Dual-hop Variable-Gain Relaying with Beamforming over 휿−흁 Shadowed Fading Channels

Dual-hop relaying with beamforming is studied under 𝜅 − 𝜇 shadowed fading
environments.
 Exact and asymptotic results for the outage probability and average capacity
are derived.
 Effects of shadowing on the system performance are analyzed in different
scenarios
 The analysis results is general that it includes many special cases.
 Dual hop relaying w. beamforming has been studied for various channels.
 𝜅 − 𝜇 and 𝜂 − 𝜇 fading are generalized fading models.
 Recently, 𝜅 − 𝜇 shadowed channel was proposed. Fits well to empirical
measurements. Very general channel model.
 It is worth to analyze DH AF relaying over 𝜅 − 𝜇 shadowed fading.
 The model incorporates fading and shadowing [13].
• Composed of multipath clusters
• Power obeys 𝜅 − 𝜇 shadowed distribution
𝑊 = ∑ 𝑋𝑖 + 𝜉𝑝𝑖
2
+ 𝑌𝑖 + 𝜉𝑞𝑖
2
- Xi,Yi ~ Gaussian
- Shadowing 𝜉 ~ Nakagami-m
• 𝝁: # of clusters
• 𝜿: ratio b/w dominant and
scattered parts 𝒅 𝟐
/𝟐𝝈 𝟐
𝝁
 Special cases
• Rician shadowed, 𝜅 − 𝜇, Rician, Nakagami-𝑚, Rayleigh
Dual-hop Variable-Gain Relaying with Beamforming
over 𝜿 − 𝝁 Shadowed Fading Channels
A. Hussain, S.-H. Kim, & S.-H. Chang
Sungkyunkwan University, Korea, Dankook University, Korea
IS-WCT: Interactive Session:
Wireless Communication Technology
Abstract
Motivation
• 𝑁𝑡 antenna S
• 𝑁𝑟 antenna D
• Single antenna R
System Model: Dual-Hop Relaying with Beamforming
The 𝜅 − 𝜇 Shadowed Fading Channel Model
Relay
(R)
1
2
1
2
tN rN
Source(S)
Destination(D)
MRT
transmission
MRC reception
Config. SR/RD channels Performance metrics References
MRT,MRC Rayleigh OP, SER [3], [4],[5]
MRT,MRC Nakagami-𝑚 SER [6]
MRT,MRC Rayleigh/Rician OP, SER [10]
MRT,MRC Nakagami-𝑚/Rician OP, SER, Avg. capacity [11]
OSTBC Nakagami−𝑚/Rayleigh OP, SER, Avg. capacity [11]
OSTBC Rician shadowed OP, SER, Aprx. capacity [9]
MRT,MRC 𝜂 − 𝜇 OP, Avg. capacity Authors

 Communication occurs b/w S and D in two steps
 First, S transmits with MRT & received signal at R:
 Then, R transmits after multiplying with a gain factor G [4]
 Received signal at D:
Instantaneous End-to-End SNR
 Received signal: Beamformer 𝐰 𝑟 is due to MRC,
 Instantaneous end-to-end SNR [1]
o 𝛾1 =
ℎ 𝑆𝑅 𝐹
2
𝑁 𝑜1
is the instantaneous SNR b/w S & R
o 𝛾2 =
𝒉 𝑅𝐷 𝐹
2
𝑁 𝑜1
is the instantaneous SNR b/w R & D
o 𝑛 𝑅 is AWGN with 𝐸 𝑛 𝑅 = 𝑁𝑜1
o 𝒏 𝐷 is the 𝑁2 × 1 noise vector AWGN with 𝐸 𝑛 𝐷 = 𝐼 𝑁 𝑟
𝑁𝑜2
o 𝒘 𝑡 =
𝒉 𝑆𝑅
𝑇 †
𝒉 𝑆𝑅 𝐹
is the transmit beamforming vector
o 𝒘 𝑟
𝑇
=
ℎ 𝑅𝐷
†
ℎ 𝑅𝐷 𝐹
is the receive beamforming vector
o (. )†
is the conjugate transpose
o ||. || 𝐹 is the Frobenius norm
Relay
(R)
1
2
1
2
SRh RDh
tN rN
Source(S)
Destination(D)
(1)T
R SR t Ry x n wh
  D (3)T
RD SR t RG x n  wDy h h n
1
2
1
(2)
hSR oF
G
N


  D (4)T T T T
D r r RD SR t R rr G x n  w w w wD= y h h n
1 2
2 2
(5)
1
t
 

 

 

 PDF of 𝛾𝑖, 𝑖 = 1,2, is given by [13, Eq. (18)]
 Its associated CDF [13]:
o 𝛾i = average SNR of the 𝑖𝑡ℎ hop
o 𝛤(. ) =Gamma function
o 𝑚 =shadowing parameter
o 𝜇 > 0 =number of multipath clusters
o 1𝐹1(. ) =confluent hypergeometric function
o 𝛷2(. , . ; . , . ) = bivariate confluent hypergeometric function
o 𝑘 > 0 = ratio of dominant component power to scattered waves
power
The 𝜅−𝜇 Shadowed Fading Model
o 𝐾=Rician-k factor
o m= Nakagami-𝑚 parameter
 
 
 
   
 
1
2
1 1
1 1
1 1
exp ; ; (6)
i i i i
i i
i
i
i i
N N m
Ni i i
i i i i i i
i i i i i
i i i i
i i i i i
m
f
m N
F N m N
m




 
 
 
   
    
  
   
   
    
   
  
   
   
  
   
   
@
1 4 4 4 4 4 4 4 2 4 4 4 4 4 4 43
1 4 2 43 1 44 2 4 43
 
 
 
   
 
1
2
1 1
1
1 1
, ;1 , , (7)
i i i i
i i
i
N N m
Ni i i
i i i i i i
i i i i i
i i i i i i i i
i i i i i
m
F
m N
m
N N m N m N
m



 
 
   
   
   
   
   
    
    
  
       
Fading distribution 𝜅 𝜇 𝑚
𝜅 − 𝜇 𝜅 𝜇 𝑚 → ∞
Rician shadowed 𝜅 = 𝐾 𝜇 = 1 𝑚 = 𝑚
Rician 𝜅 = 𝐾 𝜇 = 1 𝑚 → ∞
Nakagami-𝑚 𝜅 → 0 𝜇 =m 𝑚 → ∞
Rayleigh 𝜅 → 0 𝜇 = 1 𝑚 → ∞
One-sided Gaussian 𝜅 → 0 𝜇 = 0.5 𝑚 → ∞
Special Cases of 𝜅−𝜇 Shadowed
Exact Outage Analysis
 Exact outage probability
o SNR<threshold
o where 𝐾 𝑉(. ) is the modified Bessel function of the second kind of 𝑉𝑡ℎ order and
Γ(. ) is the Gamma function
  1 2
2 2
(8)
1tout th r thP F P
 
 
 
 
   
  
 
     
     
 
 
  
1 2
2
2
1
1 1
1 2 1 2 1
0 0 0 0 0 1 2 1
1
1 2 2
2 1 2 1
2 2 1
2 2
2
1
i!j!n!
t r
t
r
r
t
N i N i pn
t r t
i j n p s t r t
s n
N i p n si j
r th N i pnth th
p sN i n s
r
N m i N m j N
F
N i N j N m
N
N m
 




  

 
     
 
     
    
 
    
  
   
    
 
    
  
  
  
   
 
  1 2 2
1 1 22 (9)th
s n th the K  
    
  

1 2
2
1 2
1
log 1 (10)
2 1
C E
 
 
  
   
   
The capacity is given as
o where 𝑇𝑖 are functions of gamma and Meijer-G function.
o and 𝑇3 is omitted.
Capacity Analysis
 Asymptotic outage probability for the considered system
o where Π𝑖 are a function of 𝜅 and 𝜇 and 𝑚.
 Diversity order: 𝐺 𝑑 = 𝑚𝑖𝑛(𝑁1 𝜇1, 𝑁2 𝜇2)
 Diversity order is independent of the shadowing parameter 𝑚.
 Note: Asymptotic capacity can also be derived.
Asymptotic Outage Probability
 
 
 
 
 
 1 2 1 2
1 2 3
1 2 1 2
1 1 1
ln 1 ln 1 ln 1
2ln 2 2ln 2 2ln 2
T T T
C E E E                      
@ @ @
1 4 4 2 4 43 1 4 4 2 4 43 1 4 4 4 2 4 4 4 3
   
   
1 , 1, 1
1,3
3,2
1, 00
1 1
, 1, 2
!
i i
i
N nn Ni ni i i i i i
i
i i i i i in
N m n N
T G i
n N n N m

  
  

  

       
            

Special cases of Asymptotic Outage
1
2
1 2
1 1 2
1
2 1 2
2
1 2 1 2
1 2
,
, (16)
,
t
r
t r
N
th
t r
N
th
out t r
N N
th th
t r
N N
P N N
N N


 

 


 

 
 
 
  
  
  

 
    
 

             
 Rician
o 𝜅 =K, 𝜇 =1, and 𝑚 → ∞: 𝐺 𝑑 = 𝑚𝑖𝑛(𝑁1, 𝑁2)
 Nakagami-m
o 𝜅 =0, 𝜇 =m, and 𝑚 → ∞ : 𝐺 𝑑 = 𝑚𝑖𝑛(𝑁1 𝑚, 𝑁2 𝑚)
 𝜅 − μ
o 𝜅 = 𝜅, 𝜇 = 𝜇, and 𝑚 → ∞ : 𝐺 𝑑 = 𝑚𝑖𝑛(𝑁1 𝜇1, 𝑁2 𝜇2)
 Diversity order is determined by the minimum product of the number of antennas
and parameter 𝜇
 Performance improves with both the number of antennas and values of µ

Table 1. Special Special cases derived from our results for 𝜅 − 𝜇 fading channels

 Exact OP for Special cases of 𝜅 − 𝜇 shadowed Fading
 Diversity order: 𝐺 𝑑 = 𝑚𝑖𝑛(𝑁1 𝜇1, 𝑁2 𝜇2)
 Diversity order for special cases
Numerical Results: Outage Performance VS Average SNR per Hop
Parameters:
𝜅 − 𝜇 shadowed/𝜅 − 𝜇 shadowed= 𝜅2 = 10, 𝜇1 = 𝜇2 = 2, and 𝑚1 = 𝑚2 =
0.5),
𝜅 − 𝜇/ 𝜅 − 𝜇 (𝜅1= 𝜅2 = 3.5 and 𝜇1 = 𝜇2 = 2),
Rician shadowed/Rician shadowed (𝐾1= 𝐾2 = 10 and 𝑚1 = 𝑚20.5) ,
Rician/Rician (𝐾1 = 𝐾2 = 4),
Nakagami-𝑚/Nakagami-𝑚 (m1 = m2 = 2),
Rayleigh/Rayleigh fading links when 𝑁1 = 𝑁2 = 3, and 𝛾𝑡ℎ = 10𝑑𝐵.
Fading distribution 𝜿 𝝁
𝜅 − 𝜇 𝜅 μ
Rician 𝜅 = 𝐾 1
Rician Shadowed 𝜅 = 𝐾 1
Nakagami-𝑚 𝜅 → 0 m
Rayleigh 𝜅 → 0 1
One-sided Gaussian 𝜅 → 0 0.5

 Influence of shadowing parameter, 𝑚 (𝑚1, 𝑚2), with different power of
dominant component, 𝜅 (𝜅1, 𝜅2)
 Light shadowing (𝑚 = 100)
 Heavy shadowing (𝑚 = 0.5)
 From Figs and table, note that
 Under heavy shadowing, performance decreases as 𝜅 increases
 Under light shadowing, improvement occurs with 𝜅
 Average capacity in shadowing environments when 𝜅1 = 𝜅2 = 1
Parameters: 𝑁𝑡 = 𝑁𝑟 = 2, 𝜇1 = 𝜇2 = 2
 Average capacity in shadowing environments when 𝜅1 = 𝜅2 =5
• Average capacity for different values of 𝜅 and 𝑚 (𝑁𝑡 = 𝑁𝑟 = 2, 𝜇1 = 𝜇2 = 2, and 𝛾1 =
𝛾1 =10dB
Capacity performance under heavy and light shadowing environments
Parameters: 𝑁𝑡 = 𝑁𝑟 = 2, 𝜇1 = 𝜇2 = 2
Heavy Shadowing
(𝑚1= 𝑚2 = 0.5)
Light Shadowing
(𝑚1 = 𝑚2 = 100)
𝜅1 = 𝜅2 = 0 1.47 1.47
𝜅1 = 𝜅2 = 1 1.43 1.52
𝜅1 = 𝜅2 = 5 1.34 1.63
𝜅1 = 𝜅2 = 10 1.31 1.67

Performance of dual-hop relaying with beamforming is
analyzed over 𝜅 − 𝜇 shadowed fading channels
 New exact analytical and asymptotic results for outage
probability and average capacity were derived
 Our results are useful to analyze a dual-hop relaying where
fading and shadowing may co-exist in links
 These derived expressions were general and were easily
reduced to symmetric and asymmetric fading links
 System performance improves in weak shadowing condition
when parameter 𝜅 increases, while in heavy shadowing
condition, improvement in performance is negligible with 𝜅
 Interference at both the relay and/or destination is still an
open issue and to be investigated in shadowing
environments.
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Conclusions and Future Works References

Dual-hop Variable-Gain Relaying with Beamforming over 휿−흁 Shadowed Fading Channels

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