## 2 System Model  
The proposed transceiver-joint equalization technique in a MIMO system is analyzed using an electrical system model with \(n_{\text{T}}=n_{\text{R}}=2\), where \(n_{\text{T}}\) and \(n_{\text{R}}\) represent number of transmitters and receivers, respectively. In Figure 1, a \((n_{\text{R}}\times n_{\text{T}})\) MIMO transmission model with \(n_{\text{R}}=n_{\text{T}}=2\) is segregated into three parts, namely the transmitter, the channel, and the receiver. The symbols utilized in the system model are described in Table 1. The MIMO transmitter consists of a serial-to-parallel (S/P) converter and a pre-equalizer. Using the S/P converter, encoded serial binary data are transformed into \(n_{\text{T}}\) parallel data streams, where \(n_{\text{T}}\) represents the total number of MIMO inputs. The parallel symbol streams \(s_{\mu}(k)\), with \(\mu=1,2,\ldots,n_{\text{T}}\) and \(k\in\mathbb{N}\) are mapped on to \(M_{\mu}\)-ary PAM constellation. The bit-to-symbol mapping is conducted using Gray coding as it improves the system performance. The transmit symbol space of \(s_{\mu}(k)\in\{s_{\mu,0},s_{\mu,1},\ldots,s_{\mu,m}\}\), where \(m=\left[0,1,\ldots,M_{\mu}-1\right]\), consists of \(s_{\mu,m}\) symbols in accordance to the chosen \(M_{\mu}\)-ary PAM constellation. It is worth noting that the symbols in \(s_{\mu}(k)\) are independent and identically distributed. The amplitude of the symbols is defined as  
\[s_{\mu,m}=\left(\frac{m}{M_{\mu}-1}\right)\cdot P_{\text{s},\mu}^{\text{(max)} }\qquad,\text{where }M_{\mu}\
eq 1\enspace. \tag{1}\]  
In (1), \(P_{\text{s},\mu}^{\text{(max)}}\) is the maximum transmit power at the \(\mu\)-th MIMO input. The constellation diagram is modeled to provide the maximum power to the last symbol irrespective of the selected PAM format. Consequently, the parallel symbol streams are pre-equalized using the pre-equalizer \(p_{\mu}(k)\) at the \(\mu\)-th input. Therefore, \(s_{\mu}(k)\) is converted into pre-equalized multi-level signals \(b_{\mu}(k)\). The total number of distinct pre-equalized signal levels \(N_{\mu}\) at the \(\mu\)-th input is defined as \(N_{\mu}=2^{l_{\text{P}}}\), where \(L_{\text{P}}\) represents the number of pre-equalizer filter taps. The resulting pre-equalized signal \(b_{\mu}(k)=\left[b_{\mu,0},b_{\mu,1},\ldots,b_{\mu,N_{\mu}-1}\right]\) complies with the power constraint, which is given as  
\[s_{\mu,M_{\mu}-1}=b_{\mu,N_{\mu}-1}=\text{q}\cdot P_{\text{s},\mu}^{\text{( max)}}\enspace, \tag{2}\]  
Figure 1: \((2\times 2)\) MIMO system model description with a frequency-selective channel incorporated with the PPE scheme. where q is a conversion factor for converting an electrical signal into an optical signal. Although implementing the pre-equalization, a constant peak-to-average power ratio is also maintained in (2). Subsequently, the multi-level pre-equalized signal is propagated through the MIMO channel, which is an MMF link. The transmitted signal is deteriorated with the inter-symbol interference (ISI) and the inter-channel interference (ICI) originating from the MMF channel \(h_{\
u\mu}(k)\), where \(\
u=1,2,\ldots,n_{\text{R}}\) and the number of MIMO outputs is denoted by \(n_{\text{R}}\). The temporal pulse broadening and chirping occur due to the frequency-selective behavior of the MMF channel \(h_{\
u\mu}(k)\). Moreover, the indirect components of the channel \(h_{\
u\mu}(k)\), when \(\
u\
eq\mu\), are responsible for the crosstalk between the MIMO layers. An MMF channel supports several guided modes and provides easier light-launching conditions into the fiber due to the larger core diameter. Hence, the parallel data streams are generated by offset light-launching conditions to achieve a higher spectral efficiency. This technique is known as SDM using an MMF [6]. In the given system model, the first layer is created by launching the light into the center of the MMF core. Similarly, the second MIMO layer is formed by a 15 \(\upmu\)m offset launching condition into the fiber core, as shown in Figure 2.