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Abstract
The parameter-tuning stochastic resonance (SR) method can convert part of the noise energy into the signal energy to suppress the noise and amplify the signal, comparing with traditional weak periodic signal detection methods (e.g., time average method, filtering method, and correlation analysis method). In this work, the numerical calculation is conducted to find the optimal resonance parameters for applying the SR method to the wavelength modulation spectroscopy (WMS). Under the stochastic resonance state, the peak value of 2f signal (a constant concentration of CH4∼20 ppm) is effectively amplified to ∼0.0863 V, which is 3.8 times as much as the peak value of 4000-time average signal (∼0.0231 V). Although the standard deviation also increases from ∼0.0015 V(1σ) to ∼0.003 V(1σ), the SNR can be improved by 1.83 times (from ∼25.9 to ∼15.8) correspondingly. A linear spectral response of SR 2f signal peak value to raw 2f signal peak value is obtained. It suggests that the SR method is effective for enhancing photoelectric signal under strong noise background.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The laser absorption spectroscopy technology has been demonstrated in many applications, such as air quality monitoring, industrial process control, and medical diagnostic [1–6]. The precision of the measurement is important to those applications. Although laser absorption spectroscopy has many advantages in sensitive detection, it still needs a long optical path length and special measurement technology for detecting a very trace substance, with a high detection sensitivity [7–10]. For those purposes, a multi-pass cell with a long optical path is usually applied to enhance the absorption signal. However, the unexpected interference fringe, thermal noise, shot noise, electrical noise and white noise, often occur in absorption signals and seriously spoil the detection precision [11–14]. Those problems are common for laser absorption spectroscopy when using dense overlapped spot pattern multi-pass cell. Therefore, it is of great significance to effectively extract weak photoelectric absorption signals from a strong noise background. Several methods are proposed to eliminate the negative influence of the noise. The traditional weak periodic signal processing methods mainly include time average method, filtering method, and correlation analysis method [15,16]. ①The signal with a high signal-to-noise ratio (SNR) can be obtained by time average method, so the standard deviation of noise can be reduced and the signal quality can be improved. Nevertheless, the strong noise background cannot be fully eliminated by this method. ②The signal filters based on hardware and software are widely used for noise reduction, the characteristic of which is narrow bandwidth. In practical application, the desired signal and noise usually have a continuous power spectrum and wide bandwidth, but it is relatively difficult to manufacture a filter that matches the bandwidth of the signal to remove the noise. If the bandwidth of the filter is very small, the noise will be greatly attenuated. However, this may destroy the desired signal. ③The correlation detection method is used to remove the noise by the autocorrelation of the periodic signal. Its essence is to establish a very narrow bandwidth filter to filter out the noise, the frequency of which is different from that of the signal.
Compared with other weak periodic signal detection methods mentioned above, the advantage of the parameter-tuning stochastic resonance (SR) method is apparent. Even if the noise and signal have the same frequency, as long as they reach the optimal resonance matching, the SR method can convert part of the noise energy into the signal energy to suppress the noise and enhance the signal.
In this work, the SR method is applied to the wavelength modulation spectroscopy (WMS) by using the dense overlapped spot pattern multi-pass cell. first, the numerical calculation will be implemented to find the suitable parameters and evaluate the performance of the optimal SR system, and then it is verified that the SR method can effectively enhance the WMS signal by the experiments.
2. Mathematical model of stochastic resonance
The mathematical model of the SR is described by Langevin Equation:
where $u(t )= s(t )+ n(t )\; $ and $x(t )$ are the input signal and output signal of the SR system, respectively. $n(t)$ is the Gaussian white noise, and $s(t)$ is the periodic signal. $V({x,t} )$ is defined as the potential function of the SR system:When the input signal only includes the noise term $n(t )$, the particle transition rate satisfies Kramers rate in the SR system:
where D is the noise intensity. When the weak periodic signal $x(t )$ is used as the only input signal, the potential function will tilt periodically, and the particle in a potential well has a periodic property which is equal to that of the weak periodic signal $s(t ).$Based on the above stochastic resonance theory, if the noise particle transition rate matches the frequency of the weak periodic signal $s(t )$, the part of noise energy $n(t )$ will be converted to weak periodic signal $s(t )$, so that it can suppress noise and enhance the signal. It is also found that the stochastic resonance method is suitable for processing low-frequency weak signals, so it is necessary to normalize the raw input signals for converting the high-frequency signal to a low-frequency signal.
The Eq. (4) is employed to normalize the raw input signals by substituting the Eq. (4) into Eq. (1), the new normalized equation is obtained as shown below,
The frequency of normalized signal F0 is $1/a$ that of the original signal and the amplitude of the raw input signal has been reduced by a factor of $\sqrt {b/{a^3}} $. Therefore, the stochastic resonance method can be applied to the normalized raw input signal. Before practical application, the SR system parameters a and b are determined by numerical calculation. The numerical calculation follows constraint conditions as follow:
where the number of sampling data per cycle is n =${f_s}/f$, the f and fs are assumed to be the modulation frequency and sampling frequency of the raw input signal, respectively. The sampling frequency of the normalized low-frequency signal is equal to Fs = F0×n, the sampling step h=$1/{F_s}$. Before starting the numerical calculation, F0 as the frequency of the square wave is set to 0.000801 Hz. the square wave’s duty cycle is 50%. The a and $b$ are set to 0.000581 and 2, respectively. The number of sampling data n is set to 1.15×106. In order to obtain the optimal parameters a and b, the classical fourth-order Runge-Kutta algorithm is applied to solve the normalized Langevin Eq. (5). The details of the fourth-order Runge-Kutta algorithm are shown:A value ∼ 0.00625V of the ideal square wave without noise is used as the amplitude of the square wave displayed in Fig. 1 by red line. The ideal square wave superimposed with Gaussian white noise (-16dB) is regarded as the SR system low-frequency input signal exhibited in Fig. 1 by blue line, in which the ideal square wave signal has been completely submerged in Gaussian white noise. In order to obtain the square wave signal, the input signal including noise and square wave signal is processed by the SR system, the square wave signal with the same frequency as the ideal square wave can be extracted as shown in Fig. 1 by orange line. Compared with the SR raw input signal, the SNR of the SR output signal in Fig. 1 is greatly enhanced.
Fig. 1. Example of SR system response (a =0.000581; b = 2) red line: ideal square wave signal; blue line: SR input signal; orange line: SR output signal
The example of SR system response is shown in Fig. 1. The amplitude of the SR output signal which is amplified ∼218.7 times that of the ideal square wave signal, while the standard deviation is reduced from0.0401 V(1σ) to 0.0081 V(1σ), therefore, the signal-to-noise ratio has been greatly improved. As clear seen in Fig. 2, the relationship between the SR square wave amplitude and the ideal square wave amplitude is not linear, but when the ideal square wave amplitude is on a scale of 0.02 V to 0.03625 V, a linear spectral response can be observed as shown blue line in Fig. 2. The calculated R-square value for the linear fit is better than 0.92. This shows that the SR system is a linear system in a specific interval. after many parameter adjustment tests, a group of suitable parameters (a =0.000581; b = 2) is found. Based on this set of parameters, the SR system output signal has the characteristics of high signal-to-noise ratio and small distortion.
In order to further evaluate the optimal resonance state of the SR system, the spectrum maximum of the SR output square signal is compared with that of the SR input square wave signal, based on Fast Fourier Transform (FFT). The results are shown in Fig. 3. It is obvious that the SR system is in the optimal resonance state, in which the energy of the noise has been converted to the square wave because the spectrum maximum of the SR output square wave signal is magnified by 23.25times. The stochastic resonance algorithm is only effective for random noise, so only part of the high-frequency noise will be suppressed.
Fig. 3. SR system low frequency response red dot: Spectrum of SR output square wave signal; blue dot: Spectrum of SR input signal.
3. Experimental details
A schematic diagram of the experimental setup is shown in Fig. 4. A fiber-coupled 1.653µm DFB diode laser (W-1653-20-A-SM-FA, WChip) operating at center current 420 mA and room temperature of 24 °C is used as a laser source. The laser temperature and current are controlled by a commercial diode laser controller (Model LDC 501, Stanford Research Systems), and the laser frequency could span the range of 6335 cm-1 - 6338 cm-1, which is suitable for the detection of methane by probing one of the strong absorption lines located at 6046.95 cm-1. A triangular ramp wave signal (1.5V peak to peak, 20Hz) produced by a signal generator (SDG1032X, SIGLENT) and a sine-wave signal (0.06V peak to peak, 6KHz) generated from a lock-in amplifier (Healthy Photon, HPLIA) are combined through an adder and injected into the diode laser controller to scan and modulate the laser wavelength.
The laser beam is collimated with a fiber-coupled collimator (f∼4.8 mm) and subsequently injected into the dense overlapped spot pattern multi-pass cell with a base length of 8 cm and 80 times reflections (leading to an effective optical path length of 12.8 m), and the gold coated spherical mirror’s radius of curvature is 100 mm. The gold coat of the spherical mirrors has been slightly damaged, resulting in the serious etalon fringe effects and spoils the laser beam. The output laser beam is focused on an InGaAs amplified photodetector (PDA20CS-EC, Thorlabs) with a lens (f = 50 mm). A lock-in amplifier is used for demodulation of absorption signal from the photodetector at the frequency of second harmonic signal 2f (where f =6 KHz is the modulation frequency of the sine wave). The time constant of the lock-in amplifier is set to 1 ms. The demodulated signal is subsequently digitalized by a DAQ card (NI USB-6251, National Instruments) and displayed on a computer via a Labview interface.
4. Results and discussion
For evaluation of the performance of the SR system, time-series measurements of CH4 sealed in the MPC (so having a constant concentration of 20ppm) are performed. The 4000-time continuous measurements of 2f signal which is composed of 4000 sampling data points are performed to form a 4000*4000 matrix, and each column of the matrix is used as the SR input signal. The different sampling data points in each column of the matrix can be selected incrementally to obtain different SR input signals which are used to get the corresponding SR output signals. In order to better demonstrate the SR system, the 2600 data points in each column of the matrix as SR system input signal are selected to verify the performance of the SR system. On one hand, the SR system input signal is reduced by 120 times because the SR system is only suitable for small signal and then signal is biased by 0.0324V because the bias signal makes SR system signal lies in a linear amplification range from 0.02V to 0.03625V mentioned in the mathematical model of stochastic resonance section, so the parameters of the SR system for 2f signal are the same as those of the second section.
On the other hand, it is found that the peak value of the SR output signal is ∼3.8 times than that of the 4000-time average signal in Fig. 5. The noise level is determined by the standard deviation, deduced from the non-absorption wing of the CH4 spectrum and found to be∼0.003 V(1σ) and∼0.0015 V(1σ) respectively, which correspond to SNR of ∼25.9 and ∼15.8 respectively. This shows that SR system can effectively enhance the signal and improve SNR by 1.83 times.
The corresponding peak value, SD, and SNR are given in Table. As can be seen, SNR gain is about 1.83, which remains almost unchanged with the increase of concentration levels from 1 ppm to 20 ppm. However, the SR system amplifies the interference fringes distorting the baseline of 2f signal, so the SD of the SR 2f signal increases as shown in Table 1. Compared with the magnification of standard deviation, the peak value of 2f signal has a higher magnification, so the SNR is still improved.
Table 1. Peak value, SD, and SNR of the 2f signal with different CH4 concentrationa
The response of the SR 2f signal peak value with raw 2f signal peak value levels is measured and displayed. The CH4 concentrations in range of 1-10 ppm are generated by pure N2. The raw 2f signals for each calibrated CH4 concentrations are measured 4000 time. The linear response of the SR system 2f signal peak value to raw 2f signal peak value is confirmed by fitting the data with a linear slope as shown in Fig. 6.
The R squared value for the linear fit is >0.999. It suggests that the response of the SR 2f signal peak value with different CH4 concentrations is also linear because the raw 2f signal peak value with different concentrations is linear.
5. Conclusion
In this paper, a parameter-tuning stochastic resonance method is proposed, and also proved this method can effectively enhance the performance of the WMS. It is verified that this method can be applied to the photoelectric signals with the low frequency and produces the highest SNR in the optimal resonance state according to the numerical calculation and experiments. Besides, the comparison of time average method with the parameter-tuning stochastic resonance shows that the proposed method can efficiently remove the noise, amplify the signal and improve the SNR. Based on these results, the paper demonstrates that the proposed method is ideally suited for the detection of photoelectric signals under strong noise background.
Funding
Natural Science Foundation of Anhui Province (202004a07020046, 1908085QD156); National Natural Science Foundation of China (NSFC) (62105005); Natural Science Foundation of State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mines (SKLMRDPC19KF12); Research Foundation of the Institute of Environment-friendly Materials and Occupational Health of Anhui University of Science and Technology (Wuhu) (ALW2020YF04); Research Foundation of Anhui Key Laboratory of Mine Intelligent Equipment and Technology (ZKSYS202105).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data is not publicly available at this time but may be obtained from authors upon request.
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