## 3 Expert Systems for Features Classification
### Optimization of HOTA Method for IMs Fault Diagnosis  
In HOTA method, as in the other methods that develop time/frequency spaces, it is essential to adjust the proportion of window filter to maximize the resolution of the space and the desired frequencies. Depending on the proportion applied to the Gaussian window filter, the space resolution can be improved (Figure 6a,c) or be worsened (Figure 6b,d), hindering the fault detection.  
Figure 5: General scheme of the supra-system, the system generator of optimized fault diagnostic system.  
Figure 6: Examples of the time/frequency space generated with a phase current using a Gaussian window of ratio \(10/1\) (**a**) and its HOTA solution (**c**). The time/frequency space obtained with the same current but using a Gaussian window of ratio \(1/10\) is shown in (**b**), and its HOTA solution is shown in (**d**).  
Figure 6 shows an example of a motor with a broken bar fault, with the fault components at \(k=1\) and \(k=-1\); To detect properly these components in the joint time-frequency domain (t-f domain) is a challenging task, because their amplitude is much smaller than the main component amplitude. The performance of the fault diagnostic system depends critically on the analyzing window used for obtaining the t-f spectrum of the stator current. In this work, the Gaussian window has been selected, because it achieves the highest power density in the t-f domain. Nevertheless, different width to height ratios of a Gaussian window with the same minimum area in the t-f domain can alter substantially the shape of the fault harmonics. For each fault component, it can be observed that exists a specific window whose width to height ratio gives the sharpest fault components in the t-f spectrum [20], and consequently the fault components on HOTA result are also maximal (most representative fault features). Moving away from this ideal window size ratio, either in time or frequency, the resulting fault components on HOTA are less representatives for the fault. One of the goals of the supra-system presented in this paper is precisely to obtain this optimal window ratio, for generating the best t-f spectrum and the best fault representatives features in HOTA result.  
The sharpness of the fault components is calculated with the mean value of the amplitude in the frequency range between the fault components (\(k=1\) and \(k=-1\)) and the main component. In the proposed system, the search for the best window size ratio has been optimized using a binary search algorithm [21], which is more specialized than a linear search algorithm.  
As discussed above, the t-f spectrum is generated with a STFT transform. Nevertheless, as explained in [6], a SFFT transform can be used instead STFT, obtaining the same t-f spectrum. It is remarkable that the use of the SFFT in diagnostic applications reduces greatly the required computational power, regarding the STFT. This is due to the fact that the fault frequencies to be analyzed are known and the SFFT is able to generate the t-f spectrum by moving the Gaussian window in the frequency domain within a limited range close to the desired frequencies. This can be seen in Figure 7, where the results of the SFFT are shown in color, whereas the gray zone shows the results of the t-f spectrogram as obtained with the traditional STFT.