## 2 System Model
### Analytical Description of the System Model  
A frequency-selective channel-based \((n_{\text{T}}\times n_{\text{R}})\) MIMO system with \(n_{\text{R}}=n_{\text{T}}=2\), as shown in Figure 1, is analytically represented as  
\[\mathbf{r}=\mathbf{H}\cdot\mathbf{b}+\mathbf{w}\enspace. \tag{3}\]  
From (3), the pre-equalized transmitted signal is represented as \(\mathbf{b}\) with \((n_{\text{T}}\times 1)\) dimensions. The \((n_{\text{R}}\times n_{\text{T}})\) channel matrix \(\mathbf{H}\) describes the block-oriented system model, and is defined as  
\[\mathbf{H}=\begin{bmatrix}\mathbf{H}_{11}&\cdots&\mathbf{H}_{1n_{\text{T}}}\\ \vdots&\ddots&\vdots\\ \mathbf{H}_{n_{\text{R}}1}&\cdots&\mathbf{H}_{n_{\text{R}}n_{\text{T}}}\end{bmatrix}\enspace, \tag{4}\]  
where \(\mathbf{H}_{n_{\text{R}}n_{\text{T}}}\) is the SISO channel matrix, which characterizes the influence of the channel over the pre-equalized transmitted signal \(\mathbf{b}\). The \((n_{\text{R}}\times 1)\) noise vector \(\mathbf{w}\) pertains to the noise in an IM/DD system. Finally, the received signal is represented as \(\mathbf{r}\) with \((n_{\text{R}}\times 1)\) dimensions. The transmitted signal \(\mathbf{b}\) is decomposed as  
\[\mathbf{r}=\mathbf{H}\cdot\mathbf{P}\cdot\mathbf{s}+\mathbf{w}\enspace, \tag{5}\]  
where \(\mathbf{P}\) is the pre-equalizer with a size of \((n_{\text{T}}\times n_{\text{T}})\), and \(\mathbf{s}\) represents the input vector of \((n_{\text{T}}\times 1)\). For a limited increase in the complexity, the pre-equalizer \(\mathbf{P}\) is considered to be a diagonal matrix. Ultimately, the post-equalization is executed to mitigate the residual interference. The final expression of the given system model with PPE is formulated as  
\[\mathbf{y}= \mathbf{F}\cdot\mathbf{r}\enspace, \tag{6}\] \[\mathbf{y}= \mathbf{\frac{\mathbf{F}\cdot\mathbf{H}\cdot\mathbf{P}}{\text{Term 1}}}\cdot\mathbf{s}+ \mathbf{\frac{\mathbf{F}\cdot\mathbf{w}}{\text{Term 2}}}\enspace,\]  
with \(\mathbf{F}\) denoting the post-equalizer with \((n_{\text{R}}\times n_{\text{R}})\) dimensions. In (6), the first term implies that the channel influence can be mitigated with the aid of PPE. Similarly, the noise amplification due to post-equalization is evident in (6). The system performance is adversely degraded due to noise enhancement from post-equalization. Hence, a well-designed pre-equalizer will be capable of compensating for the channel along with negligible noise  
Figure 2: In (**a**), the transmitter’s configuration for launching the light into an MMF core is depicted. Measured mean power distribution with (**b**) centric (offset: 0 μm) and (**c**) eccentric (offset: 15 μm) light-launching conditions into an MMF core, which is represented by the white dashed circle.  
amplification. When a trade-off between complexity and cost-effectiveness is considered, a pre-equalizer with a few taps is preferred in conjunction to a post-equalizer.  
The quantification of the noise amplification due to the post-equalization filter taps is expressed with the noise weighting factor (NWF) \(\theta_{\mu}\) and is given as  
\[\theta_{\mu}=\sum_{i=1}^{n_{\rm R}}\sum_{k=0}^{L_{\rm f}-1}|f_{\mu\
u}[k]|^{2} \qquad\text{for}\quad\mu=1,2,\ldots,n_{\rm T}\enspace, \tag{7}\]  
where \(L_{\rm f}\) is the number of post-equalization taps and \(f_{\mu\
u}[k]\) represents the post-equalizer taps for \(\mu\)-th input and \(\
u\)-th output. From (7), it is evident that the amplitude of the post-equalization taps is responsible for the noise amplification.