5. Demodulation signal

In this section, we present the demodulation signal for the quasi-distributed sensor based on wavelength-division multiplexing. The signal processing combines the Fourier domain phase analysis (FDPA) algorithm, a bank of K filters and a band-pass filter. The FDPA algorithm was described and also applied in Refs. [3, 11]. The bank of K filters can be defined as

$$\mathcal{F}(\lambda) = \text{rect}\left(\frac{\lambda}{\Delta\lambda\_{\text{op}}}\right) \otimes \sum\_{k=1}^{K} \delta(\lambda - \lambda\_{\text{BGk}}) \tag{24}$$

rectð Þ¼ ν

frequency components Rkð Þ<sup>ν</sup> , filtering Rmð Þ <sup>λ</sup> , and its complex conjugate <sup>R</sup><sup>∗</sup>

ing Rmð Þ <sup>ν</sup>; δλ , comparison between spectrums <sup>R</sup><sup>∗</sup>

Figure 6. Digital demodulation represented schematically: R<sup>∗</sup>

complex conjugate.

position is νFPk.

1 j j ν <

Twin-Grating Fiber Optic Sensors Applied on Wavelength-Division Multiplexing and Its Numerical Resolution

8 ><

>:

The filter Fð Þν is a rect function in the frequency domain: its bandwidth is νBGk and its central

Basically, the digital demodulation consists of two stages: calibration and measurement. The calibration stage is developed once; five steps are necessary and the references are generated. The calibration considers the signal acquisition RTð Þ λ , filtering signal Rkð Þ λ , separation of

symbol \* indicates the complex conjugate. The measurement stage is developed for each measurement and eight steps are necessary. The measurement stage considers the signal acquisition RTð Þ λ; δλ , filtering Rkð Þ λ; δλ , separation of frequency components RTð Þ ν; δλ , filter-

tion; Bragg wavelength shift evaluation and an adaptive filter is applied. The adaptive filter is a set of coefficients as was described in Ref. [13]. The complete procedure is shown at Figure 6.

<sup>m</sup> <sup>¼</sup> <sup>R</sup><sup>∗</sup>

0 j j ν >

νBGk 2

νBGk 2

(28)

189

<sup>m</sup>ð Þ λ , where the

<sup>m</sup>ð Þ λ and Rmð Þ ν; δλ , 2πP ambiguity elimina-

http://dx.doi.org/10.5772/intechopen.75586

<sup>m</sup>ð Þ <sup>λ</sup> , <sup>R</sup><sup>~</sup> <sup>m</sup> <sup>¼</sup> <sup>R</sup><sup>~</sup> <sup>m</sup>ð Þ <sup>ν</sup>; δλ and the symbol \* indicates the

where the symbol ⊗ indicates the convolution operation, the rect function (Eq. (24)) is defined as

$$rect(\lambda) = \begin{cases} 1 & |\lambda| < \frac{\Delta \lambda\_{\rm op}}{2} \\ 0 & |\lambda| > \frac{\Delta \lambda\_{\rm op}}{2} \end{cases} \tag{25}$$

and δ is the Dirac delta. Invoking the Dirac delta properties, the bank of K filters is

$$\mathbf{F}(\lambda) = \sum\_{k=1}^{K} \operatorname{rect} \left( \frac{\lambda - \lambda\_{\rm BG}}{\Delta \lambda\_{\rm op}} \right) \tag{26}$$

The signal Fð Þ λ is a series of rect functions in the wavelength domain; its bandwidth is the operation range and the central positions are the Bragg wavelengths. On the other hand, Rkð Þν is the Fourier transform for the kth interference pattern. The spectrum Rkð Þν consists of three triangle functions. The component νFP<sup>0</sup> contains information from all twin-grating sensors and this signal cannot be used in the demodulation signal. The νFPk and �νFPk components contain the same information and any component can be used in the demodulation signal. Then, the band-pass filter can be defined as

$$\mathbf{F}(\nu) = \text{rect}\left(\frac{\nu - \nu\_{\text{FPk}}}{\nu\_{\text{BGk}}}\right) \tag{27}$$

where the rect function (Eq. (27)) has the next definition

Twin-Grating Fiber Optic Sensors Applied on Wavelength-Division Multiplexing and Its Numerical Resolution http://dx.doi.org/10.5772/intechopen.75586 189

$$rect(\nu) = \begin{cases} 1 & |\nu| < \frac{\mathcal{V}\_{\text{BG}}}{\mathcal{Q}} \\ 0 & |\nu| > \frac{\mathcal{V}\_{\text{BG}}}{\mathcal{Q}} \end{cases} \tag{28}$$

The filter Fð Þν is a rect function in the frequency domain: its bandwidth is νBGk and its central position is νFPk.

To illustrate, the next numerical example is presented.

5. Demodulation signal

188 Numerical Simulations in Engineering and Science

band-pass filter can be defined as

where the rect function (Eq. (27)) has the next definition

combining the wavelength-and-frequency division multiplexing.

Example 1: A broadband light source has the interval from λmin ¼ 1470 nm to λmax ¼ 1620 nm. If the operation range is selected to <sup>Δ</sup>λop <sup>¼</sup> 6 nm, the number of local sensors is <sup>K</sup> <sup>¼</sup> <sup>1620</sup>�<sup>1470</sup>

¼ 25. The quasi-distributed sensor would have 25 twin-grating sensors and the signal will have 25 wavelength channels. Here, two important points can be mentioned: 1) The parameter Δλop permits the selection of the number of local sensors; 2) the cost per sensing point can be reduced

In this section, we present the demodulation signal for the quasi-distributed sensor based on wavelength-division multiplexing. The signal processing combines the Fourier domain phase analysis (FDPA) algorithm, a bank of K filters and a band-pass filter. The FDPA algorithm was

where the symbol ⊗ indicates the convolution operation, the rect function (Eq. (24)) is defined as

8 >><

>>:

and δ is the Dirac delta. Invoking the Dirac delta properties, the bank of K filters is

k¼1

Fð Þ¼ ν rect

rect

The signal Fð Þ λ is a series of rect functions in the wavelength domain; its bandwidth is the operation range and the central positions are the Bragg wavelengths. On the other hand, Rkð Þν is the Fourier transform for the kth interference pattern. The spectrum Rkð Þν consists of three triangle functions. The component νFP<sup>0</sup> contains information from all twin-grating sensors and this signal cannot be used in the demodulation signal. The νFPk and �νFPk components contain the same information and any component can be used in the demodulation signal. Then, the

<sup>F</sup>ð Þ¼ <sup>λ</sup> <sup>X</sup> K

⊗ X K

1 j j λ <

0 j j λ >

k¼1

Δλop 2

Δλop 2

λ � λBGk Δλop � �

ν � νFPk νBGk � �

δ λð Þ � λBGk (24)

described and also applied in Refs. [3, 11]. The bank of K filters can be defined as

rectð Þ¼ λ

λ Δλop � �

Fð Þ¼ λ rect

6

(25)

(26)

(27)

Basically, the digital demodulation consists of two stages: calibration and measurement. The calibration stage is developed once; five steps are necessary and the references are generated. The calibration considers the signal acquisition RTð Þ λ , filtering signal Rkð Þ λ , separation of frequency components Rkð Þ<sup>ν</sup> , filtering Rmð Þ <sup>λ</sup> , and its complex conjugate <sup>R</sup><sup>∗</sup> <sup>m</sup>ð Þ λ , where the symbol \* indicates the complex conjugate. The measurement stage is developed for each measurement and eight steps are necessary. The measurement stage considers the signal acquisition RTð Þ λ; δλ , filtering Rkð Þ λ; δλ , separation of frequency components RTð Þ ν; δλ , filtering Rmð Þ <sup>ν</sup>; δλ , comparison between spectrums <sup>R</sup><sup>∗</sup> <sup>m</sup>ð Þ λ and Rmð Þ ν; δλ , 2πP ambiguity elimination; Bragg wavelength shift evaluation and an adaptive filter is applied. The adaptive filter is a set of coefficients as was described in Ref. [13]. The complete procedure is shown at Figure 6.

Figure 6. Digital demodulation represented schematically: R<sup>∗</sup> <sup>m</sup> <sup>¼</sup> <sup>R</sup><sup>∗</sup> <sup>m</sup>ð Þ <sup>λ</sup> , <sup>R</sup><sup>~</sup> <sup>m</sup> <sup>¼</sup> <sup>R</sup><sup>~</sup> <sup>m</sup>ð Þ <sup>ν</sup>; δλ and the symbol \* indicates the complex conjugate.
