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1. Second Order Statistics of SIR based Macro Diversity
System for V2I Communications over Composite
Fading Channels
Caslav Stefanovic
Faculty of Natural Sciences and Mathematics,
K. Mitrovica, Serbia
Stefan Panic
National Research Tomsk Polytechnic University
Tomsk, Russia
stefanpnc@tpu.ru
Stanislav Veljkovic
Faculty of Electronic Engineering
University of Nis
Nis, Serbia
ICSCCC 2018, Jalandhar, India
Srdjan Jovkovic
Collage of Applied Technical Sciences
Nis, Serbia
Mihajlo Stefanovic
Faculty of Electronic Engineering
University of Nis
Nis, Serbia
2. Outline
• Introduction
• Vehicular Communications
• Vehicle-to-vehicle (V2V)
• Vehicle-to-Infrastructure (V2I)
• Macro-diversity communications
• Channel modeling in the presence of fading for vehicular networks
• Statistical models
• Rayleigh, Rice, Nakagami-m
• genralized distributions
• Double generalized distributions
• Performance Evaluation
• First and second order performance analysis
• Cumulative distribution function (CDF) , average level crossing rate (LCR), average fade
duration (AFD)
ICSCCC 2018, Jalandhar, India
6. Channel Modeling for Vehicular
Communications
• Propagation Environment
• i) the channel behavior is highly time varying, since V2V and V2I
communications are characterized by increased vehicle mobility
• ii) low complexity requirements dominating the transceiving systems, since
many signal processing, hardware, and space limitation constraints arise,
when trying to integrate these systems in vehicles
• iii) both the transmitter and the receiver are in motion, while they are in the
same height, resulting to channel models with unique characteristics
ICSCCC 2018, Jalandhar, India
7. Vehicular Communications over Fading
Channels
• Multipath Fading is caused by physical phenomena
such as: reflection, refraction, diffraction and
scattering of radio waves.
• At the receiver it comes to the superposition of
multiple copies of the transmitted signal due to
multipath.
• The result is attenuation, delay and phase shift (constructive or
destructive interference).
• The most frequent fading models are:
• Rayleigh, Rician, Nakagami-m and Weibull.
• General feding models are:
• α-µ, k-µ, η-µ, α-k-µ
• Statistical models as a product of two RP
• Nakagami-m Nakagami-m , Rician Rician, α-µ α-µ [1]
• The application of the distribution which can be used to
describe multipath fading depends on:
• presence or absence of the line-of-sight (LOS),
• the existence of one or more clusters in the environment,
• whether the conditions of the central limit theorem are
attainable at the place of reception,
• whether the mean power of the envelope signal is variable
or constant and etc.
ICSCCC 2018, Jalandhar, India
8. Distribution PDF probability density function
Rayleigh
Rician
Weibull
Nakagami
-m
α-µ
k-µ
η-µ
ICSCCC 2018, Jalandhar, India
9. Diversity Combining
• Diversity combining can be used to
diminish the impact of the fading.
• MRC (maximum ratio combining)
• EGC (equal gain combining)
• SC (selection combining)
• SSC (switch and stay combining).
• Macro-diversity
• Switch and stay combining
ICSCCC 2018, Jalandhar, India
11. LCR and AFD
• The LCR is defined as the expected rate at which the signal envelope crosses a
specified level in a positive-going direction and can be obtained as an average
value of the first derivative of a random process.
• The LCR is relation of the time rate of change of the received signal and is
useful for designing error control codes, optimization of interleaver size and
throughput analysis.
• The AFD is defined as the average period of time for which the received signal is
below a specified level and can be obtained as the ratio of OP and LCR.
• The AFD helps determine the most likely number of signaling bits that may be
lost during a fade. AFD primarily depends upon the speed of the mobile, and
decreases as the maximum Doppler frequency fm becomes large.
ICSCCC 2018, Jalandhar, India
12. RF V2I communications with mac-div SC
reception
• The RF V2I communications is established simultaneously between
vehicle and several SC RSUs each with m branches. The outputs of the
RSUs are then processed by SC mac-div reception.
ICSCCC 2018, Jalandhar, India
13. RF V2I communications with mac-div SC
reception
• The signal to interference ratio (SIR) :
• 𝑧𝑚𝑖𝑐,𝑖𝑗 =
𝑥𝑚𝑖𝑐,𝑖𝑗
𝑦𝑚𝑖𝑐,𝑖𝑗
, 𝑖 = 1, 𝑛; 𝑗 = 1, 𝑚; (1)
• The desired signal envelopes, 𝑥𝑚𝑖𝑐,𝑖𝑗 are described with Rayleigh pdfs [24]:
• 𝑝𝑋𝑚𝑖𝑐,𝑖𝑗
𝑥𝑚𝑖𝑐,𝑖𝑗 =
2𝑥𝑚𝑖𝑐,𝑖𝑗
Ω𝑆,𝑖
𝑒
−
𝑥𝑚𝑖𝑐,𝑖𝑗
2
Ω𝑆,𝑖 , 𝑖 = 1, 𝑛; 𝑗 = 1, 𝑚; (2)
• where Ω𝑆,𝑖 present the local mean powers of 𝑥𝑚𝑖𝑐,𝑖𝑗.
• The CCI envelopes, 𝑦𝑚𝑖𝑐,𝑖𝑗 are described also with Rayleigh pdfs [24]:
• 𝑝𝑌𝑚𝑖𝑐,𝑖𝑗
𝑦𝑚𝑖𝑐,𝑖𝑗 =
2𝑦𝑚𝑖𝑐,𝑖𝑗
Ω𝐼,𝑖
𝑒
−
𝑦𝑚𝑖𝑐,𝑖𝑗
2
Ω𝐼,𝑖 , 𝑖 = 1, 𝑛; 𝑗 = 1, 𝑚; (3)
• where Ω𝐼,𝑖 present the local mean power of 𝑦𝑚𝑖𝑐,𝑖𝑗.
ICSCCC 2018, Jalandhar, India
14. First order statistics of RF V2I communications
at micro level
• The pdfs of 𝑧𝑚𝑖𝑐,𝑖𝑗, are:
• 𝑝𝑍𝑚𝑖𝑐,𝑖𝑗/Ω𝑆,𝑖Ω𝐼,𝑖
= 0
∞
𝑦𝑚𝑖𝑐,𝑖𝑗𝑝𝑋𝑚𝑖𝑐,𝑖𝑗
(𝑧𝑚𝑖𝑐,𝑖𝑗𝑦𝑚𝑖𝑐,𝑖𝑗)𝑝𝑌𝑚𝑖𝑐,𝑖𝑗
(𝑦𝑚𝑖𝑐,𝑖𝑗)𝑑𝑦𝑚𝑖𝑐,𝑖𝑗
• = 2Ω𝑆,𝑖Ω𝐼,𝑖
𝑧𝑚𝑖𝑐,𝑖𝑗
(Ω𝑆,𝑖+Ω𝐼,𝑖𝑧𝑚𝑖𝑐,𝑖𝑗
2)2 , 𝑖 = 1, 𝑛; 𝑗 = 1, 𝑚 (4)
• The cumulative density functions (cdfs) of 𝑧𝑚𝑖𝑐,𝑖𝑗 can be obtained as [24]:
• 𝐹𝑍𝑚𝑖𝑐,𝑖𝑗
𝑧𝑚𝑖𝑐,𝑖𝑗 = 0
𝑧𝑚𝑖𝑐,𝑖𝑗
𝑝𝑍𝑚𝑖𝑐,𝑖𝑗
𝑟 𝑑𝑟
• =
Ω𝐼,𝑗𝑧𝑚𝑖𝑐,𝑖𝑗
2
Ω𝑆,𝑖+Ω𝐼,𝑖𝑧𝑚𝑖𝑐,𝑖𝑗
2 , 𝑖 = 1, 𝑛; 𝑗 = 1, 𝑚; (5)
• The cdfs for i.n.i.d random processes (RPs) at the output of RF mic-div SC RSUs are:
• 𝐹𝑍𝑚𝑖𝑐,𝑖
= 𝐹𝑍𝑚𝑖𝑐,𝑖𝑗
(𝑧𝑚𝑖𝑐,𝑖)𝑚
=
Ω𝐼,𝑖
𝑚
𝑧𝑚𝑖𝑐,𝑖
2𝑚
Ω𝑠,𝑖+Ω𝐼,𝑖𝑧𝑚𝑖𝑐,𝑖
2 𝑚 , 𝑖 = 1, 𝑛; (6)
ICSCCC 2018, Jalandhar, India
15. Second order statistics of RF V2I at micro level
• The average level crossing rates (lcrs) of 𝑧𝑚𝑖𝑐,𝑖𝑗 at the inputs are:
• 𝑁𝑍𝑚𝑖𝑐,𝑖𝑗
𝑧𝑚𝑖𝑐,𝑖𝑗 = 0
∞
𝑧𝑚𝑖𝑐,𝑖𝑗 𝑝𝑍𝑚𝑖𝑐,𝑖𝑗𝑍𝑚𝑖𝑐,𝑖𝑗
𝑧𝑚𝑖𝑐,𝑖𝑗𝑧𝑚𝑖𝑐,𝑖𝑗 d𝑧𝑚𝑖𝑐,𝑖𝑗
•
2 2𝜋𝑓𝑚𝑧𝑚𝑖𝑐,𝑖𝑗Ω𝑆,𝑖
2
Ω𝐼,𝑖
2
𝛤 3
Ω𝑆,𝑖+Ω𝐼,𝑖𝑧𝑚𝑖𝑐,𝑖𝑗
2 5/2 , 𝑖 = 1, 𝑛; 𝑗 = 1, 𝑚; (7)
• The lcrs of 𝑧𝑚𝑖𝑐,𝑖 𝑖 = 1,2 … 𝑛 are:
• 𝑁𝑍𝑚𝑖𝑐,𝑖
= 𝑚𝑁𝑍𝑚𝑖𝑐,𝑖𝑗
𝑧𝑚𝑖𝑐,𝑖 𝐹𝑍𝑚𝑖𝑐,𝑖𝑗
𝑧𝑚𝑖𝑐,𝑖
𝑚−1
• =
2 2𝜋𝑓𝑚𝑧𝑚𝑖𝑐,𝑖
2𝑚−1Ω𝑆,𝑖
2
Ω𝐼,𝑖
𝑚+1
𝛤 3
Ω𝑆,,𝑖+Ω𝐼,𝑖𝑧𝑚𝑖𝑐,𝑖
2
3
2+𝑚
, 𝑖 = 1, 𝑛; (8)
ICSCCC 2018, Jalandhar, India
16. RF V2I Communications at macro level
• The random average powers ΩS,i at the inputs of n mic-div SC RSUs follow correlated joint Gamma pdf :
• 𝑝Ω𝑆,1Ω𝑆,2…Ω𝑆,𝑛
Ω𝑆,1Ω𝑆,2 … Ω𝑆,𝑛
• =
Ω𝑆,1Ω𝑆,𝑛−1
𝑐𝑆−1
2
𝑖−1
𝑛−1
𝐼𝑐𝑆 −1
2
Ω0 1−𝜌
𝜌Ω𝑆,𝑖Ω𝑆,𝑖+1
𝛤 𝑐𝑆 1−𝜌 𝑛−1𝜌
𝑛−1 𝑐𝑆−1
2 Ω𝑆,0
𝑛+𝑐𝑆 +1
× 𝑒
−
Ω𝑆,1+ 1+𝜌 𝑖=2
𝑛−1 Ω𝑆,𝑖+Ω𝑆,𝑛
Ω𝑠,0 1−𝜌
(9)
• where 𝛤 ∙ is Gamma function, 𝐼𝑣 ∙ is the first kind modified Bessel function
• of the order v, 𝜌 is correlation parameter, 𝑐𝑆 is shadowing severity and
• Ω𝑆,0 is mean value of Ω𝑆,1, Ω𝑆,2…Ω𝑆,𝑛.
• The CCI random variables Ω𝐼,𝑖 follow no correlated Gamma pdfs [24]:
• 𝑝Ω𝐼,𝑖
(Ω𝐼,𝑖) =
1
𝛤 𝑐𝐼 Ω𝐼,0
𝑐𝐼−1 Ω𝐼,𝑖
𝑐𝐼−1
𝑒
−
1
Ω𝐼,0
Ω𝐼,𝑖
𝑖 = 1, 𝑛; (10)
• where 𝑐𝐼 is Gamma interference severity parameter and Ω𝐼,0 is average power of
• Ω𝐼,1 ,Ω𝐼,2…Ω𝐼,𝑛
ICSCCC 2018, Jalandhar, India
20. Numerical Results
• Normalized LCR and AFD of RF V2I mac-div SC reception for various
values of n and m (different RSUs setups).
ICSCCC 2018, Jalandhar, India
-20 -15 -10 -5 0 5
0.01
0.1
1
LCR/f
m
zth
[dB]
n=2, m=2
n=3, m=2
n=4, m=2
n=2, m=4
n=3, m=4
n=4, m=4
n=2, m=6
n=3, m=6
n=4, m=6
-20 -10 0 10
1E-4
1E-3
0.01
0.1
1
10
100
1000
n=4, m=2
n=4, m=4
n=4, m=6
n=3, m=2
n=3, m=4
n=3, m=6
n=2, m=2
n=2, m=4
n=2, m=6
AFD
f
m
zth
21. Conclusion
• The V2I SC mac-div reception in correlated interference limited
Gamma shadowed Rayleigh multipath fading channel is considered.
• Novel, Infinite series analytical expressions for CDF, LCR and AFD for
up to four SC RSUs each with m branches are obtained.
• Numerical examples shows that:
• The significant system performance improvement can be achieved by
designing the considered model with increasing number of RSUs.
ICSCCC 2018, Jalandhar, India
22. •Thank you for your attention
IICSCCC 2018, Jalandhar, India