# Energy Efficiency Analysis of MIMO Wideband RF Front-End Receivers
## 2 Preliminaries
### Communication Model  
We consider a system as illustrated in Figure 3, with a transmitter node and a receiver node, in which we use \(N_{\rm t}\) and \(N_{\rm r}\) to represent the number of available transmitting and receiving antennas, respectively; while lower case letters \(n_{\rm t}\) and \(n_{\rm r}\) denote the active number of antennas at each node. In addition, each antenna is assembled on a single RF chain. Besides the power used to transmit the RF signal, the consumption of all building blocks at the transmitter and receiver nodes needs to be included in order to achieve the total system power budget. Thus, considering that \(n_{\rm t}\leq N_{\rm t}\) RF chains are active at the transmitter and \(n_{\rm r}\leq N_{\rm r}\) RF chains are active at the receiver, the total power consumption becomes  
\[p_{\rm total}^{\rm(sch)}=n_{\rm t}\left(P_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{\rm{\rm{\rm\rm{\rm\rm{\rm\rm\{\rm\{\rm\rm\{\rm\{ \rm\{\rm\rm\{\rm\{\rm\{\rm\{\rm\{ \rm\rm\{\rm\{ \rm\{  \rm\{         }}}}}}}}}}}}}} \right)+n_{\rm{r}}\left(P_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{       \rm{\rm{\rm{                 }}}}}}}}}}}} \rm{}}}} \right)\,, \tag{3}\]  
where the subscript \({\rm{sch}\in\{{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{ \rm{  \rm{ \rm{  \rm{   }}}}}}}}}}}}}}}}}}} \rm{{\rm{AS}}}\) indicates the transmission scheme, which we detail in Section 3,  
\[P_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{\rm{\rm{ \rm{ \rm{ \rm{ \rm\{ }}}}}}}}}}}}}}}}}}}}} \rm{=}\,P_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{ \rm{ \rm { \rm         \rm{           }}}}}}}}}}}}}}} \rm{}} \rm{{}}} \rm{}}} \right. \tag{4}\]  
is the power amplifier (PA) consumption, with \(P_{\rm{out}}\) being the transmitted power that the PA forwards to the transmitter output, and \(\eta\) represents the PA drain efficiency [21]. It is worth remarking that in  
Figure 3: RF transceiver building blocks. Figure 2: Sensitivity as a function of the power consumption of the RF front-end architectures, with \(B=20\) MHz and \(\xi_{0}=0\) dB. the case of multiple active RF chains, we assume that the transmitted power is equally distributed among all PAs. Furthermore, \(P_{\text{rx,FirstBlock}}\) is the power consumed by the first block at the receiver chain, depending on the employed LNA or LNTA architecture, according to Table 1. In addition, \(P_{\text{tx}}\) and \(P_{\text{rx}}\) are the power consumption of the other building blocks at the transmitter and receiver, respectively. Then, a communication link connecting the transmitter and receiver nodes can be expressed as  
\[\mathbf{y}=\sqrt{\kappa\,P_{\text{out}}}\,\mathbf{H}\,\mathbf{x}+\mathbf{w}, \tag{5}\]  
where \(\mathbf{y}\) is the received signal vector with dimensions \(N_{\text{r}}\times 1\), \(\kappa\) is the link budget relationship, \(\mathbf{H}\) is the channel fading matrix of size \(N_{\text{r}}\times N_{\text{t}}\), whose elements \(h_{ij}\in\mathbf{H}\), \(\forall i,j\) are independent and identically distributed random variables, whose envelops follow aNakagami-\(m\) distribution. Moreover, \(\mathbf{x}\) is the \(N_{\text{t}}\times 1\) unit energy transmitted symbol vector and \(\mathbf{w}\) is a \(N_{\text{r}}\times 1\) vector of additive white Gaussian noise (AWGN) with variance \(N_{0}/2\) per dimension. The link budget relationship is given by [22]  
\[\kappa=\frac{G_{\text{a}}\,\lambda^{2}}{(4\pi)^{2}\,d^{\text{a}}\,M_{\text{I}}}, \tag{6}\]  
where \(G_{\text{a}}\) is the product of the transmitter and receiver antenna gains, \(\lambda=\frac{\xi}{f_{\text{c}}}\) is the wavelength, \(c=3\times 10^{8}\) m/s is the speed of light, \(f_{\text{c}}\) is the carrier frequency, \(d\) is the distance between the communicating nodes, \(\alpha\) is the path loss exponent and \(M_{\text{I}}\) is the link margin.