# Harris Hawks Sparse Auto-Encoder Networks for Automatic Speech Recognition System
## 3 Methodology
### Harris Hawks Sparse Auto-Encoder Networks (HHSAE)-ASR Framework
#### 3.2.2 Signal Decomposition and Feature Extraction  
The extracted features are more useful to get the important details that improve the overall ASR systems more precisely. The feature extraction process helps maintain the robustness of the ASR system because it helps to investigate the signal \(s(n)\) in different aspects. The speech signal \(s(n)=cs(n)+ad(n)\) has the length of \(N\). Once the noise has been eliminated, \(cs(n)\) has been divided based on the trend and fluctuations by applying the wavelet transform. Here, level 4 Daubechies wavelets are utilized to extract five wavelets, such as db14, db12, db10, db8, and db2.  
Then, the level of the speech signal mapping process is illustrated as follows.  
In level 1, the speech signal \(c(n)\) is divided into the first level according to the signal length \(N/2\) of the trend \(I_{1}\) and fluctuations \(f_{1}\).  
\[c(n)\ \rightarrow\left(I_{1}|f_{1}\right) \tag{8}\]  
In level 2, the speech signal is divided by \(N/4\) length and is obtained from trend \(I_{1}\) and fluctuations \(f_{1}\), which is defined as \(I_{1}\rightarrow\left(I_{2}|f_{2}\right).\)  
\[c(n)\ \rightarrow\left(I_{2}|\ f_{2}\left|f_{1}\right) \tag{9}\]  
In level 3, the signal is calculated by dividing the \(I_{2}\) and \(f_{2}\) signals that are defined as \(I_{2}\rightarrow\left(I_{3}|f_{3}\right).\) Here, the decomposition process is carried out of length \(N/8\).  
\[c(n)\ \rightarrow\left(I_{3}|\ f_{3}|f_{2}\left|f_{1}\right) \tag{10}\]  
In level 4, the decomposition is carried out for \(N/16\) length, and it is obtained by the \(I_{3}\) and \(f_{3}\) signals that are represented as \(I_{3}\rightarrow\left(I_{4}|f_{4}\right).\)  
\[c(n)\ \rightarrow\left(I_{4}|f_{4}|\ f_{3}|f_{2}\left|f_{1}\right) \tag{11}\]  
According to the above wavelet process, 20 subsignals are obtained according to trends and fluctuations. After that, the signal entropy value \((\mathfrak{ev})\) is estimated, which helps to determine the information of the signal presented in the decomposed signals. The entropy value was obtained according to Equation (12).  
\[(\mathfrak{ev})(Q)=\sum_{i=1}^{n}p(q_{i})\log p(q_{i}) \tag{12}\]  
The entropy value \((\mathfrak{ev})\) is computed from the random phenomenon of speech signal \(Q\{q_{1},q_{2},\ldots,q_{n}\}\) and the probability value of \(p(q_{i})\) of \(Q\). Then, according to \(Q\), every subsignal entropy value is estimated using Equation (13).  
\[I_{4}\rightarrow\left\{le_{4k}\right\}\text{, }f_{j}\rightarrow\left\{ fe_{jk}\right\} \tag{13}\]  
Figure 2: Speech signal noise removal using spectral deduction method.  
The subsignal entropy value (\(\mathfrak{ev}\)) is computed from m number of frames, \(k=1,2,\ldots m\). \(j=1,2,3\). According to Equation 13, the entropy values are \(le_{4k}\) and \(fe_{jk}\). These extracted frame entropy values characterize the speech based on emotions because the fluctuations are varying when compared to the normal speaker emotion level. Then, Mel-frequency coefficient features are derived from identifying the characteristics of the speech signal.  
\[Mel(f)=2595\log\left(1+\frac{f}{700}\right) \tag{14}\]  
The \(Mel(f)\) value is obtained from the frequency value of every subsignal derived from the discrete wavelet transform process. The extracted features are trained and learned by the encoder convolution networks to train the feature to perform in any situation. The process of feature extraction is summarized in Figure 3.