# Compressed Nonlinear Equalizers for 112-Gbps Optical Interconnects: Efficiency and Stability
## Principle of Pruned Nonlinear Equalizers  
### Volterra Series and Neural Network-Based Equalizer  
The mathematical expression of \(P\)-order VE with memory length of \(M_{r}\) can be expressed as Equation (1),  
\[y(k)=W_{dc}+\sum_{r=1}^{P}\sum_{k_{1}=0}^{M_{r}-1}\cdots\sum_{k_{r}=k_{r-1}}^{ M_{r}-1}W_{r}(k_{1},k_{2},...,k_{r})\cdot x(k-k_{1})\cdots x(k-k_{r}). \tag{1}\]  
where \(x(k)\) is \(k^{th}\) sampled data from received signals and \(y(k)\) is output data through equalization. \(W_{r}\) is the \(r^{th}\)-order Volterra kernel. \(W_{dc}\) is responsible for \(DC\) component, which is not included in the final model of an AC-coupled circuit. This model presented in Equation (1) can approximate any nonlinear system in theory. However, it is physically impossible in many cases due to high computation complexity. As low-order part of Volterra series model indicates most features of nonlinear system, approximation approaches are often used to truncate theoretically infinitely long expressions. It can be seen from Equation (1), the number of coefficients, indicating computation burden of multiplication and addition, will grow very fast with increase of \(P\) and memory length \(M_{r}\). It has been known that multiplication contributes most of computation resources compared to addition and will be main reason of complexity effect of pruning algorithms [18]. Therefore, computation complexity for a \(P\)-order VE with memory length of \(M_{r}\) can be defined as the number of multiplication operations [18] and expressed as Equation (2),  
\[Complexity=M_{1}+M_{2}(M_{2}+1)+M_{3}(M_{3}+1)(M_{3}+2)/2+\cdots \tag{2}\]  
According to [3; 26], a three-order structure is sufficient for short-reach optical interconnects. By expanding Equation (1), a three-order VE, named by \(VE(M_{1},M_{2},M_{3})\) where \(M_{1},M_{2},M_{3}\) are memory lengths of first, second and third order of VE, is described by  
\[ \begin{split}& y(k)=\sum_{k_{1}=0}^{M_{1}-1}w_{1}(k_{1})x(k-k_{1}) \\ &+\sum_{k_{1}=0}^{M_{2}-1}\sum_{k_{2}=k_{1}}^{M_{2}-1}w_{2}(k_{1}, k_{2})x(k-k_{1})x(k-k_{2})\\ &+\sum_{k_{1}=0}^{M_{3}-1}\sum_{k_{2}=k_{1}}^{M_{3}-1}\sum_{k_{3} =k_{2}}^{M_{3}-1}w_{3}(k_{1},k_{2},k_{3})x(k-k_{1})x(k-k_{2})x(k-k_{3}).\end{split}  \tag{3}\]  
A basic \(VE(2,2,2)\) structure with 9 coefficients is shown in Figure 1a. In this work, a three-order VE (\(P=3\)) is applied for further experimental investigation. Artificial neural networks have been widely deployed in various areas such as image classification and natural language processing, showing much better performance than traditional algorithms. Recently, it has been intensively researched in the field of optical communication, in which nonlinear equalization is one of hottest topic [13]. To equalize PAM-N communication signals, the received analog signals are taken to a neural network with N possible outputs, representing N levels. The signal stream is first delayed through a delay array, and then input into an input layer with designed neuron number. The numbers of input neuron and layer number are also of great significance to balance the equalization performance and computation complexity. According to universal approximation theorem, a three-layer neural network will provide strong nonlinear equalization capability for the communication systems. Figure 1: The structure of (**a**) three-order \(VE(2,2,2)\) and (**b**) three-layer \(NNE\). The mathematical formula of NNE process can be expressed as Equation (4),  
\[y=argmax[softmax(f(x^{T}(k)\times W_{ih}+B_{h})\times W_{ho}+B_{o})] \tag{4}\]  
where \(x(k)\) is a sampled signal sequence, \(W_{ih}\) and \(W_{ho}\) are weight matrices of input layer to hidden layer and hidden layer to output layer respectively. \(B_{h}\) and \(B_{o}\) are bias vectors of hidden layer and output layer. \(f\) means activation function of hidden layer. Rectified linear unit (ReLU), tangent hyperbolic (Tanh) and sigmoid functions are considered in this work.