• Advanced Photonics Nexus
  • Vol. 4, Issue 2, 026007 (2025)
Xuezhi Zhang1,2,3, Shengliang Zhang1,2,3, Junfeng Jiang1,2,3,*, Kun Liu1,2,3..., Jiahang Jin1,2,3, Wenxin Bo1,2,3, Ruofan Wang1,2,3 and Tiegen Liu1,2,3|Show fewer author(s)
Author Affiliations
  • 1Tianjin University, School of Precision Instrument and Opto-Electronics Engineering, Tianjin, China
  • 2Institute of Optical Fiber Sensing of Tianjin University, Tianjin Optical Fiber Sensing Engineering Center, Tianjin, China
  • 3Tianjin University, Ministry of Education, Key Laboratory of Opto-Electronics Information Technology, Tianjin, China
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    DOI: 10.1117/1.APN.4.2.026007 Cite this Article Set citation alerts
    Xuezhi Zhang, Shengliang Zhang, Junfeng Jiang, Kun Liu, Jiahang Jin, Wenxin Bo, Ruofan Wang, Tiegen Liu, "Reducing variance of measurement in optical sensing based on self-Bayesian estimation," Adv. Photon. Nexus 4, 026007 (2025) Copy Citation Text show less

    Abstract

    In traditional sensing, each parameter is treated as a real number in the signal demodulation, whereas the electric field of light is a complex number. The real and imaginary parts obey the Kramers–Kronig relationship, which is expected to help further enhance sensing precision. We propose a self-Bayesian estimate of the method, aiming at reducing measurement variance. This method utilizes the intensity and phase of the parameter to be measured, achieving statistical optimization of the estimated value through Bayesian inference, effectively reducing the measurement variance. To demonstrate the effectiveness of this method, we adopted an optical fiber heterodyne interference sensing vibration measurement system. The experimental results show that the signal-to-noise ratio is effectively improved within the frequency range of 200 to 500 kHz. Moreover, it is believed that the self-Bayesian estimation method holds broad application prospects in various types of optical sensing.
    Is=E12+(E2δ)2+2E1E2δcos[ωt+φ(t)+φS(t)],

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    I0(t)={E12+(E2δ)2+2E1E2δcos[ωt+φ(t)]+φS(t)}·cos(ωt),

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    Q0(t)={E12+(E2δ)2+2E1E2δcos[ωt+φ(t)]+φS(t)}·sin(ωt).

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    I(t)=E1E2δcos[φ(t)],

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    Q(t)=E1E2δsin[φ(t)].

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    φ(t)=arctanQ(t)I(t),

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    I1(t)=2E1E2δ=2I(t)2+Q(t)2.

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    I1(t)=I1true+I1noise,

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    φ(t)=φtrue+φnoise.

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    xk=F·xk1+wk1,

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    yk=H·xk+vk,

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    Q=E[wwT],R=E[vvT].

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    x^k=F·x^k1,x^k,mea=H·yk.

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    x^k=x^k+G(x^k,meax^k).

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    x^k=x^k+Kk(ykH·x^k).

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    ek=xkx^k.

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    ek=xk[x^k+Kk(ykH·x^k)]=xkx^kKkHxkKkvk+KkHx^k=(IKkH)(xkx^k)Kkvk.

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    ek=xkx^k.

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    Pk=E[ekekT]=E{[(IKkH)(xkx^k)Kkvk][(IKkH)(xkx^k)Kkvk]T}=E{[(IKkH)ekKkvk][(IKkH)ekKkvk]T}=E[(IKkH)ekekT(IKkH)T(IKkH)ekvkTKkTKkvkekT(IKkH)T+KkvkvkTKkT].

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    Pk=(IKkH)E[ekekT](IKkH)T+KkE[vkvkT]KkT.

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    Pk=(IKkH)Pk(IKkH)T+KkRKkT=(PkKkHPk)(IKkH)T+KkRKkT=PkKkHPkPkHTKkT+KkHPkHTKkT+KkRKkT.

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    tr(Pk)=tr(Pk)2tr(KkHPk)+tr(KkHPkHTKkT)+tr(KkRKkT).

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    2(HPk)T+2KkHPkHT+2KkR=0.

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    Kk=PkHT(HPkHT+R)1.

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    Pk=(IKkH)Pk.

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    ek=Fxk1+wk1Fx^k1=Fek1+wk1.

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    Pk=E[(Fek1+wk1)(Fek1+wk1)T]=E(Fek1ek1TFT+Fek1wk1T+wk1ek1TFT+wk1wk1T).

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    Pk=FE(ek1ek1T)FT+E(wk1wk1T)=FPk1FT+Q.

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    x(t)=x(t1)+x(t1)·dt+x(t1)·dt2/2!+o[x(t1)]x(t)=x(t1)+x(t1)·dt+o[x(t1)].

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    {x(t)(1ωs2dt2/2)·x(t1)+dt·x(t1)x(t)(ωs2dt)·x(t1)+x(t1).

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    xk=F·xk1+wk1,yk=H·xk+vk,

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    Xuezhi Zhang, Shengliang Zhang, Junfeng Jiang, Kun Liu, Jiahang Jin, Wenxin Bo, Ruofan Wang, Tiegen Liu, "Reducing variance of measurement in optical sensing based on self-Bayesian estimation," Adv. Photon. Nexus 4, 026007 (2025)
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