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arXiv:2609.05642

Joint Average Contiguous Duration of Correlated Fading Envelopes

Muhammad Adil, Syed Junaid Nawaz, Ernestina Cianca, Wali Ullah Khan, Salman A. AlQahtani

Published Sep 4, 2026• 0 citations
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Abstract

Average contiguous duration (ACD) measures the mean time for which a fading envelope remains continuously inside a bounded amplitude interval. This letter extends the concept to two correlated observations by introducing joint average contiguous duration (JACD) and joint average fade duration (JAFD). Both metrics are expressed as joint-region occupancy probabilities divided by the corresponding inward boundarycrossing rates. For correlated Rayleigh fading, closed-form JACD and JAFD expressions are derived using the bivariate Rayleigh cumulative distribution function (CDF), Rayleigh level-crossing rates, and conditional Rician probabilities. As an application, reciprocal secret-key generation (SKG) is considered through JACD-based multilevel quantization and raw key generation rate (KGR) expressions. Numerical results show that JACD-balanced quantization improves the worst-bin joint duration, same-bin occupancy, and sample-wise raw KGR compared with marginalACD balancing.

DOI
10.48550/arxiv.2609.05642

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Average contiguous duration (ACD) measures the mean time for which a fading envelope remains continuously inside a bounded amplitude interval. This letter extends the concept to two correlated observations by introducing joint average contiguous duration (JACD) and joint average fade duration (JAFD). Both metrics are expressed as joint-region occupancy probabilities divided by the corresponding inward boundarycrossing rates. For correlated Rayleigh fading, closed-form JACD and JAFD expressions are derived using the bivariate Rayleigh cumulative distribution function (CDF), Rayleigh level-crossing rates, and conditional Rician probabilities. As an application, reciprocal secret-key generation (SKG) is considered through JACD-based multilevel quantization and raw key generation rate (KGR) expressions. Numerical results show that JACD-balanced quantization improves the worst-bin joint duration, same-bin occupancy, and sample-wise raw KGR compared with marginalACD balancing.
The wireless channel-based Secret Key Generation (SKG) algorithms aim at securing the wireless link against unauthorized eavesdropping by exploiting the channel’s randomness for generating matching secret keys at the legitimate nodes for message encryption/decryption. To counter differences in hardware and noise conditions at the legitimate nodes, which can lead to key mismatch, the SKG algorithms typically include the intermediate steps of sampling, quantization, information reconciliation, and privacy amplification. These steps collectively aim to improve the performance trade-offs between Key Generation Rate (KGR), Key Agreement Probability (KAP), and Secret Key Randomness (SKR) properties. This paper derives a closed-form expression for the Average Contiguous Duration (ACD) of Generalized Gamma (GG) fading wireless channels. The ACD is a recently introduced novel quantifier for characterizing the second-order statistics of fading channels, which includes Average Fade Duration (AFD) as its special case. The proposed GG fading ACD expression is shown to include, as its special cases, the ACD for commonly observed fading distributions such as Gamma, Nakagami- mmm , and Rayleigh. By exploiting the derived GG ACD expression, a multi-level quantization scheme for SKG is proposed that determines suitable quantization intervals for identical likelihood of an equal number of consecutive channel samples falling in each quantization interval. A comprehensive comparative analysis of the proposed ACD-based quantization for SKG is conducted in relation to conventional Uniform Quantization (UQ) and Cumulative Distribution Function (CDF)-based Non-Uniform Quantization (NUQ) schemes. The presented numerical results confirm the superior performance trade-off between KGR and KAP offered by the proposed ACD-based quantization in relation to that offered by UQ and CDF-based NUQ.
With the advent of innovative wireless technologies and the rapid growth of diverse connected devices, ensuring communication security has become a critical challenge for the future massively connected beyond 5th generation (B5G) wireless networks. The physical layer security (PLS) approach leverages the inherent physical medium characteristics of communication networks to provide secure communications. In particular, the wireless channel-based secret key generation (SKG) is a PLS approach that creates symmetric keys for message encryption/decryption by exploiting the randomness inherent in wireless channels while relying on the channel to be reciprocal between the nodes. This work proposes a novel SKG algorithm that emphasizes the pre-processing and quantization stages with the aim to increase the key generation rate (KGR) and the key agreement rate (KAR) between the legitimate nodes. The algorithm employs biasing of selective samples and sliding-window weighted averaging of the Generalized Gamma fading channel samples based on the channel’s average contiguous duration (ACD) metric, a second-order fading statistic, to enhance the channel reciprocity. The proposed scheme may also be used to improve the key randomness rate (KRR) by using a non-contiguous multi-level quantization strategy that evenly sets the quantization intervals to accommodate a similar number of channel samples across these intervals. Thus, a desired trade-off between the KGR, KAR, and KRR to meet operational requirements can be achieved with the proposed scheme. Finally, a comprehensive analysis of the performance gains of the proposed scheme in relation to other notable works is conducted by investigating the impact of several channel and algorithmic parameters on the KGR, KAR, and KRR of these schemes.
This paper presents an analytical methodology for the evaluation of two important physical layer security metrics in wiretap channels. Specifically, we first introduce the concept and the expression of average secrecy outage rate (ASOR) to quantify the average secrecy level crossing rate at a predefined secrecy threshold level. Then, we derive the expression of a new metric, namely, average secrecy outage duration (ASOD), which is a measure (in seconds) that describes how long on average the system remains in the secrecy outage status. The results are quite general and account for diversity-based systems operating over independent and identically distributed (i.i.d.) Nakagami-m fading channels. Monte Carlo simulations are conducted to confirm and discuss the analytical results. These results show that the ASOR and the ASOD are essentially affected by the diversity order, and the maximum Doppler frequency shift. In particular, and unlike the ASOD, the ASOR has a maximum value that should be considered in designing systems that are sensitive to secrecy level drops, even for short periods of time. In addition, the proposed new metric of ASOD might have large values even if the currently used metric of average fade duration shows low values on the communication link under consideration.
Two unified analytical frameworks that facilitate the evaluation of the ergodic secrecy rates of Wyner wiretap channel in generalized fading environments without/with diversity receivers are developed. In contrast to prior work that is limited to only the Rayleigh fading environment, our new formulas expressed in terms of only the moment generating function (MGF) or the cumulative distribution function (CDF) of fading signal-to-noise ratio (SNR) random variables allow us to investigate the role of fading (i.e., Effects of the amount of fading and non-identical fade distributions) between the main and the eavesdropper links as well as the impact of diversity receivers on the ergodic secrecy rate performance metric. Numerical results show that non-zero ergodic secrecy rate can be achieved even if the link quality of the main channel is worse than the eavesdropper's channel. We also show that the fundamental limit of secure communication rate can be greatly improved by understanding the role of dissimilar fading statistics and by exploiting/mitigating channel effects via diversity reception.
Abstract This paper presents a theoretical benchmark analysis of selected two-way continuous-variable quantum key distribution protocols over fast-fading channels. For uniformly fluctuating transmissivity, a non-CSI benchmark treatment evaluates the mutual information term at the worst transmissivity and averages Eve’s information over the fading distribution. Closed-form asymptotic key-rate formulas are derived for specific two-way protocols in the large-modulation regime. The framework also applies to CSI-enabled averaging and truncated log-normal weak-turbulence fading. Numerical results show that, within the adopted ideal asymptotic setting, the evaluated two-way RoR 2 benchmark retains a broader positive-key-rate region than the corresponding one-way RoR benchmark under uniform non-CSI fading. CSI-enabled averaging and the weak-turbulence extension further characterize the behavior of the selected two-way benchmarks under different channel-knowledge and fading conditions. Overall, these results characterize the performance of the selected two-way protocols across the adopted fast-fading benchmark settings.
In this paper, we study important physical layer security metrics over wiretap channels with Rician fading. We derive the average secrecy outage rate (ASOR) expression to quantify the average secrecy zero level crossing rate, and the expression of the average secrecy outage duration (ASOD), which measures (in second) how long in average the system remains in the secrecy outage status. Simulation results are conducted to confirm and discuss the theoretical derived expressions. The results show that maximum Doppler frequency shift and signal to noise ratio (SNR) at the eavesdropper are the main factors that affect the ASOR and ASOD.