Cute but Cunning: Effective Closed-Form Alternatives to the Exact Lognormal Statistics

TL;DR

This paper proposes two mathematically tractable surrogate models for the Lognormal distribution, enabling closed-form expressions for key performance metrics in wireless communications and related fields, thus overcoming its analytical intractability.

Contribution

It introduces novel surrogate models based on Nakagami variates that approximate Lognormal distributions with high accuracy and provide closed-form solutions for performance analysis.

Findings

Surrogate models closely match true Lognormal statistics.

Closed-form expressions improve analysis efficiency.

Method extends to complex cascaded fading channels.

Abstract

The Lognormal distribution is a fundamental statistical model widely used in different fields of science, including biology, finance, economics, engineering, etc. In wireless communications, it is the primary statistic for large-scale fading modeling. However, its known analytical intractability presents persistent channel characterization and performance analysis challenges. This paper introduces two effective and mathematically tractable surrogate models for the Lognormal distribution, constructed from the product of Nakagami-$m$ and Inverse Nakagami-$m$ (I-Nakagami-$m$) variates. These models yield asymptotically exact closed-form expressions for key performance metrics -- including the characteristic function, bit error rate, and Shannon's capacity -- and enable analytically tractable expressions for the probability density function and cumulative distribution function of the composite $\alpha$-$\mu$-Lognormal fading model. To facilitate implementation, a moment-matching framework is developed to map the Lognormal parameters to the surrogate model parameters. In addition, a random mixture approach is proposed to enhance convergence by exploiting the complementary approximation properties of the Nakagami-$m$ and I-Nakagami-$m$ distributions. The methodology is further extended to heterogeneous cascaded fading channels comprising arbitrary combinations of $\alpha$-$\mu$, $\kappa$-$\mu$, and $\eta$-$\mu$ variates, for which moment-based mappings to the equivalent Lognormal distributions are derived. Numerical results confirm the accuracy and efficiency of the proposed approach, positioning it as a practical and reliable alternative to exact Lognormal statistics.