## 5 The CDS case study for the OFDM-Based Long-Haul Standard Single-Mode Fiber system
### CDS Complexity  
In order to calculate the computational cost, we distinguish three types of CDS modes: (i) BER improvement mode, (ii) prediction mode, and (iii) steady state. The complexity of a CDS varies depending on the perceptor mode. When the CDS operates in the BER improvement mode, the BER converges after \(N\) OFDM frames (in Figure 12, it is about \(N=512\)) and each frame has \(K\) data-carrying subcarriers (From Table 1, it is 126). To create the histogram (i.e., to calculate \(P(Y_{n}^{k}|X_{n}^{k}))\) based on the discretized data, we need \(N\times K\) simple additions (just counting the samples within the tiny surface). To calculate \(P(X_{n}^{k}|Y_{n}^{k})\), we need \(M\) real multiplications where \(M\) is the order of QAM (in our example, \(M=16\)). The storage requirement depends on the precision factor (see Table 2). For example, with a PF of 2, we need to store a matrix of dimension \(81\times 71\times 16\). In addition, we need to store the matrix corresponding to \(P(X_{n}^{k}|Y_{n}^{k})\) which has a dimension of \(16\times 81\times 71\). When the CDS operates in the prediction mode, there is no cost associated with model extraction (i.e., to calculate \(P(Y_{n}^{k}|X_{n}^{k}))\), but we need \(M\) real multiplications to calculate \(P(X_{n}^{k}|Y_{n}^{k})\). In this mode, we need to store the matrix corresponding to \(P(X_{n}^{k}|Y_{n}^{k})\), which has a dimension of \(16\times 81\times 71\). In steady state, the CDS uses the saved \(P(X_{n}^{k}|Y_{n}^{k})\), and makes a decision symbol by symbol (see Equation (3)). These matrices can be stored in hard disk and can be loaded to RAM when needed. No multiplication or addition needs to be done in steady state. The computational cost associated with DBP is as follows. In each propagation step, \(J\log_{2}(J)\) complex multiplications are required where \(J\) is the number of samples in time domain. If there are \(B\) propagation steps, the complex multiplication is \(O(B\times J\log_{2}J)\)[20; 21]. For example, for a 20-span system, assuming 10 steps per span and with \(J=16\),384, we need roughly 45,875,200 complex multiplications! Besides, this computation needs to be done continuously for each block of data. Three complex vectors of length \(J\) (about the total size of 98,304) needs to be stored. In contrast, most of the computational complexity of the CDS is present only when there is a fluctuation in the fiber optic channel; under steady state, the computational cost is minimal. It is useful to compare the complexity of the proposed method in this paper with the digital backpropagation (DBP) for Q-factor enhancement using the simulation parameters shown in Table 1. We run both algorithms on a Microsoft Surface-Pro with Intel(r) Core(tm) i706650U CPU @ 2.20 GHz 2.21 GHz, 16 GB RAM, system type 64-bit Operating System x64-Based processor using MATLAB. For the DBP algorithm [21], the runtime is 1225.4 s (\(\sim\)20 min and 25 s). However, the runtime for the proposed CDS is 14.03 s for PF = 2. In addition, the CDS runtime reduces to 7.71 s for PF = 2 in steady state mode. However, the DBP running time always remain similar for any continuous data stream. In addition, the reported Q-factor improvement for DBP in OFDM systems is between 1 and 2.2 dB [45; 46], while it is 2.74 dB (see Figure 18) for the fiber optic link upgraded with the CDS. Additionally, the main features of the proposed CDS are the following:  
1. The CDS perceptor can extract the model (likelihood) dynamically and does not need the transmitter to send training sequences. 2. We assume that the fiber optic link parameters are unknown in an optical network. 3. The CDS can be applied to any type of fiber optic system such as single carrier, wavelength division multiplexing (WDM), single channel OFDM, or a WDM system with individual channels carrying the OFDM signal. In our approach, we do not use training sequences from the transmitter to extract the channel model \(P(Y_{n}|X_{n})\) since it would disrupt the service.