## 2 Principles
The receiving end will choose to discard or save the received data according to the set rules and send back the NACK signal to the sender to request retransmission until meeting the maximum allowable retransmission times set by the system or the data is received correctly. Therefore, HARQ technology combines the characteristics of FEC and ARQ technology, improves the decoding accuracy, and improves the overall transmission performance of the system effectively. In the type of chase combining HARQ (cc-HARQ), each retransmission block is identical to the original data [18]. The complexity of cc-HARQ is relatively low, and since each retransmission is identical, it is easy for us to combine it with other techniques, in this case, hybrid RF/FSO transmission using ADCT techniques. In order to make full use of the time division gain of HARQ under an atmospheric turbulence condition, it is necessary to study the performance of HARQ-aided hybrid FSO/RF communication systems from the perspective of information theory. Based on this, the outage and throughput performance of the system is discussed as follows. The instantaneous states of the RF link fading coefficient and the FSO link turbulence coefficient at the \(i-th\) time slot are denoted as \(H_{\text{RF},j}\) and \(H_{\text{FSO},i}\), respectively, being referred to as channel coefficients. Assume that these channel coefficients are known at the receiver. In addition, the channel gains are defined as \(G_{\text{RF},i}=|H_{\text{RF},i}|^{2}\), \(G_{\text{FSO},i}=|H_{\text{FSO},i}|^{2}\). It is assumed that the returned channel state information contains only HARQ feedback bits, and the feedback channel can be an RF, an FSO, or an FSO/RF link, being error-free and delay-free. Finally, we assumed that all the links are perfectly synchronized. For the FSO link, we considered the influence of turbulence in a large range and assumed it follows the Gamma-Gamma fading distribution in strong turbulence conditions and the Log-Normal fading distribution in weak turbulence [19]. The strong and weak turbulence are determined by refractive index structure parameter \(C_{n}^{2}\), which defines the strength of the refractive index fluctuations in the atmosphere and determines some other important parameters of atmospheric turbulence, such as the scintillation coefficient and Rytov variance. Thus, the probability density function(pdf) of the channel gain is given as follows:  
\[f_{G_{FSO}}(x)=\left\{ \begin{array}{c}\frac{1}{\sqrt{2\pi}\sigma_{\text{x}} \Sigma}\exp\left(-\frac{\left[lnx+0.5\sigma_{1}^{2}\right]^{2}}{2\sigma_{1}^{ 2}}\right),\ C_{n}^{2}<10^{-15}\\ \frac{2}{\Gamma(\alpha)\Gamma(\beta)\Sigma}\left(\frac{\alpha\beta}{\mu}x \right)x^{\frac{\alpha+\beta}{\mu}}K_{n-\beta}\left(2\sqrt{\frac{\alpha\beta}{ \mu}x}\right),\ C_{n}^{2}>10^{-15}\end{array} \right. \tag{3}\]  
Figure 3: Block diagram of the cc-HARQ scheme. where \(f_{G_{FSO}}(x)\) represents the probability density function of the _FSO_ channel gain. When \(C_{n}^{2}<10^{-15}\), we use the Log-Normal fading distribution, where \(x\) is the received light intensity, which is equivalent to the optical power in unit area, and \(\sigma_{I}\) is the scintillation index. When \(C_{n}^{2}>10^{-15}\), we use the Gamma-Gamma fading distribution, where \(K_{\rm p}(\cdot)\) is the modified Bessel function of the second order. \(\mu=E[x]\), and \(E[\cdot]\) represents the expected operator. We assumed that the FSO channel gain is normalized, e.g., \(\mu=1\). \(\alpha\) and \(\beta\) are the fading/scintillation parameters related to the atmospheric turbulence conditions, which can be expressed as functions of Rytov variance \(\sigma_{R}^{2}=1.23C_{n}^{2}k^{7/6}L^{11/6}\), \(k=2\pi/\lambda\) being the wavenumber for wavelength \(\lambda\) and \(L\) being the transmission distance. The weak, moderate, and strong turbulence conditions are characterized by \(C_{n}^{2}=10^{-15}\), \(C_{n}^{2}=10^{-14}\), and \(C_{n}^{2}=10^{-13}\), respectively [20].