# Machine Learning Applications for Short Reach Optical Communication
## 3 Machine Learning for Short-Reach Optical Communication System Signal Processing
### Machine-Learning-Based Equalizer for Short-Reach Optical Communications
#### 3.1.2 Reservoir Computing Based Equalizer  
In addition to the FFNN-based equalizers, other types of neural networks have also been studied for signal equalization in short-reach optical communications. In this section, we review the use of RC, which can be considered as a highly redundant RNN with untrained input and hidden parts of the network (reservoir) [82]. RC enables the effective training of the entire neural network by optimizing only the output weights with a single linear regression step, and hence, it avoids typical training problems such as the vanishing gradient problem of typical RNN. The typical structure of the RC based equalizer is shown in Figure 10, and it also consists of the input layer, the hidden layer and the output layer.  
Because it resembles nonlinear dynamical systems, the hidden layer is called a reservoir. The inputs \(u[n]\) are injected into the reservoir via the linear transformation \(W_{in}\) and they drive the reservoir states \(x[n]\), which are characterized by the recurrent interconnections within the reservoir \(W^{res}\) and output interconnections \(W^{out}_{res}\)[82; 116]. Therefore, the reservoir states \(x[n]\) can be expressed as:  
\[x[n]=\alpha\cdot f_{act}\left(W^{in}\cdot u[n]+W^{res}\cdot x[n-1]\right)+(1- \alpha)\cdot x[n-1] \tag{18}\]  
where \(f_{act}\) is the chosen nonlinear activation function and \(\alpha\) (\(0<\alpha<1\)) is employed to simulate an exponential decay within the reservoir to meet the echo state requirement and replicate realistic physical implementations [118]. The direct connection between the input and the output layers is characterized by \(W^{out}_{in}\). Then, the expression of output layer vector \(y[n]\) is derived as:  
\[y[n]=W^{out}_{res}\cdot x[n]+W^{out}_{in}\cdot u[n] \tag{19}\]  
RC are promising tools for channel equalization in short-reach optical communications. In [116], RC are combined with the optical pre-processing, which divides the received optical signal into small sub-bands using an arrayed waveguide grating before entering the PD. RC is then utilized for digital post-processing at the receiver after photodetection.  
Figure 10: Reservoir computing general architecture.  
The output layer of this RC model is trained through linear regression with time domain sequences. In this study, 32 GBaud/s COK modulated signal was transmitted over up to 60 km optical fiber. Compared with the more complex MLSE or FFNN based equalizers, results show that the proposed RC scheme can achieve a compensation distance increase from 10 km to 40 km under 1 dB SNR penalty threshold.  
In [117], RC has also been used to achieve channel equalization in a WDM-RoF system. In WDM-RoF systems, due to the linear fiber distortion caused by dispersion, the fiber nonlinearity due to the Kerr effect, and the inelastic scattering of signals, the adjacent channel power ratio (ACPR) performance typically becomes worse after transmission. ACPR refers to the ratio of the overall power of the adjacent channels (i.e., inter-modulation signal) to the power of the main channel (i.e., useful signal), and it is a widely used parameter to characterize the performance of the WDM-RoF system. To overcome the degraded ACPR, RC has been utilized. According to the simulation results, the RC model with 500 nodes can generate 6\(\sim\)9 dB ACPR gain in a WDM-RoF system. More significantly, the gain is directly proportional to the number of reservoir nodes within a range of 400-800 nodes. As a result, merely increasing the reservoir size can increase performance even further.