Multi-Relay Selection Design and Analysis for Multi-Stream Cooperative Communications

0.41 MB·English

by  Shunqing Zhang

TOAPPEARINIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,2010 1
Multi-Relay Selection Design and Analysis for
Multi-Stream Cooperative Communications
Shunqing Zhang, Vincent K. N. Lau, Senior Member, IEEE
Abstract
In this paper, we consider the problem of multi-relay selection for multi-stream cooperative MIMO systems with M relay
nodes. Traditionally, relay selection approaches are primarily focused on selecting one relay node to improve the transmission
reliability given a single-antenna destination node. As such, in the cooperative phase whereby both the source and the selected
relay nodes transmit to the destination node, it is only feasible to exploit cooperative spatial diversity (for example by means
1 of distributed space time coding). For wireless systems with a multi-antenna destination node, in the cooperative phase it is
1 possible to opportunistically transmit multiple data streams to the destination node by utilizing multiple relay nodes. Therefore,
0 we propose a low overhead multi-relay selection protocol to support multi-stream cooperative communications. In addition, we
2 derivetheasymptoticperformanceresultsathighSNRfortheproposedschemeanddiscussthediversity-multiplexingtradeoffas
wellasthethroughput-reliabilitytradeoff.Fromtheseresults,weshowthattheproposedmulti-streamcooperativecommunication
n
scheme achieves lower outage probability compared to existing baseline schemes.
a
J

9 I. INTRODUCTION

] Cooperativecommunicationsforwirelesssystemshasrecentlyattractedenormousattention.Byutilizingcooperationamong
T different users, spatial diversity can be created and this is referred to as cooperative diversity [1]–[3]. In [1], the authors
I consideredthecasewheremultiplerelaynodesareavailabletoassistthecommunicationbetweenthesourceandthedestination
.
s nodes using the decode-and-forward(DF) protocol, and they showed that a diversity gain of M(1−2r) can be achieved with
c
[ M relay nodes and a multiplexing gain of r. However, the advantages of utilizing more relay nodes is coupled with the

consumptionof additionalsystem resourcesandpower,so it is impractical(or eveninfeasible) to activate manyrelay nodesin
1
resource-orpower-constrainedsystems.Asaresult,variousrelayselectionprotocolshavebeenconsideredintheliterature.For
v
3 example,in[4]–[7]theauthorsconsideredcooperativeMIMOsystemswithasingle-antennadestinationnodeandtheselection
4 of onerelay nodein the cooperativephase.In [4] the authorsproposedan opportunisticrelayingprotocoland showedthatthis
6 scheme can achieve the same diversity-multiplexing tradeoff (DMT) as systems that activate all the relay nodes to perform
1 distributed space-time coding. In [5] the authors applied fountain code to facilitate exploiting spatial diversity. In [6], [7] the
.
1 authors proposed a dynamic decode-and-forward(DDF) protocol which allows the selected relay node to start transmitting as
0 soon as it successfully decodes the source message. On the other hand, there are a number of works [8]–[10] that studied
1
the capacity bounds and asymptotic performance (e.g. DMT relation) for single-stream cooperative MIMO systems with a
1
single-antenna destination node. In [8] the authors derived the DMT relation with multiple full-duplex relays and showed that
:
v the DMT relation is the same as the DMT upper bound for point-to-point MISO channels. In [9] the authors studied the
i
X network scaling law based on the amplify-and-forward (AF) relaying protocol. In all the above works, the destination node
is assumed to have single receive antenna and hence, only one data stream is involved in the cooperative phase. When the
r
a destination node has multiple receive antennas, the system could support multiple data streams in the cooperative phase and
this could lead to a higher spectral efficiency.
In this paper, we design a relay selection scheme for multi-stream cooperative systems and analyze the resultant system
performance. In order to effectively implement multi-stream cooperative systems, there are several technical challenges that
require further investigations.
• How to select multiple relay nodes to support multi-streamcooperationin the cooperativephase? Mostof theexisting
relay selection schemes are designed with respect to having a single-antenna destination node and supporting cooperative
spatial diversity. For example, in [11] the authors considered a rateless-coded system and proposed to select relay nodes for
supporting cooperative spatial diversity based on the criterion of maximizing the received SNR. In order to support multi-
stream cooperation, the relay selection metric should represent the holistic channel condition between all the selected relay
nodes and the destination node, but this property cannot be addressed by the existing relay selection schemes.
• How much additional benefit can multi-stream cooperation achieve? The spectral efficiency of the cooperative phase
can be substantially increased with multi-relay multi-stream cooperation compared to conventional schemes. However, the
This paper is funded by RGC 615609. The results in this paper were presented in part at the IEEE International Conference on Communications, May
2008.
S. Zhang was with the Department of Electronic andComputer Engineering, Hong KongUniversity ofScience and Technology, Hong Kong.He is now
withHuaweiTechnologies, Co.Ltd.,China(e-mail: [email protected]).
V. K. N. Lau is with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong (e-mail:
[email protected]).
TOAPPEARINIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,2010 2
system performancemay be bottleneckedby the source-relaylinks. Therefore,we characterizethe advantageof multi-stream
cooperation in terms of the end-to-end performance gain over conventional schemes that are based on cooperative spatial
diversity.
Weproposeamulti-streamcooperativerelayprotocol(DF-MSC-opt)forcooperativesystemswithamulti-antennadestination
node. We consider optimized node selection in which a set of relay nodes is selected for multi-stream cooperation. Based
on the optimal relay selection criterion, we compare the outage capacity, the diversity-multiplexing tradeoff (DMT), and the
throughput-reliabilitytradeoff (TRT) of the proposed DF-MSC-opt scheme against traditional reference baselines.
Notation: In the sequel, we adopt the following notations. CM×N denotes the set of complex M ×N matrices; Z denotes
the set of integers; upperand lower case letters denote matrices and vectors, respectively;(·)T denotes matrix transpose; Tr(·)
denotesmatrixtrace;diag(x ,...,x )is a diagonalmatrixwith entriesx ,...,x ;I(·) denotestheindicatorfunction;Pr(X)
1 L 1 L
denotesthe probabilityof eventX; I denotesthe N×N identity matrix;⌈·⌉ and ⌊·⌋ denotethe ceiling and floor operations,
. N .
respectively; (·)+ = max(·,0); =, ≤˙ and ≥˙ denote exponential equality and inequalities where, for example, f(ρ) = ρb if
lim log(f(ρ)) =b; E [·] denotes expectation over Y; and F(x,k) denotes the χ2 cumulative distribution function (CDF)
ρ→∞ log(ρ) Y
for value x and degrees-of-freedomk.
II. RELAY CHANNEL MODEL
Weconsiderasystemconsistingofasingle-antennasourcenode,M half-duplexsingle-antennarelaynodes,andadestination
node with N antennas. For notational convenience, we denote the source node as the 0th node and the M relay nodes as
r
the {1,2,...,M}-th node. We focus on block fading channels such that the channel coefficients for all links remain constant
throughoutthe transmission of a source message.
We divide the transmission of a source message that requires N channel uses into two phases, namely the listening phase
andcooperativephase.Inthelisteningphase,alltherelaynodeslistentothesignalstransmittedbythesourcenodeuntilK out
of the M relay nodes can decode the source message1. In the cooperativephase, the destination node chooses N nodes from
r
amongst the source node and the successfully decoding relay nodes to transmit multiple data streams to the destination node.
Specifically, let x=[x(1),x(2),...,x(N)]T ∈CN×1 denote the signals transmitted by the source node over N channeluses.
Similarly, let x = [x (1),x (2),...,x (N)]T ∈ CN×1, m = 1,...,M, denote the signals transmitted by the mth relay
m m m m
nodeoverN channeluses.Thesignalsreceivedbythemthrelaynodeisgivenbyy =[y (1),y (2),...,y (N)]T ∈CN×1,
m m m m
m=1,...,M, where
y (n)=h x(n)+z (n), (1)
m SR,m m
h is the fading channel coefficient between the source node and mth relay node, H =[h ,h ,...,h ]T ∈
SR,m SR SR,1 SR,2 SR,M
CM×1 isavectorcontainingthechannelcoefficientsbetweenthesourceandtheM relaynodes,andz (n)istheadditivenoise
m
with powernormalizedto unity.Each relay nodeattempts to decodethe source message with each receivedsignal observation
until it can successfully decode the message. The listening phase ends and the cooperative phase begins after K relay nodes
successfully decodes the source message. In the cooperative phase, let x˜ (n), k = 1,...,K, denote the signal relayed by
k
the kth successfully decoding relay node and the aggregate signal transmitted in the cooperative phase can be expressed as
x (n) = [x(n),x˜ (n),x˜ (n),...,x˜ (n)]T ∈ C(K+1)×1. Accordingly, the received signals at the destination node are given
D 1 2 K
by Y =[y(1),y(2),...,y(N)]T ∈CN×Nr, where
H x(n)+z(n) for the listening phase
y(n)= SD
H (D)Vx (n)+z(n) for the cooperative phase
D D
(cid:26)
HSD ∈ CNr×1 represents the fading channel coefficients between the source and the destination nodes, D is the set
representingtheK successfullydecodingrelaynodes,HD(D)=[HSD,H˜RD,1,H˜RD,2,...,H˜RD,K]∈CNr×(K+1) represents
the aggregate channel in the cooperative phase with H˜RD,k ∈ CNr×1 being the fading channel coefficients between kth
successfully decodingrelay and the destination node, z(n)∈CNr×1 is the additive noise with power normalizedto unity, and
V is the node selection matrix. The selection matrix is defined as V=diag(v ,v ,v ,...,v ), where v =1 if the kth node
0 1 2 K k
(k ∈{0,...,K}) is selected to transmit in the cooperative phase and v =0 if the node is not selected.
k
The following assumptions are made throughoutthe rest of the paper.
Assumption 1 (Half-duplex relay model): The half-duplex relay nodes can either transmit or receive during a given time
interval but not both.
Assumption 2 (Fading model): We assume block fading channels such that the channel coefficients H and H (D)
SR D
remain unchanged within a fading block (i.e., N channel uses). Moreover, we assume the fading channel coefficients of
the source-to-relay (S-R) links, relay-to-destination (R-D) links, and source to destination (S-D) links are independent and
identicallydistributed(i.i.d.)complexsymmetricrandomGaussianvariableswith zero-meanandvarianceσ2 ,σ2 andσ2 ,
SR RD SD
respectively.
1Therelaysystemcannotenterthecooperative phaseiflessthanK relaynodescandecodethesourcemessagewithinN channel uses.
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Assumption 3 (CSI model): Eachrelaynodehasperfectchannelstateinformation(CSI)ofthelinkbetweenthesourcenode
and itself. The destination node has perfect CSI of the S-D link and all R-D links. For notational convenience, we denote the
aggregate CSI as H= H ,H (D) .
SR D
Assumption 4 (Transmit power constraints): The transmitpower of the source nodeis limited to ρ . The transmitpower of
S
(cid:0) (cid:1)
the kth relay node is limited to ρ .
k
III. PROBLEM FORMULATION
In this section, we first present the encoding-decoding scheme and transmission protocol of the proposed DF-MSC-opt
scheme. Based on that, we formulate the multi-relay selection problem as a combinatorial optimization problem.
A. Encoding and Decoding Scheme
The proposed multi-stream cooperation system is facilitated by random coding and maximum-likelihood (ML) decoding
[12]. At the source node, an R-bit message W, drawn from the index set {1,2,...,2R}, is encoded through an encoding
function XN : {1,2,...2R} → XN. The encoding function at the source node can be characterized by a vector codebook
C ={XN(1),XN(2),...,XN(2R)}∈CNr×N. The mth codeword of codebook C is defined as
(m) (m) (m) (m)
x (1) ... x (N ) x (N +1) ... x (N)
1 1 1 1 1 1
. .
XN(m)= .. (cid:12) .. ,m=1,...,2R, (2)
(cid:12)
 x(Nmr)(1) ... x(Nmr)(N1) (cid:12)(cid:12)(cid:12) x(Nmr)(N1+1) ... x(Nmr)(N) 
 (cid:12) 
which consists of N vector symbols of dimension N ×(cid:12)1, and x(m)(n) is the symbol to be transmitted by the kth antenna
r (cid:12) k
duringthenth channeluse.ThevectorcodebookC isknowntoalltheM relaynodes(fordecodingandre-encoding)aswellas
thedestinationnodefordecoding.Atthereceiverside(relaynodeordestinationnode),the receiverdecodestheR-bitmessage
based on the observations, the CSI, and a decoding function YN :(YN ×H)→{1,2,...,2R}. We assume ML detection in
the decoding process. The detailed operation of the source node and the relay nodes in the listening and cooperative phases
are elaborated in the next subsection.
B. Transmission Protocol for Multi-Stream Cooperation
The proposed transmission protocol is illustrated in Fig. 1, and the flow charts for the processing by the source, relay, and
destination nodes are shown in Fig. 2. Specifically, the N-symbol source message codeword (cf. (2)) is transmitted to the
destinationnode overtwo phases;the listening phase thatspans the first N channeluses and the cooperativephase thatspans
1
the remainder N =N −N channel uses.
2 1
1) Listening phase: In the listening phase, the single-antennasource nodetransmits the first row2 of the message codeword
(i.e. [x(m)(1)...x(m)(N )] as per (2)) to the relay and destination nodes. Each relay node attempts to decode the source
1 1 1
message with each received signal observation. Although the source node transmits only the first row of the source message
codeword, the relay nodes can still detect the source message using standard random codebook and ML decoding argument.
Effectively, we can visualize a virtual system with a multi-antenna source node as shown in Fig. 3, and the missing rows in
the message codewordtransmittedby the sourcenodeis equivalentto channelerasurein a virtualMISO source-relaychannel.
Once a relay node successfully decodes the source message, it sends an acknowledgement ACK to the destination node
RD
through a dedicated zero-delay error free feedback link.
2) Control phase and signaling scheme: Withoutloss of generality,we assume that K relay nodes can successfully decode
the source message with N received signal observations. Upon receiving the acknowledgement from the K successfully
1
decoding relay nodes, the destination node enters the control phase and selects N nodes to participate in the multi-stream
r
cooperationphase (cf.SectionIII-C). Specifically,the destinationnodeindicatesto thesourceandrelay nodesthetransitionto
the cooperativephase aswell as the nodeselection decisionsvia an (M+1)-bitfeedbackpattern.The first bitof the feedback
pattern is used to indexthe 0th node(the source node) and the last M bits are used to indexthe M relay nodes. The feedback
pattern contains N bits that are set to 1; the mth bit of the feedback pattern is set to 1 if the corresponding node is selected
r
to participate in the cooperative phase, whereas the bit is set to 0 if the node is not selected. Note that the total number of
feedbackbits requiredby the proposedmulti-streamcooperationscheme is K ACK plusone feedbackpatternwith M+1
RD
bits, which is less than 2 bits per relay node.
2Thesourcenodehasasingletransmitantenna andhence, couldonlytransmitonerowofthevectorcodeword XN(m)duringthelistening phase.
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3) Cooperative phase: In the cooperative phase, the N selected nodes cooperate to transmit the (N +1)th to the Nth
r 1
columns of the source message codeword (cf. (2)) to the destination node to assist it with decoding the source message.
Specifically, for k=1,...,N , the node correspondingto the kth bit that is set to 1 in the feedback pattern transmits the kth
r
row of the message codeword (i.e. [x(m)(N +1)...x(m)(N)] as per (2)) to the destination node.
k 1 k
To better illustrate the proposed transmission protocol, we show in Fig. 1 an example of the proposed system with a
destination node with N =2 antennas. Suppose the feedback pattern is given as follows:
r
snooudrcee |← M relaynodes →|
Feedback Pattern 0 1 0 0 ··· 0 1
|← M+1bits →|
Thiscorrespondsto selecting the first relaynode (R ) andthe Mth relaynode (R ) to participatein the cooperativephase. In
1 M
the listening phase, the source node transmits the first row of the codeword XN(m) given by [x(m)(1)...x(m)(N )]. In the
1 1 1
cooperative phase, R and R transmit [x(m)(N +1)...x(m)(N)] and [x(m)(N +1)...x(m)(N)], respectively. Therefore,
1 M 1 1 1 2 1 2
the effective transmitted codeword can be expressed as
Transmittedbysource TransmittedbyR1
nodeinlisteningphase incooperativephase
XN(m)= x(1m)(1) ... x(1m)(N1) x(1m)(N1+1) ... x(1m)(N) .
"z }| {(cid:12)(cid:12)zx(2m)(N1+1)}|... x(2m)(N){#
(cid:12)
(cid:12) TransmittedbyRM
(cid:12) incooperativephase
| {z }
C. Problem Formulation for Multi-stream Cooperation
We focus on node selection by the destination node for optimizing outage performance (cf. Section III-B2), where the
destination node only has CSI of the S-D and R-D links H (D) (cf. Assumption 3). The proposedMulti-stream Cooperation
D
Scheme with Optimized node selection is called DF-MSC-opt and we define the outage event as follows:
Definition 1 (Outage): Outage refers to the event when the total instantaneous mutual information between the source and
the destination nodes is less than the target transmission rate R. Mathematically, outage can be expressed as
I(I (H,V)<R), (3)
DF-MSC-opt
where H=(H ,H (D)) is the aggregate channel realization and V is the node selection action given H (D).
SR D D
The total instantaneousmutualinformationof the multi-streamcooperativesystem in (3) is givenby the followingtheorem.
Theorem 1: (InstantaneousMutualInformationofMulti-StreamCooperativeSystem)Giventheaggregatechannelrealization
H = (H ,H (D)), the instantaneous mutual information I (H,V) (bits/second/channel use) between the source
SR D DF-MSC-opt
and destination nodes for multi-stream cooperative system can be expressed as
I (H,V)=1 N log(1+ρ |H |2)+N logdet(I +H (D)VΓVHH (D)H) , (4)
DF-MSC-opt N 1 S SD 2 Nr D D
where N1 and N2 are the numbers (cid:8)of channel uses of the listening phase and the cooperative phase, res(cid:9)pectively, and Γ =
diag(ρ ,ρ ,ρ ,...,ρ ) is the transmit power matrix of the source node and the M relay nodes. Note that N and N are
S 1 2 M 1 2
random variables that depend on the realization of the S-R links H .
SR
TheproofofTheorem1canbeextendedfrom[12],[13]byapplyingrandomGaussiancodebookargument.Notethatthefirst
term in the mutualinformationin (4) correspondsto the contributionfromthe sourcetransmission3. By virtueof the proposed
DF-MSC-opt scheme, multiple data streams are cooperativelytransmitted in the cooperativephase (unlike traditional schemes
wherein only a single stream is transmitted). Hence, we can fully exploit the spatial channels created from the multi-antenna
destination node and therefore achieving higher mutual information in the second term of (4).
By Definition 1, the average outage probability is given by
Pout(V)=EH I IDF-MSC-opt(H,V)<R ,
which is a functionof the node selection policyV. It fo(cid:2)llo(cid:0)wsthat, giventhe chann(cid:1)e(cid:3)lrealization H, the optimalnode selection
policy is given by V⋆=argminI I (H,V)<R , where the set of all feasible node selection actions is defined as
DF-MSC-opt
V∈Ω
Ω={Λ(cid:0) ∈{0,1}(M+1)×(M+1(cid:1))|Λ is diagonal and Tr(Λ)=Nr}. (5)
Equivalently, for any transmission rate R, the optimal node selection policy is given by
V⋆ = argmaxI (H,V). (6)
DF-MSC-opt
V∈Ω
3The mutual information contributed by the source node can also be obtained from the mutual information of the virtual multi-antenna source model in
Fig.3.
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IV. OUTAGEANALYSIS FORTHEDF-MSC-OPT SCHEME
In this section, we derive the asymptotic outage probability of the DF-MSC-opt scheme. For simplicity, we assume σ2 =
SD
σ2 = σ2 and the power constraints of all the nodes are scaled up in the same order4, i.e., lim ρm = 1 for all
RD D ρS→∞ ρS
m∈{1,...,M}. The outage probability can be expressed as
Po⋆ut =EH I IDF-MSC-opt(H,V⋆)<R (=a)EHSR Pr IDF-MSC-opt(H,V⋆)<R|HSR , (7)
where in (a) the randomnes(cid:2)s(cid:0)of I (H,V⋆) is(cid:1)(cid:3)introduced(cid:2)by(cid:0)the aggregate channel gain in(cid:1)(cid:3)the cooperative phase
DF-MSC-opt
H (D) and the optimal node selection policy V⋆. Let H denote the collection of all realizations of the S-R links H
D N1 SR
such that K out of M relay nodes can successfully decode the source message, and the listening phase ends at the (N )th
1
channel use. We define the outage probability given H and the duration of the listening phase N as P (H ,N ) =
SR 1 SR SR 1
Pr I (H,V⋆)<R|H ∈H , which includes the following cases.
DF-MSC-opt SR N1
• (cid:0)Case 1: The listening phase ends with(cid:1)in N channel uses. In this case, N1 <N, N2 =N −N1, and the outage probability
P (H ,N ) can be expressed as
SR SR 1
PCase1(H ,N )=Pr N1 log(1+ρ |H |2)+ N−N1g(H,V⋆) <R , (8)
SR SR 1 N S SD N
whereg(H,V⋆)=maxlogdet I +ρ H ((cid:2)D(cid:0))VVHH (D)H forK ≥N ,andg(H,V(cid:1)⋆)=(cid:3)logdet I +ρ H (D)H (D)H
V∈Ω Nr S D D r Nr S D D
for K <N .
r (cid:0) (cid:1) (cid:0) (cid:1)
• Case 2: The listening phase lasts for more than N channel uses. In this case, the outage event is only contributed by the
direct transmission between the source and the destination nodes. The outage probability P (H ,N ) can be expressed
SR SR 1
as
PCase2(H ,N )=Pr log(1+ρ |H |2)<R . (9)
SR SR 1 S SD
Substituting (8) and (9) into (7), the outage probability i(cid:0)s given by (cid:1)
P⋆ = N PCase1(H ,l)Pr(H ∈H )+ PCase2(H ,l)Pr(H ∈H ) (10)
out l=1 SR SR SR l l>N SR SR SR l
= N PCase1(H ,l)Pr(H ∈H )+Pr log(1+ρ |H |2)<R Pr(H ∈∪ H),
Pl=1 SR SR SR l P S SD SR l>N l
where Pr(HSR ∈Hl)Pis the probability that K out of M relay(cid:0)nodes can successfully(cid:1)decode the message at the lth channel
use. To evaluate the outage probability, we calculate each term in equation (10) as follows. First, it can be shown that
Pr(H ∈H )=Pr(H ∈∪ H)−Pr(H ∈∪ H )=Φ −Φ (11)
SR N1 SR l>N1−1 l SR l>N1 l N1−1 N1
where
Φ = K−1 M 1−exp(−2NR/l−1) M−i exp(−2NR/l−1) i, l=1,...,N, (12)
l i=0 i ρSσS2R ρSσS2R
denotestheprobabilitythatPlesstha(cid:0)nK(cid:1)(cid:0)relaynodescansucc(cid:1)essful(cid:0)lydecodethesou(cid:1)rcemessageatthelth channeluse.Second,
PCase1(H ,l) can be upper-bounded5 as shown in the following lemma.
SR SR
Lemma 1: TheoutageprobabilityfortheDF-MSC-optschemegiventhatK relaynodescansuccessfullydecodethesource
message at the lth channel use can be upper bounded as
PCase1(H ,l)≤Pr(f(α ,H,V⋆)<R), l =1,...,N, (13)
SR SR l
where α = l/N, f(α,H,V⋆) = α log(1+ρ |H |2)+(1−α ) LT log 1+ρ σ2κ(i) with L = min(N ,K +1)
l l l S SD l i=1 2 S D T r
denoting the number of transmitted streams in the cooperative phase, and κ(i), i=1,...,L , are (L out of K+1) ordered
T T
χ2-distributed variables with 2N degrees of freedom. P (cid:0) (cid:1)
r
Proof: Refer to Appendix A for the proof.
Third, when fewer than K relay nodes can successfully decode the source message within N channel uses, the source
node transmits to the destination node with the direct link only; it follows that Pr(H ∈ ∪ H ) = Φ and Pr log(1+
SR l>N l N
ρ |H |2)<R =F 2R−1;N . Therefore, the outage probability (cf. (10)) is given by
S SD ρSσD2 r (cid:0)
(cid:1) (cid:16)P⋆ =(cid:17)ΦF 2R−1;N + 1 (Φ −Φ )PCase1(H ,l)
out ρSσD2 r αl=1/N l−1 l SR SR
≤ ΦF (cid:16)2R−1;N (cid:17)+P1 (Φ −Φ )Pr(f(α ,H,V⋆)<R)
ρSσD2 r αl=1/N l−1 l l
= ΦF (cid:16)2R−1;N (cid:17)−PΦ′ Pr(f(α′,H,V⋆)<R), (14)
ρSσD2 r l′ l
(cid:16) (cid:17)
4Assuch,thereisnolossofgenerality toassumeΓ=ρSIM+1 whenstudyinghighSNRanalysis.
5Itisnon-trivial toevaluate PSCRase1(HSR,l)exactly duetothedynamics oftheoptimalnodesselection.
TOAPPEARINIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,2010 6
whereΦ′ andΦ′ denotethefirstorderderivativeofΦ withrespecttoα andevaluatedatl =α N andl′ =α′N,respectively.
l l′ l l l l
Notethatinthelaststepof(14)weapplytheMeanValueTheorem[14]whichguaranteestheexistenceofthepointα′ ∈(0,1).
l
Moreover, the outage probability upper bound of the DF-MSC-opt scheme is given by the following theorem.
Theorem 2: For sufficiently large N, the outage probability upper bound of the DF-MSC-opt scheme is given by
R R K+1
P⋆ ≤ΦF 2R−1;N −Φ′ F 2α′l−1;N F 21−α′l−1;N . (15)
out ρSσD2 r l′ ρSσD2 r ρSσD2 r
(cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19)
DirectTransmission Relay-AssistedTransmission
Proof: Refer to Appendix B|for the{pzroof. } | {z }
For systems without relays, the outage probability is given by P = F 2R−1 ;N . Note that the DF-MSC-opt
out,pp NrρSσD2 r
scheme can achieve lower outage probability by taking advantage of multi-stream(cid:16)cooperative(cid:17)transmission. We can interpret
−Φ′ as the cooperative level, which quantifies the probability that the relay nodes can assist the direct transmission. For
l′
example,−Φ′ =1 impliesthatthe DF-MSC-optschemecan befully utilized,whereas−Φ′ =0 impliesthatthe sourcenode
l′ l′
transmits to the destination node with the direct link only. As per (14), the outage probability P⋆ is a decreasing function
out
with respect to6−Φ′ , whereas −Φ′ is a decreasing function with respect to the strength of the S-R links σ2 . As shown in
l′ l′ SR
Fig. 4, when the strength of the S-R links increases from 30dB to 40dB, the cooperative level increases and hence the outage
probability decreases. On the other hand, as shown in (15) and as illustrated in Fig. 5, the outage probability decreases with
increasing number of receive antennas at the destination node.
V. DMT AND TRT ANALYSESFOR THEDF-MSC-OPT SCHEME
Inthissection,wefocusoncharacterizingtheperformanceoftheDF-MSC-optschemeinthehighSNRregime.Specifically,
we perform DMT and TRT analyses based on the outage probability P⋆ as shown in (14). There are two fundamental
out
reasons for the performance advantage of the proposed DF-MSC-opt scheme, namely multi-stream cooperation and optimal
node selection V⋆ (cf. (6)). To illustrate the contributionof the first factor,we shall compare the performanceof DF-MSC-opt
with the following baselines:
• DF-SDiv(Baseline1):TheDFrelayprotocolforcooperativespatialdiversity.Thelisteningphaseandthecooperativephase
have fixed durations, i.e., each phase consists of N/2 channel uses. The source node and all the successfully decoding relay
nodes cooperate using distributed space-time coding. At the destination node, maximum ratio combining (MRC) is used to
combine the observations from the different receive antennas.
• AF-SDiv (Baseline 2): The AF relay protocol for cooperative spatial diversity. All the relay nodes transmit a scaled version
of their soft observations in the cooperative phase. At the destination node, MRC is used to combine the observations from
different receive antennas.
• DDF(Baseline3):ThedynamicDFprotocol[6].Oncearelaynodesuccessfullydecodesthesourcemessage,itimmediately
joinsthe transmissionusingdistributedspace-timecoding.Atthe destinationnode,MRC is usedto combinetheobservations
from different receive antennas.
On the other hand, to illustrate the contributionof optimalnode selection, we shall comparethe performanceof the DF-MSC-
opt scheme with the DF-MSC-rand scheme (Baseline 4), which correspondsto a similar multi-stream cooperationscheme but
withrandomizednodeselectionVrandomlygeneratedfromthenodeselectionspaceΩ(cf.(5)).TableIsummarizesthemajor
differencesamong the DF-MSC-opt scheme and the baseline schemes. We illustrate in Fig. 6 the outage capacity versus SNR
(with P = 0.01) of the cooperative systems with N = 3, K = 3, M = 15. Note that the proposed DF-MSC-opt scheme
out r
can achieve a gain of more than 1 bit/channel use over the four baseline schemes.
A. DMT Analysis
Inorderto analyzetheDMTrelationoftheDF-MSC-optscheme,we firstderivetherelationbetweentheoutageprobability
and the multiplexing gain. In the outage probability expression (14), for a sufficiently large ρ , the asymptotic expression of
S
the first term is given by
ΦF 2R−1;N =˙ K−1 M ρ−(M−i)(1−r)+ρ−Nr(1−r)+ =. ρ−(M−K+1+Nr)(1−r)+, (16)
ρSσD2 r i=0 i S S S
and the asymptotic exp(cid:16)ression of(cid:17)Φ′ Pis given(cid:0)by(cid:1)
l′
Φ′l′ = Ki=−01 Mi (1−φl′)M−i−1(φl′)i−1((M −i)(1−φl′)′φl′ +i(1−φl′)(φl′)′)
+ + +
=˙ PK−1(cid:0)M(cid:1)ρ−(M−i) 1−αr′l ≤˙ρ−mini(M−i) 1−αr′l =ρ−(M−K+1) 1−αr′l (17)
i=0 i S S S
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
6Forasufficiently largeSNRρS,PF(cid:18)2ραSR′lσ−(cid:0)D21;(cid:1)Nr(cid:19)F(cid:18)2LTρ(1SR−σαD2′l)−1;Nr(cid:19)K+1 ismuchsmallerthanF(cid:18)N2rRρS−σ1D2 ;Nr(cid:19).
TOAPPEARINIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,2010 7
where φl′ =exp − 2rρlSαoglσ′ρS2SR−1 . Moreover, we can express the term Pr(f(α′l,H,V)<R) in (14) as
(cid:16) (cid:17)
Pr(f(α′,H,V)<R)=. Pr ρα′l(1−δ)+ LT ρ(1−α′l)(1−βi)+ <ρr
l S i=1 S S
=Pr α′l(1−δ)++ (cid:0) Li=T1(1−αQ′l)(1−βi)+ <r =. Bp(cid:1)(δ,β)dδdβ, (18)
where−δistheexponentialorderof(cid:0)σS2D,−βi isthePexponentialorderofγ(i),B(cid:1)={RδR,β :α′l(1−δ)++(1−α′l) i(1−βi)+ <
r}, and p(δ,β) is the joint probability density function of δ and β. Substituting (16)-(18) into (14), the outage probability can
P
be expressed as
P⋆ ≤˙ ρ−(M−K+1+Nr)(1−r)+ +ρ−(M−K+1)(1−αr′l)+ p(δ,β)dδdβ, (19)
out S S B
and the DMT relation for the DF-MSC-opt scheme is given by the following theorReRm.
Theorem 3: The DMT relation for the DF-MSC-opt scheme can be expressed as
d(r,K) =min(d ,d +d ), (20)
DF-MSC-opt 1 2 3
where d =(M −K+1+N )(1−r)+, d =(M −K+1)(1−2r)+, and
1 r 2 1−r
N +(N −1)K 0≤r <1/2
r r
d3 = min −N(rN+r(+NKr−−θ2)θ()K(1+−r1r−−θθ)),(Nr+−Nθr)θ(K(1+−rr1−)θ) θ+θ1 ≤r < θθ++12 (21)
 NrL(cid:16)T(1−rr) (cid:17) LLTT+1 ≤r ≤1
with θ =1,2,...,LT and LT=min(Nr,K+1).
Proof: Refer to Appendix C for the proof.
We could further optimize the parameter K in the DF-MSC-opt scheme and the resulting DMT relation is given by
d(r)⋆ = max d(r,K) =max d(r,⌊K⋆⌋) ,d(r,⌈K⋆⌉) ,
DF-MSC-opt DF-MSC-opt DF-MSC-opt DF-MSC-opt
K∈[1,2,...,M]
(cid:0) (cid:1)
where
(M+1)(2−4r+r2)+Nr(2−3r+r2) 0≤r ≤1/2
K⋆ = (Nr−1)(1−r)+r2 (22)
( (M+1)(1−r)2−Nr(3r−r2−(1−r)2θ)−θ(3θ(1−r)−1−r) θ <r ≤ θ+1
Nr(1−r)−r+(1−r)2 θ+1 θ+2
is the optimal number of successfully decoding relay nodes to wait for in the listening phase.
Inthefollowing,weshowthattheDF-MSC-optschemeachievessuperiorDMTperformancethanthetraditionalcooperative
diversity schemes. Specifically, as per [15], the AF-SDiv and DF-SDiv schemes have identical DMT relations given by
d(r) = M(1 − 2r)+ + N (1 − r)+ for 0 ≤ r ≤ 1. For the DDF protocol, the DMT relation is given by the
AF/DF-SDiv r
following lemma.
Lemma 2: The DMT relation for the DDF protocol with N receive antennas at the destination node can be expressed as
r
(M +N )(1−r) 0≤r ≤ Nr
r M+Nr
d(r)DDF = Nr+M(11−−2rr)+ MN+rNr ≤r ≤ 12 (23)
 Nr(1−rr) 21 ≤r ≤1
Proof: Refer to Appendix D for the proof.

InFig.7,wecomparetheDMTrelationsfortheDF-MSC-optschemeandthebaselineschemes.ForasystemwithM =15
relay nodes and a destination node with N =3 antennas. Note that since the DF-MSC-opt scheme (as well as the DF-MSC-
r
rand scheme) exploits multi-stream cooperation, it can achieve high diversity gain than the traditional cooperative diversity
schemes (i.e. AF-SDiv, DF-SDiv, and DDF) that exploit single-stream cooperation. Moreover, since the DF-MSC-opt scheme
optimizes the node selection policy and the number of decoding relay nodes K, it can achieve better diversity gain than the
DF-MSC-rand scheme.
B. TRT Analysis
The DMT analysis alone cannotcompletely characterize the fundamentaltradeoff relation in the high SNR regime. Specifi-
cally, the multiplexing gain gives the asymptotic growth rate of the transmission rate R at high SNR ρ and is only applicable
to scenarios where R scales linearly with logρ. Hence, there are many unique transmission rates that correspond to the same
multiplexing gain, and the DMT analysis only gives a first order comparison of the performance tradeoff at high SNR when
we have different multiplexing gains.
TOAPPEARINIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,2010 8
In orderto have a clearerpictureon the tradeoffrelations,in the followingwe study the SNR shift in the outage probability
P as we increase the transmission rate R by ∆R. We quantify the more detailed relations among the three parameters
out
(R,logρ,P (R,ρ)) by analyzing the TRT7 relation [16].
out
Consider the outage probability expression for the DF-MSC-opt scheme (14). In order to facilitate studying the TRT, we
choose the parameter K such that the outage events are dominatedby relay-assisted transmissions, i.e. K ≥⌈K⋆⌉ (where K⋆
is given by (22)). The following theorem characterizes the asymptotic relationships among R, ρ , and P⋆ for the r > 1
S out 2
case8.
Theorem 4: The TRT relation for the DF-MSC-opt scheme under K ≥⌈K⋆⌉ and r> 1 is given by
2
ρS→∞,(lRim,ρS)∈R(z)logPo⋆ut−locgDFρ-SMSC-opt(z)R =−gDF-MSC-opt(z) (24)
where
R(z), (R,ρ )|z+1> R >z for z ∈Z,0≤z <L , (25)
S (1−r)logρS T
denotes the zth operating region, c(cid:8) (z),K+1+N −(2z(cid:9)+1), and g (z),(K+1)N −z(z+1). Note
DF-MSC-opt r DF-MSC-opt r
that g (z) is defined as the reliability gain coefficient and t (z) , g (z)/c (z) is defined as
DF-MSC-opt DF-MSC-opt DF-MSC-opt DF-MSC-opt
the throughputgain coefficient.
Proof: Refer to Appendix E for the proof.
By applyingTheorem4, the SNRshiftbetweentwo outagecurveswith a ∆Rrate differenceis3∆R/t(z)dB. Fig.8shows
theoutagecurvescorrespondingto∆R=2bits/channeluseforN =3,K =3andM =15.Thisscenariocorrespondstothe
r
regionR(1)andtheTRTrelationfortheDF-MSC-optschemecanbeexpressedasg (1)=10andt (1)= 5.
DF-MSC-opt DF-MSC-opt 2
As we can see from the simulation results, the SNR shift is 2.4 dB for 2 bits/channel use increase in the transmission rate,
which matches with our analysis.
VI. CONCLUSION
In this paper, we proposed a multi-stream cooperative scheme (DF-MSC-opt) for multi-relay network. Optimal multi-relay
selection is considered and we derived the associated outage capacity as well as the DMT and TRT relations. The proposed
design has significant gains in both the outage capacity as well as the DMT relation due to (1) multi-stream transmissions in
the cooperative phase and (2) optimized relay selection.
APPENDIX A
PROOF OF LEMMA1
To find the outage probability given l, we characterize the function of g(H,V⋆) under different conditions.
• Condition 1 K <Nr: Under this condition, we allow all the successfully decoding relays to transmit in the cooperative
phase. Thus, the communication links can be regarded as a conditional MIMO link [17] and g(H,V)=logdet I +
Nr
ρ H (D)H (D)H ≤ K+1log (1+ρ σ2κ(i)), where κ(i), i=1,...,K+1, are χ2-distributedvariableswith 2N
S D D i=1 2 S D (cid:0) r
degrees of freedom.
(cid:1) P
• Condition 2 K ≥ Nr: Under this condition, we select Nr nodes out of the source and the successfully decoding
relay nodes from the node selection space Ω to participate in cooperative transmission. The analytical solution for the
term g(H,V) is in general not trivial [18]. Since in the proposed DF-MSC-opt scheme we choose the best N out of
r
K+1 transmit nodes, the upper boundcan be obtained similar to [18, (6)]. Thus, the capacity bound with transmit node
selectionisg(H,V)=maxV∈Ωlogdet INr+ρSHD(D)VVHHD(D)H ≤ Ni=r1log2(1+ρSσD2κ(i)), wheretheκ(i),
i=1,...,N , are ordered χ2-distributed variables with 2N degrees of freedom.
r r
(cid:0) (cid:1) P
Combining the two conditions above, we obtain the general form as shown in Lemma 1.
APPENDIX B
PROOF OF THEOREM2
From (14), the only unknown part is given by
Pr f(α′,H,V)<R =Pr α′log(1+ρ |H |2)+(1−α′) LT log (1+ρ σ2κ(i))<R ,
l l S SD l i=1 2 S D
7TRTanalysisal(cid:0)lowsforinvestigating m(cid:1)oregen(cid:2)eralscenarioswherethetransmissionrateRPdoesnotscalelinearlywithlogρand(cid:3)helpstostudytheSNR
gainoftheoutage probability vsSNRcurvewhenweincreasethetransmissionrateRby∆R.
8Forthe r≤ 1 case, the MIMOchannel formedbythe multiple relay nodes tothe destination nodecan onlyoperate withamultiplexing gain of1and
2
theTRTrelation istrivial.
TOAPPEARINIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,2010 9
which can be relaxed as
Pr α′log(1+ρ |H |2)+(1−α′) LT log (1+ρ σ2κ(i))<R
l S SD l i=1 2 S D
≤(cid:2)Pr(α′llog(1+ρS|HSD|2)<R)Pr P(1−α′l) Ni=r1log2(1+ρSσD2κ(cid:3)(i))<R
R
=F 2α′l−1;N Pr LT log 1+(cid:0) ρ σ2κ(Pi) < R . (cid:1)
ρSσD2 r i=1 2 S D 1−α′l
Moreover, the following rela(cid:16)tion holds f(cid:17)or α(cid:16)′P>0 (cid:0) (cid:1) (cid:17)
l
Pr LT log 1+ρ σ2κ(i) < R ≤Pr log 1+ρ σ2κ(1) < R
i=1 2 S D 1−α′ 2 S D 1−α′
l l
(cid:16)P =Pr κ(cid:0)(1)< 21−Rα′l−1(cid:1) =F 2(cid:17)1−Rα′l−1(cid:16);N (cid:0)K+1 (cid:1) (cid:17) (26)
ρSσD2 ρSσD2 r
(cid:16) (cid:17) (cid:16) (cid:17)
where in the last step we applied the results of order statistics [19]. Substituting (26) into (14), we have Theorem 2.
APPENDIX C
PROOF OF THEOREM3
TheasymptoticexpressionoftheoutageprobabilityP⋆ isgivenby(19)withB ={δ,β :α′(1−δ)++(1−α′) (1−β )+ <
out l l i i
r}.
From the first term in (19), we can obtain d =(M −K+1+N )(1−r)+. P
1 r
Since the source node is equipped with a single antenna, the maximum multiplexing gains for the S-R links as well as
the S-D and R-D links is always less than 1. Specifically, the multiplexing gain in the listening phase is r ≤ 1 and the
α′
multiplexinggain in the cooperativephase is r ≤1 or equivalently,α′ ≥max(r,1−r). Thus, we can evalulate the second
1−α′ l
term through d =(M −K+1)(1− r ) for rl<1/2 and 0 otherwise.
2 1−r
Weevaluatethethirdtermunderthefollowingcases.Whenr <1/2,α′ ≥1−randB ⊆{δ,β :r (1−δ)++ (1−β )+ <
l i i
r}. Equivalently,
(cid:0) P (cid:1)
p(δ,β)dδdβ ≤ p(δ,β)dδdβ
B (1−δ)++Pi(1−βi)+≤1
RR =. ρRR−infr′∈[0,1] Nr(K+1)−(Nr+K)r′+Nrr′ =ρ−(Nr−1)K−Nr. (27)
S S
(cid:0) (cid:1)
As a result, d = (N − 1)K + N for r < 1/2. When 1/2 ≤ r < 1, α′ ≥ r and the corresponding B = {δ,β :
3 r r l
r(1−δ)++(1−r) (1−β )+r<r}. Equivalently,
i i
P Bp(δ,β)dδdβ ≤ (1−δ)++1−rrPi(1−βi)+≤1p(δ,β)dδdβ
RR =. RρR−infr′∈[0,1] (Nr−1r−r′r)(K+1−1r−r′r)−Nrr′ (28)
S
(cid:0) (cid:1)
for 1r−r′r ∈{1,...,LT}. Thus, we can write the general form as d3 =inf0≤r′≤1 Nrr′+(Nr−θ)(K +1−θ)−(Nr+K−
2θ)(rr′ −θ) for θ ={1,2,...,L }. Since d is linear with respect to r′, we can write d =min N +(N −θ)(K+1−
1−r T 3 (cid:0) 3 r r
θ)−(N +K−2θ)( r −θ),N θ(1−r)+(N −θ)(K+1−θ) . Moreover,when r ≥L , we have rr′ −L =0 and
r (cid:1) 1−r r r r 1−r T (cid:8) 1−r T
d =N L (1−r).
3 r T r (cid:9)
Combining the three terms above, we have Theorem 3.
APPENDIX D
PROOF OF LEMMA2
Themainstepsoftheproofarebasedontheworkby[6].Denoteg tobethechannelcoefficientbetweenthejth nodeand
k,j
thekthreceiveantennaatthedestinationnode,andthereceivedsignalatthekthantennaisgivenbyy =g x +z .Following
k k,j j k
the same argument in the proof of Theorem 6 in [6], we can find that destinations ML error probability assuming error-free
decodingatallofthe relays,providesa lowerboundonthediversitygainachievedbytheprotocolandthecorrespondingPEP
for kth antenna is upper bounded by P ≤ M+1 1+ j |g |2 ρ −nj, where n is the number of symbol intervals in
k j=1 i=1 k,i j
the codeword during which a total of j nodQes are thransm(cid:0)iPtting, so that(cid:1), tihe total number of codeword length Mj=+11nj =N.
Hence,the PEPfor thedestinationto receivedthe informationis upperboundedby P ≤ Nr P . Notethat, the upperbound
k=1 k P
is nottightsince we do notconsiderthe jointdecodingamongthe receiveantennas,butit is sufficientto derivethe order-wise
relation. By changing the order of k and j, we have P ≤ M+1 Nr 1+ j |gQ|2 ρ −nj. p n is the number
j=1 k=1 i=1 k,i j=1 j
of symbol intervals that relay p has to wait, before the mutual infhormatio(cid:16)nbetween its received(cid:17)siignal and the signals that the
Q Q (cid:0)P (cid:1) P
source and other relays transmit exceeds NR. Thus, we have p n ≤ min N, NR , where h denotes
j=1 j log(1+|hp+1,1|2ρ) p+1,1
the channel condition between the relay node p and the source node. n o
P
TOAPPEARINIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS,2010 10
Define v and u as the exponential orders of g and h . We have the following relation,
k,j j,i k,i j,i
P ≤ρ−PMj=1nj(1−min{PNk=r1vk,1,...,PNk=r1vk,j})+ =ρ−{PMj=1nj(1−Nrmin{v1,...,vj})+,
where v = PNk=r1vk,i. Choose the rate R=rlogρ and the PEP bound for rate R is given by
i Nr
P ≤ρ−PMj=1nj(1−min{PNk=r1vk,1,...,PNk=r1vk,j})+ =ρ−N[{PMj=1 nNj(1−Nrmin{v1,...,vj})+−r].
Hence, the set of channel realizations that satisfy { M nj (1−N min{v ,...,v })+ ≤r results in the outage event.
j=1 N r 1 j
Define v¯ = min{v ,...,v } and we can separate the discussion in the following three cases. (Naturally, we have v¯ ≥
j 1 j 1
P
v¯ ≥...≥v¯ ).
2 M
• Case 1: v¯1 ≤1. In this case, we can follow the discussion from (62) to (71) in [6] and show that the diversity order d is
(N +M)(1−r), Nr ≥r≥0
r Nr+M
d≥ N +M(1−2r), 1 ≥r > Nr .
 rNr(1−rr1)−,r 2 1≥r >Nr12+M
• Case 2: v¯M ≥1. It’s trivial and d=M.
 N +M(1−2r), 1 ≥r ≥0
• Case3:v¯i >1≥vi¯+1.Followthesamediscussionfrom(72)to(82)in[6],weconcludethatd≥ rN (1−r1)−,r 12≥r > 1 .
(cid:26) r r 2
Combining the above cases, we have Lemma 2.
APPENDIXE
PROOF OF THEOREM4
Given r > 1 and K ≥⌈K⋆⌉, (14) can be simplified as P⋆ ≤˙ −Φ′ Pr(f(α′,H,V⋆)<R). Substituting the expression of
2 out l′ l
f(α′,H,V), we have
l
P⋆ ≤ Pr(α′log(1+ρ |H |2)+(1−α′) LT log (1+ρ σ2γ(i))<R)
out l S SD l i=1 2 S D
≤ Pr((1−r) LT log (1+ρ σ2γ(i))<R). (29)
i=1 2 S D P
Based on the above results and following thePsame lines as the proof of [16, Theorem 2], we have the result of Theorem 4.
REFERENCES
[1] J.N.LanemanandG.W.Wornell,“Distributed space-time codedprotocols forexploiting cooperative diversity inwireless networks,”IEEETrans.Inf.
Theory,vol.49,pp.2415–2425,Oct.2003.
[2] A.Sendonaris, E.Erkip, andB. Aazhang, “User cooperative diversity -part I: System description,” IEEETrans. Commun., vol.51, pp. 1927 –1938,
Nov.2003.
[3] A.Nosratinia, T.Hunter, andA.Hedayat, “Cooperative communication inwireless networks,”IEEECommun.Mag.,vol.42,pp.68–73,Oct.2004.
[4] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperative diversity method based on network path selection,” IEEE J. Sel. Areas
Commun.,vol.24,pp.659–672,Mar.2006.
[5] A.F.Molisch,N.B.Mehta,J.S.Yedidia,andJ.Zhang,“Cooperativerelaynetworksusingfountaincodes,”inProc.IEEEGLOBECOM’06,Nov.2006.
[6] K. Azarian, H. E. Gamal, and P. Schniter, “On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels,” IEEE Trans. Inf.
Theory,vol.51,pp.4152–4172,Dec.2005.
[7] K.R.KumarandG.Caire,“Codinganddecodingforthedynamicdecodeandforwardrelayprotocol,”submittedtoIEEETrans.Inf.Theory,Jan.2008.
[8] M.YukselandE.Erkip,“Multiple-antennacooperativewirelesssystems:Adiversity-multiplexingtradeoffperspective,”IEEETrans.Inf.Theory,vol.53,
pp.3371–3393,Oct.2007.
[9] H.Bo¨lcskei, R.U.Nabar, O.Oyman,andA.J.Paulraj, “Capacity scaling lawsinMIMOrelaynetworks,” IEEETrans.Wireless Commun.,vol.5,pp.
1433–1444,Jun.2006.
[10] S. Borade, L. Zheng, and R. Gallager, “Amplify-and-forward in wireless relay networks: Rate, diversity, and network size,” IEEE Trans. Inf. Theory,
vol.53,pp.3302–3318,Oct.2007.
[11] R.Nikjah andN.C.Beaulieu, “Novel rateless codedselection cooperation indual-hop relaying systems,”inProc.IEEEGLOBECOM’08,Nov.2008,
pp.1–6.
[12] T.M.CoverandA.J.Thomas,ElementsofInformation Theory. Wiley,1991.
[13] M. A.Khojastepour, A.Sabharwal, and B. Aazhang, “On capacity ofGaussian cheap relay channel,” inProc. IEEEGLOBECOM’03,Dec. 2003, pp.
1776–1780.
[14] W.Rudin,Principles ofMathematical Analysis. McGraw-Hill, 1976.
[15] A.TajerandA.Nosratinia,“Opportunisticcooperationviarelayselectionwithmininmalinformationexchange,”inProc.IEEEISIT’07,2007,pp.1926
–1930.
[16] K. Azarian and H. E. Gamal, “The throughput-reliability tradeoff in block-fading MIMO channels,” IEEETrans. Inf. Theory, vol. 53, pp. 488 – 501,
Feb.2007.
[17] L.Zhengand D.N.C. Tse,“Diversity and multiplexing: Afundamental tradeoff inmultiple antenna channels,” IEEETrans. Inf. Theory, vol. 49, pp.
1073–1096,May2003.
[18] A.F.Molisch,M.Z.Win,Y.-S.Choi,andJ.H.Winters,“Capacity ofMIMOsystemswithantennaselection,” IEEETrans.WirelessCommun.,vol.4,
pp.1759–1772,Jul.2005.
[19] M.Ahsanullah andV.B.Nevzorov, Orderedrandomvariables. NovaScience Publishers,2001.