Secrecy Outage Probability (SOP) Analysis

• Secrecy Outage Probability (SOP) is a metric that measures the likelihood that a wireless link's secrecy capacity falls below a target threshold, based on channel fading statistics.
• It employs analytical techniques such as joint PDFs, Meijer G-functions, and asymptotic expansions to evaluate performance and determine SNR penalties in various fading and turbulence scenarios.
• The metric informs practical secure system design by revealing non-monotonic impacts of channel correlation and turbulence, enabling targeted improvements in physical-layer security.

Secrecy outage probability (SOP) is a core probabilistic metric in physical-layer security that quantifies the likelihood that the instantaneous secrecy capacity of a wireless communication link falls below a target threshold, thus indicating an event where information-theoretic secrecy cannot be guaranteed. SOP has become a standard analytic tool for performance evaluation in a wide range of secure wireless systems, encompassing fading, multi-antenna, cooperative, and emerging reconfigurable intelligent surface (RIS) scenarios. Its formalism, interpretation, and calculation techniques are deeply entwined with the channel statistical properties and have important ramifications for secure system design.

1. Definition and Mathematical Foundations

At its core, SOP is defined as the probability that the instantaneous secrecy rate $C_s(\gamma_1, \gamma_2)$ is less than a target secrecy rate $R_s$: $$C_s(\gamma_1, \gamma_2) \triangleq [\ln(1 + \gamma_1) - \ln(1 + \gamma_2)]^+$$

$$\mathrm{SOP} = P\{C_s(\gamma_1, \gamma_2) < R_s\} = P\{\gamma_1 \leq \Theta \gamma_2 + \Theta - 1\}$$

where $\gamma_1$ and $\gamma_2$ are the instantaneous SNRs at the legitimate receiver (main channel) and the eavesdropper (wiretap channel), and $\Theta = \exp(R_s)$.

In practical systems, $\gamma_1$ and $\gamma_2$ are modeled as random variables dictated by the channel fading statistics (e.g., Rayleigh, Nakagami-$m$, Málaga, etc.), and may exhibit statistical correlation depending on the physical scenario (such as spatial proximity). The SOP is therefore evaluated by integrating the joint PDF $f_{\gamma_1, \gamma_2}(\gamma_1, \gamma_2)$ over the relevant region: $$\mathrm{SOP} = \iint_{\gamma_1 \leq (1+\gamma_2)\Theta - 1} f_{\gamma_1, \gamma_2}(\gamma_1, \gamma_2) d\gamma_1 d\gamma_2$$ (Ai et al., 2021).

2. Application to Málaga Turbulence Channels and Arbitrary Correlation

In free-space optical (FSO) communications, atmospheric turbulence statistics are often modeled by generalized distributions such as the Málaga ($\mathcal{M}$) distribution, which have sufficient flexibility to encompass a wide range of turbulence conditions. The joint distribution for correlated $\gamma_1$ and $\gamma_2$ under $\mathcal{M}$ fading is expressed via multi-parameter Meijer G-functions and series representations.

For arbitrarily correlated channels, the PDF $f_{\gamma_1, \gamma_2}$ incorporates the correlation coefficient $\rho$ and is encapsulated by: $$\mathrm{SOP} = \sum_{t=0}^{\infty} \mathcal{A}_t \prod_{p=1}^2 \Bigg[\frac{1}{2\sqrt{\mu_p}} \sum_{k=1}^\beta (-1)^{k-1} \binom{\beta-1}{k-1} \frac{1}{(k-1)!} \ldots \int\int \mathrm{G}(\cdot) d\gamma_1 d\gamma_2 \Bigg]$$ where the terms $\mathcal{A}_t$, $\mu_p$, $\beta$, and the Meijer G-functions encapsulate the turbulence, fading, and correlation statistics. Numerical integration is typically practical by reducing the problem to a single integral with quadrature techniques (Ai et al., 2021).

3. Asymptotic SOP Behavior and Slope Analysis

In the high-SNR regime (large mean SNR of the main channel, $\mu_1$), the SOP assumes an asymptotic form due to the dominant behavior of the leading Meijer G-function terms. By applying expansions such as Slater’s theorem, the main result is: $$\mathrm{SOP} \approx \sum_{h=1}^4 \mathcal{D}_h \,\mu_1^{-(b_h + (\alpha+1)/4)}$$ where $b_h$ are constants from the expansion and $\mathcal{D}_h$ are determined by the turbulence and correlation parameters.

The exponential slope in dB of the SOP versus SNR is determined by: $\text{slope} = \min\{\alpha/2,\, 1/2\}$ Thus, for moderate and strong turbulence, the decay rate of SOP is governed by the atmospheric parameter $\alpha$ rather than by the correlation coefficient $\rho$ (Ai et al., 2021). This provides a direct design criterion: improving turbulence conditions (larger $\alpha$) is fundamentally more effective in improving high-SNR secrecy than reducing channel correlation.

4. Non-Monotonic Impact of Channel Correlation

A distinctive insight for correlated FSO channels with Málaga fading is that SOP exhibits non-monotonic behavior as a function of the correlation coefficient $\rho$:

• Moderate $\rho$ regime: Increasing $\rho$ from a low to a moderate value degrades SOP performance. For a fixed target SOP (e.g., $10^{-3}$), a higher average SNR is needed as $\rho$ increases from $0.5$ to $0.7$, leading to an SNR penalty (e.g., $\sim10$ dB increase).
• High $\rho$ regime: Beyond a critical $\rho$ threshold (which depends on turbulence severity), further increases in $\rho$ improve the SOP, reducing the required SNR for the same outage probability.

This reversal occurs because, at high correlation, knowledge about the eavesdropper's channel can be inferred at the legitimate node due to the statistical similarity, thus enhancing security. Analytically, this behavior is a consequence of SOP expressions containing both $\rho^2$ and $(1-\rho^2)$ dependencies, with dominant terms shifting as $\rho$ crosses the critical threshold (Ai et al., 2021).

5. SNR Penalty and Practical System Design

The SNR penalty for achieving a given target SOP is not a monotonically increasing function of correlation. As shown by representative numerical results: for strong turbulence and ${\rm SOP}=10^{-3}$, increasing $\rho$ from $0.5$ to $0.7$ raises the required mean channel SNR by $\sim$10 dB, but pushing $\rho$ beyond $0.9$ can substantially lower the required SNR.

This finding supports nuanced system design: In deployments where the eavesdropper is close to the legitimate receiver and spatial correlation is strong, one might exploit this correlation (or adapt the receiver architecture) to improve secrecy performance once a high-correlation threshold is crossed.

6. Implementation Considerations and Analytical Techniques

The advanced statistical structure of $\mathcal{M}$-distributed turbulence with correlation necessitates the use of series expansions, Meijer G-functions, and numerical quadrature for exact SOP evaluation. The asymptotic results, by contrast, allow rapid assessment of large-SNR behavior and identification of the diversity order. The key implementation steps are:

• Derivation of the joint PDF $f_{\gamma_1, \gamma_2}$ with precise parameterization of turbulence and correlation.
• Reduction of the SOP double integral to a single integral using, e.g., the Gauss–Chebyshev quadrature rule.
• Asymptotic expansion in the high-SNR regime by isolating dominating terms, providing slope and diversity insights independent of correlation.
• Numerical validation to determine required SNR penalties and beneficial regimes for correlation manipulation (Ai et al., 2021).

7. Summary of Key Insights

Phenomenon	Behavior/Implication	Governing Parameter
SOP slope at high SNR	Determined by atmospheric turbulence, independent of correlation	$\min\{\alpha/2, 1/2\}$
SOP vs. correlation ($\rho$)	Non-monotonic: initial penalty, improvement past critical threshold	$\rho$
Analytical tractability	High for both exact (via Meijer G) and asymptotic (power law) forms	See (Ai et al., 2021)
System design implications	High correlation can eventually benefit security under $\mathcal{M}$-fading	Receiver/channel design

The comprehensive analytic treatment of SOP over arbitrarily correlated $\mathcal{M}$-distributed FSO channels enables accurate risk quantification and targeted design for secure wireless optical communications, accounting for real-world channel dependencies and turbulence effects (Ai et al., 2021).