## 3 Materials and Methods
### Proposed Method  
The proposed solution is a two-dimensional VAE that is trained with pairs of spectrograms of speech audios with real background noise and the original speech audios. After some convolutional operation, the encoder reduces the size to a minimal piece of information that represents its main features. These pieces of information are then placed in a latent space, which is a distribution of the encoded data obtained at the convolutional stage. The mean and variance of each input are used to fit the distribution in the latent space, this feature being the main difference from a standard encoder. For a VAE, each input represents a distribution in latent space rather than a single point. Information is recovered and upsampled from the latent space until the output has the expected dimensions. Finally, this output is compared to the spectral representation of the original audio without background. This comparison evaluates how well the background noise has been suppressed and lets the model adjust its hyperparameters and learn the information. After training with the whole dataset that comprises audios from the four use cases, the model can remove the real background noise. The input layer of the model has a size of 256 \(\times\) 256 \(\times\) 1, representing the audio of one channel, which is connected to the following layer. The input layer aims to feed the audio to the convolutional blocks. These blocks are responsible for reducing the dimensionality of the information. The convolutional stage consists of 6 convolutional blocks, where each of them obtains a feature map of the previous data. Each block has two convolutional layers with two 2-dimensional filters of size 3 \(\times\) 3, stride one, and max-pooling of 2 \(\times\) 2 (except in the last block). Convolutional layers in the same block have the same number of neurons using the Rectified Liner Unit (ReLU) as an activation function. The number of neurons from one block to another is the following: 32, 64, 128, 256, 512, and 1024. At the end of this convolutional stage, the input data are reduced to 8 \(\times\) 8 \(\times\) 1024. This small piece of information represents the input data with all the main features. Finally, a flattened layer is applied to process information as a one-dimensional vector of size, 65,536. At this point, the information goes through the bottleneck, which in the case of VAEs builds a latent space of its representation. The bottleneck entrance has a dense layer of 2048 neurons that introduces the information from the last convolutional block. Then, we have two other dense layers that manage the distribution's mean and variance, having 2048 neurons. A lambda layer is applied to choose the point representing the input data in the latent space. After getting this point, we can position it in a multivariate space and recreate the last convolution dimensions before the bottleneck, allowing the decoder to do the inverse process. To start this process, we use a dense layer that converts the point into information that can be processed in the de-convolutional stage. The stage of dimensionality upsampling, or deconvolution, now begins. This stage starts with a reshaped layer so the information can be introduced in the two dimensions' deconvolutional blocks. This process has five deconvolution blocks. Each block comprises two convolution layers, with an upsampling layer in the first position. Again, the number of neurons within the layers of the same block is the same, and all use the ReLU activation function. The number of neurons per layer of each block is 512, 256, 128, 64, and 32. These blocks increase the dimensionality of the feature map. In this deconvolution process, starting from the tiny feature map in the bottleneck stage, new audio spectrograms are created using only the essence of the original audio. In each convolution block, concatenate layers have been used with the even blocks of the encoder to speed up the training and not lose the substantial relationship between the data.