A novel and efficient parameter estimation of the Lognormal-Rician turbulence model based on k-Nearest Neighbor and data generation method

TL;DR

The paper tackles accurate estimation of the Lognormal-Rician shaping parameters $r$ and $\sigma_z^2$, which is hard due to the PDF's integral and Bessel terms. It proposes a $k$NN density approximation plus a data-generation framework to form an approximate log-likelihood (LLF) and jointly estimate $r$, $\sigma_z^2$, and $k$ by maximizing the LLF. KS tests determine the optimal $k$ (e.g., around $8$ to $15$ for typical sample sizes); gradient-descent optimization yields limited gains with more generated data, while a genetic algorithm achieves performance close to SAP/EM with reduced computation. The approach is adaptable to other fading models and reduces reliance on physical turbulence models, offering a practical balance between accuracy and complexity for free-space optical/quantum channels.

Abstract

In this paper, we propose a novel and efficient parameter estimator based on $k$-Nearest Neighbor ($k$NN) and data generation method for the Lognormal-Rician turbulence channel. The Kolmogorov-Smirnov (KS) goodness-of-fit statistical tools are employed to investigate the validity of $k$NN approximation under different channel conditions and it is shown that the choice of $k$ plays a significant role in the approximation accuracy. We present several numerical results to illustrate that solving the constructed objective function can provide a reasonable estimate for the actual values. The accuracy of the proposed estimator is investigated in terms of the mean square error. The simulation results show that increasing the number of generation samples by two orders of magnitude does not lead to a significant improvement in estimation performance when solving the optimization problem by the gradient descent algorithm. However, the estimation performance under the genetic algorithm (GA) approximates to that of the saddlepoint approximation and expectation-maximization estimators. Therefore, combined with the GA, we demonstrate that the proposed estimator achieves the best tradeoff between the computation complexity and the accuracy.

Figures (4)

• Figure 1: The optimal $k$ and KS test results for different channel conditions. (a)The optimal $k$, (b)KS test results when $k = 2$.
• Figure 2: LLFs under different channel conditions. (a)$M = 10^4$, $L = 10^4, N_{LLF}=1$, ${r}^{*}= 3.8, {\sigma_z^2}^{*} = 0.16$, (b)$M = 10^4$, $L = 10^4, N_{LLF}=20$, ${r}^{*} = 5.6, {\sigma_z^2}^{*} = 0.25$, (c)$M = 10^4$, $L = 10^6, N_{LLF}=20$, ${r}^{*} = 5, {\sigma_z^2}^{*} = 0.25$, (d) $M = 10^3$, $L = 10^6, N_{LLF}=50$, ${r}^{*} = 4.8, {\sigma_z^2}^{*}= 0.25$.
• Figure 3: MSE performance of the estimators under different Lognormal-Rician channel conditions, where $r =4$.
• Figure 4: MSE performance of the estimators under different Lognormal-Rician channel conditions, where $\sigma_z^2 = 0.25$.