# Matched Filtering for MIMO Coherent Optical Communications with Mode-Dependent Loss Channels
## 5 Numerical Simulation of Linear MIMO FSE Receiver for MDL-Impaired Optical Channel
### Channel Model  
We decide to carry out the numerical simulations of the \(\underline{H}(\omega)\) channel model described in (6) in the time domain, so that the relative delays of the different modes can be easily  
Figure 4: MIMO coherent optical communication system model with matched filter-based receiver in the absence of mode-dependent loss (MDL) (**a**) and its equivalent discrete-time system model (**b**). Figure 5: SDM communication system model with linear fractionally-spaced equalizer (FSE) MIMO receiver and integer oversampling rate \(r_{\omega 0}\). described as a time shift between them. The different mode amplitudes can also be handled simply by a diagonal matrix. The chromatic dispersion, which have a SISO frequency response \(H_{CD}(\omega)\) that does not depend on the mode \(i\in\{0,\ldots,D\}\), is represented as [7]:  
\[H_{CD}(\omega)=e^{-j\beta\frac{\omega_{2}^{2}}{\tau}}, \tag{34}\]  
where \(\beta=\bar{\beta}_{2}\ell_{tot}\), and \(\ell_{tot}=K_{amp}\ell_{span}\) when all spans are considered of equal length. The MDL effect is modeled with an amplification factor for each mode and each optical amplifier (located at the end of each span). These factors are considered time-invariant for a given channel realization in the form of a vector for the \(k\)-th span \(\underline{g}^{(k)}=[g_{1}^{(k)},g_{2}^{(k)},\ldots,g_{D}^{(k)}]\), where \(g_{i}^{(k)}\) for \(i\in\{0,\ldots,D\}\) is expressed in dB and taken from a Gaussian distribution with zero mean and standard deviation (STD) \(\sigma_{g}\). The sum of all factors \(\sum_{i=1}^{D}g_{i}^{(k)}\) is set to 0 for normalization purposes. Hence, the amplitudes matrix of the \(k\)-th span, frequency independent, is given by:  
\[\underline{\underline{A}}^{(k)}=\text{diag}\bigg{(}\bigg{[}e^{\big{(}\frac{ 1}{2}g_{1}^{(k)}\big{)}},\ldots,e^{\big{(}\frac{1}{2}g_{D}^{(k)}\big{)}} \bigg{]}\bigg{)}. \tag{35}\]  
Alternatively, for each span \(k\) of the communication link we have the delays matrix:  
\[\underline{\underline{\Delta}}^{(k)}(\omega)=\underline{\underline{A}}^{(k)} \cdot\text{diag}\bigg{(}\bigg{[}e^{-j\omega\tau_{1}^{(k)}},\ldots,e^{-j\omega \tau_{D}^{(k)}}\bigg{]}\bigg{)}, \tag{36}\]  
being \(\underline{\underline{\chi}}^{(k)}=[\tau_{1}^{(k)},\tau_{2}^{(k)},\ldots,\tau _{D}^{(k)}]\) the vector that models the MD with group delays for each mode of the \(k\)-th span. To obtain the delays, we generate the first \(D/2\) values of \(\underline{\underline{\chi}}^{(k)}\) from a Gaussian distribution with STD \(\sigma_{gd}\), and the second \(D/2\) values are taken as the opposite of these, which satisfies that \(\sum_{i=1}^{D}\tau_{i}^{(k)}=0\), since we consider that the system uses polarization multiplexing as part of the SDM [40]. The time-domain impulse response for each of the spans \(k\) is calculated by applying the inverse Fourier transform to (7) and can be expressed as:  
\[\underline{\underline{h}}^{(k)}(t)=\underline{\underline{V}}^{(k)}\underline {\underline{A}}^{(k)}\underline{\underline{d}}^{(k)}(t)\underline{\underline{ U}}^{(k)}, \tag{37}\]  
where  
\[\underline{\underline{d}}^{(k)}(t)=\text{diag}\Big{(}\Big{[}\delta(t-\tau_{1} ^{(k)}),\ldots,\delta(t-\tau_{D}^{(k)})\Big{]}\Big{)}, \tag{38}\]  
and we have used that the matrices \(\underline{\underline{A}}^{(k)},\underline{\underline{U}}^{(k)}\), and \(\underline{\underline{V}}^{(k)}\) are constant. Equation (37) describes that incoming signal at the \(k\)th span is multiplied by the unitary matrix \(\underline{\underline{U}}^{(k)}\), then each modal impulse response is delayed by \(\tau_{i}^{(k)}\), the amplification factor is set by the diagonal matrix \(\underline{\underline{A}}^{(k)}\) and the mode-mixing unitary matrix \(\underline{\underline{V}}^{(k)}\) is applied. Finally, the impulse response of the complete channel is given by:  
\[\underline{\underline{h}}_{tot}(t)=\underline{\underline{h}}^{(K_{amp})}(t)* \underline{\underline{h}}^{(K_{amp}-1)}(t)*\cdots*\underline{\underline{h}}^ {(1)}(t)*h_{CD}(t), \tag{39}\]  
where \(h_{CD}(t)=\mathcal  
to 1.