Communications and Network, 2009, 42-45
doi:10.4236/cn.2009.11006 Published Online August 2009 (http://www.scirp.org/journal/cn)
A Study of Multi-Node and Dual-Hop Collaborative Communi-
cation Performance Based on Harmonic Mean Method
Tingting YANG1, Shufang ZHANG2
College of Information Engineering, Dalian Maritime University, Dalian, China
Abstract: Closed form expressions for the PDF and MGF of the harmonic mean of two independent expo-
nential variates are cited and derived, and then applied to study the performance of cellular multi-node and
dual-hop cooperative communication systems with non-regenerative relays over flat Rayleigh fading channels.
We derive the probability density function (PDF) and asymptotic symbol error rate (SER) expression with
MRC scheme. Then we use Matlab to simulate the performance.
Keywords: harmonic mean, cooperative communication, multi-node, dual-hop, MGF
1. Introduction
MIMO technique has been regarded as the essential
technique for beyond 3G mobile cellular networks. The
Benefits of MIMO system have been extensively studied
by researchers in both academic and industry. It success-
fully meets the rapidly growing demand for high rate,
voice and especially for multimedia services [1].
In wireless cellular networks, base station can be
equipped with multiple antennas and keeps them spa-
tially separated. Unfortunately, it is hardly to fix multiple
antennas in portable mobile terminals (also called as mo-
bile terminals or users) due to insufficient antenna space,
energy, and price, etc. So, the bottleneck of capacity is
limited and the diversity technique of mobile terminals
could not be realized from the traditional end-to-end
transmission systems. In order to break this embarrassing
situation, a novel concept, namely cooperative commu-
nication (or user cooperative diversity) was introduced
by Sendonaris et al. [2].
Mazen O Hasna et al. in [3,4] firstly applied the har-
monic mean to the cellular multi-node and dual-hop co-
operative communication systems. They respectively
derived the probability density function (PDF), the cu-
mulative distribution function (CDF) and the moment
generating function (MGF) of the expression. It’s a new
train of thought of cooperative diversity scheme. This
paper applied the method of the harmonic mean to derive
PDF and MGF of two independent exponential variates
are cited and derived to study the performance of cellular
multi-node and dual-hop cooperative communication
systems with non-regenerative relays over flat Rayleigh
fading channels.
2. Harmonic Mean of Exponential Variates
2.1 Definitions
1) Harmonic Mean
Given two numbers X1 and X2, the harmonic mean of X1
and X2,


12
,
H
X
X  is defined as the reciprocal of the
arithmetic mean of the reciprocals of X1 and X2 [3] that is:

12
12
12
12
2
2
,11
 


H
X
X
XX
X
X
XX
(1)
2) Exponential RV
X follows an exponential distribution with parameterβ
> 0 if the PDF of X is given by






x
X
px eUx,
where



U is the unit step function.
2.2 Derive the Moment Generating Function
MGF of the Harmonic Mean of Two Exponential RVs
Given a RV X ∼ E(β), the PDF of Y = 1/X can be evalu-
ated with the help of [9].
Then, ),()( 2yUe
y
yp y
Y









s
y
YY
MsEe , v
K
(X)=

v
K
(X)
According to [5], we can get:

2
1
0
22[0,

 


 0]





v
x
vx
vee
xe dxKRR
  
2
00
sy sysy
y
Y
Y
M
sEepy edyeedy
y






 

=1
() 2(2)

 
y
M
ssKs
When X=


12
, 
H
X
X, we can get:

A STUDY OF MULTI-NODE AND DUAL-HOP COLLABORATIVE COMMUNICATION 43
PERFORMANCE BASED ON HARMONIC MEAN METHOD
 








12
12
12
12
11
2 012
12
12
2
121 12012
012
2
1212 112
0
12
0
2
2
12
2
1
2



  


 

 














 









 









sx sx
XX
x
sx
x
sx
sx
MsEe ee
KxKx Uxdx
eeKx Kxdx
exe Kxdx
exe
12
2
1
 
x


12
2
012



x
Kx
(2)
where K0(·) is the zero-order modified Bessel function
of the second kind defined in11].
Where K(x) is the first order modified Bessel func
n be constituted with
[
1
of the second kind defined in [11].
Here, we find the Formula (2) ca
tion
① + ②, according to [5].




1
0




vx
v
xeK xdx
211
,, ,
122
2

  





 


 

 







v
v
vv
Fv
v
Then, we can get:
①2

,

12
1
2

 ;1vs,12

 
②2

,

12
1
2


  s, 0v,12

 




12
2
12 12
1212 12
21
12 12
12 12
12 12
2
12 12
16
32
42
35
*3,;;
22 2
2
2
15
2, ; ;
22 2
X
Ms
s
s
F
s
s
s
Fs

 
 
 
 
 
 
 






  
 







  


3. Scheme of Cellular Multi-Node and Dual-
erative Communication Systems
erative communication
ource, one destination and arbi-
mobile terminals to base station).
ase station. This paper research the
M
e, where Ps=Pr=1. In the
lay and the MRC receiv-
re respectively:



Hop Coop
with Non-Regenerative Relays over Flat
Rayleigh Fading Channels
3.1 System Model
Consider an uplink wireless coop
system with only one s
trary N relays (from
The source terminal transmits Mary phase-shift keying
(M-PSK) modulation signals to destination terminal
through a direct path along with N dual-hop paths. The
channel is assumed to be quasi-static with flat fading.
Furthermore, perfect channel state information is as-
sumed available at the receivers, but the channels are
unknown at the transmitters. There are two transmission
phases. Firstly, the source broadcasts signals to destina-
tion and relays. Secondly, the relays which can success-
fully decode the signals will retransmit them to destina-
tion. Otherwise, they remain idle and do not participate
in the cooperation.
Figure 1 is the system model of multi-node and dual-
hop cooperative communication system. S is the moving
terminal, D is the b
PSK modulation of non-regenerative relay over flat
Rayleigh fading channels, and the noise is the additive
white Gaussian noise (AWGN). We suppose there are m
relays on the uplink, and the gains are Gi. Assume that
terminal S is transmitting a signal S(t) which has an av-
erage power normalized to 1.
3.2 Performance Analysis
Assume that in the EPA mod
situation of non-regenerative re
ing mode, the receiving signal a


111

R
Sthst nt





StgGhst ntnt
11
11 12d
1/
ii
Gh When can according with the harmonic
mean format, as
22
11
2
hg
00
22
11
00
 

NN
hg
NN
which can use the method of harmonic mean to calculate.
Figure 1. System model
C opyright © 2009 SciRes CN

44 A STUDY OF MULTI-NODE AND DUAL-HOP COLLABORATIVE COMMUNICATION
PERFORMANCE BASE ON HARMONIC MEAN METHOD
We define

i and

i
g
are respectively instantane-
ous SNR of the ith channels. The total SNR is:
1


m
fi
i
rr r
According to [4], we can achieve the cumulative dis-
tribution function (CDF) of the instantaneous SNR

i
of the Multi-node and dual-hop Collaborative Commu-
nication channels:
1i
g
i
()1() () 

ii
m
rh
F
rprrpr r
i
g
The probabilitydensity function (PDF) of the instanta-
neous SNR
__
11
__ __
/2[()() ]1/ 2
1
1
() 1()[(





hg
ii
iii
mrr r
rhgh
i
Frrrr eKrrr
1/2) ]
i

i of the Multi-node and dual-hop Col-
laborative Communication channels:
 
11 1
2
2



1
0
2
2










 











hg
ii ii ii
ii
ii
rr hg hg
ihg
hg
rr rr
U
rr
K
rr
The Outage Probability








 
ii
hg
m
rr
K
e


0






th
th
P utPPd o
On the assumption that the average SNR

i when

i
h =

i
g
=

i
(MGF) of th
, can obtain the moment generating
functione instantaneous SNR

i of the
Multi-node and dual-hop Collaborative Communicatio
channels[4][8]:
n

1
21
1
3
()11,2; ;
24






 



M
i
f
i
M
ss F s



0
1
1!!
12
12
sin
1
22
0
1357 21!
2
sin






 



















2
2
0
1
1!! sin
22
() 11sin
!2 1!
2




 


 

 




K
iK
N
K
KM
Ms
K
M
The upper limit of average SER of the Multi-node and
dual-hop channels is:
  
1
1
11





 

m

s
fp
i
Es
sk
g

 

 

2
22
1sin
sin sin




 





12
24
1
1!2! 1!2!
sin sin
33 44
!1!! 2 !
 







   

K
N
K
K
xpression expressed as:
MM
kK
K
d
M
This formula can’t get it’s closed-form, but we can get
the asymptotic e
22


 
 

 
 
 
 


m
ii
i
 

S
f
E
 

 

2
2
22
1
1sin sin
sin sin
1!2! 1!2!
33 4!1!! 2 !
22









 




  








 
  
 

 


m
f
m
i
i
When 1
4


i ,as (1, )

im
 
2
22
2
0
1
1sin
()
sin sin
12
sin( )3
4!
2





 


 






 


 

 
S
f
mN
miKK
k
i
K
PE
K
4. Numerical Results and Performance Analysis
In thper, we will respectively simulate and analysis
the PDF of insSNR
is pa
tantaneous

i of about different
situation the Multi-node and dual-hop Collaborative
Communication channels. We research the M-ary phase-
shift keying (M-PSK) modulation, M=4,8; the number of
relays m=4,10. We respectively simulate the situation of
high SNR and low SNR with 20 sampling nodes.
Figure 2 is the compare of PDF about different sam-
pling ranks and different relay numbers. From this figure,
C
opyright © 2009 SciRes CN

A STUDY OF MULTI-NODE AND DUAL-HOP COLLABORATIVE COMMUNICATION 45
PERFORMANCE BASED ON HARMONIC MEAN METHOD
Figure 2. The compare of PDF about different sampling
ranks and different relay numbers
Figure 3. The compare of error symbol rate (SER) abou
e different sampling ranks and different relay numbers
e performance is bette
hen the relay number is more. On the same relay no
the performance is almost the same when the high sam
pling ranks and low sampling ranks.
Figure 3 is the compare of error symbol rate (SER)
about the different sampling ranks and different rela
numbers. We respectively simulate the situation of
m=4,10, M=4,8, with 20 sampling nodes. Because
little SER, we adopt the logarithmic scale coord
method. We can find on the same SNR, the performance
better when the relay number is more. When the same
e
erformance of Multi-node and dual-hop Collaborative
C, Canada,
RYZHIK I M. Table of integrals, series, and
is
sampling nodes, the performance is better which is adopt
with the high rank modulation. It’s also approved that th
p
Communication can be improved when adopt the high
rank modulation while the user’s number is definite.
5. Conclusions
This paper applies the method of the harmonic mean to
derive PDF and MGF of two independent exponential
variates which are cited and derived to study the per-
formance of cellular multi-node and dual-hop coopera-
tive communication systems with non-regenerative relays
over flat Rayleigh fading channels.
We derive the probability density function (PDF) and
asymptotic symbol error rate (SER) expression. These
numerical results indicate that the number of relay is
more, the performance is better. It also indicates the as-
cendance performance of cooperative diversity.
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