## 2 System Model  
The spectral efficiency of the transmission link is improved by integrating the MIMO signal processing technique with HOM formats in the optical domain. In this work, a \((2\times 2)\) MIMO technique with an optical MMF link is implemented, and the corresponding system model is provided in Figure 1. This given optical MIMO transmission system model consists of two MIMO outputs \(n_{\text{R}}\) and two MIMO inputs \(n_{\text{T}}\), which are described as \((n_{\text{R}}\times n_{\text{T}})\). This model also incorporates the joint-PPE filter using an MMF channel with the PFS. The serial input data symbol \(s(k)\) is the result of concatenation of two distinct data frames D1 and D2 from two MIMO inputs \(s_{\mu}(k)\), respectively, with \(\mu=1,\ldots,n_{\text{T}}\) and \(s(k)=\big{(}s_{1}(k)\quad s_{2}(k)\ldots s_{\mu}(k)\big{)}\), where \(k\in\mathbb{N}\). Moreover, the symbol amplitudes \(s_{\mu}(k)\) at the \(\mu\)-th data frames are defined as \(s_{\mu}(k)\in\big{\{}s_{\mu,0},\ldots,s_{\mu,M-1}\big{\}}\), with \(k\in\mathbb{N}\) and \(k\) represents a discrete-time index. With the help of Gray coding, each symbol in \(s_{\mu}(k)\) is mapped to a PAM-\(M\) format, where \(M\) represents the constellation size. Overall, the functionalities of the MIMO transmitter include the concatenation of different frames into \(s(k)\), pre-equalizer \(p(k)\) implementation and serial to parallel conversion of pre-equalizer signal \(b(k)\), where \(b(k)=\big{(}b_{1}(k)\quad b_{2}(k)\cdots b_{\mu}(k)\big{)}\). One of the crucial components of the joint-PPE scheme is a pre-equalizer, which is characterized as a linear finite impulse response (FIR) filter. Consequently, \(s_{\mu}(k)\) undergoes pre-equalization, which results in generation of multi-level signals that are denoted as \(b_{\mu}(k)\) at the \(\mu\)-th input. The number of distinct amplitude levels \(N\) of \(b_{\mu}(k)\) is calculated as \(L_{\text{p}}=\log_{2}(N)\) with \(L_{\text{p}}\) corresponding to the number of the pre-equalizer filter taps. The pre-equalizer multi-level signal is defined as \(b_{\mu}(k)=\big{\{}b_{\mu,0},b_{\mu,1},\ldots,b_{\mu,N-1}\big{\}}\). The purpose of the pre-equalizer is to mitigate the adverse effects of ISI and ICI based on the estimated channel. Please note that prior to the transmission, the transmit power of the symbols with the highest amplitude at both MIMO inputs is restricted with respect to the maximum optical power \(P_{\mathrm{S,\mu}}^{\mathrm{(max)}}\) of the provided laser sources. This relation is analytically defined as  
\[s_{\mu,M-1}=b_{\mu,N-1}=\mathrm{q}\cdot P_{\mathrm{S,\mu}}^{\mathrm{(max)}}\enspace, \tag{1}\]  
with \(\mathrm{q}\) denoting the conversion factor from electrical to optical domain. While implementing HOM formats with multi-level signaling, the peak-to-average power ratio is held constant by using (1). After this power limitation, the multi-level signals \(b_{\mu}(k)\) are propagated across an MMF channel \(h_{\
u\mu}(k)\), where \(\
u=1,\ldots,n_{\mathrm{R}}\). The concept of MIMO technique is accomplished by using centric and eccentric light launching conditions using SDM, as demonstrated in Figure 2. Using the transverse spatial extent of an MMF, different mode groups of light are excited, which propagate in the fiber with distinct speeds. In Figure 2, the light is launched at the center of the optical fiber for the first MIMO input \(\mathrm{Tx}_{1}\), which results in excitation of lower-order modes. Moreover, the higher-order modes are excited at the second MIMO input \(\mathrm{Tx}_{2}\) with eccentric launching of the light using a \(15\,\upmu\) offset [14; 15]. Thereafter, the transmitted signal \(b_{\mu}(k)\) is degraded by the ISI and ICI due to the MMF channel. The frequency-selective behavior of the channel \(h_{\
u\mu}(k),\forall\,\
u,\,\mu\) produces the ISI. The independent data channels, which are created using SDM, are not ideally orthogonal. Hence, it results in the ICI, \(h_{\
u\mu}(k)\) with \(\
u\
eq\mu\).