Page 1
VariableRate TwoPhase Collaborative Communication
Protocols for Wireless Networks
Hideki Ochiai
Department of Electrical and Computer Engineering, Yokohama National University
795 Tokiwadai, Hodogayaku, Yokohama, Japan 2408501
Patrick Mitran and Vahid Tarokh
Division of Engineering and Applied Sciences, Harvard University
33 Oxford Street, Cambridge, MA 02138
Abstract
The performance of twophase collaborative communication protocols is studied for wireless
networks. All the communication nodes in the cluster are assumed to share the same channel and transmit
or receive collaboratively in a quasistatic Rayleigh flatfading environment. In addition to smallscale
fading, the effect of largescale path loss is also considered. Based on a decodeandforward approach, we
consider various variablerate twophase protocols that can achieve full diversity order and analyze the
effect of node geometry on their performance in terms of the outage probability of mutual information.
For the single relay node case, it is shown that if the collaborator node is close to the source node, a
protocol based on spacetime coding (STC) can achieve good diversity gain. Otherwise, a protocol based
on receiver diversity performs better. These protocols are also compared with one based on fixedrate
repetition coding and their performance tradeoffs with node geometry are studied. The second part deals
with multiple relays. It is known that with N relays an asymptotic diversity order of N +1 is achievable
with STCbased protocols in the twophase framework. However, in the framework of collaborative STC,
those relay nodes which fail to decode remain silent (this event is referred to as a node erasure). We
show that this node erasure has the potential to considerably reduce the diversity order and point out the
importance of designing the STC to be robust against such node erasure.
Index Terms
Collaborative (cooperative) communication, relay channel, spacetime coding, spatial diversity,
wireless networks.
The material of this paper was in part presented at IEEE Fall Vehicular Technology Conference, Los
Angels, CA, September 2004. This work was in part supported by the Telecommunications Advancement
Foundation (TAF).
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I. INTRODUCTION
In many wireless networks, the power consumption of communication nodes is a critical issue. In
addition, typical wireless channels suffer from signal fading which, for a given average transmit power,
significantly reduces communication capacity and range. If the channel is slow and flat fading, channel
coding does not help [1,2] and spatial diversity may be the only effective option that can either reduce
the average transmit power or increase communication range. Results on spacetime coding (STC) [3,
4] have shown that the use of antenna arrays at the transmitter and receiver can significantly reduce
transmit energy. However, for many applications with lowcost devices such as wireless sensor networks,
deployment of multiple antennas at each node is too costly to implement due to severe constraints on
both the size and power consumption of analog devices.
The recently proposed collaborative (or cooperative) diversity approaches [5–14] demonstrate the
potential to achieve diversity or enhance the capacity of wireless systems without deploying multiple
antennas at the transmitter. Using nearby collaborators as virtual antennas, significant diversity gains can
be achieved. These schemes basically require that the relay nodes share the information data of the source
node, and this data sharing process is generally achieved at the cost of additional orthogonal channels (in
frequency or in time). In a companion paper [15], we have shown that for a given fixed rate and under
suitable node geometry conditions, there are collaborative coding schemes that can nearly achieve the
same diversity as if all the relay node antennas were connected to the source node, without any additional
orthogonal channels or bandwidth. The construction of such codes, however, appears to be challenging.
Among many approaches in the literature, Laneman [5,6] analyzes several lowcomplexity relaying
protocols that can achieve full diversity, under realistic assumptions such as halfduplex constraint and
no channel state information (CSI) at the transmitting nodes. It has been shown that in the lowspectral
efficiency regime, the SNR loss relative to ideal transmit diversity system with the same information
rate is 1.5dB[5]. Multiplerelay cases are also considered in [6] and bandwidthefficient STCbased
collaborative protocols are proposed.
Collaborative diversity protocols are largely classified into amplifyandforward and decodeand
forward schemes [5]. In the following, we will restrict our attention to decodeandforward schemes
since these may provide some salient advantages. First, there is no error propagation if the relay transmits
information only when it decodes correctly. Otherwise, the relay remains silent and thus an unnecessary
energy transmission can be saved1. Second, the information rate per symbol does not need to be the same
for each phase. In other words, the relative duration of each phase can be changed according to node
geometry.
1Even though perfect detection of the codeword is not feasible in practice, one can design a cyclic redundancycheck (CRC)
or error detectable lowdensity paritycheck (LDPC) code such that for a given system outage probability, the effect of error
propagation is negligible. Many existing communication networks have this structure.
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R∗
T
T1
T2
R1
R2
(a)
(b)
phaseI phaseII
Fig. 1. Twophase communication. (a) baseline system. (b) twophase protocol.
It is the latter property that we shall focus on in this work. Suppose that we wish to transmit data with
information rate R∗bits per second and T is the frame period, also in seconds. Then the total information
transmitted during this period is R∗T bits (per frame). The baseline frame design that achieves this is
shown in Fig.1 (a). Alternatively, we may split the time interval into two phases of duration T1 and
T2where T = T1+ T2and each phase is operated with information rate R1and R2, respectively, as
depicted in Fig.1 (b). We assume that for both phases, the same information (but with different coding
rate) is transmitted. If R1and R2are chosen such that R1T1= R2T2= R∗T, then in principle there
is no loss of total transmission rate compared to the baseline system. Let the fraction of the relative
time period for each phase be denoted by δ1? T1/T = T1/(T1+ T2), and δ2? T2/T = 1 − δ1. Then,
the information rate during each phase is R1= R∗/δ1and R2= R∗/δ2. Therefore, during each phase,
information should be transmitted employing larger constellation sizes than the baseline system2.
For ideal AWGN and interleaved fading channels under an average signaltonoise power ratio (SNR)
constraint over the entire communication process, twophase protocols do not necessarily achieve a gain
and may even result in performance loss compared to the baseline system. However, for quasistatic or
block Rayleigh fading channels, it is not the constellation size but diversity that is the dominant factor
for the outage behavior. Thus, if additional diversity can be achieved by twophase methods, the resulting
outage probability of the mutual information may more than offset any loss due to constellation size and
yield a reduction in required SNR. (This is somewhat analogous to coded modulation which increases the
signal constellation size in order to achieve coding gain. In our case, however, we shall achieve diversity
gain.)
In practical collaborative wireless communication networks, node geometry is an important factor.
Intuitively, if the collaborative relay node is close to the source node, it may be efficient for the relay
2The fraction δ1 and δ2, or equivalently, the coding rate R1 and R2 are determined based on the node location, not on each
realization of fading channel coefficient as done in [15,16].
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to act as a transmit antenna. In this case, STC based protocols such as [6] may be efficient. On the
other hand, if the relay is close to the destination, it should operate as a receive antenna. To capture
this geometrical effect, we model the wireless network channel as an aggregate of largescale pathloss
and smallscale fading [17]. The largescale pathloss is the decay of signal power due to the transmitter
receiver separation, and is a function of the distance between the two terminals. On the other hand, the
smallscale fading is a consequence of multipath which may vary randomly according to any physical
change of surroundings. The overall system model is detailed in Section II.
In this paper, motivated by the rateflexible nature of decodeandforward protocols and the importance
of node geometry, we extend the work of Laneman [5,6] to a variablerate framework with particular em
phasis on pathloss gain effect of relay nodes, achieved due to the relay’s proximity to source/destination
nodes. Several lowcomplexity protocols are considered, including a simple multihopping protocol, the
bandwidthefficient STCbased protocol of [6], as well as its receiver diversity counterpart (SectionIII).
Their performances with a single relay node are theoretically analyzed in terms of achievable diversity
gain for a given information rate based on outage probability of mutual information. For this purpose,
convenient simple analytical tools are developed in Section IV.
The main objective of the paper is, for a given relative location of the relay node, to determine a suitable
protocol and minimize the total required power of the transmitting nodes. To that end, optimal power
control factors and relative phase durations for the relay node are derived for each protocol considered.
Associated with these protocols, closedform expressions for diversity gain are derived in SectionV, where
it is shown that by suitably choosing the protocol and controlling the transmission rate, as a function
of node geometry, the achievable diversity gain can be significantly improved. Also, it will be shown
that under severe pathloss, even a simple multihop protocol benefits relative to direct transmission. For
example, a significant gain is attained if the relay is located midway between the two communicating
nodes.
In the analysis of STCbased collaborative protocols, we presume two types of STC which we denote
as perfect and imperfect STC. A perfect STC refers to an STC with partial decodability, i.e., the (full)
information can be retrieved from a subset of the transmitting nodes, whereas an imperfect STC refers to
a system in which the receiving nodes fail to decode if any one of the transmitting nodes that constitute
the STC fails to transmit. This partial erasure of an STC antenna branch may happen if the relay nodes
fail to decode correctly. (This event will be referred to as a node erasure.) In SectionVI, we show that
the diversity order of an imperfect STC with N collaborative relays is at most 2, whereas that of a perfect
STC can achieve diversity order of N + 1 as in [6].
Throughout this paper, our main focus is on the achievable diversity gain for a given information rate.
The diversitymultiplexing tradeoff [18] of the relay channels is also of practical importance, but this is
beyond the scope of this paper. Some results in this direction are explored in [5,6,16]. As related work,
the effect of node geometry is also considered in [12,19,20], but in a fixedrate framework. Also, we do
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S
D
R
θ
HS,R
HR,D
HS,D
dS,R
dR,D
dS,D
GS
GD
Fig. 2. Twophase communication.
not address specific design issue of coding as many existing channel/STC techniques in the literature are
applicable to our framework without major modification. Note that some practical design of collaborative
codes (with implicit variablerate coding) is proposed in [21] and its outage behavior is evaluated in [13].
The use of incremental redundancy such as [22] may be of further potential in this framework. Finally,
we note that variablerate coding for multipleaccess channels has been recently studied in [23].
II. SYSTEM AND CHANNEL MODEL
Fig.2 illustrates the basic model which is considered throughout the paper. It is assumed that the three
nodes source(S), relay(R), and destination(D) are located in the two dimensional plane as in Fig.2 where
θ is the angle of the line S − R − D and dA,Bdenotes the Euclidean distance between nodes A and B.
We suppose that S wishes to transmit the message to D and that R has agreed to collaborate with S a
priori.
For simplicity, we assume that all the channel links are composed of largescale path loss and
statistically independent smallscale quasistatic frequency nonselective Rayleigh fading. Consequently,
the complex channel coefficients HS,D, HS,R, and HR,D in Fig.2 are uncorrelated and circularly
symmetric complex Gaussian random variables with zero mean and unit variance. They are assumed
to be known perfectly to the receiver sides and unknown at the transmitter sides. Perfect timing and
frequency synchronization are also assumed, even though accurate acquisition of synchronization among
distributed nodes may be challenging in practice.
The path loss between two nodes, say A and B, is modeled by
PL(A,B) = K/dα
A,B,
(1)
where K is a constant that depends on the environment and α is the pathloss exponent. For freespace
path loss, we have α = 2 and K = GtGrλ2/(4π)2, where Gtand Grare antenna gains at transmitter
and receiver, respectively, and λ is the wavelength [17]. Although the pathloss exponent and the constant
factor K may vary for each channel link, throughout the paper it is assumed that α and K are identical
for all channel links.
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For isotropic antennas, the received energy at the relay can be related to the received energy at the
destination according to
?dS,D
where EA,B denotes the average received energy between the A → B channel link, and GS is the
geometrical gain achieved by the proximity advantage of the relay node over the destination node.
Likewise, the gain at the destination node in communicating with the relay over the source is given
by
GD?ER,D
ES,D
This gain implies that if the average power of the relay is controlled in such a way that the relay transmits
its signal with the same average power as the source, then the destination node receives the relay’s signal
with a gain of GDcompared to that of the S → D channel link. By the triangle equality, we have
?dS,R
Let ζ denote the ratio of dR,Dto dS,R, i.e.,
ES,R=PL(S,R)
PL(S,D)ES,D=
dS,R
?α
ES,D? GSES,D,
(2)
=
?dS,D
dR,D
?α
.
(3)
dS,D
?2
+
?dR,D
dS,D
?2
− 2
?dS,R
dS,D
??dR,D
dS,D
?
cosθ = 1.
(4)
ζ ?dR,D
dS,R.
(5)
Then the gain GScan be expressed as a function of α, ζ, and θ:
GS=?1 + ζ2− 2ζ cosθ?α
2.
(6)
Without loss of generality, we assume 0 ≤ θ ≤ π. It is easy to observe that for a given ζ > 0, GS is
a monotonically increasing function with respect to α and θ. Note that if π/2 ≤ θ ≤ π, then the relay
node lies within the circle with diameter S → D, and θ = π corresponds to the case where the relay lies
on the line between S → D.
III. TWOPHASE PROTOCOL
There are several variations that can be considered for twophase protocols. We consider the four
specific protocols summarized in TableI. Performance analysis of these protocols in a fixedrate
framework can be found in [5,6,24]. For all protocols, it will be assumed that all component codes
are designed to have error detection capability, i.e., if the relay fails to decode the information correctly,
it knows this and remains silent in the next phase. This may lead to power savings at the transmit side
and the resulting effect is incorporated into the calculation of the SNR. In the case that a relay node is
unable to collaborate during the second phase, we denote this event as a node erasure which corresponds
to an antenna erasure in a traditional STC scenario.
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TABLE I
TWOPHASE PROTOCOLS (•: TRANSMITTING, ◦: RECEIVING, : SWITCHED OFF)
TD STD RD MH
III
S
••
R
◦•
D
◦◦
III
S
••
R
◦•
D

◦
III
S
•

R
◦•
D
◦◦
III
S
•

R
◦•
D

◦
A. Descriptions
1) Transmit Diversity (TD) Protocol: In the first protocol, during phaseI, S broadcasts its information
at rate R1 and the relay node R attempts to decode this information. Node D also receives and then
attempts to decode the information during this phase. If D is able to decode the message correctly, the
subsequent phase will be ignored.
During phaseII, both S and R (re)encode the information using an STC with rate R2similar to [6]. If
the decoding at node D after phaseI failed, node D reattempts to decode after phaseII. This approach
is referred to as the Transmit Diversity (TD) protocol.
2) Simplified Transmit Diversity (STD) Protocol: This is a simplified alternative to the TD protocol.
In this case, the destination node D is switched off during phaseI and thus ignores the signal from S.
The phaseI communication link serves only the relay R. The second phase is identical to that of the TD
protocol. The STD protocol may result in a simple receiver structure but in some cases, a performance
loss is expected compared to the TD protocol.
3) Receiver Diversity (RD) Protocol: The third scenario we consider is similar to receiver selection
diversity. In this case, during phaseI, S broadcasts information and the relay and destination decode
in the same way as the TD protocol. During phaseII, the relay reencodes the data and transmits the
data at rate R2without STC. (The source remains silent in phaseII.) This approach is referred to as the
Receiver Diversity (RD) protocol.
Further strategies such as decoding based on a combination of phaseI and phaseII data can be
considered, analogous to maximum ratio combining for receiver diversity [5,6,13,24]. Only in such an
approach can optimal performance be achieved. In our variablerate framework of decodeandforward,
such techniques may require special coding structures and thus impose additional complexity at the
receiver side. For comparison purpose, however, the performance of this approach is also studied in
Section VH.
4) MultiHopping (MH) Protocol: The effectiveness of multihopping (MH) protocols has been widely
studied3. For comparison a simple multihopping protocol will also be considered as a special case of
the RD protocol where the destination node only switches on during phaseII. This approach does not
3For example, optimal MH distances from system energy consumption efficiency perspective are discussed in [25].
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offer any diversity gain, and thus generally results in performance loss rather than gain. However, as will
be shown later, if the signal decay due to path loss is severe (α > 3), the MH protocol does offer an
SNR gain compared to direct transmission, when the relay is between the two communicating nodes.
B. Optimization Issues
An interesting question one may ask regarding twophase protocols is how we should choose the
fraction δ1. This depends on the geometrical location of the relay and the specific protocol. Intuitively, if
the relay is located close to the source node, the pathloss of the channel link S → R is relatively small
compared to that of the S → D link. Therefore, the relay receives the data with high average SNR and
thus decodes the message successfully with high probability. In this case, even small δ1(and thus high
R1) may be sufficient for successful decoding at the relay. In phaseII, one may use the TD or STD
protocols to efficiently achieve full diversity without bandwidth expansion.
On the other hand, if the relay is located close to the destination node, the situation may be reversed.
In this case, the relay and destination receive the signal with equal average power. Since the relative
path loss of the link R → D is small, the relay can transmit the received data with little power (if it
is decoded correctly), and this may add additional diversity to the destination. The overall system is
thus similar to an ideal receiver diversity system and the RD protocol may be efficient, provided that
the relay appropriately controls its transmission power.4Therefore, for practical design of relay systems,
it is important to consider the geometrical properties of the relay location, together with the choice of
appropriate protocols, relative phase durations, and power control.
IV. ANALYTICAL TOOLS FOR OUTAGE PROBABILITIES
In this section, we develop our performance criterion and analytical tools for the design and evaluation
of the above protocols. Our design criterion is based on the mutual information for a given realization
of the fading coefficients H = {HS,D,HS,R,HR,D}. Specifically, we assume that communication is
successful if the mutual information (with Gaussian code book) of the channel conditioned on H is
greater than the information rate [27]. Otherwise, an outage event follows. The probability of an outage
event defined in this way, which is commonly referred to as outage probability, not only has an analytically
convenient form but also serves as a reasonable performance indicator for practical systems; with moderate
frame length and a welldesigned STC [4], the frameerror rate may fall within a few dB of the Multiple
Input SingleOutput (MISO) channel outage probability [28,29]. Also, the outage probability can be seen
as a Complementary Cumulative Distribution Function (CCDF) of the nonergodic capacity, which is a
random variable of H [28].
4An alternative approach in this case may be a compressandforward scheme, e.g., [26], which may result in potentially
better performance. Comparisons with this scheme are beyond the scope of this paper.
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A. Asymptotic Diversity Order
Our main goal is to achieve a large diversity gain with minimum transmitted energy. We first define
the asymptotic diversity order in the high SNR regime [18].
Definition 1 (Asymptotic Diversity Order): Let Pout(SNR) denote an outage probability as a function
of the channel SNR. The asymptotic diversity order is defined as [18]
d∗(Pout(SNR)) ? − lim
SNR→∞
ln(Pout(SNR))
ln(SNR)
.
As an alternate form, in this paper we consider outage probabilities as a function of the inverse
SNR, X ?
analytically extend fout(X) in a neighborhood of X = 0. To that end, we introduce the notion of analytical
extendibility at X = 0 for a CCDF.
Definition 2 (Analytically Extendible CCDF): A CCDF f(X) is called analytically extendible at X =
0 if all the following conditions are satisfied.
1
SNR, and let fout(X) = Pout(SNR)SNR=1/X. It will also be mathematically convenient to
1) On X ≥ 0, 0 ≤ f(X) ≤ 1 and on this interval, f(X) is a nondecreasing function with f(0) = 0.
2) f(X) is analytic at X = 0. Thus, in some open interval (−ǫ,ǫ), ǫ > 0, f(X) can be expressed as
a power series centered at X = 0,
f(X) =
∞
?
n=0
anXn,an=1
n!f(n)(0),
(7)
where f(n)(X) denotes the nth derivative of f(X). Since f(0) = 0, it follows that a0= 0. When
f(X) ?= 0, we refer to the minimum value of n where an?= 0 as the order of f(X).
Since f(X) is nondecreasing on X ≥ 0, it is easy to see that am> 0 for a given minimum order m.
In the rest of this paper, we do not explicitly mention ‘at X = 0’ and simply refer to such functions
as analytically extendible CCDFs. With the above definition, we have the following lemma.
Lemma 1 (Diversity Order): If a CCDF f(X) is analytically extendible with order m, then the
asymptotic diversity order is m.
Proof: For m > 0, we have am> 0 and
?
Thus, we obtain
ln(am+?∞
Since the second term of (8) can be easily shown to be zero, we obtain d∗(f(X)) = m.
Also, the following corollary may be immediately obtained from the logarithmic property of the
diversity definition.
Corollary 1: Let gi(X), i = 1,2,..., and hj(X), j = 1,2,..., be analytically extendible CCDFs.
If f(X) =
?
f(X) = Xm
am+
∞
?
n=1
an+mXn
?
.
d∗(f(X)) = m + lim
X→0
n=1am+nXn)
lnX
.
(8)
igi(X)?
j(1 − hj(X)), then d∗(f(X)) =
?
id∗(gi(X)). Furthermore, if f(X) =
?
igi(X), then d∗(f(X)) = mini{d∗(gi(X))}.
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B. Diversity Offset Gain
We note that for an analytically extendible CCDF fout(X) of order m,
lim
X→0
fout(X)
Xn
=
am
for
n = m
0
for
n < m.
(9)
Therefore, as SNR → ∞ (X → 0), the asymptotic outage probability is given by
fout(X) ∼ amXm
where cm? a−1
lnPout(SNR) ∼ ln(am) − mlnSNR.
As addressed in [30], the asymptotic diversity order m determines the slope in a plot of the logoutage
probability versus SNR in decibels, whereas am(or cm) determines the intercept. Therefore, our design
criterion is to choose the offset term amas small as possible, thereby maximizing the gain cm. Note that
in [30], the term cmis referred to as a coding gain, but since this gain is a result of spatial diversity
rather than code structure, we refer to this relative gain as diversity offset gain (or simply, offset gain)
in the following.
or
Pout(SNR) ∼ amSNR−m= (cmSNR)−m,
(10)
m
m . Thus,
(11)
C. Ideal MISO Case
We begin by considering an ideal MISO system with m transmit antennas and let SNR denote the
total received SNR. From the literature of MIMO communications systems [28,29] the following theorem
holds.
Theorem 1 (MISO Channel Diversity Order and Offset): Consider a MISO channel with m transmit
antennas and suppose that we transmit the data with an information rate R∗. Under the assumption that
the transmitter does not know the channel coefficients and all the m channel coefficients are circularly
symmetric Gaussian random variables with zero mean and unit variance, the achievable diversity order
is m, and the offset term, denoted by ˜ am, is given by
˜ am=
1
m!(mA0)m,
(12)
where A0= 2R∗− 1.
Proof: Let the m channel coefficients be denoted by h ?
Hi are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random
variables with zero mean and unit variance by assumption. For a given m×m diagonal power allocation
matrix with nonnegative entries, denoted by P, that satisfies trace(P) ≤ 1, the mutual information
conditioned on h is defined as [29]
?
H0
H1
...Hm−1
?T, where the
C(SNR,P,h) ? log2
?1 + SNRhHPh?.
(13)
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Choosing P =
1
mI, where I is an m × m identity matrix, we obtain
C(SNR,P,h) = log2
?
1 +SNR
m
Z
?
,
(14)
where Z is a random variable which follows a central chisquare distribution with 2m degrees of freedom,
each of variance 1/2. The outage probability is given by
?SNR
where A0? 2R∗− 1 and X ?
Pout= Pr[C < R∗] = Pr
m
Z < A0
?
= Pr[Z < mA0X],
(15)
1
SNR. We thus have the outage probability as a function of X:
fout(X) = P(m,mA0X)
△
=γ (m,mA0X)
Γ(m)
,
(16)
where Γ(m) is the (complete) Gamma function and γ(m,x) is the lower incomplete Gamma function
?x
Clearly, the integrand in (17) is analytic in t and hence, so is γ(m,x) in x. Therefore, fout(X) is an
analytically extendible CCDF and since P(m,x) = e−x?∞
∂nP(m,κx)
∂xn
γ(m,x)
△
=
0
e−ttm−1dt.
(17)
k=mxk/k!, one can show that
????x=0
=
0 n < m
κn
n = m.
(18)
Hence, the asymptotic diversity order is m from Lemma 1, and we thus obtain (12).
The coefficient ˜ amin (12) may serve as a reference offset for an ideal MISO system. In the following,
we define a diversity offset gain with respect to an equivalent ideal MISO (or SISO) performance as
follows.
Definition 3: For a system with diversity order of m and offset am, the diversity offset gain with
respect to an equivalent ideal MISO (or SISO) performance is defined as
Λ(m) ? cm/˜ cm= (˜ am/am)
1
m= mA0(m!am)−1
m.
(19)
Thus, the diversity offset gain Λ(m) serves as a measure of the relative performance of a scheme with
respect to an ideal MISO system as given by the asymptotic SNR gap for a small outage probability. If
Λ(m) < 1, there is a relative loss in asymptotic SNR required to achieve the same outage probability as
a MISO system.
V. PERFORMANCE ANALYSIS WITH SINGLE RELAY
In the following, we analyze the outage probability and achievable diversity offset gain of the protocols
outlined in SectionIII, assuming an independent Rayleigh fading plus path loss channel model. We assume
that the Gaussian noise power is identical for all the channel links considered. Extensions to the cases
with variable noise power may be tedious but straightforward.
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A. Transmit SNR
For further analysis, an appropriate measure of SNR should be defined. In this paper, we shall evaluate
the system in terms of the total transmitted power (for a given noise power). Consequently, the SNR is
defined as a ratio of total transmitted signal power, which is the sum of the source and relay transmit
power, to the Gaussian noise variance, which is assumed to be constant. We will refer to this ratio
as transmit SNR and denote it by SNRt, throughout the paper. Since, in the absence of relay nodes,
the received SNR is given by SNR = PL(S,D)SNRt, by taking the path loss between the source and
destination nodes PL(S,D) to be unity, the transmit and received SNRs become identical. As opposed to
the more conventional notion of received SNR, transmit SNR is a more appropriate measure of wireless
network performance in terms of total power consumption.
1) RD and MH Protocols: Let SNRS
of phaseI, and let SNRR
The transmit SNR can be expressed as
1denote the received SNR dedicated for the communication link
2denote that of phaseII, conditioned that the relay is transmitting.
SNRt= δ1
SNRS
PL(S,D)+ δ2β
1
SNRR
PL(R,D)=
2
1
PL(S,D)
?
δ1SNRS
1+ δ2
β
GDSNRR
2
?
,
(20)
where β is an average energy consumption factor that accounts for the probability that the relay is
transmitting and thus β ≤ 1.
Now, we suppose that the relay transmits its signal with average power ∆Rtimes that of the source.
Then we may write SNRR
PL(S,D) = 1, (20) reduces to
2/PL(R,D) = ∆RSNRS
1/PL(S,D), or, SNRR
2= GD∆RSNRS
1. Thus, setting
SNRt= (δ1+ δ2β∆R)SNRS
1? lRDSNRS
1,lRD= δ1+ δ2β∆R.
(21)
2) TD and STD Protocols: In this case, we assume that the source node employs the same average
power through phases I and II for simplicity. The extension of our results to variable power cases is
straightforward. Let SNRS
we have SNRS
I and II with PL(S,D) = 1 is given by
2denote the received SNR of phaseII due to the channel link S → D. Then
1by assumption. The corresponding transmit SNR by the nodes through phases
2= SNRS
SNRt= (1 + δ2β∆R)SNRS
1? lTDSNRS
1,lTD= 1 + δ2β∆R.
(22)
B. Error Events
In order to derive the outage probability and associated diversity offset gain of various protocols, we
first define the following events: E1= Event [ Decoding at destination after phaseI fails ], ER= Event [
Decoding at relay after phaseI fails ], and E2= Event [ Decoding at destination after phaseII fails ]. If
the destination receives during two phases, we have the outage probability Pout= Pr[E1∩E2]. Otherwise,
Pout= Pr[E2]. Also, in the following,¯A denotes the complement of the event A.
DRAFT
Page 13
12
C. MultiHopping (MH) Protocol
We begin with the analysis of the MH protocol. In this protocol, since the destination listens only
during phaseII, the outage probability is given by
Pout= Pr[E2] = 1 − Pr[¯ER]Pr[¯E2¯ER].
(23)
Let C(SNR,H) denote the mutual information of the channel conditioned on channel coefficient H and
received SNR, defined as C(SNR,H) ? log2
?
1 + SNRH2?
?= Pr
1. The conditional probability Pr[¯E2¯ER] is calculated as
Pr[¯E2¯ER] = Pr?C(SNRR
where A2? 2R2− 1. Consequently, we have
. With this notation, we have
Pr[ER] = Pr?C(GSSNRS
where A1? 2R1− 1, and Pr[¯ER] = e
1,HS,R) ≤ R1
?
HS,R2≤
2R1− 1
GSSNRS
1
?
= 1 − e
−
A1
GSSNRS
1
(24)
−
A1
GSSNRS
2,HR,D) > R2
?= e
−
A2
GD∆RSNRS
1,
(25)
Pout= Pr[E2] = 1 − e
−
“
A1
GS+
A2
∆RGD
”
1
SNRS
1.
(26)
Let X ?
successfully decodes, β in (20) is given by
1
SNRt, and from (21) we obtain
1
SNRS
1= lRDX. Since the relay transmits only if the relay
β = Pr[¯ER] = 1 − Pr[ER] = e
−
A1
GSSNRS
1 = e−
A1
GSlRDX? β(X)
(27)
which is also a function of X. From (21), we may then relate lRDand X by
lRD(X) = δ1+ δ2∆Re−
A1
GSlRD(X)X.
(28)
The outage probability can then be expressed as an analytically extendible CCDF of X:
fout(X) = 1 − e−
“
A1
GS+
A2
∆RGD
”
lRD(X)X.
(29)
With careful manipulation of the analytic function fout(X), we obtain a0= 0 and
?
Hence, the asymptotic diversity order is m = 1.
In order to improve the achievable diversity offset gain, we wish to minimize a1in (30) by judiciously
choosing ∆Rand δ1. The optimization can be performed in a two step manner: first, find ∆Rfor a given
δ1, and then numerically optimize δ1.
By fixing 0 < δ1< 1 (and thus 0 < δ2< 1), the parameter ∆Rthat minimizes a1can be found by
standard calculus as ∆opt=
?rδ1/δ2, which is a function of ζ, δ1, R∗, and α from r in (30) . The
overall offset gain can be given from (19) by
??
GS
A0
a1=A1
GS
1 +
r
∆R
?
(δ1+ δ2∆R),r ?A2
A1
GS
GD
=A2
A1ζα=2R∗/(1−δ1)− 1
2R∗/δ1− 1
ζα.
(30)
ΛMH(1) =
δ1
A1
+
?
δ2
GD
A2
A0
?−2
.
(31)
DRAFT
Page 14
13
Using ζ notation of (5), we can write
ΛMH(1) =A0
?1 + ζ2− 2ζ cosθ?α
2
?√δ1A1+ ζ
α
2√δ2A2
?2
(32)
where δ1should be numerically chosen, for given R∗, ζ, and α, according to
?
1) Geometric Effect of the Relay: If the relay is located close to the source, i.e., ζ ∼ 0, the optimal
value of δ1approaches 1, which should be equivalent to direct transmission from source to destination,
and we have
δ1= arg min
0<δ1<1
δ1
?2R∗/δ1− 1?+ ζ
α
2
?
(1 − δ1)?2R∗/(1−δ1)− 1?.
(33)
lim
ζ→0ΛMH(1) =A0
A1
= 1.
(34)
Likewise, if the relay is located close to the destination, i.e., ζ ∼ ∞, the optimal value of δ1approaches
0, and we have
lim
ζ→∞ΛMH(1) =A0
A2
= 1,
(35)
which is identical to (34). Thus, for these asymptotic cases, the pathloss exponent α does not appear
in the gain expression and essentially no gain is achieved by this protocol. However, this is not the case
if the relay is located between the two nodes. For example, if the relay is located midway on the line
between the source and destination, i.e., ζ = 1, then since a function g(δ1) ?√δ1A1=
is a convex ∪ function with respect to δ1, the above gain is maximized at δ1= δ2= 1/2 and thus we
have
?2R∗− 1?(2 − 2cosθ)
Now, in this case, we observe that the offset gain is a function of pathloss exponent α, and this gain,
measured in dB, increases linearly with α provided θ >π
by increasing the pathloss exponent as a MH gain in the following.
2) Numerical Results: Fig.3(a) and (b) show the optimal δ1and corresponding achievable offset gain
ΛMH(1), respectively, with respect to the relay position ζ = dR,D/dS,Rand different values of the path
loss exponent α. The information rate is R∗= 2, and the relay is located at θ = π, i.e., on the line
between the source and destination nodes. From Fig. 3 (b), it is observed that if the pathloss exponent
is less than 4, the MH protocol always results in loss compared to a traditional SISO system and thus no
benefit is obtained. However, if α is at least 4, the MH protocol can offer some gain. (In fact, from (36),
ΛMH(1) > 1 if α > 1+log25.) Therefore, in our scenario, a positive MH gain is possible if the pathloss
exponent is sufficiently large and the relay is appropriately located5.We also observe that the maximum
?
δ1
?2R∗/δ1− 1?
ΛMH(1)ζ=1=
α
2
2(22R∗− 1)
=(2 − 2cosθ)
2(2R∗+ 1)
α
2
.
(36)
3. We shall refer to this type of gain achieved
5It is interesting to point out that the observation of suboptimality of multihopping agrees with that of [31] in terms of
system energyefficiency perspective, though the underlying performance criterion is considerably different.
DRAFT
Page 15
14
80604020020 406080
0
0.2
0.4
0.6
0.8
1
α = 2
α = 3
α = 4
α = 5
δ1
ζ [dB]
(a)
806040 2002040 60 80
4
2
0
2
4
6
α = 2
α = 3
α = 4
α = 5
ζ [dB]
Gain [dB]
(b)
Fig. 3.Optimal values for the MH protocol as a function of a relay node position ζ and different value of α. Parameters:
R∗= 2, θ = π. (a) Relative duration δ1. (b) Diversity offset gain ΛMH(1).
gain is achieved if the relay is located in the midway between the source and destination nodes with
δ1= 1/2 and thus routing data with the same information rate on each leg. This agrees with intuition
and common observation in the literature on the MH protocol (see e.g., [25]).
D. Receiver Diversity (RD) Protocol
In the RD protocol, the destination listens during both phases. The outage probability of the RD
protocol can then be expressed as
Pout= Pr[E1∩ E2] = Pr[E1]Pr[E2E1] = Pr[E1]Pr[E2].
(37)
For phaseI, we have
Pr[E1] = Pr?C(SNRS
1,HS,D) ≤ R1
?= Pr
?
HS,D2≤2R1− 1
SNRS
1
?
= 1 − e
−
A1
SNRS
1.
(38)
Also, from the result of the MH protocol, we have Pr[E2] in (26). Thus, the outage probability is given
by
1 − e−A1lRD(X)X??
Similar to the MH case, one can show that a0= 0, a1= 0, and
a2=A2
GS
∆R
where r is given in (30). The asymptotic diversity order is thus m = 2. The value of ∆Rthat minimizes
a2can be found as
??
fout(X) =
?
1 − e−
“
A1
GS+
A2
∆RGD
”
lRD(X)X?
.
(39)
1
?
1 +
r
?
(δ1+ δ2∆R)2,
(40)
∆opt=r
4
1 +8
r
δ1
δ2
− 1
?
(41)
DRAFT
Page 16
15
and the overall offset gain is given by
ΛRD(2) =
?
2GS
1 +
r
∆opt
·A0
A1
·
1
δ1+ δ2∆opt.
(42)
We now consider the following three specific cases of node geometry.
1) Relay Close to Destination: As the relay is moved near the destination, i.e., ζ → 0, we have
GS→ 1, r,∆opt→ 0. However, since limr→0r/∆opt= 0, we have
√2 ·
where the upper bound is achieved by setting δ1→ 1. Therefore, the asymptotic offset gain achieved
by this protocol, with respect to the transmit diversity bound, is 1.5 dB. Alternatively, since the receiver
diversity outperforms the transmit diversity (without CSI at the transmitter) by 3dB for the same transmit
power constraint [3], the RD protocol is asymptotically 1.5dB inferior to the twobranch receiver diversity.
It is interesting to note that this is analogous to the result in [5], where the loss of collaborative diversity
with respect to the ideal twobranch transmit diversity is shown to be 1.5dB. Since the geometrical gain
of the relay GSis unity, no MH gain is achieved in this case.
2) Relay Close to Source: If ζ is large, we have ∆opt∼ δ1/δ2and
lim
ζ→0ΛRD(2) =
2R∗− 1
2R∗/δ1− 1
1
δ1
≤
√2
(43)
ΛRD(2) ∼
A0
√2δ1δ2A1A2
?
GD.
(44)
Since GD→ 1 as the relay is moved near the destination, the maximum offset gain is achieved when
δ1= δ2= 1/2, and the corresponding gain is given by
lim
r→∞ΛRD(2) =
√2
2R∗+ 1.
(45)
Therefore, if the rate R∗is small, the loss relative to the MISO bound is small even if the relay is much
closer to source. Again, no MH gain can be achieved in this case.
3) Relay Located Midway Between the Two Nodes: In this case, let ζ ∼ 1, and we have
?
ΛRD(2)ζ=1=
2
1 + η·
A0
δ1A1+δ2
ηA2
(2 − 2cosθ)
α
4,η =
4
?
1 + 8δ1A1
δ2A2− 1
.
(46)
Therefore, in this case, the offset gain (in dB) increases linearly with α as a consequence of the MH
gain, regardless of the choice of δ1and δ2.
E. Transmit Diversity (TD) Protocol
Using the notation in the previous subsections, the outage probability can be expressed as
Pout= Pr[E1∩ E2] = Pr[ER]Pr[E1∩ E2ER] + Pr[¯ER]Pr[E1∩ E2¯ER].
The mutual information of phaseII is expressed using (13) as
(47)
C(SNRS+R
2
,P,h) = log2
?1 + SNRS+R
2
hHPh?,
(48)
DRAFT
Page 17
16
where h = [HS,D,HR,D]T, SNRS+R
phaseII provided the relay is transmitting, and P is a diagonal matrix with trace(P) ≤ 1. The total
received power with a relay power control factor ∆Ris given by
2
is the total received power (from the source and the relay) during
SNRS+R
2
= SNRS
2+ SNRR
2= (1 + ∆RGD)SNRS
1.
(49)
Since the transmitter does not know the channel coefficient, the matrix P is given by
?
22
Consequently, (48) is rewritten as
P = diag
SNRS
SNRS+R
2
,
SNRR
SNRS+R
2
?
= diag
?
1
1 + ∆RGD,
∆RGD
1 + ∆RGD
?
.
(50)
C(SNRS+R
2
,P,h) = log2
?1 + SNRS
1
?HS,D2+ ∆RGDHR,D2??.
(51)
The term Pr[E1∩ E2¯ER] in (47) is given by
Pr[E1∩ E2¯ER] = Pr[C(SNRS
1,HS,D) ≤ R1∩ C(SNRS+R
A1
SNRS
1
2
,P,h) ≤ R2]
= Pr
?
HS,D2≤
∩ HS,D2+ ∆RGDHR,D2≤
A2
SNRS
1
?
.
(52)
Let x ? HS,D2and y ? ∆RGDHR,D2. Then,
Pr[E1∩ E2¯ER] =
?min
= 1 − e
„
A1
SNRS
1,
A2
SNRS
1
«
0
?
∆RGD
∆RGD− 1e
A2
SNRS
1−x
0
e−x
1
∆RGDe−
y
∆RGDdy dx
−Amin
SNRS
1 −
−
A2
∆RGDSNRS
1
?
1 − e
−
∆RGD−1
∆RGD
Amin
SNRS
1
?
,
(53)
where Amin? min(A1,A2). The term Pr[E1∩ E2ER] in (47) is given by
Pr[E1∩ E2ER] = Pr[C(SNRS
The final expression for fout(X) can be easily found by substituting (53), (54), and (24) with
lTD(X)X into (47). We then obtain a0= 0,a1= 0, and
?
An asymptotic diversity order of m = 2 is guaranteed, and the optimum value of ∆Rcan be found as
The overall offset gain can be obtained from (19) as
?
1 +
∆opt
1 −1
We now consider the following three specific cases of node geometry.
1,HS,D) ≤ R1∩ C(SNRS
2,HS,D) ≤ R2] = 1 − e
−Amin
SNRS
1.
(54)
1
SNRS
1=
a2=A1Amin
GS
1 +
r
∆R
?
1 −1
2
Amin
A2
??
(1 + δ2∆R)2.
(55)
∆opt=r
4
?
1 −1
2
Amin
A2
?
?
?
?
?1 +8
r
1
δ2
?
1 −1
2
Amin
A2
? − 1
.
(56)
ΛTD(2) =
?
?
?
2GS
?
r
2
Amin
A2
? ·
A0
√A1Amin
·
1
1 + δ2∆opt.
(57)
DRAFT
Page 18
17
1) Relay Close to Destination: As the relay approaches destination, we have
lim
ζ→0ΛTD(2) =
√2
A0
√A1Amin
.
(58)
By choosing δ1> δ2, we have Amin= A1and thus
lim
ζ→0ΛTD(2) =
√2A0
A1
≤
√2
(59)
where the upper bound is achieved by setting δ1→ 1. Thus, the asymptotic offset gain in this case is
identical to that of the RD protocol and no MH gain is achieved.
2) Relay Close to the Source: If ζ is large, we have
lim
r→∞ΛTD(2) =
A0
?(2A2− Amin)Aminδ2
2R∗− 1
√δ2
.
(60)
By choosing δ2> δ1, we have Amin= A2and thus
lim
r→∞ΛTD(2) =
A0
√δ2A2
=
?2R∗/δ2− 1? ≤ 1
(61)
where the upper bound is achieved by setting δ2 → 1. Therefore, the TD protocol can approach the
performance of the ideal transmit diversity bound as the relay approaches the source. However, no MH
gain can be achieved in this case.
3) Relay Located in the Middle of the Two Nodes: In this case, let ζ ∼ 1, and if we set δ2> δ1, then
we have Amin= A2and
?
A1+
ΛTD(2)ζ=1,δ2>δ1=
2
1 + η1
?A1
A2
A0
δ2
2η1A2
(2 − 2cosθ)
α
4,η1=
4
?
1 + 16A1
δ2A2− 1
.
Alternatively, if we assume δ1> δ2, then we have
?
ΛTD(2)ζ=1,δ1>δ2=
2
1 + η2
A0
A1+
δ2
2η2(2A2− A1)(2 − 2cosθ)
α
4,η2=
4
?
1 + 16
A1
δ2(2A2−A1)− 1
.
Therefore, in this case, the offset gain, measured in dB, is proportional to α and the slope does not
depend on the choice of δ1and δ2.
Note that the above two equations become identical if δ1= δ2= 1/2. However, numerical calculation
in the following shows that at ζ = 1, the parameter δ2that maximizes the above gain is not equal to 1/2.
Therefore, there is a discontinuity in the optimal value of δ1around ζ = 1 (see SectionVG).
F. Simplified Transmit Diversity (STD) Protocol
In this case, we have
Pout= Pr[E2] = Pr[ER]Pr[E2ER] + Pr[¯ER]Pr[E2¯ER]
= Pr[ER]Pr[C(SNRS
2,HS,D) ≤ R2] + Pr[¯ER]Pr[C(SNRS+R
2
,P,h) ≤ R2]
(62)
DRAFT
Page 19
18
which can be calculated as
fout(X) =
?
1−e−
A1
GSlTD(X)X??
1−e−A2lTD(X)X?
+ e−
A1
GSlTD(X)X
?
1 −∆RGDe−
A2
∆RGDlTD(X)X− e−A2lTD(X)X
∆RGD− 1
?
.
We thus obtain a0= 0, a1= 0 and
a2=A1A2
GS
?
1 +1
2
r
∆R
?
(1 + δ2∆R)2.
(63)
Note that (63) is identical to (55) when Amin = A2. Therefore, when A2 ≤ A1, i.e., δ1 ≤
performance of the TD and STD protocols will be identical. The asymptotic diversity order is m = 2,
and the optimum value of ∆Rand the corresponding offset gain are given by
??
which are again identical to (56) and (57), respectively, when δ1≤ 1/2. Therefore, if the relay is closer
to the source than the destination, there is no loss by ignoring the phaseI signal (unless these two signals
are combined upon making a decision, as will be discussed in SectionVH). The offset gain achieved
by the STD and TD protocols are identical. On the other hand, if the relay is close to destination, the
performance of STD becomes inferior to that of the TD protocol. As an asymptote, we have
The above gain is maximized at δ1= 1/2 and we have
1
2, the
∆opt=r
8
1 +16
δ2r− 1
?
,ΛSTD(2) =
?
2GS
1 +
r
2∆opt
·
A0
√A1Amin
·
1
1 + δ2∆opt,
(64)
lim
ζ→0ΛSTD(2) =
√22R∗− 1
22R∗− 1=
√2
√2A0
A0
√A1A2
for δ1≥1
for δ1≤1
2
A1
2
lim
ζ→0ΛSTD(2) ≤
√2
2R∗+ 1.
It is interesting to note that this asymptotic offset gain is identical to that of RD protocol (i.e., (45))
where the relay is close to the source.
G. Numerical Comparison of the Three Protocols
In the following, we numerically evaluate the performance of the three protocols of diversity order 2,
in terms of achievable offset gains. The information rate is set to be R∗= 2.
1) Achievable Offset Gains for the RD, TD, and STD Protocols: Fig.4(a) shows the optimal δ1with
respect to the relay position ζ for the three protocols of diversity order 2, with the pathloss exponent
α = 2 and the relay location at θ = π. Changing α does not significantly affect the curves. As observed,
the optimal fraction δ1for the RD protocol ranges from 0.5 near the source node and increases as the
relay approaches the destination. Contrastingly, the optimal value of δ1for the STD protocol ranges from
0.5 near the destination node and decreases as the relay approaches the source node.
Fig.4 (b) shows the corresponding optimal offset gain Λ(2) with α = 2 and 4. It can be seen that the
offset gain of the TD protocol is identical to that of the STD protocol when ζ ≥ 0 dB, and approximates
DRAFT
Page 20
19
6040200 20 4060
0
0.2
0.4
0.6
0.8
1
RD
TD
STD
δ1
ζ [dB]
(a)
6040200204060
6
4
2
0
2
RD
TD
STD
α = 4
α = 4
α = 2
α = 2
ζ [dB]
Gain [dB]
(b)
Fig. 4.Optimal values for the three protocols as a function of a relay node position ζ. Parameters: R∗= 2, α = 2, θ = π.
(a) Relative duration δ1. (b) Diversity offset gain Λ(2). Note that the gains of the TD and STD protocols are identical and thus
their curves overlap when δ1 < 1/2.
that of the RD protocol when ζ < 0 dB. It is also observed that as the relay approaches the destination,
the offset gain of the RD and TD protocols can approach the predicted 1.5 dB bound. On the other
hand, if the relay approaches the source, the offset gain of the TD and STD protocols can approach the
predicted 0dB bound.
Consequently, in terms of minimizing complexity without sacrificing performance, the suggested
strategy is that if the relay is close to the source (ζ > 0 dB), it should employ the STD protocol
and otherwise use the RD protocol. Although the TD protocol may result in stable performance results
for both cases (which may be suitable if the exact node geometry is unknown), if the protocols are
switched appropriately, it outperforms neither.
2) Effect of Node Angle: So far, we have evaluated the performance with θ = π, which may be
optimal in the sense of the achievable diversity offset gain for a given ζ. Changing θ for a given ζ may
be expected to result in a performance loss. The SNR loss with d∗= m, relative to the case with θ = π,
can be expressed from (6) as
10log10
Λ(m)
Λ(m)θ=π
= 10log10
?
GS
GSθ=π
? 1
m
=
α
2m× 10log10
1 + ζ2− 2ζ cosθ
1 + ζ
?
α
2mL(θ).
The relative loss L(θ) [dB] defined above is plotted in Fig.5 for several instances of the geometric ratio
ζ. As observed, as ζ deviates from 1, the loss becomes small. Therefore, in many cases of interest, the
performance is not sensitive to the value of θ compared to that of ζ.
H. Comparison with TwoPhase Combining Approach
In the previous analysis, we have not exploited the fact that the information transmitted during phaseI
and phaseII are the same, and thus combining the signals of the two phases may potentially improve
performance.
DRAFT
Page 21
20
0 306090 120150180
6
5
4
3
2
1
0
ζ = ±20[dB]
ζ = ±10[dB]
ζ = ±5[dB]
ζ = 0[dB]
Relative Loss [dB]
θ [degree]
Fig. 5.Relative loss L(θ) in achievable offset gain with respect to the case with relay node position θ = π.
Therefore, in the following, we first derive an upper bound for the achievable diversity offset gain
for such a case based on the sum of mutual information method similar to [6,15] for the RD protocol.
(Similar analysis for the TD protocol may be possible, but this is not considered here for simplicity.) It
should be noted that in contrast to the previous protocols which can make use of existing channel coding
or STC and thus are attractive from a practical viewpoint, designing codes that allow combining received
codewords with different codes in an optimal manner (as this method suggests) may be challenging.
Another approach that can combine the two signals and that is much easier to implement is the use of
repetition coding [6]. In this case, the same encoding with the same information rate should be transmitted
from the relay in phaseII. In our scenario, this is possible when δ1= δ2= 1/2. We also analyze the
repetition coding in terms of diversity offset gain and then derive a bound for the variablerate case using
parallel channel coding argument.
Note that if the decision is made after combining the two phases, the decision after phaseII will be
better than that at phaseI. Hence the event that phaseI fails is a subset of the event that phaseII fails.
Hence, the outage probability is expressed as
Pout= Pr[E2] = Pr[E2¯ER]Pr[¯ER] + Pr[E1]Pr[ER]
where Pr[¯ER],Pr[ER], and Pr[E1] are identical to those of the MH protocol.
1) Parallel Channel Coding: Assuming that independent channel codes are employed for the phaseI
and phaseII, the probability of the event that the destination fails to decode conditioned that the relay
successfully decodes is given by
Pr[E2¯ER] = Pr
?
δ1log2
?
1 + SNRS
1HS,D2?
+ δ2log2
?
1 + SNRR
2HR,D2?
< R∗?
.
(65)
DRAFT
Page 22
21
By Jensen’s inequality, we obtain
Pr[E2¯ER] ≥ Pr
?
log2
?
δ1SNRS
1 + δ1SNRS
1HS,D2+ δ2SNRR
−
2HR,D2?
? Plow.
< R∗?
(66)
= 1 −δ1e
−
A0
1 − δ2GD∆Re
δ1− δ2GD∆R
A0
δ2GD∆RSNRS
1
(67)
Note that this bound becomes tight as δ1→ 1 or δ1→ 0. The outage probability is lower bounded as
Pout≥ PlowPr[¯ER] + Pr[E2ER]Pr[ER] = PlowPr[¯ER] + Pr[E1]Pr[ER].
We then have for this lower bound: a1= 0, and
a2=A2
GS
∆R
Therefore, the optimum value of ∆Ris given by (41) with r replaced by q of (68). The corresponding
offset gain is given by (42) with r replaced by q. Note that since this gain is that of the lower bound of
the outage probability, it serves as an upper bound in terms of the diversity offset gain.
Again, if ζ → 0, then we have GS→ 1 and the upper bound of the gain is expressed as
ζ→0Λ(2)UB=A0
A1
δ1
which is the same asymptotic bound of the RD protocol. On the other hand, if ζ → ∞, we obtain
∆opt= δ1/δ2and
1
?
1 +
q
?
(δ1+ δ2∆R)2,q ?
1
2δ1δ2
A2
A2
0
1
GS
GD.
(68)
lim
√21
=
2R∗− 1
2R∗/δ1− 1
√21
δ1
≤
√2
(69)
lim
ζ→∞Λ(2)UB= 1
(70)
regardless of the value of δ1. The reason that this upper bound does not depend on the value of δ1
is as follows. As the relay is located close to the source, the relay is likely to decode correctly with
high probability, and the channel links between the source and relay to the destination also becomes
equally reliable. Therefore, if we choose ∆R = ∆opt = δ1/δ2, the received energy for each phase
becomes identical, regardless of the choice of δ1and this equalenergy assignment should maximize the
mutual information for a given total received energy. Hence, this asymptotic performance should be also
equivalent to that of 2 × 1 MISO system.
Some Remarks: It is interesting to note that for the case with ζ → 0, the upper bound is 1.5dB inferior
to that of the receiver diversity bound as in (69), whereas the case with ζ → ∞ can achieve that of the
transmitter diversity bound as in (70). This is because for the latter case, due to the broadcasting nature
of the channel, the communication link between S → R is free in terms of energy and bringing δ1→ 0
cancels the loss of bandwidth efficiency required for the phaseI communication. Therefore, a virtual
transmit diversity system can be achieved without loss of efficiency. On the other hand, for the former
case, additional energy is required for the communication link between the relay and the destination,
whereas this is not required in the receiver diversity system. This accounts for the 1.5dB loss in terms
of SNR that holds regardless of the information rate.
DRAFT
Page 23
22
6040 200 204060
6
4
2
0
2
4
Repetition
Upper Bound
RD
STD
ζ [dB]
Gain [dB]
(a)
6040 200 20 4060
6
4
2
0
2
4
Repetition
Upper Bound
RD
STD
ζ [dB]
Gain [dB]
(b)
Fig. 6. Diversity offset gain of the repetition codes and parallel channel coding, as a function of a relay node position ζ.
Parameters: R∗= 2, α = 2, θ = π. (a) α = 2. (b) α = 4.
2) Repetition Coding: In this case, we set δ1= δ2= 1/2, and the received SNR is the sum of the
two phases. Thus from [6]
Pr[E2¯ER] = Pr
Consequently, we have a1= 0 and
?
log2
?
1 + SNRS
1HS,D2+ SNRR
2HR,D2?
< 2R∗?
= 1−e
−22R∗−1
SNRS
1 −GD∆Re
1 − GD∆R
−
22R∗−1
GD∆RSNRS
1
.
a2=?22R∗− 1?2?
1
GS
+
1
2∆RGD
?1
4(1 + ∆R)2.
(71)
The optimal factor ∆Rand corresponding diversity offset gain is given by
∆opt=1
8ζα??
1 + 16ζ−α− 1
?
,ΛREP=2√2GS
2R∗+ 1
1
(1 + ∆opt)
?
1 +
1
2∆optζα.
Note that as the relay node is moved closer to the source and destination, we have, respectively,
lim
ζ→∞ΛREP=
2
2R∗+ 1,lim
ζ→0ΛREP=
2√2
2R∗+ 1.
Therefore, unlike the parallel channel coding upper bound, in repetition coding the diversity offset gain
decreases rapidly as the information rate increases. This observation agrees with that in [6].
3) Numerical Results: Fig.6(a) and (b) show the diversity offset gain of the repetition codes as well
as upper bound of the parallel channel coding for pathloss exponents α = 2 and 4, respectively. Along
with these, those of the STD and RD protocols are also shown for comparison. As we can observe, if
the relay is close to the source or destination, the STD and RD protocols can approach the upper bound
of parallel channel coding. On the other hand, if the relay is located midway between the source and the
destination, the repetition coding is better. Therefore, selecting between the STD/RD/repetition protocols,
depending on the relay’s location, is an inexpensive yet powerful approach in practical design.
DRAFT
Page 24
23
VI. MULTIPLE RELAYS AND THE EFFECT OF IMPERFECT STCS ON DIVERSITY PERFORMANCE
One can extend the above ideas to the multiple relay node case. Suppose that we have N relays. For the
RD protocol, each relay must forward its data using dedicated channels, and N additional channels may
be required. On the other hand, for the TD and STD protocols, as discussed in [6], a single additional
channel suffices. For this reason, the TD and STD protocols are attractive especially when N is large.
In this section, we focus on the performance of the TD and STD protocols. In the previous analysis
of these protocols, it was assumed that even if only a subset of the transmitting nodes during phaseII
actually transmit, the destination can decode correctly provided the mutual information given this fact is
above the required rate. This implicitly assumes that the destination always knows which relay nodes are
transmitting and which are not, or equivalently, which antennas are undergoing an erasure on a frame by
frame basis. (In our scenario, a node erasure is a probabilistic event that depends on the channel link(s)
of phaseI. However, this model is also applicable to a sudden change of node status, such as a battery
failure, node failure or a change in shadowing state.) Therefore, in a practical STC system, unless the
STC is appropriately designed, the destination node may fail to correctly identify a node erasure and
thus cause a decoding error6. The effect of the node erasure may become salient, especially when the
number of relay nodes increases. Therefore, we consider the achievable diversity order of these protocols
when the STC is perfect and imperfect. Note that conventional coherent STCs such as [3,4] require the
knowledge of CSI at the receiver and thus are not necessarily perfect. On the other hand, it is easy to
see that noncoherent versions of STCs that do not require any CSI, such as [32], are perfect by nature.
A. System Model and Outage Probability
Fig.7 summarizes the system model with two relays and the associated notation. The case of three or
more relays is analogous. Following the single relay case in the previous sections, we assume that HS,Ri
and HRi,D, which denote the complex channel coefficients of each channel link, are uncorrelated and
circularly symmetric complex Gaussian random variables with zero mean and unit variance. Let GS,iand
GD,idenote the corresponding geometrical gains achieved by the ith node Ri, where i = 1,2,...,N.
Also, let ER,idenote the event that the node Rifails to decode after phaseI. For brevity we use notation
as F(1)
ii
=¯ER,i(where Fistands for the failure of ith relay).
= ER,iand F(0)
6In practice, if channel estimation is performed on a frame by frame basis, a node erasure can be easily detected at the
destination. If the relay ceases transmitting a signal for any reason then the destination will simply assume that the corresponding
channel link suffers severe fading. Conventional STCs can thus be used without modification. However, since channel fading is
slow by assumption and this generally precludes the use of short interval channel estimation, per frame channel estimation is
expensive. Furthermore, the additional overhead becomes substantial as the number of relay nodes increases.
DRAFT
Page 25
24
SD
HS,D
Ri
θi
HS,Ri
HRi,D
GS,i
GD,i
Rj
θj
HS,Rj
HRj,D
GS,j
GD,j
Fig. 7. Twophase communication with multiple relays.
In the case of the TD protocol, the outage probability is expressed as
?N
?
The above summation is the sum of 2Nterms of the product of N + 1 probability events. Consider the
specific term in which exactly n out of N relay nodes have correctly decoded the message and suppose
that their indices are i = 1,2,...,n. Then this term can be expressed as
Pout=
?
(x1,x2,...,xN)∈{0,1}N
?
i=1
Pr
?
F(xi)
i
??
Pr
?
E1∩ E2F(x1)
??
1
,F(x2)
2
,...,F(xN)
N
?
?
?Bx1x2···xN
.
(72)
B(0)n(1)N−n=
n
?
i=1
e−
A1
GS,ilTD(X)X
N
?
i=n+1
?
1 − e−
A1
GS,ilTD(X)X?
Pr
?
E1∩ E2F(0)
1,...,F(0)
n,F(1)
n+1,...,F(1)
N
?
,
where lTD= 1 + δ2
node. Note that lTD(X) is analytic about X = 0 with lTD(0) = 1+δ2
for the STD protocol can be expressed in a similar form.
?N
i=1βi∆R,i, βi= Pr[F(0)
i], and ∆R,iis the power control factor of the ith relay
?N
i=1∆R,i. The outage probability
B. Diversity Order for Multiple Relays with Perfect and Imperfect Constituent STC
We first assume that there are N collaborating relays and the STC used in the TD or STD protocol is
perfect. In this case, we have the following lemma [6].
Lemma 2 (Asymptotic Diversity Order with Multiple Relay Nodes): For the N relay node TD and
STD protocols with perfect constituent STCs, the asymptotic diversity order is d∗= N + 1.
The proof is omitted as it can be inferred from [6]. Now, as a worst case scenario, we assume that
an outage event occurs even if a single relay fails to decode. In other words, unless all the relay nodes
correctly decode the message, the phaseII link fails. Specifically, we assume that
Pr
?
E2...,F(1)
i,...
?
= 1
for any i.
(73)
DRAFT
Page 26
25
In the following discussion, the STC having this property will be referred to as an imperfect STC. The
outage performance based on this assumption may serve as an upper bound for the TD and STD protocols.
In this case we have the following theorem:
Theorem 2: For the N relay node TD and STD protocols with imperfect constituent STCs, i.e., (73)
holds, the asymptotic diversity orders are 2 and 1, respectively.
Proof: Considering the outage probability expression in (72), it is easy to see that the worstcase
terms are those with n = N − 1. Then, we have for the TD protocol
N−1
?
N−1
?
where the last equality is from Pr
E1∩ E2F(0)
1. Therefore, by Corollary1 we obtain d∗(Pout) = d∗?B(0)N−1(1)1
For the STD protocol, we have
B(0)N−1(1)1=
i=1
e
−
A1
GS,iSNRS
1
?
?
1 − e
−
A1
GS,NSNRS
1
?
??
Pr
?
E1∩ E2F(0)
1,...,F(0)
N−1,F(1)
N
?
=
i=1
e
−
A1
GS,iSNRS
1
1 − e
−
A1
GS,NSNRS
1
1 − e
−
A1
SNRS
1
?
,
(74)
?
1,...,F(0)
N−1,F(1)
N
?
= Pr
?= 2.
?
E2E1,F(0)
1,...,F(0)
N−1,F(1)
N
?
=
B(0)N−1(1)1=
N−1
?
i=1
e
−
A1
GS,iSNRS
1
?
1 − e
−
A1
GS,NSNRS
1
?
,
(75)
and thus d∗(Pout) = d∗?B(0)N−1(1)1
Therefore, in the high SNR regime, the asymptotic diversity orders of the TD and STD protocols with
imperfect STC are d∗= 2 and d∗= 1, respectively. This suggests that in the high SNR regime, it is
important that collaborative STCs for distributed nodes be designed such that the information can be
decoded with only a partial subset of the code (i.e., robustness against transmit antenna erasures in the
traditional multipleantenna STC scenario). Practical design issues in this direction are addressed in [33].
?= 1.
C. Outage Probabilities and Diversity Offset Gains for Single Relay Node Case
In the case of a single relay with imperfect STC, it is straightforward to obtain the outage probabilities
and their associated diversity offset gain expression based on the approach outlined in SectionV.
1) TD Protocol: From Theorem 2, it follows that the diversity order is 2. The corresponding outage
probability is given by
Pout= Pr[ER]Pr[E1] + Pr[¯ER]Pr[E1∩ E2¯ER].
(76)
The closedform expression can be found by using (38), (24), and (53). It follows that a0,a1= 0 and
a2=A2
GS
∆R
2
1
?
1 +
r
?
1 −1
Amin
A2
?Amin
A1
?
(1 + δ2∆R)2.
(77)
DRAFT
Page 27
26
It is observed that a2is similar to (55). In fact, if δ1> 1/2 and thus Amin= A1, the offset gain of the
TD protocol with imperfect STC, denoted by ΛIMTD(2), is equivalent to ΛTD(2). On the other hand, if
Amin= A2, the gap becomes (assuming the same ∆optis applied)
?
ΛTD(2)/ΛIMTD(2) =
A1
?
1 +A2
A1
r
2∆opt
?
/A2
?
1 +
r
2∆opt
?
,
(78)
which is significantly large if A1≫ A2(and thus δ1≪1
2) STD Protocol: In this case, we have
2).
Pout= 1 − Pr[¯ER∩¯E2] = 1 − Pr?C(GSSNRS
1,HS,R) > R1
?Pr?C(SNRS+R
2
,P,h) > R2
?,
(79)
fout(X) = 1 − e−
A1
GSlTD(X)X
?
∆RGDe−
A2
∆RGDlTD(X)X− e−A2lTD(X)X
∆RGD− 1
?
.
(80)
Consequently, we obtain a0= 0,
a1=A1
GS
(1 + δ2∆R)
and
a2=A1A2
2GS
?
r
∆R
−A1
A2
1
GS
?
(1 + δ2∆R)2.
(81)
Therefore, the asymptotic diversity order is 1, which agrees with Theorem 2. However, it should be noted
that the offset gain with respect to the SISO system is given by
ΛIMSTD(1) =
A0GS
A1(1 + δ2∆R)
(82)
and this indicates that if GSis large, one may still achieve significant gain over the baseline system. In
particular, for the SNR region where SNRt≪ ΛIMSTD(1) (X ≫ 1/ΛIMSTD(1)), the outage probability
has a local slope of order 2 since the term a1X in fout(X) is dominated by a2X2. The following section
elucidates this effect numerically.
3) Numerical Results: We numerically compare the performance of the two protocols with imperfect
and perfect STC in terms of outage probability. Fig.8 (a) and (b) show the outage probabilities of these
protocols with relay node locations ζ = 20 and 20dB, respectively. The performance of the RD protocols
is also shown as a reference. In these results, it is assumed that the relay performs the optimal power
control algorithm.
From Fig.8 (a), it is observed that the two protocols with imperfect STC are almost identical in the
low SNR region with a local slope of (diversity) order 2, but for high SNR, the bound for the STD
protocol shows a slope of order 1, whereas that of the TD protocol maintains a slope of diversity order
2. The gap between the two bounds becomes noticeable in Fig.8 (b), where the outage probability of
the TD protocol with an imperfect STC is identical to that of the ideal TD, whereas the STD protocol
with an imperfect STC is much worse than the ideal SISO bound. Therefore, if the STC is designed
imperfectly, then the use of the TD protocol can offer stable performance and is thus preferable.
DRAFT
Page 28
27
1020
SNRt[dB]
3040
105
104
103
102
101
100
MISO Bound
RD
TD
STD
SISO
TD/STD
IMTD/IMSTD
2x1 MISO
RD
Outage Probability
(a)
1020
SNRt[dB]
3040
105
104
103
102
101
100
MIMO Bound
RD
TD
STD
IMSTD
SISO
2x1 MISO
STD
RD
1x2 SIMO
TD/IMTD
Outage Probability
(b)
Fig. 8.Outage probability of the TD and STD protocols with perfect or imperfect STC. The result of the RD protocol and
associated MISO/SIMO bounds are also shown. Parameters: R∗= 2, α = 2, θ = π. (a) ζ = 20 dB. (b) ζ = −20 dB.
VII. CONCLUSION
We have analyzed the performance of various variablerate twophase collaborative diversity protocols
for wireless networks. These protocols can be implemented in a straightforward manner using standard
variablerate channel coding and STC. Theoretical analysis of the outage probability has shown that
these protocols, if properly designed based on the node geometry, can achieve full diversity order and
considerable offset gains. Our conclusion is that if the relay is close to the source and destination, the
STD and RD protocols, respectively, achieve good performance. If the relay is midway between the
source and the destination, fixedrate repetition coding with signal combining at the destination [6] is a
good candidate considering its simplicity of implementation.
It is also shown that for a system with N relays, a diversity order of N + 1 is achievable for the TD
based protocol using STC as in [6]. However, if the STC fails to be decoded whenever node erasure
occurs, their diversity order is considerably reduced and for the STD protocol with an imperfect STC, no
diversity offset gain can be achieved. Therefore, the design of STCs that are robust against node erasures
is an important area of future research.
Finally, even though perfect synchronizations are assumed throughout the paper, accurate timing and
frequency acquisitions among distributed nodes are difficult to achieve in practice. Further research in
this direction is of critical importance for implementation of these protocols.
ACKNOWLEDGMENT
The authors wish to thank the reviewers for their detailed comments and suggestions.
REFERENCES
[1] E. Malkam¨ aki and H. Leib, “Coded diversity on blockfading channels,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp.
771–782, Mar. 1999.
DRAFT
Page 29
28
[2] R. Knopp and P. A. Humblet, “On coding for block fading channels,” IEEE Trans. Inform. Theory, vol. 46, no. 1, pp.
189–205, Jan. 2000.
[3] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun.,
vol. 16, no. 8, pp. 1451–1458, Oct. 1998.
[4] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Spacetime codes for high data rate wireless communication: Performance
criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998.
[5] J. N. Laneman, D. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage
behavior,” IEEE Trans. Inform. Theory, vol. 50, pp. 3062–3080, Dec. 2004.
[6] J. N. Laneman and G. W. Wornell, “Distributed spacetimecoded protocols for exploiting cooperative diversity in wireless
networks,” IEEE Trans. Inform. Theory, vol. 49, pp. 2415–2425, Oct. 2003.
[7] M. Gastpar and M. Vetterli, “On the capacity of wireless networks: The relay case,” in Proc. IEEE INFOCOM 2002, 2002,
pp. 1577–1586.
[8] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity – Part I: System description,” IEEE Trans. Commun.,
vol. 51, pp. 1927–1938, Nov. 2003.
[9] M. Janani, A. Hedayat, T. E. Hunter, and A. Nosratinia, “Coded cooperation in wireless communications: spacetime
transmission and iterative decoding,” IEEE Trans. Signal Processing, vol. 52, pp. 362–371, Feb. 2004.
[10] M. A. Khojastepour, A. Sabharwal, and B. Aazhang, “On the capacity of ‘cheap’ relay networks,” in Proc. Conference on
Information Sciences and Systems (CISS 2003), Mar. 2003.
[11] M. Katz and S. Shamai, “Transmitting to colocated users in wireless adhoc and sensory networks,” IEEE Trans. Inform.
Theory, pp. 3540–3563, Oct. 2005.
[12] Z. Dawy, “Relay regions for the general Gaussian relay channel,” in Proc. Winter School on Coding and Information
Theory, Feb. 2003.
[13] T. E. Hunter, S. Sanayei, and A. Nosratinia, “Outage analysis of coded cooperation,” IEEE Trans. Inform. Theory, vol. 52,
no. 2, pp. 375–391, Feb. 2006.
[14] M. Yuksel and E. Erkip, “Diversity gains and clustering in wireless relaying,” in Proc. IEEE International Symposium on
Information Theory, Chicago, IL, June 2004, p. 400.
[15] P. Mitran, H. Ochiai, and V. Tarokh, “Spacetime diversity enhancements using collaborative communications,” IEEE Trans.
Inform. Theory, vol. 51, pp. 2041–2057, June 2005.
[16] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversitymultiplexing tradeoff in halfduplex cooperative
channels,” IEEE Trans. Inform. Theory, pp. 4152–4172, Dec. 2005.
[17] T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed.PrenticeHall, 2001.
[18] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multipleantenna channels,” IEEE
Trans. Inform. Theory, vol. 49, pp. 1073–1096, May 2003.
[19] E. Zimmermann, P. Herhold, and G. Fettweis, “A novel protocol for cooperative diversity in wireless networks,” in Proc.
5th European Wireless Conference, Feb. 2004.
[20] J. N. Laneman, “Network coding gain of cooperative diversity,” in Proc. IEEE MILCOM’04, 2004.
[21] T. E. Hunter and A. Nosratinia, “Performance analysis of coded cooperation diversity,” in Proc. IEEE International
Conference on Communications (ICC’03), May 2003, pp. 2688–2692.
[22] S. Sesia, G. Caire, and G. Vivier, “Incremental redundancy hybrid ARQ schemes based on lowdensity paritycheck codes,”
IEEE Trans. Commun., vol. 52, pp. 1311–1321, Aug. 2004.
DRAFT
Page 30
29
[23] G. Caire, D. Tuninetti, and S. Verd´ u, “Variablerate coding for slowly fading Gaussian multipleaccess channels,” IEEE
Trans. Inform. Theory, vol. 50, pp. 2271–2291, Oct. 2004.
[24] R. U. Nabar, H. B¨ olcskei, and F. W. Kneub¨ uhler, “Fading relay channels: Performance limits and spacetime signal design,”
IEEE J. Select. Areas Commun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004.
[25] P. Chen, B. O’Dea, and E. Callaway, “Energy efficient system design with optimum transmission range for wireless ad
hoc networks,” in Proc. IEEE International Conference on Communications (ICC’02), Apr. 2002, pp. 945–952.
[26] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacity theorems for relay networks,” IEEE Trans.
Inform. Theory, vol. 51, pp. 3037–3063, Nov. 2005.
[27] L. H. Ozarow, S. Shamai, and A. D. Wyner, “Information theoretic considerations for cellular mobile radio,” IEEE Trans.
Veh. Technol., vol. 43, no. 2, pp. 359–378, May 1994.
[28] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple
antennas,” Wireless Personal Communications, vol. 6, pp. 311–335, 1998.
[29] I. E. Telatar, “Capacity of multiantenna Gaussian channels,” European Transactions on Telecommunications, vol. 10, no. 6,
pp. 585–595, Nov./Dec. 1999.
[30] J. N. Laneman, “Limiting analysis of outage probabilities for diversity schemes in fading channels,” in Proc. IEEE
GLOBECOM’03, San Francisco, CA, 2003.
[31] R. Min and A. Chandrakasan, “Top five myths about the energy consumption of wireless communication,” ACM
SIGMOBILE Mobile Computing and Communications Review, vol. 7, no. 1, pp. 65–67, Jan. 2003.
[32] V. Tarokh and H. Jafarkhani, “Differential detection scheme for transmit diversity,” IEEE J. Select. Areas Commun., vol. 18,
no. 7, pp. 1169–1174, July 2000.
[33] P. Maurer and V. Tarokh, “Transmit diversity when the receiver does not know the number of transmit antennas,” in Proc.
WPMC’01, 2001, pp. 2688–2692.
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