Average SNR and Ergodic Capacity of Reactive DF Relaying System with Outdated Channel State Information.

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Average SNR and Ergodic Capacity
of Reactive DF Relaying System
with Outdated Channel State Information
Seokjung Kim†, Sungsoo Park, Hyunkee Min, and Daesik Hong††
Information Telecommunication Lab.(B715), School of Electrical and Electronic Engin., Yonsei Univ.,
134 Shinchondong Seodaemungu, Seoul, Korea, 120-749.
Phone: +82-2-2123-3558, Fax: +82-2-312-4887, Homepage: http://mirinae.yonsei.ac.kr
E-mail: hernetic@yonsei.ac.kr†and daesikh@yonsei.ac.kr††
Abstract—In this paper, we investigate the effects of outdated
channel state information (CSI) in opportunistic relaying systems
on the ergodic capacity. The CSI in the data transmission instant
may be different from that used in the relay selection instant due
to channel variation. When the CSI in the relay-destination link is
outdated, the average signal-to-noise ratio (SNR) and the ergodic
capacity of the reactive decode and forward (DF) relaying system
are derived. In addition, the effective ergodic capacity which is
the spectral efficiency of the successful transmissions excluding
the outage events is also derived. The numerical results show that
if the CSI in the relay-destination link is outdated, the loss in
the ergodic capacity is remarkable compared to the perfect CSI
case. In addition, the level of the inaccuracy of the CSI affects the
performance of the ergodic capacity in the reactive DF relaying
system. In other words, the capacity degradation occurs if the
CSI inaccuracy becomes much larger.
I. INTRODUCTION
Cooperative relaying techniques have attracted much in-
terest as a new way to combat fading in wireless networks.
Basically, these techniques achieve spatial diversity gain and
improve the link performance due to the transmission from
multiple relays without the help of multiple antennas [1], [2].
However, since the multiple-relay transmission supposes the
orthogonal transmission, its system throughput is limited by
the number of relays [3].
Recently, to solve this problem, Bletsas and Win proposed
an opportunistic relay-based decode and forward (DF) system
[4]. In this system, a single relay which provides the best
end-to-end signal-to-noise ratio (SNR) among several relays
is selected as the active relay. There are two different types to
select the best relay, i.e., the proactive and reactive systems.
The former selects the relay after examining the source-relay
and relay-destination link qualities simultaneously, the later
reduces the relay size, and then chooses the active relay. Also,
they analyzed their outage probabilities under a total power
constraint and observed that the opportunistic relaying system
are identical to the multiple-relay transmission from the per-
spective of the diversity-multiplexing tradeoff [4]. Considering
the source-destination link, Michalopoulos and Karagiannidis
This work was supported by Korea Science & Engineering Foundation
through the NRL Program (Grant R0A-2007-000-20043-0).
studied the outage probability of the opportunistic relaying
system [5]. In addition, Lee and Hong examined the end-to-
end average SNR and ergodic capacity of two systems [6].
In [7], the outage probability of the reactive DF relaying
system is derived when the transmission is subject to co-
channel interferences from other sources and relays. Also,
Shah and Yim studied the time and energy trade-off between
the selection and data transmission phases in the opportunistic
relaying system [8]. In those works, it is assumed that the relay
and destination use the perfect channel state information (CSI).
However, in practical relaying systems, the CSI may become
outdated if the variation in the channel over time is high
relative to the signaling duration [9], [10]. Therefore, in
the opportunistic relaying system, the CSI used at the relay
selection instant would be different from the CSI used at the
data transmission instant.
To analyze the effects of outdated CSI on the reactive
scheme, Vicario and Seco derived the outage probability in
a closed-form when the CSI of the relay-destination link is
outdated [11]. In addition, they showed that the diversity order
of the system is always limited to one regardless of the number
of relays, when imperfect channel estimation occurs in the
relay-destination link. In [12], the effects of the outdated CSI
and channel estimation errors were considered in the reactive
DF relaying system. Seyfi and Liang then analyzed the average
symbol error rate (ASER) and asymptotic diversity order.
However, all the works mentioned above only consider the
effects of outdated CSI on the outage probability and ASER
of the reactive DF relaying system. Since the capacity is one of
the important parameters to measure the system performance,
we need to investigate the effects of outdated CSI on the
ergodic capacity.
In this paper, we assume that only the CSI in the relay-
destination link is outdated. Since the relay can decide whether
it re-transmits the data, the relay selection phase is not
considered in the source-relay link. Therefore, we can suppose
that the CSI in the source-relay link is perfect. We derive
the typical and effective ergodic capacity as a closed form
where the effective ergodic capacity is defined as the spectral
efficiency of the successful transmissions excluding the outage
978-1-61284-231-8/11/$26.00 ©2011 IEEE
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event. This concept has a potential advantage of examining the
practical throughput which reflecting in the outage event [6].
Hence, we more focus on the effective ergodic capacity than
the typical ergodic capacity. Finally, we show the effects of
outdated CSI on the typical and effective ergodic capacity.
II. SYSTEM MODEL
Let us consider a half-duplex dual-hop relaying system
in which a single source (S) transmits the data to a single
destination (D) through K relays. Let us denote Rk as the
k-th relay, where k ∈ C0 = {1,2,...,K}. We assume that
there is no direct link between the source and the destination.
We also assume that the reactive scheme is exploited for the
opportunistic DF relaying systems [4]. Fig. 1 shows the relay
selection procedure of the reactive DF relaying system. In
the first phase, the source broadcasts a data to all relays, and
some of the relays successfully decode the data. A subset of
relays that satisfy the target SNR for successful decoding is
denoted as the decoding set CT, where T = 22RT−1 is a SNR
threshold1for a pre-defined end-to-end spectral efficiency RT.
In the second phase, among the relays in the decoding set CT,
only a single relay which has the maximum SNR in the Rk−D
link is selected to forward the decoded data.
In practical relaying systems, the channel used in the
relay selection may differ from the channel used in the data
transmission because of the time-varying characteristic [11].
This means that the SNRs obtained from the various channels
may be different due to channel variation. As a result, it is
implausible to guarantee that the relays in the decoding set
CT can decode the received signal from the source during the
data transmission instant. In addition, the link quality of the
selected relay among the decoding set CT may not be the
best during data transmission because of channel variation.
However, we can assume that the CSI in the S − Rklink is
perfect. This is because the relay can decide to re-transmit the
data in the distributed manner. Therefore, we only consider
the CSI accuracy in the Rk− D link.
It is assumed that each channel experiences identical and
independent Rayleigh fading. Therefore, the estimated SNRs
of the S −Rkand Rk−D links at the relay selection instant,
denoted by ˆ γ1,kand ˆ γ2,k, follow exponential distributions with
1Since the system requires two hops to transmit the data from the source
to the destination, the spectral efficiency per hop should be 2RT [4].
⁞⁞
Best relay
Phase IPhase II
Decoding
Set ( )
T
C
1 R
2
R
3 R
K
R
S
D
1 R
2 R
3 R
K
R
S
D
Decoding
Set ( )
T
C
Fig. 1.Procedure for the reactive DF relaying system
means of 1/λ1and 1/λ2, respectively [4]. In the Rk−D link,
since the instantaneous SNR γ2,kconditioned on its estimated
SNR ˆ γ2,k follows a non-central Chi-square distribution with
two degrees of freedom [11]. Hence, the probability density
function (PDF) can be represented as follows [11],
fγ2,k|ˆ γ2,k(x|y) =
λ2
(1 − ρ2
k)e
−λ2(x+y·ρ2
(1−ρ2
k)
k)
I0
?2λ2ρk√xy
1 − ρ2
k
?
,
(1)
where I0(·) is the zero-order modified Bessel function of the
first kind and ρkis the envelope of the correlation coefficient
between γ2,kand ˆ γ2,kon the Rk− D link of the k-th relay.
We assume that the correlation coefficients for all channels
are identical (i.e., ρk= ρ for ∀k ∈ C0). Under the assumption
of Jakes’ model, the correlation coefficient is given by ρ =
J0(2πfdTs), where J0(·) is the zero-order Bessel function of
the first kind, fdis the Doppler shift and Tsis the feedback
delay between the relay selection and the data transmission
instant [13].
However, since we assume that the CSI in the S −Rklink
is perfect, the instantaneous SNR of the S − Rk link, γ1,k,
is equal to its estimated SNR at the relay selection instant.
Hence, it follows an exponential distribution with mean of
1/λ1.
III. AVERAGE SNR AND ERGODIC CAPACITY ANALYSES
In this section, we analyze the end-to-end average SNR
and the ergodic capacity of the reactive DF relaying system
when the CSI of the Rk− D link is outdated. In the reactive
DF relaying system, the CDF of the instantaneous SNR,
FR−D
Γ
(t), can be represented as
FR−D
Γ
(t) =
K
?
l=1
Pl· FΓR−D
l
(t),
(2)
where Pldenotes the probability that only l relays out of the K
candidates are in the decoding set CT [4]. This is represented
as
Pl=
l
Note that Plis not affected by the correlation coefficient ρi.
This is because the decoding set CTis not changed irrespective
of the channel variation once it is determined at the relay
selection instant.
Let us define k∗
SNR of the Rk− D link among the l relays in the decoding
set CT. In addition, FΓR−D
SNR when the selected relay, k∗
decoded data. Since we assumed that the S −Rkand Rk−D
link are independent, FΓR−D
l
(t) = 1 − P?min?γ1,k∗
Note that FΓR−D
l
of γ1,k∗
S − Rk∗
?K
?
e−lλ1T?1 − e−λ1T?K−l.
(3)
las the selected relay which maximizes the
l
(t) is the CDF of the instantaneous
l, is used to forward the
(t) is expressed as
FΓR−D
l
l,γ2,k∗
l
?> t?
= 1 − P?γ1,k∗
(t) can be calculated by computing the CDFs
land γ2,k∗
land Rk∗
l> t?· P?γ2,k∗
l> t?.
(4)
l, which are the instantaneous SNRs for the
l− D links, respectively.
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At this point, let us consider the case in which the CSI
in the Rk− D link is only outdated. Using the memoryless
property of the exponential distribution, the CDF of γ1,k∗
be rewritten as
P?γ1,k∗
=
lcan
l< t?= P?γ1,k∗
l< t|ˆ γ1,k∗
l< t|ˆ γ1,k∗
l> T?
u(t − T),
= P?ˆ γ1,k∗
l> T?
?
1 − e−λ1(t−T)?
(5)
where u(t) denotes the unit step function.
Next, in the previous studies [11] and [?], the CDF of γ2,k∗
is already obtained as
?
where B = (m + 1)??1 + m?1 − ρ2
rewritten as
?
+ l
m
?
Using the rule of integration by parts, the average SNR when
the decoding set CT has l relays can be expressed by
¯ΓR−D
l
=
0
?
= T +
λ1
m=0
l
P?γ2,k∗
l< t?= l
l−1
?
m=0
l − 1
m
?(−1)m
??.
m + 1
?1 − e−tλ2B?,
(6)
2
By substituting (5) and (6) into (4), FΓR−D
l
(t) can be
FΓR−D
l
(t) =1 − e−λ1(t−T)?
l−1
?
×
u(t − T)
?(−1)m
m=0
?
l − 1
?
m + 1
?1 − e−tλ2B?
u(t − T)
1 −
1 − e−λ1(t−T)??
.
(7)
?∞
t · FΓR−D
t · fΓR−D
l
(t) dt
?∞
l−1
?
=
l
(t)
0
−
?∞
l − 1
m
?
0
FΓR−D
l
?(−1)m
1
λ1+ λ2B−
(t) dt
1
+ l
?
m + 1
?
1
λ2B−
1
λ2B
1
λ1
−T + e−λ2BT
??
,
(8)
where fΓR−D
decoding set CThas l relays. In addition, after substituting (3)
and (8) into (2), the end-to-end average SNR is
l
(t) is the PDF of the average SNR when the
¯ΓR−D=
K
?
l=1
Pl·¯ΓR−D
l
.
(9)
Note that although the Rk− D link quality is perfect (i.e.,
λ2≈ 0), the end-to-end average SNR¯ΓR−Dis still affected by
the correlation coefficient (ρ). The end-to-end average SNR is
also limited to the S−Rklink quality (1/λ1) and a predefined
threshold (T) in the high SNR regime.
Generally, the instantaneous end-to-end capacity can be
defined as C =
stantaneous end-to-end SNR. Using this identity, the CDF
of the instantaneous end-to-end capacity is represented as
FC(y) = FΓtot
1
2log2(1 + Γtot), where Γtot is the in-
?22·y− 1?. Therefore, from (7), we can obtain
the CDF of the instantaneous capacity for the Rk− D link
with the outdated CSI as
FCR−D
l
(y) =
?
1 − e−λ1(22·y−22·RT)?
l−1
?
×
· u(y − RT)
?
+ l
m=0
?
?
l − 1
m
?
?(−1)m
m + 1
1 − e−λ2B(22·y−1)?
· u(y − RT)
1 −
1 − e−1(22·y−22·RT)?
?
.
(10)
From (10), the typical capacity when the decoding set CT
has l relays is obtained as the following Lemma 1:
Lemma 1. The typical ergodic capacity when the decoding
set CT has l relays is
¯CR−D
l
=RT+eλ1(T+1)
2 · ln2E1(λ1(T + 1))
l−1
?
+ eλ1(T+1)· E1((T + 1) · (λ1+ λ2B))
−E1(λ2B (T + 1))} − RT
−eλ1(T+1)
2 · ln2
+ l
m=0
?
l − 1
m
?(−1)m
m + 1
?eλ2B
2 · ln2{E1(λ2B)
· E1(λ1(T + 1))
?
,
(11)
where E1(x) is the exponential integral function defined in
[14, ch. 8.21].
Proof: the proof is presented in Appendix A.
By using (3) and (11), the end-to-end typical capacity is
expressed as
¯CR−D=
K
?
l=1
Pl·¯CR−D
l
.
(12)
As mentioned above, when the decoding set CT is empty,
the source does not transmit the data to prevent the outage
event. However, it is possible to occur an outage event at the
destination because of the Rk−D link quality. Therefore, the
typical ergodic capacity may be different from the practical
throughput because of the outage event. In order to consider
the outage event in terms of the capacity, we evaluate the
ergodic capacity within the interval [RT, ∞). This concept
is the effective ergodic capacity defined in [6]. The effective
ergodic capacity when the decoding set has l relays,¯CR−D
is derived as the following Lemma 2:
l,eff,
Lemma 2. The effective ergodic capacity when the decoding
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set has l relays,¯CR−D
l,eff, is
¯CR−D
l, eff=RT+eλ1(T+1)
2 · ln2E1(λ1(T + 1))
l−1
?
+e(λ1(T+1)+λ2B)
2 · ln2
−RT−eλ1(T+1)
2 · ln2
+ l
m=0
?
l − 1
m
?(−1)m
m + 1
?RT· e−λ2BT
· E1((T + 1) · (λ1+ λ2B))
· E1(λ1(T + 1))
?
.
(13)
Proof: the proof is presented in Appendix B.
Therefore, from (13), the end-to-end effective ergodic ca-
pacity can be described as
¯CR−D
eff
=
K
?
l=1
Pl·¯CR−D
l, eff.
(14)
Note that as the quality of the Rk− D link is improved
(i.e., λ2→ 0), the end-to-end typical and ergodic capacities
are affected by the S − Rklink quality (1/λ1) and a prede-
fined threshold (T), respectively. This is similar to the result
obtained in [6]. In addition, for the high SNR regime, (12)
and (14) is bounded to a certain value, since the CSI accuracy
in the Rk− D link is degraded. This effect will be verified
via numerical results in the next section.
IV. NUMERICAL RESULTS
In this section, we first verify the analytical expression of
the average SNR and the ergodic capacity obtained in the
previous section via numerical results. We then study the effect
of outdated CSI caused by the channel variation on the reactive
DF relaying system.
In the simulations, the SNR thresholds and the number of
relays are fixed at T = 1.5 dB and K = 5, respectively. Also,
various values of ρ are used with a larger ρ representing a
lower mobility.
First, we evaluate how the reliability of the Rk−D link can
affect the ergodic capacity of the reactive DF relaying system.
Fig. 2 shows the end-to-end ergodic capacity for various ρ
values in the reactive DF relaying system with λ1= λ2. The
performance of the reactive DF relaying system with the ideal
case (i.e., ρ = 1) is also included in Fig. 2 as a benchmark. In
this figure, it is clear that the analytical results are in excellent
agreement with the Monte Carlo simulation results, verifying
the accuracy of the analytical expression derived in Section
III. In addition, when the CSI in the Rk−D link is outdated
(i.e., ρ ?= 1), the ergodic capacity of the reactive DF relaying
system is degraded compared to that of the ideal case. This
is because the overall link quality through the selected relay
may not be the best compared to those of the other relays.
Of interest in the results deduced from the figure is that
under the same condition with the outdated CSI in the Rk−D
link, the tendency of the ergodic capacity is similar to that
of the outage probability. From the viewpoints of the outage
probability, the system performance improves with increasing
ρ in [11], [12]. That is because the relay selection gain is lost
when the CSI is outdated [11], [12]. Similarly, the end-to-
end ergodic capacity is degraded as ρ decreases. In addition,
it is observed that the ergodic capacity becomes saturated
according to different ρ values in the high SNR regime. This
means that the level of the CSI accuracy affects the ergodic
capacity of the reactive DF relaying system if the CSI in the
Rk− D link is outdated. As mentioned in Section III, that is
because the end-to-end average SNR and ergodic capacities
are limited to the correlation coefficient (ρ) in the high SNR
regime.
In Fig. 3, the ergodic capacities of the reactive DF relaying
system with different ρ values are illustrated. As seen in Fig.
0510 15 20
0
0.5
1
1.5
2
2.5
3
3.5
SNR per Hop [dB]
End−to−End Ergodic Capacity [bits/s/Hz]
End−to−End Typical and Effective Ergodic Capacity


ρ = 1 [Typical]
ρ = 1 [Effective]
ρ = 0.9 [theoretical]
ρ = 0.9 [simulation]
ρ = 0.1 [theoretical]
ρ = 0.1 [simulation]
Typical capacity
Effective capacity
Fig. 2. Typical and effective capacity vs. SNR curves per hop of the reactive
DF relaying system. The number of relays and the required spectral efficiency
are K = 5 and RT = 1.5 bps/s/Hz, respectively, with the assumption that
λ1= λ2for various ρ values.
0.10.2 0.30.4
Correlation Coefficient [ρ]
0.50.6 0.70.80.91
0.5
1
1.5
2
2.5
3
3.5
End−to−End Ergodic Capacity [bits/s/Hz]
End−to−End Typical and Effective Ergodic Capacity


Typical
Effective
Fig. 3.
20 [dB] with RT= 1.5 bps/s/Hz
Typical and effective capacity vs. ρ curves for link SNRs of 10 and
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2, once the CSI in the Rk− D link is outdated, i.e., ρ ?= 1,
the typical and effective ergodic capacities are changed by
a varying ρ. Also, it is showed that the performance of the
reactive DF relaying system is better when the channel quality
in each link is improved.
V. CONCLUSIONS
We evaluated the performance of the reactive DF relaying
system from viewpoints of the typical and effective ergodic
capacities when the CSI in the Rk−D link is outdated. From
these results, when the CSI in the Rk− D link is outdated,
the loss in the ergodic capacity is remarkable compared to the
perfect CSI case. In addition, the level of the CSI inaccuracy
in the Rk−D link affects the ergodic capacity of the reactive
DF relaying system.
APPENDIX
A. Proof of Lemma 1
Using (10) and the rule of integration by parts, the typical
ergodic capacity when the decoding set CT has l relays can
be described as
?∞
=
y · FCR−D
0
¯CR−D
l
=
0
y · fCR−D
l
(y) dy
?∞
?
l
(y)
−
?∞
0
FCR−D
l
(y) dy,
(15)
where fCR−D
ity for the Rk−D link with the outdated CSI. In order to solve
(15), we need to compute the integration of the exponential
function in the interval [0, ∞). This integration is already
obtained by [6],
?∞
In addition, by exploiting (16), the typical capacity when the
decoding set CT has l relays can be obtained as (11).
l
(y) denotes the PDF of the instantaneous capac-
0
e−x(22·y−1)dy =
ex
2 · ln2E1(x).
(16)
B. Proof of Lemma 2
In order to preclude the outage event in terms of the
capacity, we evaluate the capacity within the interval [RT, ∞)
[6]. Using (10) and the rule of integration by parts, this can
be rewritten as
?∞
=
y · FCR−D
¯CR−D
l,eff=
RT
y · fCR−D
l
(y) dy
?
l
(y)
?∞
RT−
?∞
RT
FCR−D
l
(y) dy.
(17)
In a similar manner to the typical ergodic capacity, we can
obtain the effective ergodic capacity when the decoding set
CT has l relays as (13).
REFERENCES
[1] P. Anghel and M. Kaveh, “Exact symbol error probability of a coop-
erative network in a rayleigh-fading environment,” Wireless Communi-
cations, IEEE Transactions on, vol. 3, no. 5, pp. 1416 – 1421, Sept.
2004.
[2] Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selection amplify-
and-forward relay systems,” Communications Letters, IEEE, vol. 10,
no. 11, pp. 757 –759, Nov. 2006.
[3] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless
networks: Efficient protocols and outage behavior,” Information Theory,
IEEE Transactions on, vol. 50, no. 12, pp. 3062 – 3080, Dec. 2004.
[4] A. Bletsas, H. Shin, and M. Win, “Cooperative communications with
outage-optimal opportunistic relaying,” Wireless Communications, IEEE
Transactions on, vol. 6, no. 9, pp. 3450–3460, Sept. 2007.
[5] D. Michalopoulos and G. Karagiannidis, “Performance analysis of single
relay selection in rayleigh fading,” Wireless Communications, IEEE
Transactions on, vol. 7, no. 10, pp. 3718 –3724, Oct. 2008.
[6] S. Lee, M. Han, and D. Hong, “Average snr and ergodic capacity analysis
for opportunistic df relaying with outage over rayleigh fading channels,”
Wireless Communications, IEEE Transactions on, vol. 8, no. 6, pp.
2807–2812, June 2009.
[7] J. Si, Z. Li, and Z. Liu, “Threshold based relay selection protocol for
wireless relay networks with interference,” in Communications (ICC),
2010 IEEE International Conference on, May 2010, pp. 1 –5.
[8] V. Shah, N. B. Mehta, and R. Yim, “The relay selection and transmission
trade-off in cooperative communication systems,” Wireless Communica-
tions, IEEE Transactions on, vol. 9, no. 8, pp. 2505 –2515, Aug. 2010.
[9] R. Annavajjala, “On optimum regenerative relaying with imperfect
channel knowledge,” Signal Processing, IEEE Transactions on, vol. 58,
no. 3, pp. 1928 – 1934, March 2010.
[10] R. Mo, J. Lin, Y. H. Chew, and W. H. Chin, “Relay precoder design
for non-regenerative mimo relay networks with imperfect channel state
information,” in Communications (ICC), 2010 IEEE International Con-
ference on, May 2010, pp. 1 –5.
[11] J. Vicario, A. Bel, J. Lopez-Salcedo, and G. Seco, “Opportunistic relay
selection with outdated csi: outage probability and diversity analysis,”
Wireless Communications, IEEE Transactions on, vol. 8, no. 6, pp.
2872–2876, June 2009.
[12] M. Seyfi, S. Muhaidat, and J. Liang, “Performance analysis of relay
selection with feedback delay and channel estimation errors,” Signal
Processing Letters, IEEE, vol. 18, no. 1, pp. 67 –70, 2011.
[13] J. Vicario and C. Anton-Haro, “Analytical assessment of multi-user
vs. spatial diversity trade-offs with delayed channel state information,”
Communications Letters, IEEE, vol. 10, no. 8, pp. 588–590, Aug. 2006.
[14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and
Products, 6th ed.
New York: Academic, 2000.
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