## 2 Materials and Methods
### Reverberation Time Prediction Methodology Based on Artificial Neural Network  
To simplify the procedure for estimating the reverberation time of a room, a methodology based on the use of artificial neural networks was studied. Mathematically, the topology of an artificial neural network defines a space of hypothesis, also called the space of possibilities, in which useful representations of the data to be analyzed can be sought. A generic multilayer network is represented by a succession of layers (Figure 2): the first layer is called the input layer, the outputs of which are supplied in input to the subsequent layers which are called hidden layers. The adjective \"hidden\" derives from the fact that, in these levels, the representation of the data is transitory, and as a result hidden from the outside world, therefore we cannot properly speak of output values for them. The last layer of the network is called the output layer and contains from 1 to N neurons whose task is to provide the real network output based on the required task [28]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Type** & **Specifications** \\ \hline \multirow{3}{*}{Microphone} & Passive Sensitivity: 41 dB, 1 kHz at 1 Pa \\ \cline{2-3}  & Input gain: \(-\infty\) to 46.5 dB \\ \cline{2-3}  & Maximum sound pressure input: 136 dB SPL \\ \hline \multirow{3}{*}{Recording device} & Sampling rate: 48 KHz \\ \cline{2-3}  & Bit depth: 24 bit \\ \hline \multirow{3}{*}{Bluetooth loudspeaker} & Output power: 30 W RMS \\ \cline{2-3}  & True Wireless Stereo \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Recording and reproducing sound instrument specifications. Each layer is made up of a series of neurons. Information is stored inside each neuron and is transferred to those of the subsequent layers through a connection to which a weight is associated. Weights adjustment represents the real learning task of the entire process and takes place through an adaptation procedure based on an iterative optimization process [29]. Given a set of independent input variables \(\mathbf{x}\) = (x\({}_{1}\),..., x\({}_{n}\)), we can define with \(W\) the weights matrix, in which each element \(w_{ji}\) represents the weight of the connection between the i-th node of the input layer and j-th node of the hidden layer. Each row of the weight matrix defines the weights for a single hidden layer node, so the association between the input vector and the output of each hidden node can be illustrated through the following equation:  
\[h_{j}=f\left(\sum_{i=1}^{n}w_{ji}\times\ x_{i}+b\right) \tag{2}\]  
In Equation (2) the terms are defined as follows:  
* \(h_{j}\) is the j-th output of the hidden layer
* \(f\) is the activation function
* \(w_{ji}\) is an element of the weight matrix
* \(x_{i}\) is the i-th input
* b is the bias  
An essential feature of this technology is the activation function: Its importance lies in the fact that a neural network without an activation function essentially behaves like a normal regression model whose purpose is to provide a function that best approximates the provided data input and that, at the same time, is flexible enough to allow a forecast of a possible output value. Initially, in the history of neural network-based algorithms implementation, the first network structures were designed to perform purely linear logical operations, the function that was to approximate the input points was therefore linear [30]. With the evolution of networks, this linear function represented the distribution of points in an imprecise way. For this reason, it was decided to introduce, within the networks, an element of non-linearity, hence the sigmoidal activation function. The developed methodology aims to predict the value of the reverberation time, based on recordings of the responses of a room to a series of signals. To train the network, recordings were made in a series of rooms characterized by different reverberation times.