2088 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9
Transcript of 2088 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9
2088 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
Cooperative Wireless Multicast: PerformanceAnalysis and Power/Location Optimization
H. Vicky Zhao, Member, IEEE, and Weifeng Su, Member, IEEE
AbstractβThe popularity of multimedia multicast/broadcastapplications over wireless networks makes it critical to addressthe error-prone, heterogeneous and dynamically changing natureof wireless channels. A promising solution to combat channelfading is to explore the cooperative diversity in which usersmay help each other forward packets. This paper investigatescooperative multicast schemes that use a maximal ratio combinerto enhance the received signal-to-noise ratio (SNR), and providesa thorough performance analysis. Two relay selection schemesare considered: the distributed and the genie-aided cooperationschemes. We derive the closed-form formulation and the approxi-mations of their average outage probabilities. We also analyze theoptimal power allocation and relay location strategies, and showthat allocating half of the total transmission power to the sourceminimizes the average outage probability. Our analysis andsimulation results show that cooperative multicast gives betterperformance when more relays help forward signals. Cooperativemulticast helps achieve diversity order 2, and user cooperationcan significantly reduce the outage probability, especially inthe high SNR region. Finally, we compare the two cooperationstrategies, and show that distributed cooperative multicast ispreferred since it achieves a lower outage probability withoutintroducing extra overhead for control messages.
Index TermsβCooperative wireless multicast, cooperative re-laying, optimum power allocation, relay location optimization,outage probability.
I. INTRODUCTION
W ITH recent advance in communications, networkingand signal processing technologies, we witness the
emergence of multimedia broadcast and multicast applicationsover wireless networks, where multimedia data are delivered toa group of users simultaneously. Examples include the InternetProtocol television (IPTV) over WiMax [1] and multimediabroadcast/multicast service (MBMS) within 3GPP [2]. How-ever, due to the error-prone and dynamically changing charac-teristics of wireless fading channels, multimedia multicast overwireless medium is very challenging. To further proliferatemultimedia applications over wireless networks, it is of crucialimportance to combat inherent channel fading, path loss, andshadowing effects in wireless channels in order to providereliable and satisfactory service.
Various spatial/temporal/frequency diversity techniques pro-vide effective solutions to enhance the reliability of wireless
Manuscript received September 23, 2009; revised January 21, 2010; ac-cepted April 3, 2010. The associate editor coordinating the review of thispaper and approving it for publication was G. Abreu.
H. V. Zhao is with the Department of Electrical and Computer Engi-neering, University of Alberta, Edmonton, AB T6G 2V4 Canada (e-mail:[email protected]).
W. Su is with the Department of Electrical Engineering, State Univer-sity of New York (SUNY) at Buffalo, Buffalo, NY 14260 USA (e-mail:[email protected]).
Digital Object Identifier 10.1109/TWC.2010.06.091423
links, in which a destination receives multiple distorted ver-sions of the original signal and uses these signals collec-tively to reduce detection error rate and to improve systemperformance. A well-known approach is to use multiple-input-multiple-output (MIMO) techniques to exploit the spatialdiversity [3]. Recently, an emerging concept - cooperativecommunication - provides a new communication view, whereusers in a wireless network help each other forward packetsto improve system performance [4]. Several recent works [5],[6] addressed the information-theoretic aspects of the coop-erative communications, while many efforts have also beenfocused on the design of cooperative communication protocolsto combat severe fading in wireless channels. Specifically,various cooperation protocols were proposed for wirelessnetworks [7], [8] in which a user/node may help others toforward information serving as a relay. As users or nodes in awireless network cooperate with each other by receiving andforwarding information, each user may decode the receivedinformation and forward the decoded symbol, which results ina decode-and-forward (DF) cooperation protocol. Or it maysimply amplify and forward the message, which results in anamplify-and-forward (AF) cooperation protocol. The conceptof user cooperation was also studied in [9], [10] where aspecific two-user cooperation scheme was investigated forCDMA systems and substantial performance gain was demon-strated with comparison to the non-cooperative approach.More recently, the cooperative communication protocols werefurther analyzed and generalized to multi-node scenario (see[11]β[13] and the references therein).
The above prior works focused on optimization of coop-eration schemes for point-to-point communications where theprimary consideration is for one intended receiver/user. Therehave been works on cooperative multicast/broadcast overwireless networks, where a group of intended users receivethe same data/video service from the source [14], [15]. Toachieve user cooperation in multicast/broadcast applications,users who correctly decode the message sent by the sourceserve as relays and forward the message to others [16]. Sameas the point-to-point cooperative communications, focusing onthe 2-hop transmission of data, there are two stages (phases)in cooperative multicast/broadcast: the source (e.g., the basestation) transmits the message in the first stage, while theselected relays forward the message in stage 2. The work in[15] assumed that the selected relays transmit in orthogonalchannels, e.g., TDMA, FDMA or CDMA, and investigatedthe optimal relay scheduling and power allocation strategiesto minimize the total power consumption. The authors in[17] considered cooperative multicast of videos over wireless
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ZHAO and SU: COOPERATIVE WIRELESS MULTICAST: PERFORMANCE ANALYSIS AND POWER/LOCATION OPTIMIZATION 2089
networks. Same as in [15], [16], the relays used TDMA andtook turns to forward packets, and layered video coding wasused to provide users with different video quality depending ontheir channel conditions. The work in [17] studied the optimalrate adaptation and relay selection strategies.
Using orthogonal channels for different relays enables usersin the multicast/broadcast group to combine the signals re-ceived from all nodes and to maximize the received signal-to-noise ratio (SNR), but it reduces the spectral efficiency.To address the tradeoff between received SNR and spectralefficiency and to minimize the control overhead for relaycoordination, the work in [18] adopted the distributed co-operation strategies proposed in [19]. In [18], all users whocorrectly decode the message transmitted by the source serveas relays, and all relays forward packets simultaneously usingrandomized distributed space-time code (RDSTC). In [17]β[19], if an end user does not decode the message correctly instage 1, it uses only data received in stage 2 to decode themessage.
Note that utilizing all received signals, including the onereceived in stage 1, can help further increase the signal-to-noise ratio and improve the system performance. In this paper,we incorporate the maximal ratio combining (MRC) technique[20] into cooperative wireless multicast schemes, and combinethe signals received in both stages to jointly detect the messageat an intended receiver. We provide a thorough performanceanalysis for such MRC-based cooperative multicast schemesand optimize power allocation between source and dynamicrelays. Same as in [18], [19], we assume that all the selectedrelays transmit simultaneously in stage 2. We consider twodifferent relay selection schemes: the distributed cooperationstrategy in [18], [19], and the one used in [17] with a fixednumber of relays at predetermined locations. For each scheme,we analyze its average outage probability and derive theclosed-form formulation. We also obtain an asymptoticallytight approximation for the average outage probability to studythe asymptotic behavior of cooperative multicast schemes inthe high SNR regime. Based on the tight outage probabilityapproximation, we are able to determine the optimal power al-location and relay position strategies for cooperative multicastto minimize the average outage probability. Finally, we com-pare the performance of different multicast schemes, includingthe direct multicast without cooperation, the distributed andthe fixed cooperative multicast schemes, examine when usersshould cooperate with each other, and study their performancetradeoff.
The rest of the paper is organized as follows. Section IIintroduces the direct multicast scheme as well as the coopera-tive multicast schemes that we consider in this paper. SectionIII analyzes the average outage probability of the multicastschemes and derives the closed-form formulation. In SectionIV, we investigate the asymptotic behavior of the cooperativemulticast schemes in the high SNR region, and analyze theoptimal power allocation and relay position strategies. InSection V, we compare the performance of different multicastschemes, examine their tradeoff, and use simulation resultsto validate our theoretical analysis. Finally, conclusions aredrawn in Section VI.
II. SYSTEM MODEL
In this paper, we consider a wireless network with a circularcell of radius π 2. The base station/access point (BS/AP) islocated at the center of the cell and multicasts to π userswho are uniformly distributed in the cell. For user π (1 β€π β€ π), the joint probability density function of the userβsdistance ππ from the BS/AP and the angle ππ is π(ππ, ππ) =ππ/(ππ
22), with 0 β€ ππ β€ π 2 and 0 β€ ππ β€ 2π. The marginal
distribution of ππ is π(ππ) = 2ππ/π 22 with 0 β€ ππ β€ π 2, ππ is
uniformly distributed in [0, 2π], and ππ and ππ are independentof each other. We assume that {(ππ, ππ)}ππ=1 for different usersare independently and identically distributed (i.i.d.). We alsoassume that the wireless link between any two nodes is subjectto independent narrow-band Rayleigh fading, propagation pathloss, and additive white Gaussian noise (AWGN). All nodes inthe network work in the half-duplex mode, that is, they cannottransmit and receive in the same frequency band at the sametime.
A. Direct Multicast
In the direct multicast scheme, the BS/AP broadcasts asignal π₯ with unit power. For user π located at (ππ, ππ), itsreceived signal can be written as
π¦πππ =
βππβπ
π βππ₯+ ππ, (1)
where π is the transmission power in the direct multicastmode, βπ is the channel gain between the BS/AP and userπ, and π is the path loss parameter. Here, the subscript π is theuser index, and the superscript βncβ means no cooperation(direct multicast). βπ is modeled as a zero-mean circularlysymmetric complex Gaussian random variable with unit vari-ance, and ππ is additive white Gaussian noise with varianceπ0. Therefore, the received signal-to-noise ratio (SNR) in (1)is πΎπππ = π β£βπβ£2πβπ
π /π0, which is an exponential randomvariable with parameter ππππ = (π0π
ππ ) /π for a given ππ.
B. Cooperative Multicast
In 2-hop cooperative multicast, users receive more thanone copy of the message and explore cooperative diversityto improve system performance. It consists of two stages: inthe first stage, the BS/AP broadcasts the message; and in thesecond phase, those relays who decode the message correctlyin stage 1 forward the message to other users.1 In this work,we consider a repetition code in stage 2, i.e., all relays forwardthe same message coming from the BS/AP simultaneouslyin stage 2. Finally, for those users who decode the messageerroneously in stage 1, they use MRC to combine the signalfrom the BS/AP in stage 1 and that from relays in stage 2,and jointly decode the message. We also assume that all usersare willing to cooperate, and there is no selfish free riding ormalicious behavior.
1We assume that all relays are synchronized and the delay spread of arrivingsignals is negligible, which is valid in narrow-band wireless communications.
2090 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
1) Distributed Cooperative Multicast Scheme: In the dis-tributed cooperative multicast scheme, the BS/AP first broad-casts a unit-power signal π₯. For user π located at (ππ, ππ), itsreceived signal is
π¦π π,ππ =
βππ ,ππ
βππ βππ₯+ ππ
with SNR πΎπ π,ππ =ππ ,πβ£βπβ£2πβπ
π
π0, (2)
where ππ ,π is the transmission power used by the BS/AP inthe distributed cooperation scheme, and other parameters arethe same as in (1). Here, the subscript π is the user index andthe superscript βsd,dβ means the source-destination channelin the distributed cooperation scheme. Since βπ βΌ ππ© (0, 1),for any given ππ, πΎ
π π,ππ follows an exponential distribution with
parameter ππ π,ππ = (π0πππ ) /ππ ,π.
Assume that π (π β€ π) out of π users decode themessage correctly in stage 1 and πΆπ = {π1, β β β , ππ} denotesthe set including their indices. πΆπ = {1, 2, β β β ,π}βπΆπ ={ππ+1, β β β , ππ} contains the indices of those who decodeincorrectly in stage 1. In stage 2, all users in πΆπ serve asrelays and broadcast the message π₯ simultaneously, using thesame power ππ,π. For a user ππ β πΆπ , its received signal π¦ππ,πππin stage 2 is the superposition of all the π signals broadcastedby the relays, subject to narrow-band Rayleigh channel fading,propagation path loss and additive white Gaussian noise. Soπ¦ππ,πππ
can be written as
π¦ππ,πππ=
βππβπΆπ
βππ,π(πππ ππ)
βπβπππππ₯+ πππ ,
where (πππ ππ)2 = π2ππ + π
2ππ β 2ππππππ cos(πππ β πππ). (3)
In (3), πππ ππ is the distance between the relay ππ β πΆπ and theuser ππ β πΆπ . Here, the superscript βrd,dβ means the relay-destination channel in the distributed cooperation scheme. Weassume that the channel gains {βππππ}ππβπΆπ ,ππβπΆπ are i.i.d.following circularly symmetric complex Gaussian distributionππ© (0, 1) and πππ are additive white Gaussian noise with zeromean and variance π0. Therefore,
π¦ππ,πππ=
βππ,πβ
β²πππ₯+ πππ , (4)
where ββ²ππβ³=
πβπ=1
β(πππ ππ)
βπβππππ βΌ ππ©(0,
πβπ=1
πβπππ ππ
).
Given {(ππ, ππ)} and πΆπ = {π1, π2, β β β , ππ}, the SNR
of π¦ππ,πππis πΎππ,πππ
=(ππ,πβ£ββ²ππ β£2
)/π0, which is an
exponential random variable with parameter πππ,πππ=[
π0
(βππ=1(πππ ππ)
βπ)β1
]/ππ,π.
We assume that the channel gains βππ and ββ²ππ are known
to user ππ β πΆπ , and user ππ uses MRC to combine π¦π π,πππin
(2) and π¦ππ,πππin (3) and jointly decodes the message. Given
(ππ, ππ) and πΆπ , the SNR of the combined signal is
πΎπππ = πΎπ π,πππ+ πΎππ,πππ
=ππ ,πβ£βππ β£2πβπ
ππ+ ππ,πβ£ββ²ππ β£2
π0. (5)
For fair comparison, we let the average total transmissionpower used by the BS/AP and by all relays in the cooper-ative multicast be the same as the total power used in the
direct multicast. That is, we select ππ ,π and ππ,π such thatππ ,π + πΈ[π ]ππ,π = 2π [3], in which πΈ[π ] is the expectednumber of users who decode the message correctly in stage 1and serve as relays.
2) Genie-Aided Cooperative Multicast Scheme: The au-thors in [17] used a fixed number of relays at fixed positionsto help forward packets. Following the work in [17], weconsider a genie-aided cooperative multicast scheme in thispaper, where we can put any number of relays to help forwardthe message and the relays can be placed at optimal locationsto minimize the outage probability.
Assume that there are π β² dedicated relays located on a fixedcircle of radius π 1 with equal separation angle, and the πth(1 β€ π β€ π β²) relayβs location is (π 1, (π β 1)2π/π β²). In stage1, the BS/AP broadcasts a unit-power signal π₯, and for user πlocated at (ππ, ππ), its received signal is
π¦π π,ππ =
βππ ,ππ
βππ βππ₯+ ππ
with SNR πΎπ π,ππ =ππ ,πβ£βπβ£2πβπ
π
π0. (6)
In (6), ππ ,π is the transmission power used by the BS/APin the genie-aided cooperative multicast scheme, and the restterms are the same as in (1). Here, the subscript π is the userindex, and the superscript βsd,gβ means the source-destinationchannel in the genie-aided cooperation scheme. Given ππ, πΎ
π π,ππ
is an exponential random variable with parameter ππ π,ππ =π0π
ππ /ππ ,π . For the πth relay (1 β€ π β€ π β²), given π 1, its
received signal is
π¦π π,ππ =
βππ ,ππ
βπ1 βππ₯+ ππ
with SNR πΎπ π,ππ =ππ ,πβ£βπβ£2π βπ
1
π0. (7)
For each relay, if it correctly decodes the transmitted signalπ₯ in (7), then it will forward the decoded message in stage2. Otherwise, the relay remains idle in stage 2. Assume thatπ out of π β² relays decode the message correctly in stage1, and π πΆπ = {π1, π2, β β β , ππ} is the set including theirindices. In stage 2, relays π1, π2, β β β , ππ transmit the messageπ₯ simultaneously using the same power ππ,π . Similar to theanalysis of the distributed cooperative multicast, for user π at(ππ, ππ), in stage 2, its received signal π¦ππ,ππ is the superpositionof all π relay messages and can be written as
π¦ππ,ππ =β
ππβπ πΆπ
βππ,ππ
βππππβππππ₯+ ππ, (8)
where π2πππ = π2π +π 21 β 2πππ 1 cos
(ππ β (ππ β 1)2π
π
).
In (8), ππππ and βπππ are the distance and the channel gainbetween user π and the ππ th relay, respectively. Here, the super-script βrd,gβ means the relay-destination channel in the genie-
aided cooperation scheme. Denote ββ²πβ³=
βππβπ πΆπ
βπβππππβπππ .
We assume that all channel gains {βπππ} are i.i.d. circularlysymmetric complex Gaussian ππ© (0, 1), and ππ are additivewhite Complex Gaussian ππ© (0, π0). Therefore, given (ππ, ππ)and the indices of the relays who decode correctly in stage 1{π1, π2, β β β , ππ}, ββ²π is also a circularly symmetric complex
ZHAO and SU: COOPERATIVE WIRELESS MULTICAST: PERFORMANCE ANALYSIS AND POWER/LOCATION OPTIMIZATION 2091
Gaussian with zero mean and varianceβ
ππβπ πΆπ πβππππ
. Thus,we have
π¦ππ,ππ =βππ,πβ
β²ππ₯+ ππ
with SNR πΎππ,ππ =(ππ,πβ£ββ²πβ£2
)/π0. (9)
πΎππ,ππ in (9) follows the exponential distribution with parameter
πππ,ππ =
[π0
(βππβπ πΆπ
πβππππ
)β1]/ππ,π.
Assume that user π has perfect knowledge of the channelgains βπ and ββ²π. Given π¦π π,ππ and π¦ππ,ππ as in (6) and in (8),respectively, user π uses MRC to combine these two signals,and the SNR of the combined signal is
πΎππ = πΎπ π,ππ + πΎππ,ππ =ππ ,πβ£βπβ£2πβπ
π + ππ,πβ£ββ²πβ£2π0
. (10)
Similar to the distributed cooperative multicast, given π 1 andπ β², we select ππ ,π and ππ,π such that ππ ,π+πΈ[π ]ππ,π = 2π ,where πΈ[π ] is the expected number of relays who decodecorrectly in stage 1.
III. OUTAGE PROBABILITY ANALYSIS
In this section, we analyze average outage probabilities forthe two cooperative multicast schemes. The outage probabilityis defined as the probability that the maximum mutual infor-mation between the message transmitted by the BS/AP andthe signal received by a user is smaller than a predeterminedthreshold ππ , or equivalently, the probability that the receivedSNR is below a threshold πΎ0 [3]. If the received SNR is higherthan the threshold πΎ0, the user is assumed to be able to decodethe received message with a negligible probability of error.
A. Outage Probability Analysis for Direct Multicast
For comparison, we first derive outage probability for thedirect multicast scheme. In the direct multicast mode, foruser π at location (ππ, ππ), the maximum mutual informationbetween the input π₯ and the output π¦πππ (with i.i.d. circularlysymmetric complex Gaussian inputs) is πΌπππ = log2(1 + πΎ
πππ ),
where πΎπππ follows the exponential distribution with parameterππππ = (π0π
ππ ) /π . Thus, given a threshold ππ on the
maximum mutual information and user πβs location ππ, userπβs outage probability is
ππ [πΌπππ < ππ β£ππ] = ππ
[log2
(1 +
β£βπβ£2πβππ π
π0
)< ππ β£ππ
]
= 1β exp
{β (2ππ β 1)π0π
ππ
π
}. (11)
For a network with π uniformly distributed users, under theassumption that all channel gains {βπ} are i.i.d., the averageoutage probability of direct multicast is
π«ππ =
β«β β β
β« βππ=1 ππ [πΌ
πππ < ππ β£ππ]π
Γπ(π1)π(π2) β β β π(ππ )ππ1 β β β πππ=
β« π 2
0
ππ [πΌππ1 < ππ β£π1] π(π1)ππ1
=
β« π 2
0
(1β exp
{β (2ππ β 1)π0π
π1
π
})2π1π 2
2
ππ1
= 1β 2
ππ 22
(π
π0πΎ0π
)2/π
Ξ
(2
π,π π
2π0πΎ0ππ
), (12)
where πΎ0πβ³=2ππ β 1 and Ξ(π, π₯) =
β« π₯
0πβπ‘π‘πβ1ππ‘ is the
incomplete Gamma function. The second equality in (12) isdue to the i.i.d. assumption of usersβ locations {ππ}.
B. Outage Probability Analysis for the Distributed Coopera-tive Multicast
1) Decoding Results in Stage 1: In distributed cooperativemulticast, for user π located at (ππ, ππ), the SNR of its receivedsignal in stage 1, πΎπ π,ππ , is an exponential random variablewith parameter ππ π,ππ = π0π
ππ /ππ ,π. With i.i.d. circularly
symmetric complex Gaussian input π₯, the maximum mutualinformation between the input and the output π¦π π,ππ in (2) isπΌπ π,ππ = 1
2 log2(1+πΎπ π,ππ ), where the normalization factor 1/2
is due to the fact that the cooperative multicast scheme usestwo time slots to transmit one symbol [21]. If πΌπ π,ππ is largerthan the threshold ππ , we assume that user π can decode themessage with a negligible probability of error in stage 1. Thus,given ππ, the probability that user π decodes incorrectly in stage1 is
π«1,ππ = ππ
[πΌπ π,ππ < ππ
β£β£β£ ππ] = 1β exp
{βπΎ0ππ0π
ππ
ππ ,π
},
(13)where πΎ0π
β³=22ππ β 1. Here, the subscript π is the user
index, and the superscript β1,dβ means the stage-1 decod-ing in the distributed cooperative scheme. We assume thatall channel gains {βπ} are i.i.d., so given the locations ofthe π users {(ππ, ππ)}π=1,β β β ,π , the probability that users inπΆπ = {π1, β β β , ππ} decode correctly and users in πΆπ ={1, 2, β β β ,π}βπΆπ decode incorrectly is
π«1,π(πΆπ , πΆπ β£π1, β β β , ππ )
=βππβπΆπ
(1β π«1,π
ππ
)Γ
βππβπΆπ
π«1,πππ
=βππβπΆπ
exp
{βπΎ0ππ0π
πππ
ππ ,π
}
Γβ
ππβπΆπ
[1β exp
{βπΎ0ππ0π
πππ
ππ ,π
}]. (14)
2) Conditional Outage Probability: For user ππ β πΆπ whodecodes incorrectly in stage 1, given its location {(ππ, ππ)}and πΆπ , the SNR of the maximum-ratio combined signal isπΎπππ = πΎπ π,πππ
+ πΎππ,πππin (5), where πΎπ π,πππ
and πΎππ,πππare expo-
nential random variables with parameters ππ π,πππ= π0π
πππ/ππ ,π
and πππ,πππ= π0
(βππβπΆπ
(πππ ππ)βπ
)β1/ππ,π, respectively. For
user ππ , the maximum mutual information with i.i.d. complexGaussian inputs is [21]
πΌπππ =1
2log2
(1 + πΎπππ
)=
1
2log2
(1 + πΎπ π,πππ
+ πΎππ,πππ
). (15)
Thus, given {(ππ, ππ)} and πΆπ , for user ππ β πΆπ , its conditionaloutage probability is
π«πππ = ππ
[πΌπππ < ππ
β£β£β£ {(ππ, ππ)}, πΆπ ]= ππ
[πΎπ π,πππ
+ πΎππ,πππ< πΎ0π
β£β£β£ {(ππ, ππ)}, πΆπ ]
2092 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
= 1β πππ,πππ
πππ,πππβ ππ π,πππ
exp(βππ π,πππ
β πΎ0π)
+ππ π,πππ
πππ,πππβ ππ π,πππ
exp(βπππ,πππ
β πΎ0π). (16)
3) Average Outage Probability: Given the locations of theπ users {(ππ, ππ)}, we first average π«π
ππin (16) over all users
ππ β πΆπ and all possible decoding results (πΆπ ) in stage 1,which is shown in (17). Here, the superscript πβ² means theconditional outage probability of the distributed cooperativescheme. In (17), the fourth equality is due to the nonnegativityof πΎππ,πππ
and the fact that πΎπ π,πππ+πΎππ,πππ
< πΎ0π implies πΎπ π,πππ<
πΎ0π. The last equality comes from the i.i.d. assumption forchannel gains.
We then integrate π«πβ²in (17) with respect to user locations
{(ππ, ππ)}, and the average outage probability of the distributedcooperative multicast scheme is in (18). To analyze the averageoutage probability, we calculate the first term π΄(ππ ,π) in (18)as follows
π΄(ππ ,π) =
β« π 2
0
β« 2π
0
ππ[πΎπ π,πππ
< πΎ0π
β£β£β£ {(ππ, ππ)}] (19)
Γπ(πππ , πππ)ππππππππ=
β« π 2
0
(1β exp
{βπΎ0ππ0π
πππ
ππ ,π
})2ππππ 2
2
ππππ
= 1β 2
ππ 22
(ππ ,ππ0πΎ0π
)2/π
Ξ
(2
π,π π
2π0πΎ0πππ ,π
),
where Ξ(π, π₯) is the incomplete Gamma function. Apparently,π΄(ππ ,π) depends only on the base stationβs transmission powerππ ,π. It is the same for all possible decoding results in stage1 and for different user index ππ β πΆπ .
Then, for a given decoding result in stage 1 (πΆπ ) and userππ β πΆπ , we calculate the second term in (18), which is shownin (20). From (20), for a given πΆπ , if we consider two differentusers ππ and ππ who decode incorrectly in stage 1, we haveπ΅π (πΆπ , ππ) = π΅π (πΆπ , ππ). In addition, from (20), if we fixthe number of users who decode correctly in stage 1 as π , fortwo different decoding results (πΆ1
π , πΆ1π ) and (πΆ2
π , πΆ2π ) where
β£πΆ1π β£ = β£πΆ2
π β£ = π , we can show thatβ
ππβπΆ1ππ΅π (πΆ1
π , ππ) =βππβπΆ2
ππ΅π (πΆ2
π , ππ). Given the total number of usersπ , for afixed number of users who serve as relays π , there are a totalof
(ππ
)possible decoding results in stage 1 with β£πΆπ β£ = π .
Without loss of generality, we use πΆβ²π (π) = {1, 2, β β β , π}
and ππ = π + 1 β πΆβ²π (π) as an example, and define
π΅πβ³=π΅π(πΆβ²
π (π) = {1, 2, β β β , π}, ππ = π + 1). Therefore,we have
ββ£πΆπ β£=π
βππβπΆπ
π΅π (πΆπ , ππ) =(ππ
)(π βπ)π΅π .
Based on the above analysis, π«π in (18) can be written as
π«π = [π΄(ππ ,π)]π
+
πβ1βπ=1
π βππ
(π
π
)π΅π [π΄(ππ ,π)]
πβπβ1
= [π΄(ππ ,π)]π
+πβ1βπ=1
(π β 1
π
)π΅π [π΄(ππ ,π)]
πβπβ1 . (21)
The first term in (21) corresponds to the scenario where allusers decode incorrectly in stage 1, and thus the conditional
outage probability π«πππ
is 1 for all users. The second term in(21) calculates the average outage probability when π usersdecode correctly in stage 1 for 1 β€ π β€ π β 1. Note thatwhen π = π , that is, all users decode correctly in stage 1,the conditional outage probability is 0. Thus, (21) does notinclude the term corresponding to π =π .
4) Power Constraint: In distributed cooperative multicast,the average transmission power used by the base station andall relays are ππ ,π + πΈ[π ]ππ,π. With i.i.d. channel gains, wehave
πΈ[π ] =
πβπ=1
β« π 2
0
β« 2π
0
ππ[πΌπ ππ > π β£(ππ, ππ)
]Γπ(ππ, ππ)ππππππ
= π
β« π 2
0
exp
{βπΎ0ππ0π
π1
ππ ,π
}2π1π 2
2
ππ1
= π [1βπ΄(ππ ,π)] . (22)
For fair comparison, we select ππ ,π and ππ,π such that ππ ,π+πΈ[π ]ππ,π = 2π , where π is the transmission power usedin direct multicast to transmit one message. Therefore, giventhe BS/APβs transmission power ππ ,π and the total number ofusers π , the transmission power used by each relay is
ππ,π =2π β ππ ,π
π β [1βπ΄(ππ ,π)] . (23)
Note that ππ,π depends on the average (not the instantaneous)number of users who receive correctly in stage 1. It can beprecalculated and remains the same during the transmissions.
To summarize, for the distributed cooperative multicast,given the total number of users π and the base stationβstransmission power ππ ,π, the transmission power used by eachrelay should be determined by (23), and the average outageprobability can be calculated using (21).
C. Outage Probability Analysis for the Genie-Aided Cooper-ative Multicast
The analysis of the outage probability for the genie-aidedcooperative multicast is similar to that in the previous section,and we will present the main results by omitting some details.
1) Decoding Results in Stage 1: Similar to that in SectionIII-B1, in stage 1 of the genie-aided cooperative multicast, foruser π located at (ππ, ππ), the SNR of its received signal is πΎπ π,ππ ,which follows the exponential distribution with parameterππ π,ππ = π0π
ππ /ππ ,π.
For the π β² dedicated relays, from (7), given π 1, the SNRof the received signal by the πth relay in stage 1 πΎπ π,ππ
is an exponential random variable with parameter ππ π,ππ =π0π
π1/ππ ,π . Same as distributed cooperative multicast, we
assume that relay π can decode the message correctly whenthe maximum mutual information πΌπ π,ππ = 1
2 log2 (1 + πΎπ π,ππ )
is larger than the predetermined threshold ππ . Otherwise, therelay is assumed to decode incorrectly in stage 1. Thus, givenπ 1, the probability that relay π decodes erroneously in stage1 is
π«1,ππ = ππ [πΌπ π,ππ < ππ β£π 1]
= ππ
[1
2log2 (1 + πΎ
π π,ππ ) < ππ β£π 1
]
ZHAO and SU: COOPERATIVE WIRELESS MULTICAST: PERFORMANCE ANALYSIS AND POWER/LOCATION OPTIMIZATION 2093
π«πβ²=
1
π
βπΆπ
βππβπΆπ
π«πππ β π«1,π(πΆπ , πΆπ β£{(ππ, ππ)})
=1
π
βπΆπ
βππβπΆπ
ππ[πΎπ π,πππ
+ πΎππ,πππ< πΎ0π
β£β£β£πΆπ , {(ππ, ππ)}]Γππ
[πΎπ π,πππ
> πΎ0π βππ β πΆπ , πΎπ π,πππ< πΎ0π βππ β πΆπ
β£β£β£ {(ππ, ππ)}]=
1
π
βπΆπ
βππβπΆπ
ππ[πΎπ π,πππ
+ πΎππ,πππ< πΎ0π, πΎ
π πππ < πΎ0π, πΎ
π π,πππ
> πΎ0π βππ β πΆπ ,
πΎπ πππ < πΎ0π βππ β πΆπ and ππ β= ππβ£β£β£ {(ππ, ππ)}]
=1
π
βπΆπ
βππβπΆπ
ππ[πΎπ π,πππ
+ πΎππ,πππ< πΎ0π, πΎ
π π,πππ
> πΎ0π βππ β πΆπ ,
πΎπ π,πππ< πΎ0π βππ β πΆπ and ππ β= ππ
β£β£β£ {(ππ, ππ)}]=
1
π
βπΆπ
βππβπΆπ
ππ[πΎπ π,πππ
+ πΎππ,πππ< πΎ0π
β£β£β£πΆπ , {(ππ, ππ)}]Γ βππβπΆπ
ππ[πΎπ π,πππ
> πΎ0π
β£β£β£ {(ππ, ππ)}]Γ
βππβπΆπ ,ππ β=ππ
ππ[πΎπ π,πππ
< πΎ0π
β£β£β£ {(ππ, ππ)}] . (17)
π«π =
β«β β β
β«π«πβ²
π(π1, π1) β β β π(ππ , ππ ) ππ1ππ1 β β β ππππππ
=1
π
πβπ=0
ββ£πΆπ β£=π
βππβπΆπ
βππβπΆπ ,ππ β=ππ
β« π 2
0
β« 2π
0
ππ[πΎπ π,πππ
< πΎ0π
β£β£β£ {(ππ, ππ)}] π(πππ , πππ)πππππππποΈΈ οΈ·οΈ· οΈΈβ³=π΄(ππ ,π)
Γ
[β«β β β
β«π«ππππ(πππ , πππ )
βππβπΆπ
(ππ
[πΎπ π,πππ
> πΎ0π
β£β£β£ {(ππ, ππ)}] π(πππ , πππ)) ππ1πππ1 β β β ππππ ππππ ππππππππ]
οΈΈ οΈ·οΈ· οΈΈβ³=π΅π (πΆπ ,ππ)
=1
π
πβπ=0
ββ£πΆπ β£=π
βππβπΆπ
[π΄(ππ ,π)]πβπβ1π΅π (πΆπ , ππ). (18)
π΅π (πΆπ , ππ) =
β«β β β
β«ππ
[πΎπ π,πππ
+ πΎππ,πππ< πΎ0π
β£β£β£πΆπ , {(ππ, ππ)}] π(πππ , πππ )ΓβππβπΆπ
(ππ
[πΎπ π,πππ
> πΎ0π
β£β£β£ {(ππ, ππ)}] π(πππ , πππ)) πππ1πππ1 β β β ππππ ππππ ππππππππ . (20)
= 1β exp
{βπΎ0ππ0π
π1
ππ ,π
}οΈΈ οΈ·οΈ· οΈΈ
β³=πΆ(π 1)
. (24)
Here, the subscript π is the relay index, and the superscriptβ1,gβ means the stage-1 decoding in the genie-aided coopera-tion scheme. Assume that π out of π β² relays decode correctlyin stage 1, and π πΆπ = {π1, β β β , ππ} contains their indices.Thus, the probability that relays in π πΆπ decode correctly instage 1 is
π«1,π(π πΆπ) =β
ππβπ πΆπ
(1β π«1,πππ
)β
ππ ββπ πΆπ
π«1,πππ
= [1β πΆ(π 1)]ππΆ(π 1)
π β²βπ . (25)
2) Outage Probability: After maximum-ratio combining,for user π at (ππ, ππ), the SNR of the combined signal is
πΎππ = πΎπ π,ππ + πΎππ,ππ , where πΎπ π,ππ and πΎππ,ππ are exponentialrandom variables with parameter ππ π,ππ = π0π
ππ /ππ ,π and
πππ,ππ =
[π0
(βππβπ πΆπ
πβππππ
)β1]/ππ,π, respectively. For user
π, the maximum mutual information with i.i.d. complex Gaus-sian inputs is πΌππ = 1
2 log2 (1 + πΎππ ), and the conditional outage
probability is
π«ππ = ππ [πΌππ < ππ β£ (ππ, ππ), π πΆπ ]
= π[πΎπ π,ππ + πΎππ,ππ < πΎ0π
β£β£β£ (ππ, ππ), π πΆπ ]= 1β πππ,ππ
πππ,ππ β ππ π,ππ
exp(βππ π,ππ β πΎ0π
)+
ππ π,ππ
πππ,ππ β ππ π,ππ
exp(βπππ,ππ β πΎ0π
). (26)
2094 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
Note that when π πΆπ = β and π = 0, that is, all π β² relaysdecode incorrectly in stage 1, then πΎππ,ππ = 0 and user πβsoutage probability is
ππ[πΎπ π,ππ < πΎ0π
β£β£β£ (ππ, ππ)]= 1β exp
{βπΎ0ππ0π
ππ
ππ ,π
}= π΄(ππ ,π), (27)
where π΄(β ) is defined in (19).Averaging π«π
π over all possible decoding results in stage 1(π πΆπ ) and user πβs location (ππ, ππ), the average outage prob-ability of genie-aided cooperative multicast is in (28). In (28),πΆ(π 1) and π«π
π are defined in (24) and (26), respectively. Thesuperscript βgβ means the genie-aided cooperation scheme. In(28), the first term corresponds to the scenario where all relaysdecode incorrectly in stage 1, and the second term considersthe scenario where there are 1 β€ π β€ π β² relays that decodecorrectly in stage 1.
3) Power Constraint: Same as in Section III-B4, in thegenie-aided cooperative multicast, we adjust ππ ,π and ππ,πsuch that ππ ,π +πΈ[π ]ππ,π = 2π , where πΈ[π ] is the averagenumber of relays who decode correctly in stage 1. From (24),πΈ[π ] = π β²(1 β π«1,π
π ) = π β²(1 β πΆ(π 1)). Therefore, giventhe base stationβs transmission power ππ ,π and π 1, we have
ππ,π =2π β ππ ,π
π β² β exp (βπΎ0ππ0π π1/ππ ,π)
. (29)
Same as ππ,π in distributed cooperative multicast, ππ,π dependson the average number of relays who decode correctly in stage1, and can be precalculated.
To summarize, for the genie-aided cooperative multicast,given the total number of relays π β² and the relay locationπ 1, the transmission power used by the base station and thatused by each relay should satisfy (29), and the average outageprobability is given by (28).
IV. OPTIMUM POWER ALLOCATION AND RELAY
LOCATION
This section considers optimal power allocation for cooper-ative multicast schemes to minimize their outage probability.For genie-aided cooperative multicast, we also investigate theoptimal relay location π 1 to maximize the system perfor-mance.
A. Distributed Cooperative Multicast
In distributed cooperative multicast, given the power con-straint in (23), we look for the optimal ππ ,π that minimizesthe average outage probability in (21). Due to the complexityof (21), it is difficult to find a closed-form solution forthe optimal ππ ,π directly from (21). In the following, weconsider a high SNR scenario. We obtain an asymptoticallytight approximation of (21) to study its asymptotic behaviorin the high SNR region, and derive an asymptotically optimalpower allocation scheme.
1) Approximation of the Average Outage Probability: Wefirst analyze the term π΄(β ) in (21). From the definition in (19),
π΄(ππ ,π) =
β« π 2
0
(1β exp
{βπΎ0ππ0π
πππ
ππ ,π
})2ππππ 2
2
ππππ
ββ« π 2
0
π0πΎ0πππ ,π
ππππ2ππππ 2
2
ππππ
=π0πΎ0πππ ,π
β 2π π2
π + 2. (30)
Here, we use the first order Taylor series approximationexp(π₯) β 1 + π₯ for π₯ close to 0, which is tight at high SNRand when the ratio ππ ,π/π0 is large. Similarly, to simplify theterm π΅π in (21), given πΆπ = {1, 2, β β β , π}, with high πππ where ππ ,π/π0 and ππ,π/π0 are large, we use the followingapproximations:
ππ[πππ,ππ+1 + π
π π,ππ+1 β€ πΎ0π
β£β£β£ {(ππ, ππ)}, πΆπ ]= 1β πππ,ππ+1
πππ,ππ+1 β ππ π,ππ+1
exp(βππ π,ππ+1 β πΎ0π
)+
ππ π,ππ+1
πππ,ππ+1 β ππ π,ππ+1
exp(βπππ,ππ+1 β πΎ0π
)β 1
2πππ,ππ+1 β ππ π,ππ+1 β πΎ20π, (31)
and ππ[ππ π,ππ > πΎ0π
β£β£β£ (ππ, ππ)]= exp
{βπΎ0ππ0π
ππ
ππ ,π
}β 1β πΎ0ππ0π
ππ
ππ ,πβ 1 (32)
for 1 β€ π β€ π . Therefore, we have (33). Note that in (33),π·π depends only on π and π 2 but not other parameters.
Substituting (30) and (33) into (21), we have
π«π β(π0πΎ0πππ ,π
β 2π π2
π + 2
)π
+
πβ1βπ=1
(π β 1
π
)π2
0 πΎ20π
2ππ ,πππ,π
Γπ·π
(π0πΎ0πππ ,π
β 2π π2
π + 2
)πβπβ1
. (34)
Note that in (34), the lowest order of π0/ππ ,π and π0/ππ,πis 2 when π = π β 1. Therefore, for high πππ with largevalues of ππ ,π/π0 and ππ,π/π0, we can ignore the third andall other higher order terms and further simplify π«π as
π«π β π«π =π2
0 πΎ20π
2ππ ,πππ,ππ·πβ1, (35)
where π·πβ1 is a constant and depends only on the totalnumber of user π and the radius π 2.
2) Optimal Power Allocation: Given the above approxi-mation of the average outage probability, we can determinethe optimal power allocation between the BS/AP and therelays. From (35), we have π«π = π2
0 πΎ20π·πβ1/ (2ππ ,πππ,π),
where π·πβ1 depends only on the total number of users π .Therefore, minimization of π«π is equivalent to maximizationof the product ππ ,πππ,π under the constraint that ππ ,π and ππ,πsatisfy (23).
ZHAO and SU: COOPERATIVE WIRELESS MULTICAST: PERFORMANCE ANALYSIS AND POWER/LOCATION OPTIMIZATION 2095
π«π =
β« π 2
0
β« 2π
0
βπ πΆπ
π«ππ π«1,π(π πΆπ)π(ππ, ππ)ππππππ
=
π β²βπ=0
ββ£π πΆπ β£=π
[1β πΆ(π 1)]π[πΆ(π 1)]
π β²βπ
β« π 2
0
β« 2π
0
π«ππ
ππππ 2
2
ππππππ
= πΆ(π 1)π β²π΄ (ππ ,π) +
π β²βπ=1
[1β πΆ(π 1)]π[πΆ(π 1)]
π β²βπβ
β£π πΆπ β£=π
β« π 2
0
β« 2π
0
π«ππ
ππππ 2
2
ππππππ, (28)
π΅π ββ«
β β β β«
1
2πππ,ππ+1 β ππ π,ππ+1 β πΎ20π β π1 β β β ππ+1
(ππ 22)
π+1ππ1ππ1 β β β πππ+1πππ+1
=
β«β β β
β«1
2
π0πππ+1
ππ ,πβ π0
(βππ=1(π(π+1)π)
βπ)β1
ππ,πβ πΎ20π β π1 β β β ππ+1
(ππ 22)
π+1ππ1ππ1 β β β πππ+1πππ+1
=π2
0 πΎ20π
2ππ ,πππ,ππ·π ,
where π·πβ³=
β«β β β
β«πππ+1
[πβπ=1
(π2π + π
2π+1 β 2ππππ+1 cos(ππ β ππ+1)
)βπ/2
]β1
Γπ1 β β β ππ+1
(ππ 22)
π+1ππ1ππ1 β β β πππ+1πππ+1. (33)
To simplify the power constraint (23), we need to simplifythe denominator in (23). From (30), with high SNR and largevalues of ππ ,π/π0, we can use the approximation
1βπ΄(ππ ,π) β 1β 1
ππ ,πβ π0πΎ0π β 2π π
2
π + 2= 1β π
ππ ,π, (36)
where πβ³=π0πΎ0π2π π
2/(π + 2). Therefore, we have
ππ,π =2π β ππ ,π
π β [1βπ΄(ππ ,π)]β 2π β ππ ,π
π β (1 β π/ππ ,π) =ππ ,π(2π β ππ ,π)π(ππ ,π β π) . (37)
Consequently, to minimize the outage probability of dis-tributed cooperative multicast, we should select ππ ,π to max-
imize πΊπ(π)β³=π2(2π β π)/[π(πβ π)]. To find the optimal
ππ ,π, we let
0 =βπΊπ(π)
βπ
β£β£β£β£π=πβ
π ,π
= π2π2 β (2π + 3π)π+ 4ππ
π(πβ π)2β£β£β£β£π=πβ
π ,π
,
β 2π β2π ,π β (2π + 3π)π β
π ,π + 4ππ = 0,
β π βπ ,π =
2π + 3πΒ±β(2π + 3π)2 β 32ππ
4. (38)
With high SNR and π β« π, the optimal solution in (38) can beapproximated as π β
π ,π β π , that is, the BS/AP uses half of thetotal transmission power. The other half is evenly distributedamong relays in the statistical sense, and we use the averagenumber of relays πΈ[π ] and (23) to calculate the transmissionpower used by each relay ππ,π.
B. Genie-Aided Cooperative Multicast
To analyze the optimal power allocation and relay locationsfor the genie-aided cooperative multicast, similar to the pre-vious section, we first consider the high SNR scenario with
large transmission power and find an approximation of theaverage outage probability. Then, we use the approximatedoutage probability to find the optimal power allocation andrelay location strategies.
1) Approximation of the Average Outage Probability: Fora given π 1, for high SNR with large ππ ,π/π0, we start withthe first term corresponding to π = 0 in (28), and use (30)to approximate it as
πΆ(π 1)π β²π΄(ππ ,π)
=
[1β exp
(βπΎ0ππ0π
π1
ππ ,π
)]π β²
π΄(ππ ,π)
β(πΎ0ππ0π
π1
ππ ,π
)π β²π0πΎ0πππ ,π
β 2π π2
π + 2
= 2
(π0πΎ0πππ ,π
)π β²+1(π π β²
1 π 2)π
π + 2. (39)
To simplify the second term in (28) corresponding to the sce-nario where 1 β€ π β€ π β², we first need to find approximationsfor π«π
π in (26). Similar to the approximation of (31) in theprevious Section, for high SNRs, we can approximate π«π
π as
π«ππ = ππ
[πΎπ π,ππ + πΎππ,π<πΎ0π
π
β£β£β£ (ππ, ππ), π πΆπ ] (40)
= 1β πππ,ππ
πππ,ππ β ππ π,ππ
exp(βππ π,ππ β πΎ0π
)+
ππ π,ππ
πππ,ππ β ππ π,ππ
exp(βπππ,ππ β πΎ0π
)
β 1
2πππ,ππ ππ π,ππ πΎ20π =
1
2
π20 πΎ
20ππ
ππ
ππ ,πππ,π
ββ β
ππβπ πΆπ
πβππππ
ββ β1
.
In addition, with large ππ ,π/π0, πΆ(π 1) = 1 βexp
(β πΎ0ππ0π
π1
ππ ,π
)β πΎ0ππ0π
π1/ππ ,π << 1 and we can have
2096 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
π β²βπ=1
[1β πΆ(π 1)]π[πΆ(π 1)]
π β²βπβ
β£π πΆπ β£=π
β« π 2
0
β« 2π
0
π«ππ
1
ππ 22
ππππππ
βπ β²βπ=1
(πΎ0ππ0π
π1
ππ ,π
)π β²βπ ββ£π πΆπ β£=π
β« π 2
0
β« 2π
0
1
2
π20 πΎ
20ππ
ππ
ππ ,πππ,π
ββ β
ππβπ πΆπ
πβππππ
ββ β1
ππππ 2
2
ππππππ
=
π β²βπ=1
(πΎ0ππ0π
π1
ππ ,π
)π β²βπ1
2
(πΎ20ππ
20
ππ ,πππ,π
)ππ ,
where ππβ³=
ββ£π πΆπ β£=π
β« π 2
0
β« 2π
0
πππ
ββ β
ππβπ πΆπ
πβππππ
ββ β1
ππππ 2
2
ππππππ. (41)
1 β πΆ(π 1) β 1. Therefore, the second term in π«π can beapproximated as in (41). Note that ππ depends on the relaylocation π 1 and the relayβs decoding result in stage 1 π πΆπ ,but not the transmission powers.
Combining the results in (39) and (41), the average outageprobability of the genie-aided cooperative multicast scheme is
π«π β 2
(π0πΎ0πππ ,π
)π β²+1(π π β²
1 π 2)π
π + 2(42)
+π β²βπ=1
1
2
(πΎ20ππ
20
ππ ,πππ,π
)β (πΎ0ππ0π
π1
ππ ,π
)π β²βπ
ππ .
Same as in (34), the lowest order of π0/ππ ,π and π0/ππ,πin (42) is 2 when π = π β². Thus, for high SNR with largetransmission powers, we can further simplify the expressionin (42) by omitting the third and higher order terms, and wehave
π«π β π«πβ³=1
2
π20 πΎ
20π
ππ ,πππ,πππ β² . (43)
Here, ππ β² depends on the total number of relays π β² and theirlocations π 1 but not the transmission powers.
2) Optimal Power Allocation: To find the asymptotic op-timal power allocation scheme, for a given relay locationπ 1, similar to the analysis in Section IV-A2, we find theoptimal ππ ,π that minimizes the approximated average outageprobability π«π in (43).
Note that in (43), ππ β² does not depend on ππ ,π . Thus,to minimize π«π = (π2
0 πΎ20πππ β²)/(2ππ ,πππ,π), it is equiv-
alent to maximize the product ππ ,πππ,π under the con-straint that ππ ,π and ππ,π satisfy (29). In the high SNRregion with large ππ ,π/π0, we can have the approximationexp (βπΎ0ππ0π
π1/ππ ,π) β 1 β πΎ0ππ0π
π1/ππ ,π. Therefore,
following the same analysis as in Section IV-A2, to minimizethe outage probability, we should select ππ ,π to maximize
πΊπ(π)β³=π2(2π β π)/[π β²(π β πβ²)], where πβ²
β³=πΎ0ππ0π
π1 . That
is, the optimal π βπ ,π should satisfy βπΊπ(π)
βπ
β£β£β£π=πβ
π ,π
= 0, which
gives π βπ ,π =
2π+3πβ²Β±β
(2π+3πβ²)2β32πβ²π4 . In the high SNR
region where π β« πβ², we can further have the approximationthat π β
π ,π β π , that is, the BS/AP uses half of the totaltransmission power. The rest half is evenly distributed amongrelays in the statistical sense, and (29) is used to calculate thetransmission power used by each relay ππ,π.
TABLE IASYMPTOTIC OPTIMAL RELAY LOCATIONS FOR THE GENIE-AIDED
COOPERATIVE MULTICAST SCHEME.
Optimal RelayLocation
π β² = 2 π β² = 3 π β² = 4 π β² = 5 π β² = 6
π’β = π β1/π 2 0.485 0.665 0.735 0.765 0.785
3) Optimal Relay Location: To find the asymptotic opti-mal relay location π 1, with high SNR, following the abovediscussion on the optimal power allocation, we let ππ ,π = π ,and
ππ,π β (2π β ππ ,π)ππ ,ππ β²(ππ ,π β πβ²) =
π 2
π β²(π β πβ²) , (44)
where πβ² = πΎ0ππ0π π1 . Thus, given the number of relays π β²,
we select π 1 to minimize
π«π =1
2
πΎ20ππ20
ππ ,πππ,πππ β² β 1
2
πΎ20ππ20π
β²
π 3(π β πβ²)ππ β² , (45)
or equivalently, to minimize (π βπβ²)ππ β² . With high SNR andπ β« πΎ0ππ0π
π2 > π
β², the problem can be simplified to selectπ 1 to minimize ππ β² . With 0 β€ ππ β€ π 2 and 0 β€ π 1 β€ π 2,let π‘ = ππ/π 2 β [0, 1] and π’ = π 1/π 2 β [0, 1]. Thus, ππ β² in(41) can be rewritten as in (46), and minimization of ππ β² isequivalent to minimization of π§π β² .
Note that π§π β² is a function of the number of relays π β² andthe normalized relay position π’, but not other parameters. Foreach π β², Monte Carlo and numerical methods are used to findthe optimal π’β that minimizes π§π β² . Table I lists the optimalπ’β when π β² takes different values from 2 to 6.
V. SIMULATION RESULTS AND PERFORMANCE
COMPARISON
A. Distributed Cooperative Multicast
We first use Monte Carlo simulations to calculate the exactand the approximated average outage probabilities of thedistributed cooperative multicast, and compare them in Fig.1. We assume that the radius of the circle is π 2 = 100, thepath loss parameter is π = 2.6, and the outage threshold isππ = 4. We consider three cases with π = 10, π = 100and π = 250 users, respectively, and let π/π0 vary from75dB to 95dB. We let the transmission power used by theBS/AP equal to the average transmission power used by allrelays. We use 20,000 Monte Carlo simulations to find π΅π
ZHAO and SU: COOPERATIVE WIRELESS MULTICAST: PERFORMANCE ANALYSIS AND POWER/LOCATION OPTIMIZATION 2097
ππ β² =
β« π 2
0
β« 2π
0
πππ
β‘β£ π β²βπ=1
(π2π +π
21 β 2πππ 1 cos(ππ β (π β 1) β 2π
π β² )
)βπ/2β€β¦β1
ππππ 2
2
ππππππ
= π 2π2
β« 1
0
β« 2π
0
π‘π
β‘β£ π β²βπ=1
(π‘2 + π’2 β 2π‘π’ cos(ππ β (π β 1) β 2π
π β² )
)βπ/2β€β¦β1
π‘
ππππππ‘
οΈΈ οΈ·οΈ· οΈΈβ³=π§πβ²
. (46)
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10β5
10β4
10β3
10β2
10β1
100
P/N0 (dB)
Ave
rage
Out
age
Pro
babi
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Exact solution, M=10Approximation, M=10Exact solution, M=100Approximation, M=100M=10
M=100
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10β5
10β4
10β3
10β2
10β1
P/N0 (dB)
Ave
rage
Out
age
Pro
babi
lity
Exact solution, M=250Approximation, M=250
Fig. 1. Comparison of the exact average outage probability π«π and the approximation π«π for the distributed cooperative multicast. π 2 = 100, π = 2.6,and ππ = 4. ππ ,π = π .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
β5
10β4
10β3
10β2
10β1
100
Ξ±=Ps,d
/(2 P)
Ave
rage
Out
age
Pro
babi
lity
P/N0=75 dB
P/N0=85 dB
P/N0=95 dB
Fig. 2. Optimal power allocation in the distributed cooperative multicast.π 2 = 100, π = 2.6, ππ = 4, and π = 10. π/π0 = 75ππ΅, 85ππ΅, 95ππ΅.
The x axis is πΌβ³=ππ ,π/(2π ).
in π«π for 1 β€ π β€ π β 1 and π·πβ1 in π«π. From Fig. 1,we can see that π«π matches π«π very well, especially whenπ/π0 is large. We observe similar trend for other values of theparameters. Figure 2 plots the exact average outage probability
π«π versus different power allocation strategies πΌβ³=ππ ,π/(2π )
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Analytical results, M=10Simulation results, M=10Analytical results, M=100Simulation results, M=100Analytical results, M=250Simulation results, M=250
M=10
M=100
M=250
Fig. 3. Simulation results of the distributed cooperative multicast schemes.π 2 = 100, π = 2.6, ππ = 4, and ππ ,π = π .
in the distributed cooperative multicast scheme. In Fig. 2, weuse π = 10 with π/π0 = 75ππ΅, 85ππ΅, and 95ππ΅ as anexample, and we observe the same trend for other values ofthe parameters. From Fig. 2, setting πΌ to be around 0.5 canhelp minimize the average outage probability, especially withhigh SNR.
Next, we use simulation results to validate our analysis
2098 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
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β5
10β4
10β3
10β2
10β1
100
P/N0 (dB)
Ave
rage
Out
age
Pro
babi
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Exact solution, Nβ=2Approximation, Nβ=2
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β5
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10β3
10β2
10β1
P/N0 (dB)
Ave
rage
Out
age
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babi
lity
Exact solution, Nβ=4Approximation, Nβ=4
(a): π β² = 2 relays (b): π β² = 4 relays
Fig. 4. Comparison of the exact and the approximated average outage probabilities for the genie-aided cooperative multicast scheme. π 2 = 100, π = 2.6,and ππ = 4. ππ ,π = π , and π 1 = π 2/2 = 50.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910
β6
10β5
10β4
10β3
10β2
10β1
100
Ξ± = Ps,g
/(2P)
Ave
rage
Out
age
Pro
babi
lity
R1/R
2=0.2
R1/R
2=0.5
R1/R
2=0.8
P/N0=95dB
P/N0=85dB
P/N0=75dB
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910
β6
10β5
10β4
10β3
10β2
10β1
100
u=R1/R
2
Ave
rage
Out
age
Pro
babi
lity
2 Relays4 Relays6 Relays
P/N0=95dB
P/N0=85dB
P/N0=75dB
(a) (b)
Fig. 5. (a): Optimal power allocation in genie-aided cooperative multicast with π β² = 4. The x axis is πΌβ³=ππ ,π/(2π ). (b): Asymptotic optimal relay
positions for the genie-aided cooperative multicast scheme with ππ ,π = π and π β² = 2, 4, 6. In both (a) and (b), π 2 = 100, π = 2.6, ππ = 4, andπ/π0 = 75ππ΅, 85ππ΅, 95ππ΅.
of the distributed cooperative schemeβs outage probability.In our simulations, we randomly generate π user locationsuniformly distributed inside a circle with radius π 2 = 100.All channel gains are generated independently following thecomplex Gaussian distribution ππ© (0, 1). We let ππ ,π = πand allocate half of the total transmission power to theBS/AP. Figure 3 compares the simulation results based on 107
simulation runs with the analytical results in Section III. It canbe seen that the simulation results match our analytical resultsvery well. In addition, from Fig. 3, distributed cooperativemulticast achieves a smaller outage probability when the totalnumber of users π is larger and when more users help relaythe message in stage 2. For example, there is a 3dB gain ifπ is increased from 10 to 100, and another 1dB gain if π
is further increased to 250. Therefore, distributed cooperativemulticast performs better in denser networks with more users.
B. Genie-aided Cooperative Multicast
For the genie-aided cooperative multicast scheme, Fig. 4compares the approximated outage probability π«π with theexact π«π, both calculated using Monte Carlo simulations. Thesystem setup is similar to that in Fig. 1. We use ππ ,π = π andπ 1 = π 2/2 as an example, and we observe the same trendfor other power allocation and relay locations. We considertwo scenarios with π β² = 2 and π β² = 4 dedicated relays,respectively, and other number of π β² gives the same trend. Itcan be seen that the approximation in (43) is very close to theexact outage probability, especially in the high SNR region.
ZHAO and SU: COOPERATIVE WIRELESS MULTICAST: PERFORMANCE ANALYSIS AND POWER/LOCATION OPTIMIZATION 2099
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10β5
10β4
10β3
10β2
10β1
100
P/N0 (dB)
Ave
rage
Out
age
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babi
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Analytical, 2 relaysSimulation, 2 relaysAnalytical, 4 relaysSimulation, 4 relaysAnalytical, 6 relaysSimulation, 6 relays
Fig. 6. Simulation results of the genie-aided cooperative multicast schemes.π 2 = 100, π = 2.6, ππ = 4, and ππ ,π = π . The radius π 1 is selectedaccording to Table I.
For genie-aided cooperative multicast, Fig. 5(a) plots theexact average outage probabilityπ«π in (28) for different powerallocation strategies πΌ = ππ ,π/2π . The system setup in Fig.5(a) is similar to that in Fig. 4, and we use π β² = 4 relays withπ 1 = 0.2π 2, π 1 = 0.5π 2, and π 1 = 0.8π 2 as an example.We observe a similar trend for other values of the parameters.From Fig. 5(a), allocating half of the transmission power tothe BS/AP helps minimize the average outage probability,especially when π/π0 is large. Figure 5(b) shows the exactaverage outage probability π«π for different relay positions anddifferent SNRs. We can see from Fig. 5(b) that in the highSNR region, the selected π’β in Table I helps minimize theoutage probability. Note that we focus on the high SNR regionin this work, and our analysis of optimal power allocation andrelay position is based on our approximated outage probabilityπ«π, which matches the exact value π«π very well only in thehigh SNR region (e.g., when π/π0 β₯ 85ππ΅). Therefore, theoptimal power allocation and relay positions in the low SNRregion (e.g., when π/π0 = 75ππ΅) may be different fromour analysis in Section IV-B; while our simulation results areconsistent with our analysis in the high SNR region (e.g., whenπ/π0 = 85ππ΅ and π/π0 = 95ππ΅).
We then use simulations results to validate our analysis onthe outage probability of the genie-aided cooperative multicastscheme. Figure 6 compares the simulation results based on 107
simulation runs with the analytical results in Section III. Thesimulation setup is the same as in Fig. 3, and we follow TableI to set the relay location π 1. The simulation results match ouranalytical results very well, and the genie-aided cooperativemulticast scheme helps reduce the outage probability by alarge amount when the number of dedicated relays increases,especially in the high SNR region. In Fig. 6, we observe amaximum of 2dB gain when π β² is increased from 2 to 6.
C. Performance Comparison
We then compare the outage probability of different mul-ticast schemes with different SNRs, as shown in Fig. 7.The system setup in Fig. 7 is the same as that in Fig. 3
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Direct multicastGenieβaided, 2 relaysGenieβaided, 6 relaysDistribted, M=10Distributed, M=100Distributed, M=250
distributed, M=250
distributed, M=10
distributed, M=100
O: Genieβaided,2 relays
x: Genieβaided, 6 relays
direct multicast
Fig. 7. Performance comparison of different multicast schemes. π 2 = 100,π = 2.6, ππ = 4, and ππ ,π = ππ ,π = π . In the genie-aided cooperativemulticast, the relay positions are selected based on the results in Table I.
and 6. Comparing the performance of distributed cooperativemulticast with that of direct multicast, we observe that inthe high SNR region, user cooperation can significantly helpreduce the outage probability. For example, with π/π0 =95ππ΅, distributed cooperative multicast can help reduce theaverage outage probability from π(10β4) to π(10β6) whenthere are hundreds of users in the network. We also observethat the simulation curves of different cooperative multicastschemes go in parallel in high-SNR regions and they show ahigher diversity order than the direct multicast scheme. Thisis consistent with the theoretical analysis in (12), (35) and(43), which show that the direct multicast scheme providesonly diversity order 1 and the cooperative multicast schemesoffer diversity order 2 in the outage probability performance.Furthermore, we observe that cooperation does not alwaysgive the best performance. With the system setup as in Fig.7, for the distributed cooperation strategy, when there are tenusers in the network, direct multicast is beneficial when theSNR π/π0 is below 81dB; and with π = 100 users, usercooperation gives a smaller outage probability when SNR islarger or equal to 76dB. Similarly, the genie-aided cooperationscheme reduces the outage probability only when π/π0 islarger than 81dB.
When comparing the two cooperative multicast schemes,for a sparse network with fewer users, when the number ofdedicated relays in genie-aided cooperative multicast π β² issmall, the two cooperative multicast schemes have similar per-formance. For denser networks with large number of users, thedistributed cooperative multicast scheme achieves a smalleroutage probability and has a 1dB to 4dB gain comparedwith genie-aided cooperative multicast. Note that the genie-aided cooperative multicast scheme relies on the existence ofdedicated relays that we can put in any locations. Therefore,distributed cooperative multicast is often preferred since itgives a better performance without the help of dedicated re-lays. Another advantage of the distributed cooperation schemeis that it is easy to implement and does not introduce extracommunication overhead for control messages.
2100 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010
VI. CONCLUSIONS
In this paper, we investigated two MRC-based cooperativemulticast schemes over wireless networks, where a group ofusers receive the same data from a BS/AP and they maycooperate with each other to combat channel fading and toachieve more reliable QoS. We analyzed the outage probabilityperformance of the two cooperative multicast schemes andoptimized their power allocation.
We first analyzed the outage probabilities of the distributedand the genie-aided cooperative multicast schemes. We derivedclosed-form formulations for the outage probabilities, andprovided approximations to show the asymptotic performanceof the cooperative multicast schemes. Based on the asymptoti-cally tight outage probability approximations, we obtained theoptimal cooperation strategies. It turns out that allocating halfof the total transmission power to the BS/AP minimizes theoutage probability of cooperative multicast schemes, and theother half is evenly distributed among relays in the statisticalsense. We also determined the optimal relay locations forgenie-aided cooperative multicast.
We then compared the performance of different multicastschemes. For the distributed cooperative scheme, we observea smaller outage probability for denser networks with moreusers and a larger average number of relays. Similarly, theoutage probability of the genie-aided cooperation schemedecreases as the number of dedicated relays increases. Com-pared to the direct multicast scheme, the cooperative multicastschemes achieve diversity order 2, and user cooperation canhelp significantly improve the performance especially whenthe signal-to-noise ratio is high. Compared with the genie-aided scheme, the distributed cooperation scheme gives a 1dBto 4dB performance gain and, therefore, it is often preferredto maximize the performance without the help of dedicatedrelays and without extra overhead for control signals.
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H. Vicky Zhao (Mβ05) received the B.S. and M.S.degree from Tsinghua University, China, in 1997and 1999, respectively, and the Ph. D degree fromUniversity of Maryland, College Park, in 2004, all inelectrical engineering. She was a Research Associatewith the Department of Electrical and ComputerEngineering and the Institute for Systems Research,University of Maryland, College Park from Jan.2005 to July 2006. Since August 2006, she hasbeen an Assistant Professor with the Department ofElectrical and Computer Engineering, University of
Alberta, Edmonton, Canada.Dr. Zhaoβs research interests include information security and forensics,
multimedia social networks, digital communications and signal processing. Dr.Zhao received the IEEE Signal Processing Society (SPS) 2008 Young AuthorBest Paper Award. She co-authored the book Multimedia FingerprintingForensics for Traitor Tracing (Hindawi, 2005). She is the Associate Editor forIEEE SIGNAL PROCESSING LETTERS and Elsevierβs JOURNAL OF VISUAL
COMMUNICATION AND IMAGE REPRESENTATION.
Weifeng Su (Mβ03) received the Ph.D. degreein Electrical Engineering from the University ofDelaware, Newark in 2002. He received his B.S. andPh.D. degrees in Mathematics from Nankai Univer-sity, Tianjin, China, in 1994 and 1999, respectively.His research interests span a broad range of areasfrom signal processing to wireless communicationsand networking, including space-time coding andmodulation for MIMO wireless communications,MIMO-OFDM systems, and cooperative communi-cations for wireless networks.
Dr. Su is an Assistant Professor at the Department of Electrical Engineering,the State University of New York (SUNY) at Buffalo. From June 2002 toMarch 2005, he was a Postdoctoral Research Associate with the Departmentof Electrical and Computer Engineering and the Institute for Systems Research(ISR), University of Maryland, College Park. Dr. Su received a FellowshipAward from U.S. National Research Council (NRC) in 2010. He was therecipient of IEEE International Conference on Communications (ICC) 2010Best Paper Award. Dr. Su received the Invention of the Year Award fromthe University of Maryland in 2005. He received the Signal Processingand Communications Faculty Award from the University of Delaware in2002. Dr. Su has served as Associate Editor for IEEE TRANSACTIONS
ON VEHICULAR TECHNOLOGY and IEEE SIGNAL PROCESSING LETTERS.He has co-organized two special issues for IEEE journals in the field ofcooperative communications and networking.