# IRS-Enabled Ultra-Low-Power Wireless Sensor Networks: Scheduling and Transmission Schemes
## 3 Transmission Scheme
### Coherent Beamforming Benchmark  
Previous scheduling schemes presume no control links; hence, the IRS can only change configuration pseudo-randomly every frame \(t_{f}\), as opposed to coherently beamforming in order to increase the received signal strength at the user. In the presence of control links between the BS and the IRSs, the IRS can be configured to the optimal configuration towards the scheduled user. This leads to the coherent beamforming benchmark that we present here. In this benchmark, we assume the IRS in the ISN can find its optimal configuration vector denoted by \(\mathbf{v}_{l}^{*}=\operatorname{vec}(\mathbf{\Phi}_{l}^{*})\) at ISN \(l\) by choosing to enhance the collective channel gain, i.e., maximizing \(\|\mathbf{h}_{l}\|^{2}\) as provided in (6).  
The optimization problem is defined as  
\[(P0) \max_{\mathbf{v}_{l}} \|\mathbf{G}_{l}\mathbf{v}_{l}\|^{2}\] (26) s.t. \[|v_{l,n}|=1\forall n,\forall l, \tag{27}\]  
**Theorem 1** ([3]): _The optimal solution of \((P0)\) for the beamforming vector of IRS \(l\) is provided as_  
\[\mathbf{v}_{l}^{*}=e^{i\angle diag(\mathbf{h}_{2,l}^{H})\mathbf{b}_{l}}, \tag{28}\]  
_which can be written for each element as \(v_{l,n}^{*}=e^{i(-\angle h_{2,l,n}+\angle b_{l,n})}\), where \(h_{2,l,n}\) and \(b_{l,n}\) are the \(n\)th elements of \(\mathbf{h}_{2,l}\) and \(\mathbf{b}_{l}\), respectively._  
Here, we find the optimal configuration \(\mathbf{v}_{l}^{*}\) for each IRS, after which the BS schedules the ISN with the maximum SNR in the system:  
\[l^{*}=\underset{l=1\ldots L}{\operatorname{argmax}}\ \ |\mathbf{f}_{l}^{H} \mathbf{H}_{1,l}\mathbf{\Phi}_{l}^{*}\mathbf{h}_{2,l}|. \tag{29}\]  
In a way, this is similar to OS, except with \(\mathbf{\Phi}_{l}^{*}\) instead of a pseudo-random configuration. Next, we define an outage in an IRS-equipped WSN.  
In the benchmark case, the BS has perfect CSI and there are backhaul links between the BS and all the ISNs. Thus, we can assign \(\mathbf{\Theta}_{l}^{*}=\mathbf{\Phi}_{l}^{*}X_{l}(t)\) for a scheduled ISN \(l\). Sending one bit at a time from ISN \(l\) and assuming a continuous phase \(\theta_{n}\), we assign the same information-carrying phase shift for each element in the IRS while negating the overall cascaded channel phase uniquely for each IRS, such that  
\[\theta_{n}^{*}=-\angle h_{2,l,n}+\angle b_{l,n}+\{0,\pi\}; \tag{30}\]  
with a BPSK modulation scheme at the IRS, the common phase \(\phi_{com}\) is set to 0 for all \(N\) elements in the case of a 0 information bit and to \(\pi\) in case of 1 information bit. With an optimal configuration at the IRS and utilizing (17), the user obtains  
\[Y_{l} =\sqrt{P_{l}}(\mathbf{f}_{l}^{H}\mathbf{H}_{1,l}\mathbf{\Phi}_{l}^{*} (t)\mathbf{h}_{2,l})X_{l}(t)+Z_{l}, \tag{31}\] \[=\sqrt{P_{l}}(\mathbf{f}_{l}^{H}\mathbf{a}_{l})\mathbf{b}_{l}^{ H}\mathbf{\Phi}_{l}^{*}(t)\mathbf{h}_{2,l}X_{l}(t)+Z_{l}\] (32) \[=\sqrt{P_{l}}(\mathbf{f}_{l}^{H}\mathbf{a}_{l})\sum_{n}b_{l,n}h_ {2,l,n}e^{j\theta_{n}^{2}}X_{l}(t)+Z_{l}\] (33) \[=\sqrt{P_{l}}\|\mathbf{a}_{l}\|\sum_{n}|b_{l,n}||h_{2,l,n}|X_{l}(t)+Z_ {l} \tag{34}\]  
where \(\mathbf{f}_{l}=\frac{\mathbf{a}_{l}}{\|\mathbf{a}_{l}\|}\).