# An Extremely Close Vibration Frequency Signal Recognition Using Deep Neural Networks
## 2 Vibration Signal Detections
### Signal Analysis Methods  
#### 2.1.1 Fast Fourier Transform (FFT)  
The Fast Fourier Transform (FFT) algorithm emerges as a widely used analysis method to identify signal frequencies, especially vibrational data. The basic FFT algorithm facilitates the transformation of time-domain signals into their frequency-domain counterparts to identify the spectral content within the vibrational data. When detecting close frequencies, the basic FFT can help distinguish between different frequency components present in a signal. The resolution of the basic FFT depends on the number of samples in the input signal and the sampling rate.  
More minor differences between vibration signals will result in frequency peaks closer together or look like just a single frequency in the basic FFT output, making distinguishing them more challenging. The equation of the typical vibration signal is given below [5;34].  
\[y(t)=\sum_{i=1}^{m}b_{i}\text{cos}(2\pi f_{i}t+\varphi_{i}) \tag{1}\]  
\[\text{cos}\left(x+y\right)=\text{cos}\,x*\text{cos}\,y-\text{sin}\,x*\text{sin}\,y \tag{2}\]  
where \(x\) or \(y\) is a signal function, m is the quantity of frequency, \(f_{i}\) is the frequency function, \(\varphi_{i}\) is the phase, \(b_{i}\) is the amplitude of sine or cosine.  
The discrete nature of the basic FFT assumes a periodic and stationary signal, which may not accurately represent the complex dynamics of vibration signals characterized by extremely close frequencies, as shown in Figure 3.  
It is imperative to acknowledge the notable drawback of spectral leakage, an artifact that arises due to the fundamental process of basic Fast Fourier Transform (FFT) operations. Spectral leakage can significantly exacerbate the challenge of distinguishing between closely positioned frequency components, thereby complicating accurate analysis and interpretation of data in various applications.  
As part of the methodological approach, the study will carefully consider these limitations, addressing them through complementary techniques or alternative signal analysis methods, ensuring a comprehensive understanding of the vibrational data. The vibration signal-based FFT algorithm using sinusoidal waves was tested at various times and superposition frequencies.  
The first superposition frequency is 50.00 Hz and 52.00 Hz, shown in Figure 3a, Figure 3b, and Figure 3c, respectively. The second is 50.00 Hz and 50.20 Hz, shown in Figure 3d, Figure 3e, and Figure 3f, respectively. The third is 50.00 Hz and 50.02 Hz, shown in Figure 3g, Figure 3h, and Figure 3i, respectively. All the simulation superposition frequencies use time 0.1 s, 1 s, and 10 s and sampling frequency 1000. The \(x\)-axis of each graph represents time or frequency, and the \(y\)-axis represents amplitude. The amplitude is measured in volts (V) or decibels (dB). The different combinations of superposition signals have distinct characteristics. 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 basic FFT result, using the shortest time, 0.1 s cannot distinguish two superposition frequencies with different frequency combinations, as shown in Figure 3a,d,g. Increasing the simulation time influences the signal form so that it is far from the sinusoidal form. Using 50.00 Hz and 52.00 Hz, the basic FFT can distinguish each frequency with longer times, 1 s and 10 s, in Figure 3b,c. When changing the frequency combination to 50.00 Hz and 50.20 Hz, time 1 s cannot recognize the frequencies shown in Figure 3e but can detect two frequencies with longer time 10 s in Figure 3f. 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, as shown in Figure 3g-i. Therefore, the basic FFT is limited in recognizing the superposition of extremely close frequencies that need extended time-domain signals. The longer time means longer computational time to detect the signal frequencies.