## 3 Energy and Information Beamforming
### Strongest-Path Beamforming Scheme
#### 3.2.2 MUI Mitigation  
According to (13) and Lemma 1, the power of MUI can be expressed asymptotically in the large \(\left\{N_{x},N_{y}\right\}\) limit as:  
\[ \begin{split} Q_{l}^{\text{MUI}}&=\sum_{k=1,k\
eq l }^{L}\mathbb{E}\left[P_{k}\mathbf{w}_{Dj}^{H}\mathbf{h}_{k}\mathbf{h}_{k}^{H} \mathbf{w}_{Dj}\right]\\ &=\sum_{k=1,k\
eq l}^{L}\left(\frac{\eta\tau}{1-\tau}p_{l}\frac{ \beta_{l}K_{l}}{K_{l}+1}N_{x}N_{y}-\frac{E_{l}^{p}}{\left(1-\tau\right)T} \right)\mathbb{E}\left[\mathbf{w}_{Dj}^{H}\mathbf{h}_{k}\mathbf{h}_{k}^{H} \mathbf{w}_{Dj}\right].\end{split}  \tag{30}\]  
By exploiting the channel covariance matrix (5) and the expression of \(\mathbf{w}_{Dj}\) given by Theorem 1, \(Q_{l}^{\text{MUI}}\) can be rewritten as:  
\[ \begin{split} Q_{l}^{\text{MUI}}&=\sum_{k=1,k\
eq l }^{L}\left(\frac{\eta\tau}{1-\tau}p_{k}\frac{\beta_{k}K_{k}}{K_{k}+1}N_{x}N_{y} -\frac{E_{k}^{p}}{\left(1-\tau\right)T}\right)\\ &\times\frac{\beta_{k}}{K_{k}+1}\left(\frac{1}{N_{x}N_{y}}\prod_{ i\in\left\{x,y\right\}}\frac{\sin^{2}\left(N_{x}\pi\left(\rho_{i,l}-\rho_{i,k} \right)\right)}{\sin^{2}\left(\pi\left(\rho_{i,l}-\rho_{i,k}\right)\right)}+ \frac{1}{N_{x}N_{y}}\\ &\times\int\limits_{\rho_{k,x}^{\min}}^{\rho_{\max}^{\max}}\int \limits_{\rho_{k,y}^{\min}}^{\rho_{\max}^{\max}}S_{k}\left(\rho_{x},\rho_{y} \right)\prod_{i\in\left\{x,y\right\}}\frac{\sin^{2}\left(N_{x}\pi\left(\rho_{ i,l}-\rho_{i}\right)\right)}{\sin^{2}\left(\pi\left(\rho_{i,l}-\rho_{i} \right)\right)}\mathrm{d}\rho_{x}\mathrm{d}\rho_{y}\right).\end{split}  \tag{31}\]  
A short derivation of (31) is presented in Appendix B. Before proposing the MUI-aware sensor scheduling scheme, we first present the following lemma, which determines the scaling behavior of MUI with respect to \(\left\{N_{x},N_{y}\right\}\). **Lemma 2**. : _Consider the following two conditions on the virtual AoDs of sensors: (a) \(\left[\rho_{x,l}^{\min},\rho_{x,l}^{\max}\right]\cap\left[\rho_{x,k}^{\min}, \rho_{x,k}^{\max}\right]=\emptyset\), \(\forall l\
eq k\), which indicates that the virtual AoD regions along the \(x\)-direction of any two sensors are non-overlapping. (b) \(\left[\rho_{y,l}^{\min},\rho_{y,l}^{\max}\right]\cap\left[\rho_{y,k}^{\min}, \rho_{y,k}^{\max}\right]=\emptyset\), \(\forall l\
eq k\), which indicates that the virtual AoD regions along the \(y\)-direction of any two sensors are non-overlapping. In the large \(\left\{N_{x},N_{y}\right\}\) limit,_  
* _if the virtual AoDs of sensors satisfy only one of Conditions (a) and (b), the interference-to-useful signal ratio scales at most with_ \(\mathcal{O}\left(N_{x}^{-2}\right)\) _(if Condition (a) holds) or_ \(\mathcal{O}\left(N_{y}^{-2}\right)\) _(if Condition (b) holds)._
* _if Conditions (a) and (b) are satisfied simultaneously, the power of MUI scales with_ \(\mathcal{O}\left(N_{x}^{-1}N_{y}^{-1}\right)\)_._  
Proof. : See Appendix C.  
Lemma 2 indicates that: (1) in the low received SNR scenario, the effect of MUI can be efficiently mitigated if the sensors are scheduled to make Condition (a) or Condition (b) satisfied; (2) whereas in the high received SNR scenario, a good sensor scheduling scheme should make Conditions (a) and (b) hold simultaneously. Based on Lemma 2 and the above discussion, we can build an MUI-aware sensor scheduling scheme as follows. * In the first step, we divide the whole region of virtual AoD along the \(x\)-direction into \(B_{x}\) disjoint blocks, and the length of each block is much larger than \(\max\limits_{l}\left\{\rho_{x,l}^{\max}-\rho_{x,l}^{\min}\right\}\). Similarly, the whole region of virtual AoD along the \(y\)-direction is divided into the \(B_{y}\) disjoint blocks, and the length of each block is much larger than \(\max\limits_{l}\left\{\rho_{y,l}^{\max}-\rho_{y,l}^{\min}\right\}\), as shown in Figure 2.
* In the second step, the sensors whose virtual AoDs \(\left\{\rho_{x,l},\rho_{y,l}\right\}\) lie in the same small rectangular region in Figure 2 are gathered into one group. In this way, we will have at most \(B_{x}B_{y}\) sensor groups.