## 4 Interference-Aware Clipping Scheme
### Clipping Ratio Decision Criterion  
The clipped transmitter (TX) signal in the frequency domain, \(\widehat{X}_{t}[k]\), is the combination of message signals, \(S_{l}[k],l=1,2,\ldots,U\), and the relationship can be represented as  
\[\widehat{X}_{t}[k]=\alpha_{t}\sum_{l=1}^{U}W_{t,l}[k]S_{l}[k]+D_{t}^{\prime}[k]. \tag{16}\]  
The \(U\times N_{t}\) channel matrix, \(\mathbf{H}[k]\) can be represented as,  
\[\mathbf{H}[k]=\left( \begin{array}{cccc}H_{1,1}[k]&H_{1,2}[k]&\cdots&H_{1,N_{ t}}[k]\\ H_{2,1}[k]&H_{2,2}[k]&\cdots&H_{2,N_{t}}[k]\\ \vdots&\vdots&\ddots&\vdots\\ H_{U,1}[k]&H_{U,2}[k]&\cdots&H_{U,N_{t}}[k]\end{array} \right) \tag{17}\]  
\[=\left( \begin{array}{c}\mathbf{H}_{1,:}[k]\\ \mathbf{H}_{2,:}[k]\\ \vdots\\ \mathbf{H}_{U,:}[k]\end{array} \right),\]  
where \(\mathbf{H}_{i,:}[k]\) is the \(1\times N_{t}\) channel vector for the \(i\)th user. The \(N_{t}\times U\) precoding matrix, \(\mathbf{W}[k]\) can be represented as  
\[\mathbf{W}[k]=\left( \begin{array}{cccc}W_{1,1}[k]&W_{1,2}[k]&\cdots&W_{1,U}[ k]\\ W_{2,1}[k]&W_{2,2}[k]&\cdots&W_{2,U}[k]\\ \vdots&\vdots&\ddots&\vdots\\ W_{N_{t},1}[k]&W_{N_{t},2}[k]&\cdots&W_{N_{t},U}[k]\end{array} \right) \tag{18}\]  
\[=\left(\mathbf{W}_{:,1}[k]\quad\mathbf{W}_{:,2}[k]\quad\cdots\quad\mathbf{W} _{:,U}[k]\right),\]  
where \(\mathbf{W}_{:,t}[k]\) is the \(N_{t}\times 1\) channel vector for the \(i\)th user. Then, the effective channel at the receiver after precoding, \(\mathbf{H}[k]\cdot\mathbf{W}[k]\), can be represented as  
\[\mathbf{H}[k]\cdot\mathbf{W}[k]=\left( \begin{array}{c}\mathbf{H}_{1,:}[k] \\ \mathbf{H}_{2,:}[k]\\ \vdots\\ \mathbf{H}_{U,:}[k]\end{array} \right)\cdot\left(\mathbf{W}_{:,1}[k]\quad \mathbf{W}_{:,2}[k]\quad\cdots\quad\mathbf{W}_{:,U}[k]\right)\]  
\[=\left( \begin{array}{cccc}\mathbf{H}_{1,:}[k]\cdot\mathbf{W}_{:,1}[k]& \mathbf{H}_{1,:}[k]\cdot\mathbf{W}_{:,2}[k]&\cdots&\mathbf{H}_{1,:}[k]\cdot \mathbf{W}_{:,U}[k]\\ \mathbf{H}_{2,:}[k]\cdot\mathbf{W}_{:,1}[k]&\mathbf{H}_{2,:}[k]\cdot\mathbf{W }_{:,2}[k]&\cdots&\mathbf{H}_{2,:}[k]\cdot\mathbf{W}_{:,U}[k]\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{H}_{i,:}[k]\cdot\mathbf{W}_{:,1}[k]&\mathbf{H}_{i,:}[k]\cdot\mathbf{W }_{:,2}[k]&\cdots&\mathbf{H}_{i,:}[k]\cdot\mathbf{W}_{:,U}[k]\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{H}_{U,:}[k]\cdot\mathbf{W}_{:,1}[k]&\mathbf{H}_{U,:}[k]\cdot\mathbf{W }_{:,2}[k]&\cdots&\mathbf{H}_{U,:}[k]\cdot\mathbf{W}_{:,U}[k]\end{array} \right). \tag{19}\]  
Based on Equation (19), after compensating the \(\alpha_{t}\), the symbol received by the \(i\)th user is given by\[ \begin{array}{rcl}Y_{i}[k]&=&\sqrt{\frac{\mathbf{P}_{k}N_{t}}{K}}\mathbf{H}_{i,: }[k]\mathbf{W}_{:,l}[k]S_{i}[k]\\ &+&\sqrt{\frac{\mathbf{P}_{k}N_{t}}{K}}\sum_{l\
eq i}\mathbf{H}_{i,:}[k]\mathbf{ W}_{:,l}[k]S_{l}[k]\\ &+&\sqrt{\frac{\mathbf{P}_{k}N_{t}}{K}}\mathbf{H}_{i,:}[k]\mathbf{D}[k]+N_{i}[k],\end{array}  \tag{20}\]  
where \(\mathbf{H}_{i,:}[k]\) is the \(1\times N_{t}\) channel vector of the \(k\)-th subcarrier for the \(i\)-th user, \(\mathbf{W}_{:,l}[k]\) is the \(N_{t}\times 1\) precoding vector of \(k\)-th subcarrier for the \(i\)-th user, \(S_{i}\) is the \(k\)-th subcarrier message signal for the \(i\)-th user, \(\mathbf{D}[k]\) is the \(N_{t}\times 1\) clipping distortion vector, and \(N_{i}[k]\) is the \(k\)-th subcarrier AWGN for \(i\)-th user. The second line of Equation (20) is the IUI term.