[Paper Review] Closed-Form Expressions for Secrecy Capacity over Correlated Rayleigh Fading Channels

Xiaojun Sun, Chunming Zhao+1 more|arXiv (Cornell University)|Mar 23, 2010

Wireless Communication Security Techniques13 references3 citations

TL;DR

This paper derives closed-form expressions for the average secrecy capacity and outage probability over correlated Rayleigh fading wiretap channels under full channel state information (CSI). By leveraging the infinite-series representation of the modified Bessel function and integral transforms, the authors obtain analytically tractable expressions that generalize prior results, particularly showing significant deviation from high-SNR limiting approximations at low and moderate SNRs.

ABSTRACT

We investigate the secure communications over correlated wiretap Rayleigh fading channels assuming the full channel state information (CSI) available. Based on the information theoretic formulation, we derive closed-form expressions for the average secrecy capacity and the outage probability. Simulation results confirm our analytical expressions.

Motivation & Objective

• To derive exact closed-form expressions for average secrecy capacity and outage probability in correlated Rayleigh fading wiretap channels.
• To address the limitation of existing high-SNR approximations that fail at low and moderate SNRs.
• To provide a mathematically rigorous framework for secure communication analysis under correlated channel conditions.
• To extend prior work on independent fading channels to the more realistic correlated case using advanced special function techniques.

Proposed method

• Utilizes the infinite-series expansion of the zeroth-order modified Bessel function to express the joint probability density function (PDF) of correlated Rayleigh fading channels.
• Applies the representation of the secrecy capacity as a piecewise function depending on whether the main channel SNR exceeds the eavesdropper's SNR.
• Employs integral transforms and the incomplete gamma function to evaluate the expectation of secrecy capacity over the correlated channel distribution.
• Derives the outage probability by transforming the secrecy rate constraint into a cumulative probability involving exponential and incomplete gamma functions.
• Uses the generalized integral form involving the function $ F( heta, k, u) $, defined as $ F( heta, k, u) = u^{k+1} heta^{-k} rac{1}{k!} imes ext{integral of } u ext{ times } ext{exp}(- u u) imes u^k imes ext{log}(1 + heta u) $, to express the average secrecy capacity.
• Validates the analytical results through simulation, confirming accuracy across a range of SNR values and correlation levels.

Experimental results

Research questions

• RQ1 What is the exact closed-form expression for the average secrecy capacity over correlated Rayleigh fading channels with full CSI?
• RQ2 How does channel correlation affect the secrecy capacity, especially at low and moderate SNR regimes?
• RQ3 Can the outage probability for a target secrecy rate be expressed in closed form under correlated fading?
• RQ4 How do the proposed expressions compare to existing high-SNR limiting approximations in terms of accuracy across SNR ranges?
• RQ5 What mathematical techniques enable the derivation of these closed-form expressions from the joint PDF of correlated fading channels?

Key findings

• The paper derives a closed-form expression for the average secrecy capacity as an infinite series involving the function $ F( heta, k, u) $, which captures the impact of correlation and SNR asymmetry.
• The derived average secrecy capacity expression is exact and numerically stable, with convergence achieved by truncating the infinite series at a finite number of terms.
• The outage probability is expressed in closed form using a double summation over modified Bessel function terms and incomplete gamma functions, enabling precise performance evaluation.
• Simulation results confirm that the high-SNR limiting approximation from [14] significantly underestimates the true secrecy capacity at low and moderate SNRs, highlighting the importance of the proposed exact expressions.
• The results reduce to the independent fading case when the correlation coefficient $ ho o 0 $, confirming consistency with prior work.
• The derived expressions are applicable to practical scenarios where channel correlation is non-negligible, such as in co-located or closely spaced eavesdroppers.