## 3 Capacity Enhancement by Channel State Information
### Statistical Channel State Information  
To overcome the round-trip delay issue, we propose that the receiver computes an average of the channel that we note \(\overline{H}=\overline{U}\ \overline{\Lambda}\ \overline{V}^{\dagger}\), with \(\overline{U},\overline{V}\) are unitary matrices, \(\overline{\Lambda}=\text{diag}(\sqrt{\overline{\lambda}_{1}},...,\sqrt{ \overline{\lambda}_{M}})\) contains the long term eigenvalues of the channel. \(\overline{V}\) and \(\overline{\Lambda}\) represent the statistical channel state information. The matrix \(\overline{\Lambda}\) is used to compute the optimal powers \(\overline{p_{1}^{*}},...,\overline{p_{M}^{*}}\) per excited modes only once, and the matrix \(\overline{V}\) will be used by the transmitter to send correlated symbols vector independently of the channel variations. We refer to this capacity by \(C_{\overline{CSI}}\):  
\[C_{\overline{CSI}}=\mathbb{E}_{\lambda_{i}}\left(\sum_{i=1}^{M}\log_{2}\left( 1+\overline{p_{i}^{*}}\frac{\lambda_{i}}{2\sigma^{2}}\right)\right) \tag{8}\]  
To have an insight on the FMF channel capacity behavior, we simulate the previous capacities for the 6-mode and 10-mode fibers. Plotted results are averaged over \(10^{5}\) channel realizations. Figure 0(a) shows the complementary cumulative distribution function (CCDF) of the capacities given by Equations (5), (6) and (8) for the 6-mode fiber for \(\text{MDL}=15\ \text{dB}\). We notice that both \(C_{CSI}\) and \(C_{\overline{CSI}}\) are higher than \(C_{ep}\) and as the SNR increases the gain brought by CSI decreases. In Figure 0(b) we plotted the CSI gain defined as the ratio of \(C_{CSI}\) (resp. \(C_{\overline{CSI}}\)) to \(C_{ep}\) as a function of the SNR for different MDL values. We notice that the CSI gain increases with MDL and for higher SNRs the CSI gain vanishes. To focus on the proposed statistical CSI, we compare the average capacities of equal power allocation to the long term power allocation. The average channel \(\overline{H}\) is computed using \(10^{4}\) realizations of the instantaneous channel \(H\). Figure 1(a) shows the average capacities \(C_{ep}\) and \(C_{\overline{CSI}}\) for the 6-mode and 10-mode fibers as a function of SNR for MDL = 20 dB. We notice that the proposed statistical CSI increases the capacity for both fibers. Outage considerations are very important for the design of FMF optical systems. Since the channel changes randomly, transmitters are configured to encode for a fixed capacity that is higher than the worst case capacity called outage capacity to avoid system outage. The outage capacity \(C_{\text{out}}\) is related to an outage probability \(p_{\text{out}}\) as:  
\[\text{Pr}\left(C_{\text{inst}}(H)\leq C_{\text{out}}\right)=p_{\text{out}} \tag{9}\]  
where \(C_{\text{inst}}(H)\) is the instantaneous capacity defined withing the coherence time of the channel during which channel coefficients \(h_{ij}\) remain constants. In Figure 2b, we plotted the outage capacities for an outage probability of \(10^{-3}\) for the 6-mode and 10-mode fibers as a function of MDL for SNR = 10 dB. We compare the capacity with equal launch power to the capacity when using the long term optimal powers based on the statistical CSI. We notice that both capacities are decreasing with MDL, however \(C_{\overline{CSI}}\) is better than \(C_{ep}\). At MDL = 20 dB we have 0.3 bit/s gain for the 6-mode fiber and 0.5 bit/s gain for the 10-mode fiber. Consequently, since the availability of instantaneous CSI at the transmitter  
Figure 1: (**a**) Capacity complementary cumulative distribution function (CCDF) with channel state information (CSI) (solid), \(\overline{CSI}\) (dash-dot) and \(ep\) (dashed) for SNR = 10, 15, 20 dB and mode-dependent loss (MDL) = 15 dB; (**b**) Capacity gain as a function of SNR for MDL = 10, 20 dB for a 6-mode fiber. Figure 2: (**a**) Average capacities vs. SNR for \(M=6,10\) modes for MDL = 20 dB; (**b**) Outage capacities vs. MDL for \(M=6,10\) modes for SNR = 10 dB.