# SNR-Based Receiver-Type Decision Using Deep Learning for Multiple-Input Multiple-Output Detection
## 2 System Model  
We consider a MIMO system where the transmitter and receiver are equipped with \(N_{T}\) and \(N_{R}\) antennas, respectively. Assuming that independent \(N_{T}\) data streams are transmitted, the received signal vector \(\mathbf{y}=\left[y_{1}\cdots y_{q}\cdots y_{N_{R}}\right]^{T}\) is given by  
\[\mathbf{y}=\sqrt{\frac{E_{s}}{N_{T}}}\mathbf{H}\mathbf{x}+\mathbf{n}, \tag{1}\]  
where \(E_{s}\) denotes the average transmit energy over a symbol duration, \(\mathbf{H}\) is an \(N_{R}\times N_{T}\) MIMO channel matrix, \(\mathbf{x}=\left[x_{1}\cdots x_{p}\cdots x_{N_{T}}\right]^{T}\) is an \(N_{T}\times 1\) transmitted signal vector with \(\mathbb{E}\left[\mathbf{x}\mathbf{x}^{H}\right]=\mathbf{I}_{N_{T}}\), \(\mathbf{n}=\left[n_{1}\cdots n_{q}\cdots n_{N_{R}}\right]^{T}\) represents an \(N_{R}\times 1\) additive white Gaussian noise vector that satisfies \(\mathbb{E}\left[\mathbf{n}\mathbf{n}^{H}\right]=N_{0}N_{N_{R}}\), and \(N_{0}\) is a noise spectral density. A \((q,p)\) element of the channel matrix \(\mathbf{H}\), \(\left\{\mathbf{H}\right\}_{q,p}=h_{qp}\), is the channel coefficient between the \(p\)th transmit antenna and \(q\)th receive antenna, where \(\left\{\cdot\right\}_{q,p}\) represents the element at \(q\)th row and \(p\)th column. We assume that \(h_{qp}\) is an independently and identically distributed complex Gaussian random variable with zero mean and unit variance [1]. A linear receiver can be considered to reduce the computational complexity at the receiver in detecting the transmitted signal vector \(\mathbf{x}\), and in this paper, we adopt a ZF receiver that is one of the linear receivers. Using the ZF receiver, the post-processing signal vector \(\mathbf{z}^{ZF}\) is given by  
\[\mathbf{z}^{ZF} =\sqrt{\frac{N_{T}}{E_{s}}}\mathbf{H}^{\dagger}\mathbf{y} \tag{2}\] \[=\mathbf{x}+\sqrt{\frac{N_{T}}{E_{s}}}\mathbf{H}^{\dagger}\mathbf{ n},\]  
where \(\mathbf{H}^{\dagger}\) represents the Moore-Penrose inverse. Note that \(N_{R}\geq N_{T}\) is required for the ZF receiver to be feasible and, accordingly, \(\mathbf{H}^{\dagger}=\left(\mathbf{H}^{H}\mathbf{H}\right)^{-1}\mathbf{H}^{H}\)[18]. In (2), the demodulated signal corresponding to the \(p\)th transmitted signal, \(\hat{\xi}_{p}^{ZF}\), is obtained by the following metric:  
\[\hat{\xi}_{p}^{ZF}=\underset{s\in\mathcal{S}}{\arg\min}\Big{|}z_{p}^{ZF}-s \Big{|}, \tag{3}\]  
where \(\mathcal{S}\) represents a set of signal constellation points whose cardinality is \(\left|\mathcal{S}\right|=M\)[1]. Besides, for the given MIMO channel \(\mathbf{H}\), the symbol error probability can be expressed as  
\[\tilde{P}_{S}^{ZF}(\rho,\mathbf{H})=\frac{1}{N_{T}}\sum_{p=1}^{N_{T}}\sum_{s \in\mathcal{S}}\Pr\!\left[x_{p}=s\right]\Pr\!\left[\hat{\xi}_{p}^{ZF}\
eq x_{ p}|x_{p}=s,\mathbf{H}\right], \tag{4}\]  
where \(\rho=\frac{E_{s}}{N_{0}}\) is the average transmit SNR. Since the transmitted signal from each transmit antenna can be detected independently, the ZF receiver has a computational complexity depending on \(MN_{T}\), which grows linearly with \(N_{T}\)[1]. However, the ZF receiver can obtain only \(N_{R}-N_{T}+1\) of diversity order due to the utilization of degrees of freedom in the matrix inversion process [19]. On the other hand, we also consider applying an ML receiver, which is a well-known optimal receiver for MIMO detection. Exploiting the ML receiver, the demodulated signal vector \(\hat{\mathbf{x}}^{ML}=\left[x_{1}^{ML}\cdots\hat{x}_{p}^{ML}\cdots\hat{x}_{N_{T}} ^{ML}\right]^{T}\) is obtained by the following metric:  
\[\hat{\mathbf{x}}_{p}^{ML}=\underset{s\in\mathcal{S}_{ML}}{\arg\min}\left\| \mathbf{y}-\sqrt{\frac{E_{s}}{N_{T}}}\mathbf{H}\mathbf{s}\right\|_{F}^{2}, \tag{5}\]  
where \(\mathcal{S}_{ML}\) is a set of candidate vectors for the ML receiver whose cardinality is \(|\mathcal{S}_{ML}|=M^{N_{T}}\).