# Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels
## 4 SDM Communication System Model
### Matched Filter-Based Receiver for SDM
When the condition expressed in (27) is satisfied for all the spans of the system, we can commute the terms in (25), and therefore, we can obtain:  
\[\underline{\underline{H}}(\omega)\cdot\underline{\underline{H}}(\omega)^{H} =\prod_{k=1}^{K_{Amp}}|\underline{\Delta}^{(k)}(\omega)|^{2}=\prod_{k=1}^{K _{Amp}}e^{\big{(}\underline{s}^{(k)}\big{)}}\cdot\underline{I}=e^{\big{(} \underline{s}_{k-1}^{K_{Amp}}\,g^{(k)}\big{)}}\cdot\underline{I}. \tag{29}\]  
Revisiting (23), and plugging in (29) under the assumption of a negligible MDL in the optical channel, we can write:  
\[\underline{\underline{G}}(\omega)=\underline{\underline{Q}}(\omega)\cdot \underline{\underline{Q}}^{H}(\omega)=|P(\omega)|^{2}\cdot e^{\big{(} \underline{s}_{k-1}^{K_{Amp}}\,g^{(k)}\big{)}}\cdot\underline{I}. \tag{30}\]  
Therefore, without loss of generality, \(e^{\big{(}\underline{s}_{k-1}^{K_{Amp}}\,g^{(k)}\big{)}}=1\) can be assumed and, after sampling at the symbol rate, (30) can be written as:  
\[\underline{\underline{G}}(\Omega)=\frac{1}{T}\cdot\sum_{l=-\infty}^{\infty} \bigg{|}P\bigg{(}\frac{\Omega+2\pi l}{T}\bigg{)}\bigg{|}^{2}\cdot\underline{I }=\underline{I}, \tag{31}\]  
where we have used that \(p(t)\) defined in Equation (12) is a square-root raised cosine pulse satisfying the Nyquist criterion. Regarding the filtered noise waveforms \(z_{1}(t)\) to \(z_{D}(t)\) in Figure 4a, they have an autocorrelation function matrix \(\underline{\underline{R}}_{zz}(t)\), whose Fourier transform pair \(\underline{\underline{S}}(\omega)\) can be expressed as:  
\[\underline{\underline{S}}_{z}(\omega)=\underline{\underline{Q}}(\omega)\cdot \underline{\underline{S}}_{z}(\omega)\cdot\underline{\underline{Q}}^{H}( \omega), \tag{32}\]  
where \(\underline{\underline{S}}_{z}(\omega)\) is the Fourier transform of the autocorrelation function matrix \(\underline{\underline{R}}_{nm}(t)=\underline{I}\cdot\frac{N_{0}}{2}\cdot \delta(t)\) of the received noise vector \(\underline{u}(t)=\left[n_{1}(t),n_{2}(t),\ldots,n_{D}(t)\right]^{T}\). We remind that the noise components of the noise vector \(\underline{u}(t)\) were assumed uncorrelated with identical power in each mode equal to \(N_{0}/2\). Using (30) and (32) leads to:  
\[\underline{\underline{S}}_{z}(\omega)=\frac{N_{0}}{2}\cdot \underline{\underline{Q}}(\omega)\cdot\underline{\underline{Q}}^{H}(\omega)= \frac{N_{0}}{2}\cdot|P(\omega)|^{2}\cdot e^{\big{(}\underline{s}_{k-1}^{K_{Amp }}\,g^{(k)}\big{)}}\cdot\underline{I}. \tag{33}\]Using the previous assumptions about gains \(g^{(k)}\) and \(P(\omega)\) made before, we obtain that the sampled noise vector \(\underline{z}[n]=[z_{1}[n],\ldots,z_{D}[n]]^{T}\) has an autocorrelation matrix function \(\underline{R}_{z}[n]=\underline{l}\cdot\frac{N_{0}}{2}\cdot\delta[n]\). Therefore, we can conclude that a \(D\times D\) MIMO coherent optical communication system using a continuous-time matched filter as a receiver completely eliminates channel ISI and crosstalk when the MDL in the channel is negligible. Moreover, the equivalent discrete-time system model reduces to \(D\) discrete parallel AWGN channels as shown in Figure 3(b). Hence, there would be no loss of performance w.r.t. the AWGN channel without distortion.