# Going Deeper into OSNR Estimation with CNN
## 2 Principle of Proposed Scheme
### Design Principles of Basic CNN  
Typical structure of CNN comprises convolutional layers for feature extraction and pooling layers for downsampling, following which fully connected (FC) layers learn nonlinear mapping from high-dimensional feature spaces to corresponding value spaces.  
Despite different definitions of convolution in neural network and signal processing, the similarity in the mathematical formula reveals that the kernel in 1-D CNN could be considered as a finite impulse response (FIR) filter [25] with the taps already flipped around before doing multiplication with the signal sequence in a convolution operation.  
\[s[i]=\sum_{j=0}^{N-1}x[i-j]\omega[j] \tag{4}\]\[s[i]=f(\sum_{k=0}^{K-1}\sum_{j=0}^{N-1}x_{k}[i+j]\omega_{k}[j]+b[k]), \tag{5}\]  
where \(s[i]\) is the signal output, \(x[i]\) is the signal input, \(\omega[j]\) is the weight in the convolutional kernel, \(b[k]\) is the bias, and \(k\) represents the channel number. In an investigation into the CNN kernels in [25], where two convolutional layers were placed in serial at the front end, the first trained convolutional layer performed a few types of bandpass filters to separate the signal from the noise, while the second one worked as a low-pass filter to extract a DC component for averaging values derived from the previous convolutional layer.  
In pattern recognition, of which the aim is doing classification of the input, the pooling layer plays a role of dimension reduction with retaining basic features of the image. The pooling layer merges semantically similar features into one after computing the maximum or average of the local patch of units in feature maps [34]. Meanwhile, the pooling operation helps to make the representation approximately invariant to small translations of the input [35]. Namely, pooling brings image features invariance in translation, rotation, and scaling. The following example vividly shows the feature extraction function of pooling. In Figure 1, three different kinds of pooling are applied to the _Girl with a Pearl Earring_ (an oil painting by [PERSON] in 1665. The work has been in the collection of the Mauritshuis in The Hague since 1902). Downsampling samples the last pixel in each patch with a stride of 40 when max pooling and average pooling takes the patch's maximum and average value in every channel of the image, respectively. It is still quite easy to recognize this famous girl even when the resolution drops to only 168 pixels from 268,800 pixels for all three pooling operations.  
The success of the pooling layer applied to image recognition benefits from the sparse feature information in an image. A weak correlation between local patches also contributes to it. Specifically, features scatter in different regions of the 2D plane of an image while different regions conceal different features. However, signals in optical fiber have nothing to do with the property of sparsity. During a limited period, OSNR stays relatively unchanged and can be measured at any time of the signal. The definition of OSNR in Equation (1) also suggests that the information of OSNR is consistent at every time point and actually presents features in the frequency domain [5]. If pooling is applied in the raw signal at the very beginning, we could not derive much useful information from the time domain. The features concealed in the spectrum are damaged by pooling operation instead.  
Figure 1: Pooling operation can extract main features useful for dimension reduction.  
Multilayer feedforward networks with as little as one hidden layer are proven to have the potential of being universal approximators to any desired degree of accuracy, provided sufficient hidden units are available [36]. In many circumstances, a deeper network is more efficient to get a lower generalization error with less number of units required than a wider one when representing the desired function [35].