## 3 Signal Model
### MU-Network MIMO Signal Model  
Consider a 3-cell coordinated MU-network MIMO as shown in Figure 1b, where 3 cooperative BSs together serve a group of their \(N_{CEU}\) CEUs using the whole outer sub-band \(f_{B2}\). Assume a proper cyclic-prefix length is appended to the inverse discrete Fourier transform (IDFT) sequence such that the inter-symbol interference can be ignored. After DFT, the signal reception at subcarrier \(k\) by receiver antenna \(v\) of CEU \(u\) is:  
\[ \begin{split} Y_{k}^{(u,v)}=\sum_{b=0}^{2}\sum_{q=0}^{N_{T}-1} \lambda_{k}^{(b,u,q,v)}\left(X_{k}^{(b,u,q)}+\sum_{\begin{subarray}{c}N_{ CEU}-1\\ u^{\prime}=0\end{subarray}\end{subarray}}^{N_{CEU}-1}X_{k}^{(b,u^{\prime},d)} \right)+\\ W_{k}^{(u,v)},\;k\in\Im^{(u)},\end{split}  \tag{6}\]  
where \(\lambda_{k}^{(b,u,q,v)}\) denotes the overall channel gain at subcarrier \(k\) from transmitter antenna \(q\) of BS \(b\) to receiver antenna \(v\) of CEU \(u\). Similarly, \(X_{k}^{(b,u,q)}\) represents the pre-coded symbol transmitted from antenna \(q\) of BS \(b\) to CEU \(u\). The notation \(\Im^{(u)}\) denotes the index set of subcarriers of the outer sub-band \(f_{B2}\). \(W_{k}^{(u,v)}\) is the received noise with single-sided power spectral density \(N_{0}\). Let \(I_{k}^{(u)}\) denote the index of the \(k\)th subcarrier in \(\Im^{(u)}\) for \(k=0,1,\ldots,\left|\Im^{(u)}\right|-1\), where \(\left|\Im^{(u)}\right|\) represents the cardinality of the set \(\Im^{(u)}\). We further represent the DFT sequence of (6) as a \(\left|\Im^{(u)}\right|\times 1\) vector \(\mathbf{Y}^{(u,v)}\equiv\left[Y_{I_{0}^{(u)}}^{(u,v)},Y_{I_{1}^{(u)}}^{(u,v)},\ldots,Y_{I_{|\Im^{(u)}|-1}^{(u,v)}}\right]^{T}\), which is given by the following matrix-vector form \[\mathbf{Y}^{(u,v)}=\sum_{b=0}^{2}\sum_{q=0}^{N_{T}-1}\mathbf{\Lambda}^{(b,u,q,v)} \left(\mathbf{X}^{(b,u,q)}+\underbrace{\sum_{ \begin{subarray}{c}u^{\prime}=0\\ (u^{\prime}\
eq u)\end{subarray} }^{N_{CEU}-1}\mathbf{X}^{(b,u^{\prime},d)}}_{ \text{multi-user interference}}\right)+ \tag{7}\]  
where \(\mathbf{\Lambda}^{(b,u,q,v)}\) is a \(\left|\Im^{(u)}\right|\times\left|\Im^{(u)}\right|\) diagonal channel matrix with \(\lambda_{I_{k}^{(u)}}^{(b,u,q,v)}\) as its \(k\)th diagonal element. Both \(\mathbf{X}^{(b,u,q)}\) and \(\mathbf{W}^{(u,v)}\) are \(\left|\Im^{(u)}\right|\times 1\) vectors, whose \(k\)th entry are \(X_{I_{k}^{(u)}}^{(b,u,q)}\) and \(W_{I_{k}^{(u)}}^{(u,v)}\), respectively. For later use, we define \(\mathbf{\Lambda}^{(b,u,v)}\equiv\left[\mathbf{\Lambda}^{(b,u,0,v)},\mathbf{ \Lambda}^{(b,u,1,v)},\ldots,\mathbf{\Lambda}^{(b,u,N_{T}-1,v)}\right]\), which denotes the \(\left|\Im^{(u)}\right|\times\left|\Im^{(u)}\right|N_{T}\) aggregate channel matrix observed at the receiver antenna \(v\) of CEU \(u\) from \(N_{T}\) transmit antennas of BS \(b\). Similarly, the aggregate channel matrix observed at the antenna \(v\) of CEU \(u\) from all three cooperative BSs is given by  
\[\mathbf{\Lambda}^{(u,v)}\equiv\left[\mathbf{\Lambda}^{(0,u,v)},\mathbf{ \Lambda}^{(1,u,v)},\mathbf{\Lambda}^{(2,u,v)}\right]\in\mathbb{C}^{\left|\Im^ {(u)}\right|\times 3\left|\Im^{(u)}\right|N_{T}}, \tag{8}\]  
where \(\mathbb{C}\) represents the complex space. The aggregate signal vector sent to CEU \(u\) from \(N_{T}\) transmit antennas of BS \(b\) can be represented as  
\[\mathbf{X}^{(b,u)}\equiv\left[\mathbf{X}^{(b,u,0)^{T}},\mathbf{X}^{(b,u,1)^{T }},\ldots,\mathbf{X}^{(b,u,N_{T}-1)^{T}}\right]^{T}\in\mathbb{C}^{\left|\Im^ {(u)}\right|N_{T}\times 1}, \tag{9}\]  
To gain insight into how joint preprocessing works, we cascade all the \(\left|\Im^{(u)}\right|\times 1\) received vectors of (7) to form a \(\left|\Im^{(u)}\right|N_{R}\times 1\) received vector, as shown below:  
\[\mathbf{Y}^{(u)}\equiv\left[\mathbf{Y}^{(u,0)^{T}},\mathbf{Y}^{(u,1)^{T}}, \ldots,\mathbf{Y}^{(u,N_{R}-1)^{T}}\right]^{T}\]  
\[=\mathbf{\Lambda}^{(u)}\left(\mathbf{X}^{(u)}+\underbrace{\sum_{ \begin{subarray} {c}u^{\prime}=0\\ u^{\prime}\
eq u\end{subarray}