In a non-orthogonal multiple access (NOMA) system, the successive interference cancellation (SIC) procedure is typically employed at the receiver side, where several user’s signals are decoded in a subsequent manner. Fading channels may disperse the transmitted signal and originate dependencies among its samples, which may affect the channel estimation procedure and consequently affect the SIC process and signal detection accuracy. In this work, the impact of Deep Neural Network (DNN) in explicitly estimating the channel coefficients for each user in NOMA cell is investigated in both Rayleigh and Rician fading channels. The proposed approach integrates the Long Short-Term Memory (LSTM) network into the NOMA system where this LSTM network is utilized to predict the channel coefficients. DNN is trained using different channel statistics and then utilized to predict the desired channel parameters that will be exploited by the receiver to retrieve the original data. Furthermore, this work examines how the channel estimation based on Deep Learning (DL) and power optimization scheme are jointly utilized for multiuser (MU) recognition in downlink Power Domain Non-Orthogonal Multiple Access (PD-NOMA) system. Power factors are optimized with a view to maximize the sum rate of the users on the basis of entire power transmitted and Quality of service (QoS) constraints. An investigation for the optimization problem is given where Lagrange function and Karush–Kuhn–Tucker (KKT) optimality conditions are applied to deduce the optimum power coefficients. Simulation results for different metrics, such as bit error rate (BER), sum rate, outage probability and individual user capacity, have proved the superiority of the proposed DL-based channel estimation over conventional NOMA approach. Additionally, the performance of optimized power scheme and fixed power scheme are evaluated when DL-based channel estimation is implemented. A non-orthogonal multiple access (NOMA) system has been classified as promoting a multiple access structure for future wireless systems to boost system throughput and spectral efficacy. NOMA could utilize the present resources more effectively by resourcefully benefiting from the users’ channel environments and providing numerous users with distinct quality of service (QoS) demands. NOMA enables several users to achieve simultaneous arrival to the same time-frequency block by superpositioning them in the code or power domains . The conIn the NOMA scheme, receiver equipment will receive the multiplexing of symbols from different users in the system; therefore, elimination of interference from other users is necessary for coordinated decoding.Generally, multiuser detection (MUD) in NOMA can be managed through SIC, which was carried out in the power domain. In the SIC technique, symbols from several users are decoded consecutively based on the allocated signal power and the channel state information (CSI) . CompletIn , the autIn , the autIn , the autIn , a deep Based on the deep learning (DL) algorithm, the authors in  introducIn , the autIn , a pilotIn , the autIn , the autBased on the previous works, many deep neural networks (DNN) approaches have been proposed explicitly to address the issues associated with channel state information (CSI), channel estimation, and signal detection. To the best of authors’ knowledge, there has been no study that has investigated the combination between Deep learning (DL)-based channel estimation algorithm and the optimal power allocation scheme for multiuser detection in a downlink non-orthogonal multiple access (NOMA) system in fading channels. Most of the works are managing the issues of power optimization and deep learning-based channel estimation separately.In addition, many of the proposed DNN approaches for channel estimation require pre-training to obtain the appropriate DNN weights initialization, which may lead to an increase of the number of hidden layers with a huge number of neurons in each layer. In order to enhance the aforementioned schemes, this work proposes a framework that examine the integration between DNN-based channel estimation and the optimum power allocation scheme in a multiuser NOMA system and then inspect the system performance. Furthermore, in our proposed system structure, we manage to eliminate the need for a pre-training stage, minimize the number of DNN layers, and minimize the number of pilot symbols, and at the same time, we can realize a notable improvement in network convergence.In this work, a framework is proposed that highlights how the channel estimation-based Deep Learning (DL) and power optimization scheme are jointly utilized for multiuser (MU) detection in the PD-NOMA system.A structured and mathematical analysis is introduced to derive a non-complex analytical form for the optimum power coefficient for each user on the basis of maximizing the sum rates in a downlink NOMA system.An optimized power scheme and fixed power scheme are both evaluated and compared when the proposed Deep Learning-based channel estimation scheme is implemented.To validate the efficiency of the proposed Deep Learning channel estimation scheme, the DNN model is inspected using a Rayleigh fading channel and Rician fading channel.As a benchmark comparison, we have also conducted the simulation environment related to the work that consider the DL for joint channel estimation and signal detection and compared it with our proposed DL scheme. The simulation results emphasized that reliability can be guaranteed by our proposed DL channel estimation scheme even when cell capacity is increased.In addition, different from the aforementioned works, in our proposed DL channel estimation algorithm, we implement a minimum number of DNN layers and a minimum number of pilot symbols to achieve a remarkable improvement in system performance. Moreover, no additional interference cancelation or noise elimination factors are utilized on the receiver side.The detailed contributions of this work are outlined as follows:The rest of this paper is organized as follows. The downlink NOMA system is examined in this section, where the base station (BS) and users come across various channel gains. In this system, the NOMA cell is considered, where one BS with a single antenna is assumed to serve three users concurrently and every user equipment also has a single antenna. Typically, in NOMA scheme each user receives the combined signal sent from BS that comprise a target signal and interfering signal sent through the same time-frequency block. Consequently, multiplexing a number of signals using distinct power levels is essential to distinguish the signals and to reinforce the SIC procedure at the receiver side . In PD-Ni and k represents the path loss exponent [Users are labeled in accordance with their fading channel and the separation from base station. In the examined cell, the nearby device is indicated as near user and the device at the edge of the cell is viewed as far user. In this work, we assume that we have three users in the cell and Rayleigh fading channel is considered with zero mean. Hence, the fading channel for each user can be mathematically characterized as follows, for the near user exponent .In this paper, Additive White Gaussian Noise (AWGN) is assumed at the receiver side of each user, and the noise power is indicated as essed as ,16:(1)x=ented as :(2)yf=xhented as :(3)yf=PtThe first term in (3) represents the required signal for a far user, while the second term denotes the interference term from middle and near users. Based on Equation (3), the possible rate for far user could be expressed as :(4)Rf=loLikewise, the attainable bit rate for middle user can also be expressed as:Typically, the near user has good channel condition along with BS; thus, their signal The first term in Equation (6) represents the near user expected signal, while the second term denotes the interfering term from middle and far users. On the other hand, it can be observed from Equation (6) that the interfering term is predominant due to the additional power that can be assigned to the far user. Therefore, at the near user side, SIC is performed, where immediate decoding for far user signal essed as :(7)Rn=loThe aim here is to maximize the sum rate for users in NOMA system on the basis of optimizing the power factors for each user in accordance with the channel conditions. The sum of the above-mentioned possible rates for N users downlink NOMA network can be written as follows ,20:(8)RsIn the proposed system, the objective function and the constraints considered for the optimization problem can be clarified in the following sections.xth ser in the N-user NOMA cell.The allocated power for each user in the cell is a percentage of the overall power ies with :(9)∑x=1Nnth user and k to detect the signal of user m, where ith user in the system. By dividing both the numerator and denominator of left-hand side of Equation (11) by the noise power mth user being in outage, the following condition must be achieved:mth user in the system.To enhance user fairness, it is assumed that the weak user in each cell has a QoS requirement, which implies that a minimum rate  follows :(10)log2e system . This com→k<Rmin . The comOn the basis of the aforementioned constraints in Equations (9) and (10) and sum rate representation and the fact that there is a single antenna at the BS and user equipment, the standard optimization problem can be normally expressed as follows ,22:(14)mIn this part, the optimization analysis is realized with regards to three users in the NOMA system and the objective function can simply reformulated as follows :(15)maxαThe constraints lculated . Initial follows ,25:(19)∂At this point, a general formula can be derived for the first derivative of the objective function in terms of α ,25:(22)∂The second derivative of the objective function A general formula can also be found for the second derivative of the objective function in terms of Based on the objective function, and  maximum . The Lag maximum ,26:(27)ℒOptimality conditions can be written as follows:Slackness conditions can be represented as follows:Lagrange multipliers also need to satisfy the following:In the subsequent steps, Lagrange multipliers should be proved to be positive. This could be accomplished as follows:Based on Equations (35)–(39), this can be substituted in the optimality conditions for Lagrange as follows:Let Based on Equation (40) and after few mathematical substitutions, the following expression can be written as:Performing a few mathematical analyses and based on the fact that Similarly, For the following analysis, it can be assumed that Based on mathematical substitutions and arrangements, the closed form representation for each of the power coefficients can be derived as follows:Recurrent Neural Networks (RNNs) are regarded as a class of supervised learning procedures, and they can develop successive sequences for prediction and detection . As showIn RNNs, hidden layers are capable to act as storage for the network at a specific time; this structure enables the RNNs to handle the preceding data for a prolonged period of time; additionally, RNNs can represent time dependencies for any sequence with a lower numeral of neurons. On the other hand, traditional RNN based on backpropagation through time (BTT) experiences slow learning and a vanishing gradient problem .Therefore, RNNs will not be the best candidate for signals that may be sent through fading channels that may disperse the signal and originate long-term dependencies among its samples . Long shLSTM network can develop knowledge among time steps of the data sequence and manage the long-term dependency process of time series data. Based on their underlying design, the LSTM network includes LSTM cells, and each cell contains a set of gates that are capable of saving and gaining access to data over extended periods of time and of counteracting the error from backpropagation ,28. LSTMA framework that incorporates the LSTM model with NOMA system is discussed in this section, where the LSTM network is mainly utilized to perform training, updating, and prediction for the channel coefficients. Data-driven communication usually depends on empirical observations to determine the amount of LSTM cells in each layer and the numerals of LSTM layers that are needed in the implementation stage. Additionally, it is important to take in consideration that adding more LSTM layers may not offer a noticeable gain in learning phase, or it may not significantly affect the network convergence .L, which is the length of the input layer. The input layer includes 128 neurons, where the inputs to the network are shifted to the subsequent layer with updated weight parameters.W, the recurrent weights R, and the bias b. The third layer is a fully connected layer that processes the outputs of the LSTM layer. A fully connected layer multiplies the input by a weight matrix W and then adds a bias vector b. All neurons in a fully connected layer are connected to all the neurons in the preceding layer, and this layer bring together all of the characteristics and internal information gathered by the prior layers. In a DL-based LSTM network, the fully-connected layer behaves separately on each time step.In the second layer, one LSTM layer is implemented that includes 300 hidden units. The learnable weights of LSTM layer are the input weights MSE for regression tasks. Normalizing the responses mainly facilitates stabilizing and accelerating training of neural networks for regression. For a single observation, the mean square error can be calculated as [The last layer is the regression layer, which is responsible to improve the cell status, network weights, and biases. A regression layer can predict responses of a trained regression network and computes the lated as :(50)MSE=L is the sequence length.In case of sequence regression networks, the loss function can be represented as the half In LSTM cell, the output is generated based on the current input and the preceding cell state. In order to remember the previous cell state and determine if the prior state will be used or not, the LSTM cell consists of different types of gates, these gates are the forget gate, the input gate, and the output gate. In LSTM, there are two states, the cell state The forget gate is responsible for controlling the level of cell state that need to be reset: The input gate is responsible for controlling the level of cell state that need to be updated: The candidate state is responsible for adding information to the cell state: Updated cell state: The output gate is responsible for controlling the level of cell state added to hidden state: Estimated output coefficients: The transmitted frame involves data and pilot symbols. The applied channel model is assumed to be constant during one frame of pilot and data symbols and the channel coefficients are changing from one frame to another. In the proposed DL scheme, two major stages are implemented to achieve an effective DNN model for channel estimation. The first stage involves both training and testing, where the DNN model is trained and tested with the samples that are created based on variety of distinct Rayleigh channel coefficients ,33. In tAt the beginning of every training stage, the weights and bias values of LSTM layer are initialized, and during the training phase, weights and biases are modified according to a gradient descent procedure ,30. The On the basis of the initial channel factors generated and the pilot symbols, the size of the training and testing frames can be identified. The training model are carefully established based on the selected layers, the hidden units assigned for each layer, and the training parameters. In order to further accelerate and stabilize the training process of the training network, we choose to normalize the training data.Algorithm 1 Proposed DL Channel Estimation scheme1.W, R, b), where W is the input weights, R is the recurrent weights, and b is the bias.Initialize the learnable parameters of an LSTM layer .Throughout the training phase, the performance of the proposed DNN model-based LSTM layer is assessed using RMSE and loss functions. In the testing period, new fading coefficients will be randomly generated, such that these coefficients are not the same as those generated for training. Once the training and testing sequences are inspected using the training network, the trained model will be employed as online channel estimator for the users. The proposed DL technique for channel estimation can be outlined as shown in Algorithm 1.In this section, a description of the simulation settings and parameters is introduced. Our examined downlink NOMA cell contains one base station (BS) and three distinct users in which the BS and each user in the cell is supplied with one antenna. For the downlink NOMA scenario, the modulated signals are superimposed and transmitted by BS to the users through uncorrelated Rayleigh and Rician fading channels affected by additive white gaussian noise (AWGN), where the noise spectral density In this paper, simulations are conducted using MATLAB to simulate and to emphasize the following: first, to evaluate the effectiveness of embedding the proposed DL-based LSTM network to accurately estimate the channel parameters for each user in downlink NOMA cell. Second, to integrate the proposed DL channel estimation algorithm with the derived optimized power allocation scheme and compare it with the NOMA system when fixed power factors are considered along with the proposed DL scheme. Monte-Carlo simulations are conducted with In our simulation environment, we assume that the channel state information (CSI) is not available at the receiver side. Therefore, for the sake of comparison and in order to investigate the efficiency of the proposed DL algorithm, two methods for channel estimation are implemented at the receiver side for each user. The first method, which is the proposed scheme, uses DNN-based LSTM layer to estimate the desired channel coefficients. The gradient descent algorithm is applied in conjunction with LSTM layer, and the LSTM layer is attached to a fully connected layer, where each neuron in the former layer is fully connected to every neuron in the consequent layer. The second channel estimation scheme implemented at each receiver side is initiated based on the minimum mean square error (MMSE) . The MMSChannel taps that are employed to model the Rayleigh fading wireless channel are generated on the basis of ITU channel models. Throughout the simulations, NOMA system parameters are employed on the basis of the long-term evolution (LTE) standard . Both trInitially, different power factors are allocated for each user according to their distance from the BS and the current channel gain. Power allocation coefficients −2, the power saving achieved by DL scheme is approximately 5–8 dBm for far and middle users, while for the near user, the power saving is up to 4 dBm.In −3) when DL-based channel estimation scenario is implemented compared to the conventional estimation scheme. Likewise, the middle user with the DL estimation scheme shows more enhancement in power saving compared to the MMSE procedure, by 5–7 dBm approximately, which proves the superiority of the proposed DL estimation scheme.K defined as the ratio of the signal power in the line-of-sight component to the scattered power in other components. The other main parameter is Rician fading is a stochastic model for radio propagation, where the signal arrives at the receiver by several different paths, and hence, exhibits multipath interference. Rician fading occurs when one of the paths, typically a line of sight (LOS) signal or some strong reflection signals, is much stronger than the others. A Rician fading channel can be described by two parameters The first one is the Rician factor In In In In In In this work, the impact of the Deep Neural Network (DNN) in explicitly estimating the channel coefficients for each user in the NOMA cell is investigated, where the LSTM network is developed for complex data processing. In the proposed DL algorithm, the DNN model is trained online based on both the normalized channel statistics and the relationship between successive training sequences. The validity and efficiency of the proposed DL channel estimation scheme is emphasized by inspecting the proposed DNN model using the Rayleigh fading channel and Rician fading channel. Furthermore, we introduce a framework that investigates how the proposed channel estimation based on the DL and the power optimization scheme are jointly utilized for multiuser detection in the PD-NOMA system. To maximize the sum rate of the system users, we optimize the power coefficients allocated for each user on the basis of the overall power transmitted and the QoS constraints. A systematic mathematical analysis for the optimization problem is introduced and the Lagrange function and KKT conditions are employed to deduce the optimal power factors. The simulation results in terms of the BER, outage probability, sum rate, and individual capacity have verified that the proposed DL model-assisted NOMA can realize reliable performance compared to the conventional NOMA scheme, even when cell capacity is increased.In future work, the performance of the proposed DNN model can be further explored in terms of single input, multi output (SIMO) or multi input, single output (MISO) for different types of fading channels and modulation schemes.


To achieve accurate contour tracking of robotic manipulators with system uncertainties, external disturbance and actuator faults, a cross-coupling contour adaptive nonsingular terminal sliding mode control (CCCANTSMC) is proposed. A nonsingular terminal sliding mode manifold is developed which eliminates the singularity completely. In order to avoid the demand of the prior knowledge of system uncertainties, external disturbance and actuator faults in practical applications, an adaptive tuning approach is proposed. The stability of the proposed control strategy is demonstrated by the finite-time stability theory. Then, the developed controller combines adaptive nonlinear terminal sliding mode control (ANTSMC) of joint trajectory tracking and proportion–differentiation control of end-effector contour tracking by introducing the coupling factor between multiple axes based on Jacobian. Moreover, a unified framework of cross-coupling contour compensation and reference position pre-compensation is built. Finally, numerical simulation and experimental results validate the effectiveness of the proposed control strategy. The accuracy of contour error is one of the key concerns to ensure the quality of machined parts2. Contour error is defined as the shortest distance between the current position and the desired contour curve. However, most current control strategy of robotic manipulators focusses on optimizing the tracking performance of each joint to improve the contour machining accuracy of the end-effector6. Unfortunately, the joint tracking of high accuracy of robotic manipulators cannot effectively solve the problem of contour matching accuracy of the end-effector6. One of the main reasons is the lack of coordination of the robotic manipulator7. An integrated control strategy of cross-coupling contour error compensation based on chord error constraint, which consists of a cross-coupling controller and an improved position error compensator, was proposed by Zhang et al.8. Kommaneesang et al.9 pointed that only a few researchers concentrate on solving the contouring problem in the robotic machining system, and the contouring control problem was transformed into the regulation problem by using the method of equivalent errors. Moreover, contour accuracy control was studied in many literatures12.In the past decades, with the rapid development of modern industrial technology, robotic manipulators have been widely used in machining, laser cutting, welding, and other fields16. Thus, the system is forced to track a pre-determined trajectory of sliding mode states. The sliding mode manifold can be designed independent of object parameters and perturbations17. Aksu18 proposed SMC based on a linear sliding mode manifold which guaranteed that the system states asymptotically converge on the equilibrium point. Inducting a nonlinear term in the linear SMC, Lafrnejani19 proposed the terminal SMC (TSMC) to ensure global finite-time stability. Su20 proposed an integral sliding mode manifold and its TSMC, and manifested the global finite-time convergence of both sliding mode manifold and tracking error. The above SMC provide an effective and stable control strategy for nonlinear systems, but these rely on the prior knowledge of system uncertainties24. Besides, these SMC trajectory tracking control of the robotic manipulator only ensures the tracking accuracy of each joint, but the accuracy of contour error is not guaranteed26. In addition, there exists a singularity phenomenon near the equilibrium point caused by the negative exponent of the TSMC28. The prior information of system uncertainties is difficult to obtain in practical tasks, such as random fault parameters and disturbances29. Considering the uncertainty and control system stability analysis methods30, a cross-coupling contour adaptive nonsingular TSMC strategy is proposed to solve the strong coupling contour error problem and avoid the prior information of system uncertainties, external disturbances and actuator faults. Compared to the existing cross-coupling control of robot manipulators, the primary contributions of this paper are summarized as follows:9, the accuracy of joint error and contour error is guaranteed simultaneously. Furthermore, a unified framework of cross-coupling contour compensation and reference position precompensation is built.Considering the strong coupling between the contour error and the joint error, the coupling factor with multiple axes based on the Jacobian matrix is proposed. Compared with the method of equivalent errors in Ref.Different from the singularity is generally solved by the parameters in equivalent control law, the uncertainties of the system and the function of the actuator faults are combined with a lumped function, and an adaptive tuning algorithm is adopted to compensate for the lumped uncertainties of the system.31, the tracking errors convergence quickly, and the performance of the proposed control is improved by approximately 61% and 34%, respectively.Adaptive non-singular terminal sliding mode control with cross-coupled contour is improved, and the stability of the proposed control strategy is demonstrated by the finite-time stability theory. Compared withSMC is a particular and powerful class of variable structure control essentially which can dynamically adjust based on the current state of the systemThis paper is organized as follows: the problem formulation and motivation are indicated in “32The dynamic of For actual applications, it is difficult to obtain the precise dynamic model of the robot manipulator as the nonlinearities of the friction and the external disturbances. Hence,  can be rThe properties of  are satiConsidering the problem of actuator faults during the operating process of the robot manipulator,  can be rith state.The time profile of the faults The time profile mode of the faults is given byThe position tracking error denoted by Non-singular terminal sliding mode manifold Derivation  with resThe positive-definite Lyapunov function is given byDerivation  with resSubstituting  into 1414, we obSimplified , we obtaLet Assumed that the uncertainties and actuator faults of the robot manipulator are defined asThe upper bound of function ,, and The switch control law Thus, the NTSMC is derived asSince The estimated error is defined asDerivation  with resThe control law The positive definite Lyapunov function with estimated error is given byDerivation  with resAdaptive algorithm is given bySubstituting  into 2626, we obThe schematic of the proposed ANTSMC is shown in Fig. For finite-time stability with fast time convergence, the Lyapunov function The stability time of convergence Considering the Lyapunov function asDerivation  with resSubstituting  into 3232, we obSubstituting  into 3333, yieldAs Tsgns=s,  can be sAssumed that the uncertainties and the actuator faults changes slowly, there hasAsThenAccording to Theorem 1, the proposed adaptive non-singular terminal sliding mode control is finite time stable.When The trajectory planning interpolation method can be used to fit the trajectory contour into a straight line or a circular contour. The contour error model of the straight-line contour for an XY planar is shown in Fig. Assumed that the contour error is the shortest distance between the prevailing position and the desired contour curve. The contour error of plane line is defined asThe contour motion control with cross coupling is given byAs compensation of contour and trajectory tracking control are carried out in the task space and in the joint space, respectively, so the mapping relationship between two space is established as followsAccording to , the conThe rectifier gain of contour tracking error compensation is given byCombining  with 4141, we obFrom Eq. , one canThe schematic of the ANTSMC with cross-coupling is shown in Fig. Robotic manipulator with two-link is used to illustrate the effectiveness of the proposed control strategy, shown as Fig. Assumed that the mass of each link is concentrated. The dynamic equation of the robotic manipulator with two-link is derived as followsConsidering the influence of uncertainties and the actuator faults of robotic manipulator, Set the simulation time is Parameters of traditional PID, ANTSMC and cross coupling control are given by Table The experiment platform is built, shown as Fig. The mean square value of the tracking error is defined as20, and ANTSMC31, when the system has obvious position errors, the joint tracking errors of the proposed CCCANTSMC reach the convergence more quickly. Moreover, there are the actuator faults after 1.5 s, the control performance of PID controller decreases remarkably, and the control performance of NTSMC converges after a slight fluctuation. However, the NTSMC used adaptive algorithm can ensure the stability of the system effectively. In addition, from Fig. As shown in Figs. Table In this paper, due to sudden changes in position, velocity and acceleration at the junction inflection point of adjacent straight segments, each controller will suffer performance degradation to a certain extent at the time of 1 s and 2 s, but the adaptive tunning method of the proposed control strategy for this kind of mutation is better than the traditional PID, NTSMC and ANTSMC. However, it is necessary to perform trajectory planning processing at these sudden changes to optimize the controller performance, which will be a desired future research work.In this paper, a cross-coupling contour adaptive nonsingular terminal sliding mode control (CCCANTSMC) is proposed for the issue of precise contour tracking of the robotic manipulator in the presence of system uncertainties, external disturbance, and actuator faults. Based on the strengths of the NTSMC for driving the system state to the equilibrium point in finite time, the adaptive tuning approach is proposed. Thus, the prior knowledge of system uncertainties, external disturbance, and actuator faults is avoided and the singularity problem is eliminated. Introducing coupling factors among the multi-axes based on Jacobian, ANTSMC of joint tracking and PD control of end-effector contour tracking is combined to improve the accuracy of contour error. Furthermore, a unified framework of cross-coupling contour compensation and reference position pre-compensation is built. The experimental results are shown to prove the effectiveness of the proposed control strategy.


To rapidly obtain the complete characterization information of pulse signals and to verify the sensitivity and validity of pulse signals in the clinical diagnosis of related diseases. In this paper, an improved PNCC method is proposed as a supplementary feature to enable the complete characterization of pulse signals. In this paper, the wavelet scattering method is used to extract time-domain features from impulse signals, and EEMD-based improved PNCC (EPNCC) is used to extract frequency-domain features. The time–frequency features are mixed into a convolutional neural network for final classification and recognition. The data for this study were obtained from the MIT-BIH-mimic database, which was used to verify the effectiveness of the proposed method. The experimental analysis of three types of clinical symptom pulse signals showed an accuracy of 98.3% for pulse classification and recognition. The method is effective in complete pulse characterization and improves pulse classification accuracy under the processing of the three clinical pulse signals used in the paper. Both the pulse diagnosis in Chinese medicine and the examination of cardiovascular diseases in Western medicine attempt to extract various physiological and pathological information from the pulse wave features. However, due to the complexity and diversity of the pulse signal, enabling a complete characterization of the pulse signal has become one of the problems that many scholars need to solve.Pulse signals often contain a wealth of important information, such as the human heart and cardiovascular system. Pulse wave pressure and pulse waveform characteristics are important bases for evaluating the physiological and pathological status of the human cardiovascular system \* MERGEFORMAT2. It is of increasing interest to researchers because of its noninvasive, inexpensive, and convenient diagnostic features. The signal acquired by PPG is pulsatile, identifies relevant features of blood flow activity, and can be used to measure cardiac output4. PPG has been widely accepted by the International Organization for Standardization (ISO) and the European Committee for Oxygen Saturation Measurement. When a finger is placed on the PPG device, it can easily record a high-quality pulse signal8. Pulse signals from PPG were preprocessed to improve signal quality and estimate heart rate11.To improve life expectancy and reduce healthcare costs, PPG has emerged as a promising technique for early screening12 proposed the use of wrist pulse wave signals to analyze hemodynamic problems in hypertensive patients. The results showed that establishing pulse image analysis (PIA) in hypertensive patients can help in the early identification of target organ damage (TOD) and uniform pulse diagnosis. Hui Liu13 proposed to calculate blood flow parameters based on the characteristic quantity K of the pulse waveform and analyzed the correlation between K values and blood flow parameters. The results show that the method is simple and fast, and can use the pulse wave area characteristic quantity can accurately predict the changes of blood flow parameters. Wang14 developed a mechanical model of the pulse wave system, in which the pulse wave system is considered as a system of vital energy that propagates in the blood in the form of waves. Hsing-Chung Chen15 used an image fitting method to segment the arterial pulse wave curve, extracted its slope and period as pulse time-domain features, and compared them with the difference of the mean feature matrix. Finally, the optimal pulse period curve for physicians' reference analysis was obtained. Hadiyoso16 used the decomposition method of EEMD for the extraction of RR breathing waves. By comparing the performance of EMD and VMD, the study concluded that EEMD has better performance.In the area of pulse signal feature processing, many scholars have conducted a series of researches on the extraction and classification of pulse features. For example, Natalia17 used machine learning (ML) and deep learning (DL) to study the feasibility of the pulse signal obtained via PPG and to predict vascular aging. The results show that ML is more useful to analyze the underlying biological properties of HVA predictions when DL is computationally inexpensive. Nogueira18 computed the features of each segment of the heart sound signal by extracting a set of time-domain features and two frequency domain features and classified the heart sound signal using radial basis algorithm and finally obtained a classification accuracy of about 83.22%. Rubin19 classified the MFCC heat map by segmenting the heart sound signal and described in detail the generated neural network structure and design decisions. The MFCC method was used to transform the frequency domain features, but in the presence of external interference factors, the performance of MFCC is reduced, which affects the accuracy of the study results. Hu20 proposed a classification method based on the Shannon energy envelope, Hilbert transform, and deep convolutional neural network. The results showed that it significantly outperformed other commonly used methods when the features were unclear. Zhang21 proposed to classify about 2280 PPG signals using a 9-layer one-dimensional convolutional neural network and investigated its correlation with atherosclerosis, and finally obtained a classification accuracy of 93%.Lorenzo22 proposed to extract the features of PPG signals using CNN and normalize the features using the z-score normalization method. Finally, the overall accuracy of this network model is 94.5%. Hu23 proposed to obtain pulse features by continuous wavelet transform and finally processed by the convolutional neural network to obtain 83.81% classification accuracy. Reit Kavsao Lu24 extracted time-domain features by first and second-order derivatives of the PPG signal and ranked the contribution of biometric features based on these features. Finally, a k-NN classifier was used for validation and an average accuracy of about 90.7% was obtained. Tripti25 proposed the classification of samples by mean clustering strategy and fine Gaussian multiclass support vector machine. PPG signals from patients with various diseases were investigated and the final results showed its applicability in identifying unique heart-related diseases. In addition to screening for diseases, PPG signals have also been applied for blood pressure prediction28. Sun29 proposed the use of HHT-based convolutional neural networks for blood pressure detection and classification of PPG signals. The accuracy of classification was 98.9% for normotension, 85.8% for hypertension, and 93.54% for prehypertension. Feature extraction is performed on traditional methods. For example, feature statistics and screening based on PPG signal features33. Ramachandran34 proposed to use features such as singular value decomposition, statistical features, and wavelets and then applied softmax and Gaussian mixture model classifier to classify various risk levels of cardiovascular diseases. Finally, the classification accuracy obtained was 97.88%. Poulomi35 analyzed time-domain features and then classified patients and healthy subjects using decision trees, discriminant analysis, logistic regression, support vector machines, KNN, and augmented trees. Finally, a classification accuracy of 94% was obtained.Choon-Hian GohacfTo make a complete characterization of the pulse signal for achieving pulse classification recognition, the EPNCC-based method is used in the paper to obtain the frequency domain features and the wavelet scattering method is used to obtain the time-domain features. The time–frequency features are fused and input to a convolutional neural network for recognition. The proposed method is validated experimentally on the MIT-BIH-mimic database. We wish to provide new methods and ideas for the application of pulse signals in clinical diagnosis.The pulse long-time signal contains the internal characteristics of each cycle and the characteristic relationships between the cycles. It reflects the envelope variation of the pulse signal and is the time-varying characteristic of the pulse signal. It still has a classification effect for pulse signals when the body is in a state of motion change. Multi-period signal data has higher dimensionality than single-period signals and can contain more feature information. Therefore, in this paper, a multi-period pulse signal is selected as the data sample. More importantly, using an appropriate convolution module, it is possible to reduce the data dimensionality while still retaining the features that are not lost. The pulse period feature extraction process is shown in Fig. To facilitate the input of the convolution kernel for calculation, this article lists the corresponding pulse cycle signal as a matrix form of multiple cycles juxtaposed, satisfying:When choosing the actual number of M, a suitable value needs to be selected because when the selected value of M is too small, the variation of the pulse period cannot be obtained. However, when the value of M is chosen too large, the number of dimensions of the network input increases, which will increase the complexity of the network computation. Therefore, in this paper, the M value of the pulse period sample data is chosen to be 5. To obtain good results while reducing the computational cost, this paper first uses two convolution modules to extract the temporal characteristics of the pulse. Each convolutional module contains a convolutional layer, an activation layer, and a pooling layer.Each time the input is convolved, the convolution kernel will output a feature mapping. To prevent gradient explosion and gradient message update of multilayer network coefficients, the ReLU function is chosen as the activation function of the convolution layer in this paper. The expression is: f(x) = max. ReLU, as an activation function, can improve the training speed of the network due to its linear and non-saturated form. After the activation function processing, the maximum pooling layer is selected as the final output of each convolutional module in this paper, which further reduces the size of the data.After experimental verification, the parameter configuration of the convolution module used in this article is shown in Table 37.A wavelet scattering method was born by combining the concepts of wavelet theory and signal processing. It introduces a depth feature map in the time series to make the local time of the series invariant and stabilizes the time distortion by a cascade of signal filtering and modular operators. The expression for wavelet scattering isAmong them, In this paper, a wavelet scattering network (WS) is used to preprocess the time domain features of pulse signals. This method can well characterize the band characteristics of its low and high-frequency signals. The expression of the output characteristic coefficients of the wavelet scattering network isThe pulse period signal is processed by wavelet scattering (WS). After the parameters of the wavelet scattering decomposition frame are configured, the decomposition frame coefficients are expressed as SF, and then the entclass1pt{minimaIn the traditional PNCC method, the FFT is used for signal preprocessing, but the FFT does not work well for non-smooth signals. Therefore, we use EEMD to preprocess the signal instead of FFT to improve the PNCC method.The EEMD method is used to preprocess the PPG signal samples. the steps of the EEMD algorithm are as follows: (1) add normally distributed white noise to the original signal; (2) add the white noise to the signal as a whole and then perform EMD decomposition to obtain each IMF component; (3) repeat steps 1 and 2, adding a new normally distributed white noise sequence each time; (4) integrate and average the IMF obtained each time as the final result are integrated and averaged as the final result.After decomposing the pulse signal by EEMD, each IMF component and a residual are obtained. Due to the time-varying and random nature of the noise, the IMF components after EEMD decomposition will vary. The large differences in these IMF components are meaningless for characterizing the impulse signal properties. Therefore, before extracting the PNCC frequency domain features, a correlation analysis is performed on the EEMD decomposed IMFS components to filter out the components that can characterize the impulse signal. The correlation analysis can determine the coherence between each IMF component and the original PPG signal, and the expression formula for the coherence coefficient is:Among them, Estimate the power spectrum of each component in The power estimation is input to the Gammatone filter for filtering processing. The time-domain impulse response formula of the Gammatone filter is as follows:where b is the filter bandwidth and n is the order of the filter.After the filter filters, its power normalization (PN) process. Power normalized expression:Power function nonlinear processing, its expression is After the discrete cosine transform is performed on the nonlinearly processed signal, the EPNCC characteristic coefficient can be expressed as After the PPG signal is processed by EEMD, the components are arranged and expressed by a matrix After obtaining the EPNCC feature parameters through a series of processing, the EPNCC feature parameters are input into the convolution module for processing. The parameters of the convolution module refer to Table To fully characterize the pulse signal features, the frequency domain features and time-domain features are mixed into a convolutional neural network to identify and classify the mixed features. The general structure of the pulse signal classification network is shown in Fig. The cost function used in this paper is cross-entropy, which is defined as:When the cost function is reduced to a given acceptable error, this means that the network can be used for practical classification and recognition. For multilayer convolutional neural networks, the choice of the training algorithm is closely related to the computational complexity and classification accuracy of the network. Choosing an appropriate training rate algorithm can accelerate the convergence speed and reduce the occurrence of oscillations. The training method chosen in this paper is the momentum-driven stochastic gradient descent algorithm, which uses exponentially weighted averaging to make the computation of this gradient relevant to the previous method. As a result, the up and down oscillations of the gradient can be offset, thus accelerating the convergence rate. The specific structural parameters of the impulse classification network are shown in Table 38. The data are all ICU critical care patient pulse data from the MIMIC-||| database. These data were collected at a sampling frequency of 125 HZ. Among them, 39 groups of 1-min patient pulse data with clinical symptoms were classified into three categories: 16 groups of CHF pulmonary edema, 10 groups of respiratory failure, and 13 groups of centrally-derived shock. Twenty pulse signal samples were acquired for each group, and a total of 780 pulse data were used for network model construction. Among them, 546 were the training set and 234 were the test set.In this section, experimental analysis is performed to verify the effectiveness of the algorithm proposed in this paper. The pulse data were obtained from the MIT-BIH-mimic database provided by the Massachusetts Institute of Technology (MIT). The MIT-BIH database is one of the internationally recognized ECG databases that can be used as a standard and can effectively reflect the differences in pulse classification algorithms. In this experiment, the pulse signals of 39 patients were obtained from the simulated databaseIn this experiment, MATLAB is used as the development platform to search the pulse data in the database for the peak points, and then group and segment the process. Since the sampling frequency of the pulse signal in the database is 125 Hz, 90 points were selected as a pulse cycle by observing the waveform, and five pulse cycles with a total of 450 data points were selected as the data samples for this experiment, starting from the peak. The sample plots of the pulse waveforms of the three clinical symptoms after segmentation are shown in Fig. The network structure parameters are set as described in the previous chapter. After testing the network hyperparameters, the sgdm optimizer is selected to control the learning rate. The initial learning rate is 0.01, the learning rate reduction factor is 0.7, and the maximum iteration period is 50.The network is trained, classified, and validated by pulse period time-domain features (wavelet scattering) and the EPNCC coefficient conflation feature algorithm. The classification data in Table To demonstrate the superiority of the method in pulse classification, pulse period feature training (sppft), pulse period (wavelet scattering) training (sppwsft), EPNCC feature training (EPNCCft), pulse period feature and EPNCC feature hybrid training (spp-EPNCCft), pulse period (wavelet scattering) and EMFCC hybrid features training (sppws-EMFCCft). They are all based on 546 sets of pulse data to train the model and then validated using 234 sets of pulse test samples.By analyzing Tables When the EPNCC is used as a complement to the time-domain features and the time-domain features are fused with the frequency-domain features, it enables complete characterization of the pulse signal, thus improving all the metrics of the network classification. The f-measure data metrics can show that the method used in the paper is feasible and effective. Analyzing the three types of clinical data in the table, there are some differences in the metrics obtained from the three data classifications, but the overall deviation is not significant. This may be due to the different number of training sets for the three types of clinical data, resulting in relatively poor network learning for certain symptom data.The analysis of Table In this paper, other signal decomposition methods have also been experimented and the combination of EPNCC was proved to be more suitable for this pulse recognition model by comparing with EEMD. Because of the modal mixing in the signal decomposition process of EMD, it is not used in this paper for experiments. The decomposition methods used for comparison experiments in this paper are VMD, CEEMD, and CEEMDAN, respectively. The data obtained from the experiments are shown in Table It can be obtained from Table At present, due to the limited source of data sets, the research method in this paper is limited to the data sets used. However, at the same sampling frequency, the data points contained in each cycle are the same, except for the different shapes of the pulse signals. Therefore, the feature processing method proposed in this paper is theoretically applicable to the analysis of other pulse datasets.23 is the same as that of the pulse data used in this article. The pulse single-cycle data point is 81, which is the same as the 90 data points in this article. In terms of data input dimensions, the input dimensions of this paper are nearly half lower than those in the literature23. The final classification accuracy rate is nearly 8% higher than that in the literature23.According to the analysis of Table 24 is the same as the database used in this article, and the data points contained in each cycle are the same, which can be compared. Literature24 uses traditional feature extraction methods such as the distance between peaks and peaks to effectively extract pulse waveform features from normal healthy people, but this feature extraction method is limited in normal pulse waveforms. Abnormal pulse waveforms are not visible at every peak. There may be a situation where a certain wave peak is not obvious, which greatly increases the difficulty of traditional feature extraction methods and it is easy to ignore a certain part of the feature. The advantage of this paper is that PPG signals are processed directly through convolutional neural networks, and the EPNCC method is used to make the pulse characteristics more complete. It can be seen from the classification results that the research method is more effective than the method in the literature24.The sampling rate of pulse data in the literature36 and the pulse data in this article are from the same database. It is mentioned in the literature that single-period, multi-period, and MFCC features are used for mixed feature training, and finally a classification accuracy of 93.7% is obtained. However, this article only uses multi-period pulses for experiments and found that by adjusting the network parameters, the classification accuracy can be achieved when single-period and multi-period are mixed. Multi-period pulse data already contains the characteristics of single-period pulse data, and the superposition of the two will make the data redundant and increase the computational complexity. In this paper, the improved PNCC based on EEMD is used as a supplement to the time-domain features, and the final classification accuracy is 4.6% higher than that in the literature36. The reason may lie in the processing method in this article. The pulse signal of each frequency band is obtained through EEMD, and then after the coherence degree is screened, the PNCC processing is performed. One part of the PNCC processing is power-law nonlinear processing, which can amplify the pulse data information after processing, making the characteristics more obvious.The pulse data used in the literatureThe purpose of this study is to design a method to rapidly extract complete pulse signal features and to classify and identify them to improve the efficiency and accuracy of clinical diagnosis. This paper features the improvement of the traditional PNCC method by replacing the Fourier transform process in PNCC with EEMD and using this method to obtain feature information that is difficult to express in the time domain.The signal in each frequency band is decomposed by EEMD and then processed by PNCC so that the tiny features can be amplified to make the pulse features more obvious. It is used as a supplement to the time-domain features to obtain the complete PPG signal characteristics. In this paper, the PPG signals provided in the MIT-BIH database are used for experimental validation to classify and identify three clinical categories of pulse signals. A mixture of multi-period PPG signals and EPNCC features are trained by convolutional neural networks, and the final classification results are better. The EPNCC features are selected as a complement to the time-domain features to avoid the problem of increasing computational complexity due to feature redundancy.Due to the limited information on clinical symptoms associated with pulse signals in the MIT-BIH database. Therefore, it is difficult to construct a more complete and feasible pulse diagnosis model at this stage. However, in the future, the network model obtained in this paper can be updated and validated through migration learning. The method proposed in this paper hopes to explore more possibilities of the pulse signal in clinical diagnosis so that the pulse signal can be fully utilized to promote the development of intelligent clinical diagnosis.


In recent years, STEM education has developed students’ fundamental subject knowledge, and has allowed students to integrate STEM cross-domain knowledge. Universities emphasize innovative thinking, practice and application, problem-solving, and teamwork to nurture students while learning STEM cross-domain knowledge development in remote STEM program education. When students take online STEM programs, they may encounter unanswered questions and may give up on trying to solve them. Therefore, this study proposed a problem-based learning approach with an online programming system integrated into an online STEM programming course. To help students solve the past programming assignments, the assignments were simplified, decomposed, and reorganized. The teacher guided the students to understand the STEM programming problems and taught them to use appropriate problem-solving skills to motivate them to complete the STEM programming assignments. The experiment was conducted with students in an online STEM programming course at a university in northern Taiwan. In the experimental activities, we used a problem-based learning approach for the online STEM programming activity. The problem-based learning method can be divided into four steps, namely stating the problem, understanding the problem, developing a solution plan, and executing the plan, reflecting, and debugging. This study used a problem-based learning approach and an online programming system integrated into a STEM programming curriculum to explore the differences in students’ perceptions of STEM learning, learning outcomes, and learning behaviors. The experimental results found a significant difference between students’ prior knowledge and learning outcomes. Students showed significant gains in learning the STEM programming content using the problem-based learning approach and the online programming system. In the analysis of their STEM learning perceptions, we found that there were significant differences in students’ responses for each dimension. This shows that using the problem-based learning approach with the online programming system helped students learn the course content. The analysis of students’ behaviors in answering the STEM programming assignments indicated that some students had the habit of taking notes. This helped them to easily associate and integrate STEM cross-domain knowledge with what they had learned in the online course, and enhanced their ability to implement STEM programs. In addition, students could take the initiative and focus on repeatedly watching the teacher solve the material in the online course. Students could try different solving plans to pass the code validation of the STEM programming assignments. This revealed that students wanted to complete the STEM programming assignments to achieve good learning performance. In 1986, the United States National Research Council proposed the STEM education agenda of integrating Science, Technology, Engineering, and Mathematics. The four core elements of STEM education are all interrelated. The goal is to train students to become cross-disciplinary experts, like mathematicians, scientists, engineers, or technologists. STEM education is helpful for students to think about cross-disciplinary knowledge and to learn cross-disciplinary professional skills . For exaIn 2014, the National Science Board of America emphasized that STEM education develops students’ ability to integrate cross-disciplinary knowledge and learn hands-on problem-solving skills A . AchillePrevious studies  have attWhen students solve online programming assignments, they may encounter unanswered questions which may lead them to give up on solving the assignments. To address this problem, In this study, we proposed a problem-based learning approach and an online programming system integrated into the STEM programming activity. By integrating the problem-based learning approach with the online programming system, teachers can diversify the teaching content to enhance students’ collaboration and STEM cross-domain programming skills. In this experiment, the subjects were students from an online STEM programming course at a university in Taiwan. In the experimental activity, we used a four-step problem-based learning approach, namely stating the problem, understanding the problem, developing a solution plan, and executing the plan, reflecting, and debugging. This study explored the differences in students’ perceptions of STEM learning, learning outcomes, and learning behaviors when using the problem-based learning approach with the online programming system integrated into the online STEM programming curriculum. The study addressed the following research questions:Q1: What are the differences between students’ prior knowledge and learning performance?Q2: What are the differences between students’ learning perceptions?Q3: What are the differences between students’ behaviors in answering STEM programming assignments?In this experiment, the subjects were 20 students from a university in northern Taiwan. This study was derived from a college elective course that was open to students in third grade and above to take the online STEM programming course. All students agreed and volunteered to participate in the experimental activity. Each student had a computer to use during class, and all students had basic experience of using computers.The learning materials were developed by STEM programming experts . The leaThe course was designed as a series of learning materials. The learning materials included basic concepts of STEM education, basic programming knowledge in C/C++, and STEM programming assignments. STEM cross-domain knowledge covers the basic concepts of science, technology, engineering, and mathematics. The teacher used STEM cross-domain knowledge combined with problem-based learning to design STEM programming assignments. Based on the scientific concept and programming inquiry, we use the programming knowledge of data types and strings to develop the scientific conceptual programming assignment for the number of consecutive salt-based repetitions of genes. Based on the design of programs with new technologies, we use the programming knowledge of iterators and generalized algorithmic functions to develop the technology conceptual programming assignment for old cell phones. Based on the finished structure and engineering thinking, we applied the programming knowledge of class inheritance and virtual functions to develop the engineering conceptual programming assignment for the circuit analysis. Based on computational mathematical thinking, we used the programmatic knowledge of operator overloading to develop the mathematical conceptual programming assignment for the Inverse matrix calculator.During this experimental period, the COVID-19 outbreak occurred. The experiment was conducted for 6 weeks, once a week for 100 min, as shown in In the first week, the teacher introduced the use of the online programming system and how to conduct the experimental activity. The teacher explained the learning objectives and assignments for the activity. Next, the teacher administered a pretest to understand the students’ prior knowledge level. The teacher designed the STEM programming materials, and applied the learning materials in class.During the experimental activity, we conducted a 4-week experimental STEM programming curriculum using an online programming system and a problem-based learning approach, with each weekly session lasting 100 min. The teacher taught the cross-domain knowledge of STEM programs in the classroom, and then assigned the STEM programming assignments to the students, who completed the assignments in the classroom. These assignments included science (S), technology (T), engineering (E), and mathematics (M) concepts, such as the number of consecutive salt-based repetitions of genes, old cell phones, circuit analysis, and inverse matrix calculator. The teacher used a problem-based learning approach combined with an online programming system to implement the STEM programming assignments to develop students’ STEM cross-domain knowledge, collaboration, and STEM programming skills, as shown in As shown in The teacher implemented weekly STEM programming assignments. Students followed a step-by-step problem-solving process in the problem-based learning approach: Stating the problem, understanding the problem, developing a solution plan, and executing the plan, reflecting, and debugging.The first step was to state the problem: The teacher explained the objective of the STEM programming assignment and made sure that the students knew the key points of the assignment and the possible limits of the assignment. For example, the teacher explained the scientific conceptual programming assignment for the number of consecutive salt-based repetitions of genes. The teacher explained the main point of this assignment, which is “Please enter a DNA string consisting of four salts  and an integer n. Please find all the repetitions of this DNA string with n consecutive salt groups.”The second step was to understand the question: They needed to clarify the information in the question and understand and confirm the input and output requirements of the STEM programming assignment. For instance, the teacher explained the input and output requirements of the scientific conceptual programming assignment for the number of consecutive salt-based repetitions of genes. The input description was “Each input has a DNA string and a positive integer n.” The output description was “The number of repetitions of n consecutive salt groups according to the entered DNA string and the positive integer n.”The third step was to develop a solution plan: When the student understood the input and output requirements of the STEM programming assignment, the teacher could guide the student to develop a step-by-step solution plan and execution time, listing the goals to be achieved in each phase and a check schedule. The teacher could provide code passages for students to fix missing parts and add breakpoints to help students reduce debugging time in the development process. The plan must remain flexible enough to allow for the possibility that the solution period may be extended or shortened for some unpredictable reason.The fourth step was to execute the plan, reflect and debug: Students executed this solution plan, took a practical test, confirmed the correctness of the solution, and reflected on the correction of errors. If a student encountered a problem, the teacher gave tips on how to fix the code according to the solution plan. During this step, the teacher observed the student’s log of program debugging to understand the student’s learning, and provided timely guidance to give hints on how to solve the STEM programming assignment.At the end of the experimental activity, the teacher conducted a posttest to understand the students’ learning effectiveness and administered a learning perception questionnaire. Finally, we collected the data from the tests, questionnaires, and the system logs of the online programming system for subsequent experimental analysis.The pretest and posttest were the same test. The test was designed by experienced STEM programming teachers . The preWe used the test results to classify the low scoring group (the bottom 33%) and the high scoring group (the top 33%). Based on the low and high score groups, we calculated the difficulty and discrimination of the pretest and posttest . The difWe modified the learning perception questionnaire of previous studies  to creatThe questionnaire was administered using a 5-point Likert-type scale. We assessed the reliability of the questionnaire using the Cronbach’s alpha coefficient method . The CroThe pretest measured differences in prior knowledge among students, while the posttest measured the students’ learning effectiveness. We compared the differences in students’ prior knowledge and learning outcomes.The mean of the pretest was 51.00, and the standard deviation was 11.54. From the standard deviation in the pretest, we know that the low-scoring group of students had no experience of STEM programs. The mean of the posttest was 71.75 and the standard deviation was 12.27. This result indicated that the average score of students on the posttest was higher than the pretest.t test to compare the pretest with the posttest. This result showed that the value was-11.16 and the two-tailed test was significant (p < 0.001), rejecting the null hypothesis. We found that Cohen’s d was 0.891, which is an effective value to achieve high efficiency . Furthermore, there was a significant difference in the learning outcomes of students in the low prior knowledge group . This indicated that most students’ learning outcomes were higher than their prior knowledge.Students were divided into the high and low prior knowledge groups based on their pre-tests. The low prior knowledge group (10 students) was 50 points below the mean of the pretest, while the high prior knowledge group (10 students) was 50 points above the mean of the pretest. We used the paired-sample t test to analyze the results of students’ responses to the STEM learning perceptions, as shown in In this study, we explored students’ STEM learning perceptions using problem-based learning methods and online programming systems integrated into the online STEM programming course. We used the one sample Students submitted their STEM programming assignments using the online programming system and the total number of submissions was 707 times, as shown in According to In this study, we used a problem-based learning approach with an online programming system to integrate an online STEM programming activity. After the experimental activity, we collected the experimental data and analyzed them. We explored the effects of using the problem-based learning approach with the online program system integrated into an online STEM programming curriculum on students’ learning outcomes, STEM learning perceptions, and learning behaviors.After statistical analysis of the pre-test and post-test, we found that there were significant differences in students’ prior knowledge and learning outcomes. This means that most students used the problem-based learning method and an online programming system to help them learn STEM programs, which in turn led to a significant increase in their learning performance.Next, we divided the group into a high and a low prior knowledge group. After statistical analysis of the high and low prior knowledge groups’ learning performance, it was found that there was a significant difference between the two groups’ learning outcomes. Regardless of whether it was the high prior knowledge group or the low prior knowledge group, students in both groups had good learning performance in the STEM programming activity. In the low prior knowledge group, students had a more significant increase in learning outcomes. We found that students focused on STEM programming skills and did not care about the theoretical knowledge of STEM programming, which led to the limitation of students’ learning effectiveness. Such results are similar to the problems encountered by students as mentioned by Finally, this study used a problem-based learning approach with an online programming system to effectively enhance students’ STEM programming skills and further improve their learning outcomes. Our results are in line with The STEM learning perception questionnaire includes the organization, elaboration, metacognition, effort, attention, and reference dimensions. The results revealed that there was a significant difference in students’ responses to each dimension of the STEM learning perception questionnaire. This means that using the problem-based learning approach with the online programming system is helpful for students taking STEM programming courses. In addition, we found that the mean of the organization dimension was slightly lower than others. We think that students do not like the idea of using notes to record classroom content. Instead, they enjoy working with their hands to identify problems and solve them to complete STEM programming assignments. In addition, it was found that the reference dimension was higher than other dimensions. We believe that when students encounter a problem while working on a STEM programming assignment, they will look at the textbook again and again to find a possible solution to the problem. When students do not understand the learning material in the online course, they will actively look for other online resources to learn possible problem-solving techniques. From the results of the STEM programming assignments submitted by the students of different prior knowledge groups, we found that there were differences in the students’ behaviors in answering the STEM programming assignments. It was found that the number of wrong answers of the high prior knowledge group was higher than that of the low prior knowledge group. On the other hand, the number of correct answers was higher for the low prior knowledge group than for the high prior knowledge group. We found that the students of the low prior knowledge group repeatedly confirmed the correctness of their STEM programming assignments. Furthermore, in the high prior knowledge group, students did not give up and continued to try possible solutions to complete their STEM programming assignments.We then used the Spearman’s Rank Correlation Coefficient method to analyze students’ answering STEM programming assignment behaviors and their STEM learning perceptions. In this result, we see that there is a significant correlation between the wrong answers behavior and the Elaboration dimension of the STEM learning experience. This finding revealed that when students gave incorrect answers to the STEM programming assignments, they repeatedly looked at the learning materials in the online course. Students associated them with the problem solving skills taught by the teacher in the past, and tried different solutions until they completed the assignments. Our findings are similar to those of Under the influence of COVID-19, remote learning was widely adopted in universities. In this study, a problem-based learning approach was combined with an existing online programming system for an online STEM programming activity. However, there are still shortcomings that need to be remedied in future research. First, this study only focused on the learning behaviors associated with an existing online programming system. In future studies, learning behaviors that affect college students’ online learning of STEM programming, such as many learning behaviors related to the features of the existing online programming system, can be discovered. In addition, the sample size should be expanded and the results should be examined with a broader and more representative sample in the future. Finally, this study lacked a comparison between an experimental group and a control group in order to present the differences in learning performance between the implementation of different teaching strategies and different teaching tools in the online STEM programming course. Future research may implement different teaching strategies and different experimental teaching tools to compare students’ learning performance in the online STEM programming course.All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.This study was supported by the Ministry of Science and Technology, Taiwan, under two government grants .The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


The accurate noise parameter is essential for the Kalman filter to obtain optimal estimates. However, problems such as variations in the noise environment and measurement anomalies can cause degradation of estimation accuracy or even divergence. The adaptive Kalman filter can simultaneously estimate state and noise parameters, while its performance will also be degraded in complex noise. To address the problem of estimation accuracy degradation and result divergence of the integrated navigation system in a complex time-varying noise environment, an improved multiple-model adaptive estimation (MMAE) that combines the Sage–Husa adaptive unscented Kalman filter with the MMAE is proposed in this paper. The forgetting factor is included as an unknown parameter of MMAE so that the algorithm can adjust the value of the forgetting factor according to different system states. In addition, we improve the hypothesis testing algorithm of classical MMAE to deal with the competition problem of undesirable models that severely impacts the performance of variable-parameter MMAE and enhance the algorithm’s parameter identification capability. Simulation results show that this method enhances the system’s robustness to noises of different statistical properties and improves the estimation accuracy of the filter in time-varying noise environments. The strapdown inertial navigation system (SINS) is a navigation method in which the inertial sensors (gyroscope and accelerometer) are fixed directly to the body. In contrast to platform inertial navigation, which requires a complex mechanical and physical platform structure, the SINS adopts an algorithmic platform to solve for attitude, velocity, and position information of the body, which has the advantages of small size, light weight, strong overload resistance, simple hardware structure, and comprehensive output information, and is widely used for attitude measurement and navigation in missiles, vehicles, and aerospace ,2,3.Nevertheless, inertial navigation systems have the issue of high short-term accuracy and severe long-term drift in practice. It is hard to rely on their information for correction, which dramatically affects the accuracy of inertial navigation . TherefoMoreover, information from different sources often contains some uncertain noise. These noises cannot usually be accurately measured and simply removed, whereas the characteristic statistical parameters can be reckoned from the physical model characteristics of the system. Therefore, algorithms that can obtain helpful information in dynamic systems with uncertain noisy information and perform optimal state estimation based on the prediction of the next moment are incredibly critical .The Kalman filter (KF) is an optimal state estimation algorithm proposed by Kalman in the 1960s, which uses prior knowledge such as deterministic and statistical characteristic parameters of the system and measurement information to obtain the optimal state estimation . The staThe extended Kalman filter (EKF) is used for the state estimation of nonlinear systems. It uses the Taylor series expansion method to linearize the nonlinear model to make it conform to the Kalman filter model . HoweverIn 1995, Julier and Uhlmann proposed the unscented Kalman filter (UKF). The UKF, also known as the sigma-point Kalman filter, transforms the n-dimensional state vector and error covariance matrix to the 2n-dimensional parallel state vector by unscented transformation to evaluate the nonlinear propagation of error covariance rather than by linearization . TherefoIn theory, the Kalman filter requires an accurate system model and the exact statistical properties of the noise parameters so that the optimal estimate can be obtained. Inaccurate noise parameters can easily lead to divergence of filtering results. To solve this problem, many scholars have performed careful research on the uncertainty of filter noise parameters. Robust Kalman filtering is a method for dealing with estimation problems for uncertain systems ,14. SavkInnovation-based adaptive estimation (IAE) is based on measurement innovation sequences to correct the noise parameter matrix, but the window size predominantly affects the adaptability of the system; choosing the appropriate size of the window is a crucial problem in practice ,18. Sun In 1964, Magill proposed multiple-model adaptive estimation (MMAE), which uses a set of parallel filters to estimate systems containing unknown parameters . The modIn this paper, we focus on estimation accuracy degradation and result divergence due to improper noise parameters; aiming to improve the robustness of the filter in a complex time-varying noise environment, an improved MMAE with variable parameters model was proposed. The method combines Sage–Husa with MMAE, which greatly reduces the dependence of the multiple-model algorithm on the initial value selection of model parameters. In addition, we found in our simulations that the optimal value of the forgetting factor is not the same for different system states. Although the difference in accuracy among them is not significant, if we can adjust the forgetting factor adaptively at each moment of the algorithm operation, it will significantly improve the overall accuracy. Therefore, we incorporated the forgetting factor into the unknown parameters of the multi-model structure. In addition, the undesirable competition problem of classical MMAE can have a more severe impact on the variable-parameter model, as we detailedly analyze in The rest of this paper is structured as follows. In order to better deal with nonlinear systems, the UKF was introduced in this paper for state estimation. The UKF uses the unscented transformation to generate sigma points. The mean and variance of the sigma points are the estimated value of the state vector and the error covariance matrix, respectively, which utilizes an estimation method rather than an approximate linearized approximation and therefore has higher accuracy. The mathematical formulation of the UKF is depicted below.The process and measurement model of nonlinear discrete-time system can be described as follows:The standard UKF algorithm can be described as follows:Step 1: InitializationStep 2: Sigma-point calculationStep 3: State predictionFollowing the sigma sampling strategy in Step 2, sigma points Step 4: Measurement updateSimilarly, the sigma point is calculated using Step 5: Filtering updateMMAE uses a set of parallel filters to estimate independently, and each model corresponds to different unknown parameters. The model’s probability is calculated according to the hypothesis test algorithm, and the final state output is obtained by weighted summation of each model state estimation. MMAE enhances the system’s adaptability to noise by increasing the number of models with different unknown parameters. This approach increases the computational effort; however it is stable and generally does not suffer from divergent estimation results.The classical MMAE is given as follows.The weight of each model can be formulated as follow:The final state estimation is obtained by weighted summation of each model:In general, MMAE assigns a suitable posterior conditional probability for each model to ensure that the final estimation converges to a proper range. From the perspective of probability theory, it belongs to an optimal estimate. However, after a lot of simulation verification, we find that the classical MMAE has the following two inherent problems in practice.The classical multi-model algorithm suffers from the issue of undesirable competition of unnecessary models . SpecifiThe selection of model parameters is usually in a bounded range based on analysis of system characteristics and the significance of parameters, where fixed parameters are selected to constitute the model . In theoTherefore, in order to solve the problem of estimation accuracy degradation caused by improper noise parameter setting and limited coverage of the classical MMAE model, aiming to enhance the adaptability of the system to time-varying noise and the ability to deal with nonlinear systems, we proposed an improved structure of the MMAE, which combined Sage–Husa with MMAE. Specifically, we replace the standard Kalman filter of the classical MMAE with the Sage–Husa adaptive UKF. This variable-parameter multi-model structure reduces the system’s dependence on the selection of initial noise parameters and extends the coverage of the MMAE model.The UKF has excellent advantages for dealing with nonlinear systems; however, the model divergence problem of the UKF is also more severe when the noise parameters are inaccurate . To bettIn the filter, measurement prediction error (measurement innovation) is:The Sage–Husa covariance estimator is given as: In the standard KF, the gain calculation loop where the noise parameter matrix is located is relatively independent of the state estimation loop. In contrast, the noise parameter matrix in the Sage–Husa adaptive filter is influenced by the measurement value, thus turning the system into a complex nonlinear system that is more severely affected by the estimation anomalies. In addition, according to Equation (22), we find that measurement innovation simultaneously acts on the parameter estimation process of Classical MMAE can be affected by undesirable models with undesirable competition due to the inherent characteristics of the multiple-model algorithm. For applications where the noise parameters variation is slight, the undesirable competition only affects the convergence rate of the model without having a severe impact on the estimation accuracy. However, for environments where the noise parameters can vary, undesirable competition can seriously affect the weight assignment of the optimal model, which extends detection time and reduces the system’s decision-making capability, especially when the optimal model has changed and the innovation differences between models are delicate.When performing further analysis (Equation (17)), we noticed that MMAE assigns weights mainly based on the innovation of each model. Thus, Firstly, we define two models to perform state estimation simultaneously and at the Then the weight of model Derivative of Equation (24) with respect to It can be seen from Equation (25) that the derivative is always greater than zero; therefore, the weight of model Through analysis, we can acquire that if we enlarge the penalty value, in the same circumstances, the model that is closest to the real model is assigned a greater weight. In other words, the larger the penalty value, the optimal model will be assigned a more significant weight more quickly. However, probability jumps may occur, resulting in a decrease in estimation accuracy, especially in the early stages of the algorithm operation, in order to maintain stability during the early stages when the model parameters vary considerably, as well as to still possess the ability to adjust the model weights and alleviate the problem of undesirable competition after a period of time. Inspired by , which iIn Equation (17), we found that in addition to the exponential term, there is also an culation . What isAs mentioned above, we combined the Sage–Husa adaptive UKF with MMAE to fully exploit their advantage. The forgetting factor has an effect on the estimation results, which determines the confidence of the adaptive system for prior information. The smaller the forgetting factor is set, the stronger the adaptability of the system; however, when the estimated parameters differ from the actual parameters by a small amount, if the forgetting factor is set small, it may lead to jumps in the parameters and reduce the stability of the estimation, thus affecting the filtering accuracy. In general, the forgetting factor needs to select the optimal value for the current moment to ensure a balance between system adaptability and stability; however, it is hard to achieve in practice. Therefore, in addition to the process noise matrix, the forgetting factor is also included in the unknown parameters of the MMAE, which enables the algorithm to identify the optimal forgetting factor value among the artificially set parameters, reducing the system’s dependence on the selection of the forgetting factor. Moreover, the undesirable competition problem can seriously affect the system’s ability to respond to complex time-varying noise. In this regard, we propose MMAE with an improved hypothesis testing algorithm to overcome the problem.The specific process of improved MMAE implementation is as follows, and the structure diagram is shown in Step 1: Set up Step 2: Initialize each model to a weight of Step 3: Each filter performs state estimation independently and calculates the measurement innovations.Step 4: The model corrects the measurement noise matrix according to the innovation and the corresponding forgetting factor using the Sage–Husa algorithm.Step 5: Update the filter weights based on the measurement innovations using an improved hypothesis testing algorithm.Step 6: Weighted summation of the estimated vectors for each filter to obtain the final state output.Step 7: Update the data and repeat Steps 3–6.GPS positioning error does not drift with time and can maintain high accuracy in long-time navigation, which complements INS and becomes the first choice for inertial-based integrated navigation. The INS/GPS integrated navigation structure is shown in In this section, we applied the algorithm to INS/GPS vehicle integrated navigation and carried out three experiments. Experiment 1 is used to verify the effect of the forgetting factor on estimation accuracy under different system states. Experiment 2 is designed to test the effectiveness of the improved hypothesis testing algorithm of MMAE for mitigating competition of undesirable models. Experiment 3 by comparing the positioning accuracy of different algorithms to test the effectiveness and meliority of the proposed method. A vehicle navigation platform, as shown in In addition, we used the mean squared error (MSE) of position error to quantificationally evaluate the performance of algorithms. MSE is the mean square of the distance between the predicted and actual values, reflecting the extent of data deviating from actual values and is inversely proportional to navigation accuracy. MSE can be defined as follows:Experiment 1:In this paper, we include the forgetting factor as an unknown parameter of MMAE to improve the performance of the algorithm in time-varying noise. That is because after In 0–100 s, the adaptive system estimates the measurement noise parameters, and the filter with a smaller forgetting factor has more adaptability, achieving a better performance. While the estimation accuracy of the filter with large values of the forgetting factor is higher when the variation of the noise parameter matrix tends to stabilize. That is consistent with our previous analysis. Although the difference in accuracy is not significant, if we can select the optimal forgetting factor for each moment based on the system state while the algorithm is running, there will be a substantial improvement in the overall accuracy, especially for systems in complex noise environments.Experiment 2:For variable-parameter models, undesirable competition can affect the convergence rate of the optimal model and extend detection time, especially when the parameters of other models do not differ much from the optimal model; the optimal model could even not be assigned the maximum weights due to the influence of previous data. Therefore, we improve the hypothesis testing algorithm of the classical MMAE. To verify the effectiveness of the improved method in alleviating the undesirable competition problem, we use a set of INS/GPS navigation data and perform the combined navigation solution with the traditional MMAE and the improved method, respectively. The weight change process of the optimal model was recorded and shown in According to the result of Experiment 3:In this part, we used a set of INS/GPS vehicle navigation data collected by the vehicle navigation platform, as shown in Subsequently, we used different integrated navigation algorithms to solve the data and compare the results with the navigation results of the reference device to obtain the position errors. The position errors in the east direction for different algorithms are demonstrated in From the results, the standard UKF cannot respond well to changes in noise characteristics due to it having no adaptability. Its performance is the worst among the five algorithms during these three time periods. MMAE uses a set of parallel filters to estimate the state vectors simultaneously, each filter corresponding to a different noise parameter; the results of each filter are weighted and summed to derive the final state estimate. It expands the coverage of the noise parameter model to a certain extent, and the estimation accuracy is improved compared with the standard UKF. However, models with fixed parameters and a finite number can hardly cover the whole characteristics of varying noise. Therefore, there is only a limited improvement in accuracy.The adaptability of Sage–Husa adaptive UKF for changing noise is significantly higher than that of the standard UKF and classical MMAE. During the three observation periods, the accuracy is improved apparently. That is because it can estimate noise parameters in real-time while performing parameter estimation, which enables the system to follow the variation of the noise environment well. However, a single forgetting factor and process noise parameter matrix cannot guarantee that Sage–Husa gets the optimal estimation at every moment, which dramatically limits the performance.The exponential decay penalty value function proposed in  enables The improved MMAE takes advantage of Sage–Husa adaptive estimation and MMAE. The forgetting factor and the process noise parameter matrix are set as unknown parameters of the multi-model structure. The improved probability assignment function significantly alleviates the undesirable competition problem and enhances the ability of the algorithm to assign model weights. This makes it feasible for the proposed method to select the optimal value of the forgetting factor according to different system states. The stability and adaptability of the system are simultaneously guaranteed. The performance is better than the other four algorithms under the varying noise environment.In order to show the performance difference between the five algorithms more intuitively, the radar chart for MSE of position error as shown in In this work, an improved multiple-model adaptive estimation method was developed, taking advantage of MMAE and Sage–Husa. The performance of the proposed method in integrated navigation simulation is higher than the UKF, Sage–Husa, and classical MMAE. Furthermore, it is easy to extend the algorithm to pattern recognition, target tracking, and fault detection to reduce system dependence on noise parameter selection and improve filter robustness to time-varying noise. In addition, how to enhance filter stability when measurement anomalies and reduce adaptive system complexity will be our subject in the future.