## 4 Interference-Aware Clipping Scheme
### Clipping Ratio Decision Criterion
From Equation (20), the effective signal-to-interference and noise ratio (SINR), \(\gamma_{k}\), can be represented as follows:  
\[\gamma_{k}[k]=\frac{\frac{\mathbf{P}_{k}N_{t}}{K}|\mathbf{H}_{i,:}[k]\mathbf{ W}_{:,l}[k]|^{2}}{\frac{\mathbf{P}_{k}N_{t}}{K}\left[\,|\sum_{l\
eq i}\mathbf{H}_{i,: }[k]\mathbf{W}_{:,l}[k]|^{2}+|\mathbf{H}_{i,:}[k]|\mathbf{D}[k]|^{2}\right]+N_ {0}B}. \tag{21}\]  
Compared to the IUI, the clipping noise can be considered to be negligible in LS-MU-MIMO-OFDM, if the following condition is satisfied:  
\[|\mathbf{H}_{i,:}[k]\mathbf{D}[k]|^{2}\ll|\sum_{l\
eq i}\mathbf{H}_{i,:}[k] \mathbf{W}_{:,l}[k]|^{2}. \tag{22}\]  
The example for the validation of Equation (22) will be shown in Table 2.  
Since \(\mathbf{H}_{i,:}[k]\) has zero mean and unit variance i.i.d. Rayleigh distribution characteristics, and is independent from \(\mathbf{D}[k]\), the expectation of (22) can be approximated as  
\[E\left[|\mathbf{D}[k]|^{2}\right]\ll E\left[\,|\sum_{l\
eq i}\mathbf{H}_{i,:} [k]\mathbf{W}_{:,l}[k]|^{2}\right]. \tag{23}\]  
In the case of MF precoding, Equation (23) can be rewritten as  
\[E\left[\,|\mathbf{D}[k]|^{2}\right]\ll E\left[\,\frac{|\sum_{l\
eq i}\mathbf{ H}_{i,:}[k]\mathbf{H}_{:,l}^{H}[k]|^{2}}{N_{t}}\right], \tag{24}\]  
and it means that the \(\
u\) can be decided based on IUI and/or the channel condition. Based on the result, we can decide the decision boundary of \(\
u\). The amount of clipping distortion can be confined as  
\[\frac{M_{unit}}{N_{t}}\cdot E\left[\,|\sum_{l\
eq i}\mathbf{H}_{i,:}[k] \mathbf{H}_{:,l}^{H}[k]|^{2}\right]\leq E\left[|\mathbf{D}[k]|^{2}\right]\leq \frac{M_{max}}{N_{t}}\cdot E\left[\,|\sum_{l\
eq i}\mathbf{H}_{i,:}[k]\mathbf{ H}_{:,l}^{H}[k]|^{2}\right]. \tag{25}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(N_{t}\) & \(\tilde{\xi}\) & \(\
u\) & \(\xi\) \\ \hline
50 & 0.7352 & 3 & 0.0619 \\
100 & 0.3852 & 4 & 0.0270 \\
200 & 0.1897 & 5 & 0.0108 \\
300 & 0.1333 & 6 & 0.0038 \\
400 & 0.0897 & 7 & 0.0009882 \\
500 & 0.0776 & 8 & 0.0002922 \\
600 & 0.0646 & 9 & 0.0000276 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: An Example of a Look-UP Table (LUT) to determine \(\
u\). Note that the value of \(M_{min}\) should be decided sufficiently high for a satisfactory energy efficiency increment, and also \(M_{max}\) should be decided low enough to satisfy the error vector magnitude (EVM) requirement. We use the IUI term in the lower bound for consistency. It is obvious that the normalized clipping distortion power (when the TX power is unity), \(D^{2}\) is a function of \(\
u\), i.e., \(f(\
u)=D^{2}\), and as shown in Equation (15). The relationship between \(\
u\) and \(D^{2}\), where \(\
u\) is increased as \(D^{2}\) is decreased, will be shown in the following subsection. We can decide a relevant \(\
u\) based on Equation (25) as follows:  
\[\
u\to f^{-1}\left(\frac{\varphi}{N_{t}}\cdot E\left[|\sum_{l\
eq i}\mathbf{H} _{i,:}[k]\mathbf{H}_{,l}^{H}[k]|^{2}\right]\right). \tag{26}\]  
Note that the determination of constant \(\varphi\) is the system designer's choice. Here, we can regard the subcarrier index, \(k\), as arbitrary, since the clipping noise has fallen onto all subcarriers randomly. In a real system, Equation (26) can be implemented by using a look-up table (LUT). One example of a LUT is shown in Table 2.  
In the table, the left column shows the normalized IUI, \(\zeta\) for each \(N_{t}\), and the right column shows the normalized distortion noise, \(\zeta=D^{2}\) for each \(\
u(dB)\). A system designer determines \(\
u\) depending on the normalized IUI power \(\zeta\) and \(\zeta\). Note that the condition for satisfactory performance is \(\zeta>>\zeta\). It will be more clear if you see the performance analysis in the following section.