# Bandwidth Density Analysis of Coded Free-Space Optical Interconnects
## 5 Bandwidth Density of Coded FSOI System  
Error correction codes (ECCs) can be exploited in the FSOI systems' design to introduce correction capability and enhance the operational BD. The system correction capability reduces the SNR level needed to obtain a desired bit error rate (\(BER\)). Therefore, this capability of the system can be perceived in terms of SNR coding gain (\(G\)), which is defined as the ratio of the SNR needed without coding over the one with coding \((G=SNR_{uncodd}/SNR_{coded})\) for a given BER [18]. A variety of ECCs are available in the literature [2; 20; 34], and they are characterized by the \(G\), where a high \(G\) means good error correction capability. In this paper, we exploit Reed-Solomon (RS) ECCs of different codeword lengths to illustrate the effect of coding on the FSOI system BD. The characteristics of this type of codes are widely investigated in the literature, and the codes are exploited in many storage and communication systems [35; 36; 37]. To demonstrate the consequence of RS codes on the BD of optical interconnects, we analyze their \(G\) for a range of code rates, as shown in Figure 4. The figure is generated based on the following formulations of \(BER\)[19; 38]. \[BER_{uncoded}=\frac{1}{2}\Bigg{(}1-\text{erf}\Bigg{(}\frac{1}{2}\sqrt{\frac{ SNR_{uncoded}}{2}}\Bigg{)}\Bigg{)}, \tag{13}\]  
\[BER_{coded}=\frac{1}{2}\Bigg{(}1-\text{erf}\Bigg{(}\frac{1}{2}\sqrt{\frac{ SNR_{coded}}{2}}\Bigg{)}\Bigg{)}, \tag{14}\]  
and  
\[BER_{s}=1-(1-BER_{coded})^{m}, \tag{15}\]  
where erf in (13) and (14) represents the Gauss error function, which is given by \(\text{erf}(\alpha)=\frac{2}{\sqrt{\pi}}\int_{0}^{\alpha}\exp(-v^{2})dv\). The idea of using the \(G\) curves to enhance the bandwidth density can be explained as follows: If we consider a codeword length (n) of 127 with a code rate of 0.64, then from Figure 4, we find out that \(G\) is 2.7. This indicates that the signal-to-noise ratio level utilized to obtain the findings of Figure 2 can be decreased from 10 to 3.7 (\(10\div 2.7\)) while keeping  
Figure 4: Coding gain for six distinct codeword lengths (n) of RS-ECC at \(BER\) of \(10^{-12}\). the \(BER\) without any change. This reduction in the level of SNR is converted into a gain in the BD (\(G_{BD}\)), which is given by [18]  
\[G_{BD}=\frac{BD_{e}-BD_{u}}{BD_{uc}}, \tag{16}\]  
where \(BD_{uc}\) and \(BD_{e}\), respectively, are the uncoded BD and the effective BD (defined as the code rate multiplied by the coded bandwidth density (\(BD_{c}\))). Figure 5 demonstrates the relationship between the \(G_{BD}\) and the code rate for six different codeword lengths--7, 15, 31, 63, 127, and 255--with \(SNR_{uncoded}=10\) dB. It is clear that the coding rate is a crucial parameter that influences the \(G_{BD}\) upon coding and should be considered carefully in the design process. Low values of code rates provide a lot of redundancy, which in turn reduces the \(G_{BD}\). On the other hand, high values of code rates introduce inadequate error correction capability, which also limits the \(G_{BD}\). Therefore, a trade-off between high and low values of code rates should be considered. From Figure 5, each codeword length has an optimum code rate that needs to be selected in the design of optical interconnects to achieve the optimal \(G_{BD}\). For example, when the codeword length is 31, the optimum code rate is about 0.6, and it is about 0.68 for a codeword length of 255. One more observation from Figures 4 and 5 is that as the length of the codewords increases, the coding power improves in terms of both optimum coding gain and \(G_{BD}\). In the following, we consider the dependence of the BD of the fundamental-order Gaussian mode (TEM\({}_{00}\)) of the laser beam and the other investigated higher-order transverse modes on coding. Figure 6a demonstrates the BD of the TEM\({}_{00}\) mode before and after using Reed-Solomon error correction codes (RS-ECCs).