A Closed-Form Approximate Expression for the BEP of BDPSK Signal in Log-Normal SISO FSO Communication System

We develop an average bit error probability (BEP) expression for binary differential phase shift keying (BDPSK) signal in log-normal single-input-single-output laser free-space optical (FSO) communication system by treating the BEP under particular atmospheric turbulence as random variable and changing the integral form. By using this average BEP expression and successive polynomial approximations, we obtain an explicit three-term approximate expression of the average BEP, while the existed approximate BEP expressions are implicit. The three-term approximate BEP expression can be used to intuitively analyze the relationship between BEP and system parameters such as signal-to-noise ratio (SNR) and Rytov standard deviation. In addition, the three-term approximate BEP expression works well when the Rytov standard deviation is no larger than 0.15. In this range of Rytov standard deviation, this approximate BEP expression can accurately predict the performance of FSO communication system with the transmission distance of up to 3.9 km and the laser at a wavelength of 1550 nm. The three-term approximate BEP expression can be used directly to derive the closed-form SNR expression as a function of Rytov standard deviation. The maximum loss between closed-form and theoretical SNR expressions is only 0.50 dB.