Analytical Determination of Thresholds of LDPC Codes in Free Space Optical Channel

Free-space optical (FSO) communication is crucial for the next-generation 5G+ wireless networks. The FSO links suffer from atmospheric-turbulence-induced bit errors. For the increasing link's performance, low-density parity-check (LDPC) codes, complemented by the belief-propagation (BP) algorithm, are an excellent option. The bit error rate (BER) of the LDPC code is characterized by a parameter called the threshold. The threshold is the signal-to-noise ratio (SNR), after which the BER falls arbitrarily and becomes close to zero. We derive the threshold for the LDPC codes under the BP algorithm for an uncorrelated flat FSO channel. The determination of the FSO channel threshold is a tedious task as the density of the log-likelihood ratio from the FSO channel cannot be assumed as Gaussian and is available only in a numerical form. It, thus, requires testing different values of SNR as a possible threshold systematically. Therefore, we propose the divide and conquer algorithm. The threshold depends on the degree distributions, channel state information (CSI), and the turbulence level. When CSI is known, we obtain the threshold at an SNR of 8.10 dB in high turbulence for a regular (3, 6) LDPC code. This threshold steps up to 12.48 dB when the CSI is unknown at the receiver. We evaluate the threshold values for various degree distributions (regular and irregular LDPC codes) under high, moderate, and low turbulence levels for both channel models (CSI known and unknown at the receiver). We also confirm the derived threshold values with MATLAB simulations.