# Topological Interference Management for K-User Downlink Massive MIMO Relay Network Channel
## 5 Space-Time Relay Transmission
\tag{26}\]The corresponding transmitter 3 and 1 signals, recovered during time slot 2 from the unintended receiver 2, can be expressed as  
\[\mathbf{y}_{(2)}^{[3]} = (\mathbf{h}_{[R_{3}]}^{[3,R]^{T}}(2)+\mathbf{h}_{[R_{1}]}^{[3,R]^{T} }(2))X^{[R]}[2]+Z^{[3]}(2), \tag{27}\] \[= -\mathbf{h}_{[c_{1}]}^{[3,3]^{*}}(1)\mathbf{u}_{1}^{[3]}+\mathbf{ h}_{[c_{3}]}^{[3,3]^{*}}(3)\mathbf{u}_{2}^{[3]}+Z^{[3]}(2).\]  
\[\mathbf{y}_{(2)}^{[1]} = (\mathbf{h}_{[R_{3}]}^{[1,R]^{T}}(2)+\mathbf{h}_{[R_{1}]}^{[1,R]^ {T}}(2))X^{[R]}[2]+Z^{[1]}(2), \tag{28}\] \[= \mathbf{h}_{[c_{3}]}^{[1,1]}(3)\mathbf{u}_{1}^{[1]}+\mathbf{h}_{[ c_{1}]}^{[1,1]}(1)\mathbf{u}_{2}^{[1]}+Z^{[1]}(2).\]  
**Decoding**: The overall decoding process requires the separation of the user interference signals based on the CSIR knowledge at the receiver side. Consequently, the user 1 signal is decoded based on the relay and receiver signals from \(\mathbf{y}_{(1)}^{[1]}=\mathbf{y}_{(3)}^{[1]}+\mathbf{y}_{(2)}^{[1]}\):  
\[\mathbf{y}_{(3)}^{[1]}+\mathbf{y}_{(2)}^{[1]} = \underbrace{(\mathbf{h}_{[R_{1}]}^{[1,R]^{T}}(3)+\mathbf{h}_{[ R_{2}]}^{[1,R]^{T}}(3))X^{[R]}[3]+(\mathbf{h}_{[R_{3}]}^{[1,R]^{T}}(2)+ \mathbf{h}_{[R_{1}]}^{[1,R]^{T}}(2))X^{[R]}[2]}_{\text{recovered relay signal}} \tag{29}\] \[= \underbrace{-\mathbf{h}_{[c_{2}]}^{[1,1]^{*}}(2)\mathbf{u}_{1}^{[ 1]}+\mathbf{h}_{[c_{1}]}^{[1,1]^{*}}(1)\mathbf{u}_{2}^{[1]}+\mathbf{h}_{[c_{3 }]}^{[1,1]}(3)\mathbf{u}_{1}^{[1]}+\mathbf{h}_{[c_{1}]}^{[1,1]}(1)\mathbf{u}_ {2}^{[1]}}_{\text{recovered transmitter signal}}. \tag{30}\]  
Similarly, we can compute the received signals \(\mathbf{y}_{(2)}^{[2]}=\mathbf{y}_{(3)}^{[2]}+\mathbf{y}_{(1)}^{[2]}\) and \(\mathbf{y}_{(3)}^{[3]}=\mathbf{y}_{(1)}^{[3]}+\mathbf{y}_{(2)}^{[3]}\). After removing the corresponding interference signals, the effective channel input-output relationship for user 1 is as follows:  
\[\underbrace{ \begin{bmatrix}\mathbf{y}_{1}^{[1]}-\mathbf{y}_{2}^{[2]}\\ \mathbf{y}_{1}^{[1]}-\mathbf{y}_{3}^{[3]}\end{bmatrix} }_{\mathbf{y}_{1}^{[1]}} =\underbrace{ \begin{bmatrix}(\mathbf{y}_{(3)}^{[1]}+\mathbf{y}_{(2)}^{[1]})-( \mathbf{y}_{(3)}^{[2]}+\mathbf{y}_{(1)}^{[2]})\\ (\mathbf{y}_{(3)}^{[1]}+\mathbf{y}_{(2)}^{[1]})-(\mathbf{y}_{(1)}^{[3]}+ \mathbf{y}_{(2)}^{[3]})\end{bmatrix} }_{\mathbf{y}_{1}^{[1]}-\mathbf{y}_{2}^{[ 2]}}\underbrace{ \begin{bmatrix}\mathbf{u}_{1}^{[1]}\\ \mathbf{u}_{2}^{[1]}\end{bmatrix} }_{\mathbf{x}_{1}^{[1]}}+\underbrace{ \begin{bmatrix}\mathbf{z}_{1}^{[1]}-\mathbf{z}_{2}^{(2)}\\ \mathbf{z}_{1}^{(1)}-\mathbf{z}_{3}^{(3)}\end{bmatrix} }_{\mathbf{z}_{1}^{[1]}}. \tag{31}\]  
_Discussion_  
The proposed idea is to create a relay-aided network channel where each intended receiver experiences less interference and the unintended (additional) receiver experiences more interferences. Remarkably, the intended receivers have very weak interference, since intended receivers are affected by only one interference signal. The single unintended receiver has very strong interference because the unintended receiver is affected by two transmitter interference signals and one relay signal over the time slots. Moreover, this paper offers an efficient algorithm and a channel co-efficient for the desired, undesired, and relay signals for identifying and aligning the corresponding signals resourcefully. Subsequently, by introducing an additional (unintended) receiver at a very-strong interference region and with certain limited CSIR knowledge, the network settings are valid for recovering the combination of the three co-interference signals from transmitters and relay. By such R-STIA and Alamouti techniques, the interference signals are reconstructed at the unintended PIMAC receiver-end. The authors in [26] proposed a Quasi Orthogonal (QO)-STIA technique for a two-user \(X\) channel, where the interference cancellation and alignment is based on only two intended receivers. The decoding complexity is reasonably high, since the receiver has to decode the intended symbols over each time slots until the desired signal is recovered.