# Topological Interference Management for K-User Downlink Massive MIMO Relay Network Channel
## 5 Space-Time Relay Transmission  
Space-time relay transmission efficiently forwards the desired signals through multiple transmitters and a shared relay to the destination. In all three of the time slots considered in Figures 4-6, the receiver must use the Alamouti code to decode the interference signals. We know that the relay has a two-phase protocol and, during the first phase of the protocol, the relay receives the source broadcasted signals from multiple transmitters, while, in the second phase of the protocol, the relay transmits source signals to the destinations. By utilizing the Alamouti method for equations (13) and (14) in time slot 1, the interference signal from the relay and transmitters can be communicated as  
\[\mathbf{V}^{[3]}=\underbrace{\frac{1}{\beta} \begin{bmatrix}\mathbf{h}_{[R_{1}]} ^{[1,R]}[3]&\mathbf{h}_{[R_{2}]}^{[1,R]}[3]\\ \mathbf{h}_{[R_{1}]}^{[2,R]}[3]&\mathbf{h}_{[R_{2}]}^{[2,R]}[3]\end{bmatrix} }_ {\text{relay channel vector}}\underbrace{- \begin{bmatrix}-\mathbf{h}_{[c_{2}]}^{[1,1]^{*}} [2]&\mathbf{h}_{[c_{1}]}^{[1,1]^{*}}[1]\\ \mathbf{h}_{[c_{2}]}^{[2,2]}[1]&\mathbf{h}_{[c_{1}]}^{[2,2]}[2]\end{bmatrix} }_ {\text{transmitter channel vector}}. \tag{20}\]  
The corresponding transmitter 1 and 2 signals, recovered during time slot 1 from the unintended receiver 3 can be expressed as  
\[\mathbf{y}_{(3)}^{[1]} = (\mathbf{h}_{[R_{1}]}^{[1,R]^{T}}(3)+\mathbf{h}_{[R_{2}]}^{[1,R]^ {T}}(3))X^{[R]}[3]+Z^{[1]}(3), \tag{21}\] \[= -\mathbf{h}_{[c_{2}]}^{[1,1]^{*}}(2)\mathbf{u}_{1}^{[1]}+\mathbf{ h}_{[c_{1}]}^{[1,1]^{*}}(1)\mathbf{u}_{2}^{[1]}+Z^{[1]}(3).\]  
\[\mathbf{y}_{(3)}^{[2]} = (\mathbf{h}_{[R_{1}]}^{[2,R]^{T}}(3)+\mathbf{h}_{[R_{2}]}^{[2,R]^ {T}}(3))X^{[R]}[3]+Z^{[2]}(3), \tag{22}\] \[= \mathbf{h}_{[c_{2}]}^{[2,2]}(2)\mathbf{u}_{1}^{[2]}+\mathbf{h}_{[ c_{1}]}^{[2,2]}(1)\mathbf{u}_{2}^{[2]}+Z^{[2]}(3).\]  
Equivalently, Equations (15) and (16) are expressed for time slot 3, as mentioned below:  
\[\mathbf{V}^{[1]}=\frac{1}{\beta}\underbrace{ \begin{bmatrix}\mathbf{h}_{[R_{2}] }^{[2,R]^{T}}[1]&\mathbf{h}_{[R_{2}]}^{[2,R]}[1]\\ \mathbf{h}_{[R_{2}]}^{[3,R]}[1]&\mathbf{h}_{[R_{2}]}^{[3,R]}[1]\\ \end{bmatrix} }_{\text{relay channel vector}}\underbrace{- \begin{bmatrix}- \mathbf{h}_{[c_{2}]}^{[2,2]^{*}}[3]&\mathbf{h}_{[c_{2}]}^{[2,2]^{*}}[2]\\ \mathbf{h}_{[c_{2}]}^{[3,3]}[2]&\mathbf{h}_{[c_{3}]}^{[3,3]}[3]\end{bmatrix} }_ {\text{transmitter channel vector}}. \tag{23}\]  
The corresponding transmitter 2 and 3 signals, recovered during time slot 2 from the unintended receiver 1, can be expressed as  
\[\mathbf{y}_{(1)}^{[2]} = (\mathbf{h}_{[R_{2}]}^{[2,R]^{T}}(1)+\mathbf{h}_{[R_{3}]}^{[2,R]^ {T}}(1))X^{[R]}[1]+Z^{[2]}(1), \tag{24}\] \[= -\mathbf{h}_{[c_{3}]}^{[2,2]^{*}}(3)\mathbf{u}_{1}^{[2]}+\mathbf{ h}_{[c_{2}]}^{[2,2]^{*}}(2)\mathbf{u}_{2}^{[2]}+Z^{[2]}(1).\]  
\[\mathbf{y}_{(1)}^{[3]} = (\mathbf{h}_{[R_{2}]}^{[3,R]^{T}}(1)+\mathbf{h}_{[R_{3}]}^{[3,R]^ {T}}(1))X^{[R]}[1]+Z^{[3]}(1), \tag{25}\] \[= \mathbf{h}_{[c_{2}]}^{[3,3]}(2)\mathbf{u}_{1}^{[3]}+\mathbf{h}_{[ c_{3}]}^{[3,3]}(3)\mathbf{u}_{3}^{[2]}+Z^{[3]}(1).\]  
Similarly, Equations (17) and (18) are represented for time slot 3, as mentioned below:  
\[\mathbf{V}^{[2]}=\frac{1}{\beta}\underbrace{ \begin{bmatrix}\mathbf{h}_{[R_{3} ]}^{[3,R]}[2]&\mathbf{h}_{[R_{1}]}^{[3,R]}[2]\\ \mathbf{h}_{[R_{3}]}^{[1,R]}[2]&\mathbf{h}_{[R_{1}]}^{[1,R]}[2]\end{bmatrix} }_ {\text{relay channel vector}}\underbrace{- \begin{bmatrix}-\mathbf{h}_{[c_{1}]}^{[3,3]} [1]&\mathbf{h}_{[c_{3}]}^{[3,3]^{*}}[3]\\ \mathbf{h}_{[c_{3}]}^{[1,1]}[3]&\mathbf{h}_{[c_{1}]}^{[1,1]}[1]\end{bmatrix} }_ {\text{transmitter channel vector}}.