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21. beacon assisted spectrum access with cooperative cognitive transmitter and receiver



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  • 1. 112 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 2010 Beacon-Assisted Spectrum Access with Cooperative Cognitive Transmitter and Receiver Ali Tajer, Student Member, IEEE, and Xiaodong Wang, Fellow, IEEE Abstract—Spectrum access is an important function of cognitive radios for detecting and utilizing spectrum holes without harming the legacy systems. In this paper, we propose novel cooperative communication models and show how deploying such cooperations between a pair of secondary transmitter and receiver assists them in identifying spectrum opportunities more reliably. These cooperations are facilitated by dynamically and opportunistically assigning one of the secondary users as a relay to assist the other one, which results in more efficient spectrum hole detection. Also, we investigate the impact of erroneous detection of spectrum holes and thereof missing communication opportunities on the capacity of the secondary channel. The capacity of the secondary users with interference-avoiding spectrum access is affected by 1) how effectively the availability of vacant spectrum is sensed by the secondary transmitter-receiver pair, and 2) how correlated are the perceptions of the secondary ransmitter-receiver pair about network spectral activity. We show that both factors are improved by using the proposed cooperative protocols. One of the proposed protocols requires explicit information exchange in the network. Such information exchange in practice is prone to wireless channel errors (i.e., is imperfect) and costs bandwidth loss. We analyze the effects of such imperfect information exchange on the capacity as well as the effect of bandwidth cost on the achievable throughput. The protocols are also extended to multiuser secondary networks. Index Terms—Cognitive radio, spectrum access, cooperative diversity, opportunistic communication, channel capacity. Ç1 INTRODUCTION delectability of the unused portions of spectrum by usingC OGNITIVE radios have been introduced as a potential solution for alleviating the scarcity of frequency spec-trum in overly crowded environments [1], [2], [3], [4]. physical layer techniques, while those in the latter one, develop media access control (MAC) protocols seeking toAccording to the Federal Communication Commission maximize secondary users’ network throughput. The(FCC), a vast portion of the frequency bands is used only channel access protocols we propose in this paper fallsporadically and furthermore, the usage varies geographi- within the first category mentioned above. These protocolscally and temporally. Such inefficient utilization of spectrum, allow the secondary transmitter and receiver to coopera-as well as the increasing demand for frequency bands by the tively listen to the primary user instead of havingexisting and emerging wireless applications, motivates independent observations.opportunistic access to licensed bands by unlicensed The gains yielded for the detection of unused spectrum(secondary) users. The notion of cognitive radios makes it are due to the diversity gains introduced by the coopera-possible to accommodate self-configuring ad hoc links tion. The increase in the channel capacity is also partly duewithin currently established wireless communication infra- to the same diversity gain, but is mostly shaped by reducingstructures. For accessing vacant spectrum bands assigned to the uncertainty at the secondary transmitter and receiver nodes about each other’s observation of the spectrallicensed users, the secondary users continuously and actively activity, which is a by-product of the cooperation betweenmonitor the spectrum in order to efficiently make use of the them. As discussed in [14], discrepancy in spectral activityspectrum hole while avoiding interference with licensed awareness at the secondary transmitter and receiver, whichusers (interference-avoiding [5] or interweave paradigm [6]) is due to their spatial separation and/or imperfect channelor having controlled level of interference temperature sensing, incurs a loss in channel capacity.(interference-controlled [5] or underlay paradigm [6]). The underlying idea of the cooperation models devel- Related studies on spectrum access can be broadly oped is to have a one-time broadcast of a beacon messagecategorized as those proposing efficient methods for and a one-time relaying of this message by one of thelocating the holes in the spectrum [7], [8], [9] and those secondary users such that both enjoy a second-orderdiscussing how to optimally allocate the available unused diversity gain in detecting the beacon message. This beaconfrequency bands to secondary users [10], [11], [12], [13]. The message is a codeword within the codebook of the primaryworks in the former category aim at improving the user, reserved specifically for the purpose of informing the secondary users of the vacant channel. Besides seeking spectrum opportunities and utilizing. The authors are with the Department of Electrical Engineering, Columbia University, New York, NY 10027. E-mail: {tajer, wangx}@ee.columbia.edu. them, another major function of secondary users is to detect the return of primary users and agilely vacate theManuscript received 2 Oct. 2008; revised 3 Mar. 2009; accepted 6 May 2009; channel. The idea of deploying cooperative diversitypublished online 27 May 2009.For information on obtaining reprints of this article, please send e-mail to: schemes and their merits in detecting the return of thetmc@computer.org, and reference IEEECS Log Number TMC-2008-10-0393. primary users has also been investigated as an independentDigital Object Identifier no. 10.1109/TMC.2009.103. problem in [15], [16], [17]. 1536-1233/10/$26.00 ß 2010 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 2. TAJER AND WANG: BEACON-ASSISTED SPECTRUM ACCESS WITH COOPERATIVE COGNITIVE TRANSMITTER AND RECEIVER 1132 SYSTEM DESCRIPTIONS 3 COOPERATION PROTOCOLSWe consider a primary transmitter Tp and a pair of We assume that the secondary users are informed of thesecondary transmitter Tt and receiver Tr users. The vacancy of the channels by having the primary usersecondary users continuously monitor the channel used by broadcast a beacon message a priori known to the secondarythe primary user, seeking an opportunity to take it over users. The beacon message is a reserved codeword in thewhen it is unused. We assume that all channels between the codebook of the primary user and is dedicated to announ-primary user and the secondary users and those between the cing the availability of the channel for being accessed by thesecondary users are quasi-static wireless flat fading channels secondary users. We intend to devise a cooperation modelwhich remain unchanged during the transmission of a block such that both secondary transmitter and receiver decode theof symbols and change to independent states afterwards. beacon message with a second-order diversity gain. We denote the channels between the primary user and Cooperative diversity has been studied and developedthe secondary transmitter and receiver by p;t and p;r , extensively as a means for making communication overrespectively. The channel between the secondary nodes are wireless links more reliable [18], [19], [20], [21]. Indenoted by t;r and r;t . The physical channel between nodes cooperative communication, a point-to-point link is assistedi 2 fp; t; rg and j 2 ft; rg has an instantaneous realization by an intermediate node (relay) aiming at providing the pffiffiffiffiffiffiffi intended receiver by additional diversity gains while the i;j ¼ i;j Á hi;j ; ð1Þ overall resources (frequency bandwidth and power) remainwhere fading coefficients hi;j are assumed to be independent, unchanged compared to the noncooperative schemes.circularly symmetric complex Gaussian random variables In this paper, we first develop a novel cooperation modelCN ð0; 1Þ. The term i;j accounts for path loss and shadowing appropriate for multicast transmissions and then show howand is given by i;j ¼ Si;j dÀ , where Si;j represents the log- i;j to adopt it for building cooperative spectrum accessnormal shadowing effect, di;j is the physical distance schemes. Unlike the conventional three node cooperativebetween nodes i and j, and  > 0 is the path loss exponent. models where the objective is to achieve a second-order In [7], it is shown that compared with energy detection diversity gain at only the intended receiver, we considermethods, the detection of vacant spectrum can be signifi- broadcasting the same message to two receivers such thatcantly improved through transmitting pilot signals by the both enjoy second-order diversity gains and yet use the sameprimary user and deploying coherent detection at the amount of resources. This cooperation scheme is a combina-secondary users. Motivated by this fact, in this paper we tion of opportunistic relay assignment and bit forwarding.assume that whenever a channel is released by the primary While this protocol, like most other existing cooperationuser, it is announced to the secondary users by having the models, guarantees performance enhancement in only highprimary user broadcast a beacon message. It is noteworthy enough SNR regimes, we further modify this protocol suchthat as eventually we are evaluating the network throughput, that it exerts cooperation only if it is beneficial. Thisit is valid to assume that the secondary users are always modified protocol, compared to the noncooperative trans-willing to access the channel and are continuously monitor- mission, will ensure performance improvement over alling the channel for the beacon message by the primary user. SNR regimes and all channel realizations. We consider N consecutive channel uses for the transmis-sion of the beacon message. The beacon message, sent by the 3.1 Cooperative Spectrum Access (CSA)primary user, is denoted by x b ¼ ½xb ½1Š; . . . ; xb ½NŠŠT , and the Assume that the beacon message consists of K informa-received signals by the secondary transmitter and receiver tion bits and N À K parity bits giving rise to the codingare denoted by y t ¼ ½yt ½1Š; . . . ; yt ½NŠŠT and yr ¼ ½yr ½1Š; . . . ; rate K and requires N channel uses in a noncooperative Nyr ½NŠŠT , respectively. For the direct transmission from the transmission. We utilize the regenerative scheme of [21]primary user to the secondary users, as the baseline in our and divide the beacon message into the smaller segments 4comparisons, we have of lengths K N1 < N and N2 ¼ N À N1 , and define the 4 N1 level of cooperation as ¼ N . The transmission of the yt ½nŠ ¼ p;t xb ½nŠ þ zt ½nŠ; ð2Þ beacon message is accomplished in two steps: and yr ½nŠ ¼ p;r xb ½nŠ þ zr ½nŠ; n ¼ 1 . . . ; N; ð3Þ 1. The primary user uses only the first N1 channel useswhere z t and zr denote the zero mean additive white (as opposed to noncooperative that uses all theGaussian noise terms with variance N0 . Also we assume that N channel uses) to broadcast a reserved codebook inall transmitted signals have the same average power, i.e., its codebook (x 0b ) when it is willing to release the x 4 P channel. This reserved codeword is a weaker code-IE½jx b j2 Š Pp , and denote ¼ Np0 as the SNR without fading, xpathloss, and shadowing. Therefore, the instantaneous SNR word compared to the original beacon. Meanwhile, the secondary transmitter and receiver try to decodeis given by the first segment. SNRi;j ¼ i;j Á jhi;j j2 : ð4Þ 2. During the remaining ðN À N1 Þ channel uses, any secondary user who has successfully decoded theThroughout the paper we say that two functions fðxÞ and first segment, immediately constructs N2 additional :gðxÞ are exponentially equal, denoted by fðxÞ ¼ gðxÞ if extra bits and forwards them to the other secondary user. If both happen to decode successfully, both log fðxÞ lim ¼ 1: will try to transmit in the second phase and their x!1 log gðxÞ transmissions collide. Such collision is insignificantThe ordering operators _ and ! are defined accordingly. _ and incurs no loss as it only happens when both Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 3. 114 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 2010 secondary users have already successfully decoded As shown in the next sections, including the primary the beacon. If neither of the secondary users is transmitter in the competition has the advantage of guaran- successful in decoding the first segment of the teeing that cooperation is deployed only when it is helpful. beacon message, there will be no transmission in the A simple technique for identifying the winner in the second phase and hence ð1 À Þ portion of the time second phase in a distributed way (without having a central slot is wasted. Despite such possible waste, as controller) is to equip each primary and secondary user analyses reveal, by cooperation a better overall with a backoff timer, with its initial value set inversely performance can be achieved. As shown in Section 4, proportional to its corresponding metric t. Therefore, the the combination of opportunistic relay assignment timer of the user with the largest metric t goes off sooner and bit forwarding uses essentially the same amount and will start relaying. The idea of encoding the backoff of resources (power and bandwidth) as in the timer by channel gain has also been used in [22], [23]. noncooperative approach, and yet provides a sec- In multiuser primary networks, the central authority of ond-order diversity gain for both secondary users, the network responsible for spectrum coordinations should whereas if one of the nodes is selected a priori and be in charge of transmitting the beacon message. The reason independently of the channel conditions, this node is that individual users are not fully aware of the network will achieve only a first-order diversity gain. spectral activity and if they act independently, there existsIt is also noteworthy that the CSA protocol can be extended the possibility that some user mistakenly announces that abeyond the scope of cognitive networks and can be adopted spectrum is vacant while some other primary user isin any application in which a message is intended to be utilizing it. Such strategy will also avoid having multiplemulticast to several nodes. primary users transmit beacon messages concurrently, in which case these messages collide and are lost.3.2 Opportunistic Cooperative Spectrum Access (OCSA)Attaining a higher diversity order, which is a high SNR 4 DIVERSITY ANALYSISmeasure, regime does not necessarily guarantee perfor- In this section, we characterize the performance of themance enhancement in the low or even moderate SNR proposed protocols in terms of the probability of erroneousregimes. For instance, when either the source-relay or the detection of the beacon message, which we denote by P ðeÞ,relay-destination link is very weak, cooperation may not be at the signal-to-noise ratio . The achievable diversity gainbeneficial. When exchanging certain information between which measures how rapidly the error probability decaysthe primary and the secondary users is possible, we can with increasing SNR is given byovercome this problem by modifying the CSA protocol log P ðeÞsuch that cooperation is deployed only when deemed diversity order ¼ À lim : !1 log beneficial. Therefore, the protocol becomes opportunistic inthe sense it dynamically decides whether to allow coopera- 4.1 Noncooperative Schemetion or not. The cost of this improvement is that the primaryand the secondary users need to acquire the quality of their As a baseline for performance comparisons, we firstoutgoing channels. Acquiring such channel gains is not consider the direct transmission (noncooperative) by thenecessarily feasible in all networks, but those capable of it primary user over the channels given by (2) and (3). We denote the events that the secondary transmitter andcan benefit from deploying the OCSA protocol instead of receiver do not decode the beacon message successfullythe CSA protocol. The discussions on how the required by et and er , respectively. For a coded transmission withinformation exchange should take place are provided in coherent detection, the pairwise error probability (PEP) thatSection 6.1. The cooperation is carried out in two steps: the secondary transmitter erroneously detects the beacon 1. The first step is similar to that of the CSA protocol. codeword c in favor of the codeword ^ for the channel c 2. All the three nodes (the primary user and the pair of realization p;t is given by [24, (12.13)] secondary transmitter-receiver) step in a competition qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 for broadcasting additional party bits for x 0b during the P ðet j p;t Þ ¼ Prðc ! ^ j p;t Þ ¼ Q c c 2djp;t j2 ; ð5Þ next N2 channel uses. The competition is carried out as 4 follows: We first assign the metrics tp ¼ jp;t j2 þ jp;r j2 , where d is the Hamming distance between c and ^. c 4 2 2 4 2 2 tt ¼ jp;t j þ jt;r j , and tr ¼ jp;r j þ jr;t j to the primary Throughout the diversity order analyses, we will frequently user, the secondary transmitter, and the secondary use the following two results: receiver, respectively. Then the N2 channel uses is Remark 1. For a real value a > 0 and the random variable allocated to u > 0 with probability density function fU ðuÞ, we have a. the secondary transmitter if it successfully Z 1Z 1 pffiffiffiffiffiffi 1 2 decodes x 0b and tt > maxftp ; tr g, IEu ½Qð auފ ¼ pffiffiffiffi pffiffiffiffiffiffi eÀv =2 dv fU ðuÞ du 0 au 2 b. the secondary receiver if it successfully decodes Z 1   ð6Þ v2 1 2 x 0b and tr > maxftp ; tt g, and ¼ Pr u pffiffiffiffiffiffi eÀv =2 dv: 0 a 2 c. to the primary transmitter if either tp > maxftt ; tr g or both secondary users fail to decode x0b . Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 4. TAJER AND WANG: BEACON-ASSISTED SPECTRUM ACCESS WITH COOPERATIVE COGNITIVE TRANSMITTER AND RECEIVER 115 Z 1   2Lemma 1. For integer M > 0 and real ki > 0, v2 eÀv =2 ¼ P d1 jp;t j2 þ d2 jr;t j2 pffiffiffiffiffiffi dv 0 2 2 Z 1 YÀ M |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Àki v2 = Á 1 2 : À Á À Á 1Àe pffiffiffiffiffiffi eÀv =2 dv ¼ ÀM : ð7Þ P d1 jp;t j2 v2 2 P d1 jp;r j2 v2 2 0 i¼1 2 Z   2 1 v2 eÀv =2  P d1 jp;t j2 pffiffiffiffiffiffi dvProof. See Appendix A. u t 2 2 "0 Z   Àv2 =2 # 1 v 2 e ð13Þ As a result, from (5), (6), and (7) we can show that for  1À P d1 jp;r j2 pffiffiffiffiffiffi dv 0 2 2noncooperative transmission, we have Z 1   2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! v2 eÀv =2 þ P d1 jp;t j2 pffiffiffiffiffiffi dv NC P ðet Þ ¼ IEp;t Q 2djp;t j2 ¼ À1 ; 0 2 2 Z 1  2  Àv2 =2 v eand NC P ðer Þ ¼ . À1  P d1 jp;r j2 pffiffiffiffiffiffi dv 0 2 24.2 CSA Protocol ZThe transmissions are accomplished in N ¼ N1 þ N2 con- 1 À À v 2 ÁÀ À v 2 Á eÀv2 =2 1 À e 2d1 p;t 1 À e 2d1 p;r pffiffiffiffiffiffi dvsecutive channel uses, where N1 uses is for the direct 0 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}transmission, and N2 uses for the cooperation phase. We ¼ : 1 ðfrom Lemma 1 and Remark 1Þ 2denote d1 and d2 as the Hamming distances of the codewords Z 1 2 À À v 2 Á eÀv =2sent in the first and second phase, and we have d1 þ d2 ¼ d.  1 À e 2d1 p;t pffiffiffiffiffiffi dvIn the case only one node is successful in the first phase, we 0 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 : 1 ðfrom Lemma 1 and Remark 1Þdenote it by TR 2 fTt ; Tr g and also define TR ¼ fTt ; Tr gnTR .  ¼ Therefore, we get Z 1 À À2d vÁ eÀv2 =2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1Àe  1 pffiffiffiffiffiffi dv p;t 0 2 C P ðeR j p;R ; R;R Þ ¼ Q 2d1 jp;R j2 þ 2d2 jR;R j2 ; ð8Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}      : 1 ðfrom Lemma 1 and Remark 1Þ ¼ Z 1 2and for the case that there is no relaying, we have À À v 2 Á eÀv =2 : 1 1 : 1  1 À e 2d1 p;r pffiffiffiffiffiffi dv ¼ 2 þ 3 ¼ 2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} C P ðet j p;t Þ ¼ Q 2d1 jp;t j2 ; ð9Þ : ¼1 ðfrom Lemma 1 and Remark 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C and P ðer j p;r Þ ¼ Q 2d1 jp;r j2 : ð10Þ where the transition from (12) to (13) follows from Remark 1. Now, by comparing to the noncooperative NC : 1 case, where we have P ðet Þ ¼ , we conclude that forTheorem 1. For sufficiently large SNR values, we have C NC sufficiently large SNR, P ðet Þ < P ðet Þ. The same line C : C : 1 C NC C NC P ðetÞ¼P ðerÞ¼ 2 , P ðetÞ < P ðetÞ, and P ðerÞ < P ðerÞ. of argument holds for the secondary receiver. u tProof. We denote the success and failure of the secondary The CSA protocol ensures that for sufficiently high SNR transmitter in the first phase by Tt : S and Tt : F , values, the probability of detecting unused spectrum holes respectively. We also define Tr : S and Tr : F similarly. is boosted. However, the performance in the low SNR By using (8)-(10) and by expansion over all different regimes depends on the instantaneous channel conditions combinations of the statuses of the secondary users, for and it might happen that noncooperative spectrum access the secondary transmitter we get outperforms the cooperative scheme. C P ðet Þ 4.3 OCSA Protocol C ¼ P ðet j Tt : S; Tr : SÞ P ðTt : S; Tr : SÞ Like in the CSA protocol, the first N1 channel uses are |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} allocated for direct transmission from the primary user to ¼0 the secondary users. During the remaining time, depending C þ P ðet j Tt : S; Tr : F Þ P ðTt : S; Tr : F Þ on the instantaneous channel conditions, the primary user |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð11Þ ¼0 might transmit the additional parity bits itself or one of the C þ P ðet j Tt : F ; Tr : SÞP ðTt : F ; Tr : SÞ secondary users might take over such transmission. We þ C P ðet j Tt : F ; Tr : F Þ P ðTt : F ; Tr : F Þ denote the transmitting user in the second phase by |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} TR 2 fTp ; Tt ; Tr g. Therefore, the transmission in the second ¼1 phase is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi & R;t xR ½nŠ þ zt ½nŠ; if TR 6¼ Tt ; ¼Q 2d1 jp;t j2 þ 2d2 jr;t j2 yt ½nŠ ¼ 0; if TR ¼ Tt ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi & 2 R;r xR ½nŠ þ zr ½nŠ; if TR 6¼ Tr ; ÂQ 2d1 jp;t j  1 À Q 2d1 jp;r j2 ð12Þ and yr ½nŠ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0; if TR ¼ Tr : þQ 2d1 jp;t j2  Q 2d1 jp;r j2 As specified by the protocol, if neither of the secondary users decodes the beacon message successfully, the primary Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 5. 116 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 2010user will transmit the additional parity bits itself. In thiscase, transmission of the beacon message is similar to (2)-(3)for the entire channel uses.Theorem 2. For all values of SNR, channel realizations and level OC NC OC of cooperation, we have P ðet Þ < P ðet Þ, P ðer Þ < NC OC : ONC : 1 P ðer Þ, and P ðet Þ ¼ P ðer Þ ¼ 2 .Proof. The probability of missing the beacon message by the secondary transmitter is X OC OC OC P ðet Þ ¼ P ðet j TR ¼ Ti ÞP ðTR ¼ Ti Þ: ð14Þ i2fp;t;rg C Note that P ðet j TR ¼ Tt Þ ¼ 0. Also when TR ¼ Tr , according to the protocol we should have tr > tp , which provides that jr;t j > jp;t j. Therefore, from (14) we get OCP ðet Þ Fig. 1. Detection error probabilities and diversity gains of noncoopera- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! tive, CSA, and OCSA schemes. ¼ IEp;t Q 2d1 jp;t j2 þ 2d2 jp;t j2 P ðTR ¼ Tp Þ OC qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! performance of the proposed cooperative schemes with þ IEp;t ;r;t Q 2d1 jp;t j2 þ 2d2 jr;t j2 that of the noncooperative scheme, where it demonstrates ð15Þ that the CSA and the OCSA protocols achieve a second- OC  P ðTR ¼ Tr Þ order diversity gain. As expected theoretically, at low SNR qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! regimes the CSA protocol might not outperform the < IEp;t Q 2djp;t j2 noncooperative scheme, whereas the OCSA protocol out- OC OC NC performs the noncooperative scheme over all SNR re-  ½P ðTR ¼ Tp Þ þ P ðTR ¼ Tr ފ < P ðet Þ: gimes. In the evaluations above, we have set ¼ 0:5 and d1 and d2 are the Hamming distances of the codewords have considered channel realizations with parameters sent in the first and second phase, and we have ðp;t ; p;r ; t;r Þ ¼ ð1; 2; 3Þ. In Fig. 1, the error probability of d1 þ d2 ¼ d. By following the same line of argument, the user Ti for i 2 ft; rg in the CSA and OCSA protocols is OC NC identified by CSA-Ti and OCSA-Ti , respectively. we can accordingly show that P ðer Þ < P ðer Þ. In order to assess the diversity gain, note that 4.4 False Alarm OC OC Thus far we have examined the improvements attained in P ðet Þ ¼ P ðet j Tt : F ÞP ðTt : F Þ detecting the vacant spectrum holes. In such detection OC ð16Þ þ P ðet j Tt : SÞ P ðTt : SÞ: problems, however, another aspect that should also be |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} taken into consideration is the false alarm probability. In ¼0 our proposed spectrum access models, false alarm is the On the other hand, event that the secondary users erroneously consider a OC spectrum band to be idle, i.e., they detect the beacon P ðet j Tt : F Þ message while the channel is still used by the primary user. OC ¼ P ðet j Tt : F ; Tr : F ÞP ðTr : F Þ This leads to concurrent transmissions by the primary and OC secondary users which can potentially harm the transmis- þ P ðet j Tt : F ; Tr : SÞP ðTr : SÞ sion of the primary user. Due to stringent constraints on OC P ðTr : F Þ þ P ðet j Tt : F ; Tr : SÞ avoiding concurrent transmissions, it is imperative to study qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! the probability of false alarm as well. ð17Þ IEp;t Q 2d1 jp;r j2 For most detection problems, there exists a tension qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! between the probability of successful detection and the þ IEp;t Q 2djp;r j2 probability of false alarm, and it is crucial to maintain a good balance between these two probabilities such that : 1 1 : 1 neither of them is sacrificed in favor of the other one. ¼ þ ¼ : In this paper, we have translated a detection problem : 1 into a communication problem rested on the basis of Recalling that P ðTt : F Þ ¼ and using (16)-(17) pro- OC : 1 exchanging a beacon message. Therefore, assessing the vides that P ðet Þ ¼ 2 . A similar argument holds for OC probability of erroneously decoding a nonbeacon message P ðer Þ. u t in favor of the beacon message has the same nature as that Numerical evaluations of the probabilities of erroneous of missing a transmitted beacon and both are quantified inspectrum hole detection in different schemes are provided terms of decoding errors. More specifically, the probabilityin Fig. 1. These error probabilities for the noncooperative, that for a channel realization p;t , a nonbeacon codeword ^ cthe CSA, and the OCSA protocols are provided in (5), (13), is mistakenly decoded as the beacon message denoted by cand (15), respectively. This figure compares the detection (which is the false alarm probability), is given by Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 6. TAJER AND WANG: BEACON-ASSISTED SPECTRUM ACCESS WITH COOPERATIVE COGNITIVE TRANSMITTER AND RECEIVER 117 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 0 P ðet j p;t Þ ¼ Prð^ ! c j p;t Þ ¼ Q c 2djp;t j2 ; ð18Þ sensed the channel to be busy and idle, respectively, and Sr is defined accordingly. These state variables retain their stateswhich is equal to the probability in (5). Therefore, by for a period of Tc channel uses and vary to an i.i.d. statefollowing the same lines as in Sections 4.2 and 4.3, it is seen afterwards and as seen above, Tc only affects the lower bound.that the proposed cooperation models will also improve the For further analysis we assume that all channels (i.e.,probability of false alarm which diminishes as fast as the hp;t ; hp;r ; ht;r ) follow the same fading model and thereforeprobability of missing the beacon, enjoying a second-order the states St and Sr have the same time variations as thediversity gains. secondary channel ht;r , which means that all remain As a result, deploying the proposed cooperation schemes unchanged for Tc channel uses and change to indepen-not only incurs no loss in the probability of false alarm, but dent states afterwards. Now, corresponding to differentalso improves it. Hence, by translating the detection values of Tc we will have slow and fast fading processesproblem into a communication problem, the tension and need to look into meaningful notions of capacity forbetween the two aforementioned probabilities is resolved each of them.and they can be improved simultaneously. 1. Fast fading: Small values of Tc correspond to fast fading for which a meaningful notion of capacity is5 CAPACITY ANALYSIS given by ergodic capacity and is obtained byIn this section, we analyze the effect of erroneous detection averaging over all channel fluctuationsof unused channel and thereof missing communication 4 4opportunities on the capacity of the secondary users and Cerg ðÞ ¼ IEt;r ½C U ðt;r ފ; U Cerg ðÞ ¼ IEt;r ½C L ðt;r ފ: Lwill show how exploiting the proposed cooperative diver- ð21Þsity protocols enhances the capacity. In the OCSA protocol,each user requires to know the gains of its outgoing channelsto the other users. Acquiring such channel gains requires 2. Slow fading: Corresponding to large values of Tc ) 1,information exchange over the wireless channel. Therefore, we consider the -outage capacity C as thethe acquired channel gain estimates are not perfect in performance measure for which the bounds arepractice, which can affect the capacity. In this section, we given byassume that all channel estimates for the OCSA protocol are À Á Pr C UðLÞ ðt;r Þ < C ðÞ UðLÞ : ð22Þperfect and defer analyzing the effect of estimationinaccuracy on the channel capacity to Section 6.2. As the ergodic and outage capacities provided in (21)-(22) For fading channels, there is always a nonzero prob- depend on the probability terms PrðSt ¼ 1Þ and PrðSt ¼ability that a target rate R cannot be supported and there is Sr ¼ 1Þ, we need to assess these terms and their variationsno meaningful notion of capacity as a rate of arbitrarilyreliable communication. Therefore, we resort to looking into under different cooperation models. The states St and Srthe notions of outage capacity for slow fading channels and depend not only on the spectral activity in the vicinity of theergodic capacity for fast fading channels. secondary transmitter and receiver, but also on how Capacity of the secondary link is influenced by the successfully these users sense this spectral activity.spectral activity of the primary users and the efficiency of We introduce the random variables t ; r 2 f0; 1g tothe secondary users in detecting the unused channels. In an account for modeling the spectral activities. t ¼ 1 (r ¼ 1)earlier study in [14], it is also demonstrated how the states that no primary user is using the channel in thesecondary user capacity is affected by dissimilar perception vicinity of the secondary transmitter (receiver) and theof secondary users of primary user’s spectral activity in channel can be used by the secondary users. t ¼ 0 andtheir vicinities, where it has been shown that more r ¼ 0 are defined accordingly for busy channels. Based oncorrelated perceptions lead to higher channel capacity for the definitions above and those of St and Sr , which indicatethe secondary link. We will show that the cooperative the perception of secondary users of the activity of theprotocols are also effective in increasing such correlation. primary user, we haveLower and upper bounds on the capacity of the secondarychannel, when the received power at the secondary receiver PrðSt ¼ 1Þ ¼ Prðt ¼ 1Þð1 À P ðet ÞÞ; ð23Þis P , are given by [14]   U P PrðSt ¼ Sr ¼ 1Þ ¼ Prðt ¼ r ¼ 1ÞP ðt ; er Þ; e  ð24Þ C ðP Þ ¼ PrðSt ¼ Sr ¼ 1Þ log 1 þ ð19Þ PrðSt ¼ Sr ¼ 1Þ where et and er are the erroneous detection events. Noteand that the terms Prðt ¼ 1Þ and Prðt ¼ r ¼ 1Þ for the   cooperative and noncooperative schemes are identical and P 1 the effect of cooperation reflects in the terms P ðt Þ and e C L ðP Þ ¼ PrðSt ¼ Sr ¼ 1Þ log 1 þ À : ð20Þ PrðSt ¼ 1Þ Tc P ðt ; er Þ. In the following two lemmas, we show how the e where the states St ; Sr 2 f0; 1g, assigned to the secondary CSA and the OCSA protocols improve the probabilitytransmitter and receiver, respectively, indicate the perception PrðSt ¼ Sr ¼ 1Þ.of the secondary users about the activity of the primary user. Lemma 2. For sufficiently large SNR, we have PrC ðSt ¼St ¼ 0 and St ¼ 1 mean that the secondary transmitter has Sr ¼ 1Þ ! PrNC ðSt ¼ Sr ¼ 1Þ. Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 7. 118 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 2010Proof. We first show that for sufficiently large SNR, C NC P ðr j et Þ ! P ðr Þ as follows: e  e C CP ðr j et Þ ¼ P ðr j et ; Tr : SÞ P ðTr : SÞ e  e  |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼1 C þ P ðr j e et ; Tr : F Þ P ðTr : F Þ  |fflfflfflfflfflffl{zfflfflfflfflfflffl} Tt :S;Tr :F C ¼ P ðTr : SÞ þ P ðr j Tt : S; Tr : F ÞP ðTr : F Þ e NC ! P ðr ÞP ðTr : SÞ e ð25Þ C þ P ðr j Tt : S; Tr : F Þ P ðTr : F Þ e |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} : ¼1À 1 2 NC NC ! P ðr ÞP ðTr e : SÞ þ P ðr Þ P ðTr : F Þ e |fflfflfflffl{zfflfflfflffl} ¼ : 1À1 NC ¼ P ðr Þ: e Fig. 2. Comparing P ðt ; er Þ for different schemes. e  On the other hand, from Theorem 1 we know that for By following similar steps, we can show that C NC sufficiently large SNR, P ðt Þ > P ðt Þ, which in e e C conjunction with (25) provides that P ðt ; er Þ ¼OC e  P ðt ; er Þ ¼ P ðTt : S; Tr : SÞ e  C e  C e NC NC P ðr j et ÞP ðt Þ ! P ðr ÞP ðt Þ, which is the desired e e þ P ðTt : S; Tr : F Þ result. u t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!  1ÀQ 2d1 jp;r j2 þ 2d2 jt;r j2 ð27ÞLemma 3. For all values of SNR, we have þ P ðTt : F ; Tr : SÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! OC Pr ðSt ¼ Sr ¼ 1Þ !  1ÀQ 2d1 jp;t j2 þ 2d2 jt;r j2 : maxfPrC ðSt ¼ Sr ¼ 1Þ; PrNC ðSt ¼ Sr ¼ 1Þg: OC Comparing (26) and (27) establishes that P ðt ; er Þ ! e  OCP r o o f . W e e q u i v a l e n t l y s h o w t h a t P ðt ; er Þ ! e  C P ðt ; er Þ. Also from (26), we get e  C NC OC maxfP ðt ; er Þ; P ðt ; er Þg. By expanding P ðt ; er Þ e  e  e  OC over all different combinations of the statuses of the P ðt ; er Þ e  secondary users (success/failure), we get ! P ðTt : S; Tr : SÞ þ P ðTt : F ; Tr : F Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! OCP ðt ; er Þ e  þ P ðTt : S; Tr : F Þ 1 À Q 2djp;r j2 OC ¼ P ðt ; er j Tt : S; Tr : SÞ P ðTt : S; Tr : SÞ e  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ffl þ P ðTt : F ; Tr : SÞ 1 À Q 2djp;t j2 ¼1 OC þ P ðt ; er e  j Tt : S; Tr : F Þ P ðTt : S; Tr : F Þ ! P ðTt : S; Tr : SÞ þ P ðTt : F ; Tr : F Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} OC e ¼P ðr j Tt :S; Tr :F Þ þ P ðTt : S; Tr : F Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! OC þ P ðt ; er j Tt : F ; Tr : SÞ P ðTt : F ; Tr : SÞ e   1ÀQ 2djp;r j2 1ÀQ 2djp;t j2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} OC e ¼P ðt j Tt :F ; Tr :SÞ þ P ðTt : F ; Tr : SÞ OC þ P ðt ; er j Tt : F ; Tr : F Þ P ðTt : F ; Tr : F Þ e  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}  1ÀQ 2djp;t j 1ÀQ 2djp;r j2 NC e NC e ¼P ðt ÞP ðr Þ ð26Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ¼ P ðTt : S; Tr : SÞ þ P ðTt : S; Tr : F Þ ! 1ÀQ 2djp;t j2 1ÀQ 2djp;r j2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! NC  1ÀQ 2d1 jp;r j2 þ 2d2 maxfp;r j2 ; jt;r j2 g ¼ P ðt ; er Þ; e  þ P ðTt : F ; Tr : SÞ which completes the proof. u t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Fig. 2 illustrates numerical evaluations comparing the  1ÀQ 2d1 jp;t j2 þ 2d2 maxfp;t j2 ; jt;r j2 g term P ðt ; er Þ for different schemes given in (25)-(27). As e  þ P ðTt : F ; Tr : F Þ expected, for very low SNR regimes, CSA exhibits no gain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! over noncooperative scheme, while OCSA outperforms both 2  1ÀQ 2djp;t j 1ÀQ 2djp;r j2 : CSA and noncooperative schemes in all SNR regimes. As seen in the figure, the CSA protocol achieves considerable Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 8. TAJER AND WANG: BEACON-ASSISTED SPECTRUM ACCESS WITH COOPERATIVE COGNITIVE TRANSMITTER AND RECEIVER 119 carried out reliably via a very low-rate communication (by transmitting few pilot symbols). The information about channel gains is used only once for initializing the backoff timers, which is during the transition of the channel access from the primary user to the secondary users, and is not needed afterwards. It is noteworthy that acquiring such channel state information is not guaranteed to be feasible in all networks, in which case the secondary users can deploy the CSA protocol that does not require any information exchange. In the OCSA protocol, we need the following channel state information. 1. The secondary transmitter should know jp;t j and the secondary receiver should know jp;r j: We assume that the primary transmitter is periodically transmitting training symbols to its designated receiver forFig. 3. Lower and upper bounds on outage capacity for different training purposes. The secondary users can overhearschemes. these pilots due to the broadcast nature of the wireless channel and periodically acquire theirgain in moderate SNR regimes. For the numerical evalua- desired channel states. Obtaining these channel gainstions, we have assumed the same setup as in Fig. 1. is only possible when the primary users transmitTheorem 3. For large enough SNR values, we have training symbols to its designated receiver and these symbols are known to the secondary users a priori. U;C U;NC L;C L;NC Cerg ðÞ > Cerg ðÞ; and Cerg ðÞ > Cerg ðÞ; 2. The secondary users should know jt;r j ¼ jr;t j: This U;C C ðÞ > U;NC C ðÞ; and L;C C ðÞ > L;NC C ðÞ: channel gain can be acquired by having the secondary users exchange pilot symbols. This ex-Proof. See Appendix B. u t change should be performed while the primary user is still active. To avoid harming the communicationTheorem 4. For all SNR regimes, we have of primary users, the pilots can be exchanged using U;OC È U;C U;NC É ultrawideband (UWB) communication which allows Cerg ðÞ > max Cerg ðÞ; Cerg ðÞ ; È L;C É the secondary users to communicate the low-rate L;OC L;NC Cerg ðÞ > max Cerg ðÞ; Cerg ðÞ ; pilots well below the noise level of the primary user. U;OC È U;C U;NC É As the secondary users do not have any prior C ðÞ > max C ðÞ; C ðÞ ; È L;C É information about the time of the availability of the L;OC L;NC and C ðÞ > max C ðÞ; C ðÞ : channel, they should keep exchanging the pilots according to how static the channel t;r is. SinceProof. See Appendix C. u t exchanging pilot sequences requires a very low-rate communication, the process of channel gain estima- Fig. 3 demonstrates numerical evaluations of the outage tion can be performed reliably. This level ofcapacity achieved under different schemes. For these information exchange can be carried out in allevaluations, we have used the capacity lower and upper networks. We also assume that the secondary usersbounds given in (22) and have assumed ¼ 1 . For also exchange the estimates of jp;t and jp;r j that they 2evaluating PrðSt ¼ Sr ¼ 1Þ given in (24), we have assumed have obtained in Step 1.Prðt ¼ r ¼ 1Þ ¼ 0:7 in both cooperative and noncoopera- 3. The primary user should know jp;t j and jp;r j: We devise two time slots between the time that primary usertive schemes. We also have assumed SNR ¼ 10 dB, and for finishes the transmission of the beacon and the timethe lower bounds on the outage capacity, we have set that all users run their backoff timers. During theseTc ¼ 10. It is observed that outage capacity bound increases two time slots, the secondary users feed backas we allow higher outage probability. Also, since the gap the value of jp;t j2 þ jp;r j2 , which they have alreadybetween the lower and upper bounds for the cooperative obtained during Steps 1 and 2. Note that eachschemes is small, the bounds seem to be tight. secondary user will have feedback transmission only if it has successfully decoded the beacon, otherwise its dedicated feedback slot will be wasted. Also, the6 DISCUSSIONS primary user will not receive any feedback only if6.1 Information Exchange both secondary users fail to decode the beacon, inIn this section, we discuss how the primary and the which case its lack of knowledge about the valuesecondary users acquire the channel state information that jp;t j2 þ jp;r j2 does not harm as the secondary usersthey need in the OCSA protocol. The users need to acquire will not step in the competition phase and anychannel gains, which can be facilitated via employing random initializing of the primary users’s backofftraining-based channel estimation schemes and feedback timer suffices to ensure that the primary user will becommunication. Such estimations and feedback can be transmitting in the second phase. Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 9. 120 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 20106.2 Imperfect Channel Estimation 1. the relative capacity loss is bounded asThe OCSA protocol requires the primary to know " # ~ C OC ðt;r Þ À C OC ðt;r Þjp;t j2 þ jp;r j2 , the secondary transmitter to know jp;t j2 þ IEt;r ;p;t ;p;rjt;r j2 , and the secondary receiver to know jp;r j2 þ jr;t j2 . In C OC ðt;r Þ    !the capacity analysis in Section 5, we have assumed that 1 jtp À tt j jtp À tr jall users know their corresponding channel state informa- < IEtp ;tt ;tr Q pffiffiffiffiffiffiffi þ Q pffiffiffiffiffiffiffiffi ffi ; 3 22 22tion perfectly. In practice, however, estimating and feeding back such 2. the capacity loss for decreasing values of 2 decayschannel gains is imperfect, which can potentially affect the faster than , i.e.,channel capacity. As discussed in Section 6.1, these channel h OC ~ igains are used only for the purpose of determining which log IEt;r ;p;t ;p;r C ðt;r ÞÀC OC ðt;r Þ C OC ðt;r Þ 1user should be assigned as relay (to transmit the additional lim < :parity bits). Therefore, the effect brought about by imperfect !0 log  3channel estimate is the possibility of selecting a wrongrelay. Note that not any imperfect estimation would Proof.necessarily lead to selecting a wrong relay, as relay selection 1. We provide the analysis for the case thatis based on the relative order of ftp ; tt ; tr g and not their exact p;t ; p;r ; t;r are i.i.d. By following the same linevalues. For instance, if tt ¼ maxftp ; tt ; tr g and we denote the of argument, we can find an upper bound for the ~ ~ ~estimates of ftp ; tt ; tr g by ftp ; tt ; tr g, there is the chance that nonidentical distributions as well. When thet ~ ~ ~~t ¼ maxftp ; tt ; tr g too, in which case the estimation errors channel estimates are perfect, we denote the relaydo not affect the performance of the protocol and the selected by the OCSA by TR . The probability ofcapacity of the secondary link. selecting a wrong relay by some simple manip- In the OCSA protocol with imperfect channel estimates, ulations can be expanded as follows:for the channel realizations leading to the metrics IEtp ;tt ;tr ½Pwr ðtp ; tt ; tr ފ ¼ftp ; tt ; tr g, we denote the probability of selecting a wrong X IEtp ;tt ;tr ½P ðTR 6¼ Ti j ti ¼ maxftp ; tt ; tr gފrelay by Pwr ðtp ; tt ; tr Þ. We also denote the capacity of the :secondary link when the right relay is selected by C OC ðP Þ i2fp;t;rg 3(which is also the capacity with perfect channel estimates) ð29Þ band when a wrong relay is selected by C OC ðP Þ, where P isthe power of the received signal by the secondary receiver. On the other hand, we haveTherefore, the capacity when channel estimates are IEtp ;tt ;tr ½P ðTR ¼ Tt j tp ¼ maxftp ; tt ; tr gފimperfect is given by 4  ¼ IEtp ;tt ;tr P ðTt : SÞ P ðTr : F Þ ~ b C OC ðP Þ ¼ Pwr ðtp ; tt ; tr Þ C OC ðP Þ þ Pwr ðtp ; tt ; tr ÞC OC ðP Þ: ð28Þ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} 1ÀP ðTr :SÞ ~Note that C OC ðP Þ depends on fp;t ; p;r ; t;r g which is not  ! ~ ~  P tt > tp j tp > tt ; trexplicitly expressed in the formulations for the ease ofnotations. Errors in estimating channel gains ultimatelylead to errors in estimating ftp ; tt ; tr g which are used to set þ IEtp ;tt ;tr P ðTt : SÞP ðTr : SÞthe initial values of the backoff timers. We denote such ! ~ ~ ~ ~  P ðtt > tp j tp > tt ; tr Þ P ðtt > tr j tp > tt ; tr Þestimation errors by |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 1 4 ~ wi ¼ ti À ti ; for i ¼ fp; t; rg; ~ ~ IEtp ;tt ;tr ½P ðTt : SÞP ðtt > tp j tp > tt ; tr ފ ~where ti is the estimate of ti , and assume that wi $ N ð0; 2 Þ. ~ ~ IEtp ;tt ;tr ½P ðtt > tp j tp > tt ; tr ފTherefore, analyzing the effect of imperfect channel ~ ~ ¼ IEtp ;tt ½P ðtt > tp j tp > tt ފ;estimation on the capacity of the cognitive link translates ð30Þ ~into analyzing how C OC ðP Þ and 2 are related, which isprovided in the following theorem. We first find an upper and, similarly,bound on the relative capacity loss due to imperfect IEtp ;tt ;tr ½P ðTR ¼ Tr j tp > tt ; tr ފestimates and then show that one order of magnitude ð31Þ ~ ~ IEtp ;tr ½P ðtr > tp j tp > tr ފ:decrease in the estimation errors translates into one order ofmagnitude decrease in the capacity loss, e.g., the capacity Also, it can be readily verified that 2 1loss due to estimation noise with variance 100 is 10 of the IEtp ;tt ;tr ½P ðTR ¼ Tp j tt > tp ; tr Þcapacity loss due to the estimation noise variance with 2 . þ IEtp ;tt ;tr ½P ðTR ¼ Tr j tt > tp ; tr ފ ð32ÞTheorem 5. For i.i.d. channel realizations p;t ; p;r ; t;r and ~ ~ < IEtp ;tt ½P ðtp > tt j tt > tp ފ; imperfect channel estimates with estimation noise variance 2 , Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 10. TAJER AND WANG: BEACON-ASSISTED SPECTRUM ACCESS WITH COOPERATIVE COGNITIVE TRANSMITTER AND RECEIVER 121 and, similarly, On the other hand, we know that 8x > 0   IEtp ;tt ;tr ½P ðTR ¼ Tp j tr > tp ; tt Þ 1 1 Àx2 =2 1 2 pffiffiffiffiffiffi 1 À 2 e QðxÞ pffiffiffiffiffiffi eÀx =2 ; þ IEtp ;tt ;tr ½P ðTR ¼ Tr j tr > tp ; tt ފ ð33Þ 2x x 2x ~ ~ < IEtp ;tt ½P ðtp > tr j tr > tp ފ: which provides ~ 2  À Á 2  By noting that ti ¼ ti þ wi , where wi $ N ð0;  Þ, log pffiffiffiffix 1 À x2 eÀx =2 1 1 2 and denoting the probability density functions of lim ti by fðti Þ, (29)-(33) provide that x!1 log eÀx2 =2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼1 IEtp ;tt ;tr ½Pwr ðtp ; tt ; tr ފ  2  log QðxÞ log pffiffiffiffix eÀx =2 1 2 1 lim lim ; < IEtp ;tt ½P ðwt À wp > tp À tt j tp > tt ފ x!1 log eÀx =2 2 x!1 log eÀx2 =2 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 1 ¼1 þ IEtp ;tt ½P ðwp À wt > tt À tp j tt > tp ފ 3 or equivalently, for asymptotically large values 1 þ IEtp ;tr ½P ðwr À wp > tp À tr j tp > tr ފ of x, 3 1 : 2 þ IEtp ;tr ½P ðwp À wr > tr À tp j tr > tp ފ QðxÞ ¼ eÀx =2 : 3   Z Z 1 1 1 jtp À tt j For any given value of X, by setting x ¼ pX 2ffi , ffiffiffiffiffi ¼ Q pffiffiffiffiffiffiffiffi fðtp Þ dtp fðtt Þ dtt 2 3 0 0 22 for asymptotically large values of x (or small Z 1Z 1   1 jtp À tr j values of ) þ Q pffiffiffiffiffiffiffiffi fðtp Þ dtp fðtr Þ dtr 3 0 0 22      ! X X2 1 jtp À tt j jtp À tr j : ¼ IEtp ;tt ;tr Q pffiffiffiffiffiffiffiffi þ Q pffiffiffiffiffiffiffi ffi : Q pffiffiffiffiffiffiffi ¼ eÀ42 : ffi 3 22 22 22 ð34Þ Hence, for asymptotically small values of ,  ! h X2 i Now, from (28) and (34), we get X : " # IEX Q pffiffiffiffiffiffiffi ffi ¼ IEX eÀ42 ~ 22 C OC ðt;r Þ À C OC ðt;r Þ Z IEt;r ;p;t ;p;r ¼ 1 1 Àðx2 =42 þx=2Þ C OC ðt;r Þ ¼ e dx 2 0 < IEtp ;tt ;tr ½Pwr ðtp ; tt ; tr ފ ð35Þ pffiffiffi 2 =42      ! 1 e  : 1 1 jtp À tt j jtp À tr j ¼ Á Q pffiffiffi ¼ ; < IEtp ;tt ;tr Q pffiffiffiffiffiffiffiffi þ Q pffiffiffiffiffiffiffi ffi ;  2  2  3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 22 22 :1 ¼ which is the desired result. which, in conjunction with (37), gives rise to 2. We start by showing that  !  ! h  i jtp À tt j X : 1 jtp Àtt j IEtp ;tt Q pffiffiffiffiffiffiffi ffi ¼ IEX Q pffiffiffiffiffiffiffi ffi ¼ : ð38Þ log IEtp ;tt Q pffiffiffiffiffi ffi 22 2 2 2 2  lim ¼ 1: !0 log  Similarly, we can show that By some simplifications, we get  ! jtp À tr j : 1  ! IEtp ;tt Q pffiffiffiffiffiffiffiffi ¼ : ð39Þ jtp À tt j 22  IEtp ;tt Q pffiffiffiffiffiffiffiffi ¼ 22 Hence, " !# ð36Þ kp;r j2 À jt;r j2 j    ! IEp;r ;t;r Q pffiffiffiffiffiffiffiffi ; jtp À tt j jtp À tr j : 1 22 IEtp ;tt ;tr Q pffiffiffiffiffiffiffiffi þ Q pffiffiffiffiffiffiffiffi ¼ : ð40Þ 2 2 2 2  where we have assumed that fp;t ; p;r ; t;r g are Equations (35) and (40) provide that i.i.d. distributed as N ð0; Þ where we have h OC i 4 ~ C ðt;r ÞÀC OC ðt;r Þ defined  ¼ p;t ¼ p;r ¼ t;r . Therefore, jp;r j2 log IEt;r ;p;t ;p;r C OC ðt;r Þ 2 and jt;r j are distributed exponentially with lim !0 log  mean 2. It can be readily shown that the random h    i jtp Àtt j jtp Àtr j 4 variable X ¼ kp;r j2 À jt;r j2 j is also distributed log IEtp ;tt ;tr Q pffiffiffiffiffi þ Q pffiffiffiffiffi 2 ffi 2 2 ffi 2 1 < lim ¼ ; exponentially with mean 2. Therefore, from (36), !0 3 log  3 we get which is the desired result. u t  !  ! jtp À tt j X The numerical evaluation of jtp Àtt j IEtp ;tt ;tr ½Qð pffiffiffiffiffi Þ ffi þ jtp Àtr j Qð pffiffiffiffiffi ފ ffi IEtp ;tt Q pffiffiffiffiffiffiffiffi ¼ IEX Q pffiffiffiffiffiffiffi : ffi ð37Þ 22 22 22 22 2 versus the variance of noise estimation ( Þ is depicted in Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 11. 122 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 2010 i user Ti , the time delay due to backoff timers is TBT and the throughput is found as TCR ~ Rðp;t ; p;r ; t;r Þ ¼ i Á C OC ðt;r Þ TCR þ TFB þ TBT TCR ð41Þ ¼ ~ Á C OC ðt;r Þ;
  • 12. TCR þ TFB þ ti where we have taken into account that there is a one-time feedback transmission and backoff timer activation. Equation (41) suggests that when the duration that the secondary users access the channel is considerably longer than the time required for feedback, i.e., TCR ) TFB , the throughput loss due to feedback will be negligible. However, the same argument does not apply to the effect
  • 13. of backoff timers as for very weak channels (small ti ), ti can become a nonnegligible factor compared to TCR . In the following theorem, we assess the average throughput lossFig. 4. Relative capacity loss versus channel estimation noise accuracy. over the possible channel realizations. Theorem 6. The average throughput loss due to informationFig. 4 for three different SNR values. We have considered exchange and backoff timers is upper bounded bythe setup p;t ¼ p;r ¼ t;r ¼ 1, where we can see that when " #the noise variance is 2 ¼ 0:01, the capacity loss will be less ~ C OC ðt;r Þ À Rðp;t ; p;r ; t;r Þthan 1 percent at ¼ 0 dB, which is a negligible loss. Also, it IEt;r ;p;t ;p;r ~ C OC ðt;r Þis observed that for any fixed value of 2 , increasing the w1  w2 SNR results in less capacity loss. This is justified by noting þ ð1 þ w1 Þ e1þw1 À 1 ; 1 þ w1that for fixed noise level, more powerful signals are less whereprone to estimation errors. 4 TFB 4
  • 14. 6.3 Throughput Analysis w1 ¼ and w2 ¼ : TCR TCR p;tChannel capacity, being an intrinsic characteristic of thewireless channel, is not influenced by how effectively the Proof. First note that for the node Ti selected as the relay,channel is utilized or by how much the MAC-levelcoordinations (e.g., backoff timers and information exchange ti ! tp ¼ jt;p j2 þjt;r j2 > jt;p j2 . Therefore, from (41), we getin the OCSA protocol) cost. Therefore, in order to furnish a Rðp;t ; p;r ; t;r Þ 1fair comparison between the OCSA protocol and the ! : ~OC ðt;r Þ C 1 þ w1 þ w2 Á jh 1 j2noncooperative scheme and to incorporate the MAC-level p;tcosts of this protocol, we assess the achievable throughputs. Therefore, We have developed a throughput analysis, which rests onthe basis of our capacity analysis, to account for two types of " # Rðp;t ; p;r ; t;r Þthroughput losses incurred by the OCSA protocol. One loss IEt;r ;p;t ;p;r ~ C OC ðt;r Þis due to the time waste imposed by the backoff timers, and 2 3the other loss is due to the required feedback from the TCRsecondary users to the primary user. ! IEp;t 4 5 As running the backoff timers and exchanging informa- TCR þ TFB þ j
  • 15. j2 p;ttion take place only one time, if the secondary users access 2 3the channel for a period sufficiently larger than those of the 1 ¼ IEhp;t 4 5backoff timers and the feedback communications, the 1 þ w1 þ w2 Á jh 1 j2 p;tthroughput will be very close to the capacity and these 2 w2  3t¼1throughput losses will have a negligible effect. ð1 þ w1 ÞeÀt þ w2 e1þw1 Ei À w2 þð1þw1 Þt 1þw1 In order to formulate the throughput, which we denote ¼4 5by R, we define TCR as the duration that the secondary link ð1 þ w1 Þ2 Z 1 Àt t¼0will use the channel and we denote the time dedicated to 1 w2 ethe feedback communication by TFB . Finally, we denote the ¼ À w2 e1þw1 w dt 1 þ w1 2 t i 1þw1initial value of the backoff timer of user Ti by TBT , which is 1 w2 À À 2 Á 1 þ w1 wset as ! À w2 e1þw1 1 À e 1þw1 1 þ w1 w2 1 À w2 Á i 4
  • 16. ¼ À ð1 þ w1 Þ e 1þw1 À1 ; TBT ¼ ; for i 2 fp; t; rg; 1 þ w1 tiwhere
  • 17. is a constant and its unit depends on the unit of ti . which establishes the desired result. u tSince fti g are scalars,
  • 18. has the units of time. Therefore, for From the above result, it is consequently concluded thatany channel realization fp;t ; p;r ; t;r g, when the relay is for the appropriate choice of
  • 19. , i.e., by setting Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 20. TAJER AND WANG: BEACON-ASSISTED SPECTRUM ACCESS WITH COOPERATIVE COGNITIVE TRANSMITTER AND RECEIVER 123 network with a single pair of secondary transmitter and receiver. In this section, we provide a direction for generalizing the CSA protocol to a cognitive network consisting of multiple secondary transmitter-receiver pairs. The objective is to provide all secondary users with the diversity gain 2M for decoding the beacon message. We consider a multiuser network consisting of M pairs of secondary transmitters and receivers denoted by ðTt1 ; Tr1 Þ; . . . ; ðTtM ; TrM Þ. The physical channels between the primary user and the mth secondary transmitter and m m receiver are represented by p;t and p;r , respectively, and t;r denotes the channel between ðTtm ; Trm Þ for m ¼ 1; . . . ; M. m As in Section 2 for the channel realizations, we have qffiffiffiffiffiffiffi i;j ¼ m hm ; m i;j i;j ð42ÞFig. 5. Throughput comparison in different schemes. where hm , the fading coefficients, are independent complex i;j
  • 21. ( TCR p;t ) w2 ( 1; Gaussian CN ð0; 1Þ random variables and m represents i;jwe can make the effect of the backoff timers very negligible pathloss and shadowing effects. In the CSA protocol, awhich provides that w2 ! 0 and secondary user acts as relay based on its success in " # decoding the first segment of the beacon. The MU-CSA ~ C OC ðt;r Þ À Rðp;t ; p;r ; t;r Þ w1 IEt;r ;p;t ;p;r : protocol consists of two steps: C ~OC ðt;r Þ 1 þ w1 1. The primary user broadcasts the first segment of theAs for the effect of feedback, there will be a loss which is beacon message during the initial 0 < < 1 portion TCRupper bounded by TCR þTFB . This loss will be also marginal if of the time slot as in the CSA and OCSA protocolsthe period that the cognitive link is active is sufficiently and all the secondary transmitters and receiverslarger than the small period of the feedback communica- listen to the message and at the end of thetion, i.e., TCR ) TFB . transmission try to decode the message. Fig. 5 depicts the achievable throughput by the OCSA 2. During the remaining ð1 À Þ portion of the time slot,under different assumptions on feedback loads and all secondary users who have successfully decodedbackoff timer settings and compares them to the through- the first segment of the beacon construct the addi-put achievable via the noncooperative scheme and theCSA protocol. The solid curves represent the lower tional parity bits and broadcast them. Note that therebounds on the capacity found in Section 5. For the is the possibility that multiple secondary users havenoncooperative and the CSA schemes, due to having no been successful in the first phase. Since all these usersMAC-layer bandwidth loss, the throughput is essentially broadcast identical information packets, their concur-equivalent to the capacity, whereas for the OCSA protocol, rent transmissions are not considered as collision andthe throughput becomes equal to the capacity when we set rather can be deemed as a distributed multiple-w1 ¼ w2 ¼ 0. The dashed curves illustrate the throughput antenna transmission. For maintaining fairness inof the OCSA protocol for the different choices of w1 ; w2 . terms of the amount of resources consumed, theWe have considered a similar setup as in the numerical transmission power of individual secondary usersevaluations of Fig. 3. The results provide that for the Pp acting as relay should not exceed 2M in order to keepchoice of w1 < 0:15 and w2 < 0:15, the throughput of the the aggregate transmission power below Pp . There isOCSA scheme is higher than that of the CSA scheme, and the issue of synchronization between the successfulas w1 and w2 exceed these levels, the throughput of the relays, which they can ensure by using the beaconOCSA protocol falls below that of the CSA protocol. Notethat in practical scenarios, it is reasonable to assume that they have received as the synchronization reference. TFBw1 ; w2 < 0:15 as w1 ¼ TCR , which is the ratio of the length 7.2 Diversity Analysisof the one-time feedback packet to the length of the Next, we show that in a multiuser cognitive network with theinformation packets and is close to zero. Also, w2 dependson the constant factor
  • 22. of the timers which can be chosen MU-CSA protocol, all secondary users enjoy the diversitysuch that keeps the value of w2 arbitrarily small. gain of 2M in detecting the beacon message. We define the S ~ ~ set T ¼ fT1 ; . . . ; T2M g ¼ fTt1 ; . . . ; TtM g fTr1 ; . . . ; TrM g. Also, 47 MULTIUSER NETWORK we define M ¼ f1; ; 2Mg and denote the channel between ~ ~ the secondary users Tm and Tn by m;n , and the channel ~7.1 Multiuser CSA Protocol (MU-CSA)The cooperation protocols proposed in Section 3 and the between Tp and T ~m by p;m . The probability of missing the ~subsequent analyses in Section 4 consider a cognitive beacon message by the secondary user Tm is ~ Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 23. 124 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 2010 X C P ðeTm Þ ¼ ~ C ~ ~ P ðeTm j 8i 2 A; Ti : S; 8j 62 A; Tj : F Þ ~ AMnfmg A6¼; ~  P ð8i 2 A; Ti : S; 8j 62 A; Tj : F Þ ~ C ~ þ P ðeTm j 8i; Ti : F Þ P ð8i; Ti : F Þ ~ ~ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ffl ¼1 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 X X ¼ Q@ 2d1 j~p;m j2 þ 2d2 j~i;m j2 A AMnfmg 2M i2A A6¼; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} : 1 ¼ jAjþ1 ðfrom Lemma 1Þ Y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Q 2d1 j~p;j j2 j62A |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} : 1 ¼ 2MÀjAj ðfrom Lemma 1Þ Y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Fig. 6. Outage capacity in a cognitive network with five pairs of users.  1ÀQ 2d1 j~p;i j2 i2A |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} networks where the primary and secondary users are only 1 ðfrom Lemma 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi allowed to have orthogonal access to the channel and Y þ Q 2d1 j~p;j j2 cannot coexist concurrently. The primary user broadcasts a i2f1;...;2Mg |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} beacon message upon releasing the channel and a pair of : ¼1 ðfrom Lemma 1Þ secondary transmitter and receiver cooperatively detect the : 1 1 : 1 beacon message. The cooperation protocols are devised ¼ 2Mþ1 þ 2M ¼ 2M : such that by a single-time relaying both secondary transmitter and receiver enjoy second-order diversity gainsTherefore, all the secondary users enjoy a diversity order in detecting the beacon message. We have also quantified2M in decoding the beacon message. the effect of erroneous detection of vacant channels on the The channel capacity can be obtained similarly as inSection 5 for each secondary pair when only that pair secondary channel capacity and show the advantage of theaccesses the channel. For finding the channel capacity of a proposed protocols for achieving higher secondary channelspecific pair of secondary users, we need to find 1) the capacity. One of the proposed protocols imposes informa-probability of detecting the beacon by the secondary tion exchange and bandwidth costs, where we havetransmitter and 2) the probability that both of secondary assessed the effect of imperfect information exchange ontransmitter and receiver are successful in decoding to the capacity and the loss due to the bandwidth cost on thebeacon. The lower and upper bounds on the channel capacity achievable throughput. The cooperation models are alsoare given in (19) and (20), and the capacity for the link extended to multiuser cognitive networks as well.between ðTtm ; Trm Þ can be found by plugging the values of C PrðSTtm ¼ 1Þ ¼ PrðTtm ¼ 1ÞP ðTtm Þ e APPENDIX Aand PROOF OF LEMMA 1 4 À Á Firs t Àwe show that for the function AðM; nÞ ¼ PrðS ¼ STrm ¼ 1Þ ¼ Prð ¼ Trm ¼ C 1ÞP   e ; eTrm : PM M Á n m m¼0 m m ðÀ1Þ , AðM; nÞ ¼ 0 f o r 8M > n ! 0, a n d Ttm Ttm TtmNumerical evaluations for the lower and upper bounds on AðM; MÞ 6¼ 0. We provide the proof by induction:the secondary user capacity in a cognitive networkconsisting of five pairs of users are shown in Fig. 6, which 1. PM nM Á 0 a n d 8M > 0, M e h a v e AðM; 0Þ ¼ For À ¼ w m¼0 m ðÀ1Þm ¼ ð1 þ ðÀ1ÞÞ ¼ 0.depicts the outage capacity. For this evaluation, we have 2. Assumption: 8M > n > 0, we have AðM; nÞ ¼ 0.used the same setup as that in Section 4.3. By comparing 3. Claim: 8M > n þ 1, we show that AðM; n þ 1Þ ¼ 0.these results with those of a network with one pair ofsecondary users, it is seen that more gain in terms of X M  Mchannel capacity is achievable for the multiuser network AðM; n þ 1Þ ¼ mnþ1 ðÀ1Þm mwhich is due to opportunistic selection of the cognitive pair. m¼0As the number of secondary users pairs increases, the M À1  X  M À1quality of the best secondary user is expected to become ¼ ÀM ðm þ 1Þn ðÀ1Þm m¼0 mbetter which justifies the additional gains observed in themultiuser cognitive networks. X n M À1 M À 1 n X m ! k ¼ ÀM m ðÀ1Þ k¼0 k m¼0 m8 CONCLUSIONS X n n ¼ ÀM AðM À 1; kÞ ¼ 0:In this paper, we have proposed opportunistic cooperation k |fflfflfflfflffl{zfflfflfflfflffl} k¼0 MÀ1>n!kprotocols for supervised spectrum access in cognitive Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 24. TAJER AND WANG: BEACON-ASSISTED SPECTRUM ACCESS WITH COOPERATIVE COGNITIVE TRANSMITTER AND RECEIVER 125Now, we show that AðM; MÞ 6¼ 0. Again, by induction we APPENDIX Bhave: PROOF OF THEOREM 3 1. For M ¼ 1, Að1; 1Þ ¼ À1. We show that for any P > 0, C U;C ðP Þ > C U;NC ðP Þ and 2. Assumption: AðM; MÞ ¼ 0. 6 C L;C ðP Þ > C L;NC ðP Þ, which consequently proves the theo- 3. Claim: AðM þ 1; M þ 1Þ ¼ 0. 6 rem. First, we remark that p logð1 þ aÞ is nondecreasing in p p AðM þ 1; M þ 1Þ since M þ1    X M þ 1 @ @ a ¼ mMþ1 ðÀ1Þm C U ðaÞ ¼ p log 1 þ ! 0: m @p @p p m¼0 X M  M The inequality above holds because for the function fðuÞ ¼ 4 ¼ ÀðM þ 1Þ ðm þ 1ÞM ðÀ1Þm log u À ð1 À uÞ, fð1Þ ¼ 0 and for u ! 1, @fðuÞ ¼ u À u2 ! 0, and 1 1 1 m¼0 m @u U therefore, fðuÞ ! 0 for u ! 1. Therefore, C ðP Þ as given in X M  X M  M M ! ¼ ÀðM þ 1Þ mk ðÀ1Þm (19) is nondecreasing in the probability PrðSt ¼ Sr ¼ 1Þ. By k m k¼0 m¼0 using Lemma 2, it is concluded that for sufficiently large MÀ1   X M ! SNRC U;C ðP Þ > C U;NC ðP Þ, which, in turn, establishes the ¼ ÀðM þ 1Þ AðM; MÞ þ AðM; kÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} k¼0 k |fflfflfflfflffl{zfflfflfflfflffl} result for the upper bound. For the lower bound, as shown 6¼0 ¼0 C NC in (25), for sufficiently large SNR, P ðr j et Þ ! P ðr Þ as e  e 6¼ 0: shownKnowing the above result and considering the following C C P ðt ; er Þ e  P ðt Þ eTaylor series expansion NC ð ÞP NC ð Þ > NC P et er P ðt Þ e À Á À1 X ð2nÞ! 1 1 þ kx 2 ¼ ðÀkxÞn ; or, equivalently, n¼0 22n ðn!Þ2 R1 PrC ðSt ¼ Sr ¼ 1Þ PrC ðSt ¼ 1Þ p1 2 ffiffiffiffi 0 eÀav dv 1 À1 > NC : ð43Þand noting that 2 ¼ 2 we get 2a , 2 NC Pr ðSt ¼ Sr ¼ 1Þ Pr ðSt ¼ 1Þ Z 1 À 2 ÁM 1 2 Therefore, for the lower bound, we get that, for sufficiently 1 À eÀkv = pffiffiffiffiffiffi eÀv =2 dv 0 2 large SNR, X M  M 1 Z 1 v2   ¼ ðÀ1Þm pffiffiffiffiffiffi eÀ 2 ð1þ2km=Þ dv C L;C ðP Þ ¼ PrC ðSt ¼ Sr ¼ 1Þ log 1 þ C P À 1 m 2 0 Tc m¼0 Pr ðSt ¼ 1Þ M     1X M 2km À1=2 > PrC ðSt ¼ Sr ¼ 1Þ ¼ ðÀ1Þm 1 þ   2 m¼0 m P 1  log 1 þ C À M     1X M X ð2nÞ! 1 2km n Pr ðSt ¼Sr ¼1Þ PrNC ðSt ¼ 1Þ Tc ¼ ðÀ1Þm 2 ðÀ1Þn PrNC ðSt ¼Sr ¼1Þ 2 m¼0 m 2n n¼0 2 ðn!Þ ð44Þ   1 X Àn ð2nÞ! 1 X M M ¼ ðÀkÞn mn ðÀ1Þm 2 n¼0 2n ðn!Þ2 m¼0 m > PrNC ðSt ¼ Sr ¼ 1Þ   1 X Àn ð2nÞ! 1  log 1 þ NC P À 1 ¼ ðÀkÞn AðM; nÞ Tc ð45Þ 2 n¼0 2n ðn!Þ2 Pr ðSt ¼ 1Þ 1 X Àn ð2nÞ! 1 : ¼ C L;NC ðP Þ; ¼ ðÀkÞn AðM; nÞ ¼ ÀM ; 2 n¼M 2 n ðn!Þ2 where (44) follows from (43) and the transition from where (44) to (45) follows from the fact that p logð1 þ aÞ is ptherefore, increasing in p. Z 1 YÀ M : 2 Á 1 2 ÀM ¼ 1 À eÀmini ki v = pffiffiffiffiffiffi eÀv =2 dv APPENDIX C 0 i¼1 2 Z 1 YÀ M Á 1 PROOF OF THEOREM 4 2 2 1 À eÀki v = pffiffiffiffiffiffi eÀv =2 dv As shown in Appendix B, C U ðP Þ is nondecreasing in 0 i¼1 2 Z PrðSt ¼ Sr ¼ 1Þ. By using Lemma 3, it is readily verified 1 YÀ M Á 1 1 À eÀmini ki v 2 = 2 pffiffiffiffiffiffi eÀv =2 dv that for the function C U ðP Þ 0 i¼1 2 : C U;OC ðP Þ > maxfC U;C ðP Þ; C U;NC ðP Þg: ¼ÀM ; For the lower bound, we deploy the same approach as inwhich concludes the lemma. the proof of Theorem 3 and first show that Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.
  • 25. 126 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 1, JANUARY 2010 PrOC ðSt ¼ Sr ¼ 1Þ PrOC ðSt ¼ 1Þ [20] J.N. Laneman, D.N.C. Tse, and G.W. Wornell, “Cooperative ! ; ð46Þ Diversity in Wireless Networks: Efficient Protocols and Outage PrC ðSt ¼ Sr ¼ 1Þ PrC ðSt ¼ 1Þ Behavior,” IEEE Trans. Information Theory, vol. 50, no. 12, pp. 3062- 3080, Dec. 2004. [21] T. Hunter and A. Nosratinia, “Diversity through Coded Coopera- PrOC ðSt ¼ Sr ¼ 1Þ PrOC ðSt ¼ 1Þ and ! : ð47Þ tion,” IEEE Trans. Wireless Comm., vol. 5, no. 2, pp. 283-289, Feb. PrNC ðSt ¼ Sr ¼ 1Þ PrNC ðSt ¼ 1Þ 2006. [22] A. Bletsas, A. Khisti, D.P. Reed, and A. Lippman, “A SimpleThe above inequalities can be proven by substituting and Cooperative Diversity Method Based on Network Path Selection,” IEEE J. Selected Areas in Comm., vol. 24, no. 3, pp. 659-672, Mar.manipulating the expansions of PrC ðet Þ and PrOC ðet Þ given 2006.in Theorems 1 and 2 and the expansions of PrC ðt ; er Þ and e  [23] A. Tajer and A. Nosratinia, “Opportunistic Cooperation via Relay OC Selection with Minimal Information Exchange,” Proc. IEEE Int’lPr ðt ; er Þ in (26) and (27). After establishing the above e  Symp. Information Theory (ISIT ’07), June 2007.inequalities, the last step is similar to that of Appendix B. [24] M.K. Simon and M.-S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. Wiley, 2000.REFERENCES Ali Tajer received the BSc and MSc degrees in electrical engineering from Sharif University[1] J. Mitola, “Cognitive Radio: An Integrated Agent Architecture for of Technology, Tehran, Iran, in 2002 and Software Defined Radio,” PhD dissertation, KTH—Royal Institute 2004, respectively. Since June 2007, he has of Technology, Dec. 2000. been working toward the PhD degree in[2] S. Haykin, “Cognitive Radio: Brain-Empowered Wireless Com- electrical engineering at Columbia University, munications,” IEEE J. Selected Areas in Comm., vol. 23, no. 2, New York. In the summers of 2008 and 2009, pp. 201-220, Feb. 2005. he was a research assistant with NEC[3] N. Devroye, P. Mitran, and V. Tarokh, “Achievable Rates in Laboratories America, Princeton, New Jersey. Cognitive Radio Channels,” IEEE Trans. Information Theory, vol. 52, His research interests lie in the general areas no. 2, pp. 1813-1827, May 2006. of network information theory, cognitive radios, multiuser communica-[4] A. Jovicic and P. Viswanath, “Cognitive Radio: An Information- tion, relay networks, and statistical signal processing. He is a student Theoretic Perspective,” Proc. IEEE Int’l Symp. Information Theory, member of the IEEE. pp. 2413-2417, June 2006.[5] N. Devroye and V. Tarokh, “Fundamental Limits of Cognitive Xiaodong Wang received the PhD degree in Radio Networks,” Cognitive Wireless Networks, Springer, 2007. electrical engineering from Princeton University.[6] A. Goldsmith, S.A. Jafar, I. Maric, and S. 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Li, “Spatiotemporal Sensing in Cognitive Radio Networks,” IEEE J. Selected Areas in Comm., vol. 26, no. 1, pp. 5-12, Jan. 2008.[18] A. Sendanoris, E. Erkip, and B. Aazhang, “User Cooperation Diversity Part I: System Description,” IEEE Trans. Comm., vol. 51, no. 11, pp. 1927-1938, Nov. 2003.[19] A. Sendanoris, E. Erkip, and B. Aazhang, “User Cooperation Diversity Part II: Implementation Aspects and Performance Analysis,” IEEE Trans. Comm., vol. 51, no. 11, pp. 1939-1948, Nov. 2003. Authorized licensed use limited to: Columbia University. Downloaded on November 28, 2009 at 23:20 from IEEE Xplore. Restrictions apply.