Approximation of Finite-Length Bound for Binary Three-State Fading Channels with Applications to GNSS

This paper proposes an effective method to calculate the maximal achievable rate via a so-called normal approximation for a given block length and error rate over a binary-input three-state fading channel. The considered three-state fading model can be used to characterize large dynamic fading effects observed in land mobile-satellite services (LMSS) suitable for global navigation satellite systems (GNSSs). At first, through a finite state Markov chain at steady state conditions, we establish the corresponding channel capacity and channel dispersion of three-state fading as a linear combination of individual states. Given that the capacity and channel dispersion are in the form of double integrals, we propose a numerical technique to effectively approximate them using 2-dimensional (2-D) Gauss-Hermite quadrature formulas. Our results show that the normal approximation can be computed effectively without the need of lengthy Monte Carlo simulations. The developed method also allows us to examine the performances of existing LDPC codes used in GNSS L1C Subframes 2 and 3. Interestingly, even though these LDPC codes have been considered the most powerful codes among GNSS coding schemes, their error performances are still 2 dB away from the finite-length bound at frame-error-rate (FER) levels of 10^–4 to 10^–5 . The obtained results suggest that other novel coding schemes might be needed for an ultra-reliable performance in the next generations of GNSSs.