## 2 Proposed Methods for Speaker Recognition
#### 2.1.1 MFCC Feature Extraction  
It is challenging to observe the characteristics of the original sound signal. Therefore, in the processing of the sound signal, the original sound signal is usually converted into a spectrum signal for feature observation and processing. MFCCs are a well-known feature extraction method in the traditional sound recognition field. The MFCC is a popular feature in speech and audio processing [9]. In the early 1980s, MFCCs were adopted for speech recognition and later introduced for speaker recognition. In our proposed three methods, we use MFCC as the input feature and leverage MFCC's adoption in human auditory perception. This can effectively capture the important features in speaker recognition. The MFCC feature extraction process is shown in Figure 2. The sequential process of MFCC feature extraction is pre-emphasis, framing, Hamming windowing, fast Fourier transform (FFT), mel filter bank processing, and discrete cosine transform (DCT). First, the pre-emphasis compensates for the high frequency of the input sound signal to eliminate the radiation of the vocal cords and lips to compensate for the suppressed high-frequency part. Pre-emphasis was calculated as follows:  
\[y_{i}=x_{i}-\alpha x_{i-1}\, \tag{1}\]  
where \(x_{i}\) is the \(i\)-th input value, \(\alpha\) is the filter coefficient, and \(y_{i}\) is the \(i\)-th output after the pre-enhancement. The filter coefficient can be set between 0.9 and 1 and is usually set at 0.95. Framing divides the signal into shorter time frames after the pre-emphasis. The time length of each frame was 20-40 ms. After framing, each frame passed through a window function. The Hamming window function was used here. We use \(y_{n}\) to represent the output of the \(n\)th frame after passing through the Hamming window. This is calculated as follows:  
\[y_{n}=x_{n}-w_{n}\, \tag{2}\]  
Figure 1: LSTM-MFCC-TL method architecture diagram. Figure 2: MFCC feature extraction flowchart. where \(x_{n}\) is the input \(n\)th frame, and \(w_{n}\) is the Hamming window for the \(n\)th frame. Furthermore, \(w_{n}\) was calculated as follows:  
\[w_{n}=0.54-0.46\cos\biggl{(}\frac{2\pi n}{N-1}\biggr{)}\,\ 0\leq n\leq N-1\, \tag{3}\]  
where \(N\) denotes the total number of frames. An FFT is performed on the output signal through the Hamming window to convert the original signal from the time domain to the frequency domain. The FFT is a fast algorithm for computing the discrete Fourier transform (DFT). The output of the FFT is then converted to the mel-frequency domain through mel filter bank processing. We define \(Mel(f)\) as the mel-frequency corresponding to the general physical frequency \(f\), calculated as follows:  
\[Mel(f)=2595\log_{10}\biggl{(}1+\frac{f}{700}\biggr{)}=1125\ln\biggl{(}1+\frac {f}{700}\biggr{)} \tag{4}\]  
After obtaining the result using \(Mel(f)\), the DCT was used to calculate 12 MFCCs for each frame. In addition to these 12 MFCC coefficients, we considered the energy per frame obtained after the framing step above as one of the MFCC features. Therefore, after the sound signal of this method was extracted using MFCC features, 13 MFCC features were obtained. Next, we used the original 13 MFCC features to compute delta cepstrum, resulting in 13 delta features that represent the temporal changes in the cepstral parameters. The calculation method for delta cepstrum is as follows,  
\[\Delta C_{m}(t)=\frac{\sum_{i=-p}^{p}iC_{m}(t+i)}{\sum_{i=-p}^{p}i^{2}}\, \tag{5}\]  
where \(\Delta C\) represents the original MFCC features and \(\Delta C\) represents the delta features obtained after applying delta cepstrum. \(\Delta C_{m}(t)\) is the output of the \(m\)th MFCC feature at time \(t\), and \(p\) is the number of differential observations before and after time \(t\). Here, we use \(p=1,\ 2\) to calculate the \(\Delta C\) of the two sets of 13 features, plus the original \(C\), for 39 MFCC features. Therefore, each audio frame obtains a set of MFCC feature vectors of length 39 after the MFCC feature extraction.