## 3 ECG Biometrics using Pre-Configured CNN Models with Various Time-Frequency Representations
### Time-Frequency Representations  
Five time-frequency representations are considered here to compare the ECG biometrics, namely the spectrogram [33], log spectrogram, mel spectrogram [34], MFCC [35;36], and scalogram [37]. Figure 4: Structure of Xception with the time-frequency representations as inputs. The spectrogram used in this study is based on the short time Fourier transform (STFT). STFT is used to determine the sinusoidal frequency and phase by dividing a signal that changes over time into several regional sections. In practice, STFT splits a long time signal into shorter segments of equal length and subsequently applies the Fourier transform to each segment. For example, in the case of continuous time, the signal to be transformed is multiplied by a non-zero window function over a short period. The Fourier transform of the signal is then calculated by sliding the window along the time axis, which results in a two-dimensional representation of the signal. Mathematically, this can be expressed as Equation (2) [33]. \[\mathrm{X}(\tau,\omega)=\int\limits_{-\infty}^{\infty}\mathrm{x}(\mathrm{t}) \mathrm{w}(\mathrm{t}-\tau)\mathrm{e}^{-\mathrm{j}\omega\mathrm{t}}\mathrm{dt} \tag{2}\]  
The log spectrogram and mel spectrogram are log scaled and mel scaled versions of the spectrogram based on STFT. The log scale and mel scale have been famously used for voice applications because they emphasize the low band frequency relevant for voice analysis and deemphasize the high band frequency noise. ECG contains important information in the low band frequency [34]. The log scale and mel scale transforms can be mathematically expressed as Equations (3) and (4), with the scale mappings of the log and mel transforms shown in Figure 5.  
\[\mathrm{y}=\mathrm{log}(\mathrm{x}) \tag{3}\]  
\[\mathrm{m}=2595\log\biggl{(}1+\frac{\mathrm{f}}{700}\biggr{)} \tag{4}\]  
MFCC is a linear cosine transform of the log power spectrum at the nonlinear mel scale over short intervals of a signal. MFCC represents coefficients synthetically composed of an MFC that is derived from the cepstral representation of an audio clip. The normal cepstrum has linearly spaced frequency bands, while the mel frequency cepstrum has equally spaced frequency bands. The human auditory system, which is sensitive to low band frequencies, is similar to the mel frequency cepstrum. This frequency warping mimics the human auditory system, which is primarily sensitive to the low frequency band, to better represent low frequency signals such as ECGs. The process to calculate MFCC can be summarized as follows. Step 1: As the input signal in the time domain is constantly changing, it is divided into equally sized regions in a short time for simplicity. Step 2: The power spectrum is then calculated for each region from the divided signals. Step 3: The energies of the mel filter bank are calculated. Step 4: The energy values of the mel filter bank are logged. Step 5: The discrete cosine transform (DCT) is applied to the logged energy values. The obtained DCT coefficients represent  
Figure 5: Scale mappings of the log and mel transforms; (**a**) Log; (**b**) Mel. MFCC, and the size of the dimension can be appropriately adjusted [36]. Figure 6 shows the process used to calculate MFCC. A scalogram represents the absolute values of the coefficients obtained from the continuous wavelet transform of a signal. The wavelet transform is a time-frequency transform, which is more efficient than the cosine and Fourier transforms [38]. The Fourier transform is weak for high resolution applications as the analysis is performed on a single scale. In contrast, the wavelet transform is analyzed on a multi-scale. The Fourier transform decomposes the signal into sinusoids of different frequencies, which can subsequently be inversed. The wavelet transform decomposes the signal into shifted, scaled mother wavelets that can be inversed.