# Fourier Coefficients Applied to Improve Backscattered Signals in A Short-Range LIDAR System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiment

## 3. Results and Discussion

_{1}and second coefficient C

_{2}of Fourier from the lock-in while the backscattered pulse cross through the mist; these coefficients correspond to fundamental frequency (ω) and second frequency (2ω) when the length of the chamber is of 150 cm. The curves show the switching from on to off from the UH in order to show the transition of the mist into the chamber. Description of the process can be divided in three parts. In the first part (I) starting from t = 0 s to t = 64 s, curves show the amplitude of the FC since the chamber is empty until it is filled with mist when the UH is turned on. The second part (II), from 64 s ≤ t ≤ 129 s the chamber is filled with mist and there is constant flow through all chamber; in the third part (III) corresponding to time t ≥ 129 s the curves show the free flow of the mist when the UH is turned off. The coefficients C

_{1}and C

_{2}have a time difference at the beginning of the filling of the chamber, this is due to the fact that the lock-in takes each of the coefficients in a commutated way, but in part II, the curves show that the time of commutated is irrelevant because the chamber is filled with mist and they show the turbulence of the mist inside of the chamber. Part II of Figure 2 is analyzed in order to validate the improvement of the LIDAR signal.

_{1}is measured in the part II from Figure 2. Theoretical Fourier coefficients C

_{T1}and C

_{T2}are calculated by adjusting f(t) with a Gaussian function (f*(t), shown as a dashed red line in Figure 3 for C

_{1}).

_{Tn}decreases exponentially with the square of n. In this work, the lock-in just can measure the first two Fourier coefficients directly and, in accordance to C

_{Tn}, the first two coefficients contribute with more power to the LIDAR signal. The results are C

_{T1}= 0.032 V and in the same way is calculated C

_{T2}= 0.024 V. These results are close to the average of the experimental coefficients C

_{1}= 0.0322 V and C

_{2}= 0.0228 V correspondingly in part II from Figure 2.

_{tp}

_{,dB}) is calculated as

_{TP}is [11]

_{1}of Figure 3 has a SNR

_{tp,dB}= 13.8 dB.

_{s}) and the power of the noise (P

_{n}) i.e., [12]

_{n}shows the uncertainty of recovered coefficients of the lock-in, particularly with the time constant, hence, if τ is increased, the power of noise in the Fourier coefficient is decreased and its uncertainty is also decreased. It should be mentioned that this implies a compromise between the noise reduction of the Fourier coefficient and time of the measurement [12]. For measuring the dynamic of mist into the chamber τ = 30 s and K is obtained through Parseval theorem [13]. The SNR

_{lock-in,dB}is obtained in the same way that SNR

_{tp,dB}, therefor, for C

_{1}of the lock-in the SNR

_{lock-in,dB}= 22 dB.

_{1}) and the second Fourier coefficient (C

_{2}) of the backscattered pulse by the mist inside of the chamber corresponding to 150 and 350 cm of length. It can be notice that through the measurement of Fourier coefficients, it can be distinguished the effect of the mist density on the coefficients. As well, it is calculated the signal-to-noise ratio in decibels for a chamber length of 350 cm, the SNR

_{to,dB}of the temporal pulse is 12.3 dB and the SNR

_{lock-in,dB}for C

_{1}is 20.4 dB.

## 4. Conclusions

_{1}for a chamber length of 150 cm is 22 dB; this corresponds to an improvement of approximately of 8 dB, even more, the signal-to-noise ratio improvement was maintained with a different chamber length, this is due to the technique of phase sensitive detection can measure periodic signals into noise, similar improvement using EMD is shows by Chang, J. et al. [11]. The improvement for the two lengths of chamber shows the independence of the mist density and validate the improve of the LIDAR signal using Fourier coefficients. It should be mentioned that the OSLRF-01 system scales the speed of light approximately 881,000 times through sample time circuits (SETS) and this scaling allowed the use of a conventional lock-in as the SR530, however, in applications of short-range, atmospheric or other kind of LIDAR, the sensitive phase detection must be implemented with the correspond speed devices under the principle of SETS.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**C

_{1}and C

_{2}Fourier coefficients obtained with the lock-in from mist for a chamber length of 150 cm.

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**MDPI and ACS Style**

Gómez-Arista, I.; Dávila-Pintle, J.A.; Montalvo-Montalvo, N.; Rubin-Alvarado, A.A.; Bravo-García, Y.E.; Reynoso-Lara, E.
Fourier Coefficients Applied to Improve Backscattered Signals in A Short-Range LIDAR System. *Electronics* **2020**, *9*, 390.
https://doi.org/10.3390/electronics9030390

**AMA Style**

Gómez-Arista I, Dávila-Pintle JA, Montalvo-Montalvo N, Rubin-Alvarado AA, Bravo-García YE, Reynoso-Lara E.
Fourier Coefficients Applied to Improve Backscattered Signals in A Short-Range LIDAR System. *Electronics*. 2020; 9(3):390.
https://doi.org/10.3390/electronics9030390

**Chicago/Turabian Style**

Gómez-Arista, Iván, José A. Dávila-Pintle, Nancy Montalvo-Montalvo, Abel A. Rubin-Alvarado, Yolanda E. Bravo-García, and Edmundo Reynoso-Lara.
2020. "Fourier Coefficients Applied to Improve Backscattered Signals in A Short-Range LIDAR System" *Electronics* 9, no. 3: 390.
https://doi.org/10.3390/electronics9030390