IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 7...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 7, JULY 2008 2941

Differential Modulations for MultinodeCooperative Communications

Thanongsak Himsoon, Member, IEEE, W. Pam Siriwongpairat, Member, IEEE, Weifeng Su, Member, IEEE, andK. J. Ray Liu, Fellow, IEEE

Abstract—This paper proposes and analyzes differential mod-ulation schemes for two cooperation protocols in multinodecooperative wireless networks; namely, multinode differentialamplify-and-forward scheme (DiffAF) and multinode differentialdecode-and-forward scheme (DiffDF). In the DiffAF scheme,with knowledge of long-term average of received signals from allcommunication links, the destination efficiently combines signalsfrom direct and all multiple-relay links to improve communicationreliability. In the DiffDF scheme, by utilizing a decision thresholdat each relay-destination link, the destination efficiently combinessignals from the direct link and each relay link whose signalamplitude is larger than the threshold. For the DiffAF scheme,an exact bit error rate (BER) formulation based on optimumcombining is provided for differential -ary phase shift keying(DMPSK) modulation, and it serves as a performance benchmarkof the proposed DiffAF scheme. In addition, BER upper bounds,BER lower bounds, and simple BER approximations are derived.Then, optimum power allocation is provided to further improveperformance of the DiffAF scheme. Based on the tight BERapproximation, the optimum power allocation can be simply ob-tained through a single dimensional search. In case of the DiffDFscheme, the performance of DMPSK modulation is analyzed.First, a BER formulation for DMPSK modulation is derived.Next, an approximate BER formulation of the DiffDF scheme isobtained, and a tractable BER lower bound is derived to providefurther insights. Then, the performance of the DiffDF schemeis enhanced by jointly optimizing power allocation and decisionthresholds with an aim to minimize the BER. Finally, simulationresults under the two proposed cooperation protocols are given tovalidate their merit and support the theoretical analysis.

Index Terms—Bit error rate (BER), cooperative communi-cations, differential modulation, multinode wireless networks,virtual multiple-input–multiple-output (MIMO).

I. INTRODUCTION

RECENTLY, cooperative communications have gainedmuch attention due to the ability to explore inherent spa-

tial diversity available in relay channels by forming a virtualantenna array among cooperating nodes. In this strategy, when a

Manuscript received December 14, 2006; revised October 28, 2007. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Mounir Ghogho.

T. Himsoon and W. P. Siriwongpairat are with the Meteor CommunicationsCorporation, Kent, WA 98032 USA (e-mail: [email protected];[email protected]).

W. Su is with the Department of Electrical Engineering, State University ofNew York (SUNY) at Buffalo, Buffalo, NY 14260 USA (e-mail: [email protected]).

K. J. R. Liu is with the Department of Electrical and Computer Engineeringand Institute for Systems Research, University of Maryland, College Park, MD20742 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2008.917358

node has information to transmit, it cooperates with other nodeswhich helps forward the information to an intended destination.To explore such inherent spatial diversity, various cooperationprotocols have been proposed based on relay processing, e.g.,decode-and-forward (DF) and amplify-and-forward (AF) [1].Under the DF protocol, each relay decodes the received signalfrom the source, and then forwards the decoded informationto the destination. Under the AF protocol, on the other hand,each relay amplifies the received signal and then forwards theamplified signal to the destination. In [2] and [3], a concept ofuser cooperation has been introduced for a two-user code di-vision multiple access (CDMA) cooperation system by whichorthogonal codes are used among active users to avoid mul-tiple access interference. The work in [4] focuses on rigorousanalysis of exact symbol error rate (SER) and optimum powerallocation for the DF protocol for two-user cooperation systems.The work in [5] proposed a class of coherent multinode DFcooperation protocols with arbitrary -relay nodes in whicheach relays combines signal from the source and the previous

relays. In [6]–[8], an idea of distributedspace-time coding has been considered by which all cooperationnodes form virtual antenna array and synchronously encodeinformation using existing space-time codes.

However, most of the works in [1]–[8] assume that the desti-nation has perfect knowledge of channel state information (CSI)of all transmission links. While in some scenarios, e.g., slowfading environment, the CSI is likely to be acquired by the useof pilot symbols, it may not be possible in fast fading environ-ment. In addition, it is questionable on how the destination canobtain source-relay channel perfectly through pilot signal for-warding without noise amplification. Moreover, the computa-tional overhead for channel estimation increases in proportionto the product of number of transmit antennas at the source nodeand number of relaying nodes.

Differential modulation has been well accepted as a modula-tion technique that provides a good tradeoff between receivercomplexity and performance. In differential phase-shift keying(DPSK) [9], efficient decoding relies on constant phaseresponses of the channel from one time sample to the next.Therefore, perfect CSI is not required at the differential decoder.The merit of bypassing channel estimation makes differentialmodulation a viable candidate to be deployed in cooperativecommunication so as to reduce receiver complexity and signaloverheads. In [10], error performance of coherent/differentialmodulations for a specific two-hop relay system have beeninvestigated. In [11], a framework of noncoherent communi-cation employing frequency shift keying modulation has beenproposed for DF cooperation systems. However, the framework

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2942 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 7, JULY 2008

does not fit to a general -ary differential phase-shift keying(MDPSK) and the AF cooperation system.

The related works on differential cooperative schemes havebeen considered in [12]–[15]. In [12], a differential scheme forthe DF protocol has been investigated for a two-user coopera-tion system in which the two users transmit signal in an Alam-outi-like fashion. In [13] and [14], a differential scheme for theAF protocol and its error rate performance has been investigatedfor two-user cooperation systems. In addition, [14] provided asimple bit error rate (BER) performance formulation of the pro-posed scheme, which is derived based on the moment gener-ating function method, and this formulation is used for optimallyallocating power among nodes to further improve the systemperformance. However, the simple BER formulation is compli-cated and optimum power allocation scheme is obtained onlythrough exhaustive numerical search. In [15], a threshold-baseddifferential cooperative scheme employing the DF protocol hasbeen proposed for a two-user wireless network. A tight BERapproximation is provided, and optimum power allocation andthreshold are numerically determined to further enhance per-formance. In [16], a two-node differential DF scheme is pro-posed where power allocation at the relay is proportioned to thechannel variances at the source-relay link and the relay-desti-nation link. The power allocation scheme in [16] relies on as-sumptions that channels are quasi-static over a frame period,and the relay receives reliable feedback of channel variance ofthe relay-destination link. The work in [15], on the other hand,is applicable to more relaxed channel, which can vary symbolby symbol, and requires no channel feedback. Nevertheless,most of the existing differential cooperative schemes focus ontwo-node wireless networks.

This paper proposes differential modulation schemes for AFand DF cooperative communications in multinode cooperativenetworks. Due to their low-complexity implementations, theproposed schemes can be deployed in sensor and ad hoc net-works in which multinode signal transmissions are necessaryfor reliable communications among nodes. In this work, thedestination in the DiffAF scheme requires only long-termaverage of the received signals to efficiently combine sig-nals from all communications links. In the DiffDF scheme,each relay decodes the received signal and it forwards onlycorrectly decoded symbols to the destination. A number ofdecision thresholds that correspond to the number of relaysare used at the destination to efficiently combine receivedsignals from each relay-destination link with that from thedirect link. BER performance of both DiffAF and DiffDFschemes is analyzed and optimum power allocation is providedto further improve the system performance. In case of theDiffAF scheme, we provide an exact BER formulation basedon optimum combining weights for MDPSK modulation. Theobtained BER formulation serves as a performance benchmarkof the DiffAF scheme. In addition, BER upper bounds andsimple BER approximations are provided. One of the tightBER approximations allows us to optimize the power alloca-tion through a simple single-dimensional search. In case of theDiffDF scheme, a BER formulation with DMPSK modulationis derived. In addition, BER approximation and a tractableBER lower bound are provided. Then, power allocation and

Fig. 1. Multinode differential AF scheme.

Fig. 2. System descriptions of the multinode differential DF scheme.

thresholds are jointly optimized. Simulation results are shownto validate the merit of the proposed DiffAF/DiffDF schemesand support the theoretical analysis.

The rest of this paper is organized as follows. Section II out-lines the DiffAF and the DiffDF schemes for multinode coop-erative communications. Section III considers BER analysis forthe DiffAF and the DiffDF schemes including BER bounds andtheir BER approximations. In Section IV, optimum power al-location is determined for the DiffAF scheme where optimumpower allocation and optimum threshold are jointly determinedfor the DiffDF scheme. Simulation results and discussions aregiven in Section V. Finally, Section VI concludes this paper.

II. SIGNAL MODELS FOR MULTINODE DIFFERENTIAL SCHEMES

We consider a multinode cooperative wireless network witha source and relays as shown in Fig. 1. Each node can bea source that sends information to its intended destination,or it can be a relay that helps forward information from thesource. We consider two differential cooperation strategies,namely, differential amplify-and-forward (DiffAF) and dif-ferential decode-and-forward (DiffDF) cooperation schemes.For the DiffAF scheme, each relay amplifies each receivedsignal from the source and then forwards the amplified signalto the destination. In case of the DiffDF scheme (also knownas selective forwarding protocol [1]), as in Fig. 2, each relaydecodes each received signal and then forwards only correctlydecoded symbol to the destination. In order to take advantageof the DiffDF protocol by which only correctly decoded symbolat each relay is forwarded with a certain amount of power to thedestination, a decision threshold is used at the destinationto allow only high potential information bearing signal fromeach of the th relay link to be combined with that from thedirect link before being differentially decoded.

HIMSOON et al.: DIFFERENTIAL MODULATIONS FOR MULTINODE COOPERATIVE COMMUNICATIONS 2943

With cooperative relays in the network, signal transmis-sions for both DiffAF and DiffDF schemes comprisephases. The first phase belongs to direct transmission, and therest of phases is for signal transmission for each of therelays. The signal models for each of the transmissionphases are as follows.

In phase 1, suppose differential -ary phase shift keying(DMPSK) modulation is used, the modulated information atthe source is , where for

, and is the constellation size. The sourcedifferentially encodes by , where is thetime index and is the differentially encoded symbol to betransmitted at time . After that, the source transmits withtransmitted power to the destination. Due to the broadcastingnature of the wireless network, the information can also be re-ceived by each of the relays. The corresponding received sig-nals at the destination and the th relay, for , canbe expressed as

(1)

(2)

where and represent channel coefficients from thesource to the destination and from the source to the th relay,respectively. In this paper, and are modeled as com-plex Gaussian random variables with zero means and variances

and , respectively. The terms and are addi-tive white Gaussian noise at the destination and the th relay, re-spectively. Both of these noise terms are modeled as zero-meancomplex Gaussian random variables with variance .

In phases 2 to , depending on the cooperation protocolunder consideration, each of the relays forwards either theamplified signal or the decoded signal to the destination. Signalmodels for the DiffAF and the DiffDF schemes in phases 2 to

are presented in Section II-A and II-B, respectively.

A. Signal Model for the DiffAF Scheme

For the DiffAF scheme, the received signal in phases 2 tois given by

(3)

where represents the transmitted power at the th relayand denotes the channel coefficient at time at the threlay-destination link. We model as a zero-mean complexGaussian random variable with variance . In (3), isnormalized by , and hence the th relay requiresonly the channel variance between the source and the th relay

rather than its instantaneous value. In practice,can be obtained through long-term averaging of the receivedsignals at the th relay.

Finally, the received signals from the source and those fromall of the relays are combined at the destination, and we have

(4)

where

and

are used as combining weights for the proposed DiffAF scheme.Here, the channels and are assumed available at thedestination. Note that and can be obtained throughlong-term averaging of the received signals at the th relay andthe destination, respectively. In practice, can be forwardedfrom each of the th relay to the destination over a reliablechannel link. Accordingly, without acquiring perfect CSI, thecombined signal (4) is differentially decoded by using the de-tection rule [9]

(5)

B. Signal Model for the DiffDF Scheme

In this cooperation system, each relay forwards only correctlydecoded symbol to the destination, i.e., when the threlay decodes correctly, and , otherwise. As shown inFig. 2, decision thresholds are used at the destination to allowonly high potential information bearing signal from each of theth relay to be combined with that from the direct link before

being differentially decoded.Specifically, in phases 2 to , each of the th relay dif-

ferentially decodes the received signal from the source by usingthe decision rule [9]

(6)

Here, we assume an ideal relay that can make judgement on thedecoded information whether it is correct or not.1 If each of theth relay incorrectly decodes, such incorrectly decoded symbol

is discarded. Otherwise, the th relay differentially re-encodesthe information symbol as , where representsthe time index that the th relay correctly decodes before time .Then, is forwarded to the destination with transmitted power

. After that, is stored in a memory, represented byin Fig. 2, for subsequent differential encoding. Note that

the time index in can be any time before timedepending on the decoding result in the previous time. The

received signal at the destination in phases 2 to can beexpressed as

if relay correctly decodes

otherwise

(7)

where , denotes the channel coefficientbetween the th relay and the destination, and representsan additive noise.

1Practically, this can be done at the relay by applying a simple signal-to-noiseratio (SNR) threshold test on the received data [17], [18].


Since the perfect knowledge of CSI is not available at eachtime instant, the destination does not know when the receivedsignal from the th relay contains the information. For each threlay-destination link, a decision threshold is used at the des-tination to make the decision whether to combine with thereceived signal from the direct link. Specifically, iffor all where denotes the absolute value of , the destina-tion estimates the transmitted symbol based only on the receivedsignal from the direct link. However, if for any , thereceived signals from the source and that from the th relay arecombined for joint decoding. In this way, the combined signalat the destination can be written as

(8)where and are combining weights. In (8),

is the most recent received signal from the th relay with. It is stored in a memory, represented by in

Fig. 2, at the destination. The function in (8) repre-sents an indicator function in which if

; otherwise, . After signal combining, the desti-nation jointly differentially decodes the transmitted informationby . Note that usingdifferent combining weights ( and ) results in differentsystem performances. In this paper, we use

, which maximizes the SNR at the combiner output.

III. PERFORMANCE ANALYSIS FOR THE MULTINODE

DIFFERENTIAL SCHEMES

In this section, BER analysis for the proposed multinode dif-ferential cooperation schemes is provided. First, a BER perfor-mance for the DiffAF scheme is analyzed. Then, tight BERbounds and simple BER approximations are determined. Fi-nally, a BER analysis for the DiffDF scheme is provided, andits BER lower bound is given.

A. BER Analysis for the DiffAF Scheme

As specified in [14], the BER formulation based on ar-bitrary combining weights, i.e., and in (4), iscurrently not available in the literature. For mathemat-ical tractability, we provide an alternative BER analysisbased on optimum combining weights: and

. Thedetailed BER derivation can be found in [14] for single relaycase, and it is omitted here. Differently from [14], however,in what follows, we provide an alternative closed-form BERformulation, which allows us to analytically calculate optimumpower allocation rather than rely on numerical evaluations aspresented in [14].

With the optimum combining weights and , an in-stantaneous SNR at the combiner output can be written as

(9)

where

and

. Accordingly, the conditional BER expression for the DiffAFscheme can be approximated by the differential modulation with

-channel diversity receptions [9] as

(10)

where

(11)

and

(12)

in which

Here, , and in which andfor differential binary phase-shift keying (DBPSK) modulation,and and for differential quadra-ture phase-shift keying (DQPSK) modulation [9]. The value of

for higher constellation sizes can be found in [19]. Followingthe analysis in [14] by using the moment generating function(MGF) method to average the conditional BER (10) over theRayleigh distributed random variables, we have

(13)

where , in which. The MGF is obtained through

double integration of each over two exponential randomvariables and . After some manipulations, wehave

(14)


where

(15)

in which and. Note that the last expression in (15) is

obtained by applying results from [22, p. 358, eq. (3.352.4); p.934, eq. (8.212.1)], and represents theEuler’s constant. Hence, the average BER is

(16)

Using the same technique as in [14], the BER upper bound isobtained by substituting in (11) such that is upperbounded by . Hence, the BER upperbound can be obtained by replacing in (16) by ,and we have

(17)

where

(18)and in which

.Accordingly, a simple BER upper bound can be obtained by

focusing at high SNR region such that all 1’s in the denominatorof (17) can be discarded. After some manipulations, the simpleBER upper bound can be expressed as

(19)

where

(20)

is a constant that depends on the modulation size and the numberof relays, and and are specified in (11)and (12), respectively. The BER upper bound (19) reveals that

when relays are available in the network, the DiffAF schemeachieves diversity order of as specified in the exponentof the noise variance.

In case of BER lower bound, we first note that. By replacing in (16) by

, we obtain a BER approximation

(21)

where

(22)and

in which

. Furthermore, by ignoring all 1’s in the denominator of (21),we obtain a simple BER approximation

(23)

where and are specified in (20) and (22),respectively. We can see from the exponent of the noise variancein (23) that the achievable diversity order is . As will beshown in the simulation results, these two BER approximationsare tight at high SNR region.

B. BER Analysis for the DiffDF Scheme

BER analysis of the DiffDF scheme, as described inSection II-B, is considered in this section. First, different SNRscenarios are characterized according to the received signal

, threshold , and memories and . Then, probabilityof occurrence is provided for each of these SNR scenarios.After that, average BER is derived based on the probability ofoccurrence and the combined SNR for each scenario. Finally,a tractable BER lower bound is provided at the end of thissection.

1) Characterization of Different SNR Scenarios: At the des-tination, different combined SNRs may occur according to acomparison the received signal and the thresholdas well as the signals stored in memory and . In thisway, the destination encounters six possible SNR scenarios ateach relay-destination link, and we characterize each of them asfollows. For a given network state , we denote as an integernumber that represents an SNR scenario at the th relay-destina-tion link, i.e., . A set of joint event for


will be related to each of the scenario . Specif-

ically, when , represents a jointevent that received signals from the th relay link is not greaterthan the thresholds. We characterize

as a joint event including ,where the relay correctly decodes at time and and theinformation symbols at time and in memories and

are the same. The rest scenarios are,

, ,and . They are inter-preted in a similar way as that of .

2) Probability of Occurrence for Each SNR Scenario: Todetermine probability of occurrence for each scenario, we firstfind that the probability that the th relay forwards informationwith transmitted power due to incorrect decoding isrelated to the symbol error rate of DMPSK modulation as [21]

(24)

where represents an in-stantaneous SNR at the th relay, and

. Accordingly, theprobability of correct decoding at the th relay (or probabilityof forwarding with transmitted power ) is .Therefore, the chance that occurs is determined by theweighted sum of conditional probabilities given thator 0, and we have

(25)

where the second equality is obtained by substituting,

which is related to cumulative distribution function (CDF)of a Rayleigh-distributed random variable. The term

is related to the CDF ofRician-distributed random variable such that

where

(26)

in which is the Marcum -function [9].According to the definition of each SNR scenario in

Section III-B1, a chance that each of the scenarios tohappens is conditioned on an event that . Since

the events at time and time are independent, then theprobability that occurs is given by

(27)

The approximation in (27) is obtained by using the result in(24) and the fact that

. In this way, the first term in (27) can be cal-culated as

. In addition, the second term in (27) can be approx-imated by using the concept of conditional probability and ap-plying Bayes’ rule such that

Next, the chance that the scenario happens can be writtenas

(28)

where

(29)

Substituting (27) and (29) into (28), after some manipulations,we have

(30)

in which is defined as an expressionthat results from applying the concept of total probability [23]to

(31)

With the assumption of almost constant channels at time and, we have , i.e., scenarios and


occur with the same probability. Following the calculationsteps as used in (29), we have

(32)

Finally, the chance that scenario occurs can be determinedas

(33)

3) Approximate BER Expression for the DiffDF Scheme: Weknow from Section III-B1 that each relay contributes six pos-sible SNR scenarios at the destination. For a network withrelays, there are totally numbers of network states. We de-note as an matrix of a network state, where . Accordingly, the average BER can

be expressed as

(34)where for each is specified in (25)–(33),

represents a conditional BER for a given , anddenotes the expectation operator.

Because it is difficult to find a closed-form solution for theBER in (34), we further simplify (34) by separating a set of allpossible network states, denoted by , into two disjoint subsetsas , where denotes all possible networkstates that every element in the network state is either one ortwo, and denotes the remaining possible network states.Note that the cardinality of and is and

, respectively. In this way, we can expressthe average BER (34) as

(35)

The first term in the right-hand side of (35), , results fromthe cases where every element in the network state is eitherone or two; the second term results from the remainingcases. These two terms can be determined as follows.

First, for notational convenience, let us denote as thenumber of combining branches. By definition, we can express

as

(36)

where when , and , otherwise. Notethat the addition of 1 in (36) corresponds to the contribution ofsignal from the direct link.

Next, consider the case that every element in the network stateis either one or two, then the conditional BER

can be obtained from the multibranch differential detection ofDMPSK signals as [9]

(37)

in which and are specified in (11) and (12),respectively. The term is the SNR at the combined

output, which is given by

(38)Then, the conditional BER for the re-

maining cases can be found as follows. Since up to nowthe conditional BER formulation for DMPSK with arbi-trary-weighted combining has not been available in theliterature, cannot be exactly determined. Foranalytical tractability of the analysis, we resort to an approxi-mate BER, in which the signal from the relay is consideredas noise when any scenario from to occurs. As we willshow in the succeeding section, the analytical BER obtainedfrom this approximation is close to the simulation results.The conditional BER for these cases can be approximated as

, where

(39)

in which

and depends on as follows:

when , when

, and when . In (39), isdefined as an indicator function based on the occurrence of

such that when , and when.


From the previous results, in (35) can be approximatedas

(40)

where

(41)

in which is given in (39), and is

calculated by using (25)–(33). We can see from (40) that theevaluation of involves at most -fold integration.Although can be numerically determined, the calculationtime is prohibitively long even for a cooperation system withsmall number of relays.

Now, we determine in (35) as follows. The term

is a product of probabilities of occurrence ofscenarios 1 and 2, and it can be expressed as

(42)Substitute (37), (38), and (42) into the expression of in(35), and then average over all CSIs, resulting in

(43)

where we denote , which can be deter-mined as

(44)

in which , and, which can

be approximated as

(45)

Substituting (40) and (43) into (35), we finally obtain the av-erage BER of the multinode DiffDF scheme.

To get more insightful understanding, we further determinea BER lower bound of the multinode DiffDF scheme as fol-lows. Since the exact BER formulations under the scenarios ,

, and are currently unavailable, and the chances that thesethree scenarios happen are small at high SNR, we lower boundthe BER from these scenarios by zero. Also, we lower boundthe BER under the scenario by that under ; this allows usto express the lower bound in terms of (instead of

or ), which can be obtain without any approx-imation. In this way, the BER of multinode DiffDF scheme canbe lower bounded by

(46)

where is given in (44) and

(47)

in which

(48)

We will show through numerical evaluation that the BER lowerbound (46) is very close to the simulated performance.

IV. PERFORMANCE ENHANCEMENT BY OPTIMIZING

THRESHOLD AND POWER ALLOCATION

In this section, we provide optimum power allocation for theproposed DiffAF/DiffDF cooperation schemes. First, optimumpower allocation for the DiffAF scheme is presented. With someapproximations on the obtained BER expression, we are able toobtain a closed-form optimum power allocation for the DiffAFscheme. Then, we jointly determine the optimum threshold and


optimum power allocation for the DiffDF scheme. Numericalresults and discussions are provided.

A. Optimum Power Allocation for the DiffAF Scheme

We formulate an optimization problem to minimize the BERunder a fixed total transmitted power .Based on the simple BER approximation (23), the optimizationproblem can be formulated as (49), shown at the bottom of thepage. Although (49) can be solved numerically, it is difficultto get some insights. To further simplify the problem, we con-sider a high-SNR scenario where can be approximated as

. Then, we can rewritein (22) as

(50)

where and in which. Since the integration term

in the last expression of (50) is small compared to ,it can be neglected without significant effect on the powerallocation. Hence, can be further approximated by

(51)

As will be shown later, the obtained optimum power allocationbased on the approximated in (51) yields almost the sameperformance as that with exact as specified in (22).

Substituting (51) into (49), the optimization problem can besimplified to (52), shown at the bottom of the page. Optimizing(52) by using the Lagrangian method as given in the Appendix,the optimum solution of (52) can be obtained by finding thatsatisfies

(53)

in which

Given a specific for , we can find thecorresponding , denoted by , that satisfies (53). The op-timum power allocation can then be obtained by findingthat satisfies

(54)

The resulting optimum power allocation for the source is. Since , then the optimum power

allocation for each of the th relay is for.

1) Optimum Power Allocation for Single-Relay Systems: Forsingle relay systems, the optimization problems (A-5) and (54)are reduced to finding such that

and

(55)

which can be simply solved by any single-dimensional searchtechniques. In this way, the complexity of the optimizationproblem can be greatly reduced, while the resulting optimumpower allocation is close to that from exhaustive search in[14]. For example, Table I compares optimum power allocationfrom the exhaustive search in [14] and that from solving thelow-complexity optimization problem in (55). The results inTable I are obtained at reasonable high SNR region, e.g., 20or 30 dB. The DBPSK or DQPSK modulations are used, and

represents a vector containing channel vari-ances of the source-destination link, the source-relay link, andthe relay-destination link, respectively. We can see from thetable that the optimum power allocation based on (55) is veryclose to that from the numerical search, for any relay location.There is only about 1%–2% difference in the obtained resultsbetween these two methods.

2) Optimum Power Allocation for Multirelay Systems: Formultirelay systems, (A-5) and (54) can be used to find the op-timum power allocation. Nevertheless, the optimization basedon (A-5) and (54) involves -dimensional search because

subject to

(49)

subject to (52)


TABLE IDiffAF: OPTIMUM POWER ALLOCATION FOR COOPERATION SYSTEM WITH ONE RELAY

BASED ON EXHAUSTIVE SEARCH AND APPROXIMATE CLOSED-FORM FORMULATION (55)

TABLE IIDiffAF: OPTIMUM POWER ALLOCATION FOR COOPERATION SYSTEM WITH TWO RELAYS BASED ON EXHAUSTIVE SEARCH AND APPROXIMATE FORMULATION (56)

in (A-5) contains power allocation of each relay insidethe summation. To reduce complexity of the search space, weremove the summation inside such that the approxi-mate depends only on the of interest. Therefore, anoptimum power allocation can be approximately obtained byfinding such that

and

(56)

From (56), the optimum power allocation that involves-dimensional search is reduced to single-dimensional

search over the parameter , . Table II summarizesthe numerical search results from the multidimensional searchbased on (49) in comparison with those from approximateone-dimensional search using (56). Based on the optimizationproblem (56), the searching time for optimum power allocationcan be greatly reduced, while the obtained power allocation isvery close to that from solving (49) using multidimensionalsearch.

From the results in Tables I and II, we can also observe that,for any channel link qualities, more power should be allocatedat the source so as to maintain link reliability. This observationholds true for both DBPSK and DQPSK modulations. When thechannel link qualities between the source and the relays are good(e.g., ), the system replicatesthe multiple transmit antenna system. Therefore, almost equalpowers should be allocated at the source and all the relays.

B. Optimizing Power Allocation and Thresholds for theDiffDF Scheme

In this section, we determine the performance improvementof the DiffDf scheme through the joint optimization of powerallocation and thresholds based on the BER lower bound (46).Specifically, for a fixed total power , wejointly optimize the threshold , the power allocation at thesource , and the power allocation at each of the threlay with an objective to minimize the BER lowerbound (46)

(57)

where results from substitutingand into (46). However, joint optimiza-

tion in (57) involves -dimensional searching, which in-cludes power allocation ratios and decision thresholds.To make the optimization problem tractable and to get some in-sights on the optimum power allocation and the optimum thresh-olds, each relay is assumed to be allocated with the same trans-mitted power, and the decision thresholds are assumed the sameat the destination for each relay-destination link. Accordingly,the source is allocated with power and every relayis allocated with power . Hence, the searchspace for this optimization problem reduces to two-dimensionalsearching over and

(58)

where results from substituting ,, and into (46).

Table III summarizes the obtained power allocation andthresholds based on the optimization problem (58). TheDBPSK and DQPSK cooperation systems with two relays areconsidered, and different channel variances are used to investi-gate power allocation and thresholds for different cooperation


TABLE IIIDiffDF: OPTIMUM POWER ALLOCATION AND THRESHOLDS FOR A

COOPERATION SYSTEM WITH TWO RELAYS

network setups. From the results in Table III, even though theobtained power allocation is suboptimum, it provides someinsightful information on how much power should be allocatedto improve the system performance. In particular, as the channelquality of the relay-destination links increases, the thresholdshould be increased and more power should be allocated atthe source to maintain link reliability. For example, if all thechannel links are of the same quality, about half of the trans-mitted power should be allocated at the source and the optimumthreshold should be 0.4. On the other hand, if the channel linkbetween each relay and the destination is very good, then theoptimum power allocation at the source increases to about70% of the transmitted power, and the optimum thresholdincreases to 1.6 and 1.8 for DBPSK and DQPSK modulations,respectively.

V. SIMULATION RESULTS

We simulate the DiffAF and the DiffDF schemes withDBPSK and DQPSK modulations. We consider the scenarioswhere two or three relays ( or ) are in the networks. Thechannel coefficients follow the Jakes’ model [24] with Dopplerfrequency 75 Hz and normalized fading parameter

, where is the sampling period. The noisevariance is assumed to be one . The average BERcurves are plotted as functions of .

A. Simulation Results for the DiffAF Scheme

Fig. 3(a) shows the performance of the DiffAF scheme withDBPSK modulation for a network with two relays. The simu-lation is performed under equal channel variances, i.e.,

, and equal power allocation strategy. We can see that the exact theoretical BER

benchmark well matches the simulated BER curve. In addition,the BER upper bound, the simple BER upper bound, and the twosimple BER approximations are tight to the simulated curve athigh SNR. The BER curve for coherent detection is also shownin the figure; we observe a performance gap of about 4 dB be-tween the DiffAF scheme and its coherent counterpart at a BERof .

In Fig. 3(b), we illustrate BER performance of the DiffAFscheme with DQPSK modulation when using different numberof relays . The simulation scenario is the same as that ofFig. 3(a), and we consider two possible numbers of relays,namely, and . It is apparent that the proposedDiffAF scheme achieves higher diversity orders as increases.Specifically, as increases from 2 to 3, we observe about1.7–2-dB gain at a BER of . This observation confirmsour theoretical analysis in Section III-A. Also in this figure,

Fig. 3. DiffAF scheme with equal power allocation strategy and � �

� � � � �. (a) DBPSK: two relays. (b) DQPSK: two and three relays.

the exact theoretical BER curves for and aretight to the corresponding simulated curves. In addition, theperformance curves of the DiffAF scheme are about 4 dB awayfrom their coherent counterparts.

Fig. 4(a) shows the BER performance of the DiffAF schemewith optimum power allocation in contrast to that with equalpower allocation. We consider the DiffAF scheme with DQPSKmodulation for a network with two relays. The channel vari-ances are and , and the optimumpower allocation is (from Table II). The sim-ulated curves show that when all relays are close to the source,i.e., , the DiffAF scheme with optimum power alloca-tion yields about 0.6-dB gain over the scheme with equal powerallocation at a BER of . We observe a small performancegain in this scenario because the signal at the relays is as goodas the signal at the source. Therefore, the scheme with equalpower allocation yields almost as good performance as perfor-mance under optimal power allocation such that there is a smallroom for improvement. Also in this figure, the exact theoretical


Fig. 4. DiffAF scheme with DQPSK: two relays, optimum power allocationstrategy. (a) � � �, � � ��, and � � �. (b) � � � � � and� � ��.

BER curves are provided for both power allocation schemes,and they closely match their corresponding simulated curves.

In Fig. 4(b), we consider the BER performance of optimumpower allocation scheme for a DQPSK cooperation system withtwo relays. The channel variances are and

, which corresponds to a scenario that all relays areclose to the destination. The optimum power allocation for thisscenario is (from Table II). We observe thatthe performance with optimum power allocation is about 2 dBsuperior to that with equal power allocation at a BER of .In this scenario, we observe a larger performance gain than thecase of and in Fig. 4(a). Thereason is that using equal power allocation in this case leadsto low quality of the received signals at the relays, and thuscauses higher chance of decoding error at the destination basedon the combined signal from the cooperative links. With op-timum power allocation, more power is allocated at the source,

Fig. 5. DiffDF scheme with DQPSK: � � � � � � �, and � � �.(a) Equal power allocation. (b) Optimum power allocation.

and consequently, the quality of the received signals at the re-lays is improved. This results in more reliable combined signalat the destination, hence yielding better system performance.

B. Simulation Results for the DiffDF Scheme

Fig. 5(a) compares the BER lower bound with the simulatedperformance. We consider a DQPSK cooperation system withtwo relays. All nodes are allocated with equal power. The de-cision threshold is set at and the channel variances are

for all . We can see from this figurethat the BER lower bound yields the same diversity order as thatfrom the simulated performance even though there is a 2-dBperformance gap between these two curves. Also in this figure,the performance of the DiffDF scheme is 5 dB away from theperformance with coherent detection at a BER of . An in-teresting observation is that when the transmitted powers areoptimally allocated ( and ) at a fixedthreshold of , as shown in Fig. 5(b), the performance gapbetween the simulated performance and the BER lower boundis reduced to about 1 dB at a BER of . It is worth noting


Fig. 6. DiffDF with DQPSK. (a) Optimum power allocation (58) and exhaus-tive optimum power allocation � � � � � � � and � � �.(b) Different number of relays � � � � � � �.

that the obtained power allocation in this scenario is the sameas optimum power allocation (obtained through exhaustive nu-merical search). This is because the channel variances of the tworelays are the same.

To receive more insight and show the merit of the optimumpower allocation (58), we set up an alternative simulation sce-nario, which is similar to those in Fig. 5(a) and (b), but whererelay 2 is located closer to the source than relay 1. We repre-sent this scenario by setting variance of the link between thesource and the relay 2 as . We set the variances ofall other links to one and set the threshold to one. In this case,the optimum power allocation obtained from exhaustive searchis while the optimum power allocation obtainedfrom (58) is . Fig. 6(a) compares the BER perfor-mance of the two power allocation schemes. We can see fromthe figure that the performance with optimum power allocation(58) is almost the same as that with exhaustive optimum powerallocation.

Fig. 6(b) shows the performance of the DiffDF scheme withDQPSK modulation for different number of relays. The channel

Fig. 7. DiffDF scheme with � � � � �� . (a) DBPSK:three relays, fixed power allocation, but different thresholds. (b) DQPSK: tworelays, different power allocation and thresholds.

variances are for all . All nodesare allocated with equal power, and the threshold at the desti-nation is fixed at . We can see that the diversity orderincreases when higher numbers of relays are used. We observeabout 3.5-dB performance improvement at a BER of whenthe number of relays increases from one to two relays. An addi-tional 2-dB gain at the same BER is obtained when the systemincreases from two to three relays. We also observe a perfor-mance gap of about 5.5 dB at a BER of between the DiffDFscheme and its coherent counterpart for a cooperation systemwith three relays.

Fig. 7(a) shows the effect of using different thresholds on theperformance of the proposed scheme. We consider a DBPSKcooperation system with three relays and all nodes allocatedwith equal power. The channel variances are ,and for all . Clearly, different thresholds result indifferent performance. Specifically, the proposed scheme with

provides the best performance under this simulation sce-nario. When , not only BER deteriorates but also the di-


versity order reduces. Hence, an appropriate decision thresholdshould be employed such that the DiffDF scheme yields reason-ably good performance. Comparing the simulated performancewhen with the coherent cooperative scheme withoutthreshold, we observe about 6-dB performance gap between thetwo performance curves at a BER of . Such performancegap is large because in the DiffDF scheme, the CSIs are notavailable at the receivers, and the destination does not knowwhether the relay transmits or not.

In Fig. 7(b), we show the performance improvement whenpower allocation and decision thresholds are jointly optimized.We consider a DQPSK cooperation system with two relays. Thechannel variances are and forall . In this scenario, the optimum power allocation is

, , and , and the optimum threshold is. We can see that the performance curve with optimum

power allocation and threshold significantly improves from thatwith equal power allocation and an arbitrary decision threshold( in this case). A performance gain of 4–5 dB is observedat a BER of – . Also in this figure, we compare theperformance of optimum power allocation and threshold withthat of optimum power allocation ( , , and

) but an arbitrary threshold . We can see thatjointly optimizing power allocation and threshold leads to about1–2.5-dB gain over the scheme with optimum power allocationbut arbitrary threshold at BER ranges between and .

VI. CONCLUSION

In this paper, we propose differential schemes for multinodecooperative communications employing DiffAF and DiffDF co-operation protocols. In the DiffAF scheme, as a performancebenchmark, we provide an exact BER expression for DMPSKmodulation based on optimum combining weights. BER upperbounds and BER approximations are provided; they are tight tothe simulated performance, especially at high SNR. The theoret-ical BER reveals that the diversity order of the proposed schemeis where is the number of relays and it is confirmed bythe simulation results. We observe about 1.7–2-dB gain at a BERof when increases from 2 to 3. The BER approximationis further simplified; based on the approximate BER, we areable to optimize the power allocation using a low-complexitysingle-dimensional search. Simulation results show that whenall relays are close to the source, the proposed DiffAF schemeobtains about 0.6-dB gain over that with equal power allocationat a BER of . When all relays are close to the destination,the performance with optimum power allocation achieves about2-dB improvement over that with equal power allocation.

In case of the DiffDF scheme, we consider a multinodescenario in which each of cooperative relays forwardsonly correctly decoded symbol to the destination. Decisionthresholds are used at the destination to efficiently combinesignal from each relay-destination link with that from directlink. An approximate BER analysis for DMPSK is provided,and a low-complexity BER lower bound is derived. The BERlower bound is very close to the simulated performance undersome scenarios. While jointly optimizing power allocation andthresholds based on the BER lower bound introducesdimensional searching, the search space is reduced by assuming

that the same power is used at each relay and the same thresholdis used at the destination. Numerical results reveal that morepower should be allocated at the source and the rest shouldbe used by the relays. In addition, larger threshold should beused when the relays are close to the destination. Simulationresults show that the diversity gain of the proposed schemeincreases with the number of relays. For a DBPSK cooperationsystem, the proposed DiffDF scheme with different thresholdsleads to the performance improvement of up to 6 dB at a BERof . In case of DQPSK cooperation system, the DiffDFscheme with joint optimum power allocation and optimumthreshold achieves about 4–5-dB gain over that with equalpower allocation and a unit threshold at a BER of – .

APPENDIX

SOLUTION TO THE OPTIMIZATION PROBLEM (52) BY

LAGRANGIAN METHOD

By taking logarithm of the Lagrangian of (52) and letting, we obtain

(A-1)

in which is an vector, and denotesan vector with all ones. By differentiating (A-1) withrespect to and and equating the results to zero, we have

(A-2)

and

(A-3)

respectively. From (A-3), we can find that

(A-4)in which we denote . Observefrom (50) that can be re-expressed as

where for . Then, substi-tuting (A-4) into (A-2), we have

(A-5)


in which

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Thanongsak Himsoon (S’03–M’07) received theB.S. degree in electrical engineering from the Chula-longkorn University, Bangkok, Thailand, in 1995 andthe M.S. and Ph.D. degrees in electrical engineeringfrom the University of Maryland, College Park, in2001 and 2006, respectively.

Currently, he is a Communication Systems DesignEngineer at Meteor Communications Corporation,Kent, WA, working on algorithm and system designfor wireless communications. His research interestsinclude signal processing, wireless communications,

cooperative communications, and wireless sensor networks, with particularfocus on differential modulation systems. His research contributions en-compass differential space-time coding and modulation for MIMO systems,MIMO-OFDM, ultrawideband systems, and cooperative communications.

W. Pam Siriwongpairat (S’03–M’06) received theB.S. degree in electrical engineering from Chula-longkorn University, Bangkok, Thailand, in 1999 andthe M.S. and Ph.D. degrees in electrical engineeringfrom the University of Maryland, College Park, in2001 and 2005, respectively.

Currently, she is a Wireless Communications Spe-cialist at the Meteor Communications Corporation,Kent, WA, working on research and development ofwireless communications technology. Her researchinterests span a broad range of areas from signal

processing to wireless communications and networking, including space-timecoding for multiantenna communications, cross-layer design for wirelessnetworks, communications in mobile ad hoc networks and wireless sensornetworks, OFDM systems, and ultrawideband communications.

Weifeng Su (M’03) received the B.S. and Ph.D. de-grees in applied mathematics from Nankai Univer-sity, Tianjin, China, in 1994 and 1999, respectively,and the Ph.D. degree in electrical engineering fromthe University of Delaware, Newark, in 2002.

Since March 2005, he has been an AssistantProfessor at the Department of Electrical Engi-neering, State University of New York (SUNY),Buffalo. From June 2002 to March 2005, he was aPostdoctoral Research Associate at the Departmentof Electrical and Computer Engineering and the

Institute for Systems Research (ISR), University of Maryland, College Park.His research interests span a broad range of areas from signal processing towireless communications and networking, including space-time coding andmodulation for MIMO wireless communications, MIMO-OFDM systems,cooperative communications for wireless networks, and ultrawideband (UWB)communications.

Dr. Su received the Signal Processing and Communications Faculty Awardfrom the University of Delaware in 2002 as an outstanding graduate studentin the field of signal processing and communications. In 2005, he received theInvention of the Year Award from the University of Maryland. He has been anAssociate Editor of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, theIEEE SIGNAL PROCESSING LETTERS, and a Guest Editor of special issue on co-operative communications and networking of the IEEE JOURNAL OF SELECTED

AREAS IN COMMUNICATIONS.


K. J. Ray Liu (F’03) is the Professor and Asso-ciate Chair, Graduate Studies and Research, andDirector of Communications and Signal ProcessingLaboratory of Electrical and Computer EngineeringDepartment, University of Maryland, College Park.He leads the Maryland Signals and InformationGroup conducting research encompassing broadaspects of information technology including signalprocessing, communications, networking, infor-mation forensics and security, biomedical, andbioinformatics.

Dr. Liu is the recipient of numerous honors and awards including best paperawards from the IEEE Signal Processing Society (twice), the IEEE Vehicular

Technology Society, and EURASIP. He is the IEEE Signal Processing SocietyDistinguished Lecturer. He received the EURASIP Meritorious Service Awardand National Science Foundation Young Investigator Award. He also receivedvarious teaching and research recognitions from University of Marylandincluding university-level Distinguished Scholar-Teacher Award, Invention ofthe Year Award, and college-level Poole and Kent Company Senior FacultyTeaching Award. He is Vice President—Publications. He is on the Board ofGovernors of IEEE Signal Processing Society. He was the Editor-in-Chief ofthe IEEE SIGNAL PROCESSING MAGAZINE and the founding Editor-in-Chief ofEURASIP Journal on Applied Signal Processing.

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