## 2 Theoretical Background
### The Recovery of the Nonstationary Signal and Related Signal Processing  
The frequency spectrum's legibility in either a steady state or transient regime depends on the signal's length and sampling frequency. The fault-representing frequencies are the function of motor slip, as described in Table 1. During the transient interval, the slip is not constant, but decreases with an increase in speed as the motor goes toward a steady state regime. This changing slip results in the creation of nonstationary signals in the transient domain. With a rated supply, this transient interval is so small that those patterns do not become visible. It becomes worse when the motor operates at a low load. The transient interval can be extended by decreasing the supply voltage with a high inertia load, while  
Figure 1: The development of frequency patterns in a transient regime due to (**a**) broken bars, (**b**) dynamic eccentricity and (**c**) principal slotting harmonics. the input voltage can be changed using a variable three-phase transformer or an industrial inverter. The increasing use of the frequency converter as a substantial part of the drive system increases the test's feasibility. By reducing the supply voltage while maintaining a constant frequency, the rotor's inertia can control the transient time of the motor. Moreover, with reduced voltage, the leakage inductance also takes more time to reach the steady state point, which also supports an extended transient interval. For any reliable signal processing-based diagnostic algorithm, the role of digital filters is undeniable. Digital filters are the mathematical algorithms capable of reducing or enhancing specific parameters of a signal. They are used for either the separation of combined signals or the restoration of a distorted signal. Their diversified nature has many types, and they are being used extensively in almost every signal processing-based application. They are the potential tool to remove the strong supply-based frequency components for improving the time-frequency spectrum's legibility. This is necessary because the fundamental component is powerful in amplitude compared with the fault-based harmonics, which are very weak, particularly at the fault's embryonic stage. Those low-amplitude frequency components become barely visible in the presence of the fundamental component. To attenuate the fundamental component, the selection of an appropriate band-stop filter becomes very crucial. The infinite impulse response (IIR) band-stop filters, such as the Chebyshev II, can give promising results because of their steep transition band and low passband ripples among various digital filters. The excellent transition band and low passband ripples can reduce the filter's impact on the recovered signal. In light of the mentioned theoretical aspects, the frequency spectrum resolution for transient and steady signals is improved in the following way, as presented in Figure 2:  
* Measure the signal with the sampling frequency, meeting the Nyquist criterion. The sampling frequency can be improved later using data interpolation. This is necessary for the accurate detection of zero-crossing points. Moreover, any small constant offset due to data acquisition devices should be removed to avoid a high-amplitude 0 Hz component in the spectrum. This is achieved by subtracting the mean value of the signal from itself;
* Most signal processing techniques, such as DTFT, are sensitive to signal discontinuities and the fractional number of acquired cycles. This problem is solved by counting the integral number of cycles using zero-crossing detection. Saving the signal from the first until the last zero crossings will remove the starting and ending fractional portions of the signal;
* Each sinusoidal signal has three zero crossings, which can be exploited to get the integral number of cycles.