## 2 System Model and Problem Formulation  
The proposed hybrid transceiver for a two-hop multi-relay mm-Wave MIMO system with relay selection is shown in Figure 1, where \(\mathbf{N}_{t}\), \(\mathbf{N}_{rel}\) and \(\mathbf{N}_{r}\) are the number of antennas at source, relay, and destination, respectively. Let \(\mathbf{s}_{k}\in\mathbb{C}^{\mathbf{N}_{s}\times 1}\) be a complex information symbol vector consisting of \(\mathbf{N}_{s}\) transmitted data streams from the source to the destination through the \(k\)th selected relay from a set of \(K\) relay nodes, which is given as  
\[\mathbf{s}_{k}=\{s_{1},s_{2},\ldots,s_{\mathbf{N}_{k}}\}^{T},\ \mathbb{E}\Big{\{}\mathbf{s}_{k}\mathbf{s}_{k}^{H}\Big{\}}=\mathbf{I}_{\mathbf{N}_{s}}. \tag{1}\]The number of RF-chains at the source, relay node and destination are represented as \(N_{t}^{RF}\), \(N_{rel}^{RF}\) and \(N_{r}^{RF}\), respectively, and the essential condition that needs to be satisfied for reliable transmission of data streams, when hybrid precoding is taken into account, can be expressed as  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]  
The condition (2) clearly indicates that the number of RF chains determines the maximum number of data streams that can be transmitted reliably, and the number of RF chains is far less than the number of antennas which describes the potential of hybrid beamforming in making the system feasible on practical grounds. It is assumed that a direct link between the source and destination is not feasible due to excessive path-loss and deep fading, which is typical of mm-Wave propagation. Furthermore, each relay node follows AF relaying protocol and all communicating nodes are supposed to be in half-duplex mode. In relay-assisted MIMO systems, the signal transmission can be divided into two time-slots. In the first time-slot, source transmits \(N_{s}\) data streams to all the relay nodes. In the second time-slot, the selected \(k\)th relay where \(k\in\{1,2,\ldots,K\}\) re-transmits the signal to the end user. The transmitted signal from the source to the \(k\)th relay, after applying hybrid precoding, can be expressed as  
\[\mathbf{x}_{k}=\mathbf{V}_{RF}\mathbf{V}_{BB,k}\mathbf{s}_{k}\ \in\mathbb{C}^{N_{t}\times 1}, \tag{3}\]  
where \(\mathbf{V}_{RF}\in\mathbb{C}^{N_{t}\times N_{t}^{RF}}\) is the common analog RF beamformer and \(\mathbf{V}_{BB,k}\in\mathbb{C}^{N_{t}^{RF}\times N_{s}}\) is the digital baseband precoding matrix corresponding to the \(k\)th selected relay. The received signal at the \(k\)th relay node is given as  
\[\mathbf{y}_{r}^{k}=\mathbf{H}_{k}\ \mathbf{V}_{RF}\mathbf{V}_{BB,k}\mathbf{s}_{k}+\mathbf{z}_{r}\ \in\mathbb{C}^{N_{rel}\times 1}, \tag{4}\]  
where \(\mathbf{H}_{k}\in\mathbb{C}^{N_{rel}\times N_{t}}\), \(\mathbf{z}_{r}\in\mathbb{C}^{N_{rel}\times 1}\) are the channel matrix between the source and the \(k\)th relay node and, zero mean circularly symmetric complex Gaussian (ZMCSCG) noise vector with variance \(\sigma_{r}^{2}\) i.e., \(\mathbf{z}_{r}\ \sim\mathcal{CN}\big{(}\mathbf{0},\ \sigma_{r}^{2}\mathbf{I}_{N_{rel}}\big{)}\), respectively. Before proceeding further, it is worth mentioning that the hybrid beamforming structure at the relay node consists of two analog RF and the corresponding digital baseband processing components. One analog RF processing matrix acts as a receive beamformer for  
Figure 1: Block diagram of the proposed hybrid beamforming and relay selection for a two-hop multi-relay network. the source transmitted signal in the first time-slot. The other analog RF processing matrix is responsible for the transmit beamforming, which enables the transmission of the processed signal at the relay node to the destination in the second time-slot. Let \(\mathbf{F}_{1k}\in\mathbb{C}^{N_{rd}\times N_{rd}^{RF}}\) be the analog RF combiner and \(\mathbf{F}_{2k}\in\mathbb{C}^{N_{rd}\times N_{rd}^{RF}}\) be the analog precoder at the \(k\)th relay node.