## 2 Materials and Methods
### Deep Recurrent Neural Networks for Reverberated Signal Modeling  
Three slightly different recurrent neural network structures were compared, which could be used as candidates for a reverberation prediction model:  
* Long short-term memory (LSTM) [25];
* Bi-directional long short-term memory (BiLSTM) [26];
* Gated recurrent units (GRU) [27].  
The architecture of the recurrent neural networks (RNN) includes feedback connections, making them more suitable for modeling acoustic effects than feed-forward network structures.  
We try to predict the spectrum of the reverberated signal separately for each octave. In our study, we test all three neural network structures by training them on different frequency bands of the reverberated signal. Each prediction model consists of an input layer to which a sequence of time-varying spectral node values is sent, as well as two layers of recurrent neural network cells, one fully connected layer, and a regression layer, which generate a predicted sequence of changes in the spectral nodes over time.  
To investigate the relationship between the number of RNA cells in a layer and the accuracy of the predicted spectral band, we tested the performance of networks with three different combinations of cell numbers. We first selected 10 cells in the first layer and 20 cells in the second layer, then repeated the experiments by equalling the number of cells in the two layers to 20, and finally we performed another series of experiments by increasing the number of cells in the second layer to 40.  
In the experiments, we try to evaluate the ability of different network structures to predict a reverberated signal:  
1. In the same frequency band used in the training, but replacing the input samples with previously unknown ones;
2. In two adjacent frequency bands, when the model was trained on the middle band and tested on adjacent bands
3. In all frequency bands when the octave is divided into 12 parts. Firstly, when a separate model was trained to predict each frequency band, and secondly, when the prediction was performed by taking input data from each frequency band separately, and the reverberated signal was predicted using a model trained on only one frequency band.  
Variations of the experimental set-up were carried out to determine how flexible the prediction model can be to predict a specific band of the reverberated signal. In addition, it was necessary to see how different the model should be for neighboring frequency bands when the octave is divided into 3 or 12 parts.  
The audio used for model training and experimental testing was divided into 250 ms analysis frames. This is the maximum delay time that can be accepted in real-time auralization systems [28]. The conversion from time to frequency domain was performed using a window of 512 sample width with 256 sample overlaps.The data used to train the models were divided into three parts: training (70%) validation (15%) and testing (15%). All models were trained using the same training options: the ADAM optimizer, a constant learning rate of 0.001, shuffle of the data after each epoch from 10,000, and a small batch size of 50.