Revisiting Degrees of Freedom of Full-Duplex Systems with Opportunistic Transmission: An Improved User Scaling Law

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1 entropy Artcle Revstng Degrees of Freedom of Full-Duplex Systems wth Opportunstc Transmsson: An Improved User Scalng Law Haksoo Km 1, Juyeop Km, Sang Won Cho 3 and Won-Yong Shn 4, * ID 1 Afflated Insttute of ETRI, Daejeon , Korea; haksookm@nsr.re.kr Department of Electroncs Engneerng, Sookmyung Women s Unversty, Seoul 04310, Korea; jykm@sookmyung.ac.kr or jykm00@krr.re.kr 3 ICT Convergence Research Team, Korea Ralroad Research Insttute, Ewang , Korea; swcho@krr.re.kr 4 Department of Computer Scence and Engneerng, Dankook Unversty, Yongn 16890, Korea * Correspondence: wyshn@dankook.ac.kr; Tel.: Receved: 5 January 018; Accepted: 1 arch 018; Publshed: arch 018 Abstract: It was recently studed how to acheve the optmal degrees of freedom DoF) n a mult-antenna full-duplex system wth partal channel state nformaton CSI). In ths paper, we revst the DoF of a multple-antenna full-duplex system usng opportunstc transmsson under the partal CSI, n whch a full-duplex base staton havng transmt antennas and receve antennas supports a set of half-duplex moble statons Ss) havng a sngle antenna each. Assumng no self-nterference, we present a new hybrd opportunstc schedulng method that acheves the optmal sum DoF under an mproved user scalng law. Unlke the state-of-the-art schedulng method, our method s desgned n the sense that the schedulng role between downlnk Ss and uplnk Ss s well-balanced. It s shown that the optmal sum DoF of s asymptotcally achevable provded that the number of Ss scales faster than SNR, where SNR denotes the sgnal-to-nose rato. Ths result reveals that, n our full-duplex system, better performance on the user scalng law can be obtaned wthout extra CSI, compared to the pror work that showed the requred user scalng condton.e., the mnmum number of Ss for guaranteeng the optmal DoF) of SNR 1. oreover, the average nterference decayng rate s analyzed. Numercal evaluaton s performed to not only valdate our analyss but also show superorty of the proposed method over the state-of-the-art method. Keywords: degrees of freedom DoF); full-duplex systems; hybrd opportunstc schedulng; partal channel state nformaton CSI); user scalng law 1. Introducton 1.1. Prevous Work Wth the ncreasng demands for hgh-speed communcatons, full-duplex technologes have been taken nto account as a promsng soluton for boostng the spectral effcency n multuser wreless communcatons systems [1]. However, the potental advantage of full-duplex systems may be lmted by a new challenge the nter-termnal nterference that does not appear n half-duplex systems. The problem of nter-termnal nterference n full-duplex systems has recently been studed n the lterature n terms of degrees of freedom DoF) known as the pre-log of the sum-rate capacty n the hgh sgnal-to-nose SNR) regme) [,3]. In partcular, f channels follow the ergodc phase fadng model and full channel state nformaton at the transmtter CSIT) s avalable, then t was shown n [] that the DoF of full-duplex systems can be deally twce as large as that of half-duplex systems. Entropy 018, 0, 160; do: /e

2 Entropy 018, 0, 160 of 15 Several nter-termnal nterference cancellaton schemes for a three-termnal full-duplex system were presented n [3]. In addton, the DoF of mult-antenna full-duplex systems was recently studed n [4,5]. However, there are some practcal challenges as follows. Frst, the computatonal burden of such schemes wll ncrease steeply as the system dmensons ncrease. Second, the node cooperaton and a massve number of CSI feedback bts are requred. On the other hand, n multuser wreless communcatons systems, opportunstc transmsson technques that explot the usefulness of fadng have been wdely studed n the lterature, where one can obtan a multuser dversty gan as the number of users s suffcently large. Specfcally, opportunstc schedulng [6], opportunstc beamformng [7], and random beamformng [8] were ntroduced n sngle-cell broadcast channels. In partcular, t was ponted that the same sum-rate scalng law as the optmal drty-paper codng can be acheved for such broadcast channels va random beamformng wth far less CSI feedback [8]. oreover, scenaros explotng the multuser dversty gan were studed n cooperatve networks by applyng an opportunstc two-hop relayng protocol [9], a parallel opportunstc routng protocol [10], and an opportunstc network decouplng protocol [11] as well as n cogntve rado networks wth opportunstc schedulng [1 14]. Usng opportunstc communcatons, a certan user scalng law for achevng one DoF per user was also examned for n, K)-nterference channels [15]. In addton, such opportunsm was utlzed n mult-cell broadcast channels or, equvalently, nterferng broadcast channels) by usng mult-cell random beamformng [16,17] and opportunstc nterference algnment [18]. As a more challengng problem than the downlnk case, the optmal DoF n mult-cell multple access channels or, equvalently, nterferng multple access channels) was analyzed by presentng opportunstc nterference algnment strateges [19 ] and dstrbuted schedulng protocols [3,4]. In [16,18 1], the mnmum number of users requred to acheve the optmal DoF was nvestgated.e., the user scalng law). It s worth notng that, for achevng these DoFs, the transmtters do not requre the knowledge of the nstantaneous channel realzatons. Recently, n a full-duplex system composed of a -antenna full-duplex base statons BSs) and a large number of sngle-antenna half-duplex moble statons Ss), opportunstc beamformng and schedulng methods were proposed n [5,6]. In [5], a jont uplnk downlnk opportunstc beamformng method was employed so that uplnk and downlnk sum capactes can be acheved under a certan user scalng condton. Unlke the beamformng method n [5], the scheme n [6] took advantage of the zero-forcng ZF) recever for uplnk to acheve the full DoF snce ZF flterng at the BS s suffcent to guarantee DoF for uplnk, whch results n nfntely large sum-rates wth ncreasng SNR. In partcular, t was shown n [6] that the requred user scalng law to acheve the optmal DoF s gven by SNR 1. However, the result n [6] s pessmstc n practce n the sense that too many Ss n a cell are necessary to guarantee the DoF optmalty even f the optmal DoF under a certan user scalng law was orgnally characterzed n the full-duplex system wth partal CSIT [6]. Such a hgh user scalng law n [6] stems from the schedulng role mbalance between downlnk Ss and uplnk Ss snce a set of downlnk Ss s selected wth strong responsblty to elmnate both the downlnk nterference and S-to-S nterference, whereas a set of uplnk Ss s arbtrarly chosen. It remans an open challenge how to sgnfcantly reduce the user scalng law wthout extra CSIT n the full-duplex system usng opportunstc transmsson. oreover, there have been extensve studes on schedulng and resource optmzaton n a varety of network scenaros ncludng wreless networks wth energy harvestng [7 9] and cogntve networks [30]. 1.. an Contrbutons In ths paper, we ntroduce a new hybrd opportunstc schedulng method that acheves the optmal sum DoF of the full-duplex system addressed n Secton 1.1,.e., the full-duplex system consstng of a -antenna full-duplex BSs and N sngle-antenna half-duplex Ss, under an mproved user scalng law. We consder a practcal scenaro that the system operates n the tme-dvson

3 Entropy 018, 0, of 15 duplexng TDD) mode and the effectve channel gan nformaton s only avalable at the transmtter va offlne plot sgnalng sent durng the schedulng perod. In such a partal CSIT scenaro, how to acheve the optmal DoF s a challengng task, especally for full-duplex systems snce, wth the exstng opportunstc schedulng methods, t s not straghtforward to effectvely manage the nter-termnal nterference that does not appear n half-duplex systems. Under the partal CSIT assumpton, our method combnes the followng beamformng and schedulng strateges: ) downlnk random beamformng at the BS, ) opportunstc schedulng at both the downlnk Ss and uplnk Ss, and ) uplnk ZF beamformng at the BS. ore precsely, a set of downlnk Ss s selected n the sense that the downlnk nterference s mnmzed, and a set of uplnk Ss s selected n the sense that the S-to-S nterference s mnmzed by vrtue of utlzng the channel recprocty of the TDD system, whch s the most dstngushable feature compared to the schedulng method n [6]. We remark that our method only requres each S to feed back real values along wth the correspondng beamformng vector ndces, whch s sgnfcantly less than the full CSIT case. As our man result, when uplnk and downlnk Ss are served through our full-duplex system wth hybrd opportunstc schedulng, t s shown that the sum DoF of s achevable provded that the number of Ss, N, scales faster than SNR. That s, the full DoF s guaranteed under an mproved user scalng law wthout any extra CSI as t was shown n [6] that N need to scale faster than SNR 1 to guarantee the DoF optmalty. The nterference decayng rate, defned as the average decayng rate of the total amount of receved nterference and/or generatng nterference wth respect to the number of Ss, s also analyzed asymptotcally. In addton, numercal results are provded to valdate our analyss. It was examned that the proposed hybrd opportunstc schedulng outperforms the state-of-the-art method n [6] n terms of achevable sum-rates. Our man contrbutons are four-fold and summarzed as follows: A new hybrd opportunstc schedulng method s presented n the sense that the schedulng role between downlnk Ss and uplnk Ss s well-balanced. The DoF and user scalng law are newly derved by analyzng the dstrbutons of our schedulng metrcs. The average nterference decayng rate s also analyzed. Numercal examples are provded to not only valdate our analyss but also show superorty of the proposed method over the state-of-the-art method Organzaton The rest of ths paper s organzed as follows. Secton descrbes the system model and a performance metrc. The proposed hybrd opportunstc schedulng method s presented n Secton 3. Its DoF and user scalng laws are derved n Secton 4. Numercal evaluaton s shown va computer smulatons n Secton 5. Fnally, we conclude the paper n Secton 6.. System odel and Performance etrc In ths secton, we frst descrbe the system and channel models and then defne a performance metrc used n ths paper..1. System odel As llustrated n Fgure 1, we consder a sngle-cell mult-antenna full-duplex TDD system consstng of a full-duplex BS havng transmt antennas and receve antennas and a set of N half-duplex Ss wth a sngle antenna each, where N. Snce full-duplex operaton at the BS s assumed, uplnk and downlnk data transmsson can take place smultaneously at the BS. On the other hand, each half-duplex S can be supported by ether uplnk or downlnk, but not smultaneously,.e., S d) S u) =, where S d) and S u) denote the sets of downlnk and uplnk Ss at a gven tme, and s the empty set. oreover, we assume that S d) and S u) have the same cardnalty of,

4 Entropy 018, 0, of 15.e., S d) S = u) =. We assume that there s no self-nterference due to the full-duplex operaton at the BS,.e., self-nterference due to the full-duplex operaton at the BS s perfectly suppressed. Throughout ths paper, the operators C, E[ ], Pr{ }, and ) ndcate the feld of complex numbers, the statstcal expectaton, the probablty, and the transpose conjugate, respectvely. Unless otherwse stated, all logarthms are assumed to be to the base. We use the followng asymptotc notaton: ) f x) = Ogx)) means that there exst constants C and c such that f x) Cgx) for all gx) x > c, ) f x) = Ωgx)) f gx) = O f x)), ) f x) = ωgx)) means that lm x f x) = 0, and v) f x) = Θgx)) f f x) = Ogx)) and f x) = Ωgx)) [31]... Channel odel Fgure 1. The mult-antenna full-duplex system when = and N = 15. Now, let us turn to channel modelng. The receved sgnal for downlnk transmsson at S and the receved sgnal vector for uplnk transmsson at the BS, denoted by y d) C and y u) C 1, can be wrtten as = β d) h d) s d) + β ) j h j s u) j + n d), 1) j S u) y d) y u) = β u) h u) s u) + n u), ) S u) respectvely, where β d) h d) C 1, β u) h u) C 1, and β ) j h j C denote the channel vectors from the BS to S, from S to the BS, and channels from S j to S, respectvely. Here, the channel coeffcents β d) h d), β u) βh u), and β ) j h j consst of the large-scale path-loss component, whch s ndependent of SNR, and the small-scale complex fadng component. ore specfcally, β d), β u), and β ) j represent the nonnegatve path-loss attenuaton factor between the BS and S for downlnk, between S and the BS for uplnk, and between two Ss and j, respectvely. We assume that each element of small-scale fadng channels s ndependent and dentcally dstrbuted..d.) accordng to CN 0, 1), where the notaton CN µ, Σ) ndcates the complex Gaussan dstrbuton wth a mean vector µ and a covarance matrx Σ. The downlnk transmt sgnal vector at the BS and the uplnk sgnal at S j, denoted by s d) C 1 and s u) j C, respectvely, satsfy the average power constrants [ s E d) ] [ s u) = 1 and E j ] = 1. The addtve nose at S, denoted by n d), and each element of

5 Entropy 018, 0, of 15 the addtve nose vector at the BS, denoted by n u), are..d. complex Gaussan wth zero mean and varance of N 0, respectvely. We assume the block fadng channel model,.e., channel coeffcents are constant durng one codng or communcaton block and changes to a new ndependent value for every transmsson block. We further assume that full CSI s avalable at the recever sde, but only partal CSI effectve channel gan nformaton) s avalable at the transmtter sde, whch wll be specfed later on..3. Performance etrc As a performance metrc, we use the sum DoF, whch s defned by DoF = R u) + R d) lm SNR log SNR, 3) where R u) and R d) denote the achevable sum-rates for uplnk and downlnk, respectvely. Note that ths DoF s the pre-log of the sum-rate capacty n the hgh SNR regme. In the next secton, we descrbe our new hybrd opportunstc schedulng method for the cellular mult-antenna system wth one full-duplex BS and multple half-duplex Ss. We then show that t leads to an mproved user scalng law.e., the reduced number of Ss) for guaranteeng the optmal DoF, compared to the pror work n [6]. 3. New Hybrd Opportunstc Schedulng In the full-duplex system wth one mult-antenna BS, an opportunstc schedulng method was ntroduced n [6] by employng uplnk ZF beamformng at the BS and downlnk random beamformng at the BS. In the schedulng procedure, downlnk Ss were opportunstcally selected n the sense of mnmzng the total nterference level ncludng both downlnk nterference and S-to-S nterference, whereas uplnk Ss were arbtrarly chosen. For ths reason, the method n [6] requres plenty of Ss so that downlnk Ss who have a suffcently small amount of the schedulng metrc shown later n ths secton) are fnally selected whle achevng DoF for downlnk. That s, a strngent user scalng condton s necessary under the method n [6] due to the schedulng role mbalance between downlnk Ss and uplnk Ss. In ths secton, we propose another type of hybrd opportunstc schedulng such that both uplnk and downlnk Ss are opportunstcally selected, thereby resultng n the reduced number of Ss requred to acheve the full DoF. The overall procedure of our schedulng method s descrbed accordng to the followng steps: 1. Downlnk Random Beamformng at the BS: The BS generates orthonormal random vectors { v C 1} =1, where {v } =1 are generated accordng to the sotropc dstrbuton over the -dmensonal unt sphere. Then, the BS broadcasts ts generated beamformng vectors V = [v 1,, v ] to all Ss over the system.. Downlnk Schedulng etrc Calculaton and Feedback: We frst focus on the downlnk user schedulng process. In our proposed method, we defne the downlnk schedulng metrc of each S {1,, N} as the downlnk nterference. Let us suppose that S s served by downlnk beamformng vector v m. Then, the mth downlnk schedulng metrc of S, denoted by L,m, s expressed as L,m = k=1,k =m β d) ) h d) vk. 4) Here, S calculates the set of ts downlnk schedulng metrcs {L,1,, L, } and then feeds those values as well as ts own user ID back to the BS.

6 Entropy 018, 0, of Downlnk User Selecton: Upon recevng the sets of the downlnk schedulng metrcs from the all Ss, the BS selects π m = arg mn ) L,m, 5) {1,,N}\ {π l } m 1 l=1 whch eventually results n the set of selected downlnk Ss S d) = {π 1,, π }. Then, the BS broadcasts a short sgnalng message representng the set of selected downlnk Ss. The BS s ready for transmttng ts downlnk packets to S π m usng the beamformg vector v m, where m {1,, }. 4. Uplnk User Schedulng etrc Calculaton and Feedback: We now turn to the uplnk user schedulng process by utlzng the channel recprocty of our TDD system. The frst step of uplnk user schedulng s to defne the uplnk schedulng metrc of each S j {1,, N} \ S d) as the S-to-S nterference.e., the sum of the nterference leakage power from tself to all Ss n S d) ). Then, the uplnk schedulng metrc of S j, denoted by γ j, s represented as follows: γ j = S d) β ) j ) hj. 6) From both the feedback sgnals from the selected downlnk Ss and the short sgnalng message from the BS, each uplnk S s capable of computng the metrc n Equaton 6). Thus, S j {1,, N} \ S d) calculates ts uplnk schedulng metrc γ j and feeds ts value as well as ts own user ID back to the BS. 5. Uplnk User Selecton: Upon recevng N uplnk schedulng metrcs except for the selected downlnk Ss n S d), the BS selects uplnk Ss havng the smallest uplnk schedulng metrcs. That s, for m {1,, }, the BS selects φ m = arg mn j {1,,N}\ S d) {φ l } m 1 l=1 ) γ j, 7) whch eventually results n the set of selected uplnk Ss S u) = {φ 1,, φ }. Then, each S n S u) s ready for transmttng ts uplnk packets. 6. Uplnk ZF Beamformng at the BS: To decode uplnk packets, the BS apples ZF receve flterng by nullng out the uplnk nterference wthout CSI at the transmtter. For the proposed opportunstc schedulng method, we assume that each S j {1,, N} \ S d) can estmate the S-to-S nterference γ j by overhearng feedback sgnals sent from the downlnk Ss to report ther schedulng metrcs to the BS. oreover, t s worthwhle to address the fundamental dfferences between our approach and two dfferent types of schedulng methods for full-duplex systems as follows. Remark 1. In [5], nstead of ZF beamformng, random receve beamformng for decodng uplnk packets s employed at the BS. In [6], a set of downlnk Ss s selected to elmnate both the downlnk nterference and S-to-S nterference, whereas a set of uplnk Ss s arbtrarly chosen. 4. Analyss of DoF and User Scalng In ths secton, we frst analyze the DoF achevablty of our new hybrd opportunstc schedulng method along wth the correspondng user scalng law. We then analyze the nterference decayng rate wth respect to the number of Ss.

7 Entropy 018, 0, of User Scalng Law For uplnk transmsson, t s obvous that the sum DoF of s achevable by usng the ZF recever at the BS. Thus, we focus on analyzng how to acheve the sum DoF of for downlnk transmsson. When the sets of the selected downlnk and uplnk Ss, denoted by S d) = {π 1,, π } and S u) = {φ 1,, φ }, respectvely, are determned, the receved sgnal at S π for downlnk transmsson s rewrtten as y d) π = β d) π h d) π s d) + = β d) π h d) β ) j S u) d) π v x d) + k=1,k = π j h π js u) j β d) π h d) + n d) π d) π k v k x d) k + β ) π j h π js u) j + n d). 8) j S u) Thus, from Equaton 8), the receved sgnal-to-nterference-plus-nose rato SINR) at S π s gven by SINR d) π = SNR k=1,k = = SNR β d) π ) h d) π v I d) π +I u) π +1 SNR β d) π ) h d) π v β d) ) π h d) π vk +SNR j S u) where I d) π = SNR k=1,k = β d) π ) h d) π vk and I u) π, β ) π j = SNR j S u) ) hπ j +1 β ) π j 9) ) hπ j denote the nterference caused by other generated beams.e., the downlnk nterference) and the nterference from the selected uplnk Ss to S π.e., the S-to-S nterference), respectvely. Then, usng the receved SINR n Equaton 9), the achevable sum-rate for downlnk s gven by R d) = log 1 + SINR d) ) π. 10) =1 Now, the followng theorem establshes the DoF achevablty of the proposed hybrd opportunstc schedulng method presented n Secton 3. Theorem 1. For the mult-antenna full-duplex system n Secton, the optmal DoF of s achevable wth hgh probablty f N = ω SNR ). 11) Proof. For uplnk transmsson, t s obvous that the sum DoF of s achevable by usng the ZF recever at the BS. Thus, we focus on the achevable DoF for downlnk. Let us defne P d and P u by the probabltes that the downlnk nterference and the S-to-S nterference at all the selected downlnk Ss are less than or equal to ɛ 1 > 0 and ɛ > 0, respectvely, where ɛ 1 and ɛ are small constants ndependent of SNR. Then, P d and P u can be expressed as P d = lm Pr SNR { SNR k=1,k = β d) ) } π h d) π vk ɛ 1, {1,, } 1) and P u = lm Pr SNR SNR j S u) β ) π j ) h π j ɛ, {1,, }, 13)

8 Entropy 018, 0, of 15 respectvely. Then, the sum DoF for downlnk transmsson, denoted by DoF d, s lower-bounded by DoF d P d P u. 14) Now, let us characterze two probabltes P d and P u. Frst, P d can be rewrtten as P d { = lm SNR Pr SNR k=1,k = = lm SNR Pr { k=1,k = hd) hd) π vk ɛ 1, {1,, } π vk ɛ 1 SNR 1, {1,, } } }, 15) where ɛ 1 = ɛ 1 β d) π 1 ) 1, whch s ndependent of SNR. Here, the term k=1,k = h d) π vk corresponds to the downlnk schedulng metrc of selected S π wth no path-loss component and follows the ch-square dstrbuton wth degrees of freedom for {1,, } snce the -dmensonal downlnk channel vector h d) π s sotropcally dstrbuted. Note that the rght-hand sde of Equaton 15) ndcates the probablty that there exst at least Ss that fulflls the nequalty k=1,k = π vk ɛ 1 SNR 1. Thus, by denotng Fx) by the cumulatve densty functon CDF) of a ch-square random varable wth degrees of freedom, t follows that P d = 1 lm SNR 1 =0 N ɛ )F 1 SNR 1) 1 F ɛ 1 SNR 1)) N = 1 lm SNR 1 =0 a) b) 1 lm SNR 1 =0 1 lm SNR 1 =0 F N!!N )! N F 1 F ɛ 1 SNR 1)) N 1 F ɛ 1 SNR 1)) ɛ 1 SNR 1) 1 F ɛ 1 SNR 1)) N 1 F ɛ 1 SNR 1)) ɛ 1 SNR 1)) 1 C d,1 SNR ) N NC d, SNR ) 1 C d, SNR ), hd) 16) where and C d,1 = e 1 Γ) ɛ 1 17) C d, = 1) Γ) ɛ 1. 18) Here, Γ) = 0 t 1 e t dt s the Gamma functon; a) holds from the fact that N!!N )! N ; and b) holds from the fact that see Lemma 1 n [0]) e 1 Γ) x Fx) 1) Γ) x. 19)

9 Entropy 018, 0, of 15 Next, let us turn to characterzng P u as follows: P u lm SNR SNR lm SNR SNR lm SNR Pr { =1 j S u) j S u) =1 hπ j h π j ɛ, {1,, } hπ j ɛ } ɛ SNR 1, j Su), 0) 1, where ɛ = ɛ max{β ) π φ 1,, β ) π φ }) whch s ndependent of SNR. Snce the term hπ =1 j corresponds to the uplnk schedulng metrc γ j wth no path-loss component and s the ch-square random varable wth degrees of freedom for j S u), Equaton 1) can further be lower-bounded by P u 1 lm SNR 1 =0 N )F ) ɛ SNR 1 1 F ) ) N ɛ SNR 1 = 1 lm SNR 1 =0 1 lm SNR 1 =0 1 lm SNR 1 =0 F N )!!N )! { { N ) F ɛ SNR 1 ) ɛ 1 F SNR 1 )) N ɛ 1 F SNR 1 )) ɛ SNR 1 )} ɛ 1 F SNR 1 )) N ɛ 1 F SNR 1 )) 1 C u,1 SNR ) N N )C u, SNR } 1 C u, SNR ), 1) where C u,1 = e 1 Γ) ɛ ), ) and C u, = 1) Γ) ɛ ). 3) It s not dffcult to show that f N = ω SNR ), then two terms 1 C d,1 SNR ) N and 1 C u,1 SNR ) N decrease exponentally wth respect to SNR, whereas other two terms NC d, SNR ) { and N ) C u, SNR } ncrease polymonally for any > 0. In consequence, as SNR goes to nfnty, both P d and P u tend to one. Hence, from Equaton 14), DoF d f N = ω SNR ), whch completes the proof of ths theorem. Our man result s now compared wth the achevablty result n [6] wth respect to the user scalng law. Remark. In the mult-antenna full-duplex system consstng of a full-duplex BS havng antennas transmt and receve antennas each) and a set of N half-duplex Ss wth a sngle antenna each, t was shown

10 Entropy 018, 0, of 15 n [6] that the optmal DoF s achevable by usng opportunstc schedulng at the downlnk Ss and random selecton of the uplnk Ss, provded that N scales faster than SNR 1. In ths work, we have proposed the hybrd opportunstc schedulng method such that both the uplnk and downlnk Ss are opportunstcally selected, thereby resulng n the reduced number of Ss requred to acheve the optmal sum DoF.e., DoF). Note that our schedulng method does not utlze any further CSI at the transmtters, compared to that of [6]. 4.. Interference Decayng Rate Next, we analyze the average nterference decayng rate defned as the average decayng rate of the total amount of receved nterference and/or generatng nterference wth respect to the number of Ss, N. Ths s meanngful snce the desred user scalng law s closely related to the nterference decayng rate wth ncreasng N for gven SNR. Let I d) mn, denote the maxmum value.e., the th smallest value) among the downlnk nterference levels that selected downlnk Ss compute, whch s gven by I d) mn, = max L πm, 4) π m S d) where L πm represents the downlnk schedulng metrc of selected S π m and S d) s the set of selected downlnk Ss. In addton, let I u) mn, denote the maxmum value among the S-to-S nterference levels that selected uplnk Ss compute, whch s gven by I u) mn, = max φ j S u) γ φj, 5) where γ φj s the uplnk schedulng metrc of selected S φ j as shown n Equaton 6) and S u) s the set of selected uplnk Ss. Snce the performance of our hybrd opportunstc schedulng method s lmted manly by 1) such a selected downlnk S that receves the maxmum amount of nterference from other beams generated by the BS or ) such a selected uplnk S that generates the maxmum amount of nterference to selected downlnk Ss, t s certanly worth analyzng an asymptotc behavor of I mn, max{i d) mn,, Iu) mn, } wth respect to N. Now, we are ready to establsh [ our ] second man result, whch shows a lower bound on the average 1 nterference decayng rate E I wth respect to N. mn, Theorem. For the mult-antenna full-duplex system n Secton, the average nterference decayng rate s lower-bounded by [ ] 1 E Θ N 1/). 6) I mn, Proof. The proof essentally follows the same steps as those n Secton III-B n [3]) and Remark 1 n [6]), and thus a bref sketch of the proof s provded here. From the proof of Theorem 1 and the arkov s nequalty, t follows that { 1 Pr I mn, ɛ } SNR SNR ɛ [ E [ max{i d) = { SNR E max ɛ ) SNR = Θ N 1/ mn,, Iu) mn, } ] max π m S d) L πm, max φ j S u) γ πj for small ɛ > 0, whch tends to zero f N = ω SNR ). Here, the frst equalty holds due to Equatons 4) and 5). Ths completes the proof of Theorem. }] 7)

11 Entropy 018, 0, of 15 From the above theorem, we obtan the same scalng law as n Theorem 1. Ths mples that the faster nterference decayng rate wth respect to N, the smaller SNR exponent n the user scalng law. 5. Numercal Evaluaton In ths secton, we perform computer smulatons to valdate our analyss n Secton 4. Numercal examples are also provded to evaluate the sum-rate performance of the proposed hybrd opportunstc schedulng method for fnte parameters N and SNR. In our smulatons, each channel coeffcent n Equatons 1) and ) s generated 10 4 tmes for each system parameter. Unless otherwse stated, t s assumed that the large-scale path-loss component.e., β d), β u), and β ) j for all and j) s gven by 1 n our smulatons. The average nterference decayng rate s frst evaluated numercally accordng to the total number of Ss, N. Even f t seems unrealstc to have a large number of Ss n a cell, the range of parameter N s taken nto account to precsely see some trends of curves varyng wth N. In Fgure, the log log plot of the average nterference decayng rate versus N s shown as N ncreases for system parameter {, 3}, ndcatng the number of transmt or receve antennas at the BS. Ths numercal result reveals that the nterference decayng rate tends to decrease almost lnearly wth N, but the slopes of the curves vary accordng to. The dotted lnes are obtaned from Theorem theoretcal results) wth proper bases, and, thus, only the slopes of the dotted lnes are relevant. It s shown that the bound n Theorem s ndeed tght snce the average nterference decayng rates shown n Fgure are consstent wth the user scalng law derved n Theorem 1. oreover, t s shown that the average nterference decayng rate gets ncreased as ncreases snce the user scalng law n Theorems 1 and s expressed as an ncreasng functon of Smulaton Asymptotc analyss Interference decayng rate = = N Fgure. The average nterference decayng rate versus N when {, 3}. As shown n Fgure 3, when {, 3}, the achevable sum-rates of the proposed hybrd opportunstc schedulng method are now evaluated accordng to the receved SNR n db scale) and are compared wth the conventonal schedulng method n [6] where downlnk Ss are opportunstcally selected whle uplnk Ss are arbtrarly selected. Note that N s set to a dfferent scalable value

12 Entropy 018, 0, of 15 accordng to SNR,.e., N = SNR, to see whether the slope of a curve follows the DoF n Theorems 1. It s obvous to see that the proposed method outperforms the conventonal one n terms of sum-rates for all SNR regmes. Ths s because the DoF acheved by the method n [6] s surely lower than = 4 due to the fact that ts user scalng law N = ωsnr 1 ) s not fulflled and thus there exsts more resdual nterference at each recever sde. It ndcates that the performance gap between the two methods becomes large n the hgh SNR regme Proposed [3] Sum rates [bts/s/hz] SNR [db] a) Proposed [3] Sum rates [bts/s/hz] SNR [db] b) Fgure 3. The achevable sum-rates versus SNR. a) = ; b) = 3. In addton, the effect of the path-loss attenuaton factor on the sum-rates s examned. For convenence of an llustraton, suppose that β β d) = β u) and β ) j = 1. That s, we consder the case where both downlnk and uplnk channels experence the same degree of path-loss attenuaton.

13 Entropy 018, 0, of 15 Here, 0 < β < 1 corresponds to the case where an S and the BS are relatvely far away from each other whle most Ss are co-located. On the other hand, β > 1 corresponds to the case where the dstance between an S and the BS s relatvely close and Ss are separated by an obstacle e.g., a wall). In Fgure 4, the achevable sum-rates of the proposed hybrd opportunstc schedulng method versus the receved SNR n db scale) are evaluated for = and β {0., 0.5, 0.8, 1., 1.5}. As β decreases, both the desred sgnal power at S π and the downlnk nterference power at S π get reduced due to a more severe path-loss attenuaton between an S and the BS. From the fgure, t s shown that the sum-rates are degraded wth decreasng β, whch reveals that reducton on the desred sgnal power s more sgnfcant n determnng the sum-rate performance. Sum rates [bts/s/hz] β=0. β=0.5 β=0.8 β=1. β= SNR [db] Fgure 4. The achevable sum-rates versus SNR when = and β {0., 0.5, 0.8, 1., 1.5}. 6. Conclusons A new hybrd opportunstc schedulng method was presented n mult-antenna full-duplex systems wth partal CSIT where the effectve channel gan nformaton s only avalable at the transmtter. Unlke the pror work n [6], both the downlnk and uplnk Ss were opportunstcally selected n the proposed method, whch leads to an mproved user scalng law.e., the reduced number of Ss). It was analyzed that the proposed method asymptotcally acheves the DoF of provded that the number of Ss, N, scales faster than SNR. That s, t was shown that the full DoF s guaranteed under the mproved user scalng law wthout any extra CSIT compared to the state-of-the-art schedulng method n [6] that requres the user scalng condton of N = ωsnr 1 ). Numercal evaluaton was also shown to verfy that our method outperforms the conventonal one under realstc network condtons e.g., fnte N and SNR) wth respect to achevable sum-rates. Further nvestgaton of the numercal evaluaton n a more general setup, whch ncludes dfferent large-scale path-loss components for each lnk by consderng the spatal locaton of Ss, remans for future work. Suggestons for further research n ths area also nclude examnng the effects of S moblty on the performance va extensve computer smulatons.

14 Entropy 018, 0, of 15 Acknowledgments: Ths research was supported by the Basc Scence Research Program through the Natonal Research Foundaton of Korea NRF) funded by the nstry of Educaton 017R1D1A1A ) and by the Insttute for Informaton & communcatons Technology Promoton IITP) grant funded by the Korea government SIT) , Development of PS-LTE System and Termnal for Natonal Publc Safety Servce). Author Contrbutons: Haksoo Km wrote the smulaton code and performed experments; Juyeop Km n part desgned the system model; Sang Won Cho n part desgned the protocol; and Won-Yong Shn conceved and desgned the overall protocol, and derved the man theorem. Conflcts of Interest: The authors declare no conflct of nterest. References 1. Aryafar, E.; Khojastepour,.; Sundaresan, K.; Rangarajan, S.; Chang,. IDU: Enablng IO full-duplex. In Proceedngs of the AC Annual Internatonal Conference on oble Computng and Networkng obcom), Istanbul, Turkey, 6 August 01; pp Saha, A.; Dggav, S.N.; Sabharwal, A. On degrees-of-freedom of full-duplex uplnk/downlnk channel. In Proceedngs of the IEEE Informaton Theory Workshop ITW), Sevlla, Span, 9 13 September 013; pp Ba, J.; Sabharwal, A. Dstrbuted full-duplex va wreless sde channels: Bounds and protocols. IEEE Trans. Wrel. Commun. 013, 1, Elmahdy, A..; El-Key, A.; ohasseb, Y.; ElBatt, T.; Nafe,.; Seddk, K.G.; Khattab, T. Degrees of freedom of the full-duplex asymmetrc IO three-way channel wth uncast and broadcast messages. IEEE Trans. Commun. 017, 65, Chae, S.H.; Lm, S.H.; Jeon, S.-W. Degrees of freedom of full-duplex multantenna cellular networks. IEEE Trans. Wrel. Commun. 018, 17, Knopp, R.; Humblet, P. Informaton capacty and power control n sngle cell multuser communcatons. In Proceedngs of the IEEE Internatonal Conference on Communcatons ICC), Seattle, WA, USA, 18 June 1995; pp Vswanath, P.; Tse, D.N.C.; Laroa, R. Opportunstc beamformng usng dumb antennas. IEEE Trans. Inf. Theory 00, 48, Sharf,.; Hassb, B. On the capacty of IO broadcast channels wth partal sde nformaton. IEEE Trans. Inf. Theory 005, 51, Cu, S.; Hamovch, A..; Somekh, O.; Poor, H.V. Opportunstc relayng n wreless networks. IEEE Trans. Inf. Theory 009, 55, Shn, W.-Y.; Chung, S.-Y.; Lee, Y.H. Parallel opportunstc routng n wreless networks. IEEE Trans. Inf. Theory 013, 59, Shn, W.-Y.; a, V.V.; Jung, B.C.; Yang, H.J. Opportunstc network decouplng wth vrtual full-duplex operaton n mult-source nterferng relay networks. IEEE Trans. oble Comput. 017, 16, Ban, T.W.; Cho, W.; Jung, B.C.; Sung, D.K. ult-user dversty n a spectrum sharng system. IEEE Trans. Wrel. Commun. 009, 8, Tajer, A.; Wang, X. ultuser dversty gan n cogntve networks. IEEE/AC Trans. Netw. 010, 18, Shen, C.; Ftz,.P. Opportunstc spatal orthogonalzaton and ts applcaton n fadng cogntve rado networks. IEEE J. Sel. Top. Sgnal Process. 011, 5, Tajer, A.; Wang, X. n, k)-user nterference channels: Degrees of freedom. IEEE Trans. Inf. Theory 008, 58, Shn, W.-Y.; Jung, B.C. Network coordnated opportunstc beamformng n downlnk cellular networks. IEICE Trans. Commun. 01, E95-B, Nguyen, H.D.; Zhang, R.; Hu, H.T. ult-cell random beamformng: Achevable rate and degrees of freedom regon. IEEE Trans. Sgnal Process. 013, 14, Yang, H.J.; Shn, W.-Y.; Jung, B.C.; Suh, C.; Paulraj, A. Opportunstc downlnk nterference algnment for mult-cell IO networks. IEEE Trans. Wrel. Commun. 017, 16, Jung, B.C.; Shn, W.-Y. Opportunstc nterference algnment for nterference-lmted cellular TDD uplnk. IEEE Commun. Lett. 011, 15, Jung, B.C.; Park, D.; Shn, W.-Y. Opportunstc nterference mtgaton acheves optmal degrees-of-freedom n wreless mult-cell uplnk networks. IEEE Trans. Commun. 01, 60,

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