# Theoretical Derivation and Optimization Verification of BER for Indoor SWIPT Environments
## 3 Resource Allocation Problem Formulation  
### Quality of Service (QoS) Factor for Indoor SWIPT System  
The target for the indoor SWIPT network is providing a wireless power transfer to receivers while transferring data to the desired receiver. Thus, we can define the quality of service (QoS) measurement factor in two ways--data quality and energy efficiency. Here, we set \(H\) as the channel vectors between the desired receiver and the transmitter, \(G_{k}\) as the channel vectors between the the \(k\)th idle receiver and transmitter, and \(w\) as the corresponding beam-forming vector. The elements of \(w\) are as follows:  
\[W_{n}=I_{n}e^{i\alpha_{n}}. \tag{10}\]  
The multi-path channel with an impulse response is  
\[h_{\text{UWB}}(t)=\sum\
olimits_{i=1}^{N}\alpha_{i}\delta(t-\tau_{i}) \tag{11}\]  
where \(i\) is the index of the path, \(\tau_{i}\) is the time delay for the \(i\)th path, and \(\alpha_{i}\) is the attenuation of the path. The pulse stream is  
\[x(t)=\sqrt{E_{\text{st}}}\sum\
olimits_{l=0}^{\infty}p(t-lT_{d})d \tag{12}\]  
where \(T_{d}\) is the time span of the signal. The pulse-amplitude modulation symbols \(d\in\{\pm 1\}\) are considered to be independently identically distributed, and the Gaussian waveform can be expressed as  
\[\text{p}(t)=\frac{d^{2}}{dt}\Bigg{(}\frac{1}{\sqrt{2\pi\sigma}}\text{e}^{- \frac{l^{2}}{2\sigma^{2}}}\Bigg{)} \tag{13}\]  
where \(\sigma\) is the standard deviation. The received signal is expressed as  
\[r(t)=\Big{[}x(t)\bigotimes h_{\text{UWB}}(t)\Big{]}+N(t). \tag{14}\]  
Figure 3: Receiver architecture design for our system. The receiver samples the signal with suitably delayed references as  
\[\text{q}(\text{t})=\sum\
olimits_{n=0}^{\infty}p(t-\tau_{1}-(n-1)T_{d}) \tag{15}\]  
where \(\tau_{1}\) is the delayed time with the first revived wave. Here, the correlator output is  
\[\text{Z}(\text{n})=\int_{(n-1)T_{d}}^{nT_{d}}\biggl{\{}\sqrt{\text{E}_{\text{st }}}\sum\
olimits_{l=0}^{\infty}p(t-\varGamma_{d})d\bigotimes h_{UIWB}(t) \biggr{\}}q(t)dt+\int_{(n-1)T_{d}}^{nT_{d}}N(t)q(t)dt\;. \tag{16}\]  
Z(n) is compared with a threshold between 1 and \(-1\). The variance of the output noise is \(\sigma^{2}=\frac{N_{0}}{2}E_{st}\). The probability of error on average for one bit is calculated as  
\[P\biggl{[}Z(n)\biggr{|}\biggr{]}=\frac{1}{2}\mathit{erfc}\biggl{[}\frac{V(n)} {\sqrt{2\sigma}}d_{N}\biggr{]}. \tag{17}\]  
The data quality between the transmitter and the desired receiver for the bit error rate (BER) is as follows:  
\[\text{BER}=\sum\
olimits_{n=0}^{N}P(\vec{d_{N}}).\frac{\vec{2}}{2}\mathit{erfc }\biggl{[}\frac{V(t=nT_{d})}{\sqrt{2}\sigma}.(d_{N})\biggr{]} \tag{18}\]  
where \(\mathit{erfc}(\text{x})=\frac{2}{\sqrt{2}}\int_{x}^{\infty}e^{-y^{2}}dy\) is the complementary error function and \([\vec{d_{n}}]=\{d_{0},d_{1},\ldots d_{N}\}\) is the binary sequence. In the SWIPT system, energy is harvested by idle receivers harvested from the RF to extend their lifetime. The efficiency of energy harvesting is important in the system design of networks, and is defined as the ratio of total harvested power at the receivers and the total radiated power by the transmitter in the SWIPT system. The total harvested energy by the K idle receivers is calculated as  
\[h_{\text{UWB}}(t)=\sum\
olimits_{i=1}^{N}\alpha_{i}\delta(t-\tau_{i}) \tag{11}\]0  
where \(*\) denotes the conjugate transpose. The total energy harvested from all of the receivers is modeled as  
\[h_{\text{UWB}}(t)=\sum\
olimits_{i=1}^{N}\alpha_{i}\delta(t-\tau_{i}) \tag{11}\]1  
where we set \(\eta\) and \(\eta_{k}\) as constants, \(0\leq\eta\), \(\eta_{k}\leq 1\) for all k. This provides the conversion efficiency of the RF energy in converting the received radio signal to electrical energy by the desired receiver and K idle receiver.