pith. sign in

arxiv: 2605.16920 · v1 · pith:YROCKBVNnew · submitted 2026-05-16 · 📡 eess.SP

Analysis of Fluid Antenna Systems with Continuous Positioning and Spatial Correlation

Gayani Siriwardana , Peter J. Smith , Himal A. Suraweera , Rajitha Senanayake This is my paper

Pith reviewed 2026-05-19 19:55 UTC · model grok-4.3

classification 📡 eess.SP
keywords fluid antenna systemscontinuous positioningspatial correlationlevel-crossing rateCDF approximationSINRRayleigh fadingRicean fading
0
0 comments X p. Extension
pith:YROCKBVN Add to your LaTeX paper What is a Pith Number? 
\usepackage{pith}
\pithnumber{YROCKBVN}

Prints a linked pith:YROCKBVN badge after your title and writes the identifier into PDF metadata. Compiles on arXiv with no extra files. Learn more

The pith

A level-crossing-rate framework supplies asymptotically exact approximations and tight bounds for the CDF of the supremum performance metric in fluid antenna systems with continuous positioning under spatial correlation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a level-crossing-rate framework to analyze fluid antenna systems with continuous positioning over a track of length L, where spatial correlation renders exact performance distributions analytically intractable. This framework delivers asymptotically exact approximations and tight bounds on the cumulative distribution function of the optimized metric S*, defined as the supremum of the performance metric S(l) over all positions l in the interval from zero to L. It derives new results for single fluid antennas under Rayleigh fading for SNR, SIR and SINR, extends them to Ricean desired channels, and treats two multi-antenna receiver layouts that use maximum-ratio combining, including the case of coupled array-element and positional correlations. The analysis produces concrete insights such as linear scaling of high-threshold tail probabilities with L and substantial outage reduction from small amounts of movement.

Core claim

We develop a level-crossing-rate (LCR) framework that yields asymptotically exact approximations and tight bounds for the cumulative distribution function (cdf) of the optimized metric S* = sup_{0 <= l <= L} S(l).

What carries the argument

Level-crossing-rate (LCR) framework that tracks crossings of performance levels to approximate and bound the distribution of the supremum metric over the continuous positioning interval.

If this is right

  • High-threshold tail probabilities of the optimized metric scale linearly with the track length L.
  • The track length L required to neutralize a co-channel interferer follows directly from the framework.
  • Movement over roughly one wavelength reduces outage probability by three orders of magnitude.
  • New LCR expressions handle the case where array-element correlation and positional correlation are coupled.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same crossing-rate technique could be applied to optimize positioning trajectories in time-varying environments.
  • Bounds derived here may serve as design rules for choosing track length in practical fluid-antenna deployments.
  • Connections to level-crossing analysis in other continuous-parameter wireless problems could yield similar closed-form insights.

Load-bearing premise

The spatial correlation model renders exact performance distributions analytically intractable, so that level-crossing-rate methods are required to approximate and bound the distribution of the supremum metric.

What would settle it

Monte Carlo simulations that produce CDFs for S* deviating substantially from the LCR approximations and bounds across a range of thresholds and track lengths L would falsify the claimed accuracy.

Figures

Figures reproduced from arXiv: 2605.16920 by Gayani Siriwardana, Himal A. Suraweera, Peter J. Smith, Rajitha Senanayake.

Figure 1
Figure 1. Figure 1: System model of a single fluid antenna receiver with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Multi-antenna receiver layouts: (a) one fixed [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: SNR cdf for a single fluid antenna under Rayleigh [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SINR and SIR cdfs for a single fluid antenna of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: SNR and SIR cdfs for a single fluid antenna of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 9
Figure 9. Figure 9: SNR cdf of a correlated two-element moving array, [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Log-scale ccdf of the SNR for a receiver comprising [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: SNR cdf comparison (Rayleigh fading, no interfer [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

We analyze multi-user fluid antenna systems with continuous positioning over a track of length L under a spatial correlation model, where exact performance distributions become analytically intractable. We develop a level-crossing-rate (LCR) framework that yields asymptotically exact approximations and tight bounds for the cumulative distribution function (cdf) of the optimized metric S* = sup_{0 <= l <= L}, where S(l) denotes the performance metric at antenna position l. For a single fluid antenna, we characterize the cdfs of signal-to-noise ratio (SNR), signal-to interference ratio (SIR) and signal-to-interference-plus-noise ratio (SINR) under Rayleigh fading and extend the approach to Ricean desired channels. We further treat two multi-antenna receiver layouts with maximum-ratio combining: (i) a fluid antenna with a fixed antenna and (ii) a two-element moving array, deriving new LCR results for the practically important case where array-element correlation and positional correlation are inherently coupled. The analysis provides actionable insights: high-threshold tail probabilities scale linearly with L, we derive the required L to neutralize a co-channel interferer, and we show that about one wavelength of movement can reduce outage by three orders of magnitude. Monte Carlo results validate the accuracy across the considered scenarios and regimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript analyzes multi-user fluid antenna systems with continuous positioning over a track of length L under a spatial correlation model, where exact performance distributions are analytically intractable. It develops a level-crossing-rate (LCR) framework yielding asymptotically exact approximations and tight bounds for the CDF of the optimized metric S* = sup_{0 <= l <= L} S(l). For a single fluid antenna, the CDFs of SNR, SIR, and SINR are characterized under Rayleigh fading (with extension to Ricean desired channels). The approach is extended to two multi-antenna receiver layouts with maximum-ratio combining: (i) a fluid antenna with a fixed antenna and (ii) a two-element moving array, with new LCR derivations for the case of coupled array-element and positional correlation. Monte Carlo simulations validate the approximations, and the analysis yields insights such as linear scaling of high-threshold tail probabilities with L and substantial outage reduction from approximately one wavelength of movement.

Significance. If the LCR approximations and bounds hold, the work supplies practical analytical tools for performance evaluation in fluid antenna systems, a setting where direct CDF derivation is intractable due to spatial correlation. The explicit treatment of coupled correlations in the multi-antenna cases and the derived design insights (e.g., required L to neutralize a co-channel interferer) represent a useful extension of level-crossing techniques. Monte Carlo validation across scenarios adds credibility to the numerical accuracy of the results.

major comments (1)
  1. [LCR framework and CDF approximation (abstract and main derivation sections)] The central claim that the LCR framework supplies asymptotically exact approximations for the CDF of S* = sup S(l) (via the standard mapping P(no upcrossing) ≈ exp(−∫ LCR(·) dl) or equivalent) is load-bearing. The manuscript derives new LCR expressions for the coupled correlation case but does not state the limiting regime (high threshold or large separation) under which crossings form a Poisson point process, nor does it supply an error bound or demonstrate that the joint statistics preserve crossing independence when array-element and positional correlations are coupled. Monte Carlo agreement alone cannot establish the asymptotic property.
minor comments (2)
  1. [Abstract] The abstract refers to 'two multi-antenna receiver layouts' without naming them; adding the descriptors '(i) fluid antenna with fixed antenna and (ii) two-element moving array' would improve immediate clarity.
  2. [Numerical results] In the Monte Carlo validation sections, reporting the number of trials and any confidence intervals on the empirical CDFs would strengthen the presentation of numerical agreement.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful and constructive review. We address the major comment below and indicate where revisions will be incorporated.

read point-by-point responses

  1. Referee: The central claim that the LCR framework supplies asymptotically exact approximations for the CDF of S* = sup S(l) (via the standard mapping P(no upcrossing) ≈ exp(−∫ LCR(·) dl) or equivalent) is load-bearing. The manuscript derives new LCR expressions for the coupled correlation case but does not state the limiting regime (high threshold or large separation) under which crossings form a Poisson point process, nor does it supply an error bound or demonstrate that the joint statistics preserve crossing independence when array-element and positional correlations are coupled. Monte Carlo agreement alone cannot establish the asymptotic property.

    Authors: We appreciate the referee highlighting the need for explicit clarification on the asymptotic regime. The LCR-based CDF approximation for the supremum relies on the standard result that level crossings form a Poisson point process in the high-threshold limit for processes satisfying appropriate regularity and mixing conditions. We will revise the abstract and the main derivation sections to state this limiting regime explicitly. For the coupled-correlation case, the LCR expressions are derived from the joint distribution of the performance metric and its spatial derivative; the covariance functions used in these derivations directly incorporate both the array-element correlation and the positional correlation, so the dependence structure is accounted for in the rate. While we do not supply a new analytical error bound (which would require further extreme-value analysis beyond the present scope), the Monte Carlo validation across multiple thresholds, correlation values, and system configurations demonstrates the accuracy of the approximations in the regimes of practical interest. revision: partial

standing simulated objections not resolved
  • Supplying a rigorous error bound or formal proof that crossing independence is preserved under the coupled array-element and positional correlations.

Circularity Check

0 steps flagged

LCR framework extends established techniques to new fluid-antenna setting without reducing target CDF to self-defined or fitted quantities

full rationale

The paper applies standard level-crossing-rate methods to derive expressions for the CDF of the supremum metric S* under continuous positioning and spatial correlation, including the coupled array-element/positional case. No derivation step defines the target CDF in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose validity is internal to the present work. The claimed asymptotic exactness and bounds follow from the LCR construction applied to the new model; Monte Carlo validation is presented as external numerical support rather than part of the analytic chain. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard domain assumptions about fading and correlation; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Rayleigh fading for interfering links and Rayleigh or Ricean for the desired link
    Invoked when characterizing the CDFs of SNR, SIR and SINR.
  • domain assumption Spatial correlation model renders exact distributions analytically intractable
    Stated as the motivation for developing the LCR approximations.

pith-pipeline@v0.9.0 · 5767 in / 1489 out tokens · 57373 ms · 2026-05-19T19:55:02.217845+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Prospective multiple antenna technologies for beyond 5G,

    J. Zhang, E. Björnson, M. Matthaiou, D. W. K. Ng, H. Yang, and D. J. Love, “Prospective multiple antenna technologies for beyond 5G,” IEEE J. Sel. Areas Commun., vol. 38, no. 8, p. 1637–1660, Aug. 2020

    work page 2020

  2. [2]

    Increasing capacity in wireless broadcast systems using distributed transmission/directional reception (DTDR),

    A. J. Paulraj and T. Kailath, “Increasing capacity in wireless broadcast systems using distributed transmission/directional reception (DTDR),” US Patent 5 (599A), vol. 345, 1994

    work page 1994

  3. [3]

    Massive MIMO for next generation wireless systems,

    E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” IEEE Commun. Mag., vol. 52, no. 2, pp. 186–195, 2014

    work page 2014

  4. [4]

    A tutorial on fluid antenna system for 6G networks: Encompassing communication theory, optimization methods and hardware designs,

    W. K. New, K.-K. Wong, H. Xu, C. Wang, F. R. Ghadi, J. Zhang, J. Rao, R. Murch, P. Ramírez-Espinosa, D. Morales- Jimenez, C.-B. Chae, and K.-F. Tong, “A tutorial on fluid antenna system for 6G networks: Encompassing communication theory, optimization methods and hardware designs,” IEEE Commun. Surv. Tutor., pp. 1–1, 2024

    work page 2024

  5. [5]

    Multifunctional reconfigurable MEMS integrated antennas for adaptive MIMO systems,

    B. Cetiner, H. Jafarkhani, J.-Y. Qian, H. J. Yoo, A. Grau, and F. De Flaviis, “Multifunctional reconfigurable MEMS integrated antennas for adaptive MIMO systems,” IEEE Commun. Mag., vol. 42, no. 12, pp. 62–70, Dec. 2004

    work page 2004

  6. [6]

    A novel pixel-based reconfigurable antenna applied in fluid antenna systems with high switching speed,

    J. Zhang, J. Rao, Z. Li, Z. Ming, C.-Y. Chiu, K.-K. Wong, K.-F. Tong, and R. Murch, “A novel pixel-based reconfigurable antenna applied in fluid antenna systems with high switching speed,” IEEE Open J. Antennas Propag., pp. 1–1, 2024

    work page 2024

  7. [7]

    Liquid antennas: Past, present and future,

    Y. Huang, L. Xing, C. Song, S. Wang, and F. Elhouni, “Liquid antennas: Past, present and future,” IEEE Open J. Antennas Propag., vol. 2, pp. 473–487, 2021

    work page 2021

  8. [8]

    Low- cost 3D-printed coupling-fed frequency agile fluidic monopole antenna system,

    C. Borda-Fortuny, L. Cai, K. F. Tong, and K.-K. Wong, “Low- cost 3D-printed coupling-fed frequency agile fluidic monopole antenna system,” IEEE Access, vol. 7, pp. 95 058–95 064, 2019

    work page 2019

  9. [9]

    Movable antennas for wireless communication: Opportunities and challenges,

    L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless communication: Opportunities and challenges,” IEEE Commun. Mag., vol. 62, no. 6, pp. 114–120, June 2024

    work page 2024

  10. [10]

    Movable- antenna enhanced multiuser communication via antenna position optimization,

    L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable- antenna enhanced multiuser communication via antenna position optimization,” 2024. [Online]. A vailable: https: //arxiv.org/abs/2302.06978

    work page arXiv 2024

  11. [11]

    A survey on fluid antenna multiple access for 6G: A new multiple access technology that provides great diversity in a small space,

    A. F. M. S. Shah, M. Ali Karabulut, E. Cinar, and K. M. Rabie, “A survey on fluid antenna multiple access for 6G: A new multiple access technology that provides great diversity in a small space,” IEEE Access, vol. 12, pp. 88 410–88 425, 2024

    work page 2024

  12. [12]

    Performance limits of fluid antenna systems,

    K. K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Performance limits of fluid antenna systems,” IEEE Commun. Lett., vol. 24, no. 11, pp. 2469–2472, 2020

    work page 2020

  13. [13]

    Fluid antenna systems,

    K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1950–1962, Mar. 2021

    work page 1950

  14. [14]

    A new spatial block-correlation model for fluid antenna systems,

    P. Ramírez-Espinosa, D. Morales-Jimenez, and K.-K. Wong, “A new spatial block-correlation model for fluid antenna systems,” IEEE Trans. Wireless Commun., vol. 23, no. 11, pp. 15 829– 15 843, 2024

    work page 2024

  15. [15]

    A new analytical approximation of the fluid antenna system channel,

    M. Khammassi, A. Kammoun, and M.-S. Alouini, “A new analytical approximation of the fluid antenna system channel,” IEEE Trans. Wireless Commun., vol. 22, no. 12, pp. 8843–8858, 2023

    work page 2023

  16. [16]

    Closed-form expressions for spatial correlation parameters for performance analysis of fluid antenna systems,

    K. K. Wong, K. F. Tong, Y. Chen, and Y. Zhang, “Closed-form expressions for spatial correlation parameters for performance analysis of fluid antenna systems,” Electron. Lett., vol. 58, no. 11, pp. 454–457, 2022

    work page 2022

  17. [17]

    Copula-based performance analysis for fluid antenna systems under arbitrary fading channels,

    F. Rostami Ghadi, K.-K. Wong, F. J. López-Martínez, and K.- F. Tong, “Copula-based performance analysis for fluid antenna systems under arbitrary fading channels,” IEEE Commun. Lett., vol. 27, no. 11, pp. 3068–3072, 2023

    work page 2023

  18. [18]

    A Gaussian copula approach to the performance analysis of fluid antenna systems,

    F. Rostami Ghadi, K.-K. Wong, F. Javier López-Martínez, C.-B. Chae, K.-F. Tong, and Y. Zhang, “A Gaussian copula approach to the performance analysis of fluid antenna systems,” IEEE Trans. Wireless Commun., vol. 23, no. 11, pp. 17 573–17 585, 2024

    work page 2024

  19. [19]

    Fluid antenna multiple access,

    K.-K. Wong and K.-F. Tong, “Fluid antenna multiple access,” IEEE Trans. Wireless Commun., vol. 21, no. 7, pp. 4801–4815, July 2022

    work page 2022

  20. [20]

    On the diversity and coded modulation design of fluid antenna systems,

    C. Psomas, G. M. Kraidy, K.-K. Wong, and I. Krikidis, “On the diversity and coded modulation design of fluid antenna systems,” IEEE Trans. Wireless Commun., vol. 23, no. 3, pp. 2082–2096, 2024

    work page 2082

  21. [21]

    Fluid antenna system: New insights on outage probability and diversity gain,

    W. K. New, K.-K. Wong, H. Xu, K.-F. Tong, and C.-B. Chae, “Fluid antenna system: New insights on outage probability and diversity gain,” IEEE Transactions on Wireless Communica- tions, vol. 23, no. 1, pp. 128–140, 2024

    work page 2024

  22. [22]

    Continuous fluid antenna systems: Modeling and analysis,

    C. Psomas, P. J. Smith, H. A. Suraweera, and I. Krikidis, “Continuous fluid antenna systems: Modeling and analysis,” IEEE Commun. Lett., vol. 27, no. 12, pp. 3370–3374, Dec. 2023

    work page 2023

  23. [23]

    Fluid antenna-assisted MIMO transmission exploiting statis- tical CSI,

    Y. Ye, L. You, J. Wang, H. Xu, K.-K. Wong, and X. Gao, “Fluid antenna-assisted MIMO transmission exploiting statis- tical CSI,” IEEE Commun. Lett., vol. 28, no. 1, pp. 223–227, 2024

    work page 2024

  24. [24]

    Movable-antenna enhanced multiuser communication via antenna position op- timization,

    L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable-antenna enhanced multiuser communication via antenna position op- timization,” IEEE Trans. Wireless Commun., vol. 23, no. 7, pp. 7214–7229, 2024

    work page 2024

  25. [25]

    Fluid antennas-enabled multiuser uplink: A low- complexity gradient descent for total transmit power minimiza- tion,

    G. Hu, Q. Wu, K. Xu, J. Ouyang, J. Si, Y. Cai, and N. Al- Dhahir, “Fluid antennas-enabled multiuser uplink: A low- complexity gradient descent for total transmit power minimiza- tion,” IEEE Commun. Lett., vol. 28, no. 3, pp. 602–606, 2024

    work page 2024

  26. [26]

    An- tenna positioning and beamforming design for fluid antenna- assisted multi-user downlink communications,

    H. Qin, W. Chen, Z. Li, Q. Wu, N. Cheng, and F. Chen, “An- tenna positioning and beamforming design for fluid antenna- assisted multi-user downlink communications,” IEEE Wirel. Commun. Lett., vol. 13, no. 4, pp. 1073–1077, 2024

    work page 2024

  27. [27]

    A simple single integral represen- tation of the bivariate Rayleigh distribution,

    M. Simon and M.-S. Alouini, “A simple single integral represen- tation of the bivariate Rayleigh distribution,” IEEE Commun. Lett., vol. 2, no. 5, pp. 128–130, 1998

    work page 1998

  28. [28]

    Infi- nite series representations of the trivariate and quadrivariate Nakagami-m distributions,

    P. Dharmawansa, N. Rajatheva, and C. Tellambura, “Infi- nite series representations of the trivariate and quadrivariate Nakagami-m distributions,” IEEE Trans. Wireless Commun., vol. 6, no. 12, pp. 4320–4328, 2007

    work page 2007

  29. [29]

    Analytical insights into outage prob- ability and ergodic capacity of fluid antenna systems,

    H. Zhao and D. Slock, “Analytical insights into outage prob- ability and ergodic capacity of fluid antenna systems,” IEEE Wireless Communications Letters, vol. 14, no. 5, pp. 1581–1585, 2025

    work page 2025

  30. [30]

    R. W. Heath and A. Lozano, Foundations of MIMO Communi- cation. Cambridge University Press, 2018

    work page 2018

  31. [31]

    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. San Diego, CA: Academic Press, 2007

    work page 2007

  32. [32]

    New channel model for wireless communications: Finite-state phase-type semi-Markov channel model,

    J. Wang, J. Cai, and A. S. Alfa, “New channel model for wireless communications: Finite-state phase-type semi-Markov channel model,” in 2008 IEEE International Conference on Communications, 2008, pp. 4461–4465

    work page 2008

  33. [33]

    W. C. Jakes and D. C. Cox, Microwave Mobile Communications. Wiley-IEEE Press, 1994

    work page 1994

  34. [34]

    Level crossing rates and average outage durations of SINR with multiple co-channel interferers,

    R. Annavajjala and J. Zhang, “Level crossing rates and average outage durations of SINR with multiple co-channel interferers,” in Proc. IEEE Military Commun. Conf. (MILCOM 2010), Oct. 2010, pp. 1233–1238

    work page 2010

  35. [35]

    Second order statistics of non-isotropic mobile-to-mobile Ricean fading channels,

    X. Cheng, C.-X. Wang, D. I. Laurenson, and A. V. Vasi- lakos, “Second order statistics of non-isotropic mobile-to-mobile Ricean fading channels,” in Proc. IEEE Intl. Conf. Commun. (ICC 2009), June 2009, pp. 1–5

    work page 2009

  36. [36]

    A new twist on the Marcum Q-function and its application,

    M. Simon, “A new twist on the Marcum Q-function and its application,” IEEE Commun, Lett., vol. 2, no. 2, pp. 39–41, 1998

    work page 1998

  37. [37]

    Chapter 8: Incomplete gamma and related functions,

    R. B. Paris, “Chapter 8: Incomplete gamma and related functions,” 2026. [Online]. A vailable: https://dlmf.nist.gov/8

    work page 2026

  38. [38]

    Second-order statistics for diversity-combining techniques in Nakagami-fading channels,

    M. Yacoub, C. da Silva, and J. Vargas Bautista, “Second-order statistics for diversity-combining techniques in Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1464– 1470, Nov. 2001

    work page 2001

  39. [39]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington: US Govt. Print, 2006

    work page 2006

  40. [40]

    N. L. Johnson and S. Kotz, Continuous Univariate Distribu- tions. Boston, MA: Houghton Mifflin, 1971

    work page 1971

  41. [41]

    Level crossing rate and average fade duration of the double Nakagami- m random process and application in MIMO keyhole fading 16 channels,

    N. Zlatanov, Z. Hadzi-Velkov, and G. K. Karagiannidis, “Level crossing rate and average fade duration of the double Nakagami- m random process and application in MIMO keyhole fading 16 channels,” IEEE Commun. Lett., vol. 12, no. 11, pp. 822–824, Nov. 2008

    work page 2008

  42. [42]

    K. S. Miller, Complex Stochastic Processes: An Introduction to Theory and Application. Addison-Wesley Publishing Company, 1974

    work page 1974

  43. [43]

    N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Volume 2. John Wiley & Sons, 1995

    work page 1995

  44. [44]

    Papoulis and S

    A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes. McGraw-Hill Europe: New York, NY, USA, 2002

    work page 2002

  45. [45]

    King, Statistics for Process Control Engineers: A Practical Approach

    M. King, Statistics for Process Control Engineers: A Practical Approach. John Wiley & Sons, 2017

    work page 2017