# Topological Interference Management for K-User Downlink Massive MIMO Relay Network Channel
## 6 The \(K\)-User Relay-Aided MIMO Interference Channel  
In this section, we characterize the \(K\)-user MIMO interference channel with a MIMO relay under limited CSIR knowledge. We restrict the number of \(K\)-user communication pairs to be lesser than the number of transmitter antennas, M, where \(K\leq M\). **Time slot 1**: Let us consider \(K\)-time slots, where the transmitter sends M symbols via the relay to the corresponding M-antenna receiver; here, the \(K\)-user \(M\times 2\) MIMO interference channel with a relay and an M-antenna receiver, has local CSIR knowledge. In the \(k\)th time slot, the \((1\leq k\leq K)\) transmitter sends M desired symbols to the M-antenna receiver using the R-STIA technique; the symbols are \(\mathbf{X}^{[k]}[k]=[\mathbf{u}_{1}^{(k)},\mathbf{u}_{2}^{(k)},...,\mathbf{u} _{M}^{(k)}]^{T}\),  
\[\mathbf{y}_{j}^{(k)}(n) = \underbrace{\mathbf{h}_{j,j}^{[k,k]^{T}}(n)\mathbf{u}_{M}^{[k]}+ Z_{j}^{[k]}(n)}_{\text{desired signal + noise}}, \tag{32}\]  
where \(k\in\{1,2,3,...,K\}\)  
\[\mathbf{y}_{(j)}^{R}(n) = \underbrace{\mathbf{h}_{j}^{[R,k]}(n)\mathbf{x}_{j}^{k}(k)+Z_{j}^ {[R]}(n)}_{\text{relay signal + noise}}. \tag{33}\]  
During this time slot, the intended receiver has a weak interference and the unintended (additional) receiver has a strong interference; hence, the intended receiver has one equation, which contains M desirable symbols, whereas the unintended receiver has two equations containing M desirable symbols. **Time slot 2**: The important objective of this time slot is to provide additional information to the unintended receiver, which assists in building \((K-2)\) independent equations using the \(\mathbf{u}_{M}^{(k)}\) desired symbols. In this time slot, the receiver has CSIR knowledge for the current and past time periods because the relay forwards CSIR knowledge, such as the current and the outdated information to the receiver  
\[\mathbf{\hat{X}}^{[R]}[n]=\beta(\mathbf{V}_{k}^{[1]}(n)\hat{\mathbf{u}}_{1}^{[ k]}+\mathbf{V}_{k}^{[1]}(n)\hat{\mathbf{u}}_{2}^{[k]}+...,\mathbf{V}_{k}^{[k]}(n) \hat{\mathbf{u}}_{M}^{[k]}), \tag{34}\]\[||\mathbf{h}_{[c_{k}]}^{[K,k]^{T}}(n)||2\mathbf{\tilde{h}}_{[n-c_{k}, \perp}^{[K,k]^{T}}(n)^{T}=\beta\mathbf{h}_{[c_{k}]}^{[K,k]^{T}}(n)\mathbf{V}_{k}^ {k}(n),\ k\in K. \tag{35}\]  
**Time slot \(n\)**: where \(n\in(K+1,K+2,...,K+M-1)\) and \(||\mathbf{h}_{[c_{k}]}^{[K,k]^{T}}(n)||_{2}\) are the 2-norm of \(\mathbf{h}_{[c_{k}]}^{[K,k]^{T}}(n)\). The transmitter and relay repeatedly transmit the same symbol, until the desired symbols are received in the intended receivers. During this transmission, the unintended receiver recovers the interference signals, enabling the reconstruction of the desired signal information for the intended receivers:  
\[\mathbf{y}_{j}^{k}(n) = \mathbf{h}_{j,j}^{[k,k]^{T}}(n)\mathbf{x}_{j}^{k}(n)+\mathbf{h}_ {j,j}^{[k,R]^{T}}(n)\mathbf{x}_{j}^{R}(n), \tag{36}\] \[+\sum_{j=1,j\
eq k}^{K}\mathbf{h}_{j,j}^{[k,j]^{T}}(n)\mathbf{x}_ {i}^{j}(n)+Z_{j}^{[k]}(n).\]  
Considering that the transmitter sends \(K(K-1)\) linear combinations, \(\mathbb{F}^{[k,k]}[W_{k}]\), of multiple data symbols to all the desired users, such as  
\[\mathbb{F}^{[k,k]}[W_{k}] = \mathbf{h}_{[c_{1}]}^{[k,k]}[n]\mathbf{u}_{1}^{[k]}+\mathbf{h}_ {[c_{2}]}^{[k,k]}[n]\mathbf{u}_{2}^{[n]} \tag{37}\] \[+,...\!+\!\mathbf{h}_{[c_{k}]}^{[k,k]}[n]\mathbf{u}_{M}^{[n]},\]  
the unintended (additional) receiver assists in removing the interference and decodes the \((K-1)\) data symbols reliably at the intended receiver with a very high SNR. Hence, the intended receivers can identify and remove the corresponding interference data symbols using local CSIR knowledge and the unintended-receiver information.