# An Improved Anti-Interference Precoding of Large-Scale Fading System Based on Channel Inversion
## 3 Advanced LSFP
### Advanced LSFP  
In this paper, channel inversion technology was applied to the ZF-LSFP and O-LSFP to obtain an improved anti-interference precoding named the Advanced LSFP (A-LSFP). Channel inversion belongs to precoding technology. In the downlink communication, channel inversion can be used to eliminate data interference at a base station. In this work, channel inversion and regularized channel inversion techniques are implemented in the multi-cell LSAS. Channel inversion is similar to the zero-forcing pre-equalization technique in the single-user MIMO systems, while regularized channel inversion is similar to the MMSE pre-equalization technique in the single-user MIMO systems. In the pre-equalization technology, weighting matrix \(W\) represents the pre-equalizing weighting matrix. When the zero-forcing pre-equalization is used, the corresponding weighting matrix \(W\) is expressed as:  
\[W=\beta/H_{D}^{DL} \tag{15}\]  
\[\beta=\sqrt{\frac{M}{Tr\left[\left(H_{D}^{DL}\right)^{-1}\times\left(H_{D}^{ DL}\right)^{H}\right]}} \tag{16}\]  
where \(\beta\) denotes a constant that keeps the total power constant after pre-equalization. By applying the channel inversion technology to the LSAS and substituting \(\beta\) into the LSFP matrix, the Advanced ZF-LSFP (A-ZF-LSFP) is obtained, which can be expressed as follows:  
\[W^{k}=\gamma_{A}*\beta^{\prime}*\left(B^{k}\right)^{-1} \tag{17}\]  
\[\beta^{\prime}=\sqrt{\frac{M}{Tr\left[\left(B^{k}\right)^{-1}\times\left(B^{k} \right)^{H}\right]}} \tag{18}\]  
where \(W^{k}\) denotes a large-scale fading factor matrix for channel inversion processing and \(\gamma_{A}\) denotes a normalization factor that guarantees power limitation. \(\beta^{\prime}\) represents a constant that keeps the total power constant after pre-equalization. Substitute the ZF-LSFP (A-ZF-LSFP) to which channel inversion technology is applied:  
\[SINR^{kl}=\frac{T_{3}^{kl}}{T_{1}^{kl}+T_{2}^{kl}} \tag{19}\]  
The SINR of the A-ZF-LSFP can be expressed as:  
\[SINR^{kl}=\frac{1}{T_{1}^{kl}}=\frac{MT}{Tr[\left(W\right)^{*}P^{kl}W]} \tag{20}\]  
In the LSAS, the channel inversion technology is applied to optimize precoding. The Optimized LSFP (O-LSFP), and the Advanced O-LSFP (AO-LSFP) can be obtained:  
\[W^{\prime}=\left[ \begin{array}{cc}\left(W^{1}\right)^{T}&\cdots&\left(W^{K} \right)^{T}\end{array} \right]^{T}=\left[ \begin{array}{cc}\left(\gamma_{A}* \beta*\left(B^{1}\right)^{-1}\right)^{T}&\cdots&\left(\gamma_{A}*\beta*\left(B ^{K}\right)^{-1}\right)^{T}\end{array} \right]^{T} \tag{21}\]  
\(W^{\prime}\) denotes the A-O-LSFP precoding matrix for channel inversion processing. Applying \(W^{\prime}\) to Lemmas 1 and 2, it is easy to get the SINR expression of A-O-LSFP. The SINR of the A-O-LSFP can be expressed as:  
\[SINR^{kl}=\frac{T_{3}^{kl}}{T_{1}^{kl}+T_{2}^{kl}} \tag{22}\]  
\[T_{1}^{kl}=\frac{Tr\left[\left(W^{\prime}\right)^{*}\beta^{kl}W^{\prime} \right]}{MT} \tag{23}\]  
\[T_{2}^{kl}=\sum_{\
u\
eq l}\left|BW^{\prime}{}_{(kl+L-l),\
u}\right|^{2} \tag{24}\]  
\[T_{3}^{kl}=\left|BW^{\prime}{}_{(kl+L-l),l}\right|^{2} \tag{25}\]  
\[B=Diag\left[ \begin{array}{cc}B^{1}&\cdots&B^{K}\end{array} \right] \tag{26}\]  
\[\rho_{j}^{kln}=\left[\frac{\delta_{\
u}^{2}}{\vartheta_{f}L}+\beta_{j}^{kl} \right]\left[\frac{\delta_{d}^{2}}{\vartheta_{r}}+T\sum_{s}\beta_{j}^{\mu \mathrm{s}}\right] \tag{27}\]  
\[P^{kln}=Diag\left[ \begin{array}{cc}\rho_{1}^{kln}&\cdots&\rho_{L}^{kln} \end{array} \right] \tag{28}\]  
\[P^{kl}=Diag\left[ \begin{array}{cc}P^{kl1}&\cdots&P^{klK}\end{array} \right] \tag{29}\]  
where \(B\) represents a large-scale fading factor matrix and \(P^{kl}\) represents a matrix obtained by using a large-scale fading factor \(\beta_{j}^{kl}\) operation.