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Outage Analysis for Half-Duplex Partial Decode-
Forward Relaying over Fading Channels
Ahmad Abu Al Haija
ECE Dept., McGill University, Montreal, QC, Canada
Email: ahmad.abualhaija@mail.mcgill.ca
Mai Vu
ECE Dept., Tufts University, Medford, MA, USA
Email: maivu@ece.tufts.edu
Abstract—We analytically evaluate the outage performance
of the half-duplex partial decode-forward (PDF) relaying over
Rayleigh fading channels. In PDF relaying, the source splits
its information into cooperative and private parts, and the
relay decodes only the cooperative information then forwards
it to the destination coherently with the source. Assuming full
CSI at the receivers and limited CSI at the transmitters, we
analyze the outage performance by jointly considering outages
of the cooperative and private parts at both the relay and the
destination. In spite of additional outage at the relay and limited
CSI at the transmitters, we show that PDF relaying achieves
the full diversity order of 2at high SNR. Numerical results
confirm the analysis and show the advantage of the considered
PDF scheme, with coherent transmission and joint decoding at
the destination, over the existing DF and PDF schemes.
I. INTRODUCTION
The relay channel is considered as a basic model for
cooperative communication that can improve the performance
of multi-user communications [1] with applications in ad hoc,
sensor and future cellular networks (LTE-A and 5G). Most
research on the relay channel has focused on the transmission
scheme and achievable rate, few results exist for outage
performance. This paper analyzes the outage performance of
a partial decode-forward relaying scheme.
Several fundamental relaying techniques have been devel-
oped in [2], [3] including decode, compress and amplify-
forward techniques. Applications in wireless communication
motivate research on adapting these schemes to the half-duplex
channels [4]–[7] and analyzing their outage performances.
Existing works on relay outage performance mostly focus on
decode-forward (DF) relaying, in which the relay (R)fully
decodes and forwards the whole source (S)message [6]–[11].
Few works have considered outage for partial decode-forward
(PDF) relaying, in which Ronly decodes and forwards a
part of Smessage [12]–[14]. While PDF relaying achieve
the same transmission rate of DF in the full-duplex mode
[2], PDF achieves a higher rate in the half-duplex mode
[15] and hence a better outage performance. PDF relaying
also has additional advantages of providing privacy of some
information that is only decoded at the destination (D), and
allowing the transmission of messages with different priorities.
For half-duplex DF relaying, outage performance has been
analyzed for dual-hop and selection relaying with variable
resource allocations [6]–[9]. These works consider the optimal
resource allocation for minimum outage probability [6], [9]
and analyze the diversity order [7], [8]. Asymptotic outage
analysis shows that half-duplex DF achieves the maximum
diversity order of 2if the transmission switches between
direct and relay modes depending on the outage at R[7].
This technique, however, requires Sto have full channel state
information (CSI) of the link to Rin order to know when
outage can occur at R. The outage for PDF relaying has
been analyzed for independent [14] and superposition [12]
coding schemes. These works show that PDF can achieve the
full diversity order of 2and decreases the overall expected
distortion between the transmitted and reconstructed signals
at D[12].
These existing works, however, only formulate the outage
probability and evaluate the performance at high SNR. Fur-
thermore, they only consider non-coherent transmission from
Sand Rto Dbut not the original coherent DF schemes in [2].
In this paper, we consider the coherent PDF relaying in half-
duplex transmission as proposed in [4], [5], [16] and show that
the coherent transmission and PDF relaying can improve not
only the achievable rate but also the outage performance. The
main differences between the considered scheme and other
existing half-duplex schemes in [12], [14] lay in the coherent
transmission between Sand Rand the joint decoding of all
information parts at D. A similar analysis can be applied to
the full-duplex PDF scheme in [2].
We assume full CSI at all receivers, where each receiver
knows both the phases and amplitudes of the links to it.
Transmitters have only limited CSI where Rknows the phase
of its link to Dand Sknows the phase of its link to Dand
only the relative order between the amplitudes of its links
to Rand D. The phase knowledge at transmitters allows
coherent transmission while the link order knowledge allows
Sto switch between direct transmission and relay modes.
The outage analysis for PDF relaying takes into account
the outage of the cooperative part at Rand the outage of
both the cooperative and private parts at D. An outage for
the cooperative part leads to an outage for the private as well
because of the superposition coding structure. We show that
coherent PDF relaying achieves a full diversity order of 2and
outperforms all existing DF and PDF schemes.
II. CHANNEL MODEL
Figure 1 illustrates the half-duplex relay channel model
where the source (S)wishes to communicate with the desti-
nation (D)with help from the relay (R). Using time division,
each transmission block is divided into 2phases with channel
)UDPH
;
<
;
K
K K
<
K
<
;
Fig. 1. Half-duplex model for the Relay channel.
model as
phase 1: Y21 =h21X11 +Z21 ,Y
1=h1X11 +Z1,(1)
phase 2: Y2=h1X12 +h2X22 +Z2,
where Y21 is the signal received by Rduring phase 1;
Yk,k∈{1,2}is the signal received by Dduring the phase
k; and all the Zl,l∈{21,1,2}, are i.i.d complex Gaussian
noises (CN(0,1)).X11 and X12 are the signals transmitted
from Sduring phase 1and 2, respectively. X22 is the signal
transmitted from Rduring phase 2. Each link is affected by
Rayleigh fading and pathloss as follows:
hk=˜
hk/(dγk/2
k)=gkejθk,k∈{21,1,2}(2)
where ˜
hkis the small scale fading component and has a
complex Gaussian distribution (CN(0,1)). The large scale
fading component is captured by a pathloss model where dk
is the distance between two nodes and γkis the attenuation
factor. Let gkand θkbe the amplitude and the phase of a link
coefficient, then gk=|˜
hk|/dγk/2
khas Rayleigh distribution
while θkhas uniform distribution between [0,2π].
We consider block fading where the links stay constant in
each transmission block through the 2phases and change
independently in the next block. We assume the following
CSI: as receivers, Dknows h1and h2and Rknows h21;as
transmitters, Sand Reach knows the phase of its respective
link to Dwhich allows coherent transmission; and Sknows
if the amplitude of its link to Dis weaker or stronger than
the link to Rwhich allows Sto perform either direct or
cooperative transmission as explained in Section III.B. The
receiver CSI is a standard assumption and can be obtained
via channel estimation. The transmitter CSI can be obtained
via the reciprocity between forward and reverse channels in
time-division duplexed systems, or via feedback from D[17].
III. A COHERENT HALF-DUPLEX PDF SCHEME
In [5], [16], we proposed a time-division cooperative
scheme for the half-duplex multiple access channel based on
rate splitting, superposition coding and PDF relaying tech-
niques. The coherent PDF relaying scheme for the half-duplex
relay channel can be obtained as a special case when one
source has no information to send and only relays information
of the other source. Here, we briefly describe this coherent
PDF relaying scheme for relay channel in order to setup the
stage for the outage analysis to follow. This half-duplex PDF
relaying scheme is similar to that in [4] and can be seen as
an adaptation of the original full-duplex scheme in [2] to the
half-duplex mode using time division.
Each transmission block is divided into 2phases with
normalized durations α1and α2=1−α1. In each block,
Ssplits its information into a cooperative part (indexed by
i) and a private part (indexed by j) and encodes them using
superposition coding [5]. The private (cooperative) part is sent
at Rp(Rc)rate, so that the total transmit rate is R=Rp+Rc.
In phase 1,Ssends iand Rdecodes it. Then, in phase 2,S
sends both parts (i, j)and Rforwards i. At the end of phase
2,Dutilizes the received signals in both phases to decode
both parts using joint maximum likelihood (ML) decoding [5].
Hence, this scheme has no block decoding delay as Ddecodes
directly at the end of each transmission block.
Using Gaussian signaling, Sand Rconstruct their transmit
signals in each phase as follows:
phase 1:X11,i =√ρ11U1(i),(3)
phase 2:X12,j,i =√ρ10V1(j)+√ρ12 U2(i),
X22,i =√ρ22U2(i)
where U1,V
1and U2are i.i.d Gaussian signals (N(0,1)),
and X12 is superpositioned on U2. Note that U1and U2
convey the same information (cooperative message) but are
independent codewords sent in different phases; V1conveys
the private message. Here, ρ11 and ρ10 are the transmission
powers allocated for signals U1and V1, respectively, ρ12 and
ρ22 are the transmission powers allocated for signal U2by S
and R, respectively. These power allocation parameters satisfy
the following power constraints:
α1ρ11 +α2(ρ10 +ρ12)=P1,ρ
22 =P2/α2.(4)
where P1(P2)is the transmit power of S(R).
Using ML decoding, Rutilizes the received signal Y21 in (1)
to decode the cooperative information part (i)while Dutilizes
both Y1and Y2in (1) to jointly decode both information parts.
A. Achievable Rate
The rate constraints that ensure reliable decoding at Rand
Dlead to the following achievable rate:
Theorem 1. The achievable rate of the half-duplex coherent
PDF scheme for each channel realization is R=Rp+Rc,
where the pair (Rp,R
c)satisfies the following constraints:
Rc≤α1log 1+g2
21ρ11 =C1,(5)
Rp≤α2log 1+g2
1ρ10=C2,
R≤α1log 1+g2
1ρ11
+α2log 1+g2
1ρ10 +(g1√ρ12 +g2√ρ22 )2=C3.
for some 0≤α1≤1,α
2=1−α1and power allocation
parameters satisfying (4). Noises’ powers are normalized to 1
in these expressions as explained in (1).
Proof: Similar to the proof in [5], [16]. In (5), C1ensures
the reliable decoding of iat R;C2ensures the reliable
decoding of jat Dwhen ihas been decoded correctly; and
C3ensures the reliable decoding of both iand jat D.
B. Coherent PDF Relaying
Since the power allocation parameters are design variables,
we can coarsely adapt the scheme to the channel configuration
to obtain two transmission cases as follows.
1) Case 1(g21 ≤g1): Direct transmissions.
This case is identical to the classical point-to-point commu-
nication where Ssends its information directly to Dwithout
using R. Hence, by setting α1=1,ρ10 =ρ12 =ρ22 =0and
ρ11 =P1, the achievable rate becomes: R≤log 1+g2
1P1.
2) Case 2(g21 >g
1): PDF relaying.
In this case, Rperforms PDF relaying with all power parts
as in (4) and we obtain the rate constraints in (5).
Sneeds to know the relative amplitudes order of its links
to Dand Rin order to know which case to operate. The
combination of these two transmission cases makes up the
considered half-duplex coherent PDF scheme.
IV. PARTIAL DF OUTAGE ANALYSIS
The outage probability is an important criterion for many
practical wireless services that require a minimum target rate
(R)to be sustained. For a particular fading realization, outage
occurs if the rate supported by the channel is below R.
Let P1and P2be the outage probabilities in Case 1and 2,
respectively. For Case 1, the outage probability is obtained as
in point-to-point communication but given the condition that
g21 ≤g1. For Case 2, given that g21 >g
1, then outage can
occur at Rfor the cooperative information part or at Dfor the
private part or both parts when there is no outage at R. The
average outage probability (¯
Po)is then obtained as follows.
Theorem 2. For a given target rate Rwith a fixed rate
splitting R=Rp+Rc, the average outage probability (¯
Po)
of the half-duplex coherent PDF scheme is given as
¯
Po=P1+P2=P1+Pr+Ps+Pi,with (6)
P1=P[R>log 1+g2
1P1,g
21 ≤g1],
Pr=P[Rc>C
1,g
21 >g
1],
Ps=PR>C
3,R
c≤C1,g
21 >g
1,
Pi=PRp>C
2,R≤C3,R
c≤C1,g
21 >g
1,where
•P1is the outage probability of direct transmission when
Case 1occurs,
•Pr,Psand Piare the outage probabilities in Case 2,
–Pris the outage probability at R.
–Psis the outage probability for the sum rate at D
when there is no outage at R(region Bin Fig. 2);
–Piis the outage probability for the individual private
part at Dwhen there are no outage at Rand no
outage for the sum rate at D(region Fin Fig. 2).
The analytical expressions for P1,Pr,Psand Piare given in
Lemmas 1,2,3and 4, respectively.
Proof: see Lemmas 1—4.
Although Theorem 2 specifies the outage for fixed resource
allocation (rate splitting and power allocation), the optimal
resource allocation that minimizes ¯
Pocan be obtained numer-
ically.
A. Outage Probability for Transmission Case 1
Lemma 1. The outage probability for Case 1is given as
P1=P[R>log 1+g2
1P1,g
21 ≤g1](7)
5S
5F
$
)
&&
&
%
Fig. 2. Cooperative and private rate region (A) and outage regions (Band
F) obtained from the decoding at Dfor a given channel realization and power
allocation.
=1−exp −2R−1
μ1P1
−μ21
μ21 +μ11−exp −2R−1(μ21 +μ1)
P1μ1μ21
where μiis the mean of g2
i(μi=E[g2
i]) for i∈{21,1,2}.
Proof: Straightforward by using exponential distribution
of g2
1and g2
21.
Note that the outage probability of Case 1in (7) is different
from the scheme that always performs direct transmission
because of the correlation between the two events: (g21 ≤g1)
and (R>log 1+g2
1P1).
B. Outage Probability for Transmission Case 2
In this case, the outage can occur at Ror D. We separately
analyze the outage probability at R(Pr)and D(Pd), then
combine them to obtain the overall outage probability for this
case as follows.
Lemma 2. The outage probability for Case 2is given as
P2=P[outage at Rand D,g
21 >g
1](8)
=Pr+Pd,
where Pr=P[outage at R,g
21 >g
1],
Pd=P[outage at D,no outage at R,g
21 >g
1].
Proof: In Case 2, outage can occur at R. If outage can
also occur at Dif there is no outage at R.
To simplify the analysis, we set α1=α2=1/2in the
following sections. Note that this assumption does not affect
the diversity analysis shown later.
1) Outage at the Relay: Similar to Case 1, the outage
probability at Rcan be derived as
Pr=PRc>1
2log(1 + g2
21ρ11 ),g
21 >g
1(9)
=1−exp −η2
2
μ21 −μ1
μ21 +μ11−exp −η2
2(μ21 +μ1)
μ1μ21 ,
where η2=(22Rc−1)/ρ11.
2) Outage at the Destination: The outage at Dis consid-
ered when there is no outage at R. This outage is tied directly
with the decoding constraints (C2,C
3)at Din (5) which are
repeated here for the ease of reference:
Rp≤C2,R
c+Rp≤C3.(10)
For specific channel realization and power allocation, the rate
constraints in (10) can be plotted as in Figure 2. For a given
rate splitting, there is no outage if the split rate pair is in
region Abut there will be outage if it falls in region Bor F.
We refer to the outages pertaining to regions Band Fas the
sum outage (Ps)and individual outage (Pi)respectively, since
these regions are determined by the sum rate and individual
rate constraints in (10). From (10), given that there is no outage
at R, the sum outage probability (Ps)is obtained as
Ps=P[Rc≤C1,g
21 >g
1](11)
PR>C
3|Rc≤C1,g
21 >g
1,
where the first probability factor in (11) exists since we find
the outage at Donly when there is no outage at R. Similarly,
the individual outage probability (Pi)is given as
Pi=P[R≤C3,R
c≤C1,g
21 >g
1](12)
PRp>C
2|R≤C3,R
c≤C1,g
21 >g
1,
provided that there are no outage at Rand no sum outage at
D. Then, the overall outage probability at Dis given as
Pd=Ps+Pi=PR>C
3,R
c<C
1,g
21 >g
1,(13)
+PRp>C
2,R≤C3,R
c<C
1,g
21 >g
1.
The analytical evaluations for these outage expressions are
given in the next two Lemmas.
Lemma 3. The sum outage probability (Ps)at Dis given as
Ps=⎧
⎪
⎪
⎨
⎪
⎪
⎩
η
η2γ3
0f1(γ1,ζ,γ
3)dγ1dγ3
+ exp −η2
μ21 η
0f2(γ1,ζ)dγ1if η>η
2
exp −η2
2
μ21 η
0f2(γ1,ζ)dγ1,if η≤η2
(14)
where η2is given in (9) and
η=−P1+P2
1−ρ11(ρ10 +ρ12 )(1 −22R)
ρ11(ρ10 +ρ12 ),
ζ=1
√ρ22 22R
1+γ2
1ρ11 −(1 + γ2
1ρ10)−γ1ρ12
ρ22
,
f1(γ1,ζ,γ
3)= 4γ1γ3
μ1μ21
exp −γ2
1
μ1
+γ2
3
μ21
×1−exp −ζ2
μ2,
f2(γ1,ζ)=2γ1
μ1
exp −γ2
1
μ11−exp −ζ2
μ2 (15)
Proof: See Appendix A.
Lemma 4. The individual outage probability (Pi)at Dis
given as
Pi=⎧
⎪
⎨
⎪
⎩
P(1)
iif η2≤η≤η1,P(2)
iif η2≤η1≤η,
P(3)
iif η≤η2≤η1,P(4)
iif η≤η1≤η2,
P(5)
iif η1≤η≤η2or η1≤η2≤η
(16)
where η1=(22Rp−1)/ρ10,η
2=(22Rc−1)/ρ11,
P(1)
i=η
η2γ3
0
f3(γ1,ζ,γ
3)dγ1dγ3+ exp −η2
1
μ21 f5(γ1)
+η1
ηγ3
η
f1(γ1,ζ,γ
3)dγ1dγ3
+ exp −η2
μ21 η
0
f4(γ1,ζ)dγ1,
P(2)
i=η1
η2γ3
0
f3(γ1,ζ,γ
3)dγ1dγ3
+ exp −η2
1
μ21 η1
0
f4(γ1,ζ)dγ1,
P(3)
i= exp −η2
2
μ21 exp −η2
μ1+η
0
f4(γ1,ζ)dγ1
−μ1
μ1+μ21
exp −η2
2(μ21 +μ1)
μ21μ1
−μ21
μ1+μ21
exp −η2
1(μ21 +μ1)
μ21μ1,
P(4)
i= exp −η2
2
μ21 η
0
f4(γ1,ζ)dγ1+f5(γ1),
P(5)
i= exp −η2
2
μ21 η1
0
f4(γ1,ζ)dγ1,(17)
and f3(γ1,ζ,γ
3)= 4γ1γ3
μ1μ21
exp −γ2
1
μ1
+ζ2
μ2
+γ2
3
μ21
f4(γ1,ζ)=2γ1
μ1
exp −γ2
1
μ1
+ζ2
μ2
f5(γ1) = exp −η2
μ21 −exp −22Rp−1
ρ10μ21 (18)
Proof: Similar to that of Lemma 3 in Appendix A.
V. F ULL DF AND DIVERSITY ANALYSIS
When Shas high power, it can allocate enough power in
phase 1to send all its information to Rwithout any splitting
as in the DF scheme [15], [16]. Hence, the diversity order
of the PDF scheme is the same as the DF scheme in the
limit. This DF scheme is obtained by removing the private
information part from the PDF scheme and has the achievable
rate as follows.
Corollary 1. The achievable rate for the half-duplex coherent
DF scheme is obtained from Theorem 1 by setting Rp=
0,ρ
10 =0and R=Rc.
A. Outage Analysis
Because of the absence of the private part, the DF scheme
has a simpler outage expression and we use it to study the
asymptotic outage behavior of the PDF scheme at high SNR.
The outage for the DF scheme is obtained in a similar way to
the PDF scheme but without Pias follows.
Corollary 2. The outage probability of the half-duplex coher-
ent DF scheme is given as
PDF =P(DF)
1+P(DF)
r+P(DF)
d,where (19)
•P(DF)
1is the outage probability in Case 1,
•P(DF)
ris the outage probability at Rin Case 2,
•P(DF)
dis the outage probability at Din Case 2.
Analytical expressions for P(DF)
1,P(DF)
rand P(DF)
dare sim-
ilar to P1,Prand Psin (7), (9) and (14), respectively, with
the parameter settings in Corollary 1.
B. Asymptotic Outage Behavior and Diversity Order
Without loss of generality, assume that Sand Rtransmit
power are the same (P1=P2=P)and define a nominal SNR
as the received power at the destination in direct transmission
(SNR =μ1P1). Note that in this definition the SNR is always
proportional to P1. Next, we show that the dominant term in
each outage probability is proportional to SNR−2.
Theorem 3. The asymptotic outage probability of half-duplex
coherent PDF relaying at high SNR approaches the following
values:
P(DF)
1,∞=(2R−1)2
2μ1μ21P2,P(DF )
r,∞=(22R−1)2
2μ1μ21P2(20)
P(DF)
d,∞=δ2a12
2a22μds μdrP2,
where ρ11 =a11P1,ρ
12 =a12P1,ρ
22 =a22P2
α1a11 +α2a12 =1,α
2a22 =1,
and δ=(−1+1+a11a12 (22R−1))/(a11a12 ).
These expressions show that the diversity order is 2which is
the maximum diversity for the basic relay channel.
Proof: The formulas in (20) are obtained using Taylor
series expansion of the exponential function in (19) but we
remove the proof here because of space limitation.
Theorem 3 shows that the PDF scheme achieves the maxi-
mum diversity order of 2since the PDF scheme includes the
DF scheme as a special case. To achieve this diversity order,
Sonly needs to know the relative amplitude order between its
direct and cooperative links. This requirement is in contrast
to the existing schemes [7], [12] that require Sto have a full
knowledge of the cooperative link.
VI. NUMERICAL RESULTS
We now provide numerical results for the outage proba-
bilities of the considered coherent PDF and DF schemes in
order to verify the derived analytical solutions. The simu-
lation settings and channel configuration in all figures are:
P1=P2=P,γ=2.4,d1=20,d
2=15,and d21 =7.In
these simulations, all the links are Rayleigh fading channels
such that the average channel gain for each link is given as
μij =1
dijγ. With these settings, we define the average received
SNR in dB at Dfor the signal from Sas follows:
SNR = 10 log (μ1P/(dγ
10)) .(21)
In these simulations the power allocation parameters are varied
to obtain the best outage performance.
Figure 3 shows the outage probability versus SNR for the
considered coherent PDF and DF schemes. The simulations are
obtained using 5×105samples for each fading channel. This
0 5 10 15 20 25 30 35
10
−3
10
−2
10
−1
10
0
SNR (dB)
Outage Probability
Coherent PDF sch.
Coherent DF sch.
Simulated results
R=2 bps/Hz
R=4 bps/Hz
R=5 bps/Hz
Fig. 3. Comparison between analytical results and simulations of outage
probabilities for half-duplex coherent PDF and DF schemes versus SNR with
different R.
0 10 20 30 40 50
10
−4
10
−3
10
−2
10
−1
10
0
SNR (dB)
Outage Probability
Direct trans.
Coherent PDF scheme
Coherent DF scheme
DF scheme [7]
Selection Relaying [7]
PDF scheme [12]
Dual−hop [6]
PDF scheme [14]
Fig. 4. Comparison between half-duplex coherent PDF, DF, direct transmis-
sion and the existing schemes in [6], [7], [12], [14] with R=5bps/Hz.
number of samples is sufficient for outage probabilities above
10−3and produces the same results over different runs of the
simulation. Results show a perfect match between analytical
and simulated results. At low target rates and high SNR, S
can reliably send its full information to R; hence, PDF and
DF perform similarly. However, at high target rates and low
SNR, the PDF scheme outperforms the DF scheme.
Figure 4 compares between the considered coherent PDF
scheme and existing schemes in [6] (by analysis) [7], [12], [14]
(by simulation with 2×106channel samples) and the direct
transmission. Since the dual-hop [6], DF [7] and the PDF
[14] schemes always require decoding at R, they achieve a
diversity order of 1since the outage at Rbecomes dominant at
high SNR. The PDF scheme in [14] outperforms the selection
relaying scheme in [7] at low SNR because of rate splitting
but is worse at high SNR as the outage at Rstarts dominating
for the PDF scheme. Although rate splitting and superposition
coding are used in the PDF scheme in [12], this scheme has the
same outage performance of the selection relaying scheme [7]
since it employs sequential decoding instead of joint decoding
for the private and cooperative parts.
The selection relaying scheme in [7] and PDF scheme
in [12] both achieve a diversity order of 2. However, they
require about 4dB or more power to achieve the same outage
probability as the considered coherent relaying. One reason is
the absence of coherent transmission from Sand Rin these
two schemes. Another reason is that Dperforms sequential
decoding in the PDF scheme in [12] while it performs joint
decoding of both information parts in the considered scheme.
VII. CONCLUSION
We have analytically derived the outage probabilities for
coherent half-duplex PDF relaying over Raleigh fading chan-
nels. We assume full CSI at the receivers and limited CSI
at the transmitters. Then, we consider the outages for the
cooperative and private information parts at both the relay and
the destination. We further show that coherent PDF relaying
achieves a full diversity order of 2at high SNR. Results show
that the considered coherent PDF relaying outperforms all
existing DF and PDF schemes because of coherent source-
relay transmission and joint decoding at the destination. Fur-
thermore, PDF relaying improves the outage performance over
DF relaying especially at low SNR and high target rates.
APPENDIX A: PROOF OF LEMMA 3
Let γ1=g1,γ
2=g2and γ3=g21. Then, starting with the
expression in (13), we have
Ps=PR>C
3,R
c≤C1,g
21 >g
1(22)
=P[γ1≤min{γ3,η},γ
2≤ζ, γ3>η
2],
=∞
η2min{γ3,η}
γ1=0 ζ
γ2=0
f6(γ1,γ
2,γ
3)dγ2dγ1dγ3
=⎧
⎪
⎨
⎪
⎩
η
η2γ3
0f1(γ1,ζ,γ
3)dγ1dγ3
+∞
ηη
0f1(γ1,ζ,γ
3)dγ1dγ3,if η>η
2
∞
η2η
0f1(γ1,ζ,γ
3)dγ1dγ3,if η≤η2
where
f6(γ1,γ
2,γ
3)= 8γ1γ2γ3
μ1μ2μ21
exp −γ2
1
μ1
+γ2
2
μ2
+γ2
3
μ21 .
After the integration, (22) can be expressed as in (14).
The lower bound on γ3is obtained as follows:
0.5 log 1+γ2
3ρ11>R
c⇒γ3>η
2. The first upper bound
on γ1is obtained directly from γ1<γ
3.
The second bound on γ1and the upper bound on γ2are
obtained from the first constraint in (22) as follows:
1+g2
1ρ111+g2
1ρ10 +(g1√ρ12 +g2√ρ22 )2≤22R
⇔ρ22γ2
2+2γ1√ρ12 ρ22γ2+1+γ2
1(ρ12 +ρ10)
−22R
1+γ2
1ρ11 ≤0,⇔f(γ2)≤0(23)
The first derivative of f(γ2)is positive (∂f(γ2)
∂γ2>0) for all
values of γ1since γ1>0. Therefore, f(γ2)is an increasing
function of γ2for γ1>0. Then, (23) can be satisfied only
when f(0) <0. This condition makes f(γ2)negative for the
interval 0≤γ2≤ζ, where ζis the positive root of f(γ2).
However, the condition f(0) <0puts another bound on γ1
and leads to γ1<ηsince
f(0) ≤0⇔ρ11(ρ10 +ρ12 )γ4
1+2P1γ2
1+1−22R≤0(24)
The positive root for this quadratic equation leads to γ1<η.
Now, formula (22) is divided into two cases:
•Case 1is obtained when η>η
2. In this case, for
η2≤γ3<η, the minimum between γ3and ηis γ3
(min{γ3,η}=γ3). However, for γ3>η,(min{γ3,η}=
η).
•Case 2is obtained when η≤η2. In this case, ηis always
the minimum between γ3and η.
From these two cases, we obtain formula (14) in Lemma 3.
ACKNOWLEDGMENT
This work has been supported in part by grants from
the Fonds Quebecois de la Recherche sur la Nature et les
Technologies (FQRNT) and the Office of Naval Research
(ONR, Grant N00014-14-1-0645).
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