# Integrating Dilated Convolution into DenseLSTM for Audio Source Separation
## 3 Proposed Architecture
### DenseNet and Dilated Dense Block  
The dilated dense block architecture had the dilated block on the left and the dense block on the right, as shown in Figure 1. The dilated block can consider a wide receptive field and the dense block outputs a feature map containing more accurate target information while passing through several layers. Therefore, we placed the dense block after the dilated block in order to naturally inherit the influence considering the wide receptive field. The dense block of DenseNet concatenates the output feature maps, which is the output of the CNN filter convolving with the input data [39; 40], as shown in the right block of Figure 1 in order to exploit the advantage of efficient information transmission in the feed-forward and error back-propagation processes. The dense block is composed of several composite functions [36], and the composite function consists of a sequence of batch normalization (BN) [41], rectified linear unit (ReLU) [42], and \(3\times 3\) convolution (Conv). The equation below represents the concatenation of the dense block.  
\[x_{\ell}=H_{\ell}([x_{0},x_{1},\dots,x_{\ell-1}]), \tag{1}\]  
where \(x_{\ell}\) represents the output of the \(\ell\) layer, and \(H_{\ell}\) represents the composite function. \([x_{0},x_{1},\dots,x_{\ell-1}]\) indicates concatenating all the feature map from the 0 to \(\ell-1\) layer.  
The purpose of the dilated block is to increase the receptive field more effectively with down-sampling and up-sampling on a spectrogram. The dilated block is an architecture in which the frequency dilated convolution (FDConv), time dilated convolution (TDConv), and \(3\times 3\) standard convolution are configured in parallel after BN and ReLU, as shown in the left block of Figure 1. The dilated convolution [34] was used for the FDConv and TDConv, and the kernel size and dilation rate were adjusted appropriately to the frequency and time axes, as shown in Figure 1 [33]. In the image task, the dilation rate of dilated convolution changes the horizontal and vertical axes at the same ratio because the size of the target image changes at the same ratio in width and height according to the distance [34]. However, in the spectrogram, since the acoustic characteristics of the time axis and the frequency axis changes with independent influences, the FDConv and TDConv are arranged in parallel. For example, the speech rate affects the time axis, and the gender-dependent pitch affects the frequency axis. The FDConv broadens the receptive field of the frequency axis, and the TDConv broadens the receptive field of the time axis. The dilated block is located in front of the dense block, and this whole architecture is called a dilated dense block. The number of output feature maps in the dilated block is \(m_{0}+3k\), and the number of output feature maps in the dilated dense block is \(m_{0}+(3+L)k\). \(m_{0}\) is  
Figure 1: The architecture of dilated dense block. The [frequency, time] denotes the kernel size and dilation rate. BN: batch normalization, ReLU: rectified linear unit, FDConv: frequency dilated convolution, TDConv: time dilated convolution.  
the number of input feature maps, \(k\) is the growth rate, and \(L\) is the number of composite functions [36].