# Novel Semi-Parametric Algorithm for Interference-Immune Tunable Absorption Spectroscopy Gas Sensing

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## Abstract

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## 1. Introduction

## 2. Description of the Algorithm

#### 2.1. Steps of the Algorithm: General Description

#### 2.2. Compensate Windowing

#### 2.3. Calculate the DFT

#### 2.4. Determine Cut-Off

`N`indicates the number of points of the dataset (i.e., the finite number of experimental points),

`D_i`the ith point of the DFT of the signal ${\tilde{I}}_{tot}\left(x\right)$,

`Rsquared`the coefficient of determination ${R}^{2}$,

`FFFT`the functional form of the Fourier transform of $I\left(x\right)$, and

`Rlimit`the value which should be reached for ${R}^{2}$. This limiting value can be helpful since, beyond a certain value, even if ${R}^{2}$ continues to increase when increasing ${i}_{0}$, the quality of the fit will not improve significantly and it is not necessary to exclude more points from the DFT for the fit:

pointtoremove = 0 Rsqmin = 0 for i = 0 to N/2 remove the first i points starting from D_0 to D_{i-1} from the DFT fit the remaining points D_i to D_{N/2} to the FFFT and save the fit parameter Rsquared if (i = 0) then pointtoremove = 0 else if (Rsquared > Rsqmin) then pointtoremove = i and Rsqmin = Rsquared if (Rsqmin > Rlimit) then break loop end of for loop.

`pointtoremove`.

`Rsquared`can be chosen depending on the application. For the curves shown in this paper, the loop was stopped for

`Rlimit`$=0.99999$. An example of the evolution of ${R}^{2}$ with increasing ${i}_{0}$ is shown in Figure 3. The data refer to the third scenario described later in Section 3. Above ${i}_{0}=30$, ${R}^{2}$ still continues to slightly increase, but the statistical goodness of the fit does not improve further.

#### 2.5. Fit of DFT

#### 2.6. Algorithm Applied to a Lorentzian Line Shape

## 3. Application to Simulated Data

## 4. Experimental Results

_{2}near infrared A-band in the presence of multiple interference fringes.

#### 4.1. Experimental Setup

_{2}photodiode (FS1010, Thorlabs, Newton, NJ, USA), amplified by an adjustable-gain photodiode amplifier (PDA200C, Thorlabs, Newton, NJ, USA). The current ramp for the wavelength-sweep and the data acquisition were performed by a DAQ card (USB-6361, National Instruments Switzerland GmbH, Ennetbaden, Switzerland) using a Labview

^{TM}software. The laser current sweep was chosen so to be able to measure three oxygen absorption lines. The total distance between the laser and the detector was kept fixed at approximately 36 cm. Interference fringes of adjustable intensity were generated by inserting and tilting a glass window of known material in the optical path. By varying the thickness of the glass window, it is possible to achieve fringes with different FSR; by tilting the glass window, it is possible to adjust the amplitude of the fringes. In this work, two glass windows of BK7 of thicknesses d = 11 mm (window 1) and d = 4 mm (window 2) were used. These two windows were chosen to create interferences fringes with FSR comparable to (window 1) and greater than (window 2) the line width of the signal. These types of interferences are the most disturbing because they cannot be easily eliminated—for example, by a small jitter in the laser current or by standard post-processing filtering.

#### 4.2. Oxygen Sensing

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

TDLAS | Tunable Diode Laser Absorption Spectroscopy |

FSR | Free Spectral Range |

CFT | Continuous Fourier Transform |

DFT | Discrete Fourier Transform |

RW | Rectangular Window |

HWHM | Half Width at Half Maximum |

VCSEL | Vertical-cavity surface-emitting laser |

## Appendix A Approximation of the CFT by the Modified DFT

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**Figure 1.**Schematic flow diagram of the steps of the algorithm to extract a signal $I\left(x\right)$ from a total signal ${I}_{tot}\left(x\right)$.

**Figure 2.**Schematic representation on how to choose the width of the Tukey window W compared to the Lorentzian HWHM ${P}_{2}$. Here, it is $W=20\phantom{\rule{4pt}{0ex}}{P}_{2}$.

**Figure 3.**Evolution of ${R}^{2}$ with increasing value of the cut-off ${i}_{0}$. The data correspond to the third scenario described in Section 3.

**Figure 4.**First scenario: disturbance with an FSR comparable to the line width. (

**a**) simulated experimental total signal ${I}_{tot}\left(x\right)$ (blue line), Lorentzian line shape signal $I\left(x\right)$ (black dots) and extracted signal obtained with the algorithm (red line); (

**b**) DFT of the total signal $\left|F\right({I}_{tot}\left(x\right)\left)\left(k\right)\right|$ (red line) and DFT of the Lorentzian signal $\left|F\right(I\left(x\right)\left)\right(k\left)\right|$ (black points).

**Figure 5.**Second scenario: weak disturbance with an FSR as large as the measuring window. (

**a**) simulated experimental total signal ${I}_{tot}\left(x\right)$ (blue line), Lorentzian line shape signal $I\left(x\right)$ (black dots) and extracted signal obtained with the algorithm (red line); (

**b**) DFT of the total signal $\left|F\right({I}_{tot}\left(x\right)\left)\left(k\right)\right|$ (red line) and DFT of the Lorentzian signal $\left|F\right(I\left(x\right)\left)\right(k\left)\right|$ (black points).

**Figure 6.**Third scenario: strong multiple disturbances. (

**a**) simulated experimental total signal ${I}_{tot}\left(x\right)$ (blue line), Lorentzian line shape signal $I\left(x\right)$ (black dots) and extracted signal obtained with the algorithm (red line); (

**b**) DFT of the total signal $\left|F\right({I}_{tot}\left(x\right)\left)\left(k\right)\right|$ (red line) and DFT of the Lorentzian signal $\left|F\right(I\left(x\right)\left)\right(k\left)\right|$ (black points).

**Figure 8.**Typical absorbance for the three lines R9R9 (760.77 nm), R7Q8 (760.89 nm) and R7R7 (761.003 nm) of the O

_{2}near infrared A-band. Superimposed to the absorption features are the interferences caused by the window 1 (d = 11 mm) and window 2 (d = 4 mm). The measurement is taken at atmospheric conditions.

**Figure 9.**(

**a**) comparison of the absorbance of the R7Q8 line extracted with the algorithm (solid lines; Algorithm w1, Algorithm w2) and the expected lines from HITRAN database (points; HITRAN w1, HITRAN w2) for the window 1 (w1, d = 11 mm) and window 2 (w2, d = 4 mm); (

**b**) enlargement of the peak maximum for the same data.

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**MDPI and ACS Style**

Michelucci, U.; Venturini, F.
Novel Semi-Parametric Algorithm for Interference-Immune Tunable Absorption Spectroscopy Gas Sensing. *Sensors* **2017**, *17*, 2281.
https://doi.org/10.3390/s17102281

**AMA Style**

Michelucci U, Venturini F.
Novel Semi-Parametric Algorithm for Interference-Immune Tunable Absorption Spectroscopy Gas Sensing. *Sensors*. 2017; 17(10):2281.
https://doi.org/10.3390/s17102281

**Chicago/Turabian Style**

Michelucci, Umberto, and Francesca Venturini.
2017. "Novel Semi-Parametric Algorithm for Interference-Immune Tunable Absorption Spectroscopy Gas Sensing" *Sensors* 17, no. 10: 2281.
https://doi.org/10.3390/s17102281