## 2 Materials and Methods
### Feature Extraction  
Sound signals are the result of processes occurring in the time domain; however, it is convenient to study them in the frequency domain, both in deterministic and stochastic cases, which is why it is necessary to obtain the power spectral density (PSD) function. The input signal is time-limited, non-stationary and has overlap noise, so PSD needs to be estimated from a record of a certain amount of data. The first methods for PSD estimation were based on the estimation of the Fourier transform (FT). An important step in modern spectral estimation was the work of [PERSON] [31; 32], who established the theoretical basis for the treatment of stochastic processes. [PERSON] showed that FT relates the autocorrelation function of a stationary process and its PSD [33]. It is generally called the Wiener-Khinchin theorem. Before the introduction of the fast Fourier transform (FFT) algorithm in 1965, the usual method for estimating PSD was the implementation of the Wiener-Khinchin theorem suggested by [PERSON] and [PERSON] [34]. According to this method, the autocorrelation coefficients are estimated using a sequence of event data. To obtain the PSD, the Fourier transform of this window correlation is performed. This method is often also called the indirect method. Since the introduction of the FFT, which implied the provision of a computationally efficient algorithm for the computation of the discrete Fourier transform (DFT), the direct approximation (periodogram)  
Figure 3: Flow chart for the reverberation time prediction methodology based on artificial neural network. Two sound recordings are made: the first to record the impulse response of the room to calculate the reverberation time. This value will be used in the training phase to label values. The second recording will be used to estimate the reverberation time of the room using a method based on a supervised learning algorithm. is used intensively, which is obtained as a square quantity of the obtained DFT by FFT and applied directly to the data. The power spectrum (PS) is defined as the Fourier transform of the autocorrelation function, as in the following Equation (3):  
\[PS\left(f\right)=\sum_{n=0}^{N-1}r_{xx}(n)e^{-i2\pi/T} \tag{3}\]  
In Equation (3) \(r_{xx}\) is the autocorrelation function as defined in the Equation (4). \[r_{xx}=\sum_{k=1}^{N}x(k)\times x(k+n) \tag{4}\]  
Here \(x\) is the signal. The power spectrum returns the power distribution of a signal over a range of frequencies. The power spectrum can be evaluated by applying the Fourier transform on the entire signal. However, the averaging of the spectra is used, particularly when the available waveform is only a sample of a longer signal. In this case we speak of a periodogram [35]. The periodogram is calculated by dividing the signal into several segments, possibly overlapping, and evaluating the Fourier transform on each of these segments: On two successive data segments a shift of a number of points equal to \(D\) is performed. Each segment is made up of \(N\) points, where the i-th sequence is given by the following expression:  
\[x_{i}(n)=x(n+iD);\quad n=0,1,\ldots,N-1 \tag{5}\]  
The overlap between two consecutive segments \(x_{i}(n)\) and \(x_{i\ +1}(n)\) is equal to \(N\)-\(D\) points. If a 50% overlap is allowed, as proposed by [PERSON] [36], the number of sections K of the length L is calculated using the following equation:  
\[K=\frac{2\times N}{L-1} \tag{6}\]  
Using Equation (6), the length of the sequence does not vary. Additionally, doubling the number of periodograms to be averaged reduces the variance. The final spectrum is obtained as the average of the different spectra obtained on the segments. The frequency resolution of the spectrum is approximately 1/\(N\), where \(N\) is the number of samples per segment. Taking a small segment increases the number of segments to the mean and thus increases the smoothing in the estimated spectrum. At the same time the frequency resolution is reduced.