2011 International Conference on Advanced Technologies for Communications (ATC 2011)
Outage Probability Analysis of Cooperative
Diversity DF Relaying under Rayleigh Fading
D.T. Nguyen, Universit of Technology Sydney, Australia
Quoc Tuan Nguyen, Vietnam National University Hanoi, Vietnam
Trang Cong Chung, Vietnam National University Hanoi, Vietnam
focuses on delay-limited and non-ergodic scenarios, and evaluates
performance of cooperative relaying protocols in terms of outage
probability.
In this paper, in view of ever lowering cost, flexibility and
robustness in noise resistance of digital detection, we concentrate
only on decode-and-forward (DF) relaying protocols and ignore
the noise propagating amplify-and-forward (AF) relaying. Fixed
relaying (FR) protocols are those in which the relay is
continuously active and are normally used when channel state
information (CSI) is not available to the transmitter. Selection or
adaptive relaying (SR) protocols are designed for better
efficiency in low SNR conditions and CSI is available at the relay.
When the measured SNR falls below a threshold, the relay stops
its relaying function and the source simply continues its direct
transmission to the destination using repetition coding or other
more powerful codes. Incremental relaying (IR) protocols are
those in which the relay only transmits upon a negative feedback
(NACK) from the destination, thus avoiding wastage of
bandwidth at high SNRs. The information rate of an IR network
using DF protocol is a random variable [6] depending on how
many times the transmission requires either only one sub-block
(i.e. when the direct link is not in outage) or two sub-blocks
(when the direct link is in outage).
In order to calculate the outage capacity, because of the
complexity of the probabilistic analysis involved, most authors [3,
4, 6] resort to the max-flow min-cut theorem [2] to find an upper
bound for the outage capacity of the relay channel. The focus of
this paper, however, is the exact formulas for the outage
probability of the above three forms of DF relaying protocols. As
was pointed out in the previous paragraph, the information
capacity of relaying networks using incremental DF protocols is a
random variable depending on the number of sub-blocks being
used for transmission. The relationship between outage capacity
and outage probability is, therefore, also a statistical relationship
[6]. A simple comparison of the performance of the three DF
protocols based on outage capacity is outside the scope of this
paper.
In practical wireless sensor networks, power is limited and
SNR is usually very low, and the performance of relaying
networks in terms of energy efficiency in the low SNR regime
becomes essential. However, in the low SNR regime, the Shannon
capacity is theoretically zero as SNR→0 and is no longer a useful
Abstract: In this paper, we present exact analytical expressions for
the outage probability of cooperative diversity wireless relay
networks operating in various decode-and-forward (DF) protocols
(fixed, adaptive, and incremental relaying) under Rayleigh fading
conditions. Current works only analyze the asymptotic behavior of
these protocols, either under high signal-to-noise ratios (SNR) or
under low SNR-low rate conditions. Our analytical results are
presented in such a way that they can be used for both asymptotic
conditions.
Index Terms: Multiple relay channel, achievable rate, decode-andforward, partial decoding, linear relaying.
1. INTRODUCTION
In the slow-fading environment, once a channel is in deep
fade, message coding is no longer effective in improving
transmission reliability, and cooperative diversity transmission
has proved to dramatically improve the performance of
transmission. Upper and lower bounds of the capacity of a general
relay channel were first studied in [1] and this work forms the
theoretical foundation of most reseach work on relay networks
today. In this paper, we deal only with the classical three-terminal
relay network using low-complexity cooperative diversity
relaying protocols for ease of potential implementation. In these
protocols, relay terminals can process the received signal in
different ways, the destination terminals can use different types of
combining to achieve spatial diversity gain, and source and relay
terminals can use repetition code or other more powerful codes to
cope with low-SNR transmission under heavy fade conditions. In
slowly fading channels, the fading is assumed constant over the
length of the message block, i.e. the channel is memoryless in the
blockwise-sense, and the strict Shannon capacity of the channel is
well defined and achievable. However, when the system is
constrained by the message decoding delay T and the bandwidth
W is also limited, the requirement 2WT>>1 cannot be met and
channel parameters cannot be modeled as ergodic or
asymptotically mean stationary random variables and the strict
Shannon capacity is zero [5]. In most practical situations, the
channel is non-ergodic and capacity is a random variable, thus no
transmission rate is reliable. In this case, the outage probability is
defined as the probability that the instantaneous random capacity
falls below a given threshold, and capacity versus outage
probability is the natural information theoretic performance
measure [5]. Consequently, as with many authors, this paper
978-1-4577-1207-4/11/$26.00 ©2011 IEEE
116
measure. Therefore in [5], a more appropriate metric called
outage capacity is defined as the maximal transmission rate for
which the outage probability does not exceed. We expect that
some level of synchronization between the terminals is required
for cooperative diversity to be effective. When CSI is unavailable
to the transmitters as in most simple implementations in practice,
coherent transmission cannot be exploited, hence even full-duplex
cooperation, i.e. where terminals can transmit and receive
simultaneously, cannot improve the total Shannon capacity of the
network. Therefore, in this paper we focus on half duplex
operation.
2.
where x, y, n, and P are the normalized transmit signal, i.e.
2
E x  1 , the corresponding received signal, the additive noise
 
which is modeled as a circularly symmetric complex Gaussian
random variable with zero mean and variance σ2 at the receiver,
i.e. n~N(0,σ2), and the transmit power, respectively. The
parameters’ double subscript ij is to mean being associated with
the channel link from i to j. hij is the channel gains (or loss) from
node i to node j, being subject to frequency nonselective Rayleigh
fading, and is modeled as independent, circularly symmetric,
complex Gaussian random variables with zero mean and variance
2
µij . It is well known that under Rayleigh fading, ℎ𝑖𝑗 and its
resulting SNR at the receiver is exponentially distributed.
In the decode-and-forward (DF) relaying protocol, the relay
detects by fully decoding the entire codeword it receives from the
source in relay-receive phase, symbol by symbol, then retransmits
the signal, after recoding it, to the destination during the relaytransmit phase.
In the relay-transmit phase at time n=T/2+1, T/2+2,….T, the
relays send their signals to the destination and the source may or
may not send the signal to the destination depending on the
relaying protocol used (multiple access mode). The received
signal from the relay is
SYSTEM MODEL AND DEFINITIONS
Figure 1 shows a simple cooperative diversity relay network
using M relaying branches. Because of insufficient electrical
isolation between the transmit and receive circuitry, time-division
half-duplex operation proves to be the safest mode. In this paper,
the relays are assumed to operate in the time division mode
having two phases: the relay-receive phase and the relay-transmit
phase; each phase or sub-block is of duration T/2. There is no
correlation between the source transmit signal and the relay
transmit signal, β = E[xsxr*] = 0, i.e. asynchronous case.
M
y rd [n]   Pri hri d x ri [n]  nri d [n]
(2)
i 1
We define the instantaneous signal-to-noise ratio (SNR) in
the received signal as
2
 ij 
 ij 
Each message from the source is coded into N symbols;
each symbol occupies a transmission time unit; T/2 is the duration
of time slot reserved for each message, i.e. N=T/2. Assume that
the source and the relay each transmits orthogonally on half of the
time slots, under the power constraint 1 𝑇 𝑇𝑛=1 𝑃𝑆 𝑛 ≤ 𝑃𝑆 and
𝑇
1
𝑇 𝑛=1 𝑃𝑟𝑖 𝑛 ≤ 𝑃𝑟𝑖 , where Ps and Pri are the transmit power of
the source and of the ith relay, respectively.
In the relay-receive phase at time n=1,2,…T/2, the source
transmits the complete message (N symbols) to both the
destination and the relays (broadcast mode) in the AF case, but
only to the relays in the DF case, i.e. only (1a) applies.
(1a)
y sd [n]  Ps [n]hsd x s [n]  nsd [n]
(1b)

2
ij
2
 hij  ijAWGN
(3)
where γijAWGN is the SNR of the unfaded AWGN channel.
Under Rayleigh fading, SNR in (3) is an independent
exponential random variable with expected (average) value
Figure 1: Diagram of an M-relay cooperative diversity relaying network
.
y sri [n]  Ps [n]hsri xs [n]  nsri [n]
hij Pi
ij Pi
 ij ijAWGN
 ij2
(4)
For convenience, and to be consistent with many papers on
the subject, in this paper we simply use SNR to mean γAWGN.
In this paper, we present the calculation of exact expressions
for the cumulative distribution function (cdf) of instantaneous
channel gains of various wireless links in a cooperative diversity
relay network and the asymptotic behavior of the cdf of these
gains either at high SNRs or at low SNR – low rate conditions.
The cdf function Fhi j  x  is used to calculate the outage
t
probability, Phou
 t h  of the wireless link between two points i
ij
and j having instantaneous channel gain hij for a given outage
information rate threshold, Rth. The definition of outage is
expressed as
117
Phout
( SNR, Rth )  Pr{ hij
ij
2
  th }  Fhij ( th )
 P out (  )  1
Lim th 0  FDF th  
  th
  sr
(5)
where the channel gain threshold is defined
 th  2 ( M 1) R  1/ SNR and M is the diversity order.
as
(8)
The significance of (8) is that it shows that fixed DF relaying
does not achieve diversity gain, i.e. at high SNR its outage
probability decays as 1/SNR instead of 1/SNR2. This is because it
depends entirely on the source-to-relay link to fully decode the
source message as has been pointed out in [5].
th
There are two asymptotic behaviors associated with µth→0:
one is for very large SNRs and a given finite outage threshold, Rth,
and the other is for both SNR and Rth being very small
concurrently. In the latter case, Rth is equivalent to the ϵ-outage
capacity Cϵ which is defined as the highest transmission rate for
which outage probability stays smaller than ϵ [5]. Therefore the
limits of the cdf as µth→0 for both asymptotic cases are identical.
In power-limited applications such as ad-hoc and sensor
networks, efficient design for low SNR operation is more relevant.
At low SNRs, the popular Shannon capacity is theoretically zero
and practically difficult to quantify, and ϵ-outage capacity is more
meaningful.
In this paper, we may use either the instantaneous gain of the
fading channel, │hij│2 with its average µij, or the corresponding
2
instantaneous signal-to-noise ratio,  i j  hi j SNR , wherever is
3.2. Outage Probability of Selection DF Relaying
In the selection DF relaying protocol, when the relay is not
able to decode the source message, i.e. the source-relay link is in
outage, the source simply repeats its transmission on the direct
link. Thus the maximum average information rate in this case is
that of repetition coding. The information rate of a selection DF
relay network can be expressed as below [4]
I SDF
convenient.
1
 2 log(1  2 sd ),

 1 log(1     ),
sd
rd
 2
 sr   th
(9)
 sr   th
Its outage probability under exponential fading condition is
3.
𝑜𝑢𝑡
𝑃𝑆𝐷𝐹
𝜇𝑡ℎ = Pr ℎ𝑆𝐷𝐹 2 ≤ 𝜇𝑡ℎ
= Pr 2 ℎ𝑠𝑑 2 < 𝜇𝑡ℎ Pr ℎ𝑠𝑟 2 < 𝜇𝑡ℎ
+ Pr ℎ𝑠𝑟 2 ≥ 𝜇𝑡ℎ Pr⁡( ℎ𝑠𝑟 2 + ℎ𝑟𝑑
OUTAGE PROBABILITY CALCULATIONS
3.1. Outage Probability of Fixed DF Relaying
The maximum average mutual information between the input
and the two outputs, achieved by i.i.d complex Gaussian inputs,
of a repetition-coded fixed DF relaying network is [3].
1
1

I DF  min  log(1   sr ), log(1   sd   rd )
2
2


=
+
(6)
= 1−𝑒
−
𝜇 𝑡ℎ
𝜇 𝑠𝑟
1−
1
𝜇 𝑠𝑑 −𝜇 𝑟𝑑
𝜇𝑠𝑑 1 − 𝑒
ℎ𝑠𝑟
−
2
+ ℎ𝑟𝑑
2
𝜇 𝑡ℎ
𝜇 𝑠𝑑
– 𝜇𝑟𝑑 1 − 𝑒
𝜇 𝑡ℎ
𝜇 𝑟𝑑
𝜇 𝑠𝑑 −𝜇 𝑟𝑑
1−𝑒
𝜇𝑠𝑑 1 − 𝑒
𝜇
− 𝑡ℎ
𝜇 𝑠𝑑
− 𝜇𝑟𝑑 (1 − 𝑒
𝜇
− 𝑡ℎ
𝜇 𝑟𝑑
 P out (  ) 
   sr
1 1
1
Lim th 0  SDF 2 th  

 rd
2


2


2
 sd  sr  rd

sd
sr
sd rd
th


) (10)
(11)
3.3. Outage Probability of IR-DF Relaying
As pointed out in the Introduction, the information capacity
of relaying using incremental DF protocols is a random variable
depending on the number of sub-blocks being used for
transmission. Its information capacity is difficult to define and its
outage probability, therefore, cannot be simply defined based on a
capacity threshold. Instead, we calculate outage probability of an
IR-DF realying network directly from the definition of outage
condition. The system is in outage either when the sourcedestination and the source-relay links are both in outage, or when
the source-relay link is not in outage, i.e. able to decode-and
forward, but the accumulation of SNR at the destination of signals
> 𝜇𝑡ℎ
−
𝜇
− 𝑡ℎ
𝑒 𝜇 𝑠𝑟
< 𝜇𝑡ℎ )
−𝜇 𝑡ℎ /𝜇 𝑠𝑟
The result in the last line of (10) can be obtained from (A1) and
(A3) in the Appendix. It can then be shown that the result in [4,
equ.22] can be obtained as the second order approximation of the
exact result in (10), i.e.
The first term represents the maximum rate at which the
relay can reliably decode the source message, and the second term
represents the maximum rate at which the destination can reliably
decode the source message provided the source uses repeated
transmission. Requiring both the relay and the destination to the
message reliably results in the smaller of the two rates. This limits
the performance of a fixed DF relay to that of the link between
the source and the relay, i.e. no diversity gain can be achieved.
From (6) the corresponding probability of outage under
exponential fading condition is
𝑜𝑢𝑡
𝑃𝐹𝐷𝐹
𝜇𝑡ℎ = Pr ℎ𝐹𝐷𝐹 2 ≤ 𝜇𝑡ℎ
= 1 − Pr ℎ𝑠𝑟 2 > 𝜇𝑡ℎ Pr
1−𝑒
−𝜇 𝑡ℎ /2𝜇 𝑠𝑑
2
(7)
The result in the last line of (7) can be obtained from (A1),
(A3) and (A5) of the Appendix.
By using the first order approximation e-x≈1-x, it can be
shown that
118
from the source and the relay is not enough to exceed the outage
threshold. Thus under exponential fading the outage probability
of an IR-DF relaying wireless network is
𝑜𝑢𝑡
𝑃𝐼𝑅−𝐷𝐹
𝜇𝑡ℎ = Pr ℎ𝐼𝑅−𝐷𝐹 2 ≤ 𝜇𝑡ℎ
= Pr ℎ𝑠𝑑 2 < 𝜇𝑡ℎ Pr ℎ𝑠𝑟 2 < 𝜇𝑡ℎ
+ Pr ℎ𝑠𝑟 2 ≥ 𝜇𝑡ℎ Pr⁡
( ℎ𝑠𝑟 2 + ℎ𝑟𝑑 2 < 𝜇𝑡ℎ )
= 1−𝑒
+
𝜇
− 𝑡ℎ
𝜇 𝑠𝑑
𝜇
− 𝑡ℎ
𝑒 𝜇 𝑠𝑟
𝜇 𝑠𝑑 −𝜇 𝑟𝑑
1−𝑒
= 1−
𝜇𝑠𝑑 1 − 𝑒
− 𝜇𝑟𝑑 (1 − 𝑒
𝜇
− 𝑡ℎ
𝜇 𝑟𝑑
)
𝜇 𝑠𝑑 −𝜇 𝑠𝑟
∙
𝜇𝑠𝑑 𝑒
1
𝜇 𝑠𝑑 −𝜇 𝑟𝑑
𝜇
− 𝑡ℎ
𝜇 𝑠𝑑
𝜇𝑠𝑑 𝑒
− 𝜇𝑠𝑟 𝑒
𝜇
− 𝑡ℎ
𝜇 𝑠𝑑
𝜇
− 𝑡ℎ
𝜇 𝑠𝑟
− 𝜇𝑟𝑑 𝑒
∙
𝜇
− 𝑡ℎ
𝜇 𝑟𝑑
(17)
The result in (17) can be obtained by using (A3) and (A5) of the
Appendix. It can be easily seen that the result in [6, equ. 14] can
be obtained as the second-order approximation of the exact result
in (17), i.e.
𝜇
− 𝑡ℎ
𝜇 𝑠𝑟
𝜇
− 𝑡ℎ
𝜇 𝑠𝑑
1
 P out (  ) 
   sr
Lim th 0  CSB 2 th   rd
  th
 2 sd  sr  rd
(12)
(18)
The result in the last line of (12) can be obtained from (A1) and
(A3) of the Appendix. It can then be shown that the result in [6,
equ. 5] can be obtained as the second order approximation of the
exact result in (12), i.e..
𝑜𝑢𝑡
𝑃𝐼𝑅
−𝐷𝐹 𝜇 𝑡ℎ
2
𝜇 𝑡ℎ
𝐿𝑖𝑚𝜇 𝑡ℎ →∞
1
=𝜇
1
𝜇
𝑠𝑑 𝑠𝑟
+ 2𝜇
1
𝑠𝑑 𝜇 𝑟𝑑
2𝜇 𝑟𝑑 +𝜇 𝑠𝑟
(13)
𝑠𝑑 𝜇 𝑠𝑟 𝜇 𝑟𝑑
= 2𝜇
3.4. Cut-Set Bound on Outage probability
The max-flow min-cut theorem [2] yields the upper bound of the
capacity, i.e. lower bound of outage probability, [1][3]. It is a
valid upper bound for a general relay channel with multiple input
and multiple output, we therefore use [1, Theorem 3]
C   max min I ( X1; (Y2 , Y3 X 2 )), I (( X1 , X 2 );Y3 )
(14)
f ( X1 , X 2 )
The first term is the information capacity of the broadcast
channel, through the relay from X1 to Y2 and Y3 with given X2, i.e.
the maximum mutual information between the input X1 and the
two outputs Y2 and Y3, while the second term is the capacity of the
multiple access channel, both directly from the source to the
destination and via the relay, from X1 and X2 to Y3, i.e. the mutual
information between the two inputs X1 and X2 and the output Y3.
Thus, the upper bound for capacity, in the case of no
correlation between X1 and X2 and equal transmit power from the
source and the relay, is
1
1

C  min  log(1  ( sd   sr )), log(1  ( sd   rd ))
2
2



Figure 2: Outage probability of two different decode-and-forward
relaying protocols and their cut-set lower bound for network realization
(µsd, µsr, µrd)=(1, 2, 3).
4 RESULTS AND CONCLUSIONS
With the ever lowering cost and flexibility of implementation
of digital detection, decode-and-forward (DF) relaying protocols
have become more and more popular than their noise propagating
amplify-and-forward (AF) counterparts. In this paper, we have
successfully derived exact expressions for the outage probability
of various versions of DF protocols. Figure 2 shows the curves of
outage probability as a function of channel gain threshold µth of
two decode-and-forward relaying protocols: Selection DF from
(10) and Incremental DF from (12). The figure also shows the
cut-set lower bound of outage probability from (17) of a general
multi input-multi output relaying network. The outage probability
curve for Fixed DF from (7) is not shown in Figure 2 because its
value is about two orders larger than those of the other DF
protocols shown on the figure. We have verified all these results
using Monte Carlo simulation. A plot of the outage probability for
FDF protocol in (7) shows that it is almost a linear function of µth
(15)
Equivalently, the cut-set-bound of the end-to-end network gain is


hCSB  min ( hsd  hsr ), ( hsd  hrd )
2
2
2
2
2
(16)
The corresponding lower bound of the outage probability
under exponential fading condition is
Pout
CSB μth = 1 - Pr
=1− 1−
1−
𝜇𝑠𝑑
𝜇𝑠𝑑
hsd 2 + hsr
2
>μth . Pr
hsd 2 + hrd
2
>μth
𝜇
𝜇
1
− 𝑡ℎ
− 𝑡ℎ
𝜇𝑠𝑑 1 − 𝑒 𝜇 𝑠𝑑 − 𝜇𝑠𝑟 (1 − 𝑒 𝜇 𝑠𝑟 )
− 𝜇𝑠𝑟
∙
𝜇
𝜇
1
− 𝑡ℎ
− 𝑡ℎ
𝜇𝑠𝑑 1 − 𝑒 𝜇 𝑠𝑑 − 𝜇𝑟𝑑 (1 − 𝑒 𝜇 𝑟𝑑 )
− 𝜇𝑟𝑑
119
up to as high as 0.1. This, together with the result in (8), shows
that the FDF protocol does not achieve any spatial diversity gain,
as has already been pointed out in [5]. We can also conclude from
Figure 2 that as long as the SDF protocol incorporates a repeat
transmission of message via the direct link if the relay is in outage
(during the relay-transmit phase), its outage probability is lower
(i.e. better) than that of the IR-DF counterpart. However, this
simple comparison may be unfair to the IR-DF protocol because
its information capacity is a statistical variable and therefore the
definition of μth in (5) for IR-DF does not have the same meaning
as for SDF protocol. The IR-DF protocol may perform better on
the basis of power and bandwidth efficiencies.
What may seem to be surprising from Figure 2 is that the cutset outage probability bound lies slightly above the SDF outage
probability. This is because the cut-set bound theory in (16)
applies only to systems with continuous energy flow while in
SDF relaying network, energy flow is discontinuous conditioned
on the outage of the source-to-relay channel.
Finally, we should point out that the extension of the work in
this paper to cover M-relay case is not a difficult task, particularly
if we assume that all relays are identical.
Hence

FS (  )   f S ( x)dx 
0
e u / u
 1  P(u  μ,v  μ)  1  P(u  μ) P(v  μ)
For exponential distributions,
FM (  )  1  exp{  (
m

u v

0
u
1
v
)}
(A5)
v
 F ( )  1
1
lim  0  M







u
v
(4.13)
Note that the distribution of max( u, v) is not an exponential
random variable.
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Sons, 1991.
[3] A. Høst-Madsen and J. Zhang (June, 2005). "Capacity bounds and
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VTC 2009-Sep
(A2)
A1.2 Sum of two independent exponential random variables
Let s=u+v, where u, v are two independent exponential r.v’s
with mean μu and μv, respectively, then from the convolution
theorem


Also from (A2),
By using the approximation e x  1  x , we have
1
1
u
i.e. m is an exponential r.v. having mean μm which is
1
1
1
(A6)


FU (u )  1  e u / u
f S ( )  ( f U  f V )  
(A4)
FM (  )  1  FM (m   ) 
(A1)
 F ( )  1
lim  0  U

   u

)  u (1  e  / u ) (A3)
A1.3 Distribution of the Minimum of independent exponential
random variables
Let m  min(u, v) where u, v are independent exponential
random variables with mean μu and μv, respectively. Then the cdf
of w is
Calculation of cumulative distribution function of Combined
i.i.d. exponential random variables
A1.1 Single exponential random variable
Let u be an exponential r.v. with mean μu, then
1
  / v
v
 F ( ) 
1
lim  0  S 2  
   2 u  v
Appendix 1
u
 (1  e
By using the approximation e x  1  x  x2 / 2. , we obtain
ACKNOWLEDGEMENT
This work was supported by a research grant from Project
QG.10.44-TRIGB at the University of Engineering and
Technology, Vietnam National University Hanoi.
f U (u ) 
1
v  u
e  x / u e (   x ) /  v dx
e   / v  e   / u
v  u
120