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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
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
Outline
• Performance Evaluation for V2I communications
• RF V2I communications with mac-div SC reception
• CDF
• LCR
• AFD
• Numerical results
• Conclusion
ICSCCC 2018, Jalandhar, India
Vehicular Communications
• Types of Comm.
• V2V
• V2I
• V2X
• Standards:
• IEEE 802.11p
5.9 GHz
• Environments
• Urban
• Highway
• Rural
ICSCCC 2018, Jalandhar, India
Vehicular Communications
• V2V and V2I multi-hop vehicle
communications
vehicular networks
ICSCCC 2018, Jalandhar, India
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
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
Distribution PDF probability density function
Rayleigh
Rician
Weibull
Nakagami
-m
α-µ
k-µ
η-µ
ICSCCC 2018, Jalandhar, India
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
Performance Evaluation
ICSCCC 2018, Jalandhar, India
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
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
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
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
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
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
CDF AND LCR AT THE OUTPUT OF MAC-DIV SYSTEM
ISICSCCC 2018, Jalandhar, India
𝐹𝑍𝑚𝑎𝑐
=
0
∞
𝑑Ω𝐼,1
0
∞
dΩ𝑆,1
0
ΩS,1
dΩ𝑆,2
0
Ω𝑆,1
𝑑Ω𝑆,3 × ⋯
0
ΩS,1
dΩ𝑆,𝑛−1
0
ΩS,1
𝐹𝑧𝑚𝑎𝑐,1
(𝑧𝑚𝑎𝑐)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n
Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,1
Ω𝐼,1 dΩ𝑠,𝑛
+
0
∞
𝑑Ω𝐼,2
0
∞
dΩ𝑆,2
0
Ω𝑆,2
dΩ𝑆,1
0
Ω𝑆,2
dΩ𝑆,3 … ×
0
Ω𝑆,2
dΩ𝑆,𝑛−1
0
ΩS,2
𝐹𝑧𝑚𝑎𝑐,2
𝑧𝑚𝑎𝑐 pΩ𝑆,1Ω𝑆,2…Ω𝑆,n
Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,2
Ω𝐼,2 dΩ𝑠,𝑛 …
+
0
∞
𝑑Ω𝐼,𝑛
0
∞
dΩ𝑆,𝑛
0
Ω𝑆,𝑛
dΩS,1
0
Ω𝑆,𝑛
dΩS,2 … ×
0
Ω𝑆,𝑛
dΩS,n−2
0
Ω𝑆,𝑛
𝐹𝑧𝑚𝑎𝑐,𝑛
zmac pΩ𝑆,1Ω𝑆,2…Ω𝑆,n
Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,𝑛
Ω𝐼,𝑛 dΩ𝑠,𝑛−1
𝑁𝑍𝑚𝑎𝑐
=
0
∞
𝑑Ω𝐼,1
0
∞
dΩ𝑆,1
0
ΩS,1
dΩ𝑆,2
0
Ω𝑆,1
𝑑Ω𝑆,3 × ⋯
0
ΩS,1
dΩ𝑆,𝑛−1
0
ΩS,1
𝑁𝑧𝑚𝑎𝑐,1
(𝑧𝑚𝑎𝑐)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n
Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,1
Ω𝐼,1 dΩ𝑠,𝑛
+
0
∞
𝑑Ω𝐼,2
0
∞
dΩ𝑆,2
0
Ω𝑆,2
dΩ𝑆,1
0
Ω𝑆,2
dΩ𝑆,3 … ×
0
Ω𝑆,2
dΩ𝑆,𝑛−1
0
ΩS,2
𝑁𝑧𝑚𝑎𝑐,2
(𝑧𝑚𝑎𝑐)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n
Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,2
Ω𝐼,2 dΩ𝑠,𝑛
… +
0
∞
𝑑Ω𝐼,𝑛
0
∞
dΩ𝑆,𝑛
0
Ω𝑆,𝑛
dΩS,1
0
Ω𝑆,𝑛
dΩS,2 … ×
0
Ω𝑆,𝑛
dΩS,n−2
0
Ω𝑆,𝑛
𝑁𝑧𝑚𝑎𝑐,𝑛
(zmac)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n
Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,𝑛
Ω𝐼,𝑛 dΩ𝑠,𝑛−1
CDF AT THE OUTPUT OF MAC-DIV SYSTEM (N=4)
CDF (n=4)
𝐹𝑍𝑚𝑎𝑐 =
2𝑧𝑚𝑎𝑐
2𝑚
𝛤 𝑐𝐼 Ω𝐼,0
𝑐𝐼
1
𝛤 𝑐𝑆 1 − 𝜌 3𝜌
3
2
(𝑐𝑆−1)
Ω𝑆,0
𝑐𝑆+3
𝑖=0
∞
𝜌
Ω𝑆,0 1 − 𝜌
2𝑖+𝑐𝑆−1
1
𝑖! 𝛤 𝑖 + 𝑐𝑆
𝑗=0
∞
𝜌
Ω𝑆,0 1 − 𝜌
2𝑗+𝑐𝑆−1
1
𝑗! 𝛤 𝑗 + 𝑐𝑆
×
𝑘=0
∞
𝜌
Ω𝑆,0 1 − 𝜌
2𝑘+𝑐𝑆−1
1
𝑘! 𝛤 𝑘 + 𝑐𝑆
𝑙=0
∞
1
Ω𝑆,0 1 − 𝜌
𝑙
1
𝑗 + 𝑘 + 𝑐𝑆
𝑝=0
∞
1
𝑗 + 𝑘 + 𝑐𝑆 + 1 𝑝
1 + 𝜌 𝑝
Ω𝑆,0 1 − 𝜌
𝑝
1
𝑘 + 𝑐𝑆
×
𝑠=0
∞
1
𝑘 + 𝑐𝑆 + 1 𝑠
1
Ω𝑆,0 1 − 𝜌
𝑠
1
𝑧𝑚𝑎𝑐
2 𝑐𝐼+𝑚
𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆
Ω𝑆,0 1 − 𝜌
2 2 + 𝜌
2𝑖+2𝑗+2𝑘+𝑙+𝑝+𝑠+4𝑐𝑆+𝑐𝐼
×
𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 𝛤 𝑐𝐼 + 𝑚
𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚
2𝐹1 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼, 𝑐𝐼 + 𝑚, 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚, 1 −
Ω𝑆,0 1−𝜌
2Ω𝐼,0 2+𝜌 𝑧𝑚𝑎𝑐
2
1
𝑖+𝑗+𝑐𝑆
1+𝜌 𝑙
𝑖+𝑗+𝑐𝑆+1 𝑙
+
ICSCCC 2018, Jalandhar, India
LCR AT THE OUTPUT OF MAC-DIV SYSTEM (N=4)
LCR (n=4)
𝑁𝑍𝑚𝑎𝑐(𝑧𝑡ℎ) =
4𝑧𝑡ℎ
2𝑚−1
𝛤 𝑐𝐼 Ω𝐼,0
𝑐𝐼
𝑚 2𝜋𝑓𝑚𝛤 3
𝛤 𝑐𝑆 1 − 𝜌 3𝜌
3
2
(𝑐𝑆−1)
Ω𝑆,0
𝑐𝑆+3
𝑖=0
∞
𝜌
Ω𝑆,0 1 − 𝜌
2𝑖+𝑐𝑆−1
1
𝑖! 𝛤 𝑖 + 𝑐𝑆
𝑗=0
∞
𝜌
Ω𝑆,0 1 − 𝜌
2𝑗+𝑐𝑆−1
1
𝑗! 𝛤 𝑗 + 𝑐𝑆
×
𝑘=0
∞
𝜌
Ω𝑆,0 1 − 𝜌
2𝑘+𝑐𝑆−1
1
𝑘! 𝛤 𝑘 + 𝑐𝑆
𝑙=0
∞
1
Ω𝑆,0 1 − 𝜌
𝑙
1
𝑗 + 𝑘 + 𝑐𝑆
𝑝=0
∞
1
𝑗 + 𝑘 + 𝑐𝑆 + 1 𝑝
1 + 𝜌 𝑝
Ω𝑆,0 1 − 𝜌
𝑝
1
𝑘 + 𝑐𝑆
×
𝑠=0
∞
1
𝑘 + 𝑐𝑆 + 1 𝑠
1
Ω𝑆,0 1 − 𝜌
𝑠
1
𝑧𝑚𝑎𝑐
2 𝑐𝐼+𝑚
𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆
Ω𝑆,0 1 − 𝜌
2 2 + 𝜌
2𝑖+2𝑗+2𝑘+𝑙+𝑝+𝑠+4𝑐𝑆+𝑐𝐼
×
𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 𝛤 𝑐𝐼 + 𝑚
𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚
2𝐹1 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼, 𝑐𝐼 + 𝑚, 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚, 1 −
Ω𝑆,0 1−𝜌
2Ω𝐼,0 2+𝜌 𝑧𝑚𝑎𝑐
2
1
𝑖+𝑗+𝑐𝑆
1+𝜌 𝑙
𝑖+𝑗+𝑐𝑆+1 𝑙
+
ICSCCC 2018, Jalandhar, India
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
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
•Thank you for your attention
IICSCCC 2018, Jalandhar, India
5G Expectations
ISCCCS 2018, Jalandhar, India

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India presentation final

  • 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
  • 3. Outline • Performance Evaluation for V2I communications • RF V2I communications with mac-div SC reception • CDF • LCR • AFD • Numerical results • Conclusion ICSCCC 2018, Jalandhar, India
  • 4. Vehicular Communications • Types of Comm. • V2V • V2I • V2X • Standards: • IEEE 802.11p 5.9 GHz • Environments • Urban • Highway • Rural ICSCCC 2018, Jalandhar, India
  • 5. Vehicular Communications • V2V and V2I multi-hop vehicle communications vehicular networks 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
  • 17. CDF AND LCR AT THE OUTPUT OF MAC-DIV SYSTEM ISICSCCC 2018, Jalandhar, India 𝐹𝑍𝑚𝑎𝑐 = 0 ∞ 𝑑Ω𝐼,1 0 ∞ dΩ𝑆,1 0 ΩS,1 dΩ𝑆,2 0 Ω𝑆,1 𝑑Ω𝑆,3 × ⋯ 0 ΩS,1 dΩ𝑆,𝑛−1 0 ΩS,1 𝐹𝑧𝑚𝑎𝑐,1 (𝑧𝑚𝑎𝑐)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,1 Ω𝐼,1 dΩ𝑠,𝑛 + 0 ∞ 𝑑Ω𝐼,2 0 ∞ dΩ𝑆,2 0 Ω𝑆,2 dΩ𝑆,1 0 Ω𝑆,2 dΩ𝑆,3 … × 0 Ω𝑆,2 dΩ𝑆,𝑛−1 0 ΩS,2 𝐹𝑧𝑚𝑎𝑐,2 𝑧𝑚𝑎𝑐 pΩ𝑆,1Ω𝑆,2…Ω𝑆,n Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,2 Ω𝐼,2 dΩ𝑠,𝑛 … + 0 ∞ 𝑑Ω𝐼,𝑛 0 ∞ dΩ𝑆,𝑛 0 Ω𝑆,𝑛 dΩS,1 0 Ω𝑆,𝑛 dΩS,2 … × 0 Ω𝑆,𝑛 dΩS,n−2 0 Ω𝑆,𝑛 𝐹𝑧𝑚𝑎𝑐,𝑛 zmac pΩ𝑆,1Ω𝑆,2…Ω𝑆,n Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,𝑛 Ω𝐼,𝑛 dΩ𝑠,𝑛−1 𝑁𝑍𝑚𝑎𝑐 = 0 ∞ 𝑑Ω𝐼,1 0 ∞ dΩ𝑆,1 0 ΩS,1 dΩ𝑆,2 0 Ω𝑆,1 𝑑Ω𝑆,3 × ⋯ 0 ΩS,1 dΩ𝑆,𝑛−1 0 ΩS,1 𝑁𝑧𝑚𝑎𝑐,1 (𝑧𝑚𝑎𝑐)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,1 Ω𝐼,1 dΩ𝑠,𝑛 + 0 ∞ 𝑑Ω𝐼,2 0 ∞ dΩ𝑆,2 0 Ω𝑆,2 dΩ𝑆,1 0 Ω𝑆,2 dΩ𝑆,3 … × 0 Ω𝑆,2 dΩ𝑆,𝑛−1 0 ΩS,2 𝑁𝑧𝑚𝑎𝑐,2 (𝑧𝑚𝑎𝑐)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,2 Ω𝐼,2 dΩ𝑠,𝑛 … + 0 ∞ 𝑑Ω𝐼,𝑛 0 ∞ dΩ𝑆,𝑛 0 Ω𝑆,𝑛 dΩS,1 0 Ω𝑆,𝑛 dΩS,2 … × 0 Ω𝑆,𝑛 dΩS,n−2 0 Ω𝑆,𝑛 𝑁𝑧𝑚𝑎𝑐,𝑛 (zmac)pΩ𝑆,1Ω𝑆,2…Ω𝑆,n Ω𝑆,1Ω𝑆,2 … Ω𝑆,n pΩ𝐼,𝑛 Ω𝐼,𝑛 dΩ𝑠,𝑛−1
  • 18. CDF AT THE OUTPUT OF MAC-DIV SYSTEM (N=4) CDF (n=4) 𝐹𝑍𝑚𝑎𝑐 = 2𝑧𝑚𝑎𝑐 2𝑚 𝛤 𝑐𝐼 Ω𝐼,0 𝑐𝐼 1 𝛤 𝑐𝑆 1 − 𝜌 3𝜌 3 2 (𝑐𝑆−1) Ω𝑆,0 𝑐𝑆+3 𝑖=0 ∞ 𝜌 Ω𝑆,0 1 − 𝜌 2𝑖+𝑐𝑆−1 1 𝑖! 𝛤 𝑖 + 𝑐𝑆 𝑗=0 ∞ 𝜌 Ω𝑆,0 1 − 𝜌 2𝑗+𝑐𝑆−1 1 𝑗! 𝛤 𝑗 + 𝑐𝑆 × 𝑘=0 ∞ 𝜌 Ω𝑆,0 1 − 𝜌 2𝑘+𝑐𝑆−1 1 𝑘! 𝛤 𝑘 + 𝑐𝑆 𝑙=0 ∞ 1 Ω𝑆,0 1 − 𝜌 𝑙 1 𝑗 + 𝑘 + 𝑐𝑆 𝑝=0 ∞ 1 𝑗 + 𝑘 + 𝑐𝑆 + 1 𝑝 1 + 𝜌 𝑝 Ω𝑆,0 1 − 𝜌 𝑝 1 𝑘 + 𝑐𝑆 × 𝑠=0 ∞ 1 𝑘 + 𝑐𝑆 + 1 𝑠 1 Ω𝑆,0 1 − 𝜌 𝑠 1 𝑧𝑚𝑎𝑐 2 𝑐𝐼+𝑚 𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 Ω𝑆,0 1 − 𝜌 2 2 + 𝜌 2𝑖+2𝑗+2𝑘+𝑙+𝑝+𝑠+4𝑐𝑆+𝑐𝐼 × 𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 𝛤 𝑐𝐼 + 𝑚 𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚 2𝐹1 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼, 𝑐𝐼 + 𝑚, 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚, 1 − Ω𝑆,0 1−𝜌 2Ω𝐼,0 2+𝜌 𝑧𝑚𝑎𝑐 2 1 𝑖+𝑗+𝑐𝑆 1+𝜌 𝑙 𝑖+𝑗+𝑐𝑆+1 𝑙 + ICSCCC 2018, Jalandhar, India
  • 19. LCR AT THE OUTPUT OF MAC-DIV SYSTEM (N=4) LCR (n=4) 𝑁𝑍𝑚𝑎𝑐(𝑧𝑡ℎ) = 4𝑧𝑡ℎ 2𝑚−1 𝛤 𝑐𝐼 Ω𝐼,0 𝑐𝐼 𝑚 2𝜋𝑓𝑚𝛤 3 𝛤 𝑐𝑆 1 − 𝜌 3𝜌 3 2 (𝑐𝑆−1) Ω𝑆,0 𝑐𝑆+3 𝑖=0 ∞ 𝜌 Ω𝑆,0 1 − 𝜌 2𝑖+𝑐𝑆−1 1 𝑖! 𝛤 𝑖 + 𝑐𝑆 𝑗=0 ∞ 𝜌 Ω𝑆,0 1 − 𝜌 2𝑗+𝑐𝑆−1 1 𝑗! 𝛤 𝑗 + 𝑐𝑆 × 𝑘=0 ∞ 𝜌 Ω𝑆,0 1 − 𝜌 2𝑘+𝑐𝑆−1 1 𝑘! 𝛤 𝑘 + 𝑐𝑆 𝑙=0 ∞ 1 Ω𝑆,0 1 − 𝜌 𝑙 1 𝑗 + 𝑘 + 𝑐𝑆 𝑝=0 ∞ 1 𝑗 + 𝑘 + 𝑐𝑆 + 1 𝑝 1 + 𝜌 𝑝 Ω𝑆,0 1 − 𝜌 𝑝 1 𝑘 + 𝑐𝑆 × 𝑠=0 ∞ 1 𝑘 + 𝑐𝑆 + 1 𝑠 1 Ω𝑆,0 1 − 𝜌 𝑠 1 𝑧𝑚𝑎𝑐 2 𝑐𝐼+𝑚 𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 Ω𝑆,0 1 − 𝜌 2 2 + 𝜌 2𝑖+2𝑗+2𝑘+𝑙+𝑝+𝑠+4𝑐𝑆+𝑐𝐼 × 𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 𝛤 𝑐𝐼 + 𝑚 𝛤 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚 2𝐹1 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼, 𝑐𝐼 + 𝑚, 2𝑖 + 2𝑗 + 2𝑘 + 𝑙 + 𝑝 + 𝑠 + 4𝑐𝑆 + 𝑐𝐼 + 𝑚, 1 − Ω𝑆,0 1−𝜌 2Ω𝐼,0 2+𝜌 𝑧𝑚𝑎𝑐 2 1 𝑖+𝑗+𝑐𝑆 1+𝜌 𝑙 𝑖+𝑗+𝑐𝑆+1 𝑙 + 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
  • 23. 5G Expectations ISCCCS 2018, Jalandhar, India