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Adaptive monostable stochastic resonance for processing UV absorption spectrum of nitric oxide

Bo-Qiang Fan, Yu-Jun Zhang, Ying He, Kun You, Meng-Qi Li, Dong-Qi Yu, Hao Xie, and Bo-En Lei

Open Access Open Access

Abstract

When ultraviolet (UV) absorption spectroscopy technology is used for nitric oxide (NO) detection, the background noise will directly affect the accuracy of concentration inversion, especially in low concentrations. Traditional processing methods attempt to eliminate background noise, which damages the absorption spectrum characteristics. However, stochastic resonance (SR) can utilize the noise to extract a weak characteristic signal. This paper reports a monostable stochastic resonance (MSR) model for processing an UV NO absorption spectrum. By analyzing the characteristics of UV absorption spectrum of NO, the evaluation indexes were constructed, thereby an adaptive MSR method was designed for parameter optimization. The numerical simulation confirmed the absorbance peak can be amplified and spectral signal-to-noise ratio (SNR) can be in the stable range of the proposed method, when noise intensity increased. Finally, this experiment obtained a NO detection limit (3σ) of 1.456 ppm and the maximum relative deviation of concentration is 6.32% by this proposed method, which is satisfactory for processing of the UV NO absorption spectrum.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Nitric oxide (NO) is a colorless and odorless gas that is toxic and causes photochemical smog, which is found in vehicle exhaust, industrial waste gas and farmland emissions [1–5]. Thus, it is crucial for the environment to detect NO emission reliably and effectively. The detection of NO by ultraviolet (UV) absorption spectroscopy has been reported in recent years [6–8]. When the specific wavelength UV light is irradiated onto a detector through the measured NO, the UV spectrum can be acquired by processing detector signals. Thereby the NO concentration can be obtained by analyzing the absorption spectrum. However, the amplitude of the absorption spectrum is weak at low concentration conditions and the noise level can be compared with the absorbance peak. Therefore, how to extract the absorbance peak effectively from a strong noise background is important to study.

At present, the absorption spectrum processing methods include moving average [9], wavelet transform [10,11], empirical mode decomposition (EMD) [12] and Kalman filtering [13]. But these methods improve signal-to-noise ratio (SNR) by filtering the noise, which can also damage useful signals. Stochastic resonance (SR) [14,15] is the phenomenon of noise enhancement of weak signals in a nonlinear system and can also be applied to an aperiodic signal [16,17]. Rui Li et al. [18] proposed a SR method by introducing a constant as the external force to detect the chromatographic peak. Xiao-le Liu et al. [19] presented an adaptive SR method based on an artificial fish swarm algorithm for bearing fault detection. Ji-yong Tan et al. [20] applied SR to weak impact signal detection. In these studies the SR methods showed an enhancement of the weak characteristic signal. However, the parameters optimization of these methods is complex and the processing of an UV spectrum has different signal evaluation index from these applications.

In this paper, a monostable stochastic resonance (MSR) model was studied for processing an UV absorption spectrum of NO and the adaptive MSR method was developed by combining with a spectral evaluation index. According to the numerical simulation, the improvement of the simulated absorbance peak with an adaptive MSR method was verified. Furthermore, based on the UV NO detection platform, the absorbance peak of NO was effectively enhanced by an adaptive MSR method, which allows a quantitative analysis of the low NO concentrations.

2. Principle

The detection of NO by ultraviolet (UV) absorption spectroscopy is based on the Beer’s Lambert Law [21]:

$$\left\{ \begin{array}{l} I(\lambda ) = {I_0}(\lambda )\exp ( - \sigma (\lambda )cL)\\ {C_v} = \frac{{kT}}{p}\ast c \end{array} \right.$$
where I0(λ) is the intensity of the incident UV radiation at the wavelength of λ, I(λ) is the intensity of the UV radiation after NO absorption, σ(λ) is the absorption cross section of NO at a wavelength of λ, c is number density of NO molecules, L is the absorption path length, Cv is the volume concentration of NO, k is the Boltzmann constant, T is the temperature and P is the pressure. Figure 1 shows the NO UV absorption cross section in the 200-230 nm wavelength range, which presents three absorbance peaks at 205, 215 and 226 nm, respectively.

 figure: Fig. 1.

Fig. 1. NO UV absorption cross section in the 200-230 nm wavelength range

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When I0(λ) and I(λ) are obtained by the detector, the absorbance A(λ) and the volume concentration Cv of NO show a certain correlation. The absorbance A(λ) is defined as:

$$A(\lambda ) = \ln (\frac{{{I_0}(\lambda )}}{{I(\lambda )}}) = \frac{{\sigma (\lambda )p{C_v}L}}{{kT}}$$
By changing the wavelength λ of the incident light, the absorbance A(λ) at each wavelength can be obtained and then the NO absorption spectrum can be plotted. The NO absorption spectrum reflects the waveform, peak intensity and position for qualitative and quantitative analysis.

However, the absorbance A(λ) contains a high level of random noise, e.g., the optical noise of an optical instrument, dark current noise and electrical noise of instruments [22–24]. Thus, the actual absorbance Aactual(λ) also contains noise, which is as follows:

$${A_{\textrm{actual}}}(\lambda ) = A(\lambda ) + n(\lambda ) = \frac{{\sigma (\lambda )p{C_v}L}}{{kT}} + n(\lambda )$$
where n(λ) is the spectral noise. Therefore, the noise level directly affects the inversion of NO concentration, especially at low concentrations and it is significant for concentration inversion to decrease the noise n(λ) without affecting the absorbance A(λ).

3. Adaptive monostable stochastic resonance scheme

3.1 MSR model and evaluation index

The classical SR model can be described by using nonlinear Langevin equation [25,26]:

$$\frac{{dx}}{{dt}} ={-} \frac{{dU(x)}}{{dx}} + s(t) + n(t)$$
where x is the output. U(x) denotes the potential function of nonlinear system. s(t) denotes the input signal and n(t) denotes the zero mean Gaussian white noise. Due to the UV absorbance of NO is similar to the aperiodic impact signal, it is difficult to achieve a SR effect by forming a stable transition between potential wells of a multi-stable system. Zhang et al. [27] investigated aperiodic signal generated SR effect in the MSR model by changing the exponent of x in a potential function U(x). J. M. G. Vilar et al. [28] presented the SNR is increased with an increase of the noise intensity in monostable systems. Therefore, we denoted U(x) by a fourth-order single-well potential function which can be used in the MSR model and is expressed as:
$$U(x) = a + \frac{1}{4}b{x^4}$$
where the constants a, b are the parameters of the potential function. Figure 2 shows the monostable potential function with a=1 and b=0.001.

 figure: Fig. 2.

Fig. 2. Monostable potential function with a=1 and b=0.001

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Then, the Eq. (4) becomes:

$$\frac{{dx}}{{dt}} ={-} b\ast {x^3} + s(t) + n(t)$$
according to Eq. (3), the UV absorbance of NO is related to the wavelength λ. Therefore, the Eq. (6) becomes:
$$\frac{{d{A_{out}}(\lambda )}}{{d\lambda }} ={-} b\ast {A_{out}}^3(\lambda ) + {A_{in}}(\lambda ) + n(\lambda )$$
where Ain(λ), Aout(λ) are the input and output UV absorbance of NO, n(λ) is the input spectral noise. Equation (7) indicates that output UV absorbance Aout(λ) is obtained by calculating the differential equation. The high-frequency noise can be eliminated by integrating processes. Therefore, MSR can be as a special low-pass filter [29]. When SR takes place in a monostable system, the input extracts the surrounding noise to reach the high point of a monostable potential function to amplify the peak signal. Thus, the weak peak signal can be highlighted by a monostable system.

The absorbance Ain(λ) collected by the spectrometer is the digital signal. Thus the Eq. (7) can be expressed by fourth-order Runge-Kutta algorithm. The algorithm can be described as below:

$$\left\{ \begin{array}{l} {k_1} ={-} b{({A_{out}}[n - 1])^3} + {A_{in}}[n - 1]\\ {k_2} ={-} b{({A_{out}}[n - 1] + {k_1}/2)^3} + {A_{in}}[n - 1]\\ {k_3} ={-} b{({A_{out}}[n - 1] + {k_2}/2)^3} + {A_{in}}[n]\\ {k_4} ={-} b{({A_{out}}[n - 1] + {k_3})^3} + {A_{in}}[n]\\ {A_{out}}[n] = {A_{out}}[n - 1] + h\ast ({k_1} + 2{k_2} + 2{k_3} + {k_4})/6 \end{array} \right.$$
where Aout[n] is the discrete outputs, Ain[n] is the discrete inputs, n=1, 2, 3… N.N is the length of the inputs, b is the MSR system parameter, h=1/fs and fs is the sampling frequency. Since there is only one system parameter b, the complexity of the parameter optimization, which compares to traditional multi-stable systems, can be greatly reduced.

Traditional SR evaluation indices [30,31] are inappropriate for aperiodic signals that have an uncertain frequency. We used the spectral SNR as the evaluation index of MSR. When the spectral SNR is larger, the output has a higher resolution. In the UV spectral region the SNR is the ratio of the measured spectral absorbance peak intensity to the baseline noise intensity. The spectra SNR can be defined as:

$$SN{R_{spec}} = \frac{{{A_{peak}}}}{{\Delta {A_{base}}}} = \frac{{{A_{peak}}}}{{{A_{base\_\max }} - {A_{base\_\min }}}}$$
where ${A_{peak}}$ is the absorbance of the peak, $\Delta {A_{base}}$ is the peak-to-peak (P-P) noise which is calculated by the difference between the maximum ${A_{base\_\max }}$ and minimum ${A_{base\_\min }}$ of the background.

In mechanical fault diagnosis, the Kurtosis index is used to determine the impact state for a mechanical signal [32]. The UV absorbance of NO similarly impacts the signal. We used the Kurtosis index K to determine resolution of gas absorption. When the Kurtosis index K is larger, the output is easier to be distinguish. The Kurtosis index K is defined as:

$$K = \frac{{\frac{1}{N}\ast \sum\limits_{i = 1}^N {{{({A_{out}}[n] - \overline {{A_{out}}} )}^4}} }}{{[\frac{1}{N}\ast \sum\limits_{i = 1}^N {{{({A_{out}}[n] - \overline {{A_{out}}} )}^2}{]^2}} }}$$
where $\overline {{A_{out}}}$ is the mean of the outputs.

According to spectral SNR and the Kurtosis index K, we estimated whether the MSR can enhance the UV absorbance of NO

3.2 Adaptive MSR method

By means of the above analysis, the flow chart of the proposed adaptive MSR method for processing of the UV NO absorption spectrum is illustrated in Fig.  3.

 figure: Fig. 3.

Fig. 3. Flow chart of the proposed adaptive MSR method for processing UV absorption spectrum of NO.

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The detailed steps of the algorithm are as follows:

Step 1: According to the NO spectrum which is collected by means of a spectrometer, the absorbance of NO can be calculated. Next the absorbance is normalized in the range of 0 to 1 to guarantee all the inputs have the same intensity in a MSR system.

Step 2: Set initial system parameter b, the searching range of the system parameter b, the search step length h, the system sampling frequency fs and the range of positive Kurtosis K.

Step 3: Input the processed absorbance to the MSR system. The output of the system can be obtained by the fourth-order Runge-Kutta algorithm. Next MSR system is adjusted by means of the parameter b according to the search step-length h, circularly obtained by the output of the above algorithm, until the searching range of system parameter b is traversed. All the outputs are stored.

Step 4: Calculate the Kurtosis of the stored outputs and according to the range of positive Kurtosis and filter the stored outputs. Then calculate the spectral SNR of the filtered outputs and obtain the optimal output which has the maximum spectral SNR.

Step 5: The final the absorbance of NO can be obtained by inverse normalization of the optimal output, which is filtered through the step 4.

3.3 Numerical simulation

To analyze the feasibility of the method, the simulated UV absorbance of NO can be replaced by a Lorentzian function with noise which can be described as below:

$$\Phi (\lambda ) = \frac{{\Delta {\lambda _n}}}{{2\pi }}\ast \frac{A}{{{{(\lambda - {\lambda _0})}^2} + {{(\frac{{\Delta {\lambda _n}}}{2})}^2}}} + n(\lambda )$$
where A λ, λ0 Δλn and n(λ) contribute to the peak area, the wavelength, the wavelength of the peak, the full width at half maximum and the Gaussian white noise, respectively. We set λ0=215 nm, A=0.4, Δλn=0.4. The range of wavelength λ is from 213 nm to 217 nm and the sampling frequency fs is 1 kHz. The intensity of the noise n(λ) is from 0.005 to 0.1. Figure  3 shows the inputs and outputs of the adaptive MSR method when noise intensity is 0.005 0.02 and 0.08, respectively. Due to the fact that the simulated signal is known, we can also use the signal SNR to evaluate the signal quality. The signal SNR is defined as:
$$SN{R_{sig}} = 10\ast \lg \frac{{\sum\limits_{n = 1}^N {A_{out}^2[n]} }}{{\sum\limits_{n = 1}^N {{{({A_{ori}}[n] - {A_{out}}[n])}^2}} }}$$
Where ${A_{ori}}[n]$ is the discrete noiseless original signal. Figure 4 shows the input signal SNR and output signal SNR of the adaptive MSR method. Figure 4 illustrates that the peak of outputs can be amplified by the adaptive MSR method. Figure 5 shows the signal SNR is improved by the adaptive MSR method, especially when the noise intensity is large, the improvement is obvious. Figure 6 shows the spectral SNR is within a stable range (standard deviation σ=0.055), as the noise intensity increases. Therefore, according to the above simulation results, the adaptive MSR method can enhance the resolution of the UV absorbance of NO and efficiently extract the peak in a strong noise condition.

 figure: Fig. 4.

Fig. 4. The simulated inputs with different noise intensity and outputs: (a) (c) (e) the inputs with the noise intensity of 0.08 0.02 and 0.005, respectively; (b) (d) (f) the obtained outputs by the adaptive MSR method when the noise intensity of the inputs is 0.08 0.02 and 0.005, respectively.

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 figure: Fig. 5.

Fig. 5. The input signal SNR and output signal SNR of the adaptive MSR method

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 figure: Fig. 6.

Fig. 6. The input spectral SNR and output spectral SNR of the adaptive MSR method

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4. Experiment verification

4.1 Experiment apparatus

The NO detection system is schematically shown in Fig. 7, which is based on the UV spectrometer (Ocean Optics USB4000) and a xenon lamp (Hamamatsu, L9456-1). The standard gas of NO (503 ppm and 9.5 ppm) and N2 (99.9%) provide different mixing ratios by using the gas distribution module which is based on a mass flow-meter (Sevenstar Electronics, CS100). Next the mixed gas is pumped into the sample cell of length L = 40 cm. The xenon lamp emits a beam with a spectrum of a wide wavelength range (185-2000nm), which is directed into the sample cell using collimating. The UV spectrometer with fiber coupling collects the NO absorption spectrum and transmits the spectral data to the MCU (FriendlyARM, Smart210) for processing and storage. Finally, the concentration data and the spectral curve are send to the host computer for display.

 figure: Fig. 7.

Fig. 7. Schematic of experimental NO detection system

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4.2 Results and discussion

According to the above system, a set of NO concentrations of 2.85, 3.8, 6.65, 9.5, 25.15, 50.3, 100.6 ppm were injected into the sample cell and then the spectral data was analyzed to verify the adaptive MSR method. Figure 8(a) shows the unprocessed absorbance peak of NO with 6.65 ppm at 215 nm. The peak was almost covered by noise and it is difficult to be used for determining the inversion concentration. Figure 8(b) and Fig. 8(c) respectively show the processed absorbance peak by the exponentially weighted average [33] (weighting factor α=0.85) and the adaptive MSR method. The peak can be highlighted by two methods. After processing by exponentially weighted average, the baseline and peak are smoothed and the spectral SNR is increased by decreasing the noise intensity. For the adaptive MSR method, the baseline is elevated and the peak is also amplified simultaneously and thus the spectral SNR is increased by enhancing the peak. Therefore, compared with the exponentially weighted average, the adaptive MSR method presents the optimum peak detection performance.

 figure: Fig. 8.

Fig. 8. The absorbance peak of NO with 6.65 ppm at 215nm: (a) the unprocessed absorbance peak; (b) the processed absorbance peak by the exponentially weighted average; (c) the processed absorbance peak by the adaptive MSR method

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Figure 9 shows the absorbance peaks of NO at different concentrations can be amplified by the adaptive MSR method. The adaptive MSR method also maintains the correlation between the peak and the concentration. As shown in Table 1, the adaptive MSR method presents a better spectral SNR at low concentrations. The spectral SNR tends to be consistent by two methods in high concentration because when the unprocessed peak has been highlighted, the peak intensity has a lower effect on the spectral SNR than the p-p noise. The p-p noise of the adaptive MSR method is higher than traditional method of decreasing noise intensity. Figure 10 shows the calibration curve of concentration (from 2.85 to 100.6 ppm) and absorbance peak of NO by the adaptive MSR method, which can be obtained as c=55.07(±2.29)*a-0.05007(±1.823) (R2 = 0.9987), where a is the NO peak height at λ=215 nm and c is the NO concentration. By calculating the baseline noise, we estimated that the detection limit (3σ) is 0.02735 AU or 1.456 ppm. In order to verify the accuracy of the calibration curve by the adaptive MSR method, we prepared a set of NO concentration from 2.85 to 9.5 ppm for testing. Figure 11 shows the correlation between prepared concentrations and the calculated concentrations as well as the maximum relative deviation is 6.32%. This deviation is satisfactory to detect the NO concentration.

 figure: Fig. 9.

Fig. 9. The absorbance peaks of NO at different concentrations (A: 2.85 ppm, B: 3.8 ppm, C: 6.65 ppm, D: 9.5 ppm, E: 25.15 ppm, F: 50.3 ppm, G: 100.6 ppm): (a) the unprocessed absorbance peaks; (b) the processed absorbance peaks by the adaptive MSR method

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 figure: Fig. 10.

Fig. 10. The calibration curve of the NO concentration (from 2.85 to 100.6 ppm) and absorbance peak of NO by the adaptive MSR method

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 figure: Fig. 11.

Fig. 11. The correlation between the prepared concentration from 2.85 to 9.5 ppm and the calculated concentration

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Tables Icon

Table 1. The comparison with the spectral SNRs of the absorbance peak of NO at different concentrations by two methods

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5. Conclusion

The adaptive MSR method is developed for processing the UV NO absorption spectrum. The NO absorbance peaks of the simulation and the experiment were obtained the better spectral SNR and the higher peaks were achieved by the proposed method. The calibration curve of concentration and absorbance peaks using of the adaptive MSR method was verified to be linear. Moreover, the relative deviation of the calculated concentrations using the proposed method and the prepared concentrations was within a reasonable range. Therefore we can expect that the adaptive MSR method will be an effective tool for the obtaining the absorbance peaks in strong noise conditions.

Funding

National Key Research and Development Program of China (2016YFC0201000); Instrument and Equipment Function Development Technology Innovation of the Chinese Academy of Sciences (Y83H3y1251); Strategic Priority Research Program of the Chinese Academy of Sciences (XDA23010204).

Disclosures

The authors declare no conflicts of interest.

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Figures (11)
Fig. 1.
Fig. 1. NO UV absorption cross section in the 200-230 nm wavelength range

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Fig. 2.
Fig. 2. Monostable potential function with a=1 and b=0.001

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Fig. 3.
Fig. 3. Flow chart of the proposed adaptive MSR method for processing UV absorption spectrum of NO.

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Fig. 4.
Fig. 4. The simulated inputs with different noise intensity and outputs: (a) (c) (e) the inputs with the noise intensity of 0.08 0.02 and 0.005, respectively; (b) (d) (f) the obtained outputs by the adaptive MSR method when the noise intensity of the inputs is 0.08 0.02 and 0.005, respectively.

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Fig. 5.
Fig. 5. The input signal SNR and output signal SNR of the adaptive MSR method

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Fig. 6.
Fig. 6. The input spectral SNR and output spectral SNR of the adaptive MSR method

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Fig. 7.
Fig. 7. Schematic of experimental NO detection system

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Fig. 8.
Fig. 8. The absorbance peak of NO with 6.65 ppm at 215nm: (a) the unprocessed absorbance peak; (b) the processed absorbance peak by the exponentially weighted average; (c) the processed absorbance peak by the adaptive MSR method

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Fig. 9.
Fig. 9. The absorbance peaks of NO at different concentrations (A: 2.85 ppm, B: 3.8 ppm, C: 6.65 ppm, D: 9.5 ppm, E: 25.15 ppm, F: 50.3 ppm, G: 100.6 ppm): (a) the unprocessed absorbance peaks; (b) the processed absorbance peaks by the adaptive MSR method

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Fig. 10.
Fig. 10. The calibration curve of the NO concentration (from 2.85 to 100.6 ppm) and absorbance peak of NO by the adaptive MSR method

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Fig. 11.
Fig. 11. The correlation between the prepared concentration from 2.85 to 9.5 ppm and the calculated concentration

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Tables (1)

Tables Icon

Table 1. The comparison with the spectral SNRs of the absorbance peak of NO at different concentrations by two methods

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Equations (12)

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{ I ( λ ) = I 0 ( λ ) exp ⁡ ( − σ ( λ ) c L ) C v = k T p ∗ c
A ( λ ) = ln ⁡ ( I 0 ( λ ) I ( λ ) ) = σ ( λ ) p C v L k T
A actual ( λ ) = A ( λ ) + n ( λ ) = σ ( λ ) p C v L k T + n ( λ )
d x d t = − d U ( x ) d x + s ( t ) + n ( t )
U ( x ) = a + 1 4 b x 4
d x d t = − b ∗ x 3 + s ( t ) + n ( t )
d A o u t ( λ ) d λ = − b ∗ A o u t 3 ( λ ) + A i n ( λ ) + n ( λ )
{ k 1 = − b ( A o u t [ n − 1 ] ) 3 + A i n [ n − 1 ] k 2 = − b ( A o u t [ n − 1 ] + k 1 / 2 ) 3 + A i n [ n − 1 ] k 3 = − b ( A o u t [ n − 1 ] + k 2 / 2 ) 3 + A i n [ n ] k 4 = − b ( A o u t [ n − 1 ] + k 3 ) 3 + A i n [ n ] A o u t [ n ] = A o u t [ n − 1 ] + h ∗ ( k 1 + 2 k 2 + 2 k 3 + k 4 ) / 6
S N R s p e c = A p e a k Δ A b a s e = A p e a k A b a s e _ max − A b a s e _ min
K = 1 N ∗ ∑ i = 1 N ( A o u t [ n ] − A o u t ¯ ) 4 [ 1 N ∗ ∑ i = 1 N ( A o u t [ n ] − A o u t ¯ ) 2 ] 2
Φ ( λ ) = Δ λ n 2 π ∗ A ( λ − λ 0 ) 2 + ( Δ λ n 2 ) 2 + n ( λ )
S N R s i g = 10 ∗ lg ⁡ ∑ n = 1 N A o u t 2 [ n ] ∑ n = 1 N ( A o r i [ n ] − A o u t [ n ] ) 2
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