Automated Neural Network-Based Optimization for Enhancing Dynamic Range in Active Filter Design

Abstract

This study presents an automated circuit design approach using neural networks to optimize the dynamic range (DR) of active filters, illustrated through the design of a 7th-order Chebyshev low-pass filter. Traditional design methods rely heavily on designer expertise, often resulting in time-intensive and energy-consuming processes. Two techniques are proposed: inverse modeling and forward modeling. In inverse modeling, artificial neural networks (ANNs) predict circuit parameters to meet specific performance goals. A randomly selected subset, comprising 0.05% of the 1,953,125 possible circuit configurations, was used to train and validate the model, providing an accurate representation of the entire dataset without requiring full-scale data analysis. In forward modeling, the same subset was used to train the network, which was then used to predict DR values for the remaining dataset. This approach enabled the identification of circuit parameters that resulted in optimal DR values. The results confirm the effectiveness of these techniques, with both inverse modeling and forward modeling outperforming the standard circuit design. At 160 kHz, a critical frequency for the operation of the designed filter, inverse modeling achieved a DR of 140.267 dB and forward modeling reached 136.965 dB, compared to 132.748 dB for the standard circuit designed using the traditional approach. These findings demonstrate that ANN-based methods can significantly enhance design accuracy, reduce time requirements, and improve energy efficiency in analog circuit optimization.

1. Introduction

In modern electronic systems, efficient signal processing is crucial for applications ranging from wireless communications to biomedical devices. Electronic filters play a critical role in determining the performance and efficiency of these systems. However, while traditional analog design techniques, i.e., textbook methods [1,2,3,4] and other conventional approaches such as [5,6] commonly used in the literature, are straightforward to use, they often struggle to balance key performance metrics such as dynamic range (DR), noise, distortion, and power consumption, and this results in suboptimal designs. To obtain a near optimum design, typically, a trial-and-error process is required that heavily depends on engineering expertise, which in turn leads to a labor-expensive and time-consuming design stage.

Analog design tools, which usually require repeated simulations of the considered circuit, are not energy and time efficient, resulting in notable challenges during the design process. As a result of these shortcomings, conventional optimization techniques have been occasionally considered [7,8,9] as ineffective.

To overcome these challenges, studies have been conducted on tunable filters as a solution approach. However, their adaptability is limited by the reliance on additional components, which provide only restricted adaptive capabilities [10,11]. This highlights the need for advanced design methodologies capable of addressing these limitations.

New designs related to emerging technologies are becoming increasingly complex and competitive, making even minor improvements in the design highly valuable and capable of leading to significant enhancements. In this context, artificial neural network (ANN)-based design studies have also started to gain importance. In some of these designs, only a specific component value, such as a transistor, in the circuit is optimized [12], while in others, the focus is on optimizing the entire circuit with an ANN [13,14,15,16,17,18]. NN-based circuit designs in the literature have demonstrated improvements in gain, bandwidth, unity gain bandwidth, and phase margin [13]. However, the optimization of DR has not been explored. This study utilizes NN-based optimization to identify the circuit parameters that achieve enhanced DR performance.

ANNs have proven highly effective in deriving simplified circuit models, significantly reducing simulation times. Recent studies demonstrate that NN-based approaches not only enhance accuracy but also optimize circuit designs using transistor-level simulation data, offering superior computational efficiency compared to traditional methods [19,20]. By providing circuit-level precision with a reduced simulation overhead, ANNs have become essential in modern circuit design optimization [19,21,22,23,24,25,26].

This study aims to enhance the dynamic range of a low-pass Chebyshev filter circuit and automate the design process. The input–output relationships of Chebyshev low-pass filters can be expressed through nonlinear differential equations, and parameter optimization can be performed using these equations. However, directly solving nonlinear differential equations is often computationally intensive and complex, making the optimization process particularly challenging in high-dimensional parameter spaces. In contrast, artificial neural networks can learn from large datasets, explore a broader parameter space, and provide a fast and flexible optimization approach without requiring direct analytical solutions. Therefore, in this study, we prefer an NN-based approach instead of traditional numerical methods. The proposed method is not only limited to specific differential equations but also has the potential to generalize to different filter configurations. We present two automated design methodologies. The first approach, termed inverse modeling, utilizes artificial neural networks (ANNs) to predict circuit parameters through regression modeling. By systematically balancing these parameters, the methodology achieves a wider DR while preserving the circuit’s overall functionality. Unlike conventional techniques, inverse modeling uses the predictive ability of ANNs to precisely model and optimize component values, significantly streamlining the design process and reducing the development time. The second approach, termed forward modeling, involves training the ANN on a small dataset and employing the model to predict DR values for an extensive dataset comprising 1,953,125 circuits. The predicted results were validated using LTspice simulations. In this model, this enables the rapid identification of the circuit with the optimal DR value for a vast design space. These methodologies demonstrate the efficiency of ANN-based frameworks in automating circuit design and optimization while achieving superior performance outcomes.

The outline of this paper as follows. Section 2 describes the filter design process, including the circuit structure used in this study. Figure 1 provides the circuit, illustrating the OTA-based implementations replacing inductors and the transistor-level OTA circuits for each OTA. Section 3 defines the problem and explains the dataset configuration, detailing the process of obtaining NN training data for automatic design. Section 4 presents the proposed methodology, introducing both the inverse modeling and forward modeling approaches. It provides details on the NN model used, obtained results, and resource utilization. Section 5 presents the results, providing an analysis of the performance and resource utilization of the proposed methods. Section 6 summarizes the paper by presenting the results and suggesting potential directions for future research.

2. Filter Design Process

Automated design is particularly advantageous in integrated circuits, where physical inductors are large, expensive, and challenging to implement. In active circuit design, inductors are frequently replaced with active components to achieve equivalent inductance. Operational Transconductance Amplifier (OTA) filters offer remarkable performance in dynamic network environments due to their ability to adapt to environmental variations. With advantages such as low latency, energy efficiency, and high accuracy, this technology is preferred for next-generation communication systems. Consequently, the design and development of OTA filters hold significant importance from both academic and industrial perspectives. In particular, the optimal selection of $I_{bias}$ and $C_L$ values for each OTA is crucial for achieving the desired performance.

A 7th-order Chebyshev filter provides the best balance between steep roll-off and manageable complexity. Lower orders lack sufficient steepness, while higher orders add unnecessary complexity. Additionally, it allows efficient active filter implementation, making it the optimal choice for this study [27]. In this study, a 7th-order low-pass Chebyshev filter is employed as a representative example to illustrate the effectiveness of ANNs in circuit design. The circuit, depicted in Figure 1a, replaces the inductors with an OTA-based active inductor configuration, as shown in Figure 1b. The relationship between the equivalent inductance, the transconductance ($g_m$), and the capacitance ($C_L$) is given by the following formula:

(1)$L_{eq}={\displaystyle \frac{C_L}{g_m^2}}$

Here, $L_{eq}$ is the equivalent inductance, $g_m$ is the transconductance of the OTA, and $C_L$ is the capacitance of the connected capacitor. The detailed transistor-level circuit configuration of each OTA utilized in Figure 1b is depicted in Figure 1c. An automated circuit design methodology is developed to identify optimal bias currents $I_{bias}$ and capacitor values $C_L$ using a neural network-based model, with the aim of achieving a target dynamic range.

3. Problem Definition and Dataset Configuration

Active circuit techniques are widely used to simulate inductors because their large size, cost, and limitations make integrated circuit implementations challenging. In such designs, OTA-based structures are utilized to emulate an equivalent inductance while preserving the capacitance in the circuit. This approach allows the active circuit to replicate the behavior of the passive counterpart accurately, ensuring that the desired filter characteristics are preserved. The component values for both the standard design circuit and the standard OTA-based design circuit, as illustrated in Figure 1a, are provided in Table 1. The capacitances $C_{L3}$, $C_{L5}$, and $C_{L7}$ (in nF) and the inductances $L_3$, $L_5$, and $L_7$ (in mH) provided in Table 1 are applied using Formula (1) to compute the transconductance $g_m$. The results demonstrate that, for the standard OTA-based design circuit, $g_m$ is equal to 1 mS (1 mA/V).

To generate a dataset for training aimed at automated circuit design, a pool of capacitor values ($C_L$) and bias currents ($I_{bias}$) was generated within a specified range based on the parameters of the standard circuit variables. In designing filters with an optimal dynamic range, the circuit includes three distinct capacitors $C_{L3}$, $C_{L5}$, and $C_{L7}$, and six transconductance values, $g{m}_{31}$, $g{m}_{32}$ = $g{m}_{33}$, $g{m}_{51}$, $g{m}_{52}$ = $g{m}_{53}$, $g{m}_{71}$, and $g{m}_{72}$ = $g{m}_{73}$. A MATLAB-based automation script was developed to generate all feasible combinations of $C_L$ and $gm$ values. The code determined five equally spaced points for each $C_L$ value with a maximum tolerance of 3%, and for each $gm$ value with a maximum tolerance of 5%. Using these points, all combinations were generated. In this process, a total of 1,953,125 different combinations of $C_L$ and $gm$ values were calculated. This comprehensive dataset ensures that the design space is fully explored, covering a wide range of feasible circuit configurations. In total, 0.05% of the entire dataset was randomly selected to ensure a comprehensive representation of the design space while minimizing computational costs.

For this 0.05% of dataset combinations, LTspice netlist files were generated through MATLAB R2020b. This process resulted in the creation of a data pool consisting of 1000 circuits. Transient and noise analyses were conducted in the LTSPICE environment for the 1000 circuits. Dynamic range is typically defined as the ratio between the $V_{max}$ and $V_{min}$ of the signal. $V_{max}$ is the input voltage at the specified THD threshold and $V_{min}$ is the input referred noise value. In terms of decibels ($dB$), dynamic range is calculated using the following formula:

(2)$\mathrm{Dynamic}\mathrm{Range}\left(DR\right)=20·{log}_{10}\left({\displaystyle \frac{V_{max}}{V_{min}}}\right)dB$

In this study, a regression-based approach was adopted to train the ANN. Regression analysis in ANNs is commonly employed to model complex, nonlinear relationships between inputs and outputs, making it an appropriate choice for this application.

4. Proposed Method

The proposed method utilizes inverse modeling and forward modeling, and the details are presented in the following 4.1. Inverse Modeling and 4.2. Forward Modeling subsections.

4.1. Inverse Modeling

A neural network for inverse modeling allows for accurate adjustment of dynamic range across different frequencies. The information used to train the NN is comprised of the set of data pairs denoted as $\{X_i,P_i\}$. $X_i$ is a vector of the circuit performance metrics (dynamic range) and $P_i$ stands for the design variables, capacitor, and bias currents in the OTA circuit. ANNs were employed to predict the design variables, denoted as $\widehat{P}$. In this study, the frequencies for the standard circuit were defined as $f_1=140\mathrm{kHz}$, $f_2=150\mathrm{kHz}$, and $f_3=160\mathrm{kHz}$. AC analyses were performed on the standard circuit, and the selected frequencies were chosen to fall within the passband of the low-pass filter to ensure accurate evaluation of its performance. Figure 2 presents the neural network architecture for inverse modeling. The input data for the neural network consists of the DR values at the frequencies $f_1$, $f_2$, and $f_3$

(3)$X=[D{R}_1,D{R}_2,D{R}_3]$

and the outputs of the neural network are the circuit parameters,

(4)$P=\left[C_{L1},C_{L2},C_{L3},I_{bias31},I_{bias32},I_{bias51},I_{bias52},I_{bias71},I_{bias72}\right]$

The neural network-based model maps the input values X to the output parameters P. This relationship can be formulated as

(5)$\widehat{P}=f(X;\theta ).$

$f(X;\theta )$ is the neural network-based model, $\theta$ represents the learned parameters (weights and biases) of the model, and $\widehat{P}$ represents the predicted output vector. In this regression problem, the loss function is the MSE.

(6)$Loss(P,\widehat{P})={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(P_i-{\widehat{P}}_i\right)}^2$

The parameters of the neural network $\theta$ are optimized during training to minimize the loss function. This process allows the model to improve its ability to predict the output parameters $\widehat{P}$ as close as possible to the actual values P.

The NN parameters in Table 2 were determined through hyperparameter optimization to enhance the performance of the inverse model. Various parameters were systematically evaluated. The final set of optimized parameters, which ensures better generalization, is presented in Table 2. The comprehensive workflow is presented in Figure 3. The results were obtained by following the steps outlined in the workflow. Resource utilization for inverse modeling is presented in Table 3. Accuracy metrics of neural network predictions and $LTspice$ validation for circuit parameters in inverse modeling are presented in Table 4. Neural network-based $LTspice$ simulation results for each frequency are presented in Table 5.

4.2. Forward Modeling

Forward modeling provides a neural network-based framework for rapid performance evaluation and analysis in analog circuit design, enabling accurate prediction of performance metrics such as DR. This approach reduces reliance on computationally expensive simulations, significantly accelerating the design validation process. The input data for the neural network consists of the the circuit parameters,

(7)$X=[C_{L1},C_{L2},C_{L3},I_{bias31},I_{bias32},I_{bias51},I_{bias52},I_{bias71},I_{bias72}]$

and the output data for the neural network consists of the DR values at the frequencies $f_1$, $f_2$, and $f_3$

(8)$P=[D{R}_1,D{R}_2,D{R}_3]$

Figure 4 presents the neural network architecture for forward modeling. The neural network-based model maps the input values X to the output parameters P. This relationship can be formulated as

(9)$\widehat{P}=f(X;\theta ).$

where $f(X;\theta )$ is the neural network-based model, $\theta$ represents the learned parameters (weights and biases) of the model, and $\widehat{P}$ represents the predicted output vector. In this regression problem, the loss function is the MSE.

The parameters of the neural network $\theta$ are optimized during training to minimize the loss function. This process allows the model to improve its ability to predict the output parameters $\widehat{P}$ as close as possible to the actual values P.

(10)$Loss(P,\widehat{P})={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(P_i-{\widehat{P}}_i\right)}^2$

In the first step of forward modeling, LTspice simulations were completed for a randomly selected subset of 0.05% of the total 1,953,125 data points, and the corresponding DR values were obtained. This subset was divided into 80% training, 10% validation, and 10% testing datasets. The optimal network was achieved through hyperparameter optimization. The NN parameters in Table 6 for the forward model were optimized to maximize accuracy. Hyperparameter optimization was conducted across different learning rates, activation functions, and parameters. The final optimized parameters in Table 6 were selected to improve model accuracy and minimize overfitting. The cost data and accuracy rates for this phase are presented in Table 7, Table 8 and Table 9. Comparison of DR values for different modeling methods at 160 kHz is presented in Table 10.

In the second step, the previously trained network was utilized to predict DR values for the entire dataset of 1,953,125 data points. The best DR results were obtained for three different frequencies, with the values determined as ${\mathit{DR}}_{\mathit{1}}=139.413$ dB, ${\mathit{DR}}_{\mathit{2}}=136.971$ dB, and ${\mathit{DR}}_{\mathit{3}}=139.157$ dB. The obtained results were validated through LTspice simulations. For the third frequency, the forward modeling neural network estimated the dynamic range as ${\mathit{DR}}_{\mathit{3}}=139.157$ dB. Subsequently, the circuit was simulated in LTspice using the component values corresponding to ${\mathit{DR}}_{\mathit{3}}=139.157$ dB. The LTspice simulation results indicated a dynamic range of ${\mathit{DR}}_{\mathit{3}}=136.965$ dB. This result demonstrates that the forward modeling approach operates with an accuracy of 98.425%. This result validates the forward modeling neural network accuracy value provided for 160 kHz in Table 7. Performing LTspice simulations to calculate DR for 1,953,125 data points would have been highly time-consuming. However, with forward modeling, DR values for all data points were obtained in a significantly shorter time. Moreover, the component values corresponding to the best DR can be easily determined. This approach effectively eliminates the computational burden of simulations that would otherwise take days to complete.

5. Results

Energy consumption is a critical consideration in circuit design simulations. When operating with identical data, a comparison of energy consumption between inverse modeling and forward modeling reveals that forward modeling is more advantageous in terms of both time efficiency and energy usage. Therefore, forward modeling has also been included in this study. The resource utilization for forward and inverse modeling, based on the analysis of 0.05% of the 1,953,125 data points, is detailed in Table 11.

Forward modeling offers significant benefits in terms of energy efficiency and time savings, making it an ideal approach for analyzing large datasets and addressing design tasks that require quick results. In contrast, inverse modeling is well suited for achieving precise performance targets and plays a key role, particularly in the development of new circuit designs.

This study demonstrates the robustness and reliability of the proposed methodology, achieving accuracy levels exceeding 96% in both inverse and forward modeling approaches. The integration of advanced optimization techniques with neural network-based methodologies has significantly improved the precision in determining circuit component values for analog filter design, demonstrating notable performance enhancements over conventional approaches. Key advantages of the proposed methodology include independence from designer expertise and remarkable time efficiency. In inverse modeling, the neural network directly predicts circuit parameters based on specified performance metrics, whereas in forward modeling, DR predictions were conducted for 1,953,125 distinct circuit configurations. The accuracy of the results obtained from both modeling approaches was validated through LTspice simulations. At a frequency of 160 kHz, the DR achieved by the standard circuit was 132.748 dB, while the forward modeling approach achieved 136.965 dB, and the inverse modeling approach attained 140.267 dB. These results clearly highlight the superior performance and higher DR values delivered by the proposed methodology compared to traditional design approaches.

The NN approach used in this study provides a suboptimal solution, meaning it does not evaluate all possible options like brute force methods but instead selects one of the better solutions. The main drawback of this approach is that it does not always guarantee the absolute optimal solution. However, it can still produce sufficiently good results in most cases. While it may not always find the best possible solution, its ability to significantly reduce computational time and resource consumption makes it a practical and efficient choice, particularly for large-scale and complex problems.

6. Conclusions and Future Work

This work presents an automated and efficient approach for optimizing the DR of active filters using ANNs. The proposed methodology was demonstrated through the design of a 7th-order Chebyshev low-pass filter, revealing its ability to outperform traditional design techniques. By utilizing inverse and forward modeling approaches, this framework effectively minimized reliance on designer expertise, reduced computational demands, and accelerated the design process. In inverse modeling, ANNs predicted circuit parameters based on specified performance targets, while forward modeling enabled DR predictions for a large dataset using a trained network. Both methods utilized a small, randomly selected subset (0.05% of 1,953,125 possible configurations) for training and validation, demonstrating high accuracy and computational efficiency without requiring comprehensive data analysis. Simulation results validated the performance of the proposed techniques. At 160 kHz, a critical frequency for the designed filter, inverse modeling achieved a DR of 140.267 dB and forward modeling reached 136.965 dB, surpassing the 132.748 dB DR achieved with conventional design approaches. These findings highlight the ability of ANN-based methods to fulfill notable accuracy, time savings, and energy efficiency, making them a promising alternative for analog circuit optimization.

The proposed methodology has demonstrated promising results, but several areas for future research can further expand its applicability and performance. Expanding the approach to support more complex circuit designs and higher-order filters can increase its versatility. Incorporating multi-objective optimization techniques may also enable the simultaneous optimization of different performance criteria, addressing broader design constraints. Future work can also focus on optimizing data utilization by investigating more efficient sampling and enhancement strategies to reduce training requirements without compromising accuracy.

Author Contributions

Conceptualization, guidance, and review, S.O.; methodology, simulations, investigation, validation, formal analysis, and writing—original draft preparation, F.D. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data and code generated and analyzed during this study are not publicly available due to proprietary restrictions but can be made available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. (a) Schematic of the 7th-order low-pass Chebyshev filter circuit, (b) schematic of the OTA circuit used for inductor simulation, (c) transistor-level OTA implementation.

Figure 2. Neural network model architecture for predicting circuit parameters in inverse modeling.

Figure 3. Comprehensive workflow of co-simulation and neural network prediction in circuit parameter optimization.

Figure 4. Neural network architecture for predicting dynamic range.

Table 1

Component values for the standard design circuit and the standard OTA-based design circuit.

Standard Design Circuit		Standard OTA-Based Design Circuit
Component	Value	Component	Value
$R_1$	1 k$\Omega$	$R_1$	1 k$\Omega$
$R_L$	1 k$\Omega$	$R_L$	1 k$\Omega$
$C_2$	0.1412 nF	$C_2$	0.1412 nF
$C_3$	0.1901 nF	$C_3$	0.1901 nF
$C_4$	1.032 nF	$C_4$	1.032 nF
$C_5$	0.406 nF	$C_5$	0.406 nF
$C_6$	1.232 nF	$C_6$	1.232 nF
$C_7$	0.339 nF	$C_7$	0.339 nF
$C_8$	0.6063 nF	$C_8$	0.6063 nF
$L_3$	0.6843 mH	$C_{L3}$	0.6843 nF
$L_5$	1.094 mH	$C_{L5}$	1.094 nF
$L_7$	0.9264 mH	$C_{L7}$	0.9264 nF
		$I_{bias}$	96 μA

Table 2

Best-performing hyperparameters in inverse modeling simulation.

Parameter	Value
Initial Learn Rate	0.01
L2 Regularization	$1\times {10}^{-4}$
Epochs	100
Learn Rate Drop Factor	0.7
Learn Rate Drop Period	10
Mini Batch Size	32
Optimizer	adam

Table 3

Resource utilization in inverse modeling.

Inverse ModelingNeural Network Simulation			Inverse ModelingLTspice Verification
CPU(%)	Cumulative Power(Joule)	Time(s)	CPU(%)	Cumulative Power(Joule)	Time(s)
16.236	239.424	14.746	11.893	923.460	77.643

Table 4

Accuracy metrics of neural network predictions and LTspice validation for circuit parameters in inverse modeling.

Overall Accuracy (%)	Prediction Accuracy (%)
	${\mathit{C}}_{\mathit{L}\mathbf{1}}$	${\mathit{C}}_{\mathit{L}\mathbf{2}}$	${\mathit{C}}_{\mathit{L}\mathbf{3}}$	${\mathit{I}}_{\mathit{bias}\mathbf{31}}$	${\mathit{I}}_{\mathit{bias}\mathbf{32}}$	${\mathit{I}}_{\mathit{bias}\mathbf{51}}$	${\mathit{I}}_{\mathit{bias}\mathbf{52}}$	${\mathit{I}}_{\mathit{bias}\mathbf{71}}$	${\mathit{I}}_{\mathit{bias}\mathbf{72}}$
98.876	99.856	99.308	99.283	98.49	99.433	98.426	96.548	99.158	99.380

Table 5

Neural network-based LTspice simulation results (dB).

Neural Network-Based LTspiceSimulation Results (dB)
DR (140 kHz)	DR (150 kHz)	DR (160 kHz)
130.446	136.080	140.267

Table 6

Best-performing hyperparameters in forward modeling simulation.

Parameter	Value
Initial Learn Rate	0.01
L2 Regularization	$1\times {10}^{-4}$
Epochs	40
Learn Rate Drop Factor	0.7
Learn Rate Drop Period	10

Table 7

Accuracy metrics of neural network predictions in forward modeling.

Prediction Accuracy (%)
DR (140 kHz)	DR (150 kHz)	DR (160 kHz)
99.660	99.433	98.181

Table 8

Resource utilization in forward modeling training.

CPU (%)	Cumulative Power (Joule)	Time (s)
14.27	149.605	10.486

Table 9

Resource utilization in forward modeling for neural network simulation with 1,953,125 data points.

CPU (%)	Cumulative Power (Joule)	Time (s)
5.864	2414.027	411.680

Table 10

Comparison of DR values for different modeling methods at 160 kHz.

Method	DR (dB)
Standard model	132.748
Inverse modeling	140.267
Forward modeling	136.965

Table 11

Resource utilization metrics for forward and inverse modeling.

Inverse Modeling Neural Network Simulation			Forward ModelingNeural Network Simulation
CPU (%)	Cumulative Power(Joule)	Time(s)	CPU(%)	Cumulative Power(Joule)	Time(s)
16.236	239.424	14.746	14.27	149.605	10.486

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