## 3 Applications of the CDS for the Fiber Optic Link
### The Executive
#### 3.3.4 The CDS with a Simple Executive  
In this section, we present the CDS with a simplified executive that does only basic actions. Figure 7 shows the block diagram of the proposed executive for the long-haul fiber optic link. In Figure 7, the executive receives the \(BER_{k}\) through the main feedback channel at the \(k\)th cycle. We could define the internal reward as a function of the \(BER_{k}\) only. However, in general, as the data rate increases, the BER increases too, and higher data rates are desirable while the BER is relatively low, (i.e., under the FEC threshold). Hence, a better definition of internal reward should be a function of both BER and the current data rate. Therefore, we define internal reward as follows:  
\[rw_{j}^{k}=\frac{BER_{k}}{\left(R_{j}^{k}-R_{ref}\right)^{4}} \tag{4}\]  
\[R_{j}^{k}=R_{ref}(Gb/s)\,+j\times d(Gb/s) \tag{5}\]  
where the \(BER_{k}\) is the BER at \(k\)th PAC, \(d\) is the discretization step for changing the data rate which is set equal to 2 Gb/s in this paper, and \(j\) is equal to \(k\) before a disturbance to the fiber optic system and it is reset to 1 after the disturbance and pre-adaptive actions. The pre-adaptive actions refer to those actions which reduce data rates until the BER is under the FEC threshold. \(R_{ref}\) is a convenient reference data rate that is set equal to \(R_{1}^{k}-d\). The executive updates the \(R_{ref}\) after any disturbance using pre-adaptive actions. The internal reward \(rw_{j}^{k}\) and data rate \(R_{j}^{k}\) are further explained in Section 5.2 with an example. The actions space for this specific case study can be defined for the fiber optic link as follows:  
\[C=\left\{c_{k}|data\,rate\,R_{j}^{k}>R_{ref}\right\} \tag{6}\]  
where \(C\) is set of all possible actions in the actions space, and \(c_{k}\) is the action at \(k\)th PAC that belongs to the action space \(C\) such that the data rate \(R_{j}^{k}\) exceeds the reference data rate \(R_{ref}\). Suppose that the current action is \(c_{k}\).The executive can predict the outcome of the prospective action \(c_{k+1}\), before applying it on the fiber optic channel using the following equations:  
\[P(X_{n}^{(k+1)}|\hat{Y}_{n}^{k})=\frac{P(\hat{Y}_{n}^{k}|X_{n}^{(k+1)})P(X_{n} ^{(k+1)})}{P(\hat{Y}_{n}^{k})}, \tag{7}\]  
\[\overline{X}_{n}^{(k+1)}=\underset{t=1,2,\ldots,M}{\operatorname{argmax}}\{P( X_{n}^{(k+1)}=X_{t}|\hat{Y}_{n}^{k})\}, \tag{8}\]  
where \(X_{n}^{(k+1)}\) represents the input symbols virtually generated at the executive at (\(k\)+1)th PAC at the data rate \(R_{j+1}^{k+1}\). The BER is computed by comparing \(\overline{X}_{n}^{(k+1)}\) and \(X_{n}^{(k+1)}\). Note that there are two typesof predictions, one at the preceptor and the other at the executive. When the BER exceeds the FEC threshold, the preceptor switches to prediction mode and selects the previous model closest to the current situation. When the BER is under the FEC threshold, the executive enhances the data rate to \(R_{j+1}^{k+1}\) and predicts the BER using Equation (7) and Equation (8) (see Figure 7). The subsystems of the executive shown in Figure 7 are explained as follows. If the BER exceeds the FEC threshold, the executive runs pre-adaptive actions such as data rate reduction and sending the updated reference data rate \(R_{ref}\) to the actions space. If the BER is less than FEC threshold, the executive selects an action from the actions space. Here, the action is the data rate enhancement. The block \"prediction\" in the executive predicts the \(BER_{k+1}\) based on the statistical model received from the preceptor through the internal feedback link (see Equation (7) and Equation (8)). The block \"rewards calculation\" computes the internal reward using the predicted BER and a prospective data rate of \(R_{j+1}^{k+1}\) and Equation (4). If the predicted BER is below the FEC threshold and a lower value of the internal reward is achieved, then the executive interprets that the system is performing well with \(R_{j+1}^{k+1}\), and the data rate can be enhanced further.