Double-Shadowing Fading Channel Analysis

• Double-shadowing fading channels are wireless models characterized by two independent shadowing mechanisms superimposed on multipath fading, offering accurate empirical fits.
• They employ advanced analytical tools like Gauss–Hermite quadrature and hypergeometric series to derive tractable expressions for SNR, outage, and capacity metrics.
• These models inform system design in environments such as indoor WLANs, body area networks, and urban deployments, guiding adaptive modulation and resource allocation.

A double-shadowing fading channel is a wireless channel model that describes environments where the received signal experiences two independent and physically distinct shadowing or large-scale fluctuation mechanisms superimposed on small-scale multipath fading. This paradigm extends single-shadowing models by capturing scenarios such as indoor WLAN, body area networks, and urban deployments, where multiple layers of obstructions or reflecting/scattering agents induce additional non-Gaussian envelope fluctuations. Such models enable analytical performance analysis of capacity, error rates, and coverage, and provide physically accurate fits to measured envelope data that show compound nonstationarities or deep power fades.

1. Mathematical Definitions and Channel Models

The canonical double-shadowing channel involves the received envelope $A$ modeled as the product or convolution of two independent shadowing random variables (RVs) and a representative fading process. Various parameterizations exist based on the underlying multipath law and shadowing structure.

a. Joint Fading and Two-Path Shadowing (JFTS) Model

The JFTS model involves a convolution of a small-scale Rician fading process (parameterized by $K$ and $P_1$) and a two-wave with diffuse power (TWDP) shadowing process with parameters $S_h$, $\Delta$, and $P_2$, using a 20-point Gauss–Hermite quadrature for approximation. The probability density function (PDF) of the received envelope $A$ is given by a double sum involving Bessel and exponential terms—see [(Dey et al., 2016), eq. (1)]:

$$f_A(\alpha) = \sum_{i=1}^4 \sum_{h=1}^{20} \frac{b_i\alpha}{2P_1P_2} \mathcal{R} e^{-K-S_h-\alpha^2/(2P_2 r_h^2)} \bigg[ e^{S_h \Delta M_i} I_0\left(2\alpha\sqrt{\frac{K S_h(1-\Delta M_i)}{P_1P_2}}\right) + e^{-S_h \Delta M_i} I_0\left(2\alpha\sqrt{\frac{K S_h(1+\Delta M_i)}{P_1P_2}}\right) \bigg].$$

b. Double-Shadowed $\kappa$–$\mu$ and $\alpha$–$\kappa$–$\mu$ Models

A generalization using $\kappa$–$\mu$ or $\alpha$–$\kappa$–$\mu$ fading incorporates two multiplicative (possibly nonidentical) shadowing RVs, typically modeled with Nakagami-$m$ and inverse Nakagami-$m$ distributions. Examples include:

• Type I: Successive shadowing of the LOS component and then overall envelope
• Type II: Independent shadowing of LOS and scattered components
• Type III: Scattered clusters shadowed, followed by envelope-level shadowing

For the double-shadowed $\kappa$–$\mu$ Type I (example 1), the PDF is (for $m_t>1$, $m_d>0$) [(Simmons et al., 2018), Thm. 4]:

$$f_R(r) = \frac{2\,(m_t-1)^{m_t} m_d^{m_d} \mathcal{F}^{\mu} r^{2\mu-1}} {\Gamma(\mu)\Gamma(m_t)(m_d+\kappa\mu)^{m_d} \hat r^{2m_t}} \ _2F_1\left(m_d, m_t+\mu; \mu; -\frac{\kappa\mu r^2 \mathcal{F}}{(m_d+\kappa\mu)[(m_t-1)\hat r^2 + \mathcal{F} r^2]}\right) \left((m_t-1)\hat r^2+\mathcal{F}r^2\right)^{-(\mu+m_t)}.$$

c. Lognormal–$\kappa$–$\mu$ Shadowed Product Models

Large-scale lognormal shadowing ($H_\ell$) combined with $\kappa$–$\mu$ shadowed small-scale fading ($H_s$) yields a double-shadowed composite gain $H=H_\ell \cdot H_s$ with PDF as an integral (Chen et al., 2017):

$$f_H(h) = \sum_{i=0}^M C_i \frac{1}{\sqrt{2\pi}\sigma_s \Omega_i^{m_i} (m_i-1)!} \int_{0}^{\infty} h^{m_i-1} y^{-m_i} \exp\left(-\frac{(\ln y)^2}{2\sigma_s^2}\right) e^{-h/(y\Omega_i)} dy.$$

2. Physical Scenario and Model Taxonomies

Double-shadowing fading models account for the compound effect of multiple propagation mechanisms:

• Small-scale fading: Rician, $\kappa$–$\mu$, or $\alpha$–$\kappa$–$\mu$ multipath effects.
• Large-scale shadowing: Obstructions causing slow, lognormal (or similar) envelope variations.
• Secondary shadowing: Multiplicative shadowing applied to one or both components (LOS or multipath), representing phenomena such as blocking of dominant paths by local objects (body area, device platforms) or correlated blockage of both specular and scattered components by environmental structure (Simmons et al., 2018).

The taxonomy of double-shadowed models [(Simmons et al., 2018), Table III]:

• Type I: Dominant component first shadowed, then total envelope shadowed multiplicatively
• Type II: Independent shadowing of LOS and scattered multipath
• Type III: Clusters shadowed, entire envelope then multiplicatively shadowed

3. Statistical Properties and Performance Metrics

a. SNR Distribution

For JFTS:

$$f_\gamma(\gamma) = (\mathcal{B}/\gamma)[1 - \exp(-\mathcal{B}\gamma/\bar\gamma)]$$

where $$\mathcal{B} = \sum_{h=1}^{20} \Omega_A/(2 P_2 r_h^2)$$, $\bar\gamma$ is average SNR, and $F_\gamma(\gamma)$ involves exponential integrals (Dey et al., 2016).

b. Key Metrics

Expressions for the double-shadowed channel include:

• Outage probability $P_\text{out}(\varphi) = F_\gamma(\varphi)$ (Al-Hmood et al., 2020).
• Average BER via integration of the MGF or CDF, often with tractable Meijer-G or hypergeometric function forms (Al-Hmood et al., 2020, Simmons et al., 2018).
• Ergodic capacity $C_\text{erg} = \mathbb{E}[\log_2(1+\gamma)]$ and outage capacity $$C_\text{out}(\varepsilon) = \sup\{R : P[\log_2(1+\gamma) < R] \le \varepsilon\}$$, with explicit closed-form or infinite series for both JFTS and mixture-Gamma-based double-shadowed models (Dey et al., 2016, Al-Hmood et al., 2020).

Performance is strongly determined by the shadowing parameter set, with coverage, error, and capacity scaling as explicit functions of (for example) Nakagami-$m$, $\kappa$, $\mu$, $\sigma_s$, and associated mixture component weights.

4. Comparative Analysis and Practical Implications

a. Capacity and Diversity

For JFTS, double-shadowing leads to “heavier” envelope fluctuations, which limit the capacity even at high SNR. Unlike single-shadowing models (Nakagami–lognormal, K-distribution), capacity in the JFTS channel does not approach the non-fading Shannon bound as average SNR increases but rather is fundamentally limited by persistent, non-Gaussian specular components (Dey et al., 2016).

In the $\alpha$–$\kappa$–$\mu$ double-shadowed scenario, the asymptotic diversity order is governed by the inner shadowing index $m$ while the outer shadowing parameter $m_s$ controls the overall shadowing depth but leaves the slope untouched (Al-Hmood et al., 2020).

b. Modeling Flexibility and Empirical Fitting

The double-shadowed $\kappa$–$\mu$ model captures a wide class of fading environments, including special cases such as Rician, shadowed Rician, $\eta$–$\mu$, and single-shadowed variants by appropriate parameter selection (Simmons et al., 2018). Practical channel measurements in BANs show that double-shadowed models, especially of Type I, yield superior empirical fits to real-world envelope statistics, with AIC strongly outperforming conventional models (Simmons et al., 2018).

c. Stochastic Geometry and Cellular Coverage

In stochastic-geometry-based network analysis, the double-shadowed model allows for closed-form coverage probability through Gaussian–Hermite weighted sums of shadowed fading CCDFs. The coverage probability in Poisson cellular networks is pessimistic under double-shadowed conditions compared to models with long-term shadowing in cell selection, and the Gaussian–Hermite quadrature structure enables scalable extension to multi-tier and nonidentical interference models (Chen et al., 2017).

5. Analytical Techniques and Tractability

Double-shadowed channel laws admit integral or infinite-series representations for PDF, CDF, and MGF that, while intricate, lead to efficient evaluation. Key methods include:

• Gauss–Hermite quadrature: For numerically tractable approximations of lognormal and TWDP components (Dey et al., 2016, Chen et al., 2017).
• Series expansions in hypergeometric functions: For envelope and SNR distributions in $\kappa$–$\mu$-based models (Simmons et al., 2018).
• Mixture-Gamma representations: For exact and asymptotic analysis of $\alpha$–$\kappa$–$\mu$ double-shadowed SNR distributions (Al-Hmood et al., 2020).
• Outage and average error integrals via Meijer-G and Tricomi functions: Enabling closed-form symbolic and numerical results for error and detection probabilities (Al-Hmood et al., 2020, Simmons et al., 2018).

6. Design Guidelines and System Engineering

System design in double-shadowed fading environments should exploit transmitter CSI (to realize the full potential of OPRA), size the power amplifier to allow for a lower water-filling cut-off (reduced $\gamma_0$), and adapt modulation rates more aggressively, especially under deep, double-shadowed fades. Ergodic adaptation schemes (OPRA/ORA) are preferred whenever delay-tolerant coding is feasible, as CIFR is impractical due to infinite power requirements in such severe composite channels (Dey et al., 2016).

7. Implications for Wireless System Performance and Future Research

The double-shadowed channel framework extends the modeling toolkit for environments where simple single-layer shadowing is physically inadequate. Its impact is observed in more realistic diversity and capacity estimates—vital for dense indoor WLAN, body area networking, and heterogeneous cellular networks where deep, compound fading is commonplace. Ongoing research develops more flexible mixture-based models, further analytical tractability, and better measurement-based parameterizations, with applications in adaptive resource allocation, robust modulation, and system-level coverage prediction.

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References:

• "On the Capacity of Joint Fading and Two-path Shadowing Channels" (Dey et al., 2016)
• "On Shadowing the $\kappa$-$\mu$ Fading Model" (Simmons et al., 2018)
• "Unified Composite Distribution and Its Applications to Double Shadowed $\alpha$-$\kappa$-$\mu$ Fading Channels" (Al-Hmood et al., 2020)
• "Closed-Form Coverage Probability for Downlink Poisson Network with Double Shadowed Fading" (Chen et al., 2017)