# Channel Correlation-Based Adaptive Power Transmission for Free-Space Optical Communications
## 2 Theoretical Analysis
### Operation Principle  
Figure 2 presents the block diagram of the proposed adaptive power transmission technique. In order to visually and efficiently adjust the signal power, non-return-to-zero on-off-keying (NRZ-OOK) modulation is used, which expresses bit '1' or bit '0' by the level of light intensity. The channel correlation coefficient \(\rho\) is extracted based on the signal's fluctuation intensities after passing through the uplink and downlink turbulence channels, and \(\rho\) is given by  
Figure 1: Probability density function of the modeled turbulence channel. (**a**) \(\sigma_{I}^{2}=0.37\), (**b**) \(\sigma_{I}^{2}=0.5\).  
\[\rho=\frac{Cov(I_{u},I_{d})}{\sigma_{u}\sigma_{d}}, \tag{5}\]  
where \(I_{u}\) and \(I_{d}\) are the fluctuation intensities of the uplink and downlink transmission signal through the turbulence channels, respectively; \(Cov(I_{u},I_{d})\) is the covariance between \(I_{u}\) and \(I_{d};\sigma_{u}\) and \(\sigma_{d}\) are standard deviations of \(I_{u}\) and \(I_{d}\), respectively [17]. In a real system, a feedback path needs to be added between the transmitter and the receiver to feedback the signal's fluctuation intensities for extracting the channel correlation coefficient. Then, the power of the reverse transmission signal is adjusted, and the adjusted power \(P_{c}\) is expressed by  
\[P_{c}=\min(P_{m},\alpha\cdot\rho\cdot P_{t}), \tag{6}\]  
where \(P_{m}\) is the maximum allowable transmitting power, \(\alpha\) is an adaptation factor obtained from the fluctuation strength \(I\) of the received uplink or downlink transmission signal, \(P_{t}\) is the initial transmitting power, and \(\min(P_{m},\alpha\cdot\rho\cdot P_{t})\) is the minimum value taken from \(P_{m}\) and \(\alpha\cdot\rho\cdot P\). The modulated optical NRZ-OOK signal after a power adjustment is emitted into the turbulence channel by a 1550 nm wavelength laser diode (LD) for reverse transmission. At the receiving end of the reverse transmission, the photodiode (PD) detects the transmitted optical signal and performs photoelectric conversion. The received electrical signal after PD \(r(t)\) is represented by  
\[r(t)=I(t)s(t)+n(t), \tag{7}\]  
where \(s(t)\) is the transmitted NRZ-OOK signal after a power adjustment, and \(n(t)\) is the average noise power comprising PD and background noise. The different signal-to-noise ratio (SNR) values are used to indicate the magnitude of noise, and it is given by  
\[\mathrm{SNR}=10\lg\left(\frac{P_{s}}{P_{n}}\right), \tag{8}\]  
where \(P_{s}\) and \(P_{n}\) are the effective power of the signal and noise, respectively [18]. The FTD technique is used to identify the signal. A fixed threshold is set at the receiver side in the FTD technique compared to the ATD technique, which dynamically adjusts the decision threshold bit-by-bit based on the CSI. The bit '1' is determined when the signal level is above the set threshold, and the bit '0' is determined when it is below the set threshold. Ultimately, the BER is computed to evaluate the performance of the proposed adaptive power technique, and it is expressed by  
\[\mathrm{BER}=\int_{0}^{\infty}f(r)Q\left(r/\sqrt{2n}\right)dr, \tag{9}\]  
where \(f(r)\) is the PDF of \(r\), and \(Q(\cdot)\) is the Q-function [19].  
Figure 2: Block diagram of the proposed adaptive power transmission technique. LD: laser diode; PD: photodiode; FTD: fixed threshold decision.