# An Extremely Close Vibration Frequency Signal Recognition Using Deep Neural Networks
## 2 Vibration Signal Detections
### Signal Analysis Methods
#### 2.1.2 Short-Time Fourier Transform (STFT)  
Another signal analysis method is the Short-Time Fourier Transform (STFT), which uses a time-frequency domain. The application of time-frequency analysis enables the examination of signals in both time and frequency domains, aiding in efficiently identifying signal characteristics across diverse time intervals and frequency bands. The STFT method presents distinct advantages in signal analysis by breaking down a signal using a short-time window function. One significant benefit of STFT is its capability to capture time-varying characteristics of signals, enabling the examination of transient phenomena and dynamic frequency changes. Unlike the Fast Fourier Transform (FFT) and Discrete Fourier Transform (DFT), which analyze entire signals simultaneously, STFT offers a localized frequency analysis. This localization allows for a more precise examination of frequency components over different time intervals, enhancing the resolution of frequency content in signals with multiple components or rapid changes. Moreover, STFT adapts the window size to accommodate variations in signal characteristics, making it particularly useful for analyzing non-stationary signals. Overall, the STFT method offers a valuable tool for understanding time-varying signals with improved resolution and sensitivity compared to FFT and DFT techniques. The signal transformed by STFT can be defined as [1; 35]:  
\[w(n)=\left\{ \begin{aligned} & 1,0\leq n\leq L_{w}\\ & 0,else\end{aligned} \right. \tag{3}\]  
\[F(\,f)=\sum_{n=-\infty}^{\infty}x(n)w(n-t)e^{-\beta\pi fn} \tag{4}\]where \(f\) represents the frequency, \(x\) is a signal function, \(n\) is signal sampling, \(w\) is a window function at n on the time axis, \(L_{w}\) is window length. The vibration signal-based STFT algorithm transformed into a time-frequency domain was tried at various times, as well as superposition and sampling frequencies, as shown in Figure 4. The first superposition frequency is 50.00 Hz and 52.00 Hz, the second is 50.00 Hz and 50.20 Hz, and the third is 50.00 Hz and 50.02 Hz, respectively. The simulation superposition frequencies used times 1 s and 10 s at sampling frequency 1000, then were tried in time 10 s at sampling frequency 200. The \(x\)-axis of each graph represents time, which is in seconds, and the \(y\)-axis represents frequency in Hertz. In the first row, the superposition frequency difference is 1 Hz, the second row is 0.1 Hz, and the third row is 0.01 Hz. In the STFT result, using times 1 s and 10 s with sampling frequency 1000 and the frequency combination 50.00 Hz and 52.00 Hz cannot distinguish two superposition frequencies, as shown in Figure 4a,b. With a time of 10 s and minimizing the frequency sampling window to 200, STFT can detect the superposition frequency correctly, as depicted in Figure 4c. When changing the frequency combination to 50.00 Hz and 50.20 Hz, times 1 s and 10 s cannot recognize the frequencies shown in Figure 4d-f. For extremely close frequency combinations 50.00 Hz and 50.02 Hz, the graph detected two superposition frequencies as single frequencies in any time setting and sampling frequency window, as shown in Figure 4g-i. The STFT method, while adequate for many signal analysis tasks, presents limitations in recognizing the superposition of extremely close frequencies. One significant disadvantage lies in the trade-off between time and frequency resolution inherent in the STFT approach. When dealing with closely spaced frequencies, the window size required to  
Figure 4: Short-Time Fourier Transform (STFT). (**a**–**c**) 50.00 + 52.00 Hz with times 1 s, 10 s; (**d**–**f**) 50.00 + 50.20 Hz with times 1 s, 10 s; (**g**–**i**) 50.00 + 50.02 Hz with times 1 s, 10 s.  
achieve adequate frequency resolution may result in a loss of temporal resolution, making it challenging to distinguish between closely spaced frequency components accurately.