An Optimized User Selection Method for Cooperative Diversity Systems.
ABSTRACT Multiuser cooperative diversity is a recent techni que promising great improvement of the performance of wireless communication systems operating in fading environments. Based on combinatorial optimization theory and specifically on the socalled knapsack problem, this paper presents a method of optimizing the selection among the potential cooperating users, when amplifyandforward relays are used. In particular, two optimization problems are studied: the error probability mini mization subject to total energy consumption constraints, and the dual one, the energy consumption minimization under error performance constraints. Depending on the frequency of repea ting this selection, the above problems are categorized into short term and longterm node selection. Numerical examples verify the expected knapsack scheme's advantage of adapting the number of cooperating users, depending on the desired performance consumption tradeoff. Moreover, longterm node selection seems to lead to similar error or consumption performance compared to the shortterm one, despite its simplicity. I. INTRODUCTION In the last few years, a new concept that is being acti vely studied in multihopaugmented networks is multiuser cooperative diversity, where several terminals form a kind of coalition to assist each other with the transmission of their messages. In general, cooperative relaying systems have a source node multicasting a message to a number of cooperative relays, which in turn resend a processed version to the intended destination node. The destination node combines the signal received from the relays, taking into account the source's original signal (1) (4).

Conference Paper: Performance analysis of partial relay selection networks with multipleantenna destinations
[Show abstract] [Hide abstract]
ABSTRACT: The performance of dualhop cooperative relay networks with partial relay selection is studied. Specifically, the system model consists of a singleantenna source, singleantenna L relays and a multipleantenna destination. Both fixedgain amplifyand forward (FGAF) and decodeandforward (DF) relays are treated. The best relay is selected based on instantaneous and partial knowledge of the sourcetorelay channel state information. The cumulative distribution function of the endtoend signaltonoise ratio (SNR) is derived in closed form and used to obtain the exact outage probability and the average symbol error rate. These performance metrics are used to compare the performance of FGAF and DF relays against the channelassisted amplifyandforward (CAAF) relays. The analysis is verified throughly by using MonteCarlo simulation results.Information and Automation for Sustainability (ICIAFs), 2010 5th International Conference on; 01/2010  [Show abstract] [Hide abstract]
ABSTRACT: In the conventional dualhop cooperative communications, the relays with imbalanced channel gain to the source and the destination cannot be efficiently utilized since one of these two hops can become the bottleneck of the overall transmission. In this paper, we propose a clusterbased cooperative communication (CBCC) scheme to address this problem. In this scheme, relays are divided into two clusters, and the cooperative transmissions consist of three stages. In this way, the bottleneck between the relay and the source or the destination can be potentially broken through the relaytorelay transmissions. We focus on the relay selection problem for this clusterbased cooperation scheme, which turns out to be difficult due to the coupling of the relays between two clusters. We then resort to the design of heuristic algorithm and provide simulation results to show the performance of the proposed scheme.Communications in China (ICCC), 2012 1st IEEE International Conference on; 01/2012 
Conference Paper: Distributed relay selection for virtual MIMO in spectral efficient broadcasting networks
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ABSTRACT: Virtual multipleinput multipleoutput (VMIMO) enables the implementation of conventional MIMO on mobile devices equipped with insufficient numbers of antennas via cooperation. This paper considers a spectral efficient broadcasting network in which selected mobile devices form a VMIMO system to relay the broadcasted data to help other devices decode the source data more reliably. In particular, the relay selection problem, a fundamental issue in the construction of VMIMO, is examined. We first review existing selection schemes for users operating in the amplifyandforward (AF) mode. We then propose a distributed selection scheme based on postprocessing SNR. In the proposed scheme, each user individually finds the most favorable candidates for VMIMO construction and then all users obtain a joint decision through a voting process. Simulation results show that the proposed distributed scheme outperforms existing distributed selection schemes and achieves a nearoptimal performance with lower complexity compared to the centralized scheme.Wireless Communications and Networking Conference (WCNC), 2013 IEEE; 01/2013
Page 1
An Optimized User Selection Method for
Cooperative Diversity Systems
Diomidis S. Michalopoulos∗§, George K. Karagiannidis∗, Theodoros A. Tsiftsis†and Ranjan K. Mallik‡
∗Telecommunications Division, Department of Electrical & Computer Engineering, Aristotle University of Thessaloniki, GR54124, Thessaloniki, Greece
Email:{dmixalo, geokarag}@auth.gr
†Wireless Telecommunications Laboratory, Department of Electrical & Computer Engineering, University of Patras, Rion, GR26500, Patras, Greece
Email: tsiftsis@ee.upatras.gr
‡Department of Electrical Engineering, Indian Institute of Technology  Delhi, Haus Khas, New Delhi 110016, India (email: rkmallik@ee.iitd.ernet.in)
§The work of Mr Michalopoulos is supported by the Greek General Secretariat of Research and Technology under PENED’04
Abstract—Multiuser cooperative diversity is a recent techni
que promising great improvement of the performance of wireless
communication systems operating in fading environments. Based
on combinatorial optimization theory and specifically on the
socalled knapsack problem, this paper presents a method of
optimizing the selection among the potential cooperating users,
when amplifyandforward relays are used. In particular, two
optimization problems are studied: the error probability mini
mization subject to total energy consumption constraints, and
the dual one, the energy consumption minimization under error
performance constraints. Depending on the frequency of repea
ting this selection, the above problems are categorized into short
term and longterm node selection. Numerical examples verify the
expected knapsack scheme’s advantage of adapting the number
of cooperating users, depending on the desired performance
consumption tradeoff. Moreover, longterm node selection seems
to lead to similar error or consumption performance compared
to the shortterm one, despite its simplicity.
I. INTRODUCTION
In the last few years, a new concept that is being acti
vely studied in multihopaugmented networks is multiuser
cooperative diversity, where several terminals form a kind of
coalition to assist each other with the transmission of their
messages. In general, cooperative relaying systems have a
source node multicasting a message to a number of cooperative
relays, which in turn resend a processed version to the intended
destination node. The destination node combines the signal
received from the relays, taking into account the source’s
original signal [1] [4].
The main contribution of this paper is the proposal of
a novel nodeselection strategy for multiuser cooperative
diversity systems, according to which only a subset of the set
of available users is activated, in order to achieve the optimal
balance between error performance and total consumed energy.
More specifically, two variations of this problem are intro
duced: the endtoend error performance optimization under
total power constraints, and the dual one, the minimization
of the total consumed energy provided that the endtoend
error probability will not exceed a predefined threshold. This
is accomplished by utilizing the general concept of optimizing
the selection among the elements of a given set under specific
constraints, which was first introduced in combinatorial opti
mization theory: Given an item set, and assuming that each
item is characterized by a unique pair of profit and weight va
lues, the subset that maximizes the profit summation provided
that the weight summation does not exceed a maximum value
needs to be distinguished. These problems are widely known
as knapsack problems [5].
In this work, dualhop amplifyandforward cooperative
diversity is assumed, where a number of individual relays
amplify and retransmit the signal received from the source
node to the destination one. The structure of the studied model
is further analyzed in Section II; in Section III, the knapsack
application on cooperative diversity systems is presented. In
such applications, the gain and the energy each relay consumes
depends usually only on the fading state of its input channel,
in order to limit the output power of the relay. Moreover,
assuming maximal ratio combining (MRC) at the receiver,
the instantaneous1overall signal to noise ratio (SNR) is the
sum of the endtoend SNRs corresponding to each separate
branch. These endtoend SNRs, reflect the branches’ ability to
contribute to the total performance enhancement, and are also
determined by a combination of the fading state of the chan
nels in the input and the output of the corresponding relays. We
thus realize that two scalar metrics can be attributed to each
branch, depending only on the fading conditions corresponding
to it, and characterizing its contribution to the endtoend
performance and its energy consumption respectively. Then,
the error performance optimization problem under total energy
consumption constraints, and the dual problem of minimizing
the energy consumption provided that the received SNR is
not arbitrarily small, can be reduced to knapsack ones. This
requires that the selection is repeated on a frequent basis, at
a rate that ensures constant fading characteristics in every
selection interval. Next, this type of selection is referred as
shortterm node selection (STNS), and is further analyzed in
Section IV.
In Section V, a solution that tackles the above problems
in an average sense is presented. More specifically, since all
channels are considered ergodic, the metrics that are attributed
to the corresponding relays are determined by their fading
1In the following, the term instantaneous will be used in a loose sense,
describing the time interval in which the fading state can be considered
constant.
Page 2
statistics, and the selection problems are again reduced to
knapsack ones, optimizing the system in a longterm point of
view. For this reason, this method is referred as longterm node
selection (LTNS). Finally, in Section VI a set of numerical
examples are discussed.
II. SYSTEM AND CHANNEL MODEL
We consider a system with a source node S, communicating
with a destination node D, with the aid of L other cooperating
nodes which act as nonregenerative relays, denoted by Rj,
j = 1,...,L. Let bjstand for the system branch corresponding
to the relay Rj. Each transmission period T is divided into two
slots: in the first slot, node S communicates with the relays
and the destination terminal, while in the second slot, only
the relays communicate with node D. The above transmission
protocol was originally proposed in [6].
Let us assume that the system is operating over independent
but not necessarily identically distributed Nakagamim fading
channels, node S transmits with instantaneous power norma
lized to unity and the gain Gjof Rjis set to counterbalance
the signal degradation in the hop SRj in order to limit its
output power [3] i.e.,
G2
j(t) =
Pj,out
Sj(t) + N0,
a2
(1)
where Pj,out is the relay’s transmission power, and aSj(t),
N0are the fading amplitude and the additive white Gaussian
noise (AWGN) power of the SRj channel, respectively. In
such case, the instantaneous SNR γj, of the bjbranch is
γj=
γSjγDj
γSj+ γDj+ 1,
(2)
where γSj= a2
neous SNR of the hop SRjand RjD, respectively. Assuming
that l nodes are operating during any time interval T, the
overall instantaneous SNR at the output of the MRC at the
destination terminal during this interval can be written as
Sj/N0, γDj= Pj,outa2
Dj/N0are the instanta
γend= γ0+
l?
j=1
γSjγDj
γSj+ γDj+ 1=
l?
j=0
γj,
(3)
where γ0is the instantaneous SNR of the direct SD channel.
The system branch corresponding to the SD channel is
denoted by b0.
III. THE KNAPSACK PROBLEM AND ITS APPLICATION ON
COOPERATIVE DIVERSITY SYSTEMS
A. The Knapsack Problem
1) General Description: The wellknown zeroone knap
sack problem is defined as follows:
Given an item set N, consisting of n items with profits
pj > 0 and weights wj > 0,
capacity value Cmax, select the subset of N such that the
j = 1,...,n, and given the
total profit of the selected items is maximized and the total
weight does not exceed Cmax. In other words,
maximize
n
?
j=1
pjxj
subject to
n
?
j=1
wjxj≤ Cmax
(4)
xj∈ {0,1},j = 1,...,n.
A variation of this problem can be expressed by minimi
zing, instead of maximizing, the profit summation, under the
constraint that the total weight is greater than or equal to a
given value Cmin, i.e.,
minimize
n
?
j=1
pjyj
subject to
n
?
j=1
wjyj≥ Cmin
(5)
yj∈ {0,1},j = 1,...,n.
In what follows, we refer to the problems having the form of
(4) as traditional knapsack problems, and to the ones with the
form of (5) as minimization knapsack problems. Observing
these two versions, it is clear that although their aim is
completely different, their original concept is identical, stating
that a separation has to be made among the items, in terms
of their profits and weights combination. For this reason, both
these kind of problems are referred as knapsack ones, and the
algorithms that tackle them are similar [5].
2) Efficient Knapsack Algorithms: There is very strong
theoretical evidence that for the knapsack problem no polyno
mial time algorithm exists for computing its optimal solution
[5]. Within the general approach of knapsack problems, many
algorithms are known that come close to an optimal solution,
so as the difference between the values of the optimal and the
approximate solution is small. Thus, due to the fact that algo
rithms with high complexity are undesirable for our system,
we focus on algorithms computing a suboptimal solution in
a relatively short amount of time. One efficient suboptimal
algorithm is the wellknown Greedy one, which operates as
follows:
Algorithm 1 (Traditional Knapsack problem): “For
item j ∈ N, denote with ejthe profit to weight ratio, which
is also called the efficiency of this item, e.g.,
ej:=pj
every
wj.
(6)
Sort the items in decreasing order of efficiency, and then start
with an empty knapsack and simply go through the items
in this order adding every item under consideration into the
knapsack, if the capacity constraint is not violated thereby. ”
Algorithm 2 (Minimization Knapsack problem): “For
every item j ∈ N, denote with ej the weight to profit ratio
e.g.,
ej:=wj
pj.
(7)
Page 3
Sort the items in decreasing order of efficiency, and then start
with an empty knapsack and simply go through the items
in this order, adding every item into the knapsack unless
?L
B. Utilizing the Knapsack Approach in Cooperative Diversity
Systems
In cooperative diversity systems, the set N can be regarded
as the set of all available system branches, including b0. This
branchset is denoted by R, e.g.,
R ={b0,b1,...,bL}.
The problems that arise now are which branches to activate (if
possible) during specific time intervals, in order to optimize
system endtoend error performance under average power
constraints, or to minimize the total consumed energy provided
that the error probability will not exceed a given threshold. In
the following sections, we prove that these problems can be
formulated as traditional and minimization knapsack problems
respectively, by substituting the parameters of (4) and (5) with
the appropriate system quantities.
1) The BranchSet R: In general, R represents the set of
all the available for cooperation branches; a node is considered
available if it satisfies a number of relaying conditions, which
reflect its possibility and willingness to act as a relay. These
conditions may depend generally on the node’s energy deposit,
on its ability to forward the information in an adequate rate, on
the synchronization cost its activation may entail etc. However,
further analysis of these relaying conditions, followed by the
question of under which circumstances relaying is a worthy
technique, are beyond the scope of this paper, and are left for
future work.
2) Cooperative Diversity’s Special Knapsack Features:
Considering that, in general, the coefficients pj,wjcorrespond
to a performance and energy consumption metric respectively,
the following are the main points in which the branch selec
tion in cooperative diversity systems differs from the typical
knapsack applications:
• In cooperative diversity systems, the total weight ca
pacity does not represent a strictly fixed value with a
physical sense, as it occurs in the majority of knapsack
applications. On the contrary, it reflects the concept of
limiting the total number of cooperating nodes and the
extra energy consumed to only a single user’s avail.
• The amount of time needed for the algorithm computation
in cooperative diversity systems is very important, since
it has to be small enough in order not to cause any
significant delay in packet transmission.
• In cooperative diversity systems, the direct branch b0
is always activated since this does not entail any extra
consumed energy.
Thus, it is evident that the Greedy algorithm tackles, in a
relatively adequate amount of time, the problem of selecting
the branches with optimal compromise between contribution to
the total performance and energy consumption, provided that
the total energy consumed for a single user’s communication is
j=1wjxj≥ Cminis satisfied.”
(8)
limited. For this reason, Algorithm 1 and Algorithm 2, slightly
modified in order to always include b0, are regarded as a very
good solution in the branch selection problem.
IV. SHORTTERM NODE SELECTION (STNS)
Let Tc be the channel coherence time. Assume that each
relay Rjhas full knowledge of the CSI of the SRjand Rj
D hop, and that the relays are capable of informing the source
node about their channel state conditions, in a time much
shorter than Tc. In slow fading environments, these conditions
do not significantly change within time intervals approximately
equal to Tc. Therefore, the transmitter can decide every
Tcwhich of the relays will operate for the next interval Tc, in
order to minimize the outage probability Po2under total power
constraints, or to minimize the total consumed energy provided
that Po will not exceed a predefined value, for this interval.
Hence, STNS implies that this selection will be repeated
frequently enough, ensuring that the fading conditions of each
branch will remain constant until the next selection. This
requires advanced cooperation protocols and efficient selection
algorithms, with relatively small computation time compared
to the duration of the transmitted symbols.
A. PoMinimization Under Energy Consumption Constraints
Since γendat the output of the MRC at terminal D remains
constant during Tc, the nodeselection that minimizes Pois the
selection that maximizes γend, or equivalently the selection
that maximizes?l
Ej= (Pj,out− Pj,in)+,
where Pj,inis the signal power at the input of Rjand (·)+=
max(·,0). Eq. (9) reflects the fact that, in the case when Gj>
1 the consumed power equals the difference between Pj,out
and Pj,in, whereas in the case when Gj ≤ 1 the consumed
power is zero, since such attenuation can be achieved using
a simple voltage divider. In each case, however, the energy
consumed for any signal processing at the relays is considered
negligible. Thus, Ejcan be expressed as
??
G2
or by using (1) as
Ej=?Pj,out− a2
Considering the above, the nodeselection problem can be
formulated as a traditional knapsack one, i.e.,
j=1γj, for this interval. The energy Ejthat
the relay Rjconsumes per unit time equals to
(9)
Ej=1 −
1
j
?
Pj,out
?+
,
(10)
Sj− N0
?+.
(11)
maximize
L
?
j=0
γjxj
subject to
L
?
j=1
Ejxj≤ Cmax
(12)
xj∈ {0,1},j = 0,1,...,L.
2As outage probability Po we define the probability that the instantaneous
bit error probability exceeds a specified value [7, p. 5].
Page 4
where γj and Ej are given in (2) and (11), respectively. The
value of Cmaxin the last equation represents an instantaneous
consumed power constraint, which can be adjusted in order to
achieve the desirable tradeoff between performance and power
consumption. Recall that the L branches used in (12) are those
defined by the set R.
B. Total Energy Consumption Minimization Under Po Con
straints
Another application of the knapsack theory into cooperative
diversity systems is the minimization of the total consumed
energy, provided that the outage probability Powill not exceed
a predefined value δ (if possible3), or equivalently that the
γend is lowerbounded, for a time interval Tc. The problem
thus reduces to a minimization knapsack one, having the form
of
minimize
L
?
j=1
Ejxj
subject to
L
?
j=0
γjxj≥ Cmin
(13)
xj∈ {0,1},j = 0,1,...,L.
As in (12), the value of Cmin in (13) can be seen as a
measure of the upper bound δ of Po, taking into account the
desirable performancepower consumption tradeoff. In a BPSK
application, for example, Cminis related with δ through
erfc(Cmin) = 2δ,
(14)
with erfc(·) being the wellknown complementary error func
tion.
From a longterm point of view, in both these systems, after
a large number of selections the average bit error probability
(ABEP) is either minimized or upperbounded by a constant
C1, while the average total consumed energy is either upper
bounded by a constant C2 or minimized, respectively. The
values C1and C2are also the upper bounds of the Poand the
consumed energy in every time interval Tc. We thus realize
that STNS sets a tight constraint on the final ABEP and total
energy consumption, keeping them limited in each interval
Tc. Consequently, this type of selection does not exploit any
possible knowledge on channel statistics, which vary much
more slowly than the instantaneous channel state and can be
effectively estimated in practical applications [8] [9].
V. LONGTERM NODE SELECTION (LTNS)
Let us assume that the transmitter has full knowledge of
the fading statistics corresponding to each of the ergodic
channels SD, SRj and RjD , j = 1,...,L. The two
optimization problems described above can now be imple
mented in an average sense, formulating equivalent traditional
3Obviously, in cases when the activation of all the available relays results
in a Po which is greater than the predefined threshold the constraint cannot
be satisfied.
and minimization knapsack problems, with different optimi
zation and constraint parameters. Their main advantages are
less frequent computation and less complicated cooperation
protocols, which obviously makes them easier to implement
in most practical cases. In fact, LTNS is repeated in a rate
ensuring that the fading statistics remain constant during each
selection interval.
A. ABEP Minimization Under Energy Consumption Con
straints
Since the total energy consumption is constrained, the
weight wj of the branch bj, j = 1,...,L, is the energy
consumed by the corresponding relay, i.e.,
?
Assuming Nakagamim fading and unitary power transmitted
by S, E [Ej] can be approximated in the medium and high
SNR regime using (11) as
?1
(mSj+ 1)ΩSjΓ
wj=
E [Ej], j = 1,...,L
0,j = 0.
(15)
E [Ej] ≈
0
(1 − x)mmSj
SjxmSj−1
ΩmSj
SjΓ(mSj)exp
?
−mSjx
ΩSj
?
dx
?
= 1 − ΩSj+
mSj(mSj+ 1) Γ
?
mSj+ 1,mSj
ΩSj
Γ(mSj+ 2)
?
−
mSj,mSj
ΩSj
?
Γ(mSj+ 2)
and Γ(·), Γ(·,·) are the gamma and
,
(16)
where ΩSj= E?a2
[10, eq. (8.350.2], respectively. For the direct channel b0we
defined w0 = 0, since no extra energy is consumed for the
direct SD transmission.
The conditional bit error probability (BEP), conditioned on
the SNR γ, of the DBPSK signal modulation on an AWGN
channel equals to
Sj
?
incomplete gamma fuctions defined in [10, eq. (8.310.1)] and
Pr(E γ) = Aexp(−Bγ),
(17)
where A,B equal to 1/2 and 1, respectively. Likewise, (17)
represents an approximation of the BEP of the MPSK and M
QAM signal modulations on an AWGN channel; in such case,
A and B are derived by fitting the exact conditional BEP curve
to the approximated BEP of (17) (see [11], [8]). For instance,
for the BPSK case we found via numerical evaluations that A
and B are approximately equal to 0.2568 and 1.2 respectively,
when γ lies in the interval [0 dB, 20 dB].
Lemma 1: The coefficients (profits) in the traditional knap
sack problem (eq. (4)) that minimize the ABEP for the
DBPSK, MPSK and MQAM signal modulations are
?
where β > 14and Mγj(s) is the moment generating function
pj= Logβ
1
Mγj(−B)
?
, j = 0,1,...,L,
(18)
4The base of the logarithm in (18) can be any positive real number in the
interval (1,∞). However, small values close to 1 result in better discreteness
among the branches.
Page 5
(MGF) of γjdefined as
Mγj(s)
where fγj(·) denotes the probability density function of γj.
Proof: Please refer to [12].
?
= E [exp(sγj)] =
?∞
0
exp(sv)fγj(v)dv,
(19)
B. Total Energy Consumption Minimization Under ABEP
Constraints
Similar to (15), the coefficients pj, j = 1,...,L
represent the energy consumed by the relay corresponding to
the branch bj, i.e.,
?
where E [Ej] is given in (16).
Lemma 2: By setting the coefficients wj in the minimiza
tion knapsack problem (eq. (5)) as
?
where β > 1, we ensure that the ABEP for the DBPSK,
MPSK and MQAM signal modulations does not exceed a
predefined threshold δ (if possible5).
Proof: Please refer to [12].
in (5)
pj=
E [Ej], j = 1,...,L
0,j = 0
,
(20)
wj= Logβ
1
Mγj(−B)
?
, j = 0,1,...,L, (21)
VI. NUMERICAL EXAMPLES AND DISCUSSION
In order to illustrate the performance of the proposed
scheme, an extensive set of numerical examples is performed,
using the MGFbased approach for the ABEP given in [7, eq.
(5.3)], followed by the corresponding simulations. A BPSK
modulation scheme is used, and the fading on the SD and on
each SRjand RjD channel is considered to be independent
and Nakagamim distributed, with the fading parameter m
being a random variable (RV) uniformly distributed in the
interval [1 , 2.5]. The average values of the fading attenuation
on the direct SD, and each SRj and RjD channel are
considered continuous independent and identically distributed
(i.i.d.) lognormal RVs, with mean and standard deviation
0.25 and 0.1, respectively, for the SD, and 0.5 and 0.2,
respectively, for the SRj and RjD channels. Also, for our
example N0= 0.1, and the power transmitted by the source
node S is normalized to unity; for this reason, in the following
the term normalized will denote normalization with respect to
the average power transmitted by S.
The main advantage of the traditional and the minimization
knapsack problem utilization in multiuser cooperative diver
sity systems is presented in Tables I and II. In these Tables,
the proposed model is compared, in terms of ABEP and
normalized total power consumption, with two different long
term selection schemes: a) the conventional one, involving
5It is evident that Lemma 2 does not hold when the activation of all
the available relays leads to an ABEP which is greater than the predefined
threshold δ.
TABLE I
LTNS TRADITIONAL KNAPSACK PROBLEM: ABEP AND NORMALIZED
TOTAL POWER CONSUMPTION
ABEP / Consumption
Avail.
Users
Knapsack
Highest Eff.
All Users
51.3E3/ 1.52 8.4E2/ 0.265.2E4/ 1.96
Cmax= 1.6
106.2E4/ 1.608.4E2/ 0.26 1.6E5/ 3.91
15 4.1E4/ 1.407.5E2/ 0.201.5E7/ 6.19
202.8E4/ 1.56 7.5E2/ 0.202.2E9/ 7.90
5 5.2E4/ 1.968.4E2/ 0.26 5.2E4/ 1.96
Cmax= 6
101.6E5/ 3.90 8.4E2/ 0.261.6E5/ 3.91
153.7E7/ 5.367.5E2/ 0.201.5E7/ 6.19
20 3.4E8/ 5.47 7.5E2/ 0.20 2.2E9/ 7.90
TABLE II
LTNS MINIMIZATION KNAPSACK PROBLEM: ABEP AND NORMALIZED
TOTAL POWER CONSUMPTION
ABEP / Consumption
Avail.
Users
Knapsack
Highest Eff.
All Users
55.2E4/ 1.969.7E2/ 0.295.2E4/ 1.96
δ = 10−4
10 8.3E5/ 3.11 9.7E2/ 0.291.6E5/ 3.90
151.7E5/ 2.948.6E2/ 0.211.5E7/ 6.19
202.9E5/ 2.258.6E2/ 0.212.2E9/ 7.90
5 5.2E4/ 1.969.7E2/ 0.295.2E4/ 1.96
101.6E5/ 3.919.7E2/ 0.29 1.6E5/ 3.90
δ = 10−6
151.7E7/ 4.678.6E2/ 0.211.5E7/ 6.19
20 1.6E7/ 3.978.6E2/ 0.21 2.2E9/ 7.90
activation of all the available nodes, and b) the scheme where
a single branch bη, the one with the highest efficiency is
activated through
η = arg max
j=1,...,Lej
(22)
where ejdefined in (6) or (7).
In general, the proposed model appears to achieve the
optimal compromise between error performance and energy
consumption, performing in a way similar to the case when
a single or all the available nodes are activated. This is
determined by the energy consumption constraint Cmax, or by
the equivalent ABEP one δ, corresponding to the traditional
and the minimization knapsack problem, respectively. If, for
example, the value of Cmaxis small, or the value of δ is high
enough so as only one user is activated, the knapsack model
reduces to the scheme where the only cooperating node is
Page 6
Fig. 1.
Constraint.
Traditional Knapsack Problem: ABEP vs Energy Consumption
the one with the highest efficiency. Likewise, the knapsack
scheme can act as an ”all available users” selection model, by
setting high or small values for Cmaxor δ, respectively. Thus,
it is evident that the proposed selection method allows the
system administrator to easily adapt the system’s performance
consumption tradeoff, according to its needs.
A. ABEP Minimization Under Energy Consumption Con
straints
Fig. 1 depicts the system’s ABEP versus the normalized
value of Cmax. The error performance of the LTNS and
STNS is presented, for 15 and 20 available cooperating users.
The comparison shows that in the low energy constraint
regime STNS outperforms LTNS; however, in the high energy
constraint regime, LTNS leads to an error performance which
is generally better than the corresponding shortterm one, and
asymptotically equal to it as the energy consumption constraint
grows infinitely large. This can be intuitively explained consi
dering that STNS optimizes the error performance correspon
ding to each time interval Tc, under the same constraint with
LTNS; this acts as a STNS advantage for small values of l,6
and as a LTNS advantage as l converges to L. Indeed, since
the number of possible alternative selections decreases as l
approaches L, the STNS advantage of optimizing the selection
every Tcis counterbalanced by its inability of setting a flexible
energy constraint, as it was described in the last paragraph of
Section IV, resulting finally in worse error performance.
B. Energy Consumption Minimization Under ABEP Con
straints
Fig. 2 depicts the normalized total energy consumption
per unit time versus the ABEP upper bound δ. The energy
consumption dependence on δ presented in this figure is
6Recall that l represents the number of selected branches during the
selection interval.
Fig. 2.
bound.
Minimization Knapsack Problem: Energy Consumption vs ABEP
similar to the ABEP dependence on the energy consumption
constraint Cmax, as it is shown in Fig. 1; in general, large
number of possible alternative selections results in better
STNS performance and vice versa.
Nevertheless, the difference between STNS and LTNS per
formance is almost negligible. Hence, considering that LTNS
is simpler, in terms of algorithm repetition and protocol
complexity, it represents a more efficient solution to the branch
selection problem.
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