A gear fault diagnosis method based on maximum CGGI deconvolution filtering

Abstract

Under complex conditions such as varying operating conditions and strong noise interference, fault features in gear transmission systems are highly susceptible to being masked by noise, posing significant challenges to fault diagnosis techniques. To address the limitations of traditional blind deconvolution objective functions (OF), this paper proposes a novel blind deconvolution method based on maximizing the Cycle-embedded Generalized Gini Index (CGGI). The core innovation of this method lies in constructing an OF grounded in a new sparsity measure index, CGGI. By introducing a Generalized Weighted Envelope Spectrum (GWE), it effectively integrates the feature enhancement effects of signals under different transformation parameters. Furthermore, it inherits the superior capability of the Generalized Gini Index in quantifying signal impulsivity and cyclostationarity, thereby significantly improving the robustness of fault feature identification under complex operating conditions. Numerical analysis demonstrates that CGGI more effectively suppresses the influence of interference components compared to traditional indicators. Validations conducted on a public dataset of a two-stage parallel-shaft gearbox and a fault test bench for a high-speed train’s body-suspended transmission system show that the proposed method significantly outperforms traditional methods in terms of the Fault Characteristic Ratio (FCR) across various fault scenarios. This proves the effectiveness and superiority of the proposed method for condition monitoring and fault diagnosis of complex transmission systems in practical engineering applications.

Keywords

fault diagnosis blind deconvolution generalized gini index generalized weighted envelope spectrum gear transmission

1. Introduction

Gear transmission systems, as core components of mechanical equipment, are widely used in various critical equipment to perform the important function of power transmission. Their operational status directly affects the performance, efficiency, and safety of the entire equipment.

However, gears operating under complex conditions for extended periods are highly susceptible to damage such as tooth surface wear, pitting, tooth breakage, and cracks. Factors such as varying operating conditions, changing loads, and wheel-rail interaction interference often cause gear fault characteristics to be masked by other vibration components and noise, posing challenges for fault diagnosis.

Signal processing methods have long been mainstream techniques for vibration component denoising and decomposition (Zheng et al., 2021), and have been extensively applied for efficient extraction of multi-modal information across various fields. These include Empirical Mode Decomposition (EMD) (Huang et al., 2017), Variational Mode Decomposition (VMD) (Dragomiretskiy and Zosso, 2013), and Singular Value Decomposition (SVD) (Zhao et al., 2022), among others. However, these methods often face issues like mode mixing and requirements for prior parameters. To address these limitations, researchers have improved the algorithms by integrating concepts from optimization algorithms and machine learning.

Wang et al. (Wang et al., 2020) proposed an EMD manifold (EMDM) method, which improves upon the original EMD using manifold learning to efficiently retain fault-related transient information while suppressing noise and irrelevant components. Ma et al. (2025) addressed the selection of penalty factors and the number of decomposition modes in VMD by introducing the RIMA algorithm. Liu et al. (2024) achieved motor bearing fault diagnosis using Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN). Bagherzadeh et al. (2022) incorporated wavelet concepts into the EMD method to enhance its noise handling performance, successfully applying it to locate vibration sources within aircraft cabins. Li et al. (2025) tackled the issues of inaccurate mode decomposition and parameter selection in Symplectic Geometry Mode Decomposition (SGMD) by using dispersion entropy as a constraint for mode reconstruction and introducing a dual optimization algorithm for parameter tuning, validated on ship radiated noise type recognition and bearing fault diagnosis data.

Beyond decomposition methods, the core idea of blind deconvolution is to maximize a statistical index (the OF) of a FIR-filtered signal using classical gradient descent algorithms or eigenvector-based algorithms (optimization algorithms) (Hou et al., 2025). The choice of the OF critically determines the extraction results, prompting extensive research into its formulation.

Current research on OF primarily falls into two categories. The first involves designing OF based on time-domain features of signals, such as kurtosis (Kurt) and the Gini index (GI), which characterize the sparsity of signals in the time domain when faults occur. The second category focuses on designing OF using frequency-domain features of signals, by introducing fault characteristics into the square envelope spectrum (SES) of signals including the second-order cyclostationarity indicator (ICS2), among others. Hashim et al. (2023) proposed a blind deconvolution strategy based on spectral kurtosis and applied it to gear fault diagnosis. Wang et al. (2022) addressed the parameter optimization problem in Maximum Cyclostationarity Blind Deconvolution (CYCBD) by estimating the cyclic frequency set using the morphological envelope autocorrelation function and adaptively selecting the filter length via an equal-step search strategy, demonstrating its effectiveness for bearing fault diagnosis through simulation and experiment. Li et al. (2023) proposed to maximize the noise ratio kurtosis product (NRKP) as the objective function and construct the signal envelope cyclic kurtosis (SECK) to estimate the maximum noise ratio kurtosis product deconvolution (MNRKPD) method for fault cycles. Zhao et al. (2025) used the cyclic stable feature energy index (CCEI) as the objective function of blind deconvolution filtering to capture fault features in signals, and established an adaptive filter length determination framework based on this, solving the problem of selecting filter length. Shi et al. (2025) used a Variable-length Genetic Algorithm (VGA) to determine the optimal length and coefficients of the blind filter, combining it with existing common blind deconvolution methods. Experimental results showed the method could effectively recover bearing fault information. Gu et al. (2022) measured the sparsity of signals using the Smoothness Index and the Gini Index, introducing a knee-point driven strategy to extend the Particle Swarm Optimization algorithm for solving multi-objective blind deconvolution problems. The method was validated using railway bearing experimental data, showing excellent robustness.

Although the latest research mentioned above has effectively addressed certain challenges in blind deconvolution filtering, including aspects such as fault period identification and filter parameter selection. Difficulties still remain in fault diagnosis for gear transmission systems. Specifically, during gear degradation, the fault features generated are more susceptible to interference from gear meshing effects. Moreover, the presence of numerous complex interfering signals within gear transmission systems poses additional challenges for establishing an optimal filter that can accurately characterize fault information. In summary, the performance of blind deconvolution is primarily influenced by core issues such as dependency on prior knowledge, signal sparsity, and the OF (Yang et al., 2024).

This paper addresses the aforementioned challenges, with the main contributions as follows.

(1) A novel cycle-embedded sparsity measure is proposed based on the impulsivity and cyclostationarity characteristics of vibration signals under gear faults.

(2) This sparsity measure is used as the OF for blind deconvolution filtering, and its performance under various conditions is investigated.

(3) A generalized envelope spectrum CGGI calculation method is proposed, significantly enhancing signal sparsity.

(4) The superiority of the proposed method is demonstrated through public datasets and a scaled laboratory test rig.

This paper is structured as follows: The Introduction outlines current engineering challenges. The Theoretical Foundation reviews the theoretical background of existing methods. Subsequently, the Proposed Method elaborates on our improvements and methodology, including a performance analysis of the Optimal Filtering (OF) technique using simulated signals. The Experimental Analysis validates the proposed approach through tests on public datasets and a custom universal shaft-gearbox test rig. Finally, the Conclusion summarizes the key findings.

2. Theoretical foundation

2.1. Blind deconvolution theory

The goal of blind deconvolution is to design appropriate filter coefficients based on the known measured signals, so as to reconstruct the desired signal components through convolution operations (Cheng et al., 2018). Thus, the principle of blind deconvolution can be expressed as

s = x * h

(1)

Among them, $s$ represents the desired filtered signal with enhanced fault characteristics; $x$ is the original measured signal; $h$ denotes the filter coefficients.

Expressing the above in matrix form

s = Xh

[\begin{array}{l} s [L] \\ s [L + 1] \\ ⋮ \\ s [N] \end{array}] = [\begin{array}{c} x [L] & x [L - 1] & \dots & x [1] \\ x [L + 1] & x [L] & \dots & x [2] \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x [N] & x [N - 1] & \dots & x [N - L + 1] \end{array}] [\begin{array}{l} h [1] \\ h [2] \\ ⋮ \\ h [L] \end{array}]

(2)

Where

N

represents the length of the measured signal, and

L

denotes the filter length. The matrix form facilitates effective subsequent solving and computation.

The original measured signal $x$ is typically composed of the superposition of a signal component characterizing the fault and interference components (including: random impulse signals, modulation signals, noise, etc.), each having traversed their respective transmission paths to the sensor. It can be expressed as

x = f * g_{f} + n * g_{n}

(3)

Where

f

and

n

represent the fault component and the interference component, respectively;

g_{f}

and

g_{n}

are the transfer path functions of the corresponding signal components. Therefore, to obtain the filtered signal related to the fault component, the measured signal is filtered as follows

s = x * h = (f * g_{f} + n * g_{n}) * h

(4)

The filtered signal $s$ exhibits characteristics similar to the fault component $f$ . This process is referred to as the blind deconvolution filtering procedure because it involves designing the filter inversely based on the filtering result.

Two central issues in blind deconvolution research are finding the optimal solution for the aforementioned filter and determining the method to solve for it. The approach for solving the filter is determined by the OF of the deconvolution. Commonly used methods include the Objective Function Method (OFM), the Eigenvector Algorithm (EVA), and the Spherical Coordinate Transformation method, among others (Gu et al., 2025). Based on the OF proposed subsequently, the EVA method is selected for solving in this context. Detailed computational steps for the EVA can be found in reference (Buzzoni et al., 2018).

2.2. Cycle-embedded sparsity measures

Recently, based on the characteristics of impulsivity and cyclostationarity exhibited by vibration signals generated under rotating machinery faults, Hou et al. (2025) proposed a new type of Cycle-embedded Sparsity Measures (CESMs). CESMs are obtained by calculating the power mean of the sparsity measures corresponding to the vibration signal. The calculation process is as follows:

Step 1: Perform an envelope transform on the signal $x = [x_{1}, x_{2}, . . ., x_{n}]$ to obtain the envelope signal. This is typically achieved using the Hilbert transform or the square transform. Specifically, $H E$ and $S E$ represent the Hilbert spectrum and the square envelope spectrum of the signal $x$ , respectively, where $H (\cdot)$ denotes the Hilbert transform. For convenience in subsequent description, the envelope signal is uniformly denoted as $e = [e_{1}, e_{2}, . . ., e_{n}]$

\begin{array}{l} H E = | x + j H (x) | \\ S E = {| x + j H (x) |}^{2} \end{array}

(5)

Step 2: Segment the envelope signal $e$ into $P$ segments $[e^{(1)}, e^{(2)}, . . ., e^{(P)}]$ of equal length $l$ , $P = f l o o r (n / l)$ . Then, compute the sparsity measure for each segment of the envelope signal, and calculate the power mean of all these sparsity measures to obtain the CESMs

CESM (e) = \sqrt[α]{\sum_{i = 1}^{P} {(SM (e^{(i)}))}^{α} / P}, α \in R & α \neq 0

(6)

Here, $CESM (e)$ represents the CESMs of the envelope signal; $SM (e^{(i)})$ is the sparsity measure of the $i$ th segment signal; $a$ is the parameter for the power mean calculation. When $a = 0$ , the CESMs degenerates to the geometric mean

CESM (e) = \sqrt[P]{\prod_{i = 1}^{P} (SM (e^{(i)}))}, α = 0

(7)

The choice of the sparsity measure within CESMs is flexible and can include metrics such as kurtosis, lp/lq norm ratio, and Gini Index (GI). The literature (Hou et al., 2025) primarily focuses on the detailed study of the pq-mean and the GI, because they satisfy six desirable properties of sparsity measures and are unaffected by signal length and subtraction operations. Furthermore, CESMs possess excellent characteristics for quantifying and extracting impulsive fault signal components and can serve as the OF in various signal processing methods to characterize fault components. Therefore, in The proposed method section, based on CESMs we propose a novel sparsity measure and apply it to the OF of blind deconvolution filtering.

3. The proposed method

3.1. Cycle-embedded generalized Gini Index (CGGI) based on generalized weighted envelope

The selection of the OF has been demonstrated to be one of the critical factors affecting the performance of blind deconvolution filtering methods (Yang et al., 2024). Recently, the Generalized Gini Index (GGI) has been shown to satisfy six desirable properties of a sparsity measure. By adjusting its parameter $β$ , the convexity/concavity of the sparsity quantification curve can be altered, thus enabling different characteristics.

Hou et al. (2022) extended the definition of the GI by reformulating it as the ratio of quasi-arithmetic means based on linear and nonlinear weighting functions. This generalization replaces the linear weights in the standard GI definition with a nonlinear form incorporating a parameter p, resulting in the Generalized Gini Index (GGI)

G G I (e) = 1 - \frac{\sum_{n = 1}^{N} p_{n}^{(3)} e_{n, ↑}}{\sum_{n = 1}^{N} e_{n} / N}, p_{n}^{(3)} = \frac{w_{n}}{\sum_{n = 1}^{N} w_{n}}, w_{n} = {[2 (N - n) + 1]}^{β}

(8)

Where

e

represents the input signal, which can be the original signal or the envelope signal;

e_{n, ↑}

is the

n

th element after sorting

e

in ascending order;

p_{n}^{(3)}

and

w_{n}

are the

n

th nonlinear weighting function and nonlinear weighting coefficient, respectively;

β

is the parameter that controls the nonlinear weighting function.

As mentioned in Theoretical foundation section, the calculation of CESMs uses the envelope signals obtained through the Hilbert transform and the squared transform for sparsity measurement. Although this approach effectively enhances the sparsity of the signal, it is highly sensitive to interference components and noise in the signal. For instance, the squared envelope significantly amplifies noise, while the Hilbert envelope performs poorly for multi-component signals. To address this, a novel GWE is proposed for the computation of CESMs.

First, we extend the squared envelope to a generalized envelope via a generalized power transformation

A^{(q)} = {\begin{cases} \ln (A), & q = 0 \\ {(A)}^{q}, & q \neq 0 \end{cases}

(9)

Where

A^{(q)}

represents the generalized power transformation of the signal’s non-negative amplitude with an exponent of

A

;

q

is the transformation parameter. However, when

q < 0

, the transformation involving negative exponents can amplify noise and low-amplitude signal components present in the original signal, leading to a degradation in signal sparsity. Therefore, when

q < 0

section, this issue is addressed by performing the Hilbert transform on the signal and incorporating a regularization term to mitigate the adverse effects on signal sparsity and robustness. The generalized envelope spectrum

G E_{q} (x (n))

of the signal

x (n)

is obtained as

G E_{q} (x (n)) = {\begin{cases} \ln (| x (n) |), & q = 0 \\ \begin{array}{l} {(| x (n) |)}^{q}, \\ {(| H i l b e r t (x (n)) | + ε)}^{q}, \end{array} & \begin{array}{l} q > 0 \\ q < 0 \end{array} \end{cases}

(10)

Where

ε

is the added regularization term, which is typically related to the noise level of the signal. In the subsequent analysis of this paper, the value of

ε

is set to

ε = 10^{- 8}

The Generalized Weighted Envelope spectrum (GWE) is proposed to integrate the feature enhancement effects of envelope spectra under different transformation parameters $q$ . This approach aims to improve the capability of capturing weak features while enhancing robustness against interference components and noise. The generalized weighted envelope spectrum $G W E (x)$ is formulated as follows

G W E_{q_{1}, q_{2}} (x) = \sum_{q = q_{1}}^{q_{2}} μ_{q} G E_{q} (x)

(11)

Where

G W E_{q_{1}, q_{2}} (x)

represents the calculation of the Generalized Weighted Envelope Spectrum over the range of transformation parameters

[q_{1}, q_{2}]

;

μ_{q}

is the corresponding weight coefficient for

q

, computed via equation (12)

{\begin{cases} μ_{q} = \frac{G I (G E_{q} (x))}{\sum_{q = q_{1}}^{q_{2}} G I (G E_{q} (x))} \\ G I (x) = 1 - \frac{\sum_{n = 1}^{N} \frac{2 N - 2 n + 1}{N^{2}} x_{n, ↑}}{\sum_{n = 1}^{N} x_{n} / N} \end{cases}

(12)

In this context, to align with the subsequently proposed CGGI, the calculation of the weight coefficient

μ_{q}

incorporates the GI. Here, GI represents the proportion of the GI of the transformed signal corresponding to a specific

q

relative to the sum of GIs across the entire parameter range. Furthermore, the numerical value of GI effectively quantifies the feature enhancement effect of different

G W E_{q} (x)

. Consequently, the performance of the generalized envelope spectrum is effectively improved through this weighted summation.

Based on $G W E (x)$ and CESMs, a novel sparsity measure termed CGGI is proposed. The process begins with calculating the GGI of the GWE of the measured signal $x$

G G I (G W E (x)) = 1 - \frac{\sum_{n = 1}^{N} p_{n}^{(3)} G W E {(x)}_{n, ↑}}{\sum_{n = 1}^{N} G W E {(x)}_{n} / N}, p_{n}^{(3)} = \frac{w_{n}}{\sum_{n = 1}^{N} w_{n}}, w_{n} = {[2 (N - n) + 1]}^{β}

(13)

Then, by reformulating the GGI into the form of a CESM, the CGGI can be obtained

C G G I (x (n)) = {\begin{cases} \sqrt[α]{\sum_{i = 1}^{P} {(G G I (G W E {(x)}^{(i)}))}^{α} / P}, α \in R & α \neq 0 \\ \sqrt[P]{\prod_{i = 1}^{P} (G G I ({G W E (x)}^{(i)}))}, α = 0 \end{cases}

(14)

Equation (10) reveals that the proposed CGGI possesses the capability to measure both the impulsivity and cyclostationarity of a signal. Furthermore, the introduction of the GWE enhances the robustness of CGGI for sparsity measurement when dealing with complex signals and reduces its susceptibility to unknown frequency band components. This highlights the potential of CGGI to serve as an OF for blind deconvolution. A detailed theoretical analysis of CGGI’s specific properties will be provided in The proposed method section.

3.2. Maximum CGGI based on generalized weighted envelope blind maximum deconvolution

In this section, the proposed CGGI is employed as the OF for blind deconvolution filtering. First, according to Equation (2) and Equation (10), the generalized envelope spectrum $G E_{q} (x (n))$ is reformulated in matrix form

{GE}_{q} (s) = {\begin{cases} \ln (| s |) = \ln (| Xh |), & q = 0 \\ \begin{array}{l} {(| s |)}^{q} = {(| Xh |)}^{q}, \\ {(| H i l b e r t (s) | + ε)}^{q} = {(| H i l b e r t (Xh) | + ε)}^{q}, \end{array} & \begin{array}{l} q > 0 \\ q < 0 \end{array} \end{cases}

(15)

Where

{GE}_{q} (s) = [G E_{q} (s (L)), G E_{q} (s (L + 1)), . . ., G E_{q} (s (N))]

. Therefore, the matrix form of

G W E (s)

can be derived from the calculation process of the Generalized Weighted Envelope Spectrum according to Equations (11) and (12)

{GWE}_{q_{1}, q_{2}} (s) = μ_{q} {GE}_{q} {(s)}^{T}

(16)

Herein, ${GWE}_{q_{1}, q_{2}} (s)$ represents the matrix form of the filtered signal $s$ corresponding to $G W E (s)$ , $μ_{q} = [μ_{q_{1}}, μ_{q_{1} + 1}, μ_{q_{1} + 2}, . . ., μ_{q_{2}}]$ , ${GE}_{q} (s) = [G E_{q_{1}} (s), G E_{q_{1} + 1} (s), G E_{q_{1} + 2} (s), . . ., G E_{q_{2}} (s)]$ .

Taking $α \neq 0$ as an example, the CGGI for the filtered signal $s$ can be expressed as

C G G I (s) = \sqrt[α]{\sum_{i = 1}^{P} {(G G I (G W E {(x)}^{(i)}))}^{α} / P} = \sqrt[α]{\sum_{i = 1}^{P} {(1 - \frac{\sum_{n = 1}^{N} p_{n}^{(3)} G W E {(s)}_{n, ↑}^{(i)}}{\sum_{n = 1}^{N} G W E {(s)}_{n}^{(i)} / N})}^{α} / P}

(17)

Where

p_{n}^{(3)}

and

G W E {(s)}_{n, ↑}

represent the ascending-ordered sequences of the nonlinear weighting function and the generalized weighted envelope spectrum, respectively, which can be obtained via equation (13). The above expression is simplified and written in matrix form as follows

C G G I (s) = \frac{{‖ \frac{\sum_{n = 1}^{N - L} ϖ (n) G W E {(n)}_{↑}}{\sum_{n = 1}^{N - L} G W E {(n)}_{↑}} ‖}_{α}}{P^{1 / α}} = \frac{{‖ \frac{G^{H} diag (\frac{P}{G}) G}{G^{H} diag (\frac{1}{G}) G} ‖}_{α}}{P^{1 / α}}

(18)

Where

P = [p_{1}^{(3)}, p_{2}^{(3)}, . . ., p_{N - L}^{(3)}] = [\frac{{[2 (N - sort (1)) + 1]}^{β}}{\sum_{n = 1}^{N} {[2 (N - sort (n)) + 1]}^{β}}, . . ., \frac{{[2 (N - sort (N - L)) + 1]}^{β}}{\sum_{n = 1}^{N} {[2 (N - sort (N - L)) + 1]}^{β}}]

G = {GWE}_{q_{1}, q_{2}} (s)

can be obtained by Equations (15)–(16);

diag

represent diagonal matrix;

{‖ \cdot ‖}_{α}

is defined as the

α

-norm of the

P

-segment

G W E (s)

signal,

{‖ G G I (G W E {(x)}^{(i)}) ‖}_{α} = \sqrt[α]{\sum_{i = 1}^{P} {(G G I (G W E {(x)}^{(i)}))}^{α}}

By employing the eigenvector-based algorithm and simplifying equation (18), the CGGI of $s$ can be transformed into

C G G I (s) = \frac{{G E}_{q} {(X h)}^{H} diag ({G E}_{q} (s)) diag (\frac{P}{G}) diag ({G E}_{q} (s)) {G E}_{q} (X h)}{{G E}_{q} {(X h)}^{H} diag ({G E}_{q} (s)) diag (\frac{1}{G}) diag ({G E}_{q} (s)) {G E}_{q} (X h)} = \frac{h^{H} R_{X W_{1} X} h}{h^{H} R_{X W_{2} X} h}

(19)

Where

R_{X W_{1} X}

and

R_{X W_{2} X}

represent the weighted correlation matrices of the numerator and denominator parts, respectively. Therefore, the solution for the filter coefficients can be obtained by solving the eigenvector equation. Similarly, when

α = 0

, the process for solving the filter coefficients is analogous, yielding

C G G I (s) = \exp (\frac{1}{n} I^{T} \ln (\frac{\sum_{n = 1}^{N - L} ϖ (n) G W E {(n)}_{↑}}{\sum_{n = 1}^{N - L} G W E {(n)}_{↑}})) = \exp (\frac{1}{n} I^{T} \ln (\frac{G^{H} diag (\frac{P}{G}) G}{G^{H} diag (\frac{1}{G}) G})), α = 0

(20)

Where

I

represents a unit row vector. By applying the operational properties of exponential and logarithmic functions, a similar simplification is performed, and the solution is obtained using the eigenvector method, yielding a result analogous to equation (19). A detailed derivation is omitted here for brevity.

Therefore, following the above process of maximizing the CGGI for the filtered signal $s$ , the maximum CGGI deconvolution is constructed through an iterative procedure. The corresponding flowchart is shown in Figure 1.

Figure 1.

Flowchart of the deconvolution filtering method based on maximizing CGGI.

Step 1: Initialize the filter coefficients $h_{0}$ and compute the initial filtered signal $s_{0}$ according to equation (2). Various methods can be used to initialize the filter coefficients, such as zero initialization, random initialization, or Linear Predictive Coding (LPC) initialization. Compared to other initialization approaches, LPC models the physical significance of signal characteristics, offers pre-whitening properties, and enables fast convergence speed. This paper employs LPC initialization to compute $h_{0}$ .

Step 2: Based on Equations (15) to (19), compute the filter coefficients $s_{0}$ that maximize the CGGI value of the initial filtered signal $h_{1}$ , and obtain the corresponding filtered signal $s_{1}$

s_{1} = {X h}_{1}

(21)

Step 3: Enter the iterative process of blind deconvolution: repeat the procedure in Step 2 and determine whether the optimal deconvolution filtering result is achieved. The iteration stops and outputs the current optimal filter coefficients $h_{m}$ and filtered signal $s_{m}$ when the value of CGGI and the eigenvalues solved in equation (19) both meet the convergence tolerance criterion, or when the maximum number of iterations is reached. Based on the analysis results from subsequent simulations and experiments, the convergence tolerance criterion $δ$ in this paper is selected as $δ = 10^{- 5}$ , and the maximum number of iterations is set to $Q = 100$ .

3.3. Numerical analysis based on the proposed method

3.3.1. Performance analysis of the objective function

The performance of the OF significantly influences the results of blind deconvolution. Therefore, this section conducts a detailed numerical analysis of the proposed CGGI’s relevant performance using simulated signals. According to Equations (8) to (14), CGGI extends both the Gini index and the envelope spectrum of the signal into a generalized scope by introducing parameters. Thus, appropriate parameter selection becomes crucial for the performance of CGGI. Compared to the GI, the parameters of CGGI are mainly divided into two parts: those affecting its measurement of signal cyclostationarity (including the segment quantity $P$ and the power mean parameter $α$ ), and those affecting its measurement of signal sparsity (including the envelope parameter $q$ and the nonlinear weighting parameter $β$ ). Among these, references (Hou et al., 2022, 2025) provide a detailed analysis of parameters $P$ , $α$ , and $β$ ; this paper will primarily focus on analyzing the impact of the envelope parameter $q$ on CGGI.

First, a simulation analysis was performed on signals measured from a gearbox housing. As shown in Figure 2, the signal components constituting the gearbox housing vibration signals mainly include deterministic components (signal components generated by gear meshing), fault components (including distributed gear faults, localized gear faults, and bearing faults), and interference components (such as random impulses, modulation interference, and noise) (Schmidt and Gryllias, 2021).

Figure 2.

Gearbox simulation signal and its components: (a) measured signal $x$ ; (b) gear meshing component $x_{m}$ ; (c) distributed gear fault component $x_{d}$ ; (d) localized gear fault component $x_{l}$ ; (e) bearing fault component $x_{b}$ ; (f) random impulse component $x_{i}$ ; (g) modulation interference component $x_{M I}$ ; (h) noise component $x_{n}$ .

The envelope parameter $q$ primarily influences the performance of the GWE by defining different parameter selection ranges, thereby affecting the CGGI’s judgment of signal sparsity. To investigate this, a generalized envelope transformation under different envelope parameters $q$ is applied to the gearbox simulation signal. Figure 3(a) displays the generalized envelope spectrogram within the range of $q$ from −5 to 5. The following observations can be made: When $q < 0$ , compared to the original envelope spectrum, the amplitude corresponding to the fault characteristic frequency is significantly enhanced, but the amplitudes of some interference components are also increased. When $q > 0$ , the generalized envelope spectrum effectively enhances the fault characteristics while reducing interference components; however, some fault-related components are overlooked. Therefore, selecting an appropriate generalized envelope spectrum can effectively extract and enhance all fault components while simultaneously suppressing interference.

Figure 3.

Analysis of the generalized envelope spectrum under different envelope parameters $q$ ; (a) Generalized envelope spectrum and its magnified view; (b) Envelope parameter selection; (c) GWE.

To determine an appropriate range for the envelope parameter, the GI is used for the initial selection of the range of envelope parameter $q$ . As shown in Figure 3(b), when the value of $q$ is either too large or too small, the GI value changes only marginally, indicating that only a limited portion of effective generalized envelope spectra are covered. Therefore, a suitable threshold $ε$ is selected based on the variation of different envelope parameters to define this range. Based on the analysis of simulation and experimental results, $q \in [- 2, 4]$ is chosen in this study. According to the computational procedure of the Generalized Weighted Envelope Spectrum mentioned in The proposed method section, $G W E (x)$ is obtained as illustrated in Figure 3(c). In the spectrum, $G W E (x)$ effectively preserves all key components in the gearbox vibration signal while significantly mitigating the effects of interference and noise.

Furthermore, the performance of the OF significantly influences the results of blind deconvolution. First, as shown in Figure 4(a), a comparison is made by calculating the normalized amplitudes of the proposed CGGI and several commonly used OFs in blind deconvolution (Chen et al., 2023) under different signal components. It can be observed that most OFs (e.g., Kurtosis (Kurt), L2/L1 norm (L2/L1), Hoyer Index (HI), GI, GGI, Negentropy (NE), D-norm (DN)) are highly sensitive to the deterministic components generated by gear meshing in the gearbox. This is because the gear meshing components exhibit the highest amplitude and dominate the vibration signals. However, these OFs show comparable sensitivity to both fault components and interference components, thus they cannot effectively enhance the fault components by maximizing the OF during the blind deconvolution process. In contrast, the proposed CGGI effectively reduces sensitivity to gear meshing components and other interference components, enabling effective enhancement of fault features in the signal through the blind deconvolution process.

Figure 4.

Analysis of OFs for different types of signal components: (a) Comparison of different OFs; (b) Comparison of different envelope parameter ranges.

Furthermore, we compared the CGGI values calculated under different envelope parameter ranges, dividing the envelope parameter range into three parts: the all-negative range $(q_{2}, q_{3}) \to q_{2} = - 10 : 0.1 : - 4; q_{3} = - 6 : 0.1 : 0$ , the all-positive range $(q_{4}, q_{5}) \to q_{4} = 0 : 0.1 : 6; q_{5} = 4 : 0.1 : 10$ , and their overlapping range $(q_{6}, q_{7}) \to q_{6} = - 5 : 0.1 : 1; q_{7} = - 3 : 0.1 : 3$ . The results are shown in Figure 4(b). The CGGI calculated using $q_{2}, q_{3}$ effectively reduced sensitivity to interference components but struggled to distinguish between fault components and deterministic components. The CGGI calculated using $q_{4}, q_{5}$ was significantly affected by both interference components and deterministic components. In contrast, the calculation results based on $q_{1}, q_{6}, q_{7}$ demonstrated a better ability to distinguish and extract fault components. This conclusion also validates the correctness of the envelope parameter range selection discussed earlier. Additionally, it was observed that within the overlapping range, the calculated CGGI values from different sub-ranges exhibited varying sensitivities to different fault components. In subsequent research, the envelope parameter range can be adjusted to tailor the blind deconvolution filtering process for faults in different components.

As mentioned in The proposed method section, the input signal has an impact on the calculation of sparsity measures. Therefore, it is necessary to compare the CGGI values computed from different signals. As shown in Figure 5, the CGGI is calculated separately for the original signal, Hilbert envelope spectrum (HE), squared envelope spectrum (SE), and GWE under different noise levels. It can be observed that the CGGI values obtained from these four different types of signals exhibit a monotonic trend as the noise level changes. However, except for GWE, the CGGI values derived from the other three signals show partially pseudo-monotonic trends, as indicated within the red dashed box in Figure 5. Meanwhile, the CGGI value calculated via GWE demonstrates the least noticeable change throughout the variation in noise level. This proves that the proposed CGGI, computed based on GWE, possesses good robustness against noise interference and can serve as an excellent OF for the blind deconvolution process.

Figure 5.

Comparative results of CGGI calculations for different input signals under various noise levels.

3.3.2. Performance analysis of the blind deconvolution process

The setting of the filter length, as one of the critical initialization operations in the blind deconvolution process, significantly influences the filtering results. As shown in Figure 6(a), the time-domain and frequency-domain plots of the blind deconvolution results for the gearbox simulation signal from The proposed method section under different filter lengths are presented. When the filter length L is chosen too small (e.g., $L = 20$ and $L = 100$ ), the filtered signal obtained via blind deconvolution still contains strong noise interference, and the low-amplitude components corresponding to the fault frequencies fail to dominate the filtered signal. The gear meshing frequency $f_{m}$ and the modulation interference frequency $f_{M I}$ significantly impede the extraction of the fault characteristic frequencies. Conversely, when the filter length L is excessively large (e.g., $L \geq 300$ ), severe information loss occurs in the signal after blind deconvolution. An appropriate filter length effectively accentuates the frequency bands corresponding to the fault characteristic frequencies. The frequency bands associated with the distributed gear fault frequency $f_{d}$ , the localized gear fault frequency $f_{l}$ , and the bearing fault frequency $f_{b}$ are clearly discernible in the spectrum.

Figure 6.

Analysis of the impact of filter length on blind deconvolution: (a) time-domain and frequency-domain plots of the filtering results; (b) analysis of computational efficiency and filtering performance.

Therefore, an appropriate value of L is selected by considering both the computational time consumed by the blind deconvolution process and the FCR of the filtered signal. The calculation of FCR is as defined in equation (22)

\begin{array}{l} FCR (f_{x}) = \frac{F (f_{x})}{\sum_{f_{i} \in A} F (f_{i})} \\ A = [f_{x} - Δ f_{x} / 2, f_{x} + Δ f_{x} / 2], Δ f = δ f \end{array}

(22)

Where

f_{x}

represents the fault characteristic frequency to be identified;

Δ f

is the calculation range for the fault characteristic frequency; and

δ

is the range coefficient, selected as

δ = 0.05

in this paper. It can be observed that when the FCR value increases, the amplitude corresponding to the fault characteristic frequency dominates within the calculation range. Conversely, its amplitude remains hidden within the calculation range (Yang et al., 2023, 2024). Therefore, FCR can serve as a performance indicator for evaluating the results of blind deconvolution filtering.

As shown in Figure 6(b), the computational time of the blind deconvolution process increases significantly with larger values of L. Thus, selecting a smaller L helps improve computational efficiency. Meanwhile, for the three different fault types included in the simulation signal, Localized Gear Damage (LGD), Distributed Gear Damage (DGD), and Bearing Damage (BeD). Their FCR values under different L values are calculated to investigate the appropriate filter length selection. The results indicate that the optimal filter lengths for these faults are $L = 130$ , $L = 150$ , and $L = 120$ , respectively. This aligns with the findings from the time-domain and frequency-domain analyses. Notably, by choosing different values of L, the blind deconvolution process can be tailored to emphasize signal components corresponding to specific fault types, thereby enabling effective diagnosis of gearbox equipment under multiple fault conditions. Furthermore, when the fault type is a priori unknown in practical scenarios, the range is typically determined based on the optimal filter lengths derived from previously solved cases. Taking this section as an example, the filter lengths for each fault are $L_{1} = 120, L_{2} = 130, L_{3} = 150$ , respectively. The filter length range is then established by selecting the minimum and maximum values among these, namely $L = [0.9 L_{\min}, 1.1 L_{\max}] = [0.9 L_{1}, 1.1 L_{3}]$ . This selection strategy effectively ensures that the optimal filter lengths corresponding to all faults fall within the specified range.

4. Experimental analysis

4.1. Experiment 1: MCC5-THU gearbox fault diagnosis datasets

First, the performance of the proposed method was validated using vibration data collected from a two-stage parallel-shaft gearbox test rig. The test rig provided data corresponding to different fault levels and fault types, including healthy gears, missing teeth, wear, root cracks, and tooth breakage, as well as operational data under various rotational speeds and loads. The experimental setup, shown in Figure 7, for further details regarding the test rig, please refer to literature (Chen et al., 2024).

Figure 7.

MCC5-THU Gearbox Multi-mode Fault Datasets: (a) Test rig of the two-stage parallel-shaft gearbox; (b) Gear transmission and its fault configurations.

During the operation of a gearbox with an input shaft speed of 2000 r/min, vibration signals under two different fault conditions were collected using a data acquisition instrument with a sampling frequency of 12.8 kHz. Both faults were located on the planetary gears of the secondary gear transmission stage: missing teeth and tooth surface wear. The malfunction is obtained through manual processing. These faults were introduced separately to simulate scenarios of severe gear damage and initial gear damage. During signal acquisition, a load of 10 N·m was applied to the output terminal. The measured vibration signal is shown in Figure 8.

Figure 8.

Time-domain and frequency-domain diagrams of the experimental signals under gear fault conditions: (a) tooth surface wear, (b) missing tooth.

Analysis of the experimental signals under the two different fault conditions reveals the following: When a missing tooth fault is present in the gear transmission, distinct high-amplitude segments appear in the time-domain signal, and noticeable sidebands emerge near the first-stage meshing frequency $f_{m_{1}}$ and the second-stage meshing frequency $f_{m_{2}}$ . In contrast, when a tooth surface wear fault exists, the amplitude of the signal components characterizing the fault is significantly reduced, and the sideband information becomes obscured within the background noise and interference.

The processing results of the four blind deconvolution methods under two different fault conditions are shown in Figures 9 and 10. Identical input parameters were selected for all four blind deconvolution methods: the filter length was set to $L = 150$ , the initial filter coefficients were uniformly initialized using the LPC method, and the number of iterations was set to $Q = 100$ . The remaining parameters for SMHD, ACYCBD, and IMCKD were chosen according to references (McDonald et al., 2012; Miao et al., 2017; Nose-Filho et al., 2023), respectively. For the proposed CGGIBD method, the envelope parameter range was set to $q \in [- 2, 4]$ . The same parameter settings were adopted in the subsequent experiments.

Figure 9.

Comparison of filtering results from four blind deconvolution methods under the gear missing tooth condition: (a) SMHD; (b) ACYCBD; (c) IMCKD; (d) CGGIBD.

Figure 10.

Comparison of the results from four blind deconvolution methods under the tooth surface wear condition: (a) SMHD; (b) ACYCBD; (c) IMCKD; (d) CGGIBD.

When a tooth is missing from the gear, periodic impulse components caused by the missing tooth are clearly observed in the time-domain results of the IMCKD and CGGIBD blind deconvolution methods, whereas the results from SMHD and ACYCBD are less distinct and severely affected by noise interference. In the frequency-domain plots and the machine envelope spectrum, only CGGIBD accurately locates and enhances the frequency band corresponding to the second-stage meshing frequency $f_{m_{2}}$ , and the gear fault characteristic frequency $f_{g}$ and its harmonics appear distinctly in the envelope spectrum. In contrast, the other three methods are inevitably influenced by the first-stage transmission components of the gearbox, resulting in higher amplitudes at the input shaft frequency $f_{r_{1}}$ and its harmonics in the envelope spectrum.

In the case of tooth surface wear, the amplitude of the signal components characterizing the fault is significantly reduced. Compared with the tooth-missing fault, the weak fault information in this scenario is more challenging to accurately locate, identify, and extract. Similar to the results in Figure 9, the high-amplitude components generated by the first-stage transmission section affect the blind deconvolution process in identifying the fault information. Compared to the other three methods, the proposed CGGIBD still effectively enhances the fault information, accurately identifying the fault characteristic frequency corresponding to the tooth surface wear and its harmonics in the envelope spectrum.

Furthermore, to quantitatively compare the effectiveness of the blind deconvolution filtering, the FCR index proposed in The proposed method section is introduced. As shown in Table 1, CGGIBD achieves the best performance under different fault severity levels, indicating its significant potential for fault diagnosis and monitoring of gearboxes under complex operating conditions.

Table 1.

FCR values of the four blind deconvolution methods for gearbox fault signals.

	Algorithm	Case 1 tooth Root crack	Case 2 tooth Surface wear
FCR	SMHD	0.2609	0.1987
	ACYCBD	0.2850	0.1843
	IMCKD	0.3430	0.2740
	CGGIBD	0.5790	0.4240

4.2. Experiment 2: high-speed train’s body-suspended transmission system test rig

To verify the proposed method, we have built a fault testing device for the high-speed train body-suspended transmission system. The device includes a drive motor, a load motor, a universal shaft, a spiral bevel gearbox, an exciter, and a measurement and control system. Various faulty parts were manually processed and polished, including normal tooth count, missing teeth, tooth surface wear, gear root cracks, as well as universal joint crosshead wear, dynamic imbalance, and needle roller bearing wear. Collect operational data at different speeds and load conditions. The data acquisition system comprised an HD2000 8-channel acquisition unit, a KISTLER 8763 triaxial vibration accelerometer, and a laptop computer. The accelerometer had a measurement range of ±10 g, and the sampling frequency was set to 10 kHz. The sensors were installed on both the driving and driven gears. The experimental setup is shown in Figure 11.

Figure 11.

Fault test rig for the high-speed train body-suspended transmission system.

First, vibration signals were collected from the body-suspended transmission system under two different fault conditions and operating scenarios: a missing tooth on the pinion in the spiral bevel gear transmission and crosshead wear in the universal shaft transmission. To validate the effectiveness of the proposed method under varying working conditions, a load of 5 N·m was applied in the case of the tooth-missing fault. The time-frequency diagrams of the measured signals are shown in Figure 12.

Figure 12.

Time-frequency diagram of vibration signals under fault conditions of the body-suspended transmission system: (a) missing tooth; (b) cross shaft wear.

Analysis of the time-frequency diagrams reveals that the motor rotational frequency $f_{r}$ and its harmonics are prominent and vary with time, indicating varying operating conditions. This variability obscures the characteristic frequencies corresponding to gear and universal joint faults. Additionally, the load applied to the system intensifies the overall vibration, leading to an increase in the overall amplitude of the vibration signal and making it difficult to accurately identify and extract the fault features.

The filtering results of the four blind deconvolution methods on the measured vibration signals under the two fault conditions are shown in Figures 13 and 14. To counteract the influence of shifting fault characteristic frequencies caused by varying operating conditions, order analysis was performed by angular resampling of the signals, converting them into angular-domain signals for analysis. It can be observed that the CGGIBD method accurately identifies the fault characteristic orders $O_{g}$ and $O_{t}$ of the gear and universal joint across different fault severities and locations, whereas the other three methods fail to reliably identify these fault characteristic orders. Table 2 provides a quantitative comparison, further demonstrating the superiority of the proposed CGGIBD over the other three methods for diagnosis under variable operating conditions, as its filtering results contain richer fault information. Therefore, for high-speed trains operating under complex conditions, the proposed CGGIBD method can effectively monitor the health status of the body-suspended transmission system via vibration signals, enabling fault diagnosis of its key components.

Figure 13.

Order analysis results of different methods under the gear missing tooth condition: (a) SMHD; (b) ACYCBD; (c) IMCKD; (d) CGGIBD.

Figure 14.

Order analysis results of different methods under the universal joint crosshead wear condition: (a) SMHD; (b) ACYCBD; (c) IMCKD; (d) CGGIBD.

Table 2.

FCR values of the four blind deconvolution methods for the body-suspended transmission system fault signals.

	Algorithm	Case 1 missing tooth	Case 2 cross shaft wear
FCR	SMHD	0.1410	0.2195
	ACYCBD	0.2587	0.2575
	IMCKD	0.1633	0.3357
	CGGIBD	0.7280	0.6175

5. Conclusion

This paper focuses on the challenge of fault feature extraction in gear transmission systems under complex operating conditions. Vibration signals often contain weak fault characteristics that are masked by noise and interference. To address this, we propose a novel blind deconvolution method that adopts a newly developed sparsity measure as the objective function (OF). The core contributions include.

(1) Proposing the Cycle-embedded Generalized Gini Index (CGGI) based on the impulse and cyclostationary characteristics of gear vibration signals, using the GWE as input;

(2) Investigating the performance of CGGI as an objective function for blind deconvolution filtering and providing guidelines for selecting relevant parameters to maximize its effectiveness;

(3) Validating the superiority of the proposed method through publicly available datasets and a dedicated test platform.

Experimental results demonstrate that the method can effectively enhance fault features under various operating conditions. Currently, the performance of the proposed method under non-Gaussian noise interference remains to be studied. In practical industrial environments, non-Gaussian noise is prominently present in gear systems—such as those found in crushers and high-speed trains—and the high-amplitude impulses associated with such noise may adversely affect the performance of CGGI. Future research will focus on extending CGGI and GWE to other sparsity measures and adapting them to more complex fault diagnosis requirements.

Footnotes

ORCID iDs

Zhongshuo Hu

Jianwei Yang

Dechen Yao

Jinhai Wang

Runtao Sun

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Natural Science Foundation of China (Project NO. 52272385, 52475085) and Natural Science Foundation of Beijing Municipality (Project NO. L241033). The authors would like to thank the reviewers for their valuable comments.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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