2. System Model

waralligatorMobile - Wireless

Nov 21, 2013 (3 years and 8 months ago)

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1
/18

Statistical Channel Knowledge
-
Based Optimum Power Allocation


for Relaying Protocols in the high SNR regime

(IEEE Journal on selected area in Comm., Feb. 2007.)

1.
Introduction

2.
System Model

3.
High SNR Outage Analysis

4.
Optimal Power Allocation

5.
Coding Gain Consideration

6.
Results and Discussion

7.
Conclusion and Future Work

2
/18

1. Introduction

With perfect
CSI (Channel State Information)

at the
Tx

and Rx, the relay channel
performance can be improved significantly through optimal transmit power allocation.

Power allocation can be performed even when perfect CSIT is not available, provided
some
statistical knowledge of the channel gains
is available to the transmitting nodes.

Recently, [21] presented both
SNR maximizing
and
outage probability minimizing
optimal power allocation schemes with the knowledge of the mean channel gains.

If so, how we can derive the optimal power allocation? (Motivation)




It is to know that the optimal transmit power can be obtained by minimizing the


outage probability of the mutual information at the destination.

In this paper, at high SNR, the outage probability expressions for various protocols
are convex functions of the transmit power vector, and the optimal power allocation
depends on whether or not a direct link exists between the source and the destination.

3
/18

2. System Model

2
2 2 2
: channel gain between S and th relay,
j j
g j E g
 

 
 


channel gain Zero-mean, circularly symmet
ric, complex G. R. V
2
1 1 1
: channel gain between S and D,
g E g
 
 
 
2
3 3 3
: channel gain between th relay and D,

j j
g j E g
 

 
 
Source

Destination



Relay 1

Relay M

1
g
2
j
g
3
j
g
The total bandwidth available for the source transmission
without cooperation is denoted by
W.
Each source and relay
is available for
W/(M+1)
bandwidth
.



2
0
The noise power is single-sided P.S.D, i
n bandwidth
N
NW W












2 2
2
1 1 0 1
2
2
2 2
2
2
,
3 3
The instantaneous received SNR at each n
ode is
1 1
1
1
s s N
j j
s N
j j
r j
N
P g N W M M P g
M P g
M P g
 
 
 
   
 
 






,
,
,1,
2 2
1 1 1 2 2 2
2
3 3 3
Let us define 1,
1,1,...,
,,..., the transmit power vector
,,
s s
r j
r j
s r r M
j j j
s N s N
j j j
s N
P M P
P M P j M
P P P P
E P E P
E P
     
  
 
  
 

 
 
   
 
 
 
 
4
/18

3. High SNR Outage Analysis
-

AF Protocol

2 3
,1
1
2 3

1
j j
M
AF DL
j j
j

 
 

 
 


Output SNR at the destination
-

maximal ratio combining(MRC) [26]



2 3
,2,2 1
1
2 3
1 1
log 1 log 1
1 1 1
j j
M
AF DL AF DL
j j
j
I
M M

 
 

 
    
 
   
 


Instantaneous MI (Mutual Information) at the destination


Outage probability [26]







1
2 3
,,,1
1
2 3
Prob Prob 2 1
1
j j
M
M R
Out AF DL AF DL
j j
j
P P I R


 


 
     
 
 
 









1
1
2
,,,,2 3
1
,
1 2
1
2 1
1 1
, C, =
1!
M
M R
M
N
j
j j
Out AF DL AF DL AF DL j
M
j
s s r j
j
j
P P C
P P P
M







 

 
 
    
 
 
 
 
  
 
 








1
2
,,,,2 3 1
1
,
2
1
2 1
1 1
, C, =, 0
!
M
M R
M
N
j
j j
Out AF NDL AF NDL AF NDL j
M
j
s s r j
j
j
P P C
P P P
M






 

 
 
      
 
 
 
 

 
 


At high SNR, following the approach of [26]

5
/18

3. High SNR Outage Analysis
-

DF Protocol

,1 3

j
DF DL
j D
  

 


Instantaneous MI(Mutual Information) at the destination,
conditioned on
D


Outage probability [26]













1
,,,,1 3
Prob Prob Prob Prob Prob 2 1
M R
j
Out DF DL DF DL DF DL
D D j D
P P I R D I R D D
 


 
       
 
 
  














1
1
1
2
1
1
1
,,,,2 3
,
1
2 3
1 2
1
2 1
2 1
2 1 1 1 1 1
, C, =
1!1!
M
D
M R
M R
M D
M R
N
j
j j
Out DF DL DF DL DF DL j
k j
M
D
k D j D j D
s r j
j
j
P P C
P P
D D




 




 

  

 
 


 
 

   
     
 
 
 
 
 
 
 
 
 
 
 
  

With a DF protocol, a relay is assumed to correctly decode the source transmission if the
instantaneous MI is above the attempted transmission rate
R.















1
1 1
2 2
2
,,,,2 3
3,,
2
1 2
1
1
2 1 2 1
1 1 1 1
, C,=
!!
M M
M R M R
M D M D
j
N N
j
j j
Out DF NDL DF NDL DF DL j
M
j
M
j
D
j D j D
s r j s r j
j
j
j
P P C
P P P P
D D
 



 
 
 


   
 
   

   
    
   

 
   

 
 
 
 
 






,2,2 1 3
1 1
log 1 log 1
1 1
j
DF DL DF DL
j D
I D
M M
  

 
    
 
 
 


Output SNR at the destination













1 1
2 2
2 1 2 1
1
2 2 2 2
2
1 1 2 1
Prob Prob log 1 Prob log 1 1
1 1
M R M R
j k
M R
j k
k
j D k D j D k D k D
D R R e e
M M
 
 

 
 

 
    
   
 
 
 

   
 
 
 
         
 
   
 
 
 
 
 
   
 
 
 
 
   
    






1
1
1
1 3
1
3
2 1
1 1
Prob 2 1
1!
D
M R
M R
j
j
j D
j D
D
 







 

 
 
   
 

 


At high SNR, approximation [27]

6
/18

3. High SNR Outage Analysis


Distributed STC Protocol


Instantaneous MI(Mutual Information) at the destination,
conditioned on
D


Outage probability

Follow that is the sum of the MIs of
two independent parallel channels
. The ½ is due to the
fact that the nodes transmit in
half of the available bandwidth
.



2
2
,
,2 1 2 3 2 1 2 3
0 0
1 1 1 2 1 2
log 1 log 1 log 1 log 1
2/2 2/2 2 1 2 1
r j
j j
s
DTSC DL
j D j D
P
P
I D g g
N W N W M M
 
 
 
 
 
 
       
 
 
 
 
 
 
 
 
 
 
 
















1
2
2
,,
1
3
1
1
2 2 2
,,1 2
1
,
2 1 1
1 1
2 1
2
1
2 1, C 2 1 1/2
M
R
R
Out DSTC DL
D
j
D
k D
M D
M
M
j
R R j
DSTC DL DSTC DL N
D
D
j D j
s r j
M
P P A
C A M
P P






 

 
 
 
 
 
  
 
 
 
 
 
 
 
       
 
 
 
 



 










2 2
,,,,2
1
,
1 1
, C 2 1 1/2
!
M D
M
M
j
R j
Out DSTC NDL DSTC NDL DSTC NDL N
D
j D j
s r j
P P C M
P P
D



 
 
 
     
 
 
 
 

 




2
,,3
2
Prob Prob 2 1
1
j R
Out DSTC NDL
D j D
P P D
M


 
  
 

 
 










2
,,,,1 3
2 2
Prob Prob Prob Prob Prob 1 1 2
1 1
j R
Out DSTC DL DSTC DL DSTC DL
D D j D
P P I R D I R D D
M M
 

 
 
 
        
 
 
 
 
 
 
 
 
  






2
2 2 2 2
2
2 1 1
1 2 1 2
Prob Prob log 1 Prob log 1
2 1 2 1
2
R
k k
k
j D k D k D
M
D R R
M M
 

  
 
 
   
   
      
 
   
   
 
   
   
 
  












1
2
1
1
2 2
1 3
0
1
3
2 1 1
1
2 2 1 1 1
Prob 1 1 2 2 1 [5], A 0,
1 1 2 1!1
D
R
n
j R R
n
D
j
j D
j D
M
u u
A t du n
M M n tu
 






 
 
 
 

 
 
        
 
 
 
 
 
   
 
 
 
 



7
/18

3. High SNR Outage Analysis


Accuracy











1 2 3
1 2 3
,
Source (0,0), Destination (1,0),
Relay 1 cos 2,sin 2,3,4,6
1,cos 2,sin 2,1,2,3,4
Assuming equal power allocation, 4
Transmission rate R=1.0 bit/sec/Hz
j j
j j
j j
s r j T
j
P P P
 
       
  
 
   
   
       
   
 
8
/18

4. Optimal Power Allocation


AF Protocol

Optimal transmit power vector , minimize
s the outage probability, subject to a s
um power constraint
P


,
1
minimize subject to P
M
Out s r j T
j
P P P P

 

,
1
1
,
1 1
minimize subject to
M
M
j
s r j T
j
j
s s r j
P P P
P P P



 
  
 
 
 






,,
1
1
,
The Lagrange cost function can be writte
n as
1 1
, derivatives of , wi
th respect to ,,
M
M
j
s r j T s r j
j
j
s s r j
P P P P P P P
P P P

   


 
 
      
 
 
 
 
 






2 2
2
,,,
4 1
1 0,,1,...,
2
s j s j s j T
r j r j s j j s T r j
P P P P M
P P P PP M P j M
  
 
   
     




2 2
0 0 0 0 0,
1 1 0
4 1 2 1,,,1
M M M
j j j s T r j j T j
j j j
M P P P P
       
  
       
  




0 0
1 1
1 1 1 4, Let us assume ,1,1,...,
1 2 1
j j
M M
M j M
M M

   
 
 
 

       
 
 
 
 
 
 
 
When there is a direct link


,,,
1
,
1
M
Out AF NDL r k
k
s s k r k
P P P
P P P



 
 
 
 

 





,,
,,
,1,...,,
Out AF NDL j s
r j s j r j
P P P
j M
P P P



 

,
1
M
s r j T
j
P P P

 

,
quadratic equagion over
r j
P
0
As 0, 1
 
 
0
1
As ,
1
M
 
 



0
As 1, 1 1 2
M M M
 
   
9
/18

4. Optimal Power Allocation


AF Protocol

In the absence of a direct link
,
1
1
,
1
minimize subject to
M
M
j
s r j T
j
j
s r j
P P P
P P



 
  
 
 
 






,,
1
1
,
The Lagrange cost function can be writte
n as
1
, derivative
s of , with respect to ,,
M
M
j
s r j T s r j
j
j
s r j
P P P P P P P
P P

   


 
 
      
 
 
 
 
 




2 2
0 0 0 0 0,
1 1 0
4 2 1,,,1
M M M
j j j s T r j j T j
j j j
M P P P P
       
  
      
  


0 0
1
, Let us assume ,1,1,...,
1
j j
M j M
M
   

    

0
As 0, 1
 
 
0
As , 0
 
 
0
As , 1 1
M M
 
  
10
/18

4. Optimal Power Allocation


DF Protocol

1 1
,1 1 1
2
,1,1
ˆ
1 1 1 1 1
ˆ
minimize subject to ,2
2
s r T
s s r s s r
P P P
P P P P P P
 
 
 
     
 
 
 


1
0 1 0
1 1
2 16
1 1 1, Let us assume ,1,1
2 8
j
M

   
 
 
 
       
 
 
 

 
 
 
M=1 Relay Node
1
,1
2
,1
1
minimize subject to
s r T
s r
P P P
P P

  


0 1 0
1
1
, Let us assume 1,1
1
M
  

   



,1
D


When there is a direct link
In the absence of a direct link
11
/18

M=2 Relay Node
4. Optimal Power Allocation


DF Protocol

1 2 1 2
,1,2
3 2
,1,2 2,1,2
1 1 1 1 1
minimize subject to
2 6
s r r T
s s r r r r
P P P P
P P P P P P P
  
 
     
 
 
 






1 2 1 2
,1,2,1,2
3 2
,1,2 2,1,2
The Lagrange cost function can be writte
n as
1 1 1 1 1
, derivatives of , with respect t
o ,,,
2 6
s r r T s r r
s s r r r r
P P P P P P P P P
P P P P P P P
  
   
 
         
 
 
 






,1,2,1,2
D


1 2 1 2
4 3 2
,1,2,1,2
3 2 1 1
6
s s r r s r r
P P P P P P P
  

 
   
 
 
 
1 1 2
2 2 2
,1,1,2
1 1 1
2 6
s r s r r
P P P P P
 

 
2 1 2
2 2 2
,2,1,2
1 1 1
2 6
s r s r r
P P P P P
 

 
,1,2
s r r T
P P P P
  






3 2 2
1 2,1,2,1
1 3 12 18 0,,,2
r r s r T
P P P P P
           
          










3 2 2 2
1 2 1 2,1 1,2 1,1,2 2,1,1,2
2 2
,2,1 1 2 2,1 1,2
3 3 2 2 6 0
3 0
s s r r s r r r r r
s r r r r
P P P P P P P P P P
P P P P P
    
  
     
   
When , =1
 

,1,2
That is, 3, implying the optimality of e
qual power allocations as .
s r r T
P P P P

   
When there is a direct link
12
/18

1 2 1 2
,1,2
2
,1,2,1,2
1 1 1 1
minimize subject to
2 2
s r r T
s s r r r r
P P P P
P P P P P P
  
 
     
 
 
 






1 2 1 2
,1,2,1,2
2
,1,2,1,2
The Lagrange cost function can be writte
n as
1 1 1
, derivatives of , with respec
t to ,,,
2
s r r T s r r
s s r r r r
P P P P P P P P P
P P P P P P
  
   
 
         
 
 
 
M=2 Relay Node
4. Optimal Power Allocation


DF Protocol







,1,2,1,2
D


1 2
3 2
,1,2
3 2
s s r r
P P P P
 

 
  
 
 
 
1 1 2
2
,1,1,2
1 1
2
s r r r
P P P P
 

 
2 1 2
2
,2,1,2
1 1
2
s r r r
P P P P
 

 
,1,2
s r r T
P P P P
  






3 2 2
1 2,1,2,1
2 4 4 0,,,2
r r s r T
P P P P P
           
         










3 2 2 2
1 2 1,2 1,1,2 2,1,1,2
2 2
,2,1 1 2 2,1 1,2
2 2 0
2 0
s s r s r r r r r
s r r r r
P P P P P P P P P
P P P P P
   
  
    
   
In the absence of a direct link
When , =0
 

That is, arbitrarily small power is need
ed for the source to transmit,
instead of one-third of the total power
allocation, when .


13
/18

4. Optimal Power Allocation


DSTC Protocol

Upon comparing the outage probability expression for DSTC with DF protocol,









1
1
0
1 1
0 1
1!1!
n
n
u
A u u du
n n


  
 

In fact, the optimal power vector of the DF protocol is indeed a special case of that of the
DSTC protocol.

When , =1
 

The optimal power allocation vector of the DSTC protocol depends on the transmission rate R.

Without a direct link, ignoring the constant, the outage probability expression in the
DSTC protocol has a form very similar to that of the DF protocol. Since the optimal
power vector is not a function of the multiplicative constant of the objective
function, it then follows that the
optimal power vector of the DSTC protocol is
exactly the same as that of the DF protocol
.











1
1
1
1
,,,
,
1
2 3
2 1
2 1 1 1 1 1
,
1!1!
D
M R
M D
M R
j
Out DF DL DF DL
k j
D
k D j D j D
s r j
P P C
P P
D D


 


 

  
 

 
 

 
  
 
 
 
 
 
 
 
  










1
1
2
2 2
,,,
,
1
3
2 1 1
1 1 1
2 1 2 1,
2
M
M D
R
j
R R
Out DSTC DL DSTC DL
D D
j
D D
k D j D
s r j
M
P P A C A
P P




 
 
 
 
 
 
 
     
 
 
 
 
 
 
 
 
 
























2
3 2 2 2 2 2 2 2 2
2 1 2 2 1 2,1 1 1,2 1 1,1,2 1 2,1,1,2
2 2 2
1 1,2 2,1 2 1 2,1,2
2 1 2 1 2 1 2 2 1 2 2 1 3 0
2 1 2 1 0
R R R R R
s s r r s r r r r r
R R
r r s r r
P A P A P A P P A P P A P P P
A P P A P P P
    
  
          
     






















3 2 2 2 2 2 2 2
1 2 1 2 2 1 2
1 2 1 2 1 4 2 1 2 1 3 2 1 0,
R R R R R
A A A A A
        
           
14
/18

5. Coding Gain Considerations



2
Let us define , as the average SNR.
The outage probability can be written as
[23], where G is the so-called diversit
y gain, and
can be viewed as the asymptotic coding
gain.
d
T N
G
Out c d
c
P
P G
G


 
 










1
1
2
0
0 1,,1
1
1
0
1
2
0
0 1,,
1
1
0
,,...,, G 1!
2 1
,,...,, G!
2 1
j
M
M
j
M T c AF DL
M R
j
j j
j
M
M
j
M T c AF NDL
M R
j
j j
P P M
P P M


  
 


  
 





 
 

    
 
 
 


 
 
 
 
 

  
 
 
 


 
 
 






The coding gain ratio of a protocol with
optimal power allocation is defiened as
CGR=G Optimal Allocation Equal Allocation
c c
G




1
1
,0
1
0
1
1
M
M
j j
AF DL
j
j j
CGR M
 

 


 

 
   
 

 





1
,0
1
0
1
1
M
M
j j
AF NDL
j
j j
CGR M
 

 

 

 
   
 

 











0 1 1
,0,
2
0 0 1
1
With a single relay, the CGRs of the DF
protocol are
1 2 2 1
CGR 2,
2 1
1
DF DL DF NDL
CGR
  

 

  
 
 

15
/18

6. Results and Discussion

16
/18

6. Results and Discussion

17
/18

7. Conclusion and Future Work


At high SNR, the optimal power allocation was shown to depend not only on
the ratio of mean power gains, but also on whether or not a direct link between
the source and the destination exists.


But, with DSTC protocol the optimal power allocation also depends on the
transmission rate when a direct link exists.


Optimal power allocation yields impressive coding gains over equal power
allocation.


The coding gain gap between the AF and DF protocols can also be reduced by
the optimal power allocation.


It is possible to analyze the performance of the cooperative network if the
relays are selected.


Optimal power allocation can be provided in the network.


How can we select the cooperative relays or not?(Decision rule)

18
/18

Reference

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