Machine Learning Use Cases in the Frequency Symbolic Method of Linear Periodically Time-Variable Circuits Analysis

Abstract

This manuscript presents an analysis of machine learning (ML) usage in the Frequency Symbolic Method (FSM) to enhance the diagnosis of faults in parametric circuit analysis and optimization, with a particular focus on Linear Periodically Time-Variable (LPTV) systems. We put forth a few ML-based approaches for fault diagnosis (including anomaly detection), invisible feature detection, and the prediction of FSM output. These methodologies concentrate on identifying and diagnosing faults by evaluating particular ML techniques, extracting pertinent features, and determining the desired diagnostic outputs. The use cases of ML application considered in this paper demonstrate that machine learning can enhance fault detection and diagnosis, reduce human errors and identify previously unnoticed anomalies within the FSM framework. ML has never been used in FSM before, so the key aim of this paper is to consider possible use cases of AI application in FSM. Additionally, feature extraction, required as an input stage for the ML model, is proposed based on FSM peculiarities. This work can be considered a study of ML application in FSM.

1. Introduction

A linear time-varying (LTV) system is a non-stationary deterministic system defined by parameters that vary over time. These systems, also known as parametric systems or LTV systems, feature time-dependent coefficients, referred to as parametric functions.

A multitude of theoretical studies [1,2] and practical applications have concentrated on these systems, as they are regarded as suitable models for real-world scenarios. Systems that have been modeled by time-varying circuits are extensively utilized in modern electrical circuits [3], communication systems [4], and automatic control [5]. In particular, parametric systems are applied to the analysis of analog and digital signal processing systems [2], especially as models of sampling systems and modulation [6]. Moreover, parametric systems are used in current compensation in power grids [7], wideband low-noise amplifiers [8], filter systems and noise reduction [9,10] generators [11], and medical devices [12]. Furthermore, the modeling of circuits directly in the time domain serves as a foundation for functional testing and fault diagnosis, where time-domain excitation and/or circuit response constitute key data [13]. The implementation of LTV systems has a number of benefits, the most important of which are the improvement of system dynamics and the reduction of transients. Consequently, the modeling of parametric systems remains a highly relevant task.

It is a well-known fact in circuit theory that there are two basic ways to analyze linear and time-invariant (LTI) systems. The first method is to analyze systems in the time domain, either by solving differential equations describing the systems or by using convolution techniques. The second group of methods, called frequency analysis, is based on the use of Laplace or Fourier transforms.

LTV systems represent a direct generalization of classical linear and time-invariant (LTI) systems. Accordingly, an analysis of their dynamic and frequency properties can be performed in both the time and frequency domains. Furthermore, the concepts of impulse response, frequency response, and transmission, familiar from the theory of stationary systems, can be generalized to non-stationary systems. In the time domain, parametric systems can be described using two different methods. The first method involves representing systems in the time domain using a parametric linear differential Equation (1) [7,9,14]:

$$a_n\left(t\right)y^{\left(n\right)}\left(t\right)+a_{n-1}\left(t\right)y^{\left(n-1\right)}\left(\mathrm{t}\right)+\dots +a_1\left(t\right)y^{\prime }+a_0\left(t\right)y\left(t\right)=x\left(t\right),$$

(1)

where x(t), y(t), are the input and output signals, t is the independent variable (time), and a_i(t) are the time-varying coefficients of the differential equation, which are called parametric functions. Except for a few special cases, there are no well-established analytical methods for solving this equation. The authors’ current research focuses on a class of parametric systems with periodically varying coefficients known as LPTV, or Linear Time Periodic Systems [15].

The alternative method of describing an LTV system in the time domain, as opposed to the parametric differential equation approach, is a parametric convolution using the impulse response h(t,τ) as the kernel:

$$y\left(t\right)=\underset{0}{\overset{t}{\int }}h\left(t,\tau \right)x\left(\tau \right)d\tau .$$

(2)

It is important to note that the impulse response of LTV systems differs from that of time-invariant systems in that it depends not only on the time t, but also on the instant τ at which an excitation is applied to the input of the circuit.

In the frequency domain, two approaches may be used to describe a parametric circuit. The first is identical to that of LTI systems and is defined as the ratio of the complex output signal y(t) to the complex input signal, which is the monoharmonic function e^jωt. The second method requires knowledge of the system’s impulse response function and is expressed by the following equations:

$$W\left(j\omega ,t\right)=\mathcal{F}\left\{h(t,\tau )\right\}=\underset{0}{\overset{t}{\int }}h(t,\tau ){\mathrm{e}}^{-j\omega \tau }\mathrm{d}\tau$$

(3)

$$W\left(s,t\right)=\mathcal{L}\left\{h(t,\tau )\right\}=\underset{0}{\overset{t}{\int }}h(t,\tau ){\mathrm{e}}^{-s\tau }\mathrm{d}\tau$$

(4)

From a mathematical perspective, Formulas (3) and (4) define the classical Fourier and Laplace transforms of the impulse response with respect to the variable τ.

The main difference between the time-frequency analysis of LTV systems and that of stationary systems is the inability to proceed directly from differential equations to transfer functions. In this conventional approach, the determination of the impulse response function or the system response to an arbitrary input is of paramount importance in determining the frequency characteristics of LTV systems. The main challenge is to find fundamental solutions to the homogeneous parametric differential equations satisfying Equation (1).

Analytical solutions to differential equations are available only in specific cases, which heavily depend on the equation’s order and the form of the parametric functions.

The fundamental solutions of these equations, even for first or second order, are highly intricate and can be represented by special functions from mathematical physics, such as Bessel functions of the first and second kind with non-integral orders, as well as confluent hypergeometric functions [16]. Assuming periodic variation of the parameters of LPTV systems (with the constraint that they are at most second-order systems), well-known equations from mathematical physics, such as the Mathieu, Meissner, or Hill equations, can be used. However, the key issue remains the stability of these solutions. Depending on the choice of the coefficient variation over time, the systems can show instabilities. Furthermore, in most cases, the solutions to these equations are not available in closed form (analytically), requiring support from numerical computations in the analysis. While the exact analytical determination of this function is feasible in a limited number of cases, even then, the function is represented by special functions, rendering further analysis exceedingly challenging. The authors emphasize that while the theory of differential equations provides fundamental solutions (in some cases) through integral equations, the determination of closed-form solutions—such as the response of the system to a non-zero external forcing or an impulse response function—is often analytically intractable. Consequently, these solutions typically require the use of analytical numerical methods, and one must often rely on approximate solutions to effectively address the problem.

Given these complexities, the Frequency Symbolic Method (FSM), introduced by the authors in their earlier works [15] and briefly summarized in Section 2, becomes a valuable tool for describing and analyzing LPTV systems in the frequency domain, as it enables the frequency description of a specific class of parametric systems with periodically varying parameters in conjunction with the L.A. Zadeh equation [1]. The FSM allows direct formulation and analysis in the frequency domain, eliminating the need to solve time-domain differential equations. This approach is particularly advantageous as it avoids the intermediate step of time-domain analysis, which can be both complex and computationally expensive for systems with periodically varying parameters (LPTV). The FSM is applied using the L.A. Zadeh matrix equation, which provides a more organized and symbolic representation of LPTV systems. This matrix-based approach decomposes the problem into several independent matrix equations, making analysis easier and allowing a more methodical approach to solving the system.

Although the authors are well-versed in the frequency analysis methodology employed using this approach, there is a need for appropriate methods and computational algorithms to minimize the time required to determine the frequency response coefficients of the system. One can observe that LPTV circuits often describe circuits functioning at a variable operating point with properties that depend strictly on time-varying coefficients. Therefore, minimizing computational time is essential for the optimal selection of the variation waveform of the parametric functions. This necessity gives rise to the concept of implementing machine learning estimation, a branch of artificial intelligence (AI), in the context of the frequency-domain method of symbolic parametric circuit analysis and optimization. When implemented using specialized techniques such as d-trees, FSM offers significant improvements in both computational time and complexity management compared to standard methods such as MATLAB. This makes FSM a more practical choice for tackling large-scale problems where traditional methods may become impractical due to computational limitations.

2. About the Frequency Symbolic Method

The frequency symbolic method (FSM) is based on the following fundamental concepts [15]:

• Consider the LPTV circuit where:
  • There are one or more parametric elements, whose parameters change periodically over time with the same period T (in the simplest version),
  • There is one input with an input signal x(t) and one output with a response y(t).
• Such an LPTV circuit in the time domain can be described by the symbolic system of linear differential equations (SSLDE).
• By eliminating internal variables in the SSLDR using one of the known methods, one can obtain a differential Equation (1) describing the circuit.

  $$a_n\left(t\right)y^{\left(n\right)}+a_{n-1}\left(t\right)y^{\left(n-1\right)}+\dots +a_0\left(t\right)y=b_m\left(t\right)x^{\left(m\right)}{+b}_{m-1}\left(t\right)x^{\left(m-1\right)}+\dots +b_0\left(t\right)x,$$

  (5)

  where y(t) is the output (sought) and x(t) is the input (set) variable, t is the independent variable (time), and a_i(t), b_j(t) are parametric functions of time t.
• For an LPTV circuit described by the differential Equation (5), using the equation of L.A. Zadeh [1], we determine the differential equation describing this circuit in the frequency domain:

  $${\displaystyle \frac{1}{n!}}{\displaystyle \frac{d^{n}A(s,t)}{d{s}^n}}{\displaystyle \frac{d^{n}W(s,t)}{d{t}^n}}+...+{\displaystyle \frac{dA(s,t)}{ds}}{\displaystyle \frac{dW(s,t)}{dt}}+A\left(s,t\right)W\left(s,t\right)=B\left(s,t\right),$$

  (6)

  where:

  $$W\left(s,t\right)={\displaystyle \frac{Y\left(s,t\right)}{X\left(s,t\right)}},$$

  (7)

  is the transfer function of the LPTV circuit, and:

  $$A\left(s,t\right)=a_n\left(t\right)s^n+...+a_1\left(t\right)s+a_0\left(t\right),$$

  (8)

  $$B\left(s,t\right)=b_m\left(t\right)s^m+...+b_1\left(t\right)s+b_0\left(t\right),$$

  (9)

  are the corresponding time-dependent periodic functions of time t with period T coefficients of Equation (1), whereas Y(s,t) and X(s,t) are the images of the output and input variables in the frequency domain, respectively, and s is a complex variable.
• In general, Equation (6) does not have an exact analytical solution, so one must find an approximate solution that accounts for both the known properties of the desired transfer function W(s,t) and the specifics of the algorithm used for its determination:

  (a)

  The approximation Ŵ(s,t) of the transfer function W(s,t) must achieve a predefined accuracy level of choice,

  (b)

  Given that Ŵ(s,t) needs to undergo n-times differentiation when inserted into (6), it should be represented by a function that allows straightforward differentiation (i.e., intricate fractions, etc.). Consequently, for solving (6), one can suggest an approximation of the transfer function by a trigonometric Fourier series:

  $$W(s,t)=W_0\left(s\right)+{\sum }_{i=1}^k\left[W_{ci}\left(s\right)\cos \left(\mathsf{\Omega }t\right)+W_{si}\left(s\right)\sin \left(\mathsf{\Omega }t\right)\right], \mathsf{\Omega }={\displaystyle \frac{2\pi }{T}},$$

  (10)

  or in a complex form:

  $$\widehat{W}(s,t)=W_0\left(s\right)+{\sum }_{i=1}^k\left[W_{-i}\right(s)\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathsf{\Omega }t\right)+W_{+i}(s\left)\mathrm{e}\mathrm{x}\mathrm{p}\left(i\mathsf{\Omega }t\right)\right].$$

  (11)

The frequency symbolic method of solution to (6) finding is as follows:

Step 1. One of the expressions (11) or (13), for instance, Equation (11) is differentiated n times with respect to the variable t. The resulting derivatives, along with the original expression, are then substituted into (6).

Step 2. By transferring the right-hand side of expression (6) to the left side, one can obtain the following algebraic expression:

$$\delta (W_0,W_{c1},W_{s1},W_{c2},W_{s2},...,W_{ck},W_{sk})=0.$$

(12)

The functional δ(·) described by (13) is periodic with period T and contains (2k + 1) unknowns W₀, W_c1, W_s1, W_c2, W_s2, …, W_ck, W_sk, which need to be determined.

Step 3. As the functional δ(·) is periodic, we decompose it into a Fourier series with period T. According to (13), we equate k harmonics and the constant component of this series to zero. This results in a system of (2k + 1) linear algebraic equations, which form a symbolic system of linear algebraic equations (SSLAE) of the (2k + 1) order with (2k + 1) unknowns.

Step 4. The solution of the resulting SSLAE determines the desired unknowns of expression (8) and approximates (11) or (12).

The features of the frequency symbolic method of analyzing LPTV circuits described above are as follows:

• Regarding the choice of the number of k harmonics (step 3), it should be noted that, as a rule, the first k harmonics are typically chosen. However, there are algorithms that allow solutions to be obtained even when the number of equations is greater than the number of unknowns. In our opinion, the choice of harmonics and the number of equations in each case should be left to the specialist who designs and studies the given LPTV circuit.
• The solution obtained in step 4 of the SSLAE (in the form of M × W = P matrix) has its own peculiarities, as some or all elements of the matrix M and vector P are symbolic. Therefore, symbolic methods should be used to solve the SSLAE.
• The approximation selection of Equations (11) or (12) is not fundamentally important. However, empirical experience of using the approximation described by Equation (12), has shown that the matrix M is sparser than with approximation (11) of the same order. This is significant in the symbolic solution of the SLAP. Therefore, approximation (12) is more beneficial, in our opinion. The values of W₀(s), W_ci(s), W_si(s) or W₀(s), W_+i(s), W_−i(s) in this case are fractional rational expressions, where the denominators are identical and correspond to the determinant of the M matrix. The numerators are the determinants of the modified matrices M, where the corresponding column is replaced by the vector P.
  The peculiarity of steps 1–4 is that several (or even all) of the parameters of the analyzed circuit, including s, Ω, and other parametric element coefficients, are specified symbolically. Therefore, the differentiation of the approximating function, substitution of it and its derivatives into the equation (6), determination of k harmonics with (2k+1)-fold integration of the expression (13) products into the corresponding orthogonal functions, and solving the SSLAE in symbolically is very cumbersome. In this context, the sophisticated symbolic computation capabilities of contemporary CAD software (MATLAB 2014a) proved invaluable, enabling the calculations described in this manuscript to be carried out with a high degree of feasibility [15].
• The symbolic solution of the SLAE is performed in the system user-defined functions MAOPCs [15] using standard functions of the MATLAB environment.

Figure 1 shows the most commonly used steps of the FSM procedure. There are other possible steps depending on the input data for FSM, but they are not detailed in this manuscript. Figure 1 also presents the division of feature sets that could be extracted, which are described in Section 3.

3. Results

3.1. Consideration of Feature Extraction Required for Machine Learning Applications

This section considers the feature extraction required for machine learning (ML) applications based on the frequency symbolic method. The subsequent sections detail the specific use cases, including the inputs, outputs, and ML approaches to be employed for each case.

The basic steps in any artificial intelligence (AI) system development are as follows:

• Data Collection.
• Feature Extraction (Data Conversion for applicable inputs of the ML system).
• Target Definition (what is the target of the ML system?).
• Model Selection.
• Model Training.
• Model Evaluation.
• Parameter Tuning.

The key purpose of this work is to estimate the feature extraction and target definition steps from the list above. The data collection step could be accomplished by analyzing a set of parametric circuits with different parameters and configurations or by collecting statistics on configurations applied during FSM usage by researchers. The next steps, such as selection, training, and evaluation of the model are out of scope of this work and will be considered in further phases of the research. The feature extraction step in ML system development is devoted to converting input data into a vector of features, which later become the direct inputs to the ML model. The feature data type should typically be one of two types: numerical or categorical. Numerical values can take any continuous or discrete value within a given range. Categorical values can take one of the predefined values, such as [“Yes”, “No”] or [“Low”, “Medium”, “High”]. Figure 1 shows the feature sets that can be extracted at different steps of FSM usage.

The aforementioned feature sets are delineated and designated with unique identifiers:

• I1 (Input type 1)—features extracted from the system of differential equations.
• I2 (Input type 2)—features extracted from the transfer equation.
• M1 (Method features 1)—features extracted from the System of Linear Algebraic Equations (SLAE) obtained using FSM.
• O1 (Output features 1)—features extracted from the symbolic transfer function obtained using FSM.

In consideration of the FSM steps illustrated in Figure 1, we present the selected list of features in

Table 1. Proposed list of features set I1 extracted in machine learning application to FSM.

Feature ID and Short Name	Description	Data Type	Valid Range
I1.1 Number of LC components	Number of LC components (except parametric) in the system of differential equations describing the circuit.	Numerical	Positive integer number
I1.2 Percentage of non-zero elements	Relation of non-zero elements to all elements.	Numerical	Positive integer [0, 100]
I1.3 Count of symbolic parameters	Total number of parameters in symbolic form.	Numerical	Positive integer number
I1.4 Percentage of symbolic cells to the numerical cells	Relation of cells number in the system of differential equations containing at least one symbolic variable to the total number of cells.	Numerical	Positive integer [0, 100]
I1.5 Time-varying function type	Time-varying function type as one of single harmonic, bi-harmonic, multi-harmonic, etc.	Categorical	[Harm, biharm, multiharm]

,

Table 2. Proposed list of feature set I2 extracted in machine learning application to FSM.

Feature ID and Short Name	Description	Data Type	Valid Range
I2.1 Order of the left part	Order of the left part of the transfer equation.	Numerical	Positive integer number
I2.2 Order of the right part	Order of the right part of the transfer equation.	Numerical	Positive integer number
I2.3 Count of symbolic parameters	Total number of parameters in symbolic form.	Numerical	Positive integer number
I2.4 Complexity of expression	Numerical value representing the complexity of the symbolic expression. For example, a function depending on the number of multiplications, exponents, logarithmic operation, and so on.	Numerical	Positive integer number 1

¹ Justification: It is assumed that the assigned weights in a binary expression tree are positive integer numbers, and that the resulting complexity will also be a positive integer.

,

Table 3. Proposed list of feature set M1 extracted in machine learning application to FSM.

Feature ID and Short Name	Description	Data Type	Valid Range
M1.1 Order of the SLAE (2k + 1)	Order of the SLAE obtained using FSM.	Numerical	Positive integer number
M1.2 Complexity of SLAE	Numerical value representing the complexity of SLAE, i.e., some functions depend on the average complexity of cells in SLAE. The complexity of each SLAE cell could be calculated as binary expression tree complexity.	Numerical	Positive integer number 1

¹ Justification: It is assumed that the assigned weights in a binary expression tree are positive integer numbers, and that the resulting complexity will also be a positive integer.

and

Table 4. Proposed list of feature set O1 extracted in machine learning application to FSM.

Feature ID and Short Name	Description	Data Type	Valid Range
O1.1 Number of terms in the transfer function	Number of terms in the transfer function.	Numerical	Positive integer number
O1.2 Complexity of transfer function	Numerical value representing the complexity of the transfer function. For example, some functions depend on the average complexity of each term.	Numerical	Positive integer number 1
O1.3 Maximal complexity of one term	Numerical value representing the complexity of the most complex term in the transfer function. For example, some functions depend on the number of multiplications or exponentiations of symbolic variables.	Numerical	Positive integer number 1

¹ Justification: It is assumed that the assigned weights in a binary expression tree are positive integer numbers, and that the resulting complexity will also be a positive integer.

. As the ML application to the FSM progresses, we will provide more precise details regarding these features.

An example of the calculation of the complexity of a symbolic expression is shown in a separate chapter at the end of this article.

Target Definition. This step is appointed to determine data and its type that is desired as output of the ML system. The output data can also be of numerical or categorical type. An example of target values is shown in

Table 5. Example target definitions of the ML system applied to FSM.

Target ID and Short Name	Description	Data Type	Valid Range
T1.1 Integral deviation of obtained transfer function from desired	Integral deviation of the obtained transfer function from desired. It is obtained after substitution of the numerical values (may be also optimization applied) into the symbolic transfer function and integral comparison with the target transfer function.	Numerical	Floating-point number
T1.2 Local deviation of obtained transfer function from desired	Like above but evaluated in a specified local range of time or frequency.	Numerical	Floating-point number
T1.3 Amplification possibility	Is it possible that module of transfer function causes signal amplification?	Categorical	[“Yes”, “No”].
T1.4 Possible non-stability	Is the non-stability of the system/circuit possible?	Categorical	[“Yes”, “No”]

.

3.2. Use Case of Parametric Circuits Synthesis

This use case is mainly based on a reinforcement learning type (Figure 2). In such a case, the ML system tries to prepare a symbolic transfer function and receives a corresponding “reward” according to the level of how this transfer function fits the desired output.

The “new step” in Figure 2 could involve adding or removing an existing electric component (e.g., R, L, C) to specific nodes of the electrical circuit. The steps are as follows:

• AI tries to put electronic components in a semi-random place in the schematic considering its previous experience.
• Calculate the symbolic expression of the transfer function.
• Try to optimize component values with some constraints.
• Check the best result with the target frequency characteristic and provide the corresponding “reward”.

At each iteration, AI adjusts its coefficients and remembers the change in the schematic and corresponding impact. The Q-learning ML method could be used for remembering previous experiences.

The reward is the criterion for how well the obtained FSM output matches the desired output. For example, the targets defined in Table 5 could be used.

3.3. Use Case of Invisible Features Detection by Clusterization

This approach was appointed to find unseen by human features of FSM usage through clusterization of results based on predefined criteria. The idea, as shown in Figure 3, is to collect a large dataset of simulation results using FSM and use Unsupervised Machine Learning approach. AI will perform clusterization of the obtained results and allow for the identification of new features based on new clusters.

3.4. Use Case of Prediction of FSM Output

This idea is like the previous one but uses Supervised Machine Learning. Figure 4 shows the use case when ML is already trained. For training, the dataset consists of feature sets of type I1 and I2, corresponding to feature sets of type O1.

For example, the complexity of the transfer function is not provided as an input. Here, AI tries to predict the symbolic expression’s complexity based on previous experience.

3.5. Use Case of Assistance in Frequency Symbolic Method Usage Based on Anomaly Detection

The idea, as shown in Figure 5, is to use a trained ML system as an “Assistant” for users of FSM implementation. For training, Supervised Machine Learning should be used. First, AI is trained by collecting a dataset of symbolic expression (obtained using FSM) features to corresponding inputs. Once trained, it could be used as an “Assistant” (supervisor) for the following FSM usage. For example, the ML system can detect anomalies and inform the user. An anomaly could be a human mistake, for example, when the conductivity matrix is not as usual or a parametric element is incorrectly connected.

3.6. Use Case of Fault Diagnosis and Testing

The merging of the proposed methods with AI techniques opens new avenues for enhancing the analysis and optimization of LPTV circuits. The incorporation of AI, particularly machine learning (ML), offers significant advantages in handling the complex and computationally intensive tasks associated with the frequency symbolic method. A special case is fault diagnosis which should be differentiated from typical functional testing. It is possible for the circuit to be in a faulty state but still pass functional test, which is equivalent to behavioral test. In such a case, one or more elements can have parameters, e.g., beyond the tolerance range. This can mark the beginning of a component’s degradation process (due to aging or environmental influence) and can affect the circuit (system) lifetime (or Mean-Time Between Failure parameter, MTBF). Successful fault diagnosis can be an important part of predictive maintenance, especially in cases of a component’s soft fault (parameter beyond the tolerance range while still operating), as it can be a step before hard fault (malfunction), often leading to circuit (system) malfunction. Exemplary areas are as follows:

• Fault Diagnosis in Electrical/Electronic Circuits:
  • Predictive Maintenance: AI can be used to predict potential failures in circuits by analyzing patterns and anomalies in frequency responses. By training ML models on historical data of circuit performance, it is possible to foresee faults before they occur, allowing for proactive maintenance [13,17].
  • Anomaly Detection: AI systems can be employed to detect anomalies in circuit behavior that may indicate faults. Machine learning models can learn the normal operating conditions of circuits and flag deviations from these norms, thus identifying potential issues in real-time.
• Testing and Validation:
  • Automated Testing: AI-driven automated testing procedures can significantly reduce the time and effort required for validating circuit designs. By using reinforcement learning techniques, AI systems can optimize test scenarios to cover a wide range of operating conditions, ensuring robust performance.
  • Simulation-Based Testing (SBT): AI can enhance simulation-based testing by rapidly generating and evaluating numerous test cases, thus providing comprehensive insights into circuit behavior under various conditions. This approach can help identify edge cases and potential weaknesses in the design.
  • Parameter Tuning: AI techniques can be used to optimize the parameters of LPTV circuits. Genetic algorithms and other optimization techniques can help find the best parameter values that improve circuit performance while meeting design constraints.

By integrating AI techniques with the frequency symbolic method, the processes of circuit analysis, fault diagnosis, testing, and optimization can be improved. This integration not only enhances efficiency but also provides a deeper understanding of the complex behaviors of LPTV circuits, paving the way for more reliable electrical and electronic systems.

3.7. Example of Symbolic Transfer Function Complexity Calculation

This chapter contains an example of symbolic expression complexity calculation based on a transfer function obtained in symbolic form using the FSM. Specifically, it shows the extraction of feature O1.2, “Complexity of transfer function”. The complexity value obtained is an integer number, which could be used in a similar way to extract the features with IDs I2.4, M1.2, O1.2, and O1.3 described above. For this example, the one-circuit parametric amplifier shown in Figure 6 has been selected.

A parametric circuit with a single varying coefficient, shown in Figure 6, has been analyzed. The time-dependent parameter c(t) of the system is the capacitance, which is represented by a function:

$$c\left(t\right)=C_0\left\{1+m_c\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\Omega }t\right)\right\}, C_0=10 \mathrm{p}\mathrm{F}.$$

(13)

The parameters of the LPTV circuit shown in Figure 6 are as follows: a constant angular frequency Ω for the parametric function c(t): Ω = 2πf, where f = 200 MHz, and a modulation coefficient m_c = 0.01. The inductance is L = 253.3 nH, and the conductance values are Y₁ = 0.25 S and Y₂ = 0.4 mS. A monoharmonic input signal is a current i(t) with a time waveform:

$$i\left(t\right)=I_m\cos \left(\omega t+{\phi }_0\right), I_m=0.1 \mathrm{m}\mathrm{A}, {\phi }_0={45}^{°}, \omega =2\pi f_{in}, f_{in}=100 \mathrm{M}\mathrm{H}\mathrm{z}.$$

(14)

Three parameters remain in symbolic form: time t, constant angular frequency of the parametric function Ω, and m_c—modulation coefficient of the time-varying capacitance c. Using FSM, the transfer function was obtained for three cases: 1, 2, and 3 harmonics used to approximate the transfer function. It can be noted that the FSM makes it possible to obtain the approximation of the transfer function using Fourier series with a predetermined number of harmonics. Figure 7 (magnitude) and Figure 8 (phase) present plots of the transfer function (with 1 harmonic approximation) for presented circuit. It can be observed that there is an influence of variable capacitance at frequency f = 200 MHz.

The following formula is proposed to calculate the complexity C of an expression (transfer function):

$$C={\sum }_{k=1}^K{w}_i·N_i,$$

(15)

where:

• N_i is number of particular operations (for i-th mathematical operand),
• w_i is complexity weight of the i-th operand,
• K is total number of distinguished operands (depending on the case).

There following operands are distinguished (with abbreviations): addition (add), subtraction (sub), multiplication (mul), division (div), and exponentiation (exp). Negative numbers are not counted as subtraction (they have assumed particular hardware representation).

Deciding on the weights of each operation is a debatable topic, as it depends very much on the tools, algorithms, hardware implementation, etc. Therefore, the following assumptions have been made for this example, which may not apply to other environments:

• Every number in the expression in the generic case is a complex number and it serves as our reference.
• The addition or subtraction of two complex numbers has the same complexity as two additions/subtractions of real numbers and is considered as a unit weight with value w_add = w_sub = 1.
• Multiplication of two real numbers typically uses hardware acceleration, resulting in a weight of 1.
• Complex multiplication involves two multiplications and one addition, and based on the assumptions above, its weight is w_mul = 3.
• The division of two complex numbers uses multiplication by the conjugate, and its weight is w_div = 7.
• Binary exponentiation is calculated with complexity $O\left(2\cdot {\mathrm{l}\mathrm{o}\mathrm{g}}_{2}n\right)$. The real and imaginary parts of the complex number are stored as 32-bit values, so ${\mathrm{l}\mathrm{o}\mathrm{g}}_{2}n$ = 32, therefore the weight of the exponentiation is w_exp = 64.

Summary of weights for complexity calculation is presented in

Table 6. Assumed complexity weights for particular mathematical operations.

Operation	Weight
add	1
sub	1
mul	3
div	7
exp	64

.

Table 7. Result of transfer function complexity calculation using FSM for 1 harmonic.

Operation	Number of Operations	Cost
add	48	48
sub	33	33
mul	67	201
div	3	21
exp	44	2816
		Total 3119

shows the number of operations and the complexity calculated using expression (11) and the assumptions considered above for the transfer functions obtained using FSM with 1 harmonic approximation.

Table 8. Result of transfer function complexity calculation using FSM for 2 harmonics.

Operation	Number of Operations	Cost
add	240	240
sub	238	238
mul	433	1299
div	5	35
exp	373	23,872
		Total 25,684

and

Table 9. Result of transfer function complexity calculation using FSM for 3 harmonics.

Operation	Number of Operations	Cost
add	722	722
sub	626	626
mul	1213	3639
div	7	49
exp	1081	69,184
		Total 74,220

show the calculation complexities for transfer function calculation using 2 and 3 harmonics, respectively.

It can be observed that the required complexity increases rapidly with the number of harmonics used to find a particular transfer function.

4. Conclusions

This paper examines five instances of machine learning (ML) applications to the FSM method. For each use case, the requisite feature sets, learning type, benefits, and potential issues are analyzed. A summary is presented in

Table 10. Use cases summary.

Use Case	Benefit	Learning Type	Difficulties/Problems
Parametric circuits synthesis	Circuit could be synthesized automatically.	Reinforcement learning	Could be time consuming. Big database is needed for ML experience storage.
New features detection	Identify not visible features of the FSM method or its implementation by specific clusters occurrences.	Unsupervised learning	At the moment, it is not clear which clusters could be obtained.
Prediction	Could predict the result based on previous experience without modeling and simulation.	Supervised learning	For some not typical inputs, the result could be predicted incorrectly.
Anomaly detection	Human mistakes could be detected. This use case acts as an assistant.	Supervised learning	Could accidentally report an anomaly when the correct approach is used.

.

The initial use case is parametric circuit synthesis, in which a circuit can be synthesized automatically through reinforcement learning. The primary challenges are the time-consuming process and the necessity for a substantial database for ML experience storage. The second use case pertains to the detection of new features, which involves identifying features of the FSM method or its implementation that are not visible or cannot be discerned from specific cluster occurrences. This use case employs unsupervised learning, but it is currently unclear which clusters could be obtained. The third use case is prediction, where results can be predicted based on previous experience without the need for modeling and simulation. This use case employs supervised learning, but the result may be incorrectly predicted for atypical inputs. The fourth use case involves an assistant that could detect human mistakes through unsupervised learning.

However, there was a potential problem of inadvertently reporting an anomaly when the correct approach was used. The fifth use case was fault detection, which allowed the identification of faults not detected by standard functional tests. This use case used supervised learning but required a sufficient circuit model (simulation). There are other issues that must always be properly addressed, such as data preparation, model selection, training, and evaluation, with the aim of improving the overall efficiency of the diagnostic system, but these are all circuit-dependent and must be chosen by a test designer, engineer, or AI expert. Therefore, there is no single or best solution for a particular test case, so we have proposed a set of real-world applications that should reduce the time taken to develop a single test for a custom circuit.