Differential Modulation With Threshold-Based Decision Combining for

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 3905**Differential** **Modulation** **With** **Threshold**-**Based****Decision** **Combining** **for**Cooperative CommunicationsThanongsak Himsoon, Member, IEEE, W. Pam Siriwongpairat, Member, IEEE, Weifeng Su, Member, IEEE, andK. J. Ray Liu, Fellow, IEEEAbstract—**Differential** modulation is widely known as a practicalalternative that provides a good tradeoff between receivercomplexity and per**for**mance. However, the available differentialschemes **for** wireless relay networks require perfect synchronizationand/or provide limited transmission rates. This paper proposesa threshold-based differential decode-and-**for**ward cooperativescheme that not only alleviates the problems of synchronizationand rate limitation, but also efficiently exploits the cooperativerelay channels via the use of a predetermined decision threshold.In the proposed scheme, the source in**for**mation is **for**warded bythe relay only if it is correctly decoded. The properly designedthreshold enables the destination to decide whether the receivedsignal from the relay contains in**for**mation such that the receivedsignals from the source and the relay can be efficiently combinedand jointly decoded. Bit error rate (BER) per**for**mance analysis ofthe proposed scheme is analyzed in the case of differential M-aryPSK signals. The analysis focuses on the case when the relay isable to judge whether each decoded symbol is corrected or not.The obtained BER per**for**mance in this case serves as a per**for**mancebenchmark, and the proposed differential decode-and-**for**wardscheme can achieve it in an ideal situation that the errorpropagation at the relay can be negligible. A tight BER approximationis first derived, and then BER upper bound and lowerbound are determined. **Based** on the tight BER approximation,joint optimum decision threshold and power allocation are numericallyevaluated. Both analytical and simulation results reveal thatthe decision threshold and the power allocation depend on qualitiesof the communication channels. Interestingly, when the linkquality between relay and destination is very good, the effect ofthe threshold dominates the effect of the power allocation at highsignal-to-noise ratio. Extensive simulation results are provided tovalidate the merit of the proposed scheme and confirm the theoreticalanalysis.Index Terms—Bit error rate (BER), cooperative diversity, decode-and-**for**wardprotocol, differential modulation, per**for**manceanalysis, virtual multiple-input multiple-output (MIMO), wirelessnetworks.Manuscript received August 16, 2005; revised November 15, 2006. This workwas supported in part by the U.S. Army Research Laboratory under CooperativeAgreement DAAD 190120011. The associate editor coordinating the review ofthis manuscript and approving it **for** publication was Dr. Ananthram Swami.T. Himsoon and W. P. Siriwongpairat are with the Department of Electricaland Computer Engineering and the Institute **for** Systems Research, Universityof Maryland, College Park, MD 20742 USA, and also with Meteor CommunicationsCorporation, Kent, WA 98032 USA (e-mail: khimsoon@meteorcomm.com; psiriwongpairat@meteorcomm.com).W. Su is with the Department of Electrical Engineering, State University ofNew York (SUNY) at Buffalo, Buffalo, NY 14260 USA (e-mail: weifeng@eng.buffalo.edu).K. J. R. Liu is with the Department of Electrical and Computer Engineeringand the Institute **for** Systems Research, University of Maryland, College Park,MD 20742 USA (e-mail: kjrliu@eng.umd.edu).Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2007.894268I. INTRODUCTIONDIFFERENTIAL detection, together with diversity combining,is widely known as an attractive alternative tocoherent detection that provides a good tradeoff betweenreceiver complexity and per**for**mance in conventional communicationsystems. The technique is also useful **for** sufficientlysevere environment where phase and amplitude tracking **for**coherent demodulation is difficult if not impossible. In differentialphase-shift keying (DPSK) systems [1], the transmitterdifferentially encodes input in**for**mation in adjacent phasedifference of the two transmitted symbols be**for**e transmittingthe encoded symbol over fading channel. While perfect channelstate in**for**mation is not required, an efficient differential decodingat the receiver relies on constant fading from one timesample to the next. Such a slow fading assumption appears tobe realistic in many practical applications, and it is crucial **for**per**for**mance analysis.No need **for** channel estimation has attracted many researchersto deploy differential detection in multiple-inputmultiple-output (MIMO) systems [2], [3]. The differentialschemes in [2] and [3] omit the burden of multichannel estimation,while provide significant per**for**mance improvementwithout compromising bandwidth efficiency over that of conventionalsingle antenna system. However, the deployment ofmultiantenna terminals may be difficult in some applicationsbecause the mobile terminals are practically small. Recently,an idea of resource sharing among users in wireless networkshas been introduced as cooperative communication paradigm[4], [5]. By taking advantage of the broadcasting nature of thewireless networks, the cooperative communications is able toexplore inherent spatial diversity through relay channels byutilizing different kinds of cooperation protocols.In [4] and [6], two possible cooperation protocols wereintroduced according to relay processing: amplify-and-**for**ward(AF) and decode-and-**for**ward (DF) protocols. In the case of theAF protocol, while one user acts as a source node and transmitsits in**for**mation, other users in the networks can act as relaynodes. The relay nodes amplify the signal that they receive fromthe source and **for**ward the amplified signal to the destination.For the DF protocol, on the other hand, the relays decodethe received signal and then **for**ward the decoded in**for**mationsymbol to the destination. Also, in [4] and [6], the outageprobability of the two cooperation schemes were analyzed. Theproposed works in [7] and [8] show an idea of user cooperation**for** a two-user CDMA systems. In these schemes, both users1053-587X/$25.00 © 2007 IEEE

3906 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007simultaneously transmit in**for**mation by using an orthogonalcode such that the effect of multiple access interference can bereduced. Another cooperation technique called coded cooperationwas proposed in [9], in which the existing channel codesare incorporated into the cooperation. The coded cooperationachieves higher diversity by transmitting two portions of thecoded symbol separately. Specifically, the first encoded portionis transmitted by the source itself, while the other portion isdelivered by its partners or relay nodes. Recently, rigorousanalysis on exact symbol error rate (SER) and optimum powerallocation **for** the DF protocol have been proposed in [10] **for**two-user cooperation systems. The exact SER **for**mulation andoptimum power allocation **for** the DF protocol **for** a more generalmultihop scenario have been further investigated in [11]. In[5], [12], and [13], an idea of distributed space-time coding hasbeen considered by which all cooperation nodes **for**m virtualantenna array and synchronously encode in**for**mation usingexisting space-time codes.Most of the works in [4]–[13] assume that the destinationhas perfect knowledge of channel state in**for**mation (CSI) **for** alltransmission links. While, in some scenarios, e.g., slow fadingenvironment, the CSI is likely to be acquired by the use ofpilot symbols, it may not be possible in the fast fading environment.In addition, it is questionable as to how the destinationobtains source-relay channel perfectly through pilot signal **for**wardingwithout noise amplification. Moreover, the computationaloverhead **for** channel estimation increases proportional tothe product of the number of transmit antennas and the numberof relaying nodes.To overcome such problems and reduce receiver complexity,various cooperative noncoherent/differential schemes [14]–[16]have been proposed. In [14], symbol error rate analysis based onthe moment generating function method was provided **for** a specifictwo-hop relay systems with coherent detection and differentialdetection. In [15], a framework of noncoherent cooperativediversity has been proposed **for** the DF protocol employingfrequency shift keying signals. However, the framework doesnot fit to the DPSK modulation. Recently, a specific two-userdifferential cooperation scheme was proposed in [16]. Nevertheless,the scheme relies on synchronization among users in thenetworks, and the scheme provides limited transmission rate.In this paper, we propose a threshold-based differential cooperativescheme employing the DF protocol. The proposedscheme not only alleviates the above-mentioned problems ofsynchronization and rate limitation, but also efficiently exploitsinherent spatial diversity through relay channels by the use of apredetermined decision threshold. In particular, the relay helps**for**ward the source symbol only if the symbol is correctly decoded.At the destination, the received signal from the relay iscombined with that from the source only if its amplitude is largerthan the threshold; otherwise only the received signal from thesource is used **for** the detection. A properly designed thresholdallows the destination to make a judgment whether the signalfrom the relay contains the in**for**mation such that the signalsfrom the source and the relay can be efficiently combined andjointly decoded. We analyze bit error rate (BER) per**for**manceof the proposed threshold-based differential DF scheme employingdifferential M-ary phase shift keying (DMPSK) modulation.We consider an ideal case where the relay is able tojudge whether each decoded symbol is correct or not. The per**for**mancein this case serves as a per**for**mance benchmark, andthe proposed scheme can achieve it in an ideal situation that theerror propagation at the relay can be negligible. First, a tightapproximate BER **for**mulation is derived. Then, **for** better understanding,the proposed scheme per**for**mance, we also provideBER upper bound and BER lower bound. **Based** on thetight BER approximation, we jointly determine optimum decisionthreshold and power allocation **for** different scenarios. Simulationresults are shown to validate our proposed scheme andsupport our analytical analysis.The rest of the paper is organized as follows. Section II outlinesthe proposed threshold-based differential DF scheme. InSection III, we analyze BER per**for**mance and then derive atight BER approximation of the proposed scheme under the caseof perfect judgment on the decoding symbol at the relay. BERlower bound and BER upper bound of the proposed schemeare given in Section IV. In Section V, optimum threshold andoptimum power allocation based on the tight BER approximationare jointly determined. Simulation results and discussionsare provided in Section VI. Finally, Section VII concludes thepaper.II. PROPOSED THRESHOLD-BASED DIFFERENTIALDECODE-AND-FORWARD SCHEMEThis section presents detailed descriptions of the proposedthreshold-based differential scheme **for** DF cooperation systems.As shown in Fig. 1, we consider a two-user cooperationsystem in which signal transmission involves two transmissionphases. A user who sends in**for**mation directly to the destinationis considered as a source node. The other user who helps**for**ward the in**for**mation from the source node is a relay node.In Phase I, the source differentially encodes its in**for**mation andthen transmits the encoded symbol to the destination. Due to thebroadcasting nature of the wireless networks, the relay is alsoable to receive the transmitted symbol from the source. In PhaseII, while the source is silent, the relay differentially decodes thereceived signal from the source. If the relay correctly decodesthe transmitted symbol, the relay differentially re-encodes thein**for**mation, and then **for**wards the encoded symbol to thedestination. Otherwise, the relay does not send or remains idle.In both phases, we assume that all users transmit their signalsthrough orthogonal channels by the use of existing schemessuch as time division multiple access (TDMA), frequencydivision multiple access (FDMA), or code division multipleaccess (CDMA).As will be shown in simulation results, the proposed schemeachieves higher diversity gain than the noncooperative transmission,and the obtained diversity is the same as that of thedistributed space-time coding scheme [12], [13]. Compared tothe distributed space-time coding scheme, the proposed schemedoes not require synchronization among nodes since the signalsfrom the source and the relay are not transmitted in the samechannel. However, the bandwidth efficiency of the proposedscheme is less than that of the distributed space-time codingscheme.In Phase I, a set of in**for**mation symbols to be transmitted bythe source is given by where is a set of

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3907Fig. 1. **Threshold**-based differential scheme **for** decode-and-**for**ward cooperative communications.in**for**mation phases. In the case of differential M-ary phaseshift keying (DMPSK), is specified as **for**. In this way, the source differentiallyencodes the in**for**mation symbol into the phase differencebetween two consecutive symbols aswhere is the time index, and is the differentially encodedsymbol to be transmitted at time . After that, the source sendsout with transmitted power to the destination and therelay. The corresponding received signals at the destination andthe relay can be expressed aswhere and are fading coefficients at the source-destinationlink and the source-relay link, respectively, and andare additive noise. Both channel coefficients andare modeled as zero-mean complex Gaussian random variableswith variances and , i.e., and ,respectively. Each of the additive noise terms is modeled aswhere represents the Gaussian noise variance.In Phase II, the relay differentially decodes the transmittedsymbol from the source. Two consecutive received signals,and , are required to recover the transmitted in**for**mation ateach symbol period. By assuming that the channel coefficientis almost constant over two symbol periods, the relay differentiallydecodes based on the decision rule [18]by which the CSI is not required. In the relay-cooperation mode,the relay decides whether to **for**ward the received in**for**mation ornot according to the quality of the received signal. For mathematicaltractability, we assume that the relay can judge whether(1)(2)(3)(4)the decoded in**for**mation is correct or not. 1 If the relay incorrectlydecodes the received signal, such an incorrectly decodedsymbol is discarded, and the relay does not send any in**for**mation.Otherwise, the relay differentially re-encodes the correctlydecoded in**for**mation symbol and **for**wards it to the destination.In conventional differential detection, successful differentialdecoding requires that the encoder differentially encodes eachin**for**mation symbol with the previously transmitted symbol. Inthis way, if the in**for**mation symbols are sent every time slot,then the in**for**mation symbol to be transmitted at time is differentiallyencoded with the transmitted symbol at time .However, in the proposed differential DF scheme, the in**for**mationsymbols at the relay are transmitted only if they are correctlydecoded. There**for**e, the transmission time of the previouslytransmitted symbol can be any time be**for**e the currenttime . We denote such previous transmission time as **for**, i.e., is the latest time that the relay correctly decodesthe symbol be**for**e time . In order to per**for**m successful differentialen/decoding, we let a memory at the relay (see Fig. 1)store the transmitted symbol at time . Note that having amemory does not increase the system complexity comparedto the conventional differential system. The difference is that thememory in our proposed scheme stores the transmitted symbolat time instead of time as does the conventionaldifferential scheme. The differentially re-encoded signal at therelay in Phase II can be expressed aswhere is the differentially encoded symbol at the relay attime . We can see from (1) and (5) that the differentially encodedsymbols at the relay and at the source convey thesame in**for**mation symbol . However, the two encoded symbolscan be different since the relay differentially encodes thein**for**mation symbol with the symbol in the memory which1 Practically, this can be done at the relay by applying a simple SNR thresholdtest to each received symbol. However, this may lead to some error propagation;the effect of error propagation on the system per**for**mance is illustrated by simulationexamples in Section VI.(5)

3908 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007need not be as used at the source. After differential re-encoding,the relay sends the symbol to the destination withtransmitted power , and then stores the transmitted symbolin the memory **for** subsequent differential encoding. Inthis way, the received signal at the destination from the relay inPhase II can be expressed asif relay correctly decodesif relay incorrectly decodes(6)where is the channel coefficient from the relay to the destinationand is an additive noise. We assume that isdistributed, and is distributed.The received signal at the destination comprises the receivedsignal from the source in Phase I and that from the relay inPhase II. As discussed previously, in Phase II, the relay may**for**wards the in**for**mation or remains idle. **With**out knowledgeof the CSI, the destination is unable to know whether the receivedsignal from the relay contains the in**for**mation or not. Toallow the destination make decision whether to combine the signalsfrom the source and the relay, we propose to use a decisionthreshold at the destination node (see Fig. 1). In this way,the received signal at the relay-destination link with amplitudegreater than the threshold is considered as a high-potentialin**for**mation bearing signal to be used **for** further differentialdetection.Particularly, if the amplitude of the received signal from therelay is not greater than the decision threshold, i.e., ,the destination estimates the transmitted symbol based only onthe received signal from the direct link. On the other hand, if, the received signal from the source and that fromthe relay are combined together, and then the combined outputis jointly differentially decoded. Note that, **for** successfullydecoding, the differential detector requires previously receivedsignal to serve as a CSI estimate. Since the received signal fromthe relay may contain the transmitted symbol or may be onlynoise, we propose to use a memory at the destination asshown in Fig. 1 to store the previously received signal at therelay-destination link that tends to contain the in**for**mation. Anideal situation is to let the memory stores the received signalonly when the signal contains the transmitted symbol;however, such in**for**mation is not available since the destinationdoes not have the knowledge of the CSI. In order to efficientlydecode the received signal from the relay, the memoryis used to store the received signal whose amplitude isgreater than the decision threshold. If the threshold is properlydesigned, then the signal in the memory corresponds to thereceived signal from the relay that carries the encoded symbolstored in the memory . **With** an assumption that the channelcoefficients stay almost constant **for** several symbol periods,the signal in the memory serves as a channel estimate ofthe relay-destination link and it is used **for** efficient differentialdecoding at the destination. **Based** on the multichanneldifferential detection in [1], the combined signal be**for**e beingdifferentially decoded isifif(7)where and are combining weights, and representsthe time index of the latest signal in memory , i.e.,is the most recent received signal from the relay whose amplitudeis larger than the threshold. Note that different combiningweights, and , result in different system per**for**mances.In Section III, we determine the BER per**for**mance in the caseof and ; these two combiningweights maximize the signal-to-noise ratio (SNR) of the combineroutput when the destination is able to differentially decodethe signals from both source and relay. Although, these combiningweights are not optimum in general, they are optimum **for**a case that signals from both source-destination and relay-destinationlinks contain in**for**mation and the corresponding channelvariances **for** these two links are the same. **Based** on the combinedsignal in (7), the decoder at the destination jointly differentiallydecodes the transmitted in**for**mation symbol by usingthe following decision rule:III. BER ANALYSISIn this section, we analyze the BER per**for**mance of theproposed threshold-based differential DF scheme employingDMPSK modulation. We focus on an ideal case when the relayis able to judge whether each decoded symbol is correct. Thisleads to a per**for**mance benchmark, and the proposed schemecan achieve it in an ideal situation that the error propagation atthe relay can be negligible. First, we classify different scenariosthat correspond to different instantaneous SNRs at the combineroutput of the destination. Next, the probability that each scenariooccurs is determined. Then, we evaluate an average BERper**for**mance by taking into account all the possible scenarios.A. Classification of Different ScenariosWe classify in this section all different scenarios that result indifferent SNRs at the combiner output. Recall that, if the amplitudeof received signal from the relay is larger than the decisionthreshold, then the destination jointly decodes thereceived signals from the source and the relay . Otherwise,only the received signal from the source is used **for**differential detection. There**for**e, we can classify the scenariosinto two major groups, namely the scenarios thatand . In the case that , the SNR can besimply determined based on the received signal from the directlink. On the other hand, if , the SNR at the combineroutput depends not only on the received signals from the directlink but also on that from the relay link. According to the decisionrule in (7), if , the per**for**mance of the differentialdecoder relies on the received signals from the source at the currenttime and the previous time as well asthe received signal from the relay at the current time andthat stored in the memory . The received signals andfrom the relay may or may not contain the in**for**mation, dependingon the correctness of the decoded symbol at the relayat time and time . If the relay decodes the symbol correctly,then the relay sends the encoded symbol with transmittedpower , otherwise, the transmitted power is zero. Thus, basedon the transmitted power at the relay at time and time ,(8)

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3909Fig. 2. Two possible differential detection techniques at the destination; single-channel differential detection or two-channel differential detection, with six possiblescenarios based on the currently received signal and the signal stored in memory M .we can further classify the scenarios into four different categories,namely (a) and , (b) and, (c) and , as well as (d)and . In the case that and ,the received signals and convey the symbols and, respectively. Since the relay differentially encodes the in**for**mationsymbol from the source with the symbol in thememory , the SNR at the combiner output also depends onwhether the received signal used **for** decoding correspondsto the symbol used **for** encoding. There**for**e, the scenariosunder and can be further separated intotwo cases, namely and . All of these six possiblescenarios are summarized in Fig. 2.In order to facilitate the BER analysis in the subsequent section,we define six different scenarios byasfollows. We let the first scenario be the scenario that the amplitudeof the received signal is not larger than the threshold ,i.e.,When occurs, the destination does not combine the receivedsignal from the relay with that from the source. The second scenariois a scenario that consists of three joint events, namelythe amplitude of the received signal is larger than the threshold,the relay transmitted powers at both time and are equal to, and the received signal stored in memory conveysthe in**for**mation symbol stored in memory . Specifically,can be written as(9)(10)The scenario is similar to the scenario excepts that thereceived signal does not contain the symbol , i.e.,. We express this scenario as. The scenarios to correspondto the scenarios that and either or is zero.Under the scenario , the transmitted power is whereasthe transmitted power is 0. We express the scenarioas. Under the scenario, the transmitted power is 0 whereas the transmittedpower is , i.e.,. The scenario corresponds to the scenario that bothand are zeros. We define the scenario as.For subsequent derivations, we denote as the conditionalBER given a scenario and a set of channel realizationsincluding the source-destination link, the source-relay link, andthe relay-destination link. We also denote as the chancethat the scenario occurs given a set of channel realizations.Accordingly, the conditional BER of the proposed differentialDF scheme can be expressed as(11)By averaging (11) over all channel realizations, the average BERof the proposed scheme is given by(12)where represents the expectation operation.In the following sections, we determine the chance that eachscenario occurs , the conditional BER , andfinally obtain the average BER of the proposed differentialDF scheme.B. Probability of OccurrenceTo determine the chance that each scenario occurs, we firstnote that the amplitude of the received signal depends on therelay transmitted power or the correctness of the decodedsymbol at the relay. **With** DMPSK signals, the chance of incorrectdecoding at the relay, i.e., , can be obtained fromthe conditional SER [17] as(13)

3910 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007where(14)and is the zeroth-order modified Bessel function of thefirst kind. By substituting (18) and (19) into (17), we can expressthe chance that occurs asis the instantaneous SNR per symbol at the relay due to thetransmitted symbol from the source, and(15)Similarly, the chance that the relay **for**wards the symbol withtransmitted power is determined by the chance ofcorrect decoding at the relay, hence(16)where is specified in (13).Consider the scenario in which the amplitude of the receivedsignal is not greater than the decision threshold. Byusing (13)–(16), the chance that occurs can be written as(22)The rest of the scenarios, to , are related to the situationwhen the amplitude of the received signal from the relay,, is greater than the decision threshold . In these scenarios,both the currently received signal and that stored in thememory are used **for** differential detection at the destination.In the proposed scheme, the memory stores onlythe received signal from the relay whose amplitude is largerthan the threshold. This implies that the amplitude islarger than the threshold. There**for**e, the chance that each of thescenarios to happens is conditioned on the event that. From (10), the chance that the scenario occursis given by(17)The conditional probabilities,andcan be obtained from thecumulative distribution function (CDF) of the random variable. The received signal is a complex Gaussianrandom variable with mean and variance , i.e.,.If which results fromincorrect decoding at the relay, the received signal is azero-mean complex Gaussian random variable with variance, and its amplitude is Rayleigh distributed.Hence, the conditional probability that is notgreater than the decision threshold given that the relay does notsend in**for**mation can be expressed as [1](18)Ifwhich corresponds to the case of correct decodingat the relay, then is Gaussian distributed with meanand variance . In this case, is a Ricean-distributedrandom variable. The conditional probability thatis not greater than the decision threshold, given that therelay sends the in**for**mation with transmitted power , can beobtained based on the results in [1] as(23)Since the events at time are independent of the events attime can be written as a product of the probabilities(24)The first term on the right hand side of (24) represents the probabilitythat the relay transmits the decoded symbol with powerand the received signal from the relay is larger than thethreshold. This term can be reexpressed as(25)In (25), the chance that the amplitude of the received signal fromthe relay is larger than the threshold given that the relay sendsthe in**for**mation can be obtained from (19) as(26)There**for**e, using the results in (16) and (26), (25) can berewritten aswherein which is the Marcum -function [18](19)(20)(21)(27)Next, we consider the second term on the right hand side of (24)which represents the chance that the relay transmits with power, and the received signal stored in the memoryconveys the in**for**mation symbol stored in the memory. By using the concept of conditional probability [20], wecan find that

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3911(28)By using the result in (27), the termin (28) can be evaluatedas. If the channels stayalmost constant **for** several time slots, then it can be furtherapproximated by. Note that thederivation in (28) assumes an infinite packet size; however, theimpact of packet size on the per**for**mance is negligible becausethe error propagation stops whenever one symbol is correctlydecoded at the relay and the destination. In addition, a chanceof having large is very small, as it corresponds to the casewhen all consecutive symbols are incorrectly decoded atthe relay. By applying Bayes’ rule [20] and using the results in(13) and (18), we can express the product term in (28) aswhere. Since the signals at time and time areindependent, we can expressas(34)By applying Bayes’ rule, the second term on the right hand sideof (34) is given by(35)The numerator on the right hand side of (35) is in the same **for**mas (27) with replaced by , whereas the denominator canbe calculated by using the concept of total probability [20](29)where the resulting approximation comes from approximatingby **for** all . Accordingly,we can approximate (28) as(36)In (36), the chance that the received signal is greater thanthe decision threshold given that the relay does not send in**for**mationcan be obtained from (18) as(37)Substitute (13), (27), and (37) into (36) resulting in(30)Hence, by substituting (27) and (30) into (24), the chance thatthe scenario happens can be approximated by(31)Next, we consider the scenario which is similar to thescenario except that the received signal stored in thememory at the destination does not convey the symbolstored in the memory at the relay. The chance thatthe scenario happens is given by(32)Observe that the scenarios and are disjoint. Thus, thechance that the scenario happens can be obtained fromas(38)By assuming that the channel coefficients**for** anytransmission link from to , then from (27), (36), and (38), wecan determine the chancein (34) as(39)Now the chance that the scenario happens can be obtainedby substituting (31) and (39) into (33). We can express the probabilityas(33)

3912 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007(40)Following the same steps as used to determinein (34)–(38), the chance that the scenario occurs can be expressedasscheme. The analysis is based on the assumption of perfectjudgment on each decoded symbol at the relay; the obtainedaverage BER will serve as the per**for**mance benchmark that theproposed scheme is able to achieve when the error propagationis negligible at the relay.1) Conditional BER of Each Scenario: When the scenariooccurs, the destination estimates the transmitted symbolby using only the received signal from the source. Letrepresents the instantaneous SNR given that the scenario occurs.**Based** on the conditional BER **for**mulation **for** DMPSKsignals with single-channel reception [18], the conditional BER**for** the scenario is(41)By substituting (27) and (38) into (41), we arrive at (42), shownat the bottom of the page. Similarly, the chance that the scenariohappens can be determined as (43), shown at the bottom ofthe page. **With** an assumption that the channels at time andtime are almost the same, we can see from (41) and (43)that the chances that the scenarios and happen are equal.There**for**e, can be expressed in the same **for**m as (42).Last, the chance that the scenario occurs can be determinedasSubstituting (13) and (37) into (44), we have(44)(45)C. Average BER FormulationWe provide in this section the conditional BER given theoccurrence of each scenario **for** . Using theobtained conditional BER and the probability of occurrenceof each scenario from the previous section, we then derivethe average BER **for**mulation **for** the proposed differential DFwhere the SNR is specified as [18]and(46)(47)(48)(49)in which is the constellation size. In (48) and (49), the parameteris a constant whose value depends on constellationsize. For example, and **for** DBPSKmodulation, and and **for** DQPSKmodulation [18]. The values of and **for** larger constellationsizes can be obtained by using the result in [1].The scenarios to correspond to the case that the destinationcombines the received signal from the source andfrom the relay. The conditional BER **for** these scenarios dependson the combining weights and [see (7)]. Under thescenario , the received signals from the source and the relaycan be expressed as(50)(51)where and are the additive noise terms, and eachof them is zero-mean Gaussian distributed with variance .Since the noise terms of the received signals from both directlink and relay link have the same mean and variance, the SNR(42)(43)

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3913of the combiner output under the scenario can be maximizedby choosing the combining weights .These combining weights are not optimum in general. However,they are optimum in a scenario when signals from both directand relay links contains in**for**mation and their correspondingchannel variances have the same values, i.e., . **With**combining weights, the conditional BERcan be determined by using the conditional BER **for**mulation **for**DMPSK signals with two-channel reception [18]. Thus, we canexpress ascan be ob-independent, the average BER **for** the scenariotained by using (22) and (46) such thatwhere(52)Averaging over the Rayleigh fading channels, the average BER under the scenario is given by(56), and(53)and is specified in (49). The instantaneous SNR is givenby(54)For the remaining scenarios, namely to , the destinationalso combines the received signal from the source and that fromthe relay. However, the use of the two-channel differential detection**for** these four cases are not guaranteed to be optimumsince either the received signals or from the relay containonly noise. Up to now, the conditional BER **for**mulation**for** DMPSK with arbitrary-weighted combining has not beenavailable in the literature. For analytical tractability of the analysis,we resort to an approximate BER, in which the signal fromthe relay is considered as noise when to occur. As wewill show in the succeeding section, the analytical BER obtainedfrom this approximation is very close to the simulationresults. The approximate conditional BER **for** the scenarios, are , where(55)in which, and(57), in which is a function of .According to (31) and (52), the average BER under the scenariocan be approximated as(58)in which represents the noise power that comes from therelay link given that the scenario occurs. The noise powerdepends on the received signal from the relay at the currenttime and that stored in the memory . Under thescenario is given by . Under thescenarios and , the relay does not send the in**for**mationat either time or time . These two scenarios result in thesame noise power such that. Thelast scenario, i.e., , corresponds to the case when bothand does not contain any in**for**mation symbol. The noisepower is equal to the noise variance .2) Average BER: In what follows, we determine the averageBER**for** each scenario. Assumingthat the fading channels of different transmit-receive links areBy averaging over all channel coefficients, we havewhere(59)(60)

3914 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007The average BER under the scenario can be approximatedby . From (31),(39), and (55), we can approximate the average BER under thescenario asBy averaging over all channel coefficients, we obtainwhere(65)(66)**Based** on the chance that the scenario occurs and the conditionalbit error probability , we can see that the approximateBER under the scenario is the same as that underthe scenario , and, hence, can be expressed as in (64).From (45) and (55), the average BER under the scenario canbe approximate as(61)Average (61) over all channel coefficients, resulting inwhere(62)(67)By averaging over all channel coefficients, we havewhere(68)Under the scenarioas(63), the approximate BER can be expressed(64)(69)To this end, we have completed the derivation of the averageBER **for** all possible scenarios. Finally, the average BER, ,can be determined by summing together the average BER**for**(70)in which are specified in (57), (59), (62), (65), and (68).The average BER in (70) is based on an assumption of perfectjudgment on each decoded symbol at the relay. As will be shownlater, the BER in (70) is the best BER per**for**mance that the proposedscheme is able to achieve.IV. BER UPPER BOUND AND BER LOWER BOUNDWe provide in this section BER upper bound and BER lowerbound of the proposed threshold-based differential DF scheme.

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3915Then, we show through computer simulation that, when thepower allocation and the decision threshold are properly designed,the BER upper bound and the BER lower bound areclose to the simulated BER.To obtain a BER upper bound, we first note that the conditionalBER **for** each scenario, , is at most . In addition,if the threshold is properly designed, then the chancesthat the scenarios to happen are small compared to thechances that the scenarios and happen. There**for**e, theBER upper bound can be obtained by bounding the conditionalBER , and by 1/2, wehave(71)where and are determined in (57) and (59), and, and , are given in (32), (42), (43), and(45), respectively.To find a BER lower bound, we found that it is hard to obtainthe exact BER of and because the exact expressionsof the chances that the scenarios and occur involve the approximationthat the channel coefficients are constant **for** severalsymbol periods. In addition, the BER **for**mulation given that thescenario happens is currently unavailable. However, if thepower ratio and the threshold are properly designed, the chancethat the scenario occurs tends to be small compared to thechance that the scenario occurs. Moreover, the conditionalBER under the scenariois larger than that underthe scenario. There**for**e, the BER lower boundcan be obtained by bounding the conditional BER with. In this way, the BER under the scenarios andcan be lower bounded as(72)where, which is evaluated in (39), is the chancethat the scenarios and occur. According to (31) and (52),the average BER in (72) can be expressed asBy averaging over the channel realization, we obtain(73)(74)where(75)Since the exact BER **for**mulations under the scenarios 4, 5, and 6are currently unavailable, and the chances that these three scenarioshappen are small at high SNR, we further lower boundthe BER , and by 0. As a result, the BER ofthe proposed differential DF scheme can be lower bounded by(76)where is determined in (57).Fig. 3(a)–(f) compares the BER approximation (70), theBER upper bound (71), and the BER lower bound (76) withthe simulated per**for**mance in the case of DQPSK modulation.In Fig. 3(a)–(c), we consider the differential DF cooperationsystem with, i.e., all the channel linkshave the same qualities. We can see from the figures that theapproximate BER closely matches with the simulated BER, andboth the approximate BER and the simulated BER lie betweenthe BER upper bound and the BER lower bound. Moreover, thesystem per**for**mance depends on the power allocation and thethreshold. By choosing proper power allocation and threshold,not only the BER per**for**mance improves, but also the lowerbound is closer to the simulated per**for**mance. For example,considering a system with threshold of and total transmitpower where and are transmitted powerof the source and the relay, respectively. By changing thepower allocation from to , the BERper**for**mance is improved by 1 dB at a BER of , whilethe per**for**mance gap between the simulated BER and the BERlower bound is reduced by 2 dB at the same BER. This resultfollows the fact that when the threshold is appropriatelychosen, the scenarios to occur with much smaller probabilitythan the scenarios and . Even though the BERunder each of the scenarios to is larger than that underthe scenarios or , the average BERare smaller. The BER is dominated by the BER underscenarios and . There**for**e, the per**for**mance gaps betweenthe bounds and the approximate BER is small if the thresholdis properly designed.We have the same observations in Fig. 3(d)–(f) **for** a systemwith , and . At a threshold of, the per**for**mance can be improved by allocating morepower at the source and less power at the relay. This is in agreementwith the results in [10] which illustrate that the channellink between source and relay and the channel link betweenrelay and destination should be balanced in order to achievea per**for**mance diversity of two. Interestingly, the per**for**mancecan be significantly improved by choosing a proper thresholdregardless of the power allocation. For instance, increasing thethreshold from 1 to 2, the per**for**mance of the proposed schemewith equal power allocation improves 4 dB at a BER of .At the threshold of , by changing the power allocation

3916 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007Fig. 3. DQPSK cooperation system: =1; =1. (a) P =0:5P; P =0:5P; =1; =1. (b) P =0:8P; P =0:2P; =1; =1.(c) P =0:8P; P =0:2P; =2; =1. (d) P =0:5P; P =0:5P; =1; =10. (e) P =0:5P; P =0:5P; =2; =10. (f) P =0:8P; P =0:2P; =2; =10.from to , the system per**for**mance isfurther improved by only 0.5 dB at the same BER. This impliesthat the effect of the threshold dominates; the per**for**mance doesnot severely depend on the power allocation after the thresholdis properly designed.Another observation obtained from Fig. 3(a)–(f) is that thethreshold depends on the channel link qualities. To be specific,the threshold should be increased as the link quality betweenthe relay and the destination increases. For example, under thescenario , the threshold results in superiorper**for**mance in the case of while the thresholdleads to better per**for**mance in the case of .This observation can be explained as follows. When the linkquality between the relay and the destination is good, i.e., the

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3917channel variance is high, the received signal from the relay tendsto have large energy if it carries the in**for**mation. As a result, byincreasing the threshold from 1 to 2, we reduce the chance thatthe received signal whose amplitude is larger than the thresholdcontains no in**for**mation. Thus, with threshold of 2, the receivedsignals from the relay and the destination are efficiently combinedand, hence, resulting in better per**for**mance.V. OPTIMUM THRESHOLD AND POWER ALLOCATIONSimulation results in the previous section show that the choiceof power allocation and threshold significantly affects the per**for**mance.In this section, we determine an optimum decisionthreshold and an optimum power allocation **for** the proposeddifferential DF cooperation scheme based on the tight BER approximationin (70). To simplify the notation, let us denoteas the power ratio of the transmitted power at the sourceover the total power . For a fixed total transmittedpower , we jointly optimize the threshold andthe power ratio such that the tight BER approximation in (70)is minimized. The optimization problem can be **for**mulated as(77)whererepresents the BER approximation as specifiedin (70) with and . Note thatthe optimum power allocation and the optimum threshold arejointly determined based on channel variances, not instantchannel coefficients. The destination needs to feedback theobtained optimum power allocation in a **for**m of a few in**for**mationbits through a control channel. The feedback in**for**mationis updated only when the network topology or the channelschange dramatically.Fig. 4(a)–(f) shows the BER per**for**mance of the proposedscheme with DQPSK signals as a function of power allocationand threshold. In Fig. 4(a)–(c), we consider the case when thechannel variances of all communication links are equal, i.e.,. The BER approximation is plottedin Fig. 4(a) and its cross sections are shown in Fig. 4(b) and (c)together with the simulated BER curves. **Based** on the approximateBER in Fig. 4(a), the jointly optimum power allocationand decision threshold are and . Fig. 4(b) comparesthe cross-sectional curves of the approximate BER withthe simulated BER in the case of . We can see that the approximateBER is close to the simulated BER. Furthermore, theproposed scheme with any threshold less than 1.5 yields almostthe same BER per**for**mance, and the per**for**mance significantlydegrades as the threshold increases more than 1.5. The reason isthat when the threshold increases, the chances that the scenariosto occur increases, and, hence, the average BER is dominatedby the BER under these scenarios. Fig. 4(c) depicts theapproximate and simulated BER curves as functions of powerallocation in the case of the decision threshold . We canobviously see that the power ratio of results in the optimumper**for**mance.In Fig. 4(d)–(f), we consider the case of channel variances. Fig. 4(d) depicts the BER ofthe proposed scheme as a function of the power allocationand the decision threshold . In this scenario, we can see thatthe jointly optimum power allocation and decision threshold areand . Fig. 4(e) shows cross-sectional curvesof the approximate BER and the simulated BER per**for**manceunder the same power allocation at . We can see thatthe approximate BER closely matches to the simulated BER **for**every threshold value. According to both the simulated BER andthe approximate BER, the optimum threshold **for** this case isabout 1.7. We show in Fig. 4(f) a comparison of the approximateBER and the simulated BER with decision thresholdunder different power allocation. Clearly, the approximate BERfollows the same trend as the simulated BER, and the optimumpower allocation is **for** the differential DF system withdecision threshold .Fig. 5(a)–(c) shows simulated and theoretical BER curveswhen the channel variances arewhich correspond to the case when the relay is located near thesource. From Fig. 5(a), we observe that the joint optimum powerallocation and optimum threshold are and ,respectively. Fig. 5(b) plots the BER curve as a function ofthreshold at the optimum power allocation. It confirms our expectationthat in this case the threshold does not significantly influencethe system per**for**mance. This is because when the relayis close to the source, the relay receives almost all the originalin**for**mation from the source and, hence, **for**wards most of thesource in**for**mation to the destination.VI. SIMULATION RESULTSWe per**for**m computer simulations of the two-user cooperationsystems employing the DF protocol. The DQPSK modulationis used in all simulations. The channel fading coefficientsare modeled according to the Jakes’ model [21] with theDoppler frequency Hz and normalized fading parameterwhere is the sampling period. The noisevariance is . We generate channel realizations, andthe BER is computed **for** each SNR value when at least 100 erroneousbits are detected. The BER curves are plotted as functionsof , where is the total transmitted power. The powerallocation **for** the source node and the relay node are fixed at.Fig. 6(a) compares the per**for**mance of the proposedthreshold-based differential DF scheme to that of the differentialDF scheme with ideal relay but without thresholdand that of the differential DF scheme in which the relayalways **for**wards the decoded symbol to the destination andno threshold is used. In Fig. 6, the ideal relay refers to thecase that the relay is able to perfectly judge whether eachdecoded symbol is correct and **for**ward only the correctlydecoded symbols to the destination. We consider the systemwith power allocation, and channelvariances , and . From Fig. 6(a),the proposed differential DF scheme outper**for**ms the other twoschemes. The reason is that a decoding error at the relay tendsto result in an error at the destination. Hence, the per**for**manceof the differential DF scheme with always-**for**ward relay andwithout threshold is worse than that of the proposed scheme.In particular, the per**for**mance degradation of 11 dB can be observedat a BER of . Adding a threshold at the destinationcan reduce the chance that the incorrectly decoded signal from

3918 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007Fig. 4. DQPSK: Comparison of theoretical BER curves and simulated curves. (a)–(c) = = =1. (d)–(f) =1; =1; =10.(a) Jointly optimum threshold and power allocation. (b) Varying threshold, P =0:7P and P =0:3P . (c) Varying power allocation, =1. (d) Jointly optimumthreshold and power allocation. (e) Varying threshold, P =0:8P and P =0:2P . (f) Varying power allocation, =1:7.the relay is combined to the signal from the source. There**for**e,the proposed scheme yields superior per**for**mance to thedifferential DF scheme with ideal relay but without threshold.As shown in Fig. 6(a), the proposed scheme yields about 4-dBgain at a BER of compared to the scheme with idealrelay but without threshold. In Fig. 6(a), we also compare theper**for**mance of the proposed differential DF scheme employingDQPSK signals with that of direct transmission schemes employingdifferential and coherent BPSK signals. All schemeshave the same bandwidth efficiency; however, the proposeddifferential DF scheme outper**for**ms the direct transmissionschemes with differential detection and with coherent detection

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3919Fig. 6. DQPSK with different cooperation schemes, P = P =0:5P; =1; =1, and =10. (a) Comparison of differentcooperation schemes. (b) Effect of decoding error at relay.Fig. 5. DQPSK: Comparison of theoretical BER curves and simulated curves. =1; =10, and =1. (a) Jointly optimum threshold and powerallocation. (b) Varying threshold, P =0:5P and P =0:5P . (c) Varyingpower allocation, =0:4.when the SNRs are larger than 12 and 16 dB, respectively. Thisis because the proposed differential DF scheme obtains higherdiversity gain but requires larger signal constellation size thanthe direct transmission; the effect of noise to the system per**for**mancedominates at low SNR while the effect of diversity gaindominates at high SNR. We also show the per**for**mance of coherentDF cooperative QPSK scheme with ideal relay, which isconsidered as the coherent counterpart of the proposed differentialDF scheme with ideal relay. Note that in the case of coherentdetection, the destination is able to determine whether the relaysends an in**for**mation or not from a channel estimation process,and, hence, it does not require any threshold test at the destination.At a BER of , the differential DF scheme with idealrelay but without threshold losses about 9 dB from the coherentcounterpart, while the proposed scheme is only 5 dB away fromthe coherent counterpart.Fig. 6(b) shows simulated per**for**mance in the case of nonidealrelay, i.e., the judgment on the correctness of each decodedsymbol at the relay is not perfect. The simulation scenario inFig. 6(b) is the same as that in Fig. 6(a). In the simulation, theprobability that the relay makes incorrect judgment is at 2%,

3920 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 20075%, 10%, and 15% of the total decoded symbols. It is obviousfrom the figure that the per**for**mance degrades as the relay makesmore incorrect judgment. A small per**for**mance error floor isalso observed in this case. We can see that **for** an incorrect judgmentof 2% of the total decoded symbols, the per**for**mance curveis slightly different from the ideal relay case at medium SNRrange. The effect of error propagation is quite obvious at highSNR range. Nevertheless, the obtained per**for**mance is betterthan the per**for**mance of the differential scheme in which therelay always **for**wards each decoded symbol and no thresholdis used at the destination. In this case, we observe more than8-dB per**for**mance improvement at a BER of . The resultin Fig. 6(b) indicates that it is also important to design a detectionscheme at the relay that achieves small judgment errors.The problem of optimizing a threshold test at the relay and thecorresponding per**for**mance analysis of the system with imperfectjudgment at the relay are further investigated in our futurework.Fig. 7(a) and (b) illustrates the BER per**for**mances of the proposedscheme with different thresholds. In Fig. 7(a), the powerallocation is and , and the channel variancesare. We can see that the proposedscheme achieves the per**for**mance diversity of two at highSNR **for** any threshold. However, the per**for**mance degrades asthe threshold increases. Fig. 7(b) shows the BER per**for**mancein the case of power allocation and ,and channel variances , and . Obviously,different thresholds result in different BER per**for**mances.The threshold of provides the best per**for**mance in thisscenario. Furthermore, if the threshold is too small, e.g., ,not only the BER per**for**mance degrades but also the diversityorder is less than two. This is because when the threshold issmall, the destination tends to combine the signals from boththe relay and the destination. As a result, the incorrect decodingat the relay leads to significant per**for**mance degradation at thedestination.In Fig. 8(a) and (b), we study the effect of power allocationon the BER per**for**mance **for** the proposed scheme with a fixedthreshold. In Fig. 8(a), we consider the cooperation system withchannel variancesand a threshold. We can see that the power ratios and0.7 yield almost the same per**for**mances. When the power ratioincreases to , the per**for**mance degradation is about 2 dBat BER of compared to the equal power allocation scheme.This is due to the fact that at , small power is allocatedat the relay. Consequently, even though the received signal fromthe relay carries an in**for**mation, there is high chance that itsamplitude is smaller than the threshold, and, hence, the detectionis based only on the received signal from the direct link. Fig. 8(b)depicts the per**for**mance in the case of channel variances, and , and a threshold . We can seethat the per**for**mance improves as the power ratio increases fromto . The reason is that the relay-destination linkis of high quality while the threshold is small. **With** only smallpower at the relay, the amplitude of the received signal from therelay can be larger than the threshold. There**for**e, by allocatingmore power at the source, we not only increase the chance ofcorrect decoding at the relay, but also increase the SNR of theFig. 7. DQPSK: Different thresholds with fixed power allocation. (a) P =0:7P; P =0:3P; = = =1. (b) P =0:8P; P =0:2P; = =1; =10.combiner output. **Based** on the numerical results in Fig. 4(d),the optimum power ratio **for** this scenario is at the SNRofdB. Clearly, the simulation results in Fig. 8(b)agree with the numerical results at the SNR of 16 dB. Moreover,Fig. 8(b) illustrates that the power ratio of results inoptimum per**for**mance **for** the entire SNR range. At the threshold, the proposed scheme with optimum power allocationachieves about 5-dB improvement over that with equal powerallocation at a BER of .Fig. 9(a) and (b) compares the per**for**mances of the proposeddifferential DF scheme with different power allocations and decisionthresholds. We consider the case ofin Fig. 9(a), and the case of , and inFig. 9(b). From both figures, it is clear that the proposed schemewith jointly optimum power allocation and optimum thresholdyield the best per**for**mance over the entire SNR range. In the

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3921Fig. 8. DQPSK: Different power allocations with fixed threshold. (a) =2; = = =1. (b) =1; = =1; =10.Fig. 9. DQPSK: Different power allocations and different thresholds.(a) = = =1. (b) = =1; =10.case of equal link qualities, i.e., ,optimum power allocation and optimum threshold yields 2-dBper**for**mance improvement at a BER of compared to thescheme with equal power allocation and without threshold. Inaddition, if the power allocation is optimum, the scheme withoutthreshold yields almost the same per**for**mance as that with optimumthreshold. When the quality of the relay-destination linkis very good, e.g., , the use of optimum thresholdis more important than the use of optimum power allocation athigh SNR. Specifically, by properly choosing the threshold, theproposed differential DF scheme achieves almost the same per**for**mance**for** any power allocation at high SNR. As we can seefrom Fig. 9(b), in the case of equal power allocation, using theoptimum threshold leads to more than 5-dB improvement gainover the scheme without threshold at a BER of . **With** optimumthreshold, the per**for**mance difference between the proposedscheme with optimum power allocation and that withequal power allocation is only about 0.5 dB at a BER of .VII. CONCLUSIONWe propose, in this paper, a threshold-based differential decode-and-**for**wardscheme **for** two-user cooperative communicationssystems. By allowing the relay to **for**ward only the correctlydecoded symbols and introducing a decision threshold atthe destination, the proposed scheme efficiently combines thesignals from the direct and the relay links. We provide BERanalysis of the proposed scheme with DMPSK modulation byclassifying six different scenarios which lead to different instantaneousSNRs at the combiner output of the destination. Theanalysis focuses on the case when the relay is able to judge thecorrectness of each decoded symbol. A tight approximate BERexpression is derived, and it serves as a per**for**mance benchmarkthat the proposed scheme can achieve when error propagationis negligible at the relay. The corresponding BER lower boundand BER upper bound are **for**mulated. The approximate BER is

3922 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007very close to the simulated BER curve, and it lies between theobtained BER lower bound and BER upper bound. **Based** onthe tight BER approximation, we determine the optimum decisionthreshold and power allocation numerically. Both theoreticaland simulation results reveal that the optimum threshold andoptimum power allocation rely on the qualities of the channellinks. When the quality of the relay-destination link is muchlarger than the other links, i.e., and ,then the decision threshold is more important than the powerallocation at high SNR. For instance, in the case of DQPSK signalswith equal power allocation, using the optimum thresholdresults in more than 5-dB improvement over the scheme withideal relay but without threshold at a BER of . By furtherusing the optimum power allocation, the per**for**mance improvementis about 0.5 dB at the same BER. Simulation resultsalso show that the proposed scheme with DQPSK signals provides11-dB per**for**mance improvement over the differential DFscheme with always-**for**ward relay and without threshold at aBER of .REFERENCES[1] J. G. Proakis, Digital Communications, 4th ed. New York, NY: Mc-Graw-Hill, 2000.[2] B. M. Hochwald and W. Sweldens, “**Differential** unitary space-timemodulation,” IEEE Trans. Commun., vol. 48, no. 12, pp. 2041–2052,Dec. 2000.[3] B. L. Hughes, “**Differential** space-time modulation,” IEEE Trans. Inf.Theory, vol. 46, no. 11, pp. 2567–2578, Nov. 2000.[4] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversityin wireless networks: Efficient protocols and outage behavior,” IEEETrans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.[5] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols**for** exploiting cooperative diversity in wireless networks,” IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 2415–2525, Oct. 2003.[6] J. N. 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Norwell,MA: Kluwer, 2001.Thanongsak Himsoon (S’03–M’07) received theB.S. degree in electrical engineering from the ChulalongkornUniversity, Bangkok, Thailand, in 1995,and the M.S. and Ph.D. degrees in electrical engineeringfrom the University of Maryland, CollegePark, in 2001 and 2006, respectively.He is currently a communications systems designengineer at Meteor Communications Corporationworking on algorithm and system design **for** wirelesscommunications. His research interests includesignal processing, wireless communications, cooperativecommunications, and wireless sensor networks, with particular focus ondifferential modulation systems. His research contributions encompass differentialspace-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 ChulalongkornUniversity, Bangkok, Thailand, in 1999,and the M.S. and Ph.D. degrees in electrical engineeringfrom the University of Maryland, CollegePark, in 2001 and 2005, respectively.She is currently a wireless communications specialistwith Meteor Communications Corporation,working in the research and development of wirelesscommunications technology. From January 2005to May 2005, she was a Postdoctoral ResearchAssociate with the Department of Electrical and Computer Engineering andthe Institute **for** Systems Research (ISR), University of Maryland, CollegePark. Her research interests span a broad range of areas from signal processingto wireless communications and networking, including space-time coding**for** multi-antenna communications, cross-layer design **for** wireless networks,communications in mobile ad hoc networks and wireless sensor networks,OFDM systems, and ultrawideband communications.Weifeng Su (M’03) received the B.S. and Ph.D.degrees in mathematics from Nankai University,Tianjin, China, in 1994 and 1999, respectively, andthe Ph.D. degree in electrical engineering from theUniversity of Delaware, Newark, in 2002.He is an Assistant Professor at the Department ofElectrical Engineering, State University of New York(SUNY) at Buffalo. From June 2002 to March 2005,he was a Postdoctoral Research Associate with theDepartment of Electrical and Computer Engineeringand the Institute **for** Systems Research (ISR), Universityof Maryland, College Park. His research interests span a broad rangeof areas from signal processing to wireless communications and networking,

HIMSOON et al.: DIFFERENTIAL MODULATION WITH THRESHOLD-BASED DECISION COMBINING 3923including space-time coding and modulation **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 serves as anAssociate Editor **for** the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.K. J. Ray Liu (F’03) is a Professor and AssociateChair of Graduate Studies and Research,Electrical and Computer Engineering Department,University of Maryland, College Park. His researchcontributions encompass broad aspects of wirelesscommunications and networking, in**for**mation **for**ensicsand security, multimedia communications andsignal processing, bioin**for**matics and biomedicalimaging, and signal processing algorithms andarchitectures.Dr. Liu is the recipient of numerous honors andawards, including best paper awards from the IEEE Signal Processing Society(twice), the IEEE Vehicular Technology Society, and EURASIP; the IEEESignal Processing Society Distinguished Lecturer, the EURASIP MeritoriousService Award, and National Science Foundation Young Investigator Award.He also received various teaching and research awards from University ofMaryland, including the Poole and Kent Company Senior Faculty TeachingAward and the Invention of the Year Award. He is Vice President—Publicationsand on the Board of Governor of the IEEE Signal Processing Society. He wasthe Editor-in-Chief of IEEE Signal Processing Magazine and the foundingEditor-in-Chief of the EURASIP Journal on Applied Signal Processing.