## 1. Introduction

With the fast development of electronic science and technology, fault diagnosis and testing as fundamental tasks in preventive maintenance of electronic systems play a vital role in reliability of the product and promoting industrial development [

1,

2]. It is estimated that testing covers one third of the cost of the product, and majority of the testing is due the testing of the analog parts of the mixed signal circuits [

3,

4]. Due to continuous parameter and tolerance of analog components, and lack of test nodes, the diagnostics approaches of analog circuits are far less advanced, comparing with well-developed automatic fault diagnosis methodologies for digital circuits. Consequently, there is a pressing need to explore effective fault diagnosis and testing approaches to prevent fault enlargement and guarantee analog electronic system reliable operation.

Faults in analog circuits can be categorized into soft faults and hard faults. Soft faults result in system performance degradation where the parameters of components only deviate from the normal values exceeding the tolerance range. The causes for soft faults mainly include: the aging of an electronic system, fabrication tolerance, electromagnetic interfere and effect of ambient temperature [

5]. Conversely, hard faults mainly happen in short- and open- circuit, or they are caused by the larger parameter variation of components [

6]. The majority examples of hard faults involve the structural failure in bipolar junction transistor (BJT) and metallic oxide semiconductor field effect transistor (MOSFET) and the parameter deviation of key components in filter circuits.

Currently, there are many diagnosis approaches aiming at the two kinds of analog circuit faults. The vast majority of these methods are only implemented for field failure in factory production processes. However, the implementation of component-level diagnosis is challenging [

2,

6]. With respect to analog circuits, it is mainly due to the complex and changing operation conditions and external environment, such as strong electromagnetic interference, high-temperature and complicated failure mechanisms. Therefore, it is necessary to investigate an effective diagnosis method for component failure in analog electronic systems.

The rest of this paper is organized as follows.

Section 2 contains a survey of the related work. In

Section 3, fault feature extraction based on cross-wavelet singular entropy and parametric t-SNE is introduced. In

Section 4, the algorithm and implementation procedures of the proposed PSO for parameter selection of SVM are provided. Further, fault diagnosis test in two experimental circuits is performed in

Section 5 to verify the effectiveness of the proposed method. In

Section 6, a discussion based on Shannon, Rényi and Tsallis entropies is presented. Finally, some conclusions are drawn in

Section 7.

## 2. Related Works

Traditionally, analog circuit fault diagnoses are classified into two broad approaches: Simulation After Test (SAT) and Simulation Before Test (SBT). Compared with SAT approach, the SBT approach is more suitable for diagnostics of analog circuits as it only implements once off-line simulation process, removing on-line computation before testing and running [

7]. Among SBT, data-driven diagnostic methods are based on the case that features of the system relatively changed when a fault happens. They extract features from output signals, then apply pattern recognition techniques such as neural networks (NNs) and support vector machines (SVMs) to locate a fault [

8]. Meanwhile, the data-driven techniques do not need to construct an explicit model. Hence, the data-driven approaches have been applied to fault diagnosis in many relative works [

9,

10].

Technically, a data-driven approach can be divided into two phases: feature extraction and classifier application [

11,

12,

13]. Obviously, feature extraction is the vital steps. To date, increasing numbers of feature extraction tools have been utilized in fault diagnosis, and they can be summarized into three categories: time-domain analysis, frequency-domain analysis, and time-frequency analysis [

9,

14]. Signals collected from the testing nodes of faulty circuits always carry interference components that probably overwhelm useful information. Thus, it is difficult to effectively recognize the defects of analog electronic systems when only considering the features of time-domain or frequency-domain [

15]. As a typical time-frequency domain analysis, wavelet transform (WT) can reveal overlaps in time-frequency domains by decomposing the signal into a set of wavelet coefficients that vary continually over time [

10]. Nevertheless, in practice, the measured signals of analog circuits commonly contain random noise, which may lead to misclassification. Therefore, it is necessary to take actions to minimize the impact of random noise. Noise removal can be executed by setting a threshold when computing wavelet coefficients [

10]. However, there are some limitations: The threshold needs to be set manually, and the calculation process is time-consuming. Recently, cross-wavelet transforms (XWT) has been employed to handle partial discharge pulses and ECG signal [

16,

17]. Moreover, XWT has an outstanding ability in extracting time-frequency characteristics of signal and restraining noise. Consequently, XWT is applied to process the fault signals of analog circuits.

However, there are still several open issues that need to be addressed for XWT. In practical application, XWT is limited to being imported into classifiers directly because the transformed result is a high-dimension matrix. Therefore, it is necessary to combine XWT with other feature extraction techniques to reduce information abundance.

As a description of disorder or randomness of matter, entropy is capable of providing rich information about signals, which is fit for feature extraction [

18,

19,

20]. Many scholars have devoted themselves to the field of feature extraction with use of entropy techniques. Approximative maximum entropy (Apen) has been used to diagnosis faults [

9,

21]. However, a bad performance could be obtained when processing the short data-set. Moreover, the Apen is sensitive to noise. Because sample entropy (Samp) is insensitive to data length and immune to noise, it can be employed as an input vector of classifiers [

20,

22]. However, because the Heaviside step function of sample entropy entails discontinuity at the boundary, negative results are possible. In view of this, many scholars adopt Fuzzy entropy (Fen) that vary smoothly and continuously to estimate data complexity [

23]. Unfortunately, the membership function in Fen is usually difficult to determine. Some achievements in fault detection have been made using cross entropy and Rényi’s entropy [

24,

25], but the faulty components have not been located. Moreover, none of these techniques are used to extract features with wavelet transform. The utilization of wavelet Shannon entropy (Wse) in feature extraction is proposed, achieving a desirable performance [

26]. Nevertheless, the XWT manifests a non-extensive character because of energy leakage and aliasing in the phase of wavelet operation, while Shannon entropy belongs to extensive entropy.

Based on the above, a novel feature extraction technique based on XWT and Tsallis entropy is proposed for fault diagnosis. Owning to its ability of regulating non-extensiveness, Tsallis entropy is employed to construct the feature set, denoting the complexity of fault signals [

27,

28]. Furthermore, to improve the efficiency of fault pattern recognition, a feasible feature reduction approach needs be implemented. A manifold learning technique is able to unearth intrinsic information embedding in highly dimensional datasets via mapping them into a low-dimensional space and retaining the local neighborhood information. Parametric t-stochastic neighbor embedding (t-SNE) has a good capability in mapping the data with high-dimension into low-dimension representation. It maintains the conditional probability distribution of data associated with the pairwise similarity from the high-dimension space to the feature subspace [

29]. Therefore, it is utilized to extract discriminative features between different fault patterns.

To locate the faults, a support vector machine is employed as the classifier. SVM has advantages of high training speed and distinctive generalization ability by finding the optimal hyper-plane [

30,

31]. However, in practical application, it is difficult to assign the free parameter. To address this issue, various intelligent optimization algorithms, such as genetic algorithm and simulated annealing, have been utilized to determine hyper-parameters of SVM. Owing to high speed of converge and good quality of computation, quantum-behaved particle swarm optimization (QPSO) is adopted to obtain the optimal parameters [

32].

## 6. Discussion

Compared with Tsallis entropy, other entropy techniques, such as Rényi and Shannon entropies have already been applied to many diverse practical problems [

48,

49]. Therefore, a discussion based on Shannon, Rényi and Tsallis entropy is described in this section.

For given two probabilities

${p}_{1}$ and

${p}_{2}({p}_{2}=1-{p}_{1})$, the plots of Shannon, Rényi and Tsallis entropies are shown in

Figure 14,

Figure 15 and

Figure 16. Here, the Rényi entropy is defined as

${I}_{q}=\frac{1}{1-q}\mathrm{log}({\sum}_{i=1}^{n}{p}_{i}^{q})$.

As shown in the figures, with the increase of q, the statistical range of Rényi entropy and Tsallis entropy will change, and the entropy values of the probability events will decrease correspondingly. However, with Shannon entropy, the statistical value of the probability events remains unchanged due to the equal weights in the entropy computation. For Tsallis entropy, the events with high probability contribute more than lower probabilities. The Rényi entropy with higher q parameter is determined by events with higher probabilities and the lower values of q coefficients weigh the events more equally.

For a signal containing noise components, the low energy components which can be used to characterize may be corrupted by the background noise that is relative to the events with small probability. In this context, Rényi and Tsallis entropies can achieve better results in extracting features by selecting appropriate

q parameter to minimize noise as compared to Shannon entropy. Furthermore, Tsallis entropy is a much more sensitive function than Rényi entropy with respect to changes in

q value, which is conducive to determine the proper

q parameter. Besides, Tsallis entropy has been found to possess non-extensive property, which is helpful to deal with non-extensive character of XWT transform [

50]. Based on the above advantages of the Tsallis entropy, it is applied to the fault feature extraction of analog circuits in this work.