# Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels
## 3 Long-Haul Optical Link MIMO Channel Model  
In this section we describe the multi-section optical channel model used in this work. The effect of the channel noise is discussed separately in Section 4. The relationship between the input vector \(\underline{x}(\omega)=[x_{1}(\omega),x_{2}(\omega),\ldots,x_{D}(\omega)]^{T}\) of complex electric field amplitudes of each of the \(D\) modes propagating along the fiber, and the corresponding output vector \(\underline{y}(\omega)=[y_{1}(\omega),y_{2}(\omega),\ldots,y_{D}(\omega)]^{T}\) can be modeled, after neglecting non-linear effects, as a multiple-input multiple-output linear system \(\underline{H}_{tot}(\omega)\)[2]:  
\[\underline{y}(\omega)=\underline{H}_{tot}(\omega)\underline{x}(\omega), \tag{4}\]  
where \(\underline{H}_{tot}(\omega)\) is a \(D\times D\) matrix that models the signal propagation along the channel. For \(D=2\), the system is equivalent to a classical PDM over a SMF, and \(\underline{H}_{tot}(\omega)\) takes the form of the Jones matrix [8]. For \(D>2\), extensions to the Jones matrix have been proposed to be adequate for the SDM model [19; 20]. In the case of long-haul systems, \(\underline{H}_{tot}(\omega)\) can be further modeled as a concatenation of \(K_{amp}\) spans composed of the optical fiber and an optical amplifier [2; 18; 33; 34]. Hence, the whole channel transfer function can be written as:  
\[\underline{H}_{tot}(\omega)=H_{\text{CD}}(\omega)\cdot\underline{H}(\omega), \tag{5}\]  
where \(H_{\text{CD}}(\omega)=e^{\left(-\frac{j}{2}\omega^{2}\beta_{2}\ell_{tot}\right)}\) is a single-input single-output (SISO) term that models the mode-averaged distortion due to CD, \(\bar{\beta}_{2}\) represents the mode-averaged CD per unit length, and \(\ell_{tot}\) denotes the total link length. The matrix \(\underline{H}(\omega)\) includes inter-mode cross-talk, MDL and MD effects of the complete MIMO system. Equation (5) can be written as a product over the \(K_{amp}\) spans:\[\underline{\underline{H}}(\omega)=\prod_{k=1}^{K_{\text{amp}}}\underline{\underline{ H}}^{(k)}(\omega), \tag{6}\]  
where \(\underline{\underline{H}}^{(k)}(\omega)\) is the channel response of the \(k\)th span. We use \(k\in\{1,\ldots,K_{amp}\}\) to index the spans in the optical channel. We can write out \(\underline{\underline{H}}^{(k)}(\omega)\) as [20; 34]:  
\[\underline{\underline{H}}^{(k)}(\omega)=\underline{\underline{V}}^{(k)} \underline{\underline{\Delta}}^{(k)}(\omega)(\underline{\underline{U}}^{(k)} )^{H}, \tag{7}\]  
where the diagonal matrix \(\underline{\underline{\Delta}}^{(k)}(\omega)\) for a given span \(k\) includes the MDL effects and the MD of each mode w.r.t. the mode-averaged value [33], and can be expressed as:  
\[\underline{\underline{\Delta}}^{(k)}(\omega)=\text{diag}\bigg{(}\bigg{[}e \big{(}\{g_{1}^{(k)}-j\omega\tau_{1}^{(k)}\}\big{)},\ldots,e^{\big{(}\frac{1}{ g_{D}^{(k)}}-j\omega\tau_{D}^{(k)}\}\big{)}\bigg{]}\bigg{)}, \tag{8}\]  
being \(\underline{g}^{(k)}=[g_{1}^{(k)},g_{2}^{(k)},\ldots,g_{D}^{(k)}]\) the uncoupled modal gains and \(\underline{\underline{\tau}}^{(k)}=[\tau_{1}^{(k)},\tau_{2}^{(k)},\ldots, \tau_{D}^{(k)}]\) uncoupled modal group delays. We assume that the uncoupled modal group-velocity dispersion is equal to zero for all the \(k\) spans [33]. The \(k\)-th span mode coupling is modeled by the frequency-independent \(\underline{\underline{V}}^{(k)}\) and \(\underline{\underline{U}}^{(k)}\) matrices.