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Joint Average Contiguous Duration of Correlated Fading Envelopes
Muhammad Adil, Syed Junaid Nawaz, Ernestina Cianca, Wali Ullah Khan, Salman A. AlQahtani
TL;DR
Marginal duration does not capture how long two correlated observations remain simultaneously usable, as required when both SKG observations must stay in a common quantization interval. The paper defines joint persistence metrics through occupancy and inward boundary crossings, then uses JACD-balanced multilevel quantization. Compared with marginal-ACD balancing, reported results improve the weakest-bin joint duration, same-bin occupancy, and sample-wise raw KGR.
Problem
Marginal duration is insufficient when two correlated channel observations must remain usable simultaneously, as in reciprocal SKG where a usable run ends if either observation leaves its quantization interval.
Source: §I · p. 1Method
The paper defines JACD and JAFD from joint-region occupancy and inward boundary-crossing rates, then applies JACD-balanced multilevel quantization to reciprocal SKG.
Source: §I · p. 1, 1) JACD-Balanced Multilevel Quantization · p. 4Results
JACD balancing improves the worst bin by 21.2%, raises Psame from 0.6138 to 0.6410, and increases R(s) KGR/fs from 1.2276 to 1.2820 bits/sample.
Source: §V · p. 5Takeaways & Limitations
For reciprocal SKG, JACD balancing improves the weakest joint-duration bin while revealing a tradeoff between longer runs and excursion refresh rate.
Source: §VI · p. 5
Abstract
from arXiv · showhide
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.
I. Introduction
Marginal duration does not capture how long two correlated channel observations remain simultaneously usable, as reciprocal secret-key generation requires. The paper introduces joint duration metrics and develops their analysis for correlated fading, with SKG as an application.
I. Introduction
- ACD measures persistence within an arbitrary amplitude range, connecting fading statistics to sampling, quantization, encoding, and physical-layer security.This contrasts with average fade duration, which is tied to a lower-tail event.Source: §I · p. 1
- Two-observation applications need a duration measure for simultaneous usability because a joint run ends when either observation leaves its prescribed interval.The paper names these intervals joint contiguous-duration instances and their stationary mean the joint average contiguous duration (JACD).Source: §I · p. 1
- The paper introduces JACD for joint persistence and JAFD for simultaneous fades, addressing two-dimensional amplitude events not captured directly by existing correlated crossing and fade-duration statistics.JACD relates joint-region occupancy to inward boundary-crossing rates; the framework is specialized to correlated Rayleigh fading and applied to reciprocal SKG.Source: §I · p. 1
II. Joint Average Contiguous Duration for Arbitrary
JACD is the stationary mean length of the maximal intervals in which both envelopes remain inside their prescribed ranges. It is obtained by dividing joint-region occupancy by the rate of inward entries; the joint-fade case gives JAFD.
II. Joint Average Contiguous Duration for Arbitrary
- JACD averages the maximal temporal intersections of Alice’s and Bob’s scalar contiguous-duration instances within their respective amplitude ranges.A joint instance starts when the envelope pair enters the joint region and ends when either component exits its interval.Source: §II · p. 2
- Joint-region occupancy divided by the total inward crossing rate gives the mean persistence per joint entry.Occupancy is the fraction of time both envelopes lie in their intervals; the inward rate counts new joint excursions.Source: §II · p. 2
- JAFD applies the same occupancy–crossing–duration structure to the region where both envelopes are below their fade thresholds.A joint fade ends at the first exit of either component from that region.Source: §II · p. 2
III. Closed-Form Correlated Rayleigh JACD
For correlated Rayleigh fading under the Jakes model, the section derives closed-form JACD expressions for arbitrary node intervals from joint occupancy and inward boundary flux. Correlation affects both simultaneous occupancy and the chance that the other envelope remains admissible at a crossing.
III. Closed-Form Correlated Rayleigh JACD
- Under the adopted jointly proper Gaussian model, zero derivatives of the relevant correlations at zero lag make the instantaneous channel vector independent of its derivative vector.The temporal model uses Jakes autocorrelation, with J0 and each link's maximum Doppler frequency specifying its lag dependence.Source: §III · p. 3
- The correlated-Rayleigh JACD for arbitrary node intervals is closed form, combining joint-region occupancy with inward boundary fluxes evaluated using Rayleigh LCRs and conditional Rician interval probabilities.The occupancy uses the bivariate Rayleigh CDF; derivative independence under the adopted model simplifies the boundary fluxes.Source: §III · p. 3
- Correlation changes both the probability of simultaneous occupancy and the probability that the other envelope remains admissible when a boundary crossing occurs.Source: §III · p. 3
- For identical links, ΓJF(0; rf)=ΓF(rf)/2, while limρ→1 ΓJF(ρ; rf)=ΓF(rf); the corresponding entry-rate limits are 2N(rf)F(rf) and N(rf).Source: §III · pp. 3–4
- The independent- and perfectly correlated-link entry-rate limits are equal when F(rf)=1/2, explaining the correlation-dependent crossing described for the numerical results.Source: §III · p. 4
IV. Applications
JACD characterizes how long two correlated observations remain in the same prescribed amplitude region, supporting same-bin persistence analysis for reciprocal SKG. JACD-balanced quantization equalizes joint bin durations, while independent and perfectly correlated limits provide reference cases.
IV. Applications
- Joint occupancy, entry rate, and duration distinguish how often simultaneous runs begin from how long they persist, informing sampling, threshold, guard-region, and minimum-run choices.The metrics also support estimating jointly usable sample-run lengths and characterizing simultaneous deep-fade runs.Source: §IV · p. 4
- JACD describes persistent same-bin observations and new key-generation opportunities, while JAFD characterizes jointly observed deep-fade runs in deep-fade-based SKG.The two observations must remain in the same quantization interval for a jointly usable run.Source: §IV · p. 4
- For identical links, joint duration ranges from ΞI/2 under independence to ΞI under perfect correlation because independent crossings terminate runs separately.Under perfect correlation, boundary crossings coincide.Source: §IV · p. 4
- JACD balancing equalizes the joint durations across quantization bins and maximizes the shortest bin duration, removing the shortest-persistence bottleneck.At sufficiently fine sampling, the mean joint sample-run length is approximately the bin’s JACD divided by the sampling interval.Source: 1) JACD-Balanced Multilevel Quantization · p. 4
1) JACD-Balanced Multilevel Quantization:
Joint occupancy and entry rate do not by themselves specify persistence: bins with the same occupancy may contain many short runs or fewer long ones.
1) JACD-Balanced Multilevel Quantization:
- Equal joint occupancy can correspond to many short runs or fewer long runs, so occupancy alone does not capture persistence.For bin ℓ, occupancy is PJC,ℓ, entry rate is ΛJC,ℓ, and mean duration is ΞJC,ℓ, related by PJC,ℓ = ΛJC,ℓΞJC,ℓ.Source: 2) Persistence versus Occupancy · p. 4
- This distinction matters for SKG schemes that accept only sufficiently persistent runs.Source: 2) Persistence versus Occupancy · p. 4
- Guard bands, unequal thresholds, and asymmetric quantizers can be handled using the corresponding Alice–Bob rectangle in the general JACD formulation.Source: 2) Persistence versus Occupancy · p. 4
2) Persistence versus Occupancy:
Raw key-generation rates distinguish counting every jointly usable sample from retaining one label per joint excursion. They are pre-reconciliation rates, not secret-key rates.
2) Persistence versus Occupancy:
- The raw KGR expression counts every jointly usable sample, whereas an alternative retains one label per joint excursion.Source: 2) Persistence versus Occupancy · p. 4
- For JACD-balanced bins, sample-wise KGR is bL P_same/ΨJC; longer JACD can increase usable same-bin samples while reducing distinct excursion starts.Source: 2) Persistence versus Occupancy · p. 4
- The displayed lower-tail figure compares normalized JAFD and joint fade-entry rate across unequal thresholds at ρ = 0.84 and equal thresholds across five correlations.For equal thresholds, JAFD increases with threshold and correlation; the joint fade-entry rate is non-monotonic.Source: Figure 2 · p. 5
- Both rates are pre-reconciliation raw rates; a secret-key rate additionally requires reconciliation leakage and the eavesdropper’s observation statistics.Source: 2) Persistence versus Occupancy · p. 5
V. Numerical Results √
Numerical results validate the joint-duration analysis and show that JACD-balanced quantization improves the weakest bin, same-bin occupancy, and sample-wise raw KGR relative to marginal-ACD balancing, alongside a lower excursion-based KGR rate.
V. Numerical Results
- Simulations use complex Gaussian fading sequences generated by an FFT implementation of the Jakes Doppler spectrum, with fs=200fm.Analytical results use equations (10), (14)–(15), (27), and (31).Source: §V · p. 5
- JAFD increases with threshold and correlation; at ρ = 0, the joint fade-entry rate ΓJF(0; rf) equals half the single-envelope rate ΓF(rf), and as ρ approaches 1 it converges to ΓF(rf).Source: §V · p. 5
- At ρ=0, fmΞJC=0.10824 is half of fmΞI=0.21648, and the JACD approaches scalar ACD as ρ→1.The reported JACD is for I=(0.5, 1].Source: §V · p. 5
- JACD balancing equalizes normalized bin durations fmΞJC,ℓ at approximately 0.16396, using thresholds [0, 0.5103, 1.1146, 1.7567, 2.5].For comparison, marginal-ACD balancing uses [0, 0.5056, 1.0330, 1.6221, 2.5].Source: §V · p. 5
- 21.2%: at L=4, ρ=0.84, JACD balancing gains worst-bin duration, raises Psame from 0.6138 to 0.6410, and raises R(s) KGR/fs from 1.2276 to 1.2820 bits/sample.Fig. 3 compares the balanced-bin results and reports the worst-bin gain.Source: Figure 3 · p. 5, §V · p. 5
- R(e) KGR/fm falls from 8.3291 to 7.8190 under JACD balancing, while the worst-bin gain peaks near ρ=0.84.Fig. 3(d) confirms theoretical bin-averaged joint sample-run lengths for L ∈ {2, 4, 8, 16, 32}.Source: §V · p. 5, Figure 3 · p. 5
VI. Conclusion
The conclusion presents JACD and JAFD as joint-duration extensions for correlated fading, with closed-form expressions for correlated Rayleigh fading. For reciprocal SKG, JACD-balanced quantization improves the weakest joint-duration bin while exposing a tradeoff between longer runs and excursion refresh rate.
VI. Conclusion
- The framework derives JACD for arbitrary amplitude intervals and JAFD for simultaneous fades by relating joint-region occupancy to inward boundary-crossing rates.Closed-form expressions are obtained for correlated Rayleigh fading.Source: §VI · p. 5
- Independent and perfectly correlated limits provide exact consistency checks, while JACD distinguishes joint-event occupancy from persistence.The metrics also predict average jointly usable sample runs.Source: §VI · p. 5
- For reciprocal SKG, JACD-balanced quantization improves the weakest joint-duration bin but reveals a tradeoff between longer runs and excursion refresh rate.Source: §VI · p. 5
- JAFD retains the classical AFD–LCR interpretation in a two-envelope setting.Source: §VI · p. 5