A Fuzzy Comprehensive Evaluation Framework Integrating Time–Frequency Features and Combined Weighting for Matching Impact Signals with Multi-Layer Penetration Response Signals

Abstract

In impact testing, evaluating multiple-impact signals is critical for verifying whether a test setup can reproduce penetration response signals and ensure reliable results. To overcome the limitations of traditional methods, including incomplete indicator coverage, subjective weighting, and poor consistency, this study proposes a fuzzy comprehensive evaluation (FCE) framework based on time–frequency features and combined weighting. Using multi-layer penetration response signals as the matching target, a multidimensional indicator system covering time-domain features, frequency-domain features, and signal quality and stability is established. A combined weighting method integrating AHP, EWM, and CRITIC is then developed, and subjective and objective weights are fused using the geometric mean method. A fuzzy comprehensive evaluation model is used to quantify the matching degrees of multiple sets of multiple-impact signals, and robustness is verified through weight consistency tests and sensitivity analysis. The results show that the evaluated signal sets are rated “Excellent”. Under reasonable weight combinations, the probability of obtaining an “Excellent” result reaches 99.94%, and the maximum variation caused by a ±10% perturbation in a single indicator weight is only 0.0087. The proposed framework provides a practical tool for evaluating multi-layer penetration response simulations and can be extended to other complex dynamic signal-matching problems.

1. Introduction

In modern engineering, repeated impact loads have an increasingly significant effect on the performance and reliability of electromechanical systems. Such loads are commonly encountered in mechanical vibration, collision, and multi-layer target penetration by munitions [1,2,3,4]. More broadly, transient dynamic signals generated under impact or impact-like loading conditions are also important in structural monitoring and digital-twin-assisted system validation [5,6], as well as in advanced manufacturing and structural diagnostics, where complex engineering systems may operate under severe dynamic environments and potential impact conditions [7,8]. In these applications, reliable signal feature extraction and quantitative evaluation are essential for distinguishing true physical response characteristics from measurement uncertainty, noise interference, and structural variability. Among these scenarios, multi-layer target penetration represents one of the most severe operating conditions, because the control components of a projectile must withstand extreme transient impacts, placing stringent requirements on impact signal identification and control [9]. Owing to the inherent limitations of live-fire penetration tests, such as difficulties in data acquisition, high costs, and strong site dependence, multiple-impact test setups have become important tools for reproducing complex impact environments. Although these setups have improved in loading control and impact sequence simulation [10,11,12], systematic methods for evaluating whether the generated impact signals can reproduce the penetration response signal remain insufficient. At present, unified standards for impact signal matching evaluation are still lacking. Traditional single-indicator evaluation methods have several limitations, including incomplete indicator coverage, subjective weighting, and insufficient stability assessment. As a result, parameter adjustment often depends on experience, and the comparability of test results is limited. Therefore, a scientific evaluation framework is needed to improve the simulation accuracy and evaluation reliability of multiple-impact test setups.

Currently, methods for evaluating the matching degree of impact signals mainly include single weighting methods, hybrid weighting methods, and multi-criteria decision-making (MCDM)-based evaluation methods. Single weighting methods are typically represented by AHP [13] and the entropy weight method (EWM) [14]. AHP can incorporate expert experience and engineering judgment, but its results may be affected by subjectivity. In contrast, EWM determines weights according to the dispersion of indicator data and is more objective, but it does not consider the correlation or conflict among indicators. Hybrid weighting methods [15] are used in attempts to integrate subjective and objective information; however, they may still face challenges such as weight allocation uncertainty, indicator redundancy, and insufficient robustness. MCDM-based methods [16] provide a structured framework for processing multidimensional indicators, assigning weights, and generating comprehensive evaluation results. Therefore, they are suitable for impact signal matching, where signal similarity is jointly determined by time-domain characteristics, frequency-domain characteristics, signal quality, and stability.

Within the MCDM framework, combined weighting methods are often used to balance subjective judgment and objective data information. AHP can reflect expert experience and engineering judgment, whereas EWM determines weights according to the dispersion of indicator data. CRITIC further considers the contrast intensity of indicators and the conflict among them. Therefore, these methods can provide complementary weighting information from different perspectives. For impact signal matching evaluation, such a combined weighting strategy is useful because the matching degree is affected by multiple time-domain, frequency-domain, and signal quality indicators, and no single weighting method can fully represent their relative importance.

In recent years, multi-criteria decision-making (MCDM) methods have shown considerable potential in the evaluation of complex multi-indicator systems and have been widely applied in engineering design, risk assessment, infrastructure planning, site selection, supplier quality evaluation, sustainable manufacturing, and life-cycle-based assessment [17,18,19,20,21,22,23,24,25,26,27,28]. To improve evaluation reliability, researchers have increasingly combined subjective weighting methods with objective weighting methods and decision-making models. For example, AHP has been integrated with CRITIC, EWM, VIKOR, DEMATEL, GIS, neural networks, and machine learning to address complex engineering decision problems involving multidimensional indicators and uncertain data [20,21,22,23,24,25,26]. Other studies have applied MCDM to sustainable manufacturing and life-cycle-based building assessment, demonstrating its ability to balance multiple criteria and improve decision consistency under different engineering preferences [27,28].

These studies indicate that current MCDM research is developing in three main directions. First, MCDM is increasingly used as a general evaluation framework for complex engineering systems. Second, MCDM models are being combined with spatial, intelligent, and data-driven methods to improve decision support under heterogeneous information. Third, combined weighting, fuzzy or uncertain modelling, and robustness verification are increasingly adopted to improve interpretability and stability under scattered or uncertain data. These trends support the use of an MCDM-based framework for multiple-impact signal matching evaluation, where different time–frequency indicators must be integrated under experimental scatter and indicator uncertainty. However, such hybrid weighting strategies have rarely been applied to multiple-impact signal matching evaluation, and a complete framework integrating indicator construction, combined weighting, fuzzy comprehensive evaluation, and robustness verification has not yet been fully established.

To address the aforementioned issues, the aim of this study is to develop a fuzzy comprehensive evaluation framework for quantitatively assessing the matching degree between multiple-impact signals generated by a multiple-impact test setup and the penetration response signal. Specifically, the proposed framework is designed to solve the problem that traditional impact signal evaluation methods cannot provide a comprehensive, weighted, and robust assessment of signal matching performance. The framework consists of a multidimensional time–frequency feature indicator system, an AHP-EWM-CRITIC combined weighting method, and a fuzzy comprehensive evaluation model. Through this framework, the dynamic characteristics of impact signals can be comprehensively described, the subjective and objective information of indicators can be integrated, and the matching degree between multiple-impact signals and the penetration response signal can be quantitatively classified. This enables a more objective evaluation of the performance of multiple-impact test setups and provides a quantitative basis for assessing the matching degree between impact signals and the penetration response signal, as well as the simulation accuracy of the test setup. In addition, the proposed framework can provide a methodological reference for the evaluation of other complex dynamic signals in structural monitoring, system validation, and impact-like operating conditions. The main contributions of this study are summarized as follows:

(1) A multidimensional time–frequency feature indicator system is constructed for impact signal matching evaluation. The system covers time-domain features, frequency-domain features, and signal quality and stability indicators, enabling the characterization of impact signal intensity, duration, spectral distribution, and stability.

(2) An AHP-EWM-CRITIC combined weighting method is developed to integrate expert experience and objective data information. AHP is used to obtain subjective weights, EWM and CRITIC are used to extract objective weights, and the geometric mean method is used to obtain comprehensive weights. This combined weighting strategy serves as the weight-assignment module of the MCDM-based evaluation framework and provides more balanced indicator weights for the subsequent fuzzy comprehensive evaluation.

(3) The proposed framework is validated using impact test data and the penetration response signal. Weight consistency tests and sensitivity analysis are conducted to verify the rationality of the comprehensive weights and the robustness of the evaluation results.

2. Materials and Methods

2.1. Data Source

The experimental data used in this study to validate the fuzzy comprehensive evaluation framework for impact signal matching were obtained using an in-house-developed multiple-impact test setup [11], as shown in Figure 1. Figure 1a shows the schematic diagram of the setup, while Figure 1b,c presents photographs of the key experimental components, including the impact body and the target before and after impact. The setup mainly consists of a driving system, a high-mass rotating arm, an impact body, a test specimen, a layered target structure, an acceleration measurement system, a rotating electrical interface, and a data acquisition system. During the test, the impact body is fixed on the rotating arm, and the test specimen is mounted inside the impact body. The driving system drives the rotating arm to rotate, and the rotating arm then carries the impact body to penetrate the layered target structure, thereby generating successive impact loads. The acceleration response of the specimen is measured by an accelerometer mounted inside the test specimen and recorded by an oscilloscope (MSO5354, RIGOL Technologies Co., Ltd., Suzhou, China) through the rotating electrical interface.

The previous study [11,29] introduced the design of the setup and verified its ability to generate multiple-impact acceleration signals with identifiable impact peaks and repeatable response characteristics. Therefore, the setup provides a feasible experimental basis for obtaining the multiple-impact signals used in the present matching evaluation. In this study, the multiple-impact signals generated by this setup were used as the evaluated signals, while the measured penetration response signal was used as the reference signal.

With the support of the aforementioned multiple-impact test setup, multiple-impact tests were conducted, and the corresponding impact test data were collected.

2.2. A Fuzzy Comprehensive Evaluation Framework for Impact Signals Based on Time–Frequency Features and Combined Weighting

2.2.1. Overall Framework

To evaluate the matching degree between impact signals generated by multiple-impact test setups and multi-layer penetration response signals, this study develops a fuzzy comprehensive evaluation framework for impact signals based on time–frequency features and combined weighting. The overall workflow of the proposed framework is shown in Figure 2. The framework is built on a multidimensional time–frequency feature index system, uses the AHP-EWM-CRITIC combined weighting method as its core, and incorporates a fuzzy comprehensive evaluation model to achieve quantitative matching assessment. Among these components, combined weighting plays a central role in the proposed framework. Single-weighting methods have inherent limitations: AHP can incorporate expert experience but is highly subjective; EWM is more objective but does not consider correlations among indicators; and CRITIC objectively considers indicator correlations but does not incorporate expert judgment. In this study, the geometric mean method is used to integrate these three weighting methods, thereby preserving the rationality of expert experience while fully utilizing the objective information contained in the data. As a result, a composite weight vector that balances subjective and objective information is obtained. The fuzzy comprehensive evaluation (FCE) model [30] can handle uncertainty and fuzziness among indicators, but its effectiveness depends on the rationality of the weights and membership degrees. Therefore, the composite weights obtained from the combined weighting method are integrated with the indicator membership matrix, enabling the combination of subjective and objective information with fuzzy evaluation. In this way, the matching degree between multiple-impact signals and penetration response signals can be accurately quantified.

The complete implementation process of the proposed framework consists of the following four stages:

• Construction of a multidimensional time–frequency feature index system for impact signals: Based on the time-domain features, frequency-domain features, and quality and stability characteristics of impact signals, a multidimensional time–frequency feature index system is established. This system includes amplitude, pulse width, root mean square (RMS), total power, spectral flatness, dominant frequency, signal-to-noise ratio (SNR), noise power, and time–frequency entropy, thereby providing a comprehensive characterization of signal properties.
• Indicator normalization: To eliminate differences in units, dimensions, and value ranges, all indicators are normalized to ensure comparability under a unified scale.
• Combined weight calculation: Subjective weights determined by experts using AHP are combined with objective weights derived from EWM, while CRITIC is employed to analyze the correlations and conflicts among indicators. The comprehensive weights are then obtained using a geometric mean-based combination method.
• Fuzzy comprehensive evaluation: By combining the comprehensive weights with the membership matrix, a fuzzy comprehensive evaluation vector is constructed. This vector is then applied to multiple-impact signal datasets to calculate the evaluation scores and membership degrees of the matching degree for different datasets, thereby achieving a quantitative evaluation of the matching degree among multiple-impact signals.

2.2.2. Construction of a Multidimensional Time–Frequency Feature Index System

Impact signals generated under medium-velocity penetration [31] and multiple-impact conditions usually exhibit typical transient characteristics, including high-amplitude impact peaks, limited pulse duration, frequency concentration, and noise-induced fluctuations. These characteristics are closely related to the dynamic response of the tested system and the damage behavior of the target material [32]. Therefore, the evaluation of impact signal matching should consider not only the intensity and duration of the response, but also the spectral distribution, signal quality, and stability.

Based on this consideration, a multidimensional evaluation indicator system was established from three aspects: time-domain features, frequency-domain features, and signal quality and stability. The time-domain indicators, including amplitude, pulse width, and root mean square (RMS), are used to characterize the peak impact intensity, the duration of the main impact process, and the overall response level of the signal [33]. The frequency-domain indicators, including total power, spectral flatness, and dominant frequency, are used to describe the signal energy distribution, spectral concentration, and dominant vibration component [33,34]. The signal quality and stability indicators, including signal-to-noise ratio (SNR), noise power, and time–frequency entropy, are introduced to quantify noise interference, waveform disturbance, and the complexity of the signal energy distribution [33,35,36]. Among them, time–frequency entropy provides a measure of the non-stationary characteristics of the impact signal in the time–frequency domain [36].

Compared with high-dimensional joint time–frequency representations, the selected indicators are physically interpretable, computationally stable, and suitable for normalization, combined weighting, and fuzzy comprehensive evaluation. Therefore, they are appropriate for the objective of this study, which is to quantify the overall matching degree between multiple-impact signals and the penetration response signal rather than to reconstruct the complete localized time–frequency evolution of the response signal. The structure of the multidimensional evaluation indicator system is shown in Figure 3.

• Development of multidimensional indicators based on time-domain features:

• Amplitude

Amplitude represents the maximum instantaneous value of an impact signal and reflects its transient response characteristics. It is calculated as follows:

$$\begin{array}{cc}{A}_P=\max (\left|x\left(i\right)\right|),& i=1,2,\dots ,N\end{array}$$

(1)

where A_p denotes the amplitude of the impact signal, x(i) represents the sampled value of the signal at the i-th sampling point, and N represents the total number of sampling points. Amplitude characterizes instantaneous impacts, mechanical collisions, and overload conditions during system operation, reflecting the transient impact force generated by the interaction between the impact body and the target structure. Therefore, it serves as a key indicator for evaluating impact intensity, material resistance, dynamic response, and penetration depth.

• Pulse width

Pulse width is defined as the duration for which an impact pulse remains at a high level. For an ideal rectangular pulse, it can be expressed as follows:

$$W_p=t_{off}-t_{on}$$

(2)

where t_on and t_off represent the start and end times of the pulse, respectively. Pulse width reflects the duration of the impact load and the continuity of energy transfer. A shorter pulse width is usually associated with a high-intensity transient impact, which may cause brittle failure in the surface layer. In contrast, a longer pulse width indicates more sustained energy transfer, which may lead to deep plastic deformation or global failure.

• Root mean square (RMS)

The root mean square (RMS) characterizes the effective intensity and overall energy-related characteristics of an impact signal. It is calculated as follows:

$$\begin{array}{cc}\mathrm{R}\mathrm{M}\mathrm{S}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^{N}x{\left(i\right)}^2}},& i=1,2,\dots ,N\end{array}$$

(3)

The RMS effectively reflects the overall energy-related characteristics of the impact signal and should satisfy the preset minimum energy requirement.

2.

Development of multidimensional indicators based on frequency-domain features:

• Total power

Total power is defined as the integral of the power spectral density (PSD), and it can be expressed as follows:

$$\mathrm{P}\mathrm{S}{\mathrm{D}}_{total }={\displaystyle {\int }_0^{f_{\max }}X}(f)df$$

(4)

where PSD_total denotes the total power of the impact signal, X(f) represents the power spectral density (PSD) of the signal, and f_max denotes the upper frequency limit. The PSD can be obtained from the frequency-domain representation X(f) = F{x(t)}, where f is the frequency. Total power reflects the overall energy-related intensity of the impact signal. It can be used to quantify signal strength, thereby supporting the evaluation of its matching degree with the penetration response signal.

• Spectral flatness

Spectral flatness measures the uniformity of the spectrum of an impact signal and indicates the degree to which the signal approaches white noise [34]. It is calculated as follows:

$$\mathrm{SF}={\displaystyle \frac{e^{\frac{1}{M}{\displaystyle \sum _{f=0}^{M-1}\ln \left(X\left(f\right)\right)}}}{\frac{1}{M}{\displaystyle \sum _{f=0}^{M-1}X(f)}}}$$

(5)

where M represents the total number of frequency points in the spectrum. A higher spectral flatness indicates a more uniform spectrum, whereas a lower value indicates the presence of distinct spectral peaks.

• Dominant frequency

Dominant frequency refers to the frequency component with the highest amplitude in the spectrum of an impact signal and is typically extracted using the fast Fourier transform (FFT). The closer the dominant frequency of the impact signal is to that of the penetration response signal, the more similar the two signals are in the frequency domain. Therefore, dominant frequency is an important indicator for evaluating signal matching.

3.

Development of multidimensional indicators based on signal quality and stability:

• Signal-to-noise ratio (SNR)

The signal-to-noise ratio (SNR) measures the ratio of useful signal power to noise power and reflects the clarity and identifiability of an impact signal. It is calculated as follows:

$$\mathrm{SNR}=10{\log }_{10}\left({\displaystyle \frac{P_c}{P_b}}\right)$$

(6)

where P_c represents the useful signal power after noise reduction, and P_b represents the noise power. A higher SNR indicates better signal quality.

• Noise power

Noise power is used to quantify the noise intensity in an impact signal and is typically obtained by integrating the noise power spectral density over a specified frequency band:

$$P_{\mathrm{noise}}={\displaystyle {\int }_{f_{\mathrm{low}}}^{f_{\mathrm{high}}}{S}_{\mathrm{noise}}}(f)df$$

(7)

where t_on and t_off represent the start and end times of the pulse, respectively. Pulse width where S_noise(f) represents the noise power spectral density, f_high represents the upper limit of the noise frequency band, and f_low represents the lower limit of the noise frequency band.

• Time–frequency entropy

Time–frequency entropy is used to measure the complexity and information content of impact signals. By integrating the time-domain and frequency-domain characteristics of a signal, it reflects the distribution pattern of signal energy in the time–frequency domain [35,36].

First, the acquired impact signal is transformed into the time–frequency domain to obtain its time–frequency representation X(f,t). The normalized time–frequency energy distribution is calculated as follows:

$$P(i,j)={\displaystyle \frac{|X(f,t){|}^2}{{\displaystyle \sum _{i,j}|}X(f,t){|}^2}}$$

(8)

where P(i,j) denotes the normalized energy probability distribution of the impact signal at the i-th time point and the j-th frequency point. The sum of all probability values satisfies $\sum P(i,j)=1$.

After the normalized time–frequency distribution P(i,j) is obtained, the time–frequency entropy H is calculated as follows:

$$H=-{\displaystyle \sum _{i,j}P}(i,j)\log (P(i,j))$$

(9)

A higher entropy value indicates a broader frequency distribution and a more complex signal structure. Conversely, a lower entropy value indicates a more concentrated energy distribution and stronger regularity in the impact signal.

The consistency of these indicators between the impact signal and the penetration response signal is used to quantify the matching degree.

2.2.3. Indicator Normalization

In this study, the penetration response signal was used as the reference target for matching evaluation. Because the nine secondary indicators differ in units, value ranges, and optimization directions, normalization was performed before weighting and fuzzy comprehensive evaluation. The indicators were classified as positive, negative, or interval-type indicators according to their matching relationship with the penetration response signal. For positive indicators, larger original values indicate better matching and correspond to larger normalized values. For negative indicators, smaller original values indicate better matching after normalization. For the interval-type indicator, the optimal matching state is reached within a specified frequency range. Therefore, the normalization criteria were used to convert the original indicator values of multiple-impact signals into comparable matching scores relative to the penetration response signal.

Based on the above analysis, the normalization formula for positive indicators, including pulse width, root mean square (RMS), total power, spectral flatness, noise power, and time–frequency entropy, is expressed as follows:

$$S_{std}=\left\{\begin{array}{cc}1,& S\ge S_{\max }\\ {\displaystyle \frac{S-S_{\min }}{S_{\max }-S_{\min }}},& S_{\max }>S\ge S_{\min }\\ 0,& S<S_{\min }\end{array}\right.$$

(10)

where S_std represents the normalized value, S represents the original value of the indicator to be normalized, and S_max and S_min represent the upper and lower bounds of the indicator, respectively.

For negative indicators, including amplitude and signal-to-noise ratio (SNR), the normalization formula is expressed as follows:

$$S_{std}=\left\{\begin{array}{cc}1,& S<S_{\min }\\ {\displaystyle \frac{S_{\max }-S}{S_{\max }-S_{\min }}},& S_{\max }>S\ge S_{\min }\\ 0,& S\ge S_{\max }\end{array}\right.$$

(11)

For the dominant frequency, an interval-type normalization method is adopted. Specifically, the dominant frequency is treated as a positive indicator in the range of 0–500 Hz and as a negative indicator in the range of 2000–5000 Hz, ensuring that the normalized value accurately reflects the matching degree between the impact signal and the penetration response signal.

It should be noted that the positive or negative attribute of each indicator was determined according to its matching relationship with the penetration response signal rather than by general signal quality criteria.

2.2.4. Comprehensive Weight Fusion Using the Geometric Mean Method

In impact signal matching evaluation, a single weighting method has difficulty balancing subjective experience and objective information because of indicator diversity, feature complexity, and the involvement of expert judgment. In addition, correlations and redundancies may exist among indicators, which can affect the rationality of the weights and the robustness of the evaluation results. To address these issues, an AHP-EWM-CRITIC combined weighting method is adopted in this study, and the geometric mean method is used to integrate the three types of weights. In this way, a comprehensive weighting scheme that balances subjective rationality and objective information is established. This scheme provides a reliable weighting basis for subsequent fuzzy comprehensive evaluation and supports the quantitative and robust assessment of the matching degree of multiple-impact signals. The specific process is described as follows:

• Calculation of subjective weights using AHP

AHP was used to determine subjective weights based on expert judgments [37]. The relative importance of the primary and secondary indicators was scored using the 1–9 scale, and judgment matrices were constructed to calculate the corresponding weight vectors.

Several relevant technical experts were trained using the 1–9 scoring scale, and simulated expert scoring was then conducted. Based on the relative importance of each indicator, scores were assigned to the primary indicators, including time-domain features, frequency-domain features, and signal quality and stability indicators, as well as to the secondary indicators, including amplitude, pulse width, root mean square (RMS), total power, spectral flatness, dominant frequency, signal-to-noise ratio (SNR), noise power, and time–frequency entropy. Judgment matrices were then constructed for indicators at all levels. Subsequently, according to the AHP procedure, the subjective weight vectors of the impact signal matching indicators at each level were obtained, as shown in Equation (13).

Table 1. AHP judgment matrices for the evaluation criteria of impact signal matching.

Primary Indicators	Time-Domain Features	Frequency-Domain Features	Quality and Stability	Secondary Indicators	Amplitude	Pulse Width	RMS
Time-domain features	1			Amplitude	1
Frequency-domain features		1		Pulse width		1
Quality and stability			1	RMS			1
Secondary indicators	Total power	Spectral flatness	Dominant frequency	Secondary indicators	SNR	Noise power	Time– frequency entropy
Total Power	1			SNR	1
Spectral flatness		1		Noise power		1
Dominant frequency			1	Time–frequency entropy			1

presents the scoring results for the primary and secondary indicators.

$$\left\{\begin{array}{l}{W}^{\mathrm{AHP}}=(W_1^{\mathrm{AHP}},W_2^{\mathrm{AHP}},W_3^{\mathrm{AHP}})\\ W_1^{\mathrm{AHP}}=(w_1^{\mathrm{AHP}},w_2^{\mathrm{AHP}},w_3^{\mathrm{AHP}})\\ W_2^{\mathrm{AHP}}=(w_4^{\mathrm{AHP}},w_5^{\mathrm{AHP}},w_6^{\mathrm{AHP}})\\ W_3^{\mathrm{AHP}}=(w_7^{\mathrm{AHP}},w_8^{\mathrm{AHP}},w_9^{\mathrm{AHP}})\end{array}\right.$$

(12)

where W^AHP denotes the AHP weight vector of the primary indicators, including time-domain features, frequency-domain features, and signal quality and stability. $W_i^{AHP}$ denotes the local AHP weight vector of the secondary indicators under the i-th primary indicator, where i = 1, 2, 3. $w_j^{AHP}$ denotes the local AHP weight of the j-th secondary indicator, where j = 1, 2, …, 9.

• Calculation of objective weights using EWM

EWM was used to calculate objective weights according to the dispersion and information entropy of the normalized indicator data [38]. Indicators with greater dispersion provide more effective information and are assigned larger weights. The specific process is described as follows:

• Based on the normalized indicator values, the proportion of each indicator in all samples is calculated to form a probability distribution matrix;
• The information entropy formula is used to calculate the entropy value of each indicator, which reflects the inherent information dispersion or uncertainty of the indicator;
• The information effectiveness and weighting contribution of each indicator are measured by calculating its redundancy, which is defined as 1 minus the entropy value;
• The redundancy values are normalized to obtain the final objective weight vector, which represents the relative importance of each indicator in the matching evaluation.

The objective weight vectors of indicators at different levels calculated using the EWM method are expressed as follows:

$$\left\{\begin{array}{l}{W}^{\mathrm{EWM}}=(W_1^{\mathrm{EWM}},W_2^{\mathrm{EWM}},W_3^{\mathrm{EWM}})\\ W_1^{\mathrm{EWM}}=(w_1^{\mathrm{EWM}},w_2^{\mathrm{EWM}},w_3^{\mathrm{EWM}})\\ W_2^{\mathrm{EWM}}=(w_4^{\mathrm{EWM}},w_5^{\mathrm{EWM}},w_6^{\mathrm{EWM}})\\ W_3^{\mathrm{EWM}}=(w_7^{\mathrm{EWM}},w_8^{\mathrm{EWM}},w_9^{\mathrm{EWM}})\end{array}\right.$$

(13)

The symbols have the same meanings as those defined for AHP, with the superscript replaced by EWM.

• Calculation of objective weights using CRITIC

CRITIC was used to determine objective weights by considering both indicator variation and inter-indicator conflict [39]. This method assigns larger weights to indicators with greater variability and lower redundancy. The specific process is described as follows:

• The standard deviation of the normalized indicator data is calculated to reflect the dispersion degree and information content of each indicator;
• The Pearson correlation coefficients between pairs of indicators are calculated to measure the redundancy among indicators;
• The conflict degree of each indicator is determined based on its correlations with other indicators;
• The standard deviation and conflict degree are combined and normalized to obtain the CRITIC objective weight vector of each indicator, reflecting its relative importance in the fuzzy comprehensive evaluation of impact signal matching.

The weight vectors of indicators at different levels calculated using the CRITIC method are expressed as follows:

$$\left\{\begin{array}{l}{W}^{\mathrm{CRI}}=(W_1^{\mathrm{CRI}},W_2^{\mathrm{CRI}},W_3^{\mathrm{CRI}})\\ W_1^{\mathrm{CRI}}=(w_1^{\mathrm{CRI}},w_2^{\mathrm{CRI}},w_3^{\mathrm{CRI}})\\ W_2^{\mathrm{CRI}}=(w_4^{\mathrm{CRI}},w_5^{\mathrm{CRI}},w_6^{\mathrm{CRI}})\\ W_3^{\mathrm{CRI}}=(w_7^{\mathrm{CRI}},w_8^{\mathrm{CRI}},w_9^{\mathrm{CRI}})\end{array}\right.$$

(14)

The symbols have the same meanings as those defined for AHP, with the superscript replaced by CRITIC.

• Geometric mean-based weight combination

To integrate the subjective and objective weighting results, the geometric mean method was adopted in this study. AHP reflects expert judgment and engineering experience, EWM reflects the information dispersion of indicator data, and CRITIC reflects both the contrast intensity and conflict among indicators. Therefore, the three weighting methods provide complementary information. Compared with the arithmetic mean, the geometric mean has a multiplicative aggregation characteristic, which can reduce the dominance of any single weighting method. An indicator can obtain a relatively large comprehensive weight only when it receives relatively high weights from AHP, EWM, and CRITIC simultaneously. Conversely, if an indicator is assigned a high weight by only one method but low weights by the other methods, its comprehensive weight will be moderated during the integration process. Therefore, the geometric mean method provides a compromise weighting strategy for balancing expert judgment, data dispersion, and inter-indicator conflict.

Since three weighting methods are integrated, the unnormalized comprehensive weight of the j-th indicator is calculated as the cubic root of the product of the three weights:

$$w_j={\displaystyle \frac{\sqrt[3]{w_j^{\mathrm{AHP}}{w}_j^{\mathrm{EWM}}{w}_j^{\mathrm{CRI}}}}{{\displaystyle {\sum }_{j=1}^9\sqrt[3]{w_j^{\mathrm{AHP}}{w}_j^{\mathrm{EWM}}{w}_j^{\mathrm{CRI}}}}}}$$

(15)

where w_j represents the comprehensive weight of the j-th secondary indicator. The calculated comprehensive weights are grouped according to the primary indicators to form the following weight vectors:

$$\left\{\begin{array}{l}W=(W_1,W_2,W_3)\\ W_1=(w_1,w_2,w_3)\\ W_2=(w_4,w_5,w_6)\\ W_3=(w_7,w_8,w_9)\end{array}\right.$$

(16)

where W denotes the overall comprehensive weight vector of the nine secondary indicators. W_j denotes the comprehensive weight vectors of the secondary indicators under the i-th primary indicator, and w_j denotes the comprehensive weight of the j-th secondary indicator.

2.2.5. Fuzzy Comprehensive Evaluation

By combining the AHP-EWM-CRITIC combined weighting method with the fuzzy comprehensive evaluation model, a quantitative analysis of the matching degree of impact signals is conducted. This method can handle uncertainty and fuzziness among indicators while considering the effects of indicator weights on the evaluation results, thereby enabling a robust assessment of impact signal matching.

The factor set and evaluation set are constructed as follows:

• Primary indicator set: U = {u₁, u₂, u₃} (time-domain features, frequency-domain features, and signal quality and stability);
• Secondary indicator set: u₁ = {u₁₁, u₁₂, u₁₃}, u₂ = {u₂₁, u₂₂, u₂₃}, u₃ = {u₃₁, u₃₂, u₃₃};
• Evaluation set: V = {v₁, v₂, v₃} (Excellent, Satisfactory, and Unsatisfactory).

For each secondary indicator, the membership degree corresponding to each evaluation grade is calculated using the membership function. Thus, the fuzzy evaluation matrices R₁, R₂, and R₃ are constructed as follows:

$$R_1=\left(\begin{array}{ccc}{i}_{v1}(u_{11})& i_{v2}(u_{11})& i_{v3}(u_{11})\\ i_{v1}(u_{12})& i_{v2}(u_{12})& i_{v3}(u_{12})\\ i_{v1}(u_{13})& i_{v2}(u_{13})& i_{v3}(u_{13})\end{array}\right)$$

(17)

$$R_2=\left(\begin{array}{ccc}{i}_{v1}(u_{21})& i_{v2}(u_{21})& i_{v3}(u_{21})\\ i_{v1}(u_{22})& i_{v2}(u_{22})& i_{v3}(u_{22})\\ i_{v1}(u_{23})& i_{v2}(u_{23})& i_{v3}(u_{23})\end{array}\right)$$

(18)

$$R_3=\left(\begin{array}{ccc}{i}_{v1}(u_{31})& i_{v2}(u_{31})& i_{v3}(u_{31})\\ i_{v1}(u_{32})& i_{v2}(u_{32})& i_{v3}(u_{32})\\ i_{v1}(u_{33})& i_{v2}(u_{33})& i_{v3}(u_{33})\end{array}\right)$$

(19)

where i_vk(u_ij) represents the membership degree of the secondary indicator u_ij to the evaluation grade v_k, with I = 1, 2, 3, j = 1, 2, 3, and k = 1, 2, 3.

The fuzzy comprehensive evaluation vector for each primary indicator is calculated as follows. The comprehensive weight vector of the secondary indicators under each primary indicator is multiplied by the corresponding fuzzy evaluation matrix to obtain the primary-level fuzzy comprehensive evaluation vector:

$$P_i=W_i\times R_i$$

(20)

where P_i represents the fuzzy comprehensive evaluation vector of the i-th primary indicator, W_i represents the comprehensive weight vector of the secondary indicators, and R_i represents the corresponding fuzzy evaluation matrix.

The final comprehensive evaluation vector is then calculated by aggregating the primary-level fuzzy evaluation vectors according to the primary-level comprehensive weight vector P:

$$P=W\left(\begin{array}{l}{P}_1\\ P_2\\ P_3\end{array}\right)$$

(21)

Based on the maximum membership principle, the matching quality of multiple-impact signals is determined by the evaluation grade with the largest membership degree in P. Thus, both qualitative and quantitative analyses can be conducted, and the final matching quality of the impact signal can be classified as Excellent, Satisfactory, or Unsatisfactory.

3. Experimental Study on Matching Degree Evaluation Based on the Fuzzy Comprehensive Evaluation Framework

In this section, the proposed framework is applied to three sets of multiple-impact signals. Time–frequency features are extracted, normalized, and weighted using the AHP-EWM-CRITIC combined weighting method, and then evaluated using the fuzzy comprehensive evaluation model. All algorithms are implemented using Python 3.8.10 (Python Software Foundation, Beaverton, OR, USA).

3.1. Extraction of Time–Frequency Features from Multiple-Impact Signals

Using the multiple-impact test setup, three sets of impact tests were conducted under the following conditions: three target plates made of AISI 1020 steel were used, with thicknesses of 4 mm, 4 mm, and 5 mm, respectively, and all impactors had the same design. The experimental results of the three sets of multiple-impact signals are shown separately in Figure 4a–c.

Time–frequency features were extracted from the impact signals of the three impact test groups, covering nine indicators in three dimensions: time-domain features, frequency-domain features, and signal quality and stability. The signals were sampled at 20 kHz, and the data range extended from 2 ms before the first waveform to 2 ms after the last waveform, as shown in Figure 4. The parameters of each indicator were extracted within this range and are summarized in

Table 2. Extracted time–frequency feature indicators for matching impact signals with penetration response signals.

Test Case	Amplitude (g)	Pulse Width (ms)	RMS (g)	Total Power (g2)	Spectral Flatness	Dominant Frequency (Hz)	SNR (dB)	Noise Power (g2)	Time– Frequency Entropy
Penetration response signal	−18,343.30	0.90	5413.89	4.62 × 1011	0.16	611.00	9.11	3.89 × 1010	5.08
Test 1 impact signal	−14,552.00	0.63	3532.65	1.25 × 1011	0.42	497.51	0.75	5.05 × 1010	5.51
Test 2 impact signal	−14,858.30	0.87	3922.59	1.54 × 1011	0.34	523.73	1.75	5.81 × 1010	5.34
Test 3 impact signal	−16,269.70	0.85	4550.73	2.07 × 1011	0.22	510.21	5.09	4.63 × 1010	4.64
AISI 1020-1	−19,288.00	1.09	5774.99	5.39 × 1011	0.09	443.46	9.87	3.12 × 1010	4.52
AISI 1020-2	−19,195.33	0.79	5422.98	4.61 × 1011	0.09	443.46	11.27	1.78 × 1010	4.61
AISI 1020-3	−20,287.67	0.90	5234.76	4.42 × 1011	0.10	443.46	11.75	1.65 × 1010	4.75
AISI 1040-1	−25,606.33	0.59	5639.66	4.15 × 1011	0.23	554.32	8.68	3.05 × 1010	5.42
AISI 1040-2	−23,561.33	0.72	6723.55	7.06 × 1011	0.07	554.32	10.78	3.28 × 1010	4.60
AISI 1040-3	−23,743.00	0.76	5441.15	4.05 × 1011	0.10	554.32	11.45	1.58 × 1010	5.00
AISI 1060-1	−22,776.49	0.82	8253.75	1.29 × 1012	0.05	554.32	9.83	6.76 × 1010	4.61
AISI 1060-2	−20,744.95	0.84	6802.72	8.74 × 1011	0.06	554.32	12.28	2.21 × 1010	4.70
AISI 1060-3	−21,873.65	0.94	6826.82	8.39 × 1011	0.06	443.46	12.23	2.27 × 1010	4.58
2A12-1	−10,861.00	0.65	2896.14	1.20 × 1011	0.19	443.46	7.12	1.11 × 1010	5.50
2A12-2	−11,097.00	0.55	2550.71	8.95 × 1010	0.19	443.46	6.97	9.07 × 109	5.42
2A12-3	−12,782.00	0.83	3948.09	2.47 × 1011	0.11	443.46	9.24	1.43 × 1010	4.83
7020-1	−12,141.67	0.51	3305.02	1.39 × 1011	0.24	443.46	6.69	1.54 × 1010	5.71
7020-2	−15,065.67	0.50	3825.85	2.18 × 1011	0.14	443.46	8.26	1.63 × 1010	5.35
7020-3	−14,169.33	0.76	3998.13	2.57 × 1011	0.11	443.46	8.28	1.78 × 1010	4.82
7075-1	−14,939.67	0.41	4021.14	2.47 × 1011	0.11	443.46	9.69	1.38 × 1010	4.89
7075-2	−16,343.33	0.76	4929.22	4.01 × 1011	0.11	443.46	9.49	2.19 × 1010	4.93
7075-3	−16,099.67	0.60	4820.40	3.89 × 1011	0.09	443.46	10.22	1.83 × 1010	4.70
AZ31B-1	−6563.00	0.67	1741.77	3.79 × 1010	0.15	554.32	5.73	4.79 × 109	5.36
AZ31B-2	−7345.33	0.58	1877.25	5.44 × 1010	0.14	443.46	3.94	8.83 × 109	5.32
AZ31B-3	−5739.67	0.53	1312.31	2.26 × 1010	0.11	554.32	8.64	1.43 × 109	5.22
H62-1	−18,599.33	0.94	4415.73	2.86 × 1011	0.15	443.46	9.89	1.51 × 1010	5.28
H62-2	−19,088.67	0.98	5723.02	5.49 × 1011	0.07	443.46	10.83	2.27 × 1010	4.52
H62-3	−18,096.00	0.79	4332.04	2.87 × 1011	0.07	554.32	17.83	2.80 × 109	4.77
S30153-1	−16,508.67	0.72	4498.14	3.28 × 1011	0.14	554.32	7.50	2.91 × 1010	5.12
S30153-2	−18,894.67	0.73	5200.97	3.88 × 1011	0.13	554.32	9.31	2.80 × 1010	4.80
S30153-3	−16,259.00	0.69	4270.97	2.52 × 1011	0.14	443.46	11.40	1.10 × 1010	5.19

Note: The data in this table were obtained from experimental tests performed in this study. The penetration response signal was obtained from a medium-velocity penetration test in which a projectile penetrated a three-layer concrete target at a velocity of 770 m/s, and it was used as the reference signal for matching evaluation. The multiple-impact signals and the 27 groups of material test results were obtained using the self-developed multiple-impact test setup [29]. The suffixes “-1”, “-2”, and “-3” indicate the first, second, and third tests conducted using the corresponding material, respectively. The curves of the penetration response signal and the 27 groups of material test results are provided as Figures S1 and S2 in the Supplementary Materials, respectively.

.

Table 2 also includes the penetration response signals and the test results of 27 groups with different materials obtained using this setup. These material groups involved nine materials, namely AISI 1020 steel, AISI 1040 steel, AISI 1060 steel, 2A12 Al alloy, 7020 Al alloy, 7075 Al alloy, AZ31B Mg alloy, H62 brass, and S30153 stainless steel, with three repeated tests conducted for each material. These additional 27 groups of material test data were used to construct the sample set for objective weight calculation and normalization, thereby improving the representativeness of the EWM and CRITIC weighting results. The test data comprise a database for model validation.

3.2. Indicator Normalization and Parameter Determination

Using the multiple-impact test setup, three sets of impact tests were conducted under the following conditions: three target plates made of AISI 1020 steel were used, with thicknesses of 4 mm, 4 mm, and 5 mm, respectively, and all impactors had the same design. The experimental results are shown in Figure 4.

In this study, the penetration response signal was used as the reference signal for the matching evaluation. The purpose of the evaluation was to determine whether the multiple-impact signals generated by the test setup could reproduce the main time–frequency characteristics of the penetration response signal. Therefore, the normalization parameters, including the indicator type, S_min, and S_max, were determined according to the extracted features of the penetration response signal, the distribution range of the experimental samples, and expert experience, as shown in

Table 3. Normalization parameter settings for the evaluation indicators.

No.	Evaluation Indicator	Indicator Type	Smin	Smax
1	Amplitude (g)	Negative indicator	−16,000	−10,000
2	Pulse width (ms)	Positive indicator	0.20	0.80
3	RMS (g)	Positive indicator	1000	5000
4	Total power (g2)	Positive indicator	0	3.00 × 1011
5	Spectral flatness	Positive indicator	0	0.3
6	Dominant frequency (Hz)	Positive indicator	0	500
		Negative indicator	2000	5000
7	SNR (dB)	Negative indicator	0	10
8	Noise power (g2)	Positive indicator	2.00 × 1010	5.00 × 1010
9	Time–frequency entropy	Positive indicator	2	5

.

Because the nine secondary indicators have different units, value ranges, and optimization directions, normalization was performed before weight calculation and fuzzy comprehensive evaluation. Positive indicators indicate that a larger normalized value corresponds to better matching performance, whereas negative indicators indicate that a smaller original value is more favorable. For interval-type indicators, the normalized score is determined according to whether the original value falls within the desired matching range. Therefore, Table 3 provides the normalization criteria used to transform the original indicator values into comparable matching scores.

It should be noted that the parameters in Table 3 are not final evaluation results, but normalization criteria for different indicators. These criteria ensure that indicators with different physical meanings and units can be converted into dimensionless scores within a comparable range. In this way, the normalized values can be used consistently in the subsequent AHP-EWM-CRITIC weighting process and fuzzy comprehensive evaluation. For the dominant frequency indicator, the two parameter ranges in Table 3 correspond to the piecewise normalization rule used to describe the preferred and unfavorable frequency ranges.

The three sets of multiple-impact signals obtained from the impact tests were normalized to obtain the normalized values of each secondary indicator, as shown in

Table 4. Normalized results of the secondary indicators.

Test No.	Amplitude (g)	Pulse Width (ms)	RMS (g)	Total Power (g2)	Spectral Flatness	Dominant Frequency (Hz)	SNR (dB)	Noise Power (g2)	Time–Frequency Entropy
1	0.76	0.72	0.63	0.42	1.00	1.00	0.93	1.00	1.00
2	0.81	1.00	0.73	0.51	1.00	1.00	0.83	1.00	1.00
3	1.00	1.00	0.89	0.69	0.73	1.00	0.49	0.88	0.88

. In the table, values closer to 1 indicate that the corresponding indicator is closer to the ideal matching state. These normalized results effectively reflect the matching performance of the impact signals under the current test conditions and provide a quantitative basis for evaluating the matching degree of impact signals.

Table 4 shows that the three test signals have different matching characteristics after normalization. Test 1 has better performance in signal quality and stability indicators, but its RMS and total power values are relatively low. Test 2 shows a more balanced performance among time-domain, frequency-domain, and signal quality indicators. Test 3 has higher normalized values for amplitude, pulse width, RMS, dominant frequency, noise power, and time–frequency entropy, but its total power and SNR indicators are relatively lower. Therefore, the normalized results in Table 4 not only reflect the differences among the three impact signals, but also provide the direct input data for the subsequent combined weighting and fuzzy comprehensive evaluation.

3.3. Combined Weighting Based on AHP-EWM-CRITIC

3.3.1. Subjective Weighting Based on AHP

To improve the reliability of the subjective weighting results, 20 relevant technical experts were invited to perform pairwise comparisons of the primary and secondary indicators using the AHP 1–9 scale. A total of 20 scoring forms were collected.

Table 5. Scoring form for primary indicators in impact signal matching evaluation.

Primary Indicator	Time-Domain Features	Frequency-Domain Features	Quality and Stability
Time-domain features	1	5	7
Frequency-domain features	1/5	1	2
Quality and stability	1/7	1/2	1

presents one representative scoring form for the primary indicators to illustrate the construction of the AHP judgment matrix. The same procedure was applied to all scoring forms for both primary and secondary indicators.

In Table 5, each element represents the relative importance of the row indicator compared with the column indicator. The diagonal elements are equal to 1, and the reciprocal relationship a_ji = 1/a_ij is satisfied. Based on Table 5, the judgment matrix A for the primary indicators is constructed as follows:

$$A=\left[\begin{array}{ccc}1& 5& 7\\ 1/5& 1& 2\\ 1/7& 1/2& 1\end{array}\right]$$

(22)

where the rows and columns of A correspond to time-domain features, frequency-domain features, and quality and stability indicators, respectively. The weight vector of the primary indicators was calculated using the maximum eigenvalue method. The obtained weight vector was W = (0.7396, 0.1666, 0.0938), with a maximum eigenvalue of 3.0143 and a consistency index of CI = 0.0071. Since the number of primary indicators is n = 3, the random index is RI = 0.58 according to the standard random consistency index table. The consistency ratio is calculated as CR = CI/RI = 0.0123. Since CR ≤ 0.1, the judgment matrix satisfies the consistency requirement. All 20 expert scoring forms were then processed using the same procedure, and the corresponding weight vectors for the primary and secondary indicators were obtained, as shown in

Table 6. Weight calculation results for primary indicators in impact signal matching evaluation.

No.	Time-Domain Features	Frequency-Domain Features	Quality and Stability	$\mathit{C}\mathit{R}$	Consistency Test
1	0.74	0.17	0.09	0.01	Passed
2	0.68	0.18	0.14	0.05	Passed
3	0.54	0.10	0.36	0.08	Passed
4	0.49	0.45	0.06	0.01	Passed
5	0.58	0.08	0.33	1.28	Failed
6	0.62	0.29	0.09	0.00	Passed
7	0.73	0.20	0.07	0.08	Passed
8	0.58	0.34	0.08	0.03	Passed
9	0.44	0.47	0.08	0.00	Passed
10	0.41	0.26	0.33	0.05	Passed
11	0.50	0.07	0.43	0.02	Passed
12	0.68	0.18	0.14	0.63	Failed
13	0.56	0.37	0.07	0.07	Passed
14	0.79	0.11	0.10	0.02	Passed
15	0.51	0.08	0.41	0.05	Passed
16	0.61	0.06	0.33	0.01	Passed
17	0.51	0.41	0.08	0.05	Passed
18	0.46	0.48	0.06	0.00	Passed
19	0.63	0.11	0.26	0.03	Passed
20	0.69	0.13	0.17	0.07	Passed
Average Weight	0.58	0.24	0.18

,

Table 7. Weight calculation results for secondary indicators of time-domain features.

No.	Amplitude	Pulse Width	RMS	$\mathit{C}\mathit{R}$	Consistency Test
1	0.69	0.13	0.17	0.07	Passed
2	0.32	0.62	0.07	0.00	Passed
3	0.61	0.17	0.22	0.08	Passed
4	0.44	0.49	0.08	0.01	Passed
5	0.41	0.50	0.09	0.03	Passed
6	0.49	0.45	0.06	0.01	Passed
7	0.71	0.14	0.14	0.00	Passed
8	0.50	0.37	0.14	0.08	Passed
9	0.66	0.07	0.27	0.04	Passed
10	0.46	0.42	0.13	0.01	Passed
11	0.67	0.26	0.07	0.63	Failed
12	0.52	0.36	0.12	0.09	Passed
13	0.77	0.14	0.09	0.05	Passed
14	0.47	0.44	0.08	0.00	Passed
15	0.69	0.19	0.13	0.08	Passed
16	0.49	0.44	0.07	0.01	Passed
17	0.58	0.36	0.06	0.05	Passed
18	0.59	0.34	0.06	0.02	Passed
19	0.44	0.50	0.06	0.02	Passed
20	0.29	0.62	0.09	0.00	Passed
Average Weight	0.53	0.35	0.11

,

Table 8. Weight calculation results for secondary indicators of frequency-domain features.

No.	Total Power	Spectral Flatness	Dominant Frequency	$\mathit{C}\mathit{R}$	Consistency Test
1	0.18	0.09	0.74	0.00	Passed
2	0.08	0.57	0.36	0.05	Passed
3	0.42	0.07	0.51	0.03	Passed
4	0.42	0.09	0.48	0.02	Passed
5	0.07	0.19	0.74	0.15	Failed
6	0.21	0.08	0.72	0.02	Passed
7	0.14	0.16	0.71	0.02	Passed
8	0.17	0.19	0.63	0.01	Passed
9	0.14	0.11	0.75	0.06	Passed
10	0.10	0.17	0.73	0.03	Passed
11	0.13	0.17	0.69	0.07	Passed
12	0.10	0.09	0.81	0.00	Passed
13	0.07	0.43	0.50	0.02	Passed
14	0.13	0.42	0.46	0.01	Passed
15	0.18	0.14	0.68	0.05	Passed
16	0.43	0.10	0.47	0.00	Passed
17	0.07	0.56	0.37	0.07	Passed
18	0.08	0.17	0.75	0.09	Passed
19	0.06	0.56	0.38	0.09	Passed
20	0.14	0.06	0.80	0.09	Passed
Average Weight	0.17	0.22	0.61

and

Table 9. Weight calculation results for secondary indicators of signal quality and stability.

No.	SNR	Noise Power	Time–Frequency Entropy	$\mathit{C}\mathit{R}$	Consistency Test
1	0.68	0.10	0.22	0.00	Passed
2	0.41	0.08	0.51	0.05	Passed
3	0.47	0.05	0.47	0.00	Passed
4	0.45	0.09	0.45	0.00	Passed
5	0.75	0.17	0.08	0.09	Passed
6	0.48	0.16	0.36	1.18	Failed
7	0.68	0.07	0.25	0.01	Passed
8	0.60	0.20	0.20	0.00	Passed
9	0.63	0.09	0.29	0.01	Passed
10	0.14	0.09	0.77	0.05	Passed
11	0.63	0.19	0.17	0.01	Passed
12	0.37	0.08	0.55	0.07	Passed
13	0.59	0.16	0.25	0.05	Passed
14	0.59	0.05	0.36	0.03	Passed
15	0.44	0.08	0.47	0.00	Passed
16	0.57	0.07	0.36	0.05	Passed
17	0.33	0.41	0.26	0.05	Passed
18	0.36	0.06	0.58	0.05	Passed
19	0.47	0.43	0.10	0.00	Passed
20	0.37	0.41	0.22	0.32	Failed
Average Weight	0.49	0.14	0.37

.

After data that failed the consistency test were excluded, the AHP weights of the primary and secondary indicators were calculated, as shown in

Table 10. AHP weighting results for indicators at different levels.

Primary Indicator	Weight	Secondary Indicator	Local Weight	Global Weight
Time-domain features	0.5825	Amplitude	0.5334	0.3107
		Pulse width	0.3544	0.2064
		RMS	0.1122	0.0654
Frequency-domain features	0.2382	Total power	0.1706	0.0406
		Spectral flatness	0.2225	0.0530
		Dominant frequency	0.6069	0.1446
Quality and stability	0.1793	SNR	0.4922	0.0883
		Noise power	0.1373	0.0246
		Time–frequency entropy	0.3705	0.0664

.

3.3.2. Objective Weighting Based on the EWM

The EWM weighting results are shown in

Table 11. EWM weighting results for indicators at different levels.

Primary Indicator	Weight	Secondary Indicator	Global Weight
Time-domain features	0.4029	Amplitude	0.1252
		Pulse width	0.1416
		RMS	0.1361
Frequency-domain features	0.4099	Total power	0.1332
		Spectral flatness	0.1319
		Dominant frequency	0.1448
Quality and stability	0.1871	SNR	0.0418
		Noise power	0.0005
		Time–frequency entropy	0.1448

.

3.3.3. Objective Weighting Based on the CRITIC Method

The CRITIC weighting results are shown in

Table 12. CRITIC weighting results for indicators at different levels.

Primary Indicator	Weight	Secondary Indicator	Global Weight
Time-domain features	0.4728	Amplitude	0.2442
		Pulse width	0.0721
		RMS	0.1565
Frequency-domain features	0.2830	Total power	0.1800
		Spectral flatness	0.1001
		Dominant frequency	0.0029
Quality and stability	0.2443	SNR	0.1325
		Noise power	0.0854
		Time–frequency entropy	0.0264

.

3.3.4. Calculation of Comprehensive Weights

Using the geometric mean method described in Section 2.2.4, the AHP, EWM, and CRITIC weights were integrated to obtain comprehensive weights. The results are shown in

Table 13. Comprehensive weighting results for impact signal matching evaluation.

Primary Indicator	Weight	Secondary Indicator	Local Weight	Global Weight
Time-domain features	0.5434	Amplitude	0.4689	0.2548
		Pulse width	0.2839	0.1543
		RMS	0.2472	0.1343
Frequency-domain features	0.2734	Total power	0.4363	0.1193
		Spectral flatness	0.3907	0.1068
		Dominant frequency	0.1730	0.0473
Quality and stability	0.1832	SNR	0.5173	0.0948
		Noise power	0.0668	0.0122
		Time–frequency entropy	0.4159	0.0762

.

3.4. Fuzzy Comprehensive Evaluation and Classification

3.4.1. Determination and Construction of Membership Functions

To evaluate the multiple-impact signals generated by the multiple-impact test setup, three membership functions were constructed in this study, corresponding to the evaluation grades “Excellent,” “Satisfactory,” and “Unsatisfactory.” An ascending half-Cauchy-type function, a symmetric Cauchy-type function, and a descending half-Cauchy-type function were used for these three grades, respectively. These functions were adopted because they can describe nonlinear and gradual transitions among adjacent evaluation grades, which is suitable for the fuzzy classification of impact signal matching quality.

In this study, the normalized evaluation score $u$ was expressed on a scale of 0–100. According to the engineering interpretation of the matching degree under the current medium-velocity penetration scenario, u < 60 was regarded as a low-matching region, 60 ≤ u ≤ 80 as a transition region, and u ≥ 80 as a high-matching region. Therefore, 60 and 80 were used as the main grade boundaries. In addition, 70 was introduced as the transition threshold for the “Excellent” membership function, and 65 was used as the center of the “Satisfactory” membership function. The coefficients in the Cauchy-type functions were used to control the transition rate between adjacent grades, so that the membership degree could reflect gradual changes in signal matching quality rather than a simple binary classification.

The thresholds and shape parameters were determined according to the normalized indicator range, the characteristics of the penetration response signal, the distribution of the experimental samples, and expert experience under the current experimental conditions. It should be noted that these membership parameters are scenario-specific rather than universal constants. When the proposed framework is applied to other impact conditions, such as different material properties, projectile geometries, target configurations, impact velocities, or sensor installation conditions, the indicator ranges and membership function parameters should be redefined using representative data from the new application scenario.

The membership functions are defined as follows:

$$i_{v1}(u)=\left\{\begin{array}{cc}0,& 0\le u<70\\ {\displaystyle \frac{1}{1.25+0.0018{(u-80)}^2}},& 70\le u<80\\ {\displaystyle \frac{u}{100}},& 80\le u\le 100\end{array}\right.$$

(23)

$$i_{v2}(u)={\displaystyle \frac{1}{1+0.0085{(u-65)}^2}}$$

(24)

$$i_{v3}(u)=\left\{\begin{array}{cc}1,& 0\le u<60\\ {\displaystyle \frac{1}{1+0.01{(u-60)}^2}},& 60\le u<80\\ 1-{\displaystyle \frac{u}{100}},& 80\le u\le 100\end{array}\right.$$

(25)

3.4.2. Fuzzy Comprehensive Evaluation Results and Classification

Based on the membership functions defined above, the membership degrees of each normalized secondary indicator to the three evaluation grades were calculated. The fuzzy relation matrices of the three primary indicators were then constructed. Taking Test 1 as an example, the fuzzy relation matrices R₁, R₂, and R₃ were obtained as follows:

$$R_1=\left(\begin{array}{ccc}0.7808& 0.4989& 0.2842\\ 0.7273& 0.7256& 0.4234\\ 0& 0.9766& 0.9007\end{array}\right)$$

(26)

$$R_2=\left(\begin{array}{ccc}0& 0.1777& 1\\ 1& 0.0876& 0\\ 0.9950& 0.0900& 0.0050\end{array}\right)$$

(27)

$$R_3=\left(\begin{array}{ccc}0.9250& 0.1346& 0.0750\\ 1& 0.0876& 0\\ 1& 0.0876& 0\end{array}\right)$$

(28)

The fuzzy comprehensive evaluation vectors P₁, P₂, and P₃ are calculated by combining the comprehensive weight vectors W₁, W₂, and W₃ with the corresponding fuzzy relation matrices R₁, R₂, and R₃, respectively:

$$\left\{\begin{array}{l}{P}_1=W_1\times R_1=\left(\begin{array}{ccc}0.5726& 0.6813& 0.4761\end{array}\right)\\ P_2=W_2\times R_2=\left(\begin{array}{ccc}0.5628& 0.1273& 0.4372\end{array}\right)\\ P_3=W_3\times R_3=\left(\begin{array}{ccc}0.9612& 0.1119& 0.0388\end{array}\right)\end{array}\right.$$

(29)

Finally, the fuzzy comprehensive evaluation vectors of the three primary indicators are aggregated to obtain the final evaluation vector P for the matching evaluation of the impact signal:

$$P=\left(\begin{array}{ccc}0.6411& 0.4256& 0.3854\end{array}\right)$$

(30)

For test 1, the final fuzzy comprehensive evaluation vector was P = (0.6411, 0.4256, 0.3854). According to the maximum membership principle, the impact signal was classified as “Excellent”. This indicates that the impact signal from Test 1 achieved a high matching degree with the penetration response signal.

Using the same method, the impact signals from the remaining two test groups were evaluated, and the fuzzy comprehensive evaluation vector of each group was obtained. The final evaluation results of the three test groups are summarized in

Table 14. Final fuzzy comprehensive evaluation results for impact signals in Tests 1–3.

Test No.	PExcellent	PSatisfactory	PUnsatisfactory
1	0.6411	0.4256	0.3854
2	0.7818	0.2741	0.2340
3	0.7337	0.2816	0.2249

.

The fuzzy comprehensive evaluation results of the three test groups show that the matching degrees of all three impact signals reached the “Excellent” grade according to the maximum membership principle. However, noticeable differences in matching quality were observed among the three tests. Test 2 obtained the highest membership degree for the “Excellent” grade, with a value of 0.7818, indicating the best matching performance and the closest agreement with the penetration response signal. Test 3 ranked second, with an “Excellent” membership degree of 0.7337, showing good matching performance. Test 1 also reached the “Excellent” grade, with an “Excellent” membership degree of 0.6411, but its membership degrees for the “Satisfactory” and “Unsatisfactory” grades were relatively high, at 0.4256 and 0.3854, respectively. This indicates that the matching result of Test 1 was less distinct than those of Tests 2 and 3.

Further analysis of the fuzzy relation matrices for Test 1, as shown in Equations (26)–(28), indicates that the membership degree of the pulse width indicator for the “Satisfactory” grade was close to that for the “Excellent” grade. For the RMS indicator, the membership degrees followed the order “Satisfactory” > “Unsatisfactory” > “Excellent”, whereas for the total power indicator, the order was “Unsatisfactory” > “Satisfactory” > “Excellent”. These results indicate that pulse width, RMS, and total power still have certain deficiencies in Test 1, which explains why its “Satisfactory” and “Unsatisfactory” memberships remained relatively high despite its final classification as “Excellent”.

Overall, the three impact signals all satisfied the “Excellent” matching criterion, but the matching quality followed the order Test 2 > Test 3 > Test 1. This result shows that the proposed fuzzy comprehensive evaluation framework can not only classify the matching grade of impact signals, but also distinguish differences in matching quality among signals with the same evaluation grade.

4. Discussion

Although the proposed framework achieved an “Excellent” evaluation result for all three test groups, the rationality of the comprehensive weights and the robustness of the evaluation results should be further verified. Therefore, consistency tests and sensitivity analyses were conducted in this section to evaluate the stability and reliability of the proposed framework.

4.1. Consistency Test for Comprehensive Weights

To verify the rationality of the comprehensive weights, consistency tests were conducted by comparing the comprehensive weight vector with the weight vectors obtained from AHP, EWM, and CRITIC. Spearman’s rank correlation coefficient was used to evaluate ranking consistency, and Euclidean distance was used to evaluate numerical similarity.

4.1.1. Spearman’s Rank Correlation Coefficient

Spearman’s rank correlation coefficient [40] was used to measure the ranking consistency between the comprehensive weight vector and each original weight vector. It is calculated as follows:

$$\rho =1-{\displaystyle \frac{6{\displaystyle {\sum }_{j=1}^n{d}_j^2}}{n(n^2-1)}}$$

(31)

where d_j represents the difference between the ranks of the j-th indicator in two weight vectors, and n represents the total number of indicators.

The specific steps are described as follows:

• The nine comprehensive weight values w_j in the comprehensive weight vector W are ranked. A smaller rank indicates a larger weight, where rank 1 represents the most important indicator and rank 9 represents the least important indicator. If two or more weights are equal, their average rank is used;
• The same ranking procedure is performed for the weight values obtained from other weighting methods, including AHP, EWM, and CRITIC;
• The rank difference d_j of each indicator between two weighting systems is then calculated. For example, the rank difference between the comprehensive weight and the AHP weight of the j-th indicator is calculated as follows:

$$d_j=\mathrm{Rank}\left(w_j\right)-\mathrm{Rank}\left(w_j^{\mathrm{AHP}}\right)$$

(32)

4.

The squared rank differences $d_j^2$ are summed and substituted into Equation (31) to obtain Spearman’s rank correlation coefficient.

The subjective weights, objective weights, and comprehensive weights obtained using AHP, EWM, and CRITIC were ranked according to indicator importance, and the results are shown in

Table 15. Importance rankings based on AHP, EWM, CRITIC, and comprehensive weights.

Indicator	$\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{k} \left({\mathit{w}}_{\mathit{j}}^{\mathbf{A}\mathbf{H}\mathbf{P}}\right)$	$\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{k} \left({\mathit{w}}_{\mathit{j}}^{\mathbf{E}\mathbf{W}\mathbf{M}}\right)$	$\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{k} \left({\mathit{w}}_{\mathit{j}}^{\mathbf{C}\mathbf{R}\mathbf{I}}\right)$	$\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{k} \left({\mathit{w}}_{\mathit{j}}\right)$
1	1	7	1	1
2	2	3	7	2
3	6	4	3	3
4	8	5	2	4
5	7	6	5	5
6	3	1	9	8
7	4	8	4	6
8	9	9	6	9
9	5	2	8	7

.

The calculation results show that the Spearman’s rank correlation coefficient between the AHP subjective weights and the comprehensive weights was 0.48, whereas the coefficient between the CRITIC objective weights and the comprehensive weights was 0.63. These results indicate that the comprehensive weights maintain a certain degree of consistency with the importance rankings obtained from AHP and CRITIC. In contrast, the Spearman’s rank correlation coefficient between the EWM objective weights and the comprehensive weights was only 0.03, indicating a relatively weak ranking consistency. This low rank correlation is mainly attributed to the small variation among the EWM weights, which weakens its ability to distinguish the relative importance of different indicators in terms of ranking. However, this result only reflects the ranking relationship and does not necessarily indicate a large numerical deviation between the EWM weights and the comprehensive weights. Therefore, the numerical similarity among different weight vectors is further examined using Euclidean distance.

4.1.2. Euclidean Distance

Euclidean distance [41] was used to measure the numerical similarity between the comprehensive weight vector and each original weight vector. By treating each weight vector as a point in a nine-dimensional space, the distance between W and W^P can be calculated as follows, where P ∈ {AHP, WEM, CRI}:

$$D_E\left(W,W^P\right)=\sqrt{{\displaystyle \sum _{j=1}^9{\left(w_j-w_j^P\right)}^2}}$$

(33)

The calculation results show that the Euclidean distances between the comprehensive weight vector W and the weight vectors W^AHP, W^EWM, and W^CRI are 0.17, 0.19, and 0.15, respectively. These results indicate that the comprehensive weight vector maintains a relatively close numerical relationship with all three original weight vectors. Among them, the comprehensive weight vector is closest to the CRITIC weight vector, followed by the AHP weight vector, while the distance from the EWM weight vector is also within a small range.

Further calculations show that the Euclidean distances among the three original weight vectors are 0.26, 0.27, and 0.28, with an average value of 0.27. In contrast, the average distance between the comprehensive weight vector and the three original weight vectors is 0.17. Since this value is smaller than the average distance among the three original weight vectors, the comprehensive weight vector can be considered to be located near the center of the three original weighting schemes. This indicates that the proposed combined weighting method effectively integrates the subjective information from AHP, the data-dispersion information from EWM, and the inter-indicator conflict information from CRITIC.

It should be noted that the Spearman correlation coefficient and Euclidean distance reflect different aspects of similarity. Spearman’s rank correlation focuses on the consistency of indicator importance rankings, whereas Euclidean distance measures the numerical closeness of weight vectors. Therefore, although the rank correlation between the EWM weights and the comprehensive weights is relatively low, their Euclidean distance remains small. This may be attributed to the relatively small variation among EWM weights, which weakens its ranking discrimination ability but does not necessarily lead to a large numerical distance from the comprehensive weight vector.

In summary, the analyses based on Spearman’s rank correlation coefficients and Euclidean distances show that the comprehensive weighting method achieves a reasonable integration of multiple weighting schemes. The comprehensive weights preserve a certain ranking consistency with AHP and CRITIC while maintaining numerical closeness to all three original weight vectors. These results verify the rationality and integration performance of the proposed AHP-EWM-CRITIC combined weighting method.

4.2. Weight Sensitivity Analysis

Uncertainties arising from subjective judgment and data dispersion may affect the weight determination process. In the proposed AHP-EWM-CRITIC combined weighting scheme, AHP weights are mainly influenced by expert judgment, whereas EWM and CRITIC weights are related to the dispersion and correlation structure of the experimental data. Therefore, sensitivity analysis was conducted to examine whether weight uncertainty would affect the final fuzzy comprehensive evaluation result. In this section, the results of Test 1 are used as an example. OAT-based local perturbation, AHP weight perturbation, and Dirichlet-based global sampling are used to analyze the robustness of the evaluation results from different perspectives.

4.2.1. Local Perturbation Analysis and Tornado Plot Based on the OAT Method

The one-at-a-time (OAT) method [42] is used to evaluate the marginal effect of a single weight w_j on the target output P_Excellent by applying a relative perturbation of ±ρ, such as 10%, to the weight:

$${\tilde{w}}_j^{\left(\pm \right)}=w_j(1\pm \rho )$$

(34)

To ensure that the sum of all weights remains equal to 1, the remaining weights are normalized as follows:

$${\tilde{w}}_k=w_k\cdot {\displaystyle \frac{1-{\tilde{w}}_j}{1-w_j}}, k\ne j$$

(35)

The change in the target output P_Excellent under the positive and negative perturbations of ${\tilde{w}}_j$ is then calculated as follows:

$$\Delta P_{\mathrm{Excellent}}^{(+)}=P_{\mathrm{Excellent}}({\tilde{w}}_j^{\left(+\right)})-P_{\mathrm{Excellent}}(w_j), \Delta P_{\mathrm{Excellent}}^{(-)}=P_{\mathrm{Excellent}}({\tilde{w}}_j^{\left(-\right)})-P_{\mathrm{Excellent}}(w_j)$$

(36)

The maximum absolute change is used to construct the tornado plot:

$$L_j =\max (\left|\Delta P_{\mathrm{Excellent}}^{(+)}\right|,\left|\Delta P_{\mathrm{Excellent}}^{(-)}\right|)$$

(37)

The local sensitivity coefficient can be approximated using the central difference method:

$${\left.S_j={\displaystyle \frac{\partial P_1}{\partial w_j}}\right|}_{w_j}\approx {\displaystyle \frac{P_{\mathrm{Excellent}}({\tilde{w}}^{(+)})-P_{\mathrm{Excellent}}({\tilde{w}}^{(-)})}{2\rho w_j}}$$

(38)

This coefficient can be positive or negative, reflecting the direction and magnitude of the response of P_Excellent to changes in w_j. In addition, whether the evaluation grade changes under different perturbation conditions should be examined to assess the stability of the model output.

Taking P_Excellent as the target output, the OAT sensitivity analysis results for ρ = 10% were calculated. The maximum impact and local sensitivity coefficient of each weight are presented in

Table 16. Results of the OAT-based weight sensitivity analysis. (ρ = 10%).

Weight	Initial Weight	Lj	$\Delta {\mathit{P}}_{\mathbf{E}\mathbf{x}\mathbf{c}\mathbf{e}\mathbf{l}\mathbf{l}\mathbf{e}\mathbf{n}\mathbf{t}}^{(+)}$	$\Delta {\mathit{P}}_{\mathbf{E}\mathbf{x}\mathbf{c}\mathbf{e}\mathbf{l}\mathbf{l}\mathbf{e}\mathbf{n}\mathbf{t}}^{(-)}$	Sj *
w3	0.1343	0.0087	−0.0085	0.0087	−0.6410
w4	0.1193	0.0077	−0.0076	0.0077	−0.6413
w5	0.1068	0.0039	0.0038	−0.0039	0.3589
w1	0.2548	0.0037	0.0035	−0.0037	0.1398
w9	0.0762	0.0028	0.0027	−0.0028	0.3589
w7	0.0948	0.0027	0.0027	−0.0027	0.2840
w6	0.0473	0.0017	0.0017	−0.0017	0.3540
w2	0.1543	0.0014	0.0013	−0.0014	0.0862
w8	0.0122	0.0004	0.0004	−0.0004	0.3579

* The sign and magnitude of S_j indicate the direction and strength of the local sensitivity.

. The results are sorted in descending order according to the maximum impact L_j, and the weight numbers correspond to the order of the weights in the vector.

As shown in Figure 5, the two weights with the greatest influence on P_Excellent are w₃ and w₄. However, the maximum change in P_Excellent caused by either of these weights is only 0.0087. Moreover, none of the ± 10% perturbations applied to any single weight caused a change in the evaluation grade, indicating that local variations in the weights do not alter the final evaluation conclusion.

4.2.2. AHP Weight Perturbation Analysis Based on the OAT Method

Since the AHP weights are derived from expert judgment, fluctuations in expert scoring or judgment consistency may affect the subjective weight vector and further propagate to the final comprehensive weights. To examine the influence of subjective weighting uncertainty, an AHP weight perturbation analysis was performed. Different from the OAT analysis in Section 4.2.1, which directly perturbs the final comprehensive weights, this analysis perturbs the AHP weight vector before combined weighting. The EWM and CRITIC weights were kept unchanged, and the comprehensive weights were then recalculated using the geometric mean method.

Following the perturbation and normalization procedure described in Section 4.2.1, each AHP weight was perturbed by ±5% and ±10%, respectively. After each perturbation, the AHP weight vector was renormalized to satisfy the unit-sum constraint. The recalculated comprehensive weights were then substituted into the fuzzy comprehensive evaluation model, and the change in P_Excellent was used to evaluate the influence of AHP weight uncertainty on the final matching result.

The sensitivity analysis results under AHP weight perturbations are shown in

Table 17. Sensitivity analysis results under AHP weight perturbations.

ρ	AHP Weight No.	Initial Weight	$\Delta {\mathit{P}}_{\mathbf{E}\mathbf{x}\mathbf{c}\mathbf{e}\mathbf{l}\mathbf{l}\mathbf{e}\mathbf{n}\mathbf{t}}^{\mathbf{A}\mathbf{H}\mathbf{P}(+)}$	$\Delta {\mathit{P}}_{\mathbf{E}\mathbf{x}\mathbf{c}\mathbf{e}\mathbf{l}\mathbf{l}\mathbf{e}\mathbf{n}\mathbf{t}}^{\mathbf{A}\mathbf{H}\mathbf{P}(-)}$	${\mathit{L}}_{\mathit{j}}^{\mathbf{A}\mathbf{H}\mathbf{P}}$	Evaluation Grade
5%	1	0.3107	0.0006	−0.0006	0.0006	Excellent
	2	0.2064	0.0002	−0.0002	0.0002	Excellent
	3	0.0654	−0.0014	0.0015	0.0015	Excellent
	4	0.0406	−0.0013	0.0013	0.0013	Excellent
	5	0.0530	0.0006	−0.0007	0.0007	Excellent
	6	0.1446	0.0003	−0.0003	0.0003	Excellent
	7	0.0883	0.0004	−0.0005	0.0005	Excellent
	8	0.0246	0.0001	−0.0001	0.0001	Excellent
	9	0.0664	0.0004	−0.0005	0.0005	Excellent
10%	1	0.3107	0.0011	−0.0012	0.0012	Excellent
	2	0.2064	0.0004	−0.0005	0.0005	Excellent
	3	0.0654	−0.0028	0.0030	0.0030	Excellent
	4	0.0406	−0.0025	0.0026	0.0026	Excellent
	5	0.0530	0.0012	−0.0013	0.0013	Excellent
	6	0.1446	0.0005	−0.0006	0.0006	Excellent
	7	0.0883	0.0009	−0.0009	0.0009	Excellent
	8	0.0246	0.0001	−0.0002	0.0002	Excellent
	9	0.0664	0.0009	−0.0009	0.0009	Excellent

. As shown in the table, the maximum change in P_Excellent caused by the ±5% AHP weight perturbation was 0.0015, and the maximum change caused by the ±10% perturbation was 0.0030. In both cases, the final evaluation grade remained “Excellent”, indicating that moderate fluctuations in the AHP subjective weights did not change the final matching conclusion. Compared with the direct perturbation of the comprehensive weights, the AHP perturbation analysis reflects the propagation effect of subjective judgment uncertainty through the combined weighting process. The small variation in the final evaluation result indicates that the AHP-EWM-CRITIC combined weighting scheme can effectively reduce the influence of subjective weight uncertainty on the final fuzzy comprehensive evaluation result.

4.2.3. Global Perturbation and Probabilistic Analysis Based on Dirichlet Sampling

In practical applications, uncertainty in weight values often exists. Dirichlet sampling [43] is used to simulate this uncertainty, thereby evaluating the probability distribution and robustness of the model output.

Samples are generated from a Dirichlet distribution using the initial weight vector w₀ and a concentration parameter κ > 0. A larger value of κ indicates that the sampled weights are more concentrated around w₀:

$$w~\mathrm{Dirichlet}(\kappa w_0)$$

(39)

The Dirichlet samples can also be generated using the Gamma distribution as follows:

$$w_j=X_j/{\displaystyle {\sum }_{i=1}^n{X}_i}$$

(40)

For each sampled weight vector, the corresponding membership degree P_Excellent and evaluation grade are calculated. Then, the probability of each evaluation grade, as well as the mean, standard deviation, and quantiles of P_Excellent, are computed. The results are shown in Figure 6 and Figure 7.

The results show that the probability of obtaining an “Excellent” grade is as high as 99.94%, with a mean P_Excellent value of 0.6411 and a standard deviation of 0.0378. The 5th, 25th, 50th, 75th, and 95th percentiles are 0.5770, 0.6157, 0.6425, 0.6671, and 0.7009, respectively. This indicates that, even when weight uncertainty is considered, the model output consistently remains at the “Excellent” grade, and the P_Excellent values are concentrated within the range of 0.5770–0.7009, with only slight variation.

Both OAT and Dirichlet analyses confirm that the proposed framework is insensitive to weight perturbations. The evaluation grade remained “Excellent” under both local and global weight variations, demonstrating the robustness of the evaluation conclusion.

4.3. Comparison with Conventional Matching and Weighting Methods

To further evaluate the effectiveness of the proposed fuzzy-combined weighting model, a comparison with conventional multi-criteria matching methods was conducted. Since the evaluation framework in this study is constructed based on normalized time–frequency indicators, each impact signal was represented as a nine-dimensional normalized feature vector, and the penetration response signal was used as the target feature vector. Three conventional multi-criteria matching methods were selected for comparison, including grey relational analysis (GRA), technique for order preference by similarity to ideal solution (TOPSIS), and VIKOR, which is commonly used for multi-criteria optimization and compromise ranking [44,45,46]. GRA evaluates the relational closeness between the evaluated signal and the target signal, TOPSIS measures the relative closeness to the ideal solution, and VIKOR provides a compromise ranking based on the overall deviation and the maximum individual deviation from the target feature vector.

In the implementation of TOPSIS and VIKOR, the penetration response feature vector was used as the positive ideal or target vector. Because all indicators had been normalized into benefit-type matching scores, the zero vector was used as the negative ideal vector. The comparison results are summarized in

Table 18. Comparison between conventional multi-criteria matching methods and the proposed fuzzy comprehensive evaluation method.

Test No.	GRA	TOPSIS	VIKOR Q	Proposed PExcellent	Grade
Test 1	0.5863	0.6524	1.0000	0.6411	Excellent
Test 2	0.6809	0.7088	0.7468	0.7818	Excellent
Test 3	0.8182	0.8249	0.0000	0.7337	Excellent

Note: For GRA, TOPSIS, and P_Excellent, larger values indicate better matching; for VIKOR Q, smaller values indicate better matching.

.

As shown in Table 18, the conventional multi-criteria matching methods provide quantitative comparison results based on the normalized feature vectors. GRA, TOPSIS, and VIKOR all identify Test 3 as the closest to the penetration response signal from the perspective of feature-vector closeness. Specifically, Test 3 obtained the highest GRA value of 0.8182 and the highest TOPSIS value of 0.8249, while its VIKOR Q value was 0.0000, indicating the best compromise ranking among the three tests. This suggests that Test 3 has the smallest overall deviation from the target feature vector according to these conventional methods.

However, conventional multi-criteria matching methods usually provide only a single closeness or ranking score. They cannot directly describe the grade-level uncertainty of the matching result. In contrast, the proposed fuzzy comprehensive evaluation method integrates multidimensional indicators, AHP-EWM-CRITIC combined weights, and fuzzy membership classification. Therefore, it provides not only P_Excellent, but also the membership degrees corresponding to the grades “Excellent,” “Satisfactory,” and “Unsatisfactory.” In the proposed method, Test 2 obtained the highest P_Excellent value of 0.7818, followed by Test 3 and Test 1. This indicates that the proposed method evaluates the comprehensive matching quality from a fuzzy multi-criteria decision-making perspective, rather than only measuring the overall closeness between feature vectors.

The difference between the conventional methods and the proposed method indicates that feature-vector closeness and fuzzy grade membership describe different aspects of signal matching. Conventional methods are useful for identifying the signal closest to the target feature vector, whereas the proposed method further evaluates whether the weighted time-domain, frequency-domain, and signal quality indicators jointly satisfy the predefined matching grade. Therefore, the proposed method provides a more interpretable evaluation result for impact signal matching under experimental scatter.

To further compare the influence of different weighting schemes, the fuzzy comprehensive evaluation results obtained using equal weighting, AHP, EWM, CRITIC, and the proposed AHP-EWM-CRITIC combined weighting method are shown in

Table 19. Comparison of fuzzy comprehensive evaluation results under different weighting schemes.

Method	Test 1 PExcellent	Test 2 PExcellent	Test 3 PExcellent	Penetration Response PExcellent
Equal-weigh FCE	0.7142	0.8203	0.7107	0.6667
AHP-FCE	0.7623	0.8684	0.8396	0.8341
EWM-FCE	0.6608	0.8014	0.7595	0.8258
CRITIC-FCE	0.5804	0.7110	0.6314	0.6820
Proposed FCE	0.6411	0.7818	0.7337	0.7862

. The penetration response signal was also evaluated using the same membership functions to provide a reference grade-membership benchmark. It should be noted that P_Excellent of the penetration response signal is not a waveform similarity value equal to 1, but the membership degree of the reference signal to the “Excellent” grade under the same fuzzy evaluation criteria.

The results show that different weighting schemes lead to different P_Excellent distributions. Equal-weight FCE does not consider the relative importance of indicators, whereas AHP-FCE is more strongly influenced by expert judgment. EWM-FCE and CRITIC-FCE introduce objective information from data dispersion and indicator conflict, but each single objective method emphasizes only one aspect of the data structure. In contrast, the proposed FCE combines expert judgment, data dispersion, and inter-indicator conflict through AHP-EWM-CRITIC combined weighting.

For the proposed FCE, the P_Excellent value of Test 2 is 0.7818, which is very close to that of the penetration response signal, 0.7862, with an absolute difference of only 0.0044. This indicates that Test 2 has the closest grade-membership behavior to the penetration response signal under the proposed comprehensive evaluation framework. Compared with conventional single-score matching methods, the proposed method provides a more interpretable evaluation result because it simultaneously gives the matching grade, the grade membership degrees, and the influence of multidimensional weighted indicators.

Overall, the comparison results show that conventional multi-criteria matching methods are useful for describing the overall closeness between the impact signal feature vector and the penetration response feature vector. However, they provide only single-score results and cannot directly reflect the contribution of different indicator groups or the uncertainty of grade classification. The proposed fuzzy-combined weighting model provides a more comprehensive evaluation by combining feature-level comparison, indicator weighting, and fuzzy grade membership. Therefore, it is more suitable for impact signal matching evaluation under experimental scatter, where multiple signal characteristics jointly influence the final matching conclusion.

4.4. Scalability and Limitations of the Proposed Framework

The proposed fuzzy comprehensive evaluation framework was established and validated under the current medium-velocity penetration signal matching scenario. The framework integrates multidimensional time–frequency indicators, combined weighting, and fuzzy comprehensive evaluation, and the results demonstrate its effectiveness and robustness for the present test conditions. However, several limitations should be noted when the framework is extended to other impact scenarios or complex dynamic signal evaluation tasks.

First, the feature representation used in this study focuses on physically interpretable and computationally stable indicators. The selected time-domain, frequency-domain, and signal quality indicators can characterize the main response intensity, duration, spectral distribution, and stability of impact signals. However, these indicators may not fully capture localized energy–frequency variations during short-duration transient stages. Advanced joint time–frequency methods, such as short-time Fourier transform, wavelet transform, or Hilbert-Huang transform, can provide more detailed descriptions of local time-varying spectral characteristics. Therefore, these advanced features can be incorporated into the proposed framework in future work by defining additional time–frequency indicators, followed by normalization, combined weighting, and fuzzy comprehensive evaluation.

Second, the membership functions used in this study were established for the current medium-velocity penetration signal matching scenario. Their thresholds and shape parameters were determined based on the normalized indicator range, the measured penetration response signal, the distribution of the experimental samples, and expert experience. Therefore, these parameters should not be regarded as universal constants. When the framework is applied to other impact scenarios, the membership function parameters may be affected by material properties, projectile geometry, target configuration, impact velocity, sensor installation, and data acquisition conditions. For example, changes in target material or projectile structure may alter the amplitude, pulse width, dominant frequency, and time–frequency entropy of the response signal, thereby changing the appropriate grade boundaries. Therefore, representative data from the new application scenario should be used to redefine the indicator ranges and membership function parameters.

Third, the objective weights obtained using EWM and CRITIC may be affected by the quality of the experimental data. Both methods rely on the dispersion characteristics of the indicator data, and CRITIC further considers the correlation and conflict among indicators. Therefore, abnormal fluctuations caused by signal-to-noise ratio degradation, high-frequency electrical noise, sensor drift, or unstable data acquisition conditions may increase the apparent dispersion of noise-related indicators, such as SNR, noise power, and time–frequency entropy. This may lead to an overestimation of the objective weights associated with signal quality and stability indicators, thereby introducing bias into the comprehensive evaluation. In the present study, the OAT perturbation, AHP weight perturbation, and Dirichlet sampling analyses show that the final evaluation grade remains unchanged under moderate weight perturbations. However, when severe noise degradation or sensor drift occurs, signal preprocessing, denoising, sensor calibration, and abnormal-sample screening should be performed before objective weighting. In addition, the objective weights should be recalculated using representative data from the updated experimental condition to improve the reliability of the comprehensive evaluation.

These limitations do not affect the general structure of the proposed framework, because the indicator system, combined weighting strategy, and fuzzy comprehensive evaluation procedure can be retained, while the feature indicators, indicator ranges, and membership thresholds can be adjusted according to different engineering applications. In future work, the scalability of the proposed framework will be further validated under a wider range of impact conditions, including different target materials, projectile geometries, multi-layer target configurations, and impact velocities.

5. Conclusions

To address the difficulties in multiple-impact signal evaluation, including indicator diversity, feature complexity, and the influence of subjective judgment, this study proposed a fuzzy comprehensive evaluation framework based on time–frequency features and AHP-EWM-CRITIC combined weighting. Traditional evaluation methods often have difficulty balancing subjective and objective information, which may lead to unreasonable weight assignment and insufficient robustness of the evaluation results. The proposed framework was designed to quantitatively assess the matching degree between multiple-impact signals and the penetration response signal. The main conclusions are summarized as follows:

• A multidimensional time–frequency feature indicator system was established, covering time-domain features, frequency-domain features, and signal quality and stability. This system consists of three primary indicators and nine secondary indicators, enabling the comprehensive quantification of the intensity, duration, frequency distribution, and noise-related characteristics of impact signals. It effectively addresses the limitations of traditional single-indicator evaluation methods, which cannot fully characterize the complex dynamic properties of impact signals;
• A fuzzy comprehensive evaluation model integrated with AHP-EWM-CRITIC combined weighting was proposed. In this model, expert experience is incorporated through the analytic hierarchy process (AHP), objective information is extracted from the data using the entropy weight method (EWM), and correlations and conflicts among indicators are considered using the CRITIC method. The geometric mean method is then used to integrate the three types of weights and obtain comprehensive weights. The weight consistency tests show that the Spearman’s rank correlation coefficients between the comprehensive weights and the AHP and CRITIC weights are 0.48 and 0.63, respectively, indicating that the comprehensive weights maintain a certain degree of ranking consistency with AHP and CRITIC. The Euclidean distance analysis further shows that the average distance between the comprehensive weight vector and the three original weight vectors is 0.17, which is smaller than the average distance among the three original weight vectors. These results demonstrate that the proposed method achieves an effective balance between subjective and objective information and improves the rationality of weight assignment.
• The validity and robustness of the proposed framework were experimentally verified. Based on case studies using three sets of multiple-impact signals and the penetration response signal, the results show that the matching degrees of all three test groups reached the “Excellent” grade, with “Excellent” membership degrees of 0.6411, 0.7818, and 0.7337, respectively. Among them, Test 2 achieved the best matching performance. Further robustness analysis showed that the maximum change in the “Excellent” membership degree caused by a ±10% perturbation in a single indicator weight was only 0.0087, and no change in the evaluation grade occurred. The Dirichlet-based global sampling analysis indicated that, under reasonable weight combinations, the probability of obtaining an “Excellent” evaluation result was as high as 99.94%. These results demonstrate that the proposed framework is insensitive to weight perturbations and that the evaluation conclusions are stable and reliable under the current medium-velocity penetration signal matching scenario.

Compared with previous single-indicator evaluation methods and conventional multi-criteria matching approaches, the proposed fuzzy comprehensive evaluation framework provides a more comprehensive and interpretable method for quantifying the matching degree between multiple-impact signals and penetration response signals. By integrating time–frequency indicators, AHP-EWM-CRITIC combined weighting, fuzzy membership classification, and robustness verification, the framework enables a more objective evaluation of the simulation performance of multiple-impact test setups. However, the method still has some limitations. The evaluation process is more complex than traditional single-score approaches because it requires indicator construction, combined weight calculation, and membership function setting. In addition, the evaluation results may be affected by the selected indicators, membership function parameters, and data quality. Since the present validation was conducted under the current medium-velocity penetration signal matching scenario, the feature indicators, indicator ranges, and membership function parameters should be redefined when the framework is applied to different impact conditions. Future work will further validate the scalability of the proposed framework under broader impact conditions, including different target materials, projectile geometries, multi-layer target configurations, impact velocities, and noise levels. After scenario-specific adjustment, the proposed framework may also provide a methodological reference for complex dynamic signal evaluation in structural monitoring, system validation, and other impact-like operating conditions.