## 3 Background of Study
### Proposed Intelligent Monitoring Framework
#### 3.2.1 FFT-Based Spectral Isolation  
In addition to providing more reliable paradigms for both stationary and non-stationary signals, the invention of various frequency-domain (and time-frequency-domain) techniques provides a solution to the limitations of time-domain methods of digital signal processing (DSP). Named after [PERSON] (21 March 1768-16 May 1830), the Fourier transform (and its variants) forms the basis for most frequency-domain signal processing techniques [7; 15]. As DSP tools, the FFT and PSD provide a reliable avenue for analyzing the spectral behavior of signals. While FFTs can flourish reliably under relaxed assumptions because most signals are comprised of a complex synthesis of Sine and Cosine functions, the PSD offers an even more reliable alternative to the FFT due to its comparatively higher sensitivity to spectral changes in a signal. This is because the FFT has high sensitivity and outputs a wide range of frequencies (including insignificant ones) that constitute a signal, and due to this high sensitivity, it is sometimes inefficient in monitoring conditions accurately. In addition, the PSD computes the energy densities of the constituent frequencies, thereby exaggerating the relevance of high-energy signal components while suppressing the effects of the lower-energy constituents.  
Given a time-record (one-dimensional sensor signal) \(f(x)=\{x_{1},x_{2},\ldots,x_{m}\}\), the Fourier transform of a function \(f(x)\) is traditionally denoted \(F(k)\) and is computed using Equation (1) below:  
\[F(\omega)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}dx \tag{1}\]  
Figure 2: Proposed monitoring framework.  
where \(f(x)\) is the input signal, \(w\) is the length of the transform, and \(F(w)\)s is the corresponding frequency-domain output of the signal.  
The idea behind spectral isolation is to emphasize the critical frequency bands while moderately reducing the interference of noise and other irrelevant frequency sources. This requires knowledge of the target system's spectral behaviors and identifying the critical frequency bands peculiar to the equipment's operating conditions. Isolating these critical frequencies entails that the interfering frequency bands are muted (reduced to zero magnitudes).  
Suppose one wants to retain the critical frequency segment ranging from \(\omega_{a}\) to \(\omega_{b}\) (where \(0\leq\omega_{a}\leq\omega_{b}\leq|\omega|\)) by creating a filter that eliminates a range of frequency bands--\(\omega_{c}\in|\omega|\
ot\in\{\omega_{a}\rightarrow\omega_{b}\}\), we will specify a box-shaped frequency response with cutoff frequency \(\{\omega_{a}\rightarrow\omega_{b}\}\) such that:  
\[F_{a\to b}(\omega)=\begin{cases}1&\{\omega_{a}\rightarrow\omega_{b}\}\\ 0&\omega_{c}\end{cases} \tag{2}\]  
Subsequently, the inverse FFT (iFFT) will be applied to transform the frequency segment (critical frequencies) back to a time domain. The iFFT of \(F_{a\to b}(\omega)\) is then computed using Equation (3) below:  
\[f(x_{a\to b})=\frac{1}{2\pi}\int_{-\infty}^{\infty}F_{a\to b}( \omega)e^{i\omega x}d\omega \tag{3}\]  
where \(\omega\) ranges between \(\omega_{a}\) to \(\omega_{b}\).  
Finally, the iFFT-transformed signals in the time domain, which only preserves the critical frequency segment, will be used as the most discriminating signals (inputs) for diagnostic modeling.