Shadowed-Rician Channels in SAGIN

• Shadowed-Rician SAGIN Channels are parametric fading models that combine deterministic LoS and fluctuating multipath components with random shadowing to capture complex wireless propagation.
• The model unifies classic fading distributions such as Rician, Rayleigh, and κ–μ shadowed by tuning parameters, enabling tractable evaluation of outage, capacity, and BER across SAGIN links.
• Analytical expressions and closed-form results support adaptive resource allocation and RIS/relay-based designs, improving performance in satellite-to-ground, UAV-to-ground, and multi-antenna scenarios.

A Shadowed-Rician channel in a Satellite–Aerial–Ground Integrated Network (SAGIN) is a parametric small-scale fading model combining specular Line-of-Sight (LoS) and fluctuating multipath diffuse components, further modulated by random shadowing effects. This model provides a rigorous framework for capturing the nonstationary, temporally correlated, and spatially diverse nature of wireless propagation across space, aerial, and ground segments. It unifies multiple canonical fading distributions, enables tractable analysis of multi-hop and multi-antenna channels, and underpins both theoretical and applied research in SAGIN system performance—including outage probability, capacity, bit-error-rate, secrecy, interference, and adaptive resource allocation.

1. Parametric Formulation of Shadowed-Rician Channels

The canonical Shadowed-Rician model expresses the received baseband channel as $h_c = A e^{j\zeta} + Z e^{j\xi}$, where $A e^{j\zeta}$ models non-LoS Rayleigh scattering and $Z e^{j\xi}$ captures a LoS component undergoing Nakagami-$m$ shadowing; power normalization $2b_0+\Omega=1$ ensures consistent mean. The squared envelope $\alpha^2=|h_c|^2$ follows a generalized or "shadowed-Rician" law, with PDF and CDF expressed as finite sums or confluent hypergeometric functions for integer $m$:

$f_{\alpha^2}(x)=\sum_{k=0}^{m-1}\binom{m-1}{k}\frac{K_{\text{Sct}}^{m-k-1}K_{\text{LoS}}^k k!}{(K_{\text{Sct}}+K_{\text{LoS}})^m} \left(\frac{x}{K_{\text{Sct}}+K_{\text{LoS}}\right)^{k}e^{-\frac{x}{K_{\text{Sct}}+K_{\text{LoS}}}$

where $K_{\text{Sct}}=2b_0$ and $K_{\text{LoS}} = \Omega/m$ (Zhang et al., 25 Jan 2026).

In single-input single-output (SISO) and multi-input multi-output (MIMO) contexts, generalizations yield the Gamma–Wishart distribution for the Gram channel matrix $Y=H^HH$, where the channel realization $H$ comprises a Gaussian scattering component plus an independent Nakagami–Gamma shadowed LoS term (Moreno-Pozas et al., 2014). All relevant statistical characterizations—PDF, CDF, moment generating function—are available in closed-form, including reductions to the $\kappa$–$\mu$ shadowed, Rayleigh, Rician, Nakagami-$m$, and more (Paris, 2013).

2. Unification and Applicability Across Fading Scenarios

The Shadowed-Rician model subsumes multiple classical distributions by tuning its parameters:

• Rician fading: $m \to \infty$, eliminates shadowing (noncentral Wishart).
• Rayleigh fading: vanishing specular ($\Omega \to 0$), captures pure scattering (central Wishart).
• $\kappa$–$\mu$ shadowed: generalizes to arbitrary clusters and shadowed LoS terms, specializing to SISO/SAGIN links by proper reduction.
• Alternate Rician-Shadowed (ARS) models: treat the LoS state as a two-component mixture, useful in environments with intermittent blockage (Wang et al., 2020).

The model is thus suitable for LEO-to-ground, UAV-to-ground, and terrestrial links with different LoS probabilities, Doppler regimes, and environmental shadowing. Satellite-to-aerial and aerial-to-ground segments are mapped onto $(\kappa,\mu,m)$ via elevation angle, obstacle statistics, and shadowing variance (Moreno-Pozas et al., 2014).

3. Analytical Performance Metrics and Channel Statistics

Closed-form expressions exist for key metrics in Shadowed-Rician channels:

• Outage Probability: $$P_{\text{out}}(\gamma_{\text{th}}) = F_\gamma(\gamma_{\text{th}})$$ using CDF formulations in terms of ${}_1F_1$ and Horn/Marcum $Q$ functions (Paris, 2013, Nguyen et al., 2024, Wang et al., 2020).
• Ergodic Capacity: $$C = \frac{1}{\ln2}\int_0^\infty \frac{1-F_\gamma(x)}{1+x}dx$$ or via MGF-based expressions (Moreno-Pozas et al., 2014, Wang et al., 2020).
• Bit-Error-Rate (BER): For QAM under LS estimation, exact formulas are derived using joint bivariate Gamma approximations and Gauss–Chebyshev quadrature (Zhang et al., 25 Jan 2026).
• Maximum-eigenvalue distributions (MRC/SNR): via matrix hypergeometric functions and determinants of auxiliary matrices (Moreno-Pozas et al., 2014).
• Secrecy metrics: ASC, SOP, PNZ in ARS/Shadowed-Rician channels are accessible via Meijer $G$ and Fox $H$ functions (Wang et al., 2020).
• Interference modeling: In LEO multi-beam systems, desired and interfering powers are scaled SSR random variables, yielding tractable SNR, INR, SIR, SINR distributions (Kim et al., 2021).

All closed-form results require hypergeometric functions, complex gamma functions, and, in MIMO, determinants and traces of kernel matrices, allowing rapid performance evaluation or Monte-Carlo simulation.

4. Physical Interpretation and Parameter Mapping in SAGIN

The Shadowed-Rician parameters $(K, m, \Omega)$ capture the link's physical environment:

• $K$: Ratio of LoS/specular to diffuse (scattering) power. Large $K$ implies strong deterministic LoS (satellite direct-link, UAV optimal path).
• $m$: Nakagami shadowing parameter regulating LoS fluctuations. Higher $m$ denotes less severe shadowing—typical for clear-sky, low-obstacle LEO-to-ground links.
• $\Omega$: Total received mean power.

In SAGIN, each segment (satellite–air, air–ground, ground–ground) exhibits distinct $(K, m)$ associated with topology, elevation angle, object density, and path geometries. End-to-end channels may be modeled as a cascade: either as independent product distributions or through parameter aggregation by moment matching (Moreno-Pozas et al., 2014, Kim et al., 2021).

A plausible implication is that RIS-aided or relay-aided architectures can mitigate deep fades, as reflected paths accumulate central-limit advantages with increased reflectors—verified for $N=256$ element RISs compensating near-complete blockage (Nguyen et al., 2024).

5. Fading Correlation, Doppler, and Synchronization Effects

Time-varying and spatially correlated Shadowed-Rician channels are critical in dynamic SAGIN environments. Models integrate:

• Residual Doppler spreads: Jakes’ spectrum for residuals post-LoS compensation, affecting BER through time-correlation coefficients $\rho_J(\tau)$ (Zhang et al., 25 Jan 2026).
• Synchronization delays: The impact of pilot overhead and timing skew can be quantified via bivariate Gamma correlation coefficients; increased delay yields pronounced BER degradation, especially under severe Doppler (Zhang et al., 25 Jan 2026).
• Orbital mechanics: Realistic elliptical orbits versus circular affect link fading rates and net relativistic time delays, typically below microsecond levels (Zhang et al., 25 Jan 2026).

A plausible implication is that high-elevation passes (increased $K$) with fine synchronization and Doppler compensation are required for LEO–ground links to achieve low BER.

6. Numerical, Algorithmic, and Design Insights for SAGIN

Empirical parameters for SSR/Shadowed-Rician fits in LEO and UAV contexts reveal:

• Light to moderate shadowing: $(b, m, \Omega) = (0.158, 19.4, 1.29)$ to $(0.126, 10.1, 0.835)$ yield interference- to noise-limited transitions.
• Heavy shadowing: $(0.063, 0.739, 8.97\times10^{-4})$ drives deep fades even in central beam locations, with a shift toward noise limitation (Kim et al., 2021).

Closed-form expressions allow efficient evaluation of outage/BER, enabling real-time adaptive algorithms (phase-shift optimization, resource allocation) under Markovian blockage and multi-element RIS (Nguyen et al., 2024). All key formulas involve truncated hypergeometric series and/or determinants, implementable with standard numerical libraries.

In conclusion, the Shadowed-Rician (and associated unifications) supply a comprehensive, analytically rigorous, and physically interpretable modeling toolkit for characterizing and optimizing wireless propagation in SAGIN platforms, integrating environmental, orbital, and protocol-level phenomena across space, air, and ground domains (Moreno-Pozas et al., 2014, Kim et al., 2021, Nguyen et al., 2024, Wang et al., 2020, Zhang et al., 25 Jan 2026, Paris, 2013).