## 2 Methodology
### Stimulus-Based Improved EWT  
In this section, an improved EWT method with a new boundary condition based on stimuli is proposed. In the SBT method, stimuli are very important for simulation, and they must be capable of transmitting fault information between one or more nodes [18]. At present, the two most prevalent excitation signals in analog circuit fault diagnosis are pulse signals [9; 10; 12; 13] and sinusoidal sweep signals [8; 11; 19; 20; 21]. Fault information can be transmitted by pulse signals and sweep signals, and fault diagnosis can be accomplished by analysing circuit response signals. However, the impulse response only contains time domain information; therefore, it is difficult to analyse in the frequency domain. The sweep response of the circuit is a kind of unsteady signal, which increases the difficulty of frequency domain analysis. Previous studies [22; 23] combined multiple sinusoidal signals of different frequencies as the excitation of the CUT and verified the effectiveness of multifrequency analysis in analog circuit fault diagnosis. Inspired by these articles, this paper takes advantage of the characteristic that the square wave signal consists of multiple frequency components of fundamental and odd harmonics, and a square wave is used as the stimulus at the input of the CUT. Assuming that the square wave signal \(x(t)\) is of frequency \(f\) and amplitude E, (9) is the Fourier series expansion of \(x(t)\), where \(\Omega_{0}=2\pi f\). Using the square wave as the excitation of the CUT can not only transmit fault information and meet the needs of multiple frequencies but is also more convenient for the time and frequency domain analysis of the square wave response of the CUT. \[x(t)=\frac{2E}{\pi}(\sin\Omega_{0}t+\frac{1}{3}\sin 3\Omega_{0}t+\frac{1}{5} \sin 5\Omega_{0}t+\cdots+\frac{1}{m}\sin m\Omega_{0}t+\cdots) \tag{9}\]  
The main idea of EWT is to construct a bandpass filter bank to obtain AM-FM components centred on a specific frequency. The traditional EWT arranges the local maximum values in the spectrum in descending order and takes the centre of two continuous local maximum values as the boundary of the spectrum segmentation. Thus, (1) and (2) can be used to construct a bandpass filter bank to obtain component signals. Analog circuits are particularly complex, and most of them are nonlinear. When a square wave is used as a stimulus, the components of the circuit response are affected by different faults; therefore, the local maxima of the spectrum may be inconsistent in different faults. As a result, the centre frequency of the AM-FM components of the response of the same CUT in different faults is not necessarily the same, which will result in the inability to accurately analyse the impact of different faults on the components, thus limiting the improvement of the analog circuit fault diagnosis accuracy. Therefore, for the stimulus selected in this article, the traditional EWT used for the preprocessing of the response signal at the output of the analog circuit will no longer be applicable. To accurately reflect the effect of different analog circuit faults on the components, this paper uses a priori knowledge on the frequency of the excitation to determine the spectrum splitting boundaries in the improved EWT. Suppose that the Fourier axis of \(x(t)\) is normalised to \([0,2\pi]\), and the fundamental frequency of the square wave can be denoted as follows:  
\[\omega^{\prime}_{0}=\frac{\Omega_{0}}{f_{s}} \tag{10}\]  
where \(f_{s}\) denotes the sampling frequency of \(x(t)\). According to (9), the square wave is mainly composed of fundamental and odd harmonics. If the boundary of the splitting spectrum is determined at even times, \(\omega^{\prime}_{0}\), the components with odd times, \(f\), as the centre frequency can be accurately extracted.