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Abstract
A dual-optimized adaptive Kalman filtering (DO-AKF) algorithm based on back propagation (BP) neural network and variance compensation was developed for high-sensitivity trace gas detection in laser spectroscopy. The BP neural network was used to optimize the Kalman filter (KF) parameters. Variance compensation was introduced to track the state of the system and to eliminate the variations in the parameters of dynamic systems. The proposed DO-AKF algorithm showed the best performance compared with the traditional multi-signal average, extended KF, unscented KF, KF optimized by BP neural network (BP-KF) and KF optimized by variance compensation (VC-KF). The optimized DO-AKF algorithm was applied to a QCL-based gas sensor system for an exhaled CO analysis. The experimental results revealed a sensitivity enhancement factor of 23. The proposed algorithm can be widely used in the fields of environmental pollutant monitoring, industrial process control, and breath gas diagnosis.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
This work focuses on filtering techniques for laser spectroscopy, which has been extensively used in atmospheric environmental monitoring, industrial process control, and medical diagnosis [1–3]. Exhaled breath analysis is an emerging field that offers a non-invasive, painless, and online means for medical diagnosis with a fast response time. However, the sensitivity and resolution of measurements are affected by intrinsic noises and interferences induced by electrical components and background changes [4–5]. Therefore, a highly precise processing of the measured spectra data is crucial.
Several filtering techniques based on hardware have been recently developed to process the measured spectra [6–7]. However, their practical application has been limited by their complex system structure, high cost, and poor compatibility. Moreover, the intrinsic noises, including flicker, shot, and detector noises, are difficult to remove [8–9]. Therefore, a filtering technique based on software has become an ideal choice due to its simple system structure and low cost. Although numerous filtering techniques based on software have been applied to process gas absorption spectra, only few techniques have been considered effective for the online filtering of these spectra. Although widely used, the multi-signal averaging filtering (MAF) is a time-consuming method that can only suppress white noise [10–11]. Meanwhile, the wavelet transform denoising technique allows for an efficient online filtering [12–13], but depends on a large number of parameters and involves a complex denoising calculation. The discontinuity of wavelet coefficients near the threshold also limits the filtering performance of this approach [14–15]. An adaptive Savitzky–Golay filter algorithm has been recently proposed for real-time infrared gas detection and has demonstrated an excellent performance in optimizing the filtering process. However, this approach can only filter out high-frequency noise [16–17].
Kalman filtering (KF), as an efficient tool for reducing the noise of polluted signals, has recently attracted the interest of researchers and has been successfully applied in the fields of GPS positioning and navigation, satellite measurement and control, biomedicine, and signal processing [18–20]. KF does not need to store a large number of measured data and demonstrates a fast processing speed. Therefore, KF is a suitable filtering technique for real-time exhaled breath detection [21–22]. However, in a nonlinear system, KF causes a serious deviation in the spectral absorption part. To address this problem, an extended KF (EKF) algorithm has been proposed. This approach can achieve an optimal prediction and control of the nonlinear model, thereby ensuring its applicability [23–24]. Although this approach obtains a good effect in nonlinear systems, EKF has an uncertain system model and filtering parameters for a fuzzy system. Recently, an adaptive KF (AKF) algorithm that combines system identification and filtering estimation has been proposed. This approach continuously evaluates the dynamics of the system and then estimates and modifies the model parameters and noise statistical characteristics to reduce the actual error of filtering [25–26]. AKF shows an excellent performance in dynamic systems and has been applied to measure non-stationary signals, such as instantaneous power and time-varying spectra [27–28]. The filtering deviation of the nonlinear system in AKF is usually corrected by selecting the sub-optimal filtering results that reduce the efficiency of the algorithm.
In this paper, a dual-optimized adaptive KF (DO-AKF) algorithm is developed based on back propagation (BP) neural network and variance compensation to enhance the measurement system performance of KF. The BP neural network is used to optimize the KF parameters, and variance compensation is applied to obtain an accurate system state equation and filtering result. Compared with KF, DO-AKF can realize “self-adjustment” and “tracking feedback” to achieve optimal signal-to-noise ratios (SNRs) of the spectra. The construction of DO-AKF is discussed in section 2. The simulation of the absorption spectrum filtering process and the experimental setup of the exhaled CO gas detection are described in section 3. The denoising results of several spectral filtering algorithms are compared in section 4, and the proposed DO-AKF demonstrates the best performance.
2. Mathematical model and algorithm theory
KF is an optimal state estimation algorithm that is widely used in linear systems. However, this approach may cause a serious deviation in nonlinear signal processing. The system adaptability can be improved by introducing the BP neural network.
The BP neural network has a hierarchical feed forward network architecture. The output of each layer is sent directly to each neuron in the above layer. The entire process requires a minimum of three layers, namely, the input layer, hidden layer, and output layer [29]. The structure of the BP neural network is shown in Fig. 1.
The error and correlation of the dynamic system may influence the stability of KF. To improve filtering stability and accuracy, the filtering variance matrix needs to be optimized by using variance compensation.
Variance compensation is coupled with the spectral vector and prediction residual [30]. The spectral noise variance matrix is estimated in real time. To compensate for the limitations in spectral noise variance and covariance estimation, the spectral filtering vector needs to be predicted and corrected continuously. The proposed DO-AKF is constructed by combining KF theory with the BP neural network and variance compensation algorithm, for which flow chart is shown in Fig. 2.
2.1. Spectral system process model
In laser spectroscopy, the measurement equation and the state equation in the state space model can be characterized by the spectral state and measurement vector in the spectral filtering system. The spectral filtering system and measurement vector are defined as
where k is the discrete time index, $X(k)$ is the spectral state vector, $U(k)$ is the process control vector of the system, $Z(k)$ is the measurement vector. A, $B$, and $H$ are the spectral state transition matrixes. $W(k)$, $V(k)$ are the process and measurement noises, respectively. All of these parameters A, $B,\, H,\, W(k)$ and $V(k)$ are the hyperparameters of the state space model in the meanwhile.The spectral state vector $X(k)$ and measurement vector $Z(k)$ can be substituted by the estimated and detected values of spectral data, respectively. As for the hyperparameters in the state space model, it is generally assumed that the parameters of the state space model are known and constants for the convenience of calculation. However in the proposed spectral filtering system, BP neural network is structured to traverse the optimal values of these hyperparameters or the state transition matrixes A, $B$, and H instead of defining them as constants as usual. The process and measurement noises are represented by $W(k)$ and $V(k)$, where the expected value, $E[{W(k)} ]= E[{V(k)} ]= 0$. And their covariance matrices are represented by $Q(k)$ and $R(k)$, respectively. The covariance $Q(k)$ represents the process noise introduced by the actual optical path, while $R(k)$ represents the measurement noise introduced by the sensor.
The spectral vectors are not constant in the actual system. Therefore, the KF algorithm cannot accurately reflect the time-varying characteristics of the spectral model. The BP neural network is used to optimize the real-time system states and vectors, and each matrix is updated in each step of KF.
1) Time recursive state variables
According to the system model, the predicted state vector can be obtained recursively as
where $X(k|k - 1)$ is the predicted spectral state vector, and $X(k - 1|k - 1)$ is the previous state vector.2) Forward calculation of error covariance
The corresponding predicted covariance matrix P is obtained as
where ${A^T}$ means the transpose operation of A. $P(k|k - 1)$ is the corresponding predicted covariance matrix, and $P(k - 1|k - 1)$ is the previous covariance matrix.2.2. Spectral parameters update equation of KF
The original absorption spectra includes two parts, i.e. the non-absorption baseline part (corresponding to laser intensity background) and the molecular absorption peak part. In this case, the non-linear laser intensity background makes large errors and even diverging problems when a common KF algorithm model is used. Therefore, BP neural network is innovatively proposed to traverse their optimal values, instead of defining them as constants in a common KF algorithm model. The variance compensation is applied to compensate for the inadequacy of dynamic noise equation and covariance estimation. Combining with the precise state transition parameters and the variance compensation, the improved KF algorithm model can adapt to the instability of the nonlinear system (i.e. the original spectroscopy signal).
2.3. BP neural network and variance compensation optimization
Based on the variance compensation principle, the prediction residual $F(k)$ is defined as
where $L(k)$ and $\hat{L}(k|k - 1)$ denote the measured and best predicted vectors, respectively. The prediction residuals are used to estimate and correct the covariance vector of dynamic noise and to compensate for the deficiencies of the dynamic noise equation or covariance estimation in the filtering process.The BP neural network is constructed through the spectral state vector, and the following parameters are selected for the input layer of the BP neural network: (a). the difference between the spectral state and predicted state vectors ($X(k) - X(k|k - 1)$), (b). the gap between the measured vector and forecast measured data ($Z(k) - HX(k|k - 1)$), and (c). KF gain filtering $Kg$. After optimizing the BP neural network, the filtering compensation vector $\Delta X(k)$ is defined as
where D is the theoretical vector after filtering. By optimizing the BP neural network and variance compensation, DO-AKF can be used to improve the spectral filtering accuracy.3. Experimental details
The spectral data is usually polluted by various noises, which are categorized into two sources: interference noises due to spectral and optical interferences, and electronic noises from detectors, lasers, and additional electronics. Therefore, it is significant to perform data preprocessing to obtain high SNR data. In this study, a dual-optimized adaptive DO-AKF algorithm based on BP neural network and variance compensation was developed for high-sensitivity trace gas detection. To demonstrate the applicability for this technique, the simulated spectra by referring to the HITRAN database [31] and experimental CO data were used to evaluate the proposed algorithm, and detailed comparison with other common algorithms are also presented.
3.1 Optimization of the BP neural network
Similar to other digital signal processing techniques, the effectiveness of KF largely depends on the system state equation. The selection of an appropriate system state equation is essential to achieve a trade-off between reducing noise and avoiding bias. In this work, a reasonable BP neural network is constructed to optimize the system state equation and correct the filtering result.
The number of neurons in the input and output layers of the BP neural network is determined by the dimensional data of the input and output vectors. Given that the input and output data are composed of wave number and transmittance, the number of neurons in the input and output layers is 2. The BP neural network with a single hidden layer can theoretically approximate any continuous function in a closed interval. The number of hidden layer units can be determined by $P\mbox{ = }\sqrt {N + M} \mbox{ + }a$, where $N$ and $M$ denote the number of input and output layer units, respectively [32]. However, this equation only provides a rough estimation, and both convergence and learning efficiency need to be improved. In practical applications, the optimization of the hidden layer number and number of hidden layer units is crucial to achieve the best SNR. The simulated spectra shown in section 4.1 are used for constructing the optimal network structure to find the optimal SNR and optimizing the filter. The number of hidden layers and hidden layer units ranges between 1 and 10, and the filtering results of different hidden layer and unit numbers are compared. The SNRs under different layer and unit numbers are shown in Fig. 3, and the results for the optimal number of hidden layer units under each hidden layer are shown in Table 1. Figure 3 shows that the effectiveness of final denoising largely depends on the hidden layers and hidden layer units. The optimal effect is achieved under the combination of four hidden layers and seven hidden layer units.
Table 1. Optimal hidden layer unit number and filtered SNR statistics for different hidden layer numbers (ranging from 1 to 10).
Fig. 3. SNR distribution of filtered signals with different combinations of hidden layer numbers and hidden layer units (ranging from 1 to 10).
3.2 Experimental setup
To demonstrate the performance of DO-AKF, this algorithm is applied to process the absorption spectroscopy of the exhaled CO. The experimental setup is illustrated in Fig. 4. A thermoelectrically cooled CW room temperature QCL (ALPES, Switzerland) emitting at ∼4.6 µm is used as the optical source with a maximum peak output power of 30 mW. The laser output power and wave number are controlled by temperature (Starter Kit, Alpes) and current controllers (LDX 3232, ILX Light wave), respectively. The linewidth of laser is estimated to be less than 0.001 cm−1, and the beam of the QCL laser passes through a CaF2 mirror to combine the tracer laser with the trace laser. The astigmatic Herriott multi-pass cell with an effective optical path length of 62.5 m is used as gas cell. A ZnSe beam splitter is used to co-align a visible red light (632.8 nm) for beam tracing purposes. The exiting beam is then focused into a thermoelectric cooled mercury cadmium telluride detector (Vigo, VI-4TE-5) with a 75 mm focal length plano-convex lens. This detector does not require liquid nitrogen cooling and can be used for long-term operations. The laser frequency is scanned from 2188.5 cm−1 to 2191.5 cm−1 by using a triangular wave at a typical frequency of 100 Hz. The detector signal is subsequently acquired by using a DAQ card (National Instruments, USB-6259), and the data are processed and saved by a LabVIEW program. A polytetrafluoroethylene filter (0.22 µm) for dust retention and an integrating mass flow controller (Alicat, USA) are connected to the inlet of the multi-pass cell. The system pressure is maintained by using a diaphragm pump and pressure controller to avoid potential upstream pressure transients.
4. Analysis and results
4.1 Results and analysis of simulated data
In the range of 2188.5 cm−1 to 2191.5 cm−1 and pressure of 1 atm, the spectrum of CO satisfying the Viogt line-shape function at a temperature of 296 K is obtained by using the HITRAN database. An optical path length of 62.5 m is used and the CO mixing ratio is 1.4 ppm. The spectral data are superimposed with the Gaussian noise (white noise) and non-Gaussian noise, respectively. Under the background of Gaussian noise (white noise, the mean value is 0, the variance is 1 and the standard deviation is 0.009), the SNR of the spectral data is 7.9162 dB. Noise pollution greatly reduces the accuracy of the spectrum. The MAF, EKF, unscented KF (UKF), variance compensation optimized KF (VC-KF), BP neural network optimized KF (BP-KF), and DO-AKF are used to process the simulated spectrum. The filtered results are shown in Fig. 5, and the SNRs of different filtering methods are shown in Table 2.
Fig. 5. Simulation of the CO gas absorption spectra of the six filtering algorithms (under the background of Gaussian noise).
Table 2. Comparison of the SNR and RMSE for the filtered signal obtained by different filtering methods.
As shown in Fig. 5 and Table 2, the SNRs have increased from 7.9162 dB to 20.2685 dB, and the root mean square error (RMSE) has decreased from 5.255×10−5 to 2.401×10−6. The filtering effect of MAF is not obvious, whereas the spectra after the application of EKF, UKF, VC-KF, BP-KF, and DO-AKF are relatively smooth. The VC-KF shows a more effective filtering effect than EKF, and UKF has higher filtering SNR than BP-KF. The DO-AKF shows the best performance and obtains the highest SNR among the six filtering algorithms.
The frequency response of the filter is an important property of filtering method. The residual calculated from the difference between the filtered data and the origin data without noise is used to evaluate the performance of the proposed filter algorithm. The frequency domain distribution of the residual noise is analyzed by Fast Fourier Transformation (FFT) before and after applying various filters, as shown in Fig. 6. As can been seen, MAF and EKF have poor noise filtering performance, which still retain the frequency distribution characteristics of the original noise. UKF, VC-KF, BP-KF, DO-AKF can remove more high frequency noise. DO-AKF shows the best filtering performance, and the amplitude of low-frequency noise can be reduced to one third compared to UKF, VC-KF and BP-KF algorithm.
Indeed, the simulated noised in Fig. 5 is Gaussian noise (white noise). In order to demonstrate the influence of the non-Gaussian noise on the proposed DO-AKF algorithm. The non-Gaussian background noise (exponential distribution noise, mean parameter = 0.005) is simulated for test, and the filtered results are shown in Fig. 7, and the SNRs of different filtering methods are summarized in Table 3.
Fig. 7. Simulation of the CO gas absorption spectra of the six filtering algorithms. (under the background of non-Gaussian noise).
Table 3. Comparison of the SNR and RMSE for the filtered signal obtained by different filtering methods.
As shown in Fig. 7 and Table 3, in the case of non-Gaussian noise, a spectral data with a SNR of 8.2634 dB is simulated. After the application of DO-AKF filtering algorithm, the SNR has been improved from 8.2634 dB to 21.1207 dB, and the RMSE has decreased from 4.913×10−5 to 2.286×10−6. Although UKF and EKF are both non-linear filtering methods, UKF has better filtering effect (the second highest filtering SNR) than EKF. And DO-AKF also shows the best performance and obtains the highest SNR under these non-Gaussian noise among the six filtering algorithms.
The transmission spectrum and residual of EKF and VC-KF are compared in Fig. 8. VC-KF obtains a lower residual than EKF, whereas the SNR significantly increases from 14.2457 dB (EKF) to 16.1512 dB (VC-KF). Variance compensation is used to optimize the filtering process in real time. The variance is more flexible in noise reduction and can compensate for the filtering variance based on the current filtering state.
Fig. 8. Comparison of transmission spectra and residual distributions for the EKF and VC-KF algorithms. (a) EKF and residual of the CO gas absorption spectra; and (b) VC-KF and residual of the CO gas absorption spectra
Given the aforementioned factors, DO-AKF is identified as the most suitable filtering algorithm for extracting a gas absorption signal from a contaminated signal. Depending on the optimal traversal of BP neural network to the filtering parameters and the optimal compensation of variance to the filtering covariance, DO-AKF can be adapted to both Gaussian and non-Gaussian noise and maintain the optimal filtering performance.
4.2 Results and analysis of experimental data
To demonstrate the effectiveness of DO-AKF in processing the spectral signal, the measured and filtered spectra of exhaled CO at a pressure of 1atm and temperature of 296 K are shown in Fig. 9(a). In our case, measurement time resolution of 1 Hz is commonly selected for practical application, which including data acquisition, signal average, filtering process and spectral fitting, data storage. We restricted the spectral data records to 1024 sample points, sampling rate of data acquisition board is set to 200 kHz. Higher responsivity can also be realized by using high-speed data acquisition card. The absorption characteristic is partially submerged in noise. The filtered transmission spectrum has been significantly smoothed, thereby facilitating the identification and fitting of the absorption spectrum. The measured transmission spectrum of exhaled CO is converted into a gas absorption spectrum and fitted by using the Voigt function as shown in Fig. 9(b). The Doppler broadened HWHM ωG is a function of temperature T and molecular weight M, which can be written as:
Fig. 9. (a) Typical example of experimentally determined and filtered CO absorption spectra at a pressure of 1 atm; and (b) absorption spectra of CO and fitted spectra with a Voigt function centered at ∼2190.017 cm−1.
The bulk matrix of breath is a mixture of N2, O2, CO2, H2O and other compounds, such as CO and N2O. The main constitutes of human breath, N2 and O2, usually do not interfere with the spectra of CO. Therefore, the predominant interference specie for breath CO detection is H2O and CO2. The line intensity of CO is several orders of magnitude higher than H2O and CO2. Generally, by reducing sample gas pressure, the spectral interference of H2O and other species can be effectively reduced. To verify the universality of the DO-AKF algorithm for spectral data at different pressures, a series of spectra are measured at different pressures (30, 50, 80, and 100 mbar) near 2190.01 cm−1, and the filtered results of the measured spectral data after DO-AKF filtering process are shown in Fig. 10. At the same time, the top-left panels and top-right panels show the filtering details near absorption peaks. The algorithm can effectively eliminate the interference of background noise.
Fig. 10. The filtered results of the real-time measured exhaled CO gas spectral data (at four different pressures) after the filtering process of DO-AKF.
Allan variance is used to investigate the performance of the sensor system. The custom CO gas is injected into the sample cell for a time scale of several minutes, and a pressure of 150 mbar is maintained. The CO concentrations are calculated from the fitted peak area of the acquired spectra. The real-time measured CO concentration levels are shown in Fig. 11(a), and the Allan deviation is shown in Fig. 11(b). The average and standard deviations of the measured data are 457.45 ppb and 18.20 ppb, respectively, while the average and standard deviations of the filtered data are 457.27 ppb and 0.86 ppb, respectively. As shown in Fig. 11(b), the optimum sensitivity of 0.81 ppbv is reached at 181 s.
Fig. 11. (a) Real-time measured CO concentrations; and (b) the Allan deviation plot of the measured and filtered data.
5. Conclusion
In this paper, the DO-AKF algorithm was developed based on BP neural network and variance compensation to achieve a high-sensitivity trace gas detection in laser spectroscopy. The BP neural network was used to optimize the KF parameters, and variance compensation was introduced to eliminate the parameter variation and divergence of dynamic systems. The developed algorithm was then applied to a QCL based gas sensor system to measure exhaled CO, and a sensitivity enhancement factor of 23 was obtained. DO-AKF showed two main advantages over the other algorithms, including MAF, EKF, UKF, VC-KF, and BP-KF. First, DO-AKF showed a significant filtering effect. Second, in dynamic systems, DO-AKF demonstrated universality by using variance compensation and BP neural network. Therefore, DO-AKF can be widely used in the fields of environmental pollutant monitoring, industrial process control, and breath gas diagnosis.
Funding
University Natural Science Research Project of Anhui Province (KJ2018A0034); National Basic Research Program of China (973 Program) (2016YFC03022022016YFC0301900); National Natural Science Foundation of China (41875158, 61705002, 61905001); Natural Science Foundation of Anhui Province (1808085QF198, 1908085QF276).
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