# Deep Neural Network Equalization for Optical Short Reach Communication
## 3 Experimental Verification and Discussion
### Measurement Results over an Optical Channel with 600 Gb/s/\(\lambda\)
#### 3.2.2 Deep Neural Network Equalizer  
While the LS-solver calibrates (MP-)VNLE kernels in a single shot, the weights and biases for DNN-NLE are trained in numerous iterations (epochs). The weights and biases of the DNN-NLE are updated by shifting the previous values towards the negative direction of the gradient. Therefore, the mini-batch backpropagation algorithm, which computes the gradients of the cost function in respect to the network parameters, and the stochastic gradient descent optimizer ADAM are applied. The hyper parameters are constant for all structures. The mini-batch size is set to 1% of the numbers of symbols of the training frame, respectively to \(10^{3}\) symbols. The learning rate of the ADAM optimizer is set to \(10^{-4}\), which is a common value. In order to prevent overfitting, the performance of the DNN is repetitively validated on testing data during the training phase. The design options for DNN-NLEs are numerous and interrelated. There are no rules for the number of layers or neurons per layer to model a given problem. The more neurons the network interconnects, the finer the modeling capabilities become. More complexity, however, makes DNNs prone to overfitting [19]. Figure 9 shows two DNN equalizer architectures with multiple hidden layers and extra input nodes in order to consider the channel memory effects. On the left hand side, four separate DNNs are depicted with one output node for handling the inphase and quadrature components and polarization independently. On the right hand side, a DNN structure is shown that handles the inphase and quadrature for each polarization jointly. In principle, the joint option could learn as well nonlinear phase impairments in addition to nonlinearities. In both cases, the input nodes feed \(x_{I}(n)\) and \(x_{Q}(n)\) as well as \(y_{I}(n)\) and \(y_{Q}(n)\). Memory effects of channel and components are considered by adding time delayed versions of the input signal. The memory depth is defined by the parameter \(M\) and is set to 4, equal to the previous setting of the Volterra equalizers. The number of hidden layers layers and the corresponding number of neurons per layer of the particular networks were varied during the examination. In order to compare learning speed and performance. Table 1 list the different structures, where the last column represents the total number of independent hardware multpliers to process the data stream of two polarizations. Regardless of the structure, for the hidden neurons as well as for the output neurons a tanh has been chosen as activation functions. The design e.g., \(5|20|30|40|1\) stands for 5 input neurons (input signal + 4 memory taps) followed by three hidden layers with 20, 30 and 40 neurons, feeding into 1 output node. The higher the number of layers and neurons, the higher the computational effort and hence the complexity. Figure 8: optical back-to-back 600 Gb/s/\(\lambda\) MP-VNLE preFEC measurement results. Figure 10 shows the learning process of the four different independent I&Q-DNN equalizer structures for the optical BtB 600 Gb/s measurements. The blue and the red dashed curves indicate the cost functions, given by Equation (14), of the inphase and quadrature components for one polarization, while the green curves represent the preFEC BERs related to numbers of epochs. It can be observed, that the most complex design, shown in Figure 9(b) converges the fastest. The learning speed of the structures depicted in Figure 9(a),c,d is slightly slower, especially at the beginning. However, all options need at least around \(10^{5}\) epochs for learning the component nonlinearities. The long training process is one disadvantage of the neural network equalizer and calls for further improvement and research.