## 2 System Model
### Quality Criteria  
The BER is considered to be the key indicator to evaluate the performance of an MMF channel. The transmission quality \(\varrho_{\mu}\) of the \(\mu\)-th MIMO input is defined as  
\[\varrho_{\mu}=\frac{U_{\rm A,\mu}^{2}}{\widetilde{P}_{\rm n,\mu}}\enspace, \tag{8}\]  
where \(U_{\rm A,\mu}\) represents the half-vertical eye-opening of the \(\mu\)-th input and the noise power after the post-equalization is defined with \(\widetilde{P}_{\rm n,\mu}\). From (7) and (8), the relationship between \(\theta_{\mu}\) and \(\widetilde{P}_{\rm n,\mu}\) is given by  
\[\widetilde{P}_{\rm n,\mu}=\theta_{\mu}\cdot P_{\rm n}\enspace. \tag{9}\]  
By substituting (9) into (8), the resulting transmission quality at the \(\mu\)-th input is  
\[\varrho_{\mu}=\frac{U_{\rm A,\mu}^{2}}{\widetilde{P}_{\rm n,\mu}}=\frac{U_{ \rm A,\mu}^{2}}{\theta_{\mu}\cdot P_{\rm n}}\enspace. \tag{10}\]  
In (10), the noise amplification factor is inversely proportional to the transmission quality. Typically, for an SDM-based MIMO system, the BER performance \(P_{\rm BER}^{(\mu)}\) of the \(\mu\)-th input is  
\[P_{\rm BER}^{(\mu)}=\frac{1}{\log_{2}(M_{\mu})}\left(\frac{M_{\mu}-1}{M_{\mu}} \right)\text{erfc}\!\left(\sqrt{\frac{\varrho_{\mu}}{2}}\right)\enspace, \tag{11}\]  
with \(\text{erfc}\!\left(\cdot\right)\) denoting the complimentary error function, the constellation size and the transmission quality at the \(\mu\)-th input is represented by \(M_{\mu}\) and \(\varrho_{\mu}\), respectively.  
By combining (10) with (11), the final BER performance is evaluated as  
\[P_{\rm BER}^{(\mu)}=\frac{1}{\log_{2}(M_{\mu})}\left(\frac{M_{\mu}-1}{M_{\mu}} \right)\text{erfc}\!\left(\sqrt{\frac{U_{\rm A,\mu}^{2}}{2\cdot\theta_{\mu} \cdot P_{\rm n}}}\right)\enspace. \tag{12}\]  
In (11), when an error is encountered during the transmission, it is assumed that the error will only be limited to its neighboring symbol, i.e., only a one-bit error due to the use of Gray coding. However, more-than-one-bit errors can occur at low received power due to the high noise variance. Second, the noise variance of each symbol is considered to be identical in (11). In contrast, the noise variances are different in the shot noise dominant region. Hence, the area integration method is adapted to accurately estimate the BER incorporating probability density function (PDF). The symbol-specific BER \(P_{\rm BER}^{(s_{\mu,m})}\) is calculated as  
\[P_{\rm BER}^{(s_{\mu,m})}=\frac{1}{\log_{2}(M_{\mu})}\sum_{i=0}^{M_{\mu}-1}h_{ \mu,i}^{(n)}\cdot\int\limits_{\widetilde{R}_{i}}p(y_{\
u,m})\,\mathrm{d}y\enspace, \tag{13}\]where \(h_{m,i}\) represents the Hamming distance between the symbols \(s_{\mu,m}\) and \(s_{\mu,i}\) at \(\mu\)-th input. The PDF of the received symbol \(y_{\
u,m}\) is described as \(p(y_{\
u,m})\) and \(R_{i}\) is the decision region. The overall BER is defined as  
\[P_{\text{BER}}^{(\mu)}=\frac{1}{M_{\mu}}\sum_{m=0}^{M_{\mu}-1}P_{\text{BER}}^{(s _{\mu,m})}\enspace. \tag{14}\]  
In summary, the system performance will be degraded due to the noise amplification originating from the extensive utilization of the post-equalizer. Theoretically, this deterioration in the system performance is shown in (12). From an analytical perspective, it is worth investigating, in order to practically estimate the usefulness of shifting the post-equalizer taps to the transmitter side with the given system model. The challenges of designing the optimal PPE filter and evaluating the system performance are addressed in the following sections.