaki, “Temperature-dependent Sellmeier coefficients and chromatic dispersions for some optical fiber glasses,” J. Lightwave Technol. 12, 1338–1342 (1994).
[CrossRef]

Jackson, D. A.

C. Boulet, M. Hathaway, and D. A. Jackson, “Fiber-optic-based absolute displacement sensors at 1500 nm by means of a variant of channeled spectrum signal recovery,” Opt. Lett. 29, 1602–1604 (2004).
[CrossRef]

T. Liu, M. Wu, Y. Rao, D. A. Jackson, and G. F. Fernando, “A multiplexed optical fibre-based extrinsic Fabry–Perot sensor system for in-situ strain monitoring in composites,” Smart Mater. Struct. 7, 550–556 (1998).
[CrossRef]

S. Taplin, A. G. Podoleanu, D. J. Webb, and D. A. Jackson, “Displacement sensor using channelled spectrum dispersed on a linear CCD array,” Electron. Lett. 29, 896–897 (1993).
[CrossRef]

Jiang, Y.

Kay, S. M.

S. M. Kay, Modern Spectral Estimation: Theory and Application (Prentice-Hall, 1999).

Khomenko, A. V.

R. Cortés, A. V. Khomenko, A. N. Starodumov, N. Arzate, and L. A. Zenteno, “Interferometric fiber-optic temperature sensor with spiral polarization couplers,” Opt. Commun. 154, 268–272 (1998).
[CrossRef]

Kim, D. W.

Kit Wing, C.

H. C. So, C. Kit Wing, Y. T. Chan, and K. C. Ho, “Linear prediction approach for efficient frequency estimation of multiple real sinusoids: algorithms and analyses,” IEEE Trans. Signal Process. 53, 2290–2305 (2005).
[CrossRef]

Kobayashi, S.

Lally, E.

Liu, T.

T. Liu and G. F. Fernando, “A frequency division multiplexed low-finesse fiber optic Fabry–Perot sensor system for strain and displacement measurements,” Rev. Sci. Instrum. 71, 1275–1278 (2000).
[CrossRef]

T. Liu, M. Wu, Y. Rao, D. A. Jackson, and G. F. Fernando, “A multiplexed optical fibre-based extrinsic Fabry–Perot sensor system for in-situ strain monitoring in composites,” Smart Mater. Struct. 7, 550–556 (1998).
[CrossRef]

T. Liu, D. Brooks, A. R. Martin, R. A. Badcock, and G. F. Fernando, “Design, fabrication, and evaluation of an optical fiber sensor for tensile and compressive strain measurements via the use of white light interferometry,” Proc. SPIE 2718, 408–416 (1996).
[CrossRef]

Ma, C.

Ma, Q.

H. C. So, Y. T. Chan, Q. Ma, and P. C. Ching, “Comparison of various periodograms for sinusoid detection and frequency estimation,” IEEE Trans. Aerosp. Electron. Syst. 35, 945–952 (1999).
[CrossRef]

Majumdar, A.

Martin, A. R.

T. Liu, D. Brooks, A. R. Martin, R. A. Badcock, and G. F. Fernando, “Design, fabrication, and evaluation of an optical fiber sensor for tensile and compressive strain measurements via the use of white light interferometry,” Proc. SPIE 2718, 408–416 (1996).
[CrossRef]

May, R. G.

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

Murphy, K. A.

V. Bhatia, M. B. Sen, K. A. Murphy, and R. O. Claus, “Wavelength-tracked white light interferometry for highly sensitive strain and temperature measurements,” Electron. Lett. 32, 247–249 (1996).
[CrossRef]

V. Bhatia, K. A. Murphy, R. O. Claus, T. A. Tran, and J. A. Greene, “Recent developments in optical-fiber-based extrinsic Fabry–Perot interferometric strain sensing technology,” Smart Mater. Struct. 4, 246–251 (1995).
[CrossRef]

K. A. Murphy, M. F. Gunther, A. M. Vengsarkar, and R. O. Claus, “Quadrature phase-shifted, extrinsic Fabry–Perot optical fiber sensors,” Opt. Lett. 16, 273–275 (1991).
[CrossRef]

Musa, S. M.

S. M. Musa, “Real-time signal processing and hardware development for a wavelength modulated optical fiber sensor system,” Ph.D. dissertation (Virginia Tech, 1997).

Peng, W.

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

Pickrell, G. R.

M. Han, Y. Zhang, F. Shen, G. R. Pickrell, and A. Wang, “Signal-processing algorithm for white-light optical fiber extrinsic Fabry–Perot interferometric sensors,” Opt. Lett. 29, 1736–1738 (2004).
[CrossRef]

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

Podoleanu, A. G.

S. Taplin, A. G. Podoleanu, D. J. Webb, and D. A. Jackson, “Displacement sensor using channelled spectrum dispersed on a linear CCD array,” Electron. Lett. 29, 896–897 (1993).
[CrossRef]

Qi, B.

B. Qi, G. R. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, “Novel data processing techniques for dispersive white light interferometer,” Opt. Eng. 42, 3165–3171 (2003).
[CrossRef]

Rao, Y.

T. Liu, M. Wu, Y. Rao, D. A. Jackson, and G. F. Fernando, “A multiplexed optical fibre-based extrinsic Fabry–Perot sensor system for in-situ strain monitoring in composites,” Smart Mater. Struct. 7, 550–556 (1998).
[CrossRef]

Rao, Y. J.

Rife, D.

D. Rife and R. Boorstyn, “Single tone parameter estimation from discrete-time observations,” IEEE Trans. Inf. Theory 20, 591–598 (1974).
[CrossRef]

Schwider, J.

Sen, M. B.

V. Bhatia, M. B. Sen, K. A. Murphy, and R. O. Claus, “Wavelength-tracked white light interferometry for highly sensitive strain and temperature measurements,” Electron. Lett. 32, 247–249 (1996).
[CrossRef]

Shen, F.

So, H. C.

H. C. So, C. Kit Wing, Y. T. Chan, and K. C. Ho, “Linear prediction approach for efficient frequency estimation of multiple real sinusoids: algorithms and analyses,” IEEE Trans. Signal Process. 53, 2290–2305 (2005).
[CrossRef]

H. C. So, Y. T. Chan, Q. Ma, and P. C. Ching, “Comparison of various periodograms for sinusoid detection and frequency estimation,” IEEE Trans. Aerosp. Electron. Syst. 35, 945–952 (1999).
[CrossRef]

Starodumov, A. N.

R. Cortés, A. V. Khomenko, A. N. Starodumov, N. Arzate, and L. A. Zenteno, “Interferometric fiber-optic temperature sensor with spiral polarization couplers,” Opt. Commun. 154, 268–272 (1998).
[CrossRef]

Takeda, M.

Taplin, S.

S. Taplin, A. G. Podoleanu, D. J. Webb, and D. A. Jackson, “Displacement sensor using channelled spectrum dispersed on a linear CCD array,” Electron. Lett. 29, 896–897 (1993).
[CrossRef]

Tran, T. A.

V. Bhatia, K. A. Murphy, R. O. Claus, T. A. Tran, and J. A. Greene, “Recent developments in optical-fiber-based extrinsic Fabry–Perot interferometric strain sensing technology,” Smart Mater. Struct. 4, 246–251 (1995).
[CrossRef]

Tsai, H.-L.

Vengsarkar, A. M.

Wang, A.

C. Ma, E. Lally, and A. Wang, “Toward eliminating signal demodulation jumps in optical fiber intrinsic Fabry–Perot interferometric sensors,” J. Lightwave Technol. 29, 1913–1919 (2011).
[CrossRef]

C. Ma, B. Dong, J. Gong, and A. Wang, “Decoding the spectra of low-finesse extrinsic optical fiber Fabry–Perot interferometers,” Opt. Express 19, 23727–23742 (2011).
[CrossRef]

C. Ma and A. Wang, “Multimode excitation-induced phase shifts in intrinsic Fabry–Perot interferometric fiber sensor spectra,” Appl. Opt. 49, 4836–4845 (2010).
[CrossRef]

M. Han and A. Wang, “Mode power distribution effect in white-light multimode fiber extrinsic Fabry–Perot interferometric sensor systems,” Opt. Lett. 31, 1202–1204 (2006).
[CrossRef]

F. Shen and A. Wang, “Frequency-estimation-based signal-processing algorithm for white-light optical fiber Fabry–Perot interferometers,” Appl. Opt. 44, 5206–5214 (2005).
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Figures (9)

Fig. 1.
Fig. 1.

Schematic of the fiber-optic WLI FP sensing system.

Fig. 2.
Fig. 2.

Performance evaluation of the FFT, LR, and PT methods: (a) OPD estimation and (b)  φ 0 estimation. Plotted together in both figures are the standard deviations of both the TYPE I and TYPE II estimators. The CRBs for the corresponding variances are coplotted. Insets are zoomed-in views of the lines, which provide better visibility of the algorithms’ performance.

Fig. 3.
Fig. 3.

Absolute value of OPD bias versus cavity length plotted for TYPE I LR, FFT, and windowed FFT (Blackman–Harris) estimators. The result for the TYPE II estimator using FFT (Blackman–Harris) is coplotted, which demonstrates superior bias suppression.

Fig. 4.
Fig. 4.

Performance comparison of the LR, FFT, and windowed FFT (Blackman–Harris). The rms error includes the contributions from both the estimator variance and bias. The windowed FFT for both TYPE I and TYPE II estimations manifests superior performance in bias reduction, at the expense of a 3 dB increase in the rms error.

Fig. 5.
Fig. 5.

Measured computation complexity in terms of execution time, plotted as a function of data length ( N ). The FFT, PT, and LR methods are compared to demonstrate a linear relationship with N . The complexity of FFT is the highest, while the complexities of PT and LR are almost identical.

Fig. 6.
Fig. 6.

Computer-simulated phase term φ 0 caused by material dispersion (solid curve) and FPN (dashed curve). The dispersion of a 70 μm thick silica wafer was modeled by the temperature-dependent Sellmeier model. WLI System I, together with Blackman–Harris windowed FFT, was used for signal demodulation. For the simulation of FPN, the applied white Gaussian noise yields SNR = 12 dB .

Fig. 7.
Fig. 7.

Experimentally obtained PDF of φ 0 estimation.

Fig. 8.
Fig. 8.

Reduction of jump probability by phase calibration. (a) PDF of φ 0 estimation plotted with OPD, σ p = 0.158 π . The area between the solid lines is the assigned phase range assuming constant phase, and dashed lines are the boundaries of the OPD-calibrated range with linear fitting. (b) The corresponding probability of jump. Solid curve: OPD-dependent jump probability for the constant range scheme. Dashed curve: jump probability for the calibrated range scheme.

Fig. 9.
Fig. 9.

Computer-simulated variance and covariance terms in Eq. (A1). The corresponding CRBs are plotted together. An important observation is that the variance terms and the covariance term cancel to yield a significantly reduced variance for the total phase estimation.

Tables (1)

Tables Icon

Table 1. List of Key Parameters of the WLI Systems Used in the Research

Equations (34)

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I ( Φ ) = 2 R 1 cos ( Φ ) 1 + R 2 2 R cos ( Φ ) ,
I ( Φ ) = 2 R [ 1 cos ( Φ ) ] ,
I ( Φ ) = R 1 + ν 2 2 ν cos ( Φ ) 1 + R 2 ν 2 2 R ν cos ( Φ )
I ( Φ ) = a 0 + n = 1 a n cos ( n Φ )
a n + 1 / a n = R ν ,
I norm ( Φ ) = 1 + 2 ν 1 + ν 2 cos ( k · OPD + π ) .
I norm ( Φ ) = 1 + 2 ν 1 + ν 2 cos ( k · OPD + φ 0 ) ,
ω n = OPD · Δ k / π ,
I n = A cos ( OPD · k n + φ 0 ) + W n ,
k c = ( k 1 + k N ) / 2 k 0 + N Δ k / 2 ,
Φ ^ = k c · OPD ^ + φ ^ 0 ,
OPD ^ tot = Φ ^ / k c .
E [ ( θ ^ θ ) 2 ] = var [ θ ^ ] + bias 2 [ θ ^ ] ,
var [ OPD ^ ] = 12 S · ( Δ k ) 2 N ( N 2 1 ) .
var [ OPD ^ ] = 1 S · ( Δ k ) 2 Y .
var [ φ ^ 0 ] = 12 Y S · N 2 ( N 2 1 ) .
var [ φ ^ 0 ] = 1 S · N .
var [ OPD ^ tot ] = 1 S · N · k c 2 .
G = 2 3 ( n 0 N + 1 2 ) .
f ( φ ^ 0 ; φ 0 , σ p 2 ) = 1 σ p 2 π exp [ ( φ ^ 0 φ 0 ) 2 2 σ p 2 ] .
P jump 2 F ( f ; π ) = 2 π f ( φ ^ 0 ; 0 , σ p 2 ) d φ ^ 0 ,
N = λ c 2 δ OPD · Δ λ ,
| A ( ω n ) | = | n = 0 N 1 I n exp ( i · π n ω n ) | ,
OPD ^ = ω ^ n π / Δ k ,
φ ^ 0 = arg [ exp ( i OPD ^ · k 0 ) A ( ω ^ n ) ] .
I n = A cos ( OPD · k n + φ 0 ) + W ¯ n ,
I ˜ k = FFT k ( I n ) = S k + w ˜ k ,
Δ φ 0 ( d O P D d t ) · k c 2 f s N Δ k ,
var [ Φ ^ ] = k c 2 · var [ L ^ ] + 2 k c · cov [ L ^ , φ ^ ] + var [ φ ^ ] ,
J ( 1 , 2 ) = J ( 2 , 1 ) = 6 ( 2 n 0 + N 1 ) S · Δ k · N ( N 2 1 ) .
var [ Φ ^ ] 12 k c 2 N 12 k c Δ k ( 2 n 0 + N 1 ) N + 12 Δ k 2 Y S · Δ k 2 · N 2 ( N 2 1 ) .
Y Δ k 2 [ 1 + 1 12 ( N n 0 ) 2 ] k c 2 N = ( 1 + C 0 ) k c 2 N .
var [ Φ ^ ] 1 S · N ,
var [ OPD ^ tot ] 1 S · ( Δ k ) 2 Y ,

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