## 3 System Model  
This paper examines a common BC system, which includes a semi-passive BD, a reader as a legitimate receiver, and an RF source, as illustrated in Figure 1. The RF source and the passive BD both have one antenna, while the reader has an M-element antenna array for detecting the backscattered signals. As the RF source operates through broadcasting, its signal is received by both the reader and the BD simultaneously. Even though the BD is considered a passive low-power device, it can transmit its binary BD symbols by deciding whether to reflect the RF signals back to the reader. In this scenario, the reader can interpret the BD symbols by detecting the variations in the received signals. Assuming the baseband discrete-time signal model, the BD's received signal can be expressed as follows:  
\[y_{B}=\sqrt{P_{S}}h_{SB}+n_{B} \tag{1}\]  
where \(y_{B}\), \(P_{S}\), \(h_{SB}\), and \(n_{B}\) denote the received signal at BD, the source's power, the channel coefficient between the RF source and BD, and the additive white Gaussian noise (AWGN) at BD with zero mean and variance \(\delta_{B}^{2}\), respectively. Now, according to Equation (1) and Figure 1, the reader's received signal can be expressed as follows:  
\[y_{R}=\sqrt{P_{S}}S(t)h_{SB}h_{BR}+\sqrt{P_{S}}h_{SR}+n_{R} \tag{2}\]  
where \(y_{R}\), \(S(t)\), \(h_{BR}\), \(h_{SR}\), and \(n_{B}\) denote the received signal at the reader, the information signal backscattered from BD (assuming that BD is emitting with a unit power), the channel coefficient between BD and the reader, the channel coefficient between the RF source and the reader, and the AWGN at the reader with zero mean and variance \(\delta_{R}^{2}\), respectively. Here, \(t\in\{1,\cdots,T\}\) is the \(t\)-th BD symbol period. It is worth noting that the noise power produced by the BD's antenna is significantly smaller than the signal from the RF source, thus it is disregarded in Equation (2) according to [42]. According to Equation (2), the received SNR at the reader can be obtained as follows:  
\[\Gamma_{R}=\frac{\left|\sqrt{P_{S}}(h_{SB}h_{BR}+h_{SR})\right|^{2}}{\delta_{R}^ {2}} \tag{3}\]  
By assuming that the channel coefficient, \(h\), follows Rayleigh distribution, then \(g=|h|^{2}\), as the channel gains will follow the exponential distribution. Then, the SNR at the reader can be rewritten as follows:  
\[\Gamma_{R}=\frac{P_{S}\delta_{SB}\delta_{BR}}{\delta_{R}^{2}d_{SB}^{X}d_{BR}^{X }}+\frac{P_{S}\delta_{SR}}{\delta_{R}^{2}d_{SR}^{X}} \tag{4}\]  
where \(d_{SB}^{X}\), \(d_{BR}^{X}\), \(d_{SR}^{X}\), and \(\chi\) denote the distance between the RF source and BD, the distance between the BD and reader, the distance between the RF source and reader, and the channel path-loss exponent, respectively. Additionally, the ratio of the average channel gains between the direct link and the backscattered link in the studied BC system model is determined as their relative coefficient, which is expressed as follows:  
\[\Psi_{R}=\frac{E(g_{SB}\delta_{BR})}{E(g_{SR})} \tag{5}\]  
where \(E(5)\) denotes the probability of statistical expectation. Now, let us define the source-to-BD ratio (SBR), which represents the number of RF source symbols in a single BD symbol period, as equal to \(N\). This means that each BD data symbol in the \(t\)-th BD symbol period (\(S(t)\)) remains constant over \(N\) periods of RF source symbols. Therefore, for \(\forall t\), Equation (2) can be rewritten as follows:  
\[y_{R}=\mu_{t}\sqrt{P_{S}}S(t)h_{SB}h_{BR}+\sqrt{P_{S}}h_{SR}+n_{R} \tag{6}\]  
where \(\mu_{t}\in\mathrm{M}=\{0,1\}\) is used to represent the data symbol for the \(t\)-th BD in the context of binary on-off keying modulation. In other words, when \(\mu_{t}=0\), it means that BD is not backscattering any data symbol, and when \(\mu_{t}=1\), it means that BD chooses to reflect the RF source together with its information. The objective of the BD signal detector is to retrieve  
Figure 1: The studied wireless BC system model with a single BD and reader case.