## 5 Feature Extraction
### Frequency-Domain Features  
This section presents the background of Fourier analysis techniques used for feature extraction for road surface anomaly detection and classification. Moreover, the studies that used this signal representation are presented and analyzed. Additionally, the common features that have been computed based on the FT are also listed. The FT is the technique used to generate a frequency representation of a signal defined in the time-domain. The FT's basic concept is to create an orthogonal basis of sine and cosine functions with increasing frequency. The mathematical representation of the FT can be appreciated in the equation below [57]. \[F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt \tag{4}\]  
where \(f(t)\) is a time-domain function multiplied with a complex exponential of frequency omega (\(\omega\)) that corresponds to the term \(e^{-i\omega t}\). Nonetheless, the FT on discrete data vectors must be defined when computing or operating with real data. The Discrete Fourier Transform (DFT) is a discretized Fourier sequence for data vectors. For this purpose, the mathematical representation of the DFT is presented below. \[F(k)=\sum_{n=0}^{N-1}f[n]e^{\frac{-2\omega\omega t}{N}} \tag{5}\]  
The DFT is practical to approximate and compute the FT of data vectors, but it does not perform well with huge data vectors since the computational complexity increases. In this case, the computational complexity of the DFT is \(O(N^{2})\). The FFT was developed to reduce the computational complexity of the DFT. The FFT scales the computational complexity of the DFT to the order of \(O(Nlog(N))\). As \(N\) becomes very large, the \(log(N)\) component grows slowly, and the algorithm approaches linear scaling [58]. Frequency analysis is a crucial feature extraction technique; the magnitude of the FT is used to calculate the feature that will be used for the classification tasks. Common features that are derived from the magnitude of FT are listed below as proposed by [PERSON] et al. [18], [PERSON] et al. [56], and [PERSON] et al. [37]. * The **Spectrum Energy** of the signal is equivalent to the squared sum of the FT coefficients;
* The **Median Frequency** refers to the frequency that divides the FT magnitude into two partitions of equal size;
* The **Peak Magnitude** refers to the maximum value of the FT magnitude;
* The **Minimum Magnitude** refers to the minimum value of the FT magnitude;
* The **Mean Power** refers to the FT magnitude power average;
* The **Total Power** is the aggregate of the signal power;
* The **Discrete Cosine Component** refers to the first component of the magnitude of the FT;
* The **Mean Frequency** refers to the average frequency in the signal's magnitude of the FT;
* The **Maximum Frequency** refers to the highest frequency in the signal's magnitude of the FT. In addition, FT is a crucial step in computing other types of features, such as the power spectral density (PSD), Mel Frequency Cepstral Coefficients (MFCCs), and the perceptual linear prediction coefficients (PLP) [59]. The PSD of a signal analyzes the distribution of power along all the frequency ranges. The primary purpose of the PSD is to compute the spectral density estimation of a given signal [60]. MFCCs is a feature extraction method widely used in speech recognition tasks that focuses its resolution analysis at low frequencies [61]. PLP is a frequency-based feature extraction technique used for speech recognition. A feasible engineering approximation of various well-known hearing characteristics is used in the PLP technique, and an autoregressive all-pole model is used to mimic the resulting auditory-like spectrum of speech [62]. MFCCs and PLP have been used for road condition monitoring as presented in the study of [PERSON] et al. [63]. Otherwise, in Refs. [26; 42] the PSD was computed to extract features for road anomaly recognition.