Wang C, et al. Sci China Inf Sci June 2014 Vol. 57 062301:3
2Systemmodel
The cooperative network presented in Figure 1 is considered in this paper, which is made up of a single
source node S, N relays, i.e., R
k
, 1 k N and a single destination node D. All relay nodes operate
in the half-duplex AF mode. We suppose the channel coefficient between any two nodes is quasi-static
Rayleigh fading. We model the channel coefficients of two hops denoted by h
sr
k
and h
r
k
d
respectively for
k =1,...,N as zero mean, independent, circular-symmetric complex Gaussian random variables with
variances σ
2
sr
k
and σ
2
r
k
d
, i.e., h
sr
k
∼CN
0,σ
2
sr
k
, h
s
k
r
∼CN
0,σ
2
s
k
r
and remain unchanged over the
durations of data transmissions. In addition, each node employs the same transmitted power P
0
and the
additive noise at relays and destination is modeled as a zero-mean, complex Gaussian random variable
with the variance N
0
. Since the intermediate wall exists between the source and destination, direct S −D
link is blocked. Fortunately, the relays can assist the source in transmitting data to destination. Under
these assumptions, the average system SNR is P
0
/N
0
and the instantaneous SNRs of S −R
k
and R
k
−D
link can be expressed as P
0
r
sr
k
/N
0
and P
0
r
r
k
d
/N
0
respectively, where r
sr
k
= |h
sr
k
|
2
and r
r
k
d
= |h
r
k
d
|
2
are the channel gains.
Among N relays, the “best” relay is selected based on a certain criterion to forward the signal trans-
mitted by the source with AF protocol. The selection is performed by the destination, which collects all
information and feeds the selection decision back to the relays.
Because of the outdated CSI, we denote outdated channel coefficients of the S − R
k
and R
k
− D link
which are the ones at relay selection by
h
sr
k
and
h
r
k
d
respectively, which is an outdated version of h
sr
k
and h
r
k
d
at the time of data transmission. h
sr
k
and
h
sr
k
are jointly Gaussian distribution. Conditioned on
h
sr
k
, h
sr
k
obeys a complex Gaussian distribution [7]: h
sr
k
|
h
sr
k
∼CN(ρ
sr
k
h
sr
k
, (1 −ρ
2
sr
k
)σ
2
sr
k
). Likewise,
h
r
k
d
conditioned on
h
r
k
d
follows a Gaussian distribution: h
r
k
d
|
h
r
k
d
∼CN(ρ
r
k
d
h
r
k
d
, (1 − ρ
2
r
k
d
)σ
2
r
k
d
). The
correlation coefficient between h
sr
k
and
h
sr
k
denoted by ρ
sr
k
is determined by the model of fading process
[14, Table 2.1], so is ρ
r
k
d
. Here, the correlation coefficients are assumed to be constant. It is worth noting
that although the CSI in relay selection is outdated, the CSI of the destination node is assumed to be
perfect thanks to the pilots in the packet for receiver channel estimation. In CSI-assisted AF mode of
operation, the equivalent instantaneous end-to-end SNR of the link from the source to the destination
via the kth relay is Υ
k
, which can be calculated by [13]
Υ
k
=
P
0
r
sr
k
N
0
P
0
r
r
k
d
N
0
1+
P
0
r
sr
k
N
0
+
P
0
r
r
k
d
N
0
, for k =1,...,N. (1)
3 Opportunistic relay selection with outdated and statistical CSI
In the absence of outdated CSIs, best relay selection is decided by max
k
{Υ
k
}. The performance analysis
of this relay selection strategy is difficult, even if the exact analysis formula can be obtained, which may
be too complicated to work out. Fortunately Υ
k
has a tight upper bound [12]
Υ
ub
k
= min (P
0
/N
0
r
sr
k
,P
0
/N
0
r
r
k
d
) Υ
k
. (2)
It is worth pointing out that, (2) is a good approximation for (1) and coincides with (1) approximately
at medium and high SNR values. When the average system SNR (P
0
/N
0
) is given, the upper bound of
the equivalent end-to-end channel gain is r
ub
k
= min(r
sr
k
,r
r
k
d
). Therefore, for analytical simplicity, given
system SNR = P
0
/N
0
,weuseγ
ub
k
instead of γ
k
in the following analysis. A similar approximation was
also made in many literatures, such as [13].
In this paper, the outdated versions of instantaneous SNR in the S −R
k
and R
k
−D links are denoted
by P
0
r
sr
k
/N
0
and P
0
r
r
k
d
/N
0
where r
sr
k
= |
h
sr
k
|
2
and r
r
k
d
= |
h
r
k
d
|
2
. When the outdated CSI problem
is encountered, no accurate CSIs can be utilized, some papers [7,9,11–13], choose the best relay with
outdated CSIs, i.e., max
k
(r
ub
k
)wherer
ub
k
= min(r
sr
k
, r
r
k
d
). In the following, we refer to the strategy for
choosing the relay only with outdated CSI as maximum outdated channel gain (MCG) strategy. Although