Artificial Intelligence Enabling Denoising in Passive Electronic Filtering Circuits for Industry 5.0 Machines

Abstract

The paper proposes an innovative model able to predict the output signals of resistance and capacitance (RC) low-pass filters for machine-controlled systems. Specifically, the work is focused on the analysis of the parametric responses in the time- and frequency-domain of the filter output signals, by considering a white generic noise superimposed onto an input sinusoidal signal. The goal is to predict the filter output using a black-box model to support the denoising process by means of a double-stage RC filter. Artificial neural networks (ANNs) and random forest (RF) algorithms are compared to predict the output of noisy signals. The work is concluded by defining guidelines to correct the voltage output by knowing the predictions and by adding further RC elements correcting the distorted signals. The model is suitable for the implementation of Industry 5.0 Digital Twin (DT) networks applied to manufacturing processes.

1. Introduction

Resistance and capacitance (RC) low-pass filters are adopted in many application fields such as hydraulic switching controlling valves and actuators [1], electrooculography systems [2], automatic gain control [3], and continuous time auto-tuning filtering [4]. Specifically, low-pass filtering is an important function for electronic energy management systems [5,6], positioning systems [7,8], and for the predictive control of power [9]. Filtering processes are generally influenced by noises disturbing machine control. Signal noise reduction is a fundamental process to improve filtering actions [10,11]. A method to optimize filtering is to analyze the lower and the upper envelopes of the noisy signal [12] in order to apply corrective actions adjusting and correctly filtering the output signal. In this direction, artificial intelligence (AI) tools could estimate the envelope trend of the noisy output signal to suppress the noise [13,14]. Artificial neural networks (ANNs) and random forest (RF) algorithms are suitable and robust for signal classification [13,15]. A possible implementation of AI tools is the modeling of circuits behaving as ‘black boxes’ [13,16] to predict or classify the signal only knowing the inputs and outputs of the analyzed circuit [17]. ANN is suitable to predict amplified signals disturbed by additive noises having specific carriers as characteristics [13]. When white noise is considered, the noisy signal classification and the denoising process become more difficult. In this direction, a technique adopted for the denoising process to remove the residual white noise is the complex self-adaptive wavelet packet technique [18,19,20] which does not consider the prediction to anticipate the future trend of an output signal affected by a noise. The white noise is a random signal with an even frequency distribution. It is generally studied for predictive maintenance of manufacturing processes and for fault detection in industrial machinery processes adopting machine-learning algorithms [21,22,23]. More precisely, the white noise, namely also background noise, is an electrical noise which can be generated by different sources such as power electronic devices, control circuits, arcing equipment, electromagnetic interferences, and switching power supplies [24]. In electronic devices, the white noise is a thermal noise (Johnson noise) or a shot noise (Schottky noise) [25]: the thermal noise is generated by the random Brownian motion of electrons increasing with temperature in resistive elements, while the shot noise is due to the not smooth current flowing through the electronic junctions (fluctuations in electrical current). The goal of the proposed study is the formulation of a black-box model predicting RC filtered signals disturbed by white noise to provide automatically corrective actions through an RC compensatory network. The challenge of the proposed work is to formulate a procedure to select dynamically possible RC parameters arranged in cascade layouts and optimizing filtering by means of the prediction of the noisy signal. This challenge is hard for the problem complexity due to the random nature of the white noise effect. The work is structured in the following sections:

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The Material and Methods Section discussing the circuital low-pass filtering model and the AI-ANN/RF workflow adopted for the data processing; the section provides the proof of concept of the AI controlled black box double-stage RC filter influenced by white noise, information about the adopted tools, and a description of the basic voltage signals;

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The Results Section describing circuit simulation through a parametric analysis modeling different effects of the white noise, and the AI results and performances; the section highlights the parametric circuital responses of the analyzed filter by modulating the white noise amplitude and predicting by means of the ANN and RF algorithms the voltage output enabling a possible switching condition on a further RC stage (second stage) to denoise the voltage output; furthermore, the section provides performance results proving the correct choice of the ANN and RF algorithms for the proposed study;

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The Discussion Section describing the advantages, perspectives, disadvantages, and limitations of the proposed model, and the new protocol to follow in order to use a double-stage low-pass filtering network; the protocol allows us to understand better the proof of concept of the switching circuit selecting the single or double RC filtering stage;

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The Conclusions Section highlighting the results.

2. Materials and Methods

The basic concept of the model adopted in the proposed work is to integrate into a black box (see Figure 1a) both the RC circuit and the white noise signal. The analyzed double-stage RC circuit model automatically optimizing the filtering process is sketched in Figure 1b. The model takes into account at the input port a pure sinusoidal signal disturbed by white noise. The effect of the white noise signal on the input signal is modulated by the R_x resistance.

The circuit model is characterized by considering the R_x-parametric analysis simulating the different effects of the white noise at the output port. A first RC low-pass filtering network constituted by the electrical resistance R_f and the capacitance C_f enables the main denoising process (first RC filter stage). Based on the predicted results, an AI controller is able to select only the RC first stage or to further connect in cascade another RC filtering network (compensatory network) constituted by other elements (resistance R_c and capacitance C_C) able to improve the final filtering result according with the predicted envelopes of the output voltage. The circuit is modeled by different switches controlled by AI triggering the command ‘0’ to select only the first RC stage, or the command ‘1’ to select the whole cascade network including the compensatory filtering stage. The goal of the proposed model is to use the AI noisy signal prediction to decide the selection of the second RC stage dynamically optimizing the filtering process.

The electrical signals of the circuit model of Figure 1 are estimated by the LTSpice circuit of Figure 2. LTSpice is an open-source ‘SPICE-based’ analog electronic circuit simulator [26] (Version x64: 24.0.12) suitable for time-domain and frequency-domain simulations. The circuit model of Figure 2 takes into account the coupling of white noise with the input signal defined as follows:

$$V_{Input\_signal}=sin\left(2\pi t\ast {10}^2\right)$$

(1)

The carrier of Equation (1) is f₀ = 100 Hz. The time-domain simulation is performed by setting a transient analysis with the following parameters: stop time = 100 ms, time to start saving data = 0 s, maximum time step = 0.5 ms. The coupling between signal and noise is modulated through the R_x resistance. A parametric analysis versus the variable R_x is performed to analyze the different noise modulation effects: the parametric analysis is executed by assigning the values of R_x as 60 Ω, 100 Ω, 140 Ω, 180 Ω, and 220 Ω. The first RC stage is constituted by the electrical resistance R₄ = 1 kΩ and the capacitance C₂ = 100 nF as fixed parameters. The second stage behaves as a further compensatory filtering network constituted by other R and C parameters (variable parameters) to assign the function of the predicted trends of the voltage output envelopes (ANN and RF predicted trend). White noise is commonly used to describe a random noisy process as a random process X(t) having a flat power spectral density S_X(f):

$$S_X\left(f\right)=constant={\displaystyle \frac{N_0}{2}} for all frequencies$$

(2)

In the proposed simulations, the white noise intensity is variable and modeled by the R_x resistance of Figure 2 modulating the noise amplitude (parametric simulation changing the R_x value). All electronic and electrical elements intrinsically generate noise which could be modeled by random white noise. The noise for semiconductor devices includes all resistors’ noise effects such as thermal, shot, and avalanche noises. The thermal noise (named the Johnson noise) characterizes all passive resistive elements and is generated by the random Brownian motion of electrons increasing with temperature in electrical resistances. The thermal noise is defined by the following power density:

$$v_n^2=4K_{B}TBR$$

(3)

where v_n is the noise voltage (in Volts rms units) which is the value that would be measured driving a totally noise-free band-pass filter (with bandwidth B) with a voltage generated by a resistor at temperature T, K_B is the Boltzmann’s constant (expressed in Joules per Kelvin), T is the resistor’s absolute temperature (expressed in Kelvin), R is the resistor’s value (expressed in Ohms), and B = Δf is the band. The shot noise (named the Schottky noise), is generated in active devices for charge flowing as in transistors and diodes. The noise is due to the random arrive of individual electrons flowing through the junction generating microscopic current pulses. This noise can be modeled by a Gaussian white spectral density:

$$I_n^2=2q{I}_{DC}B$$

(4)

where I_n is the noise current (in Ampere rms units) estimating the electrical current fluctuations, q is the electron charge (1.6 × 10⁻¹⁹ C), I_DC is the direct current (DC), and B = Δf is the band. Finally, the avalanche noise is typical in p-type and n-type (PN) junctions as Zener diodes, working in reverse breakdown mode: the current generated during the avalanche breakdown process is made by random distributed noise spikes behaving as shot noises. In the proposed study, the frequency analysis is performed by means of the fast Fourier transform (FFT) algorithm. All the white noise typologies are characterized by intensity depending on the effective physical effect. In the discussed model, the noise intensities are changed by adding in series to the white noise generator an R_x resistance having the role of modulating the noise amplitude (see Figure 2).

The AI prediction of the noisy signal is implemented by the Konstanz Information Miner (KNIME) workflow of Figure 3. KNIME is an open-source tool [27] adopting graphical interfaces for the setting of the hyperparameters of the AI algorithms. The KNIME workflow is structured into the following three main sequential parts:

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Input data: input dataset for the training and testing of the AI supervised algorithms;

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Data pre-processing: data cleaning and manipulation providing the final dataset to process by the AI algorithms;

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Data processing and reporting: AI data processing, data visualization, and algorithm performance.

The KNIME workflow graphical user interfaces (nodes) are as follows:

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‘File Reader’: data input importing the output voltages of the parametric analysis (output of the first RC filtering stage) changing with the R_x noise parameter;

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‘Normalizer’: scaling of the input voltage data from values ranging from 0 to 1 (decrease in the computational error);

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‘Column Appender’: merging operation of the all input data into an unique data table to be processed by the ANN and RF algorithms;

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‘Row Filter’: filter preparing the best database to process (the final record number to process is 2409);

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‘Partitioning’: block able to structure the training and the testing dataset (randomly distributed with 70% in the training dataset and the remaining 30% in the testing dataset);

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‘RProp MLP Learner’: block implementing an ANN multilayer perceptron (MLP) learning algorithm using the RProp optimizer;

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‘MultiLayer Perceptron Predictor’: node adopted for the ANN testing data processing;

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‘Random Forest Learner’: RF algorithm learner block;

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‘Rando Forest Predictor’: RF algorithm testing block;

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‘Numeric Scorer’: node estimating algorithm performances;

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‘Line Plot’: dashboard visualizing ANN and RF results;

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‘X-Partitioner’: node used for the cross-validation loop; the dataset is divided into different groups named folds (for the specific case, 10 folds are considered);

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‘X-Aggregator’: at the end of the loop, the ‘X-Aggregator’ node aggregates the results from each iteration; all nodes between the ‘X-Partitioner’ and the ‘X-Aggregator’ node are iteratively executed; the block compares the predicted classes with the real classes for all rows providing iteration statistics.

The protocol defining the denoising process using AI is designed by a Business Process Modeling and Notation (BPMN) workflow (see Section 4). The BPMN is an international standard (ISO/IEC 19510:2013 standard [28]) useful to define all the steps to follow for a measurements and actuation process suitable for the modeling of manufacturing production processes.

Figure 3. KNIME workflow adopted for RF and ANN signal prediction.

Figure 3. KNIME workflow adopted for RF and ANN signal prediction.

3. Results

This section proposes three sub-sections: the first describes the white noise effect by adopting the LTSpice simulator; the second discusses the ANN and RF results defining predictive specifications to optimize the filtering process and algorithm efficiency; finally, the third sub-section allows the LTSpice simulator to prove the efficiency of the second filtering stage based on predictive results.

3.1. White Noise and RC Filtering: LTSpice Simulation

The time-domain input signal is plotted in Figure 4a. The FFT signal of Figure 4b highlights the carrier component at f₀ = 100 Hz. White noise is added to the input signal following the schemes of Figure 1 and Figure 2. The time-domain white noise signal and the related FFT response are illustrated in Figure 5a,b, respectively.

The random nature of the noisy signal is observed in Figure 5, where no carriers are observed in a broad band. The white noise influences the input signal, distorting the latter as observed in Figure 6 representing the simulation of a parametric analysis modeling the noise modulation effect: by changing the parameter R_x of the white noise (R_x values of 60 Ω, 100 Ω, 140 Ω, 180 Ω, and 220 Ω), it is possible to observe the ripple behavior intensity, the signal amplitude variation, and the signal shifting. Specifically, for lower values of R_x a greater decrease is observed in the input signal voltage amplitude; otherwise, for a higher value of R_x, a higher value of the input signal voltage amplitude is observed.

The basic circuital behavior of a single RC stage and a double RC stage is described in Appendix A: this study allows us to better comprehend the effect of each electrical component facilitating an understanding of the whole circuit behavior. As observed in Appendix A, the first RC stage, without the noise, generates a slow shifting of the output signal. The effects of the C₁ parameter, also in the presence of white noise, are the shifting of the output voltage and the amplitude decrease (the output signal amplitude decreases with an increase in the C₁ parameter).

3.2. ANN and RF Noisy Signal Prediction: Predicted Signal Trends and Algorithm Performances

The proposed model is able to learn the ANN and RF supervised algorithms through the output signals of the first RC filtering stage. The learning models allow us to predict the signal voltages at the output of the first filtering stage by considering all the signals (noise plus RC signals) enclosed in a black box. In Figure 7a–d, different comparisons between the voltage outputs (using different R_X values) and the ANN predictions are illustrated. The ANN simulations highlight the trend overlapping between the output voltage signal and the predicted ones. We note that for certain noise intensities, the predicted signal trends present differences if compared with those of the output signals: this is due to the learning of the algorithm (history of different signals with different noise intensities) and not to the algorithm performances which appear to be rather good (see

Table 1. Comparison between ANN and RF algorithms: MSE and MAE performance parameters. The ANN approach shows a slightly better performance.

Method	MSE	MAE
ANN	0.008	0.06
RF	0.01	0.065

).

The plots of Figure 7 are characterized by different time steps according with the LTSpice sampling. The main result of Figure 7 is the prediction of the noisy signal trend marked by the red dashed lines. The analysis of the envelope trend supports the decision making about the application of the second filtering stage (compensatory filtering network of Figure 2) and the choice of the parameters C₁ and R₃ of Figure 2 able to optimize the denoising process. The RF results are very similar to ANN ones. In Figure 8, a comparison between the ANN and RF trend of the normalized predicted output voltage is illustrated. The algorithm performance is estimated by the mean absolute error (MAE) and the mean squared error (MSE) defined as follows:

$$MAE={\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^n\left|y_i-x_i\right|}{n}}$$

(5)

$$MSE={\displaystyle \frac{1}{n}}{{\displaystyle \sum }}_{i=1}^n{\left(Y_i-{\widehat{Y}}_i\right)}^2$$

(6)

where |y_i − x_i| = e_i is the arithmetic average of the absolute error, y_i is the predicted value and x_i is the true value, n is the number of the predicted results, Y_i is the vector of the observed values, and Ŷ_i is the vector of the predicted values. The MAE is the absolute deviation between the observed and predicted values. The MSE is the squared deviation between the observed and predicted values. Both the y_i and Ŷ_i values refer to the points of the ANN/RF predictions estimated at precise epochs, while both the x_i and Y_i values refer to the voltage output considered as the class to predict and estimated at the same epochs (the column containing the target variable).

The method followed to fix the hyperparameters of the ANN algorithm is summarized in the following steps, specifically it is as follows:

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Fix the minimum number of neurons per layer (in the proposed case, the minimum number is 10 corresponding to the number of the input nodes);

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Fix the number of epochs enough to complete the algorithm’s iterations (in the analyzed case the epochs are 400);

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Vary the number of hidden layers, finding the minimum error condition (see Figure 9, finding the minimum condition for 18 hidden layers).

In Table 1, the performance of the ANN algorithm is compared with the RF one in terms of the MSE and MAE parameters: a slow variation is observed, validating both the ANN and RF approaches for the prediction analysis. In

Table 2. ANN 10-fold cross-validation test: MSE versus the different folds constructed for the testing.

Fold	MSE
Fold0	0.012
Fold1	0.01
Fold2	0.014
Fold3	0.013
Fold4	0.015
Fold5	0.015
Fold6	0.008
Fold7	0.011
Fold8	0.013
Fold9	0.012

, the MSE values of the cross-validation test of the ANN algorithm are listed. The cross-validation test is executed for the optimized ANN-MLP algorithm: the test considers ten number of validations adopting the stratified sampling to structure the ten folds. For the analyzed case, the best estimated MSE is 0.008. The similar performance results in Table 2 prove that the adopted dataset is rather balanced and validate the correct choice of the hyperparameters. In Appendix B, the best ANN architecture providing the minimum ANN error is illustrated.

3.3. Denoising Optimization through the Second Compensatory RC Stage

The predicted envelopes of Figure 7 allow the choice of the electrical parameter of the RC compensatory filtering stage of Figure 2: specifically, good parameters for adjusting the signal trend are C₁ = 1000 nF and R₃ = 1 kΩ. The choice of network parameters is related to the compensation of the predicted worst condition (greater difference between upper and lower envelopes) indicating the worst condition of noise ripple. The LTSpice tool is adopted again to check this compensation.

The simulation of the parametric analysis shows at the output of the first RC stage a noisy signal (see Figure 10a) which is further denoised by the compensatory RC filtering stage, cleaning the final signal (see Figure 10b) and proving the correct choice of the C₁ and R₃ parameters.

By observing a predicted voltage amplitude decrease, it is preferable to apply a further corrective action by adding an operational amplifier at the circuit output. Furthermore, a prediction of a signal shift requires a corrective action based on signal shifting (see the phase shift effect in Appendix A). The clearer frequency response behavior of the compensatory network is also observed by the FFT parametric analysis of Figure 11 (Figure 11a,b), showing the parametric frequency responses at the output of the first RC stage and at the end of the compensatory circuit varying R_x, respectively. Figure 11 highlights that the frequency response is also denoised using the compensatory filtering stage (simultaneous check between the time-domain and frequency-domain responses).

4. Discussion

White noise is triggered by random processes involving signal coupling, turbulence or vibration effects, random impacts, electronic defects, or electromagnetic interferences due to charge/discharge processes in a motor, cabling, electrical/electronic junctions, etc., and it particularly occurs in machinery control systems. White noise generalizes the presence of noisy sources which are not easy to identify. For this purpose, the proposed black-box model is suitable not to classify the noise type but to predict the output signal trend of a specific machinery environment affected by disturbances. The application of the proposed model with random noise as white noise allows us to generalize the application field for different typologies of noises including those with carrier components and additive or multiplicative noises.

The white noise and AI implementation are characterized by different advantages and disadvantages as listed in

Table 3. Advantages and disadvantages of the white noise and AI implementations.

Proposed Modeling in Different Methodologies of Analysis	Advantages of Implementing White Noise	Disadvantages of Implementing White Noise
Generic signal processing	In signal processing, white noise is applied to check machinery systems and the frequency behavior of electronic circuits and to prevent quantization errors (a typical application is in high voltage transformers [29]). AI could facilitate the understanding of the signal behaviors in the presence of white noise through the classifications of noisy signals.	Signal processing requires an accurate check system measuring the noise effect with a high sensitivity (high sensitivity also in measuring low-intensity noises). In this direction, the white noise could strongly amplify the error, distorting the system response, and a high sensitivity model may not be suitable for the denoising process. Furthermore, AI algorithms could be useless when the noise completely covers the signal.
Time-series analysis	In time-series analysis, white noise is used to estimate residuals and errors, by considering residuals as random fluctuations in machine circuits. Furthermore, AI could predict the time-domain signal behavior to prevent possible corrective actions, thus decreasing the risk of machine breakdown.	Regarding this methodology, a main disadvantage is the different time sampling of the electrical signals according with the adopted circuit simulation tool: this could reduce the number of records to be used for the training model and could require dataset manipulation to obtain a homogeneous dataset to process (filtering of empty records).
Statistics	White noise is a baseline in many statistical models to compare other signals. Statistics can be compared with AI results finding matching results supporting the whole analysis.	AI is unnecessary for data processing requiring simple static approaches to study white noise responses.
Interference analysis	White noise is adopted to study interferences in an analogic systems and to prevent crosstalk [30]. AI could be suitable to classify the different interferences influencing machines.	The origins of circuit interferences are difficult to determine. In this direction, the AI-based black-box approach is appropriate to describe the circuital behavior without the knowledge of the interference which can be modeled in a generic way by white noise.
Machine control systems	Additive white noise and disturbances characterized by specific carriers are useful to analyze disturbances in the Proportional–Integral–Derivative (PID) circuits controlling machines. AI algorithms could support corrective actions as for passive RC filtering networks coupled with PID systems.	There are limitations in the identification of possible white noise effects in each element of the PID control system (individually affecting the P or I or D element). AI algorithms could be useless to differentiate the noise effect on each individual element.
Digital Twin (DT) simulations	White noise and AI are essential to structure a DT model able to simulate the real behavior of machines working in manufacturing processes by predicting breakage risks.	DT models should be accurately tested for the setting of efficient testbeds to associate with a specific machine or production line.

. Specifically, Table 3 addresses the proposed modeling in different methodologies of analysis such as generic signal processing, time-series analysis, statistics, interference analysis, machine control systems, and Digital Twin (DT) simulations, by highlighting possible advantages and disadvantages of implementing white noise.

The limitations of the proposed approach are mainly in the use of passive RC circuits. Possible future developments could be in the design of a similar filtering network using the same procedure as the proposed paper but adopting active elements. In

Table 4. Limitations and perspectives of the proposed approach.

Technological Limits	Technology Description	Technological Perspectives
Application for RC filtering network	The methodology is applied for specific double-stage RC circuits controlling machines. More complex systems could be controlled by PID systems combined with appropriate filtering networks.	Extension of application fields for more complex machines or components including drones [31], smart building [32], and more complex optoelectronic systems [33,34].
Passive noise control	The proposed approach takes into account a passive network and not considers electrically powered equipment or components. An active network uses electrically powered equipment or components to route the signal and could be more efficient for the whole denoising process.	Possible perspectives are the adoption of active systems for active noise control [35], CNN active noise control [36] (as for acoustic signal processing), generative fixed-filters [37], and virtual sensing active noise control system [38].
Technique to command switching circuits	The proposed approach is based on the concept to control circuits by switches controlled by commands (‘0’ or ‘1’ values).	Advanced control switching systems could be achieved by adopting innovative approaches as switching noise suppression with Multisampled Pulsewidth Modulation technique (MS-PWM) [39].
Difficulty to distinguish the origins of the white noise influencing the machine control circuits	Usually the origin of the white noise is unknown. The causes can be casual or intentional.	In the case of intentional causes as for cybersecurity applications, the proposed model can be applied to find possible analog Trojans modifying the electrical behavior of the control system [40,41,42], or in general to detect attacks in radio frequency environments.

, these aspects are detailed.

The proposed data processing model is a good alternative to non-AI methods such as the overall filtering algorithm [43], Weighted Nuclear Norm Minimization (WNNM) [44], the Unbiased Finite Impulse Response (UFIR) filtering algorithm [45], the Mixed Gaussian and Random-Valued Impulse Noise (RVIN) removal approach [46], and the bias compensated Maximum Complex Correntropy Criterion (MCCC) [47].

The protocol indicating the usage of the proposed methodology is illustrated in the BPMN workflow of Figure 12. The workflow is described by the following steps numbered in the figure:

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Step 1: the first RC stage is able to read possible noisy voltage outputs; the outputs are adopted for the AI training and testing processes (creation of the historical dataset);

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Step 2: ANN/RF predictions provide the envelopes of the predicted voltages and the predicted signal trend;

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Step 3b: if the envelopes are regular, no further filtering is required and the machine is correctly controlled (the ‘0’ command of the circuit of Figure 1);

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Step 3a/Step 4a: if the envelopes are different from the expected voltage envelopes (voltage envelopes without the noise effect), then the circuit simulation is executed to find the best R and C component of the second stage filtering compensatory network (Step 3a); the best R and C components are selected dynamically, and the further compensatory RC filtering network is applied (the ‘1’ command of the circuit of Figure 1).

Figure 12. BPMN workflow of the guidelines for the use of the circuit model of Figure 1.

Figure 12. BPMN workflow of the guidelines for the use of the circuit model of Figure 1.

The proposed work is a full upgrade of a previous work [13] adopting AI in electronic systems. The main differences and novelties are in the functionalities listed in

Table 5. Functionalities and novelties of the proposed model compared with the functionalities of [13].

Functionalities	Proposed Model	Model Proposed in Reference [13]
AI prediction of sinusoidal signals	√	√
Black-box modeling	√	√
Model applicable for different topologies of circuits	√	× (only for amplifier)
Multi stage circuit model	√	×
White noise implementation	√	×
Feedback principle defining the switching control condition of an intelligent circuit denoising the output signal	√	×
AI setting for the prediction of random noise generalizing a generic noise effect	√	×
Estimation of the signal output envelopes to compensate automatically the noise effect (denoising process)	√	×
Multiplicative noise modeling and simulation	√	×
Modeling and simulation of the modulation of the noise intensity (parametric analysis)	√	×

.

In Appendix C, an example is described of the application of the ANN model considering experimental data [48]. The proposed model is a first step to realize a real testbed Digital twin (DT) including as hardware a microcontroller, and as software we use an interface implementing the workflow of Figure 12. A scheme of the testbed architecture is illustrated in Figure 13.

5. Conclusions

The paper investigates an innovative approach to filter generic white noise disturbing a control signal. Specifically, a double RC stage network is able to filter into one or two steps a sinusoidal control signal based on the predicted envelopes of the voltage outputs. Specifically, ANN and RF supervised algorithms have been applied to predict the output voltage trend. The SPICE-based circuit model allowed us, through a parametric analysis, to understand the RC output behavior simulating the noise modulation effect by means of a resistance element. Based on the output trend, the proposed model is able to intelligently select or not select a further compensatory RC filtering second stage optimizing the denoising process. Specifically, the AI model optimizes the choice of the resistance and capacitance values of the second filtering stage according with the predicted voltage output. Furthermore, the AI prediction provides further information about other possible corrective actions introducing other circuital elements to achieve signal gaining or response shifting. As for practical applications in industrial environments influenced by random noises, the black-box model is the best choice to parametrically simulate the output signals when different modulation effects of the white noise are considered and to efficiently train the machine-learning algorithms. Finally, the paper defines by a BPMN workflow the guidelines to use for the proposed model of the denoising filtering process. The whole discussed model is addressed in Industry 5.0 application fields including advanced manufacturing control systems requiring signal denoising. The model is adaptable to other typologies of electronic filtering stages such as active filters and a higher order of low-pass, band-pass, and high-pass filters. The purpose of the proposed research is to define a methodological procedure structuring a filtering Digital Twin (DT) model suitable for the automatic denoising process by means of circuit and AI simulations.

The main novelties of the research are the modeling of a generic white noise effect applicable to generic electronic circuits including multi-stage filtering networks able to improve the noise filtering. The implementation of the pseudocode implementing the AI controller automatism is under investigation and will be the subject of a further publication. Future works will also be addressed on the implementation of the automatic control providing the switching commands of the proposed network.