【正文】 T(τ) is a crosscorrelation function weighted with the coherence of the signals and can detect the time delay more precisely and robustly than the simple crosscorrelation function. The maximum of RSCOT(τ) is the best estimate _t of the time delay between two FBG sensors. The measured velocity v meas is then calculated from the following simple equation: t?? smeas dv (2) where ds is the distance between two sensors. Fig. 2 illustrates the schematic diagram of the whole system. We used an amplified spontaneous emission (ASE) as an optical source of the system. The ASE has output power of 22 dBm and full width at half maximum (FWHM) of 50 nm at Cband. The light from the ASE is separated by an optical 3 dB coupler and then illuminates two FBG sensors installed to the PVC pipe whose inner diameter is 20 mm. The light reflected by the FBG sensors is fed to Mach–Zehnder interferometers, which are prised of a 22 and a 33 couplers, with the optical path differences of and mm. In the 33 coupler, three fibers are arranged in a triangular array. These interferometers are used as wavelengthshift detectors for interferometric detection. The incident light is phasemodulated by the Mach–Zehnder interferometer and then converted to voltage signals by photodetectors. Six output signals are simultaneously digitized by an A/D converter with a resolution of 16 bit and sampling frequency of 10 kHz, and the detected signals are then processed to obtain the time delay. The FBG reflects the light wave with a certain wavelength λB called Bragg wavelength and the wavelength is then expressed as follows: ?? nB 2? ，（ 3） where n is the effective refractive index of the FBG and Λ is the modulation pitch of the refractive index of the FBG. The Bragg wavelength λB changes by longitudinal strain εz applied to the FBG, and the Bragg wavelengthshift δλB is expressed as follows [3]: ? ? ZBB P ???? 121?? (4) where p12 is the elastooptic constant of the optical fiber and is approximately . This yields the strain sensitivity of pm/_ε. To obtain the shift δλB, the outputs of the interferometer are used, and the outputs Vm (m = 1, 2, 3) are expressed as follows [9]: Vm = αmVin + Re[Γ(τ)] = αmVin[1 + γ cos(θMZI + θm)], （ 5） where Γ (τ) is the autocorrelation function of light wave reflected by the FBG sensor, Vin is the voltage corresponding to optical power reflected by the FBG sensor, and αm is the coefficient pensating differences of photodetector sensitivities and obtained from preliminary experiments. If the split ratios of the 2 2 and 3 3 couplers are 1:1 and 1:1:1, respectively, one can obtain θ1 = 0, θ2 = 2π/3 and θ3 = ?2π/3, and the outputs V1, V2 and V3 are derived as follows: ?????????????????)]32c o s (1[)]32c o s (1[)c o s1(312111???????????M Z IinM Z IinM Z IinVVVVVV （ 6） The signal θMZI can be then calculated using the following equation: 132321 2 )(3t a n2 VVV VVLBM Z I ???? ???? (7) where L is the optical path difference of the interferometer. The relationship between the signal variation δθMZI and the shift δλB is expressed as follows: ? ? BBBBBBBBBM Z I LLLL ?????????? ?????? ??? 22222 ???????? (8) where the term (λB + δλB) was assumed to be nearly equal to λB because the shift δλB is significantly smaller than λB. An accidental loss of optical power while measurement, which causes problems in optical intensity modulation type sensors, is admissible in some measure because the wavelengthtophase sensitivity (=?2πL/λ2B) depends on only the path difference L. 3. Noise estimation of the FBG sensor with interferometric detection There are some reports about the noise estimation of FBG sensors with interferometric detection. However the noise estimation reports about the interferometric detection using a 2 2 and a 3 3 couplers have not been presented. . Noise of the photodetector Fig. 3 shows the circuit diagram of the photodetector. The noise of the photodetector is defined by the noise of a photodiode and a transimpedance amplifier. The noise of the photodetector is determined by thermal noises due to the feedback resistance Rf of the transimpedance amplifier Rf and shunt resistance Rsh of the photodiode and by shot noise due to the output current im (=Vm/Rf ) of the photodiode. The equivalent input noise voltage and current of the OpAmp are neglected because they are a few orders smaller than the other noises. The RMS of the noise voltage VN,m (m = 1, 2, 3) can be then written as follows: fWBWshWBfmN R TBBqiR TBRV k42k4 m, ??? （ 9） where kB is the Boltzmann constant ( 10?23 J/K), T is the absolute temperature (300 K), Bw is the equivalent noise bandwidth of the photodetector ( kHz) and q is the electronic charge ( 10?19 C). In order to calculate the shot noise by im, the visibility γ should be known. Although the spectral profile of a FBG is expressed as a function of hyperbolic sine and cosine, for ease of calculation of the visibility γ, we assumed that the FBG had a Gaussian spectral profile S(υ) given as follows [10]: ? ? ? ? ??????? ??? 222ln4e xp2ln2 BBBinm vvvv VvS ? ? (10) where υ is the frequency of the light wave, υB is the Bragg frequency, _υB is FWHM of the FBG in the frequency domain. From Eq. (10) andWiener–Khintchine theorem, the visibility γ can be derived as follows: ,ln4e 222??? ?????? Lvcxp B?? (11) where c is the speed of the light wave in vacuum. From Eqs. (6) and (11), we can obtain the dc value of im and calculate the shot noise by im. . Quantization noise during A/D conversion Quantization noise should be taken into consideration because the system shown in Fig. 2 uses digitized signals for demodulation of the signal θMZ