## 2 Generative Adversarial Networks (GANs)  
Generative adversarial networks (GANs) are architectures of deep neural networks widely used in the generation of synthetic images [37]. This architecture is composed of two neural networks that face each other, called the generator and discriminator [38]. In Figure 2, a general diagram of a GAN architecture is presented. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Year** & **Author** & **Previous Method** \\ \hline
1992 & [PERSON] et al. [16] & Exponentially damped sinusoidal model \\ \hline
1995 & [PERSON] et al. [17] & S1 and S2 as transient–linear chirp signals \\ \hline
1998 & [PERSON] et al. [18] & Sum of Gaussian-modulated sinusoidal \\ \hline
2000 & [PERSON] et al. [19] & S1 and S2 as transient–nonlinear chirp signal \\ \hline
2009 & [PERSON] et al. [20] & A heart-sound-like chaotic attractor \\ \hline
2011 & [PERSON] et al. [21] & A dynamical model based in electrocardiogram (ECG) signal \\ \hline
2012 & [PERSON] et al. [22] & Amplitude and width modification of S1 and S2 sounds from real heart sounds, and combining them with noise \\ \hline
2013 & [PERSON] et al. [23] & Coupled ordinary differential equations \\ \hline
2018 & [PERSON] [24] & Estimation of the second heart-sound split using windowed sinusoidal models \\ \hline
2018 & [PERSON] et al. [25] & Sum of almost periodically recurring deterministic “wavelets”; S1 and S2 are modeled by two sinusoidal pulses of Gaussian modulation. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Timeline of methods for the generation of synthetic heart sounds. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Year** & **Author** & **Previous Method** & **Synthetic Signal** \\ \hline
2015 & [PERSON] et al. [32] & WaveNet: Probabilistic and autoregressive model based on deep neural networks (DNNs) & Music; text-to-speech \\ \hline
2017 & [PERSON] et al. [33] & WaveNet and autoencoders & Musical notes \\ \hline
2017 & [PERSON] et al. [34] & Generative adversarial network (GAN) & Glottal waveform \\ \hline
2018 & [PERSON] et al. [35] & Hidden Markov model & Text-to-speech \\ \hline
2019 & [PERSON] et al. [36] & WaveGAN: GANs unsupervised & Intelligible words \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Previous methods for generation of synthetic audio. The generator (counterfeiter) needs to learn to create data in such a way that the discriminator can no longer distinguish it as false. The competition between these two networks is what improves their knowledge, until the generator manages to create realistic data. As a result, the discriminator must be able to correctly classify the data generated by the generator as real or false. This means that their weights are updated to minimize the probability that any false data will be classified as belonging to the actual data set. On the other hand, the generator is trained to trick the discriminator by generating data as realistically as possible, which means that the weights of the generator are optimized to maximize the probability that the false data it generates will be classified as real by the discriminator [38]. The generator is an inverse convolutional network--that is, it takes a random noise vector and converts it into an image--unlike the discriminator, which takes an image and samples it to produce a probability. In other words, the discriminator (\(D\)) and generator (\(G\)) play the following two-player min/max game with value function \(\mathrm{L}(G,D)\), as described in Equation (1):  
\[\min_{G}\max_{D}\mathrm{L}(D,G)=\mathbb{E}_{x\sim\rho_{data}(x)}[\log D(x)]+ \mathbb{E}_{z\sim\rho_{data}(z)}[\log(1-D(G(z))] \tag{1}\]  
where \(D(x)\) represents the probability that \(x\) is estimated by the discriminator, \(z\) represents the input of random variables from the generator, and \(\rho_{data}(x)\) and \(\rho_{data}(z)\) denote the data distribution and the distribution of samples from the generator, respectively.