# Topological Interference Management for K-User Downlink Massive MIMO Relay Network Channel
## 2 Channel Model and Formula Definitions
#### Key Definitions  
**Definition 1**.: _Alamouti code (space time block code)._  
_It is a technique used in wireless communication to transmit multiple data streams using massive antennas at the BS and a relay to exploit several received versions of data for improving the data transfer consistency and desired signal information [5; 19]._  
_Example: Alamouti code for a \(2\times 2\) MIMO wireless system model is given by_  
\[\begin{bmatrix}\mathbf{y}_{[jj]}\\ \mathbf{y}_{[ji]}\\ \mathbf{y}_{[ii]}^{*}\\ \mathbf{y}_{[ii]}^{*}\\ \end{bmatrix}=\begin{bmatrix}\mathbf{h}_{[jj]}&\mathbf{h}_{[ii]}\\ \mathbf{h}_{[ij]}&\mathbf{h}_{[ii]}\\ \mathbf{h}_{[ii]}^{*}&-\mathbf{h}_{[ij]}^{*}\\ \mathbf{h}_{[ii]}^{*}&-\mathbf{h}_{[ij]}^{*}\\ \end{bmatrix}\begin{bmatrix}\mathbf{x}_{j}^{k}[n]\\ \mathbf{x}_{i}^{k}[n]\\ \mathbf{x}_{i}^{k}[n]\\ \end{bmatrix}+\begin{bmatrix}\mathbf{Z}_{[jj]}\\ \mathbf{Z}_{[ji]}\\ \mathbf{Z}_{[ii]}^{*}\\ \end{bmatrix}, \tag{3}\]  
_where \([\mathbf{y}_{jj},\mathbf{y}_{ii}]^{T}\), and \([\mathbf{y}_{ij},\mathbf{y}_{ji}]^{T}\) denote the received, desired, and interference signal vectors. \(\mathbf{h}_{jj}\), \(\mathbf{h}_{ii}\), and \(\mathbf{h}_{ji}\), \(\mathbf{h}_{ij}\) are the desired and interference channel matrices, where \(\mathbf{x}_{j}^{k}[n]\), \(\mathbf{x}_{i}^{k}[n]\) denote the transmitted signals; \([\mathbf{Z}_{ji},\mathbf{Z}_{ji}]^{T}\), \([\mathbf{Z}_{ij},\mathbf{Z}_{ii}]^{T}\) denote the zero-mean additive white Gaussian Noise (AWGN)._  
**Definition 2**.: _(Degree of freedom and achievable sum-rate)._  
_The degree of freedom (DoF) region for this network topology is \(d_{1}+d_{2}+,...,+d_{n}\leq 1\) and, for a symmetric network, it is \(d_{sym}=\frac{1}{n}\)[3; 20]_  
\[\mathsf{d}_{sym}=\limsup_{\mathrm{P}\to\infty}\sup_{(\mathrm{R}, \ldots,\mathrm{R})\in\mathcal{C}}\frac{\mathrm{R}\sum(\mathrm{P})}{\mathrm{logP}},\] \[\mathrm{D}=\{(d_{1},...,d_{k})\in\mathbb{R}_{+}:d_{i}= \limsup_{\mathrm{P}\to\infty}  
_The achievable sum-rate for the proposed K-user MIMO relay interference channel model [16; 21] is expressed as_  
\[\mathrm{R}_{\mathrm{K}}\leq\log\left(1+\frac{\parallel\mathbf{h}_{j, j}^{[k,k]^{\mathsf{T}}}(n)\parallel^{2}P+\parallel\mathbf{h}_{R_{j}}^{[k,R]^{ \mathsf{T}}}(n)\parallel^{2}(\frac{P_{\mathrm{R}}}{2})}{\parallel\mathbf{h}_ {j,l}^{[k,j]^{\mathsf{T}}}(n)\parallel^{2}P+1}+\frac{2\alpha\parallel\mathbf{h }_{j,j}^{[k,k]^{\mathsf{T}}}(n)\parallel\parallel\mathbf{h}_{R_{j}}^{[k,R]^{ \mathsf{T}}}(n)\parallel\sqrt{P(\frac{P_{\mathrm{R}}}{2})}}{\parallel\mathbf{h}_ {j,l}^{[k,j]^{\mathsf{T}}}(n)\parallel^{2}P+1}\right). \tag{6}\]  
All the users (nodes) are subject to an average power-constraint \((P)\) is defined as \(P=SNR\times\sigma^{2}\) and \(R^{[k]}(P)\) indicate the achievable rate, where (SNR) represents signal-to-noise ratio [22]. The transmitted signals, \(E(\|\mathbf{X}^{[k]}[n]\|^{2})\leq P\) and the relay signals, \(E(\|\mathbf{X}^{[k]}[n]\|^{2})\leq P_{R}\), subject to the average power constraint are briefly described, where the coefficient \(\alpha\) represents the in-phase addition of signals coming from the transmitters and the relay. The fraction of the interference power for the desired signal is defined as follows:  
\[\mathrm{q}_{arg} = \frac{1}{k}\sum_{k=1}^{k}\sigma_{k}. \tag{7}\]