Optimal Pricing for Interference Control in Time-reversal Device-to-Device Uplinks

Transcription

1 05 IEEE Global Conference on Sgnal and Informaton Processng (GlobalSIP) Optmal Prcng for Interference Control n Tme-reversal Devce-to-Devce Uplns Qny Xu, Yan Chen, and K. J. Ray Lu, Abstract The Devce-to-Devce (DD) communcaton s a promsng technque to empower local wreless communcatons. However, wthout proper management t may generate nterference to the exstng networ and degrade the overall performance. By treatng each multpath as a vrtual antenna, tme-reversal (TR) sgnal transmsson n a rchscatterng envronment produces a spatal-temporal resonance whch effcently suppresses the nter-user nterference (IUI) whle boostng the sgnal power at the target recever. In ths wor, we desgn a TRbased DD hybrd networ, where both prmary users (PUs) and DD pars share the same tme-frequency resources and use TR focusng effect to combat nterference. Wth the purpose of enhancng DD performance whle provdng a performance protecton to PUs, an effcent optmal prcng algorthm s proposed to dynamcally control nterference through TR focusng strength control. Index Terms Tme reversal, Devce-to-Devce communcaton, power control, optmal prcng, QoS protecton. I. INTROCTION To meet the growng demand for hgher data rate wreless access, Devce-to-Devce (DD) communcaton as a new relable, scalable and green paradgm has been proposed. Amng to revolutonze the conventonal communcaton method, DD communcaton s defned as the drect communcaton between two devces wthout or wth lmted access pont (AP) nvolvement, and operates as an underlay n the tradtonal networ [] [3]. By utlzng proxmty, DD communcaton becomes a promsng technque n that t brngs n advantages such as hgher networ spectral effcency, energy effcency and flexblty. However, t may also ntroduce nterference to the exstng networ and sgnfcantly degrade ts performance wthout proper nterference management. As a part of resource control, power control s a essental method to tacle nterference n DD networs. Power control over DD lns was frstly proposed to ncrease the networ throughput n [3], [4] by consderng the dstance between users n a DD par and ntroducng a fxed booster factor and a bacoff factor. Amng at mprovng the performance of DD pars whle mantanng a desred performance of networ, an nterference-aware resource allocaton scheme was desgned usng the power measurement feedbac from DD pars durng upln [5]. In [6], the resource allocaton was consdered under the assumpton that the cellular base statons are capable of selectng the best resource allocaton scheme among non-orthogonal sharng, orthogonal sharng and relayng mode for both PUs and DD connectons. Moreover, a sequental second prce aucton game was desgned n [7] to allocate each spectrum resources to mprove DD performance n downlns. A scheme that reuses the underutlzed upln spectrums and avods near-far nterference to DD transmssons was proposed n [8]. Recently, a dstrbuted resource allocaton algorthm to mprove the DD throughput n an OFDM system was proposed n [9]. The tme reversal (TR) based sgnal transmsson technque treats each path of a mult-path channel n a rch scatterng envronment as a dstrbuted vrtual antenna. It creates a hgh-resoluton spataltemporal resonance, commonly nown as focusng effect, va smply Fg. : The TR-based DD hybrd upln networ. transmttng bac a tme-reversed conjugate verson of the channel mpulse response (CIR) [0] [3]. As a result, TR effectvely elmnates the nter-user nterference (IUI) n a sngle-carrer multuser communcaton networs. Meanwhle the sgnal-to-nose rato (SNR) of the receved sgnal s boosted due to the nherent nature of TR that fully collects energy of mult-path propagaton. In the aforementoned analyss of DD communcaton networs, the exstng models were bult upon a mult-carrer system where prmary users n the exstng networ are assgned wth orthogonal channels shared wth at most one DD par. Inspred by the study n [0] [3], we propose a novel DD communcaton framewor ncorporatng the TR technque, where all DD pars and PUs transmt wthn the same frequency band and tme slot. Together wth the rch-scatterng envronment, TR technque generates a unque spatal-temporal resonance, whch s a natural way that suppresses the IUIs and nter-ter nterference (ITI) between DD pars and PUs (or access ponts (APs)). In the proposed TR-based hybrd DD networs, the nterference s managed by the means of TR focusng strength control. The rest of ths paper s organzed as follows. The system model for TR-based DD hybrd networ s ntroduced n Secton II, and n Secton III the focusng strength optmzaton problem s analyzed. In Secton IV, an optmal prcng algorthm s desgned to smplfy and solve the DD optmzaton problem. The system performance s evaluated n Secton V wth conclusons n Secton VI. II. SYSTEM MODEL Accordng to the TR upln dagram n [3], the receved sgnal at the AP sde s the combnaton of all sgnals comng from both PUs and DD users (s) that can be wrtten as N s AP P () PU H() AP PU x() PU + M H(m) AP x(m) +n, m () where N s the number of prmary users (PUs), M s the number of DD pars, P () PU s the transmt power of the th PU wth /5/$ IEEE 096

2 05 IEEE Global Conference on Sgnal and Informaton Processng (GlobalSIP) transmtted symbol x () PU, s the transmt power of the mth DD transmtter, and n represents the whte Gaussan nose at the AP wth zero mean and varance σ. Moreover, H () AP PU s the Toepltz matrx formed by the channel from the th PU to the AP, smlar to H (m) AP. The receved sgnal s AP n () wll pass through a TR flter ban {g PU, } to extract nformaton and suppress nterference. Here, the TR sgnature s defned as g PU [] h () AP PU [L L ]/ h () AP PU [l], where denotes the conjugate, h () AP PU l0 s the channel delay profle wth length L. The receved sgnal at the m th DD recever s smlar to that one at AP, and due to the lmted space detals are omtted. III. PROBLEM FORMULATION Based upon the analyss, the system upln SINR for the th PU and the m th DD par are gven n () and (3). In (), R AP PUj H (j)h AP PU H(j) AP PU, R(0) AP PU H ()H H () wth upscrpt H denotng Hermtan and H () beng the L th row of toepltz matrx H () AP PU, ˆR AP PU H ()H AP PU H() AP PU R(0) AP PU. Other channel correlaton matrces R AP etc. are defned smlarly. Moreover, ˆR AP PU g PU, P (j) PU gh PU j R AP PUj g PUj and P () gh R AP g, represent ISI, IUI and ITI respectvely. For smplcty, we assume the varances of channel nose are all the same at dfferent recevers. We ntroduce matrx D as a nonnegatve dagonal matrx wth [D] /g H PU AP PU g PU. The nter-ter nterference matrx A representng the nterference caused by DD users s defned as [A] j g H PU j R AP g PUj. The crosstal matrx Φ for the upln prmary users whose elements correspond to the ISI and IUI terms as [Φ] j g H PU j R AP PU g PUj f j and g H PU j ˆR AP PUj g PUj, otherwse. Then the upln SINR for PU n () can be rewrtten as γ PU SINRPU UL PU D (Φ T PPU+AT P +σ ). (4) Smlarly, for each DD par we can defne D as [ D ] ] /g H DA g, nter-ter nterference matrx [Ã Ã as j g H j R DAj PU g j, and crosstal matrx Φ Φ Φ as [ Φ] j g H PU j R DAj g PUj f j and [ Φ] j g H PU j ˆR DAj j g PUj, otherwse. The upln SINR for the m th DD par n (3) can be rewrtten as m j SINR UL m. (5) D mm( Φ T mp +ÃT mp PU+σm) In order to jontly optmze the resources utlzed by PUs and s, we formulate our problem as a DD throughput mzaton problem wth a servce protecton to the PUs. Gven the ndvdual PU power constrants P PU, DD power constrants P and SINR threshold for PUs γpu, th the mzaton problem can be wrtten as P PU, P m s.t. P PU 0, P 0, α mlog ( + m ) P PU P PU, P P. γ PU γ th PU, (6) where {α m} M m denotes the weghted factor n sum rate. A. Two-stage Optmzaton I: Prmary Feasblty Problem The optmzaton problem n (6) can be decomposed nto a two stage problem. The frst subproblems s a feasblty problem for PUs n upln where we want to mze the mnmum among all the receved SINR γ PU P PU,.e., mn,,n γpu (7) s.t. P PU 0, P PU P PU, Let us denote the γpu as the optmal value towards the problem n (7) wth optmal varable P PU. Then f γpu γpu, th the optmal value of problem n (6) s 0 and the ultmate optmal powers are P PU and P 0. Otherwse, the threshold SINR for PUs s achevable, and the frst subproblem s feasble. It s easy to show that n the soluton to (7) all users wll have the same SINRs. Based on (4) and supposng P s nown and fxed, the correspondng feasble PU power allocaton vector that acheves γpu th for all PUs s a functon of P as P PU γpu(i th γpudφ th T ) D(A T P + σ). Let P o denote the power vector for PUs to acheve γpu th wthout consderng DD pars, then P o γpu(i γ th PUDΦ th T ) Dσ. Matrx F represents the power-boost matrx ntroduced by the nterference of DD pars, F γpu(i th γpudφ th T ) DA T. Hence, the optmal power assgnment for PUs n (6) s decoupled nto two parts as P PU P o + FP. B. Two-stage Optmzaton II: DD Throughput Maxmzaton Once the frst subproblems s feasble, we can obtan a meanngful par of P o and F to address the second-stage problem that s defned a DD throughput mzaton as α mlog ( + γm ) P c m (8) s.t. P 0, P P, P o + FP 0, P o + FP P PU. In (8), the objectve functon s not concave wth many nequalty constrants whch mae the searchng space for optmal P too complcated. To address the nonconvexty, we convert the problem n (8) nto an equvalent DD SINR allocaton problem. An effcent algorthm s proposed n Secton IV that borrows the prncple of the stacelberg game to relax the problem nto an optmal prcng problem. IV. OPTIMAL PRICING FOR JOINT POWER CONTROL IN HETEROGENEOUS UPLINKS In the proposed algorthm, the AP acts as the leader that decdes the prce. Each DD par ndependently chooses a best response that mzes ther utltes wth the prce from AP. The mathematcal formulaton s descrbed as follows. In the leader s problem, AP wants to fnd a prce c that mzes hs utlty, whch s the revenue from followers that depends on both AP s prce and DD users choces: u AP (c, )c s.t. ρ(dag{ } DˆΦ T ) <, P ( ) P, P o+fp ( ) 0, P o+fp ( ) P PU, (9) 097

3 05 IEEE Global Conference on Sgnal and Informaton Processng (GlobalSIP) SINR UL SINR UL m PU AP PU g PU ˆR AP PU g PU +, () P (j) PU gh PU j R AP PUj g PUj + M P () gh R AP g +σ j gh m ˆR DAm m g m + m gh m DA m m g m P () gh R DAm g + N R DAm PU g PU +σ. (3) where ρ( ) denotes the spectral radus. The feasblty condton P 0 n (8) s equvalent to ρ(dag{ } DˆΦ T ) <. Wth the prce c from AP, each DD par forms a noncooperatve best response problem as u DD ( )α log ( + γ ) cγ (0) s.t. 0. In ths prce optmzaton problem, the total gan s obtaned as U gan(c, ) M m αmlog( + γ m ). Even though the optmal prcng problem has the same objectve as the one n (8), due to reducton of the dmenson of searchng space by utlzng a scalar prce c as the varable, the optmal value of the total throughput s lower than the optmum obtaned n (8) n some cases. A. Best Strateges for DD Users n the Followers Game In order to fnd the best strategy for both AP and DD users, we apply the bacward nducton approach n game theory to our problem. Gven the prce AP select, each DD par ndependently chooses ther best response γm (c) that mzes hs utlty n (0). Because of the convexty of (0), the optmal pont can be easly obtaned through KKT condtons as ( ) + γm αm (c) c ln, () where (X) + {0,X}. The () s a typcal waterfllng soluton. When c < αm, ln γ m (c) s a decreasng and convex functon n c, and γm (c) 0for any c αm ln. B. Best Prcng Strategy n the Leader Game Substtutng the best response functon n () nto the utlty functon of AP n (9), we have our upper problem become c u AP (c) ( α ln c ) + s.t. same constrants n(9). () The tas of AP s to fnd the optmal prce c such that hs utlty n () s mzed w.r.t the constrants. Although the upper problem s stll not a convex optmzaton problem, a low-complex and effcent algorthm can be appled to fnd the optmal c. Proposton : The utlty functon u AP (c) s a pecewse affne, contnuous and decreasng functon n c. Proposton : The DD best response power allocaton functon P m ( (c))m and the best response SINR γm (c) are contnuous and nonncreasng n c. Proposton 3: The spectral radus ρ(dag{ (c)} DˆΦ T ) s a monotoncally nonncreasng functon n c. Algorthm Optmal Prce Search Step Solve the feasblty problem n (7). Chec f the solutons γ γ th, f so P PU P PU, P 0 and stop. Otherwse, go to Step. Step Calculate P o and F, set count 0, and obtan the searchng nterval [c mn,c ] through the Feasble Prce Interval Search Algorthm. Go to Step 3. Step 3 Wthn the nterval obtaned n Step : set c c c mn, and chec f c s feasble, namely ρ(dag{ (c)} DˆΦ T ) <, P ( (c)) < P and P PU(c) < P PU. Ifso,c c, otherwse c mn c. Go to Step 4. Step 4 If c c mn ɛ and count MaxIter, repeat Step 3. Otherwse, c c. Algorthm Feasble Prce Interval Search Intal Defne v m αm. O ln [v(),, v ( M) ] s a ascendng permutaton of {v m} M m wth repeated value dscarded and M M. I low, I up M. Step Chec f O(I low ) s nfeasble. If so, c mn 0, c O(I low ), return. Otherwse, go to Step. Step Che f O(I up) s nfeasble. If so, c mn O(I up), c and return. Otherwse, go to Step 3. Ilow +I Step 3 I curr up, and c O(I curr). Chec f c s feasble. If so, I up I curr, otherwse I low I curr. IfI low I up, go to Step 4. Otherwse repeat Step 3. Step 4 c mn O(I low ), c O(I up), return. Based upon the Proposton, and 3, we can have our followng searchng algorthm for optmal prce c. The bsecton method n Algorthm ensures the convergence and the propertes n the propostons guarantee that the lmt s the optmum c. The Feasble Prce Interval Search Algorthm s descrbed n Algorthm. V. SIMULATION RESULT To evaluate the proposed framewor, several experments are conducted and the effcency of the proposed optmal prcng algorthm s tested through prce-of-anarcy (PoA) [4]. In the smulaton, we set the pathloss factor γ 3 and bandwdth to be 5 MHz. Moreover, each PUs are unformly dstrbuted n a crcular ndoor envronment wth radus to be 5 meters, whereas each DD pars unformly locate n a concentrc rng that s meter away from the center. A. Performance Smulaton Here, we consder a upln networ wth 3 PUs and pars of DD transcevers. PUs transmt drectly to AP wth ndvdual power 098

4 IEEE Global Conference on Sgnal and Informaton Processng (GlobalSIP) Sum Rate (bps/hz) sum rate for PU (w TR & DD) sum rate for (w TR & DD) networ throughput (w TR & DD) networ throughput (wo TR & w DD) sum rate for (wo TR & w DD) sum rate for PU (wo TR & w DD) Sum Rate (bps/hz) sum rate for PU (w TR & DD) sum rate for (w TR & DD) networ throughput (w TR & DD) sum rate for PU (w TR & wo DD) sum rate for (w TR & wo DD) networ throughput (w TR & wo DD) algorthm could acheve almost the same result as the exhaustve search method when the number of DD pars s small and SNR s low. Even when the the number of DD pars and SNR ncrease, the POA value s upper bounded by.5. Hence we can conclude that the proposed optmal prcng algorthm reduce the computatonal complexty greatly wthout degradng much the optmal system performance. 0 P/σ (a) TR v.s. drect transmsson. POA P/σ 0 0 (a) equal weght P/σ (b) DD v.s. conventonal communcaton. Fg. : Sum rate Comparson 3 4 Number of Users 5 POA P/σ Fg. 3: Prce of Anarchy Number of Users (b) random weght. constrants p and a hgher QoS prorty. In the DD-scenaro uplns, DD senders drectly communcate to ther recevers wth a power constrant 0.p and a lower prorty. In the TR conventonal uplns, the DD senders transmt to AP wth a lower prorty, power constrant p and TR sgnature. The upln performance comparson between TR technque and drect transmsson under DD scenaro s shown n Fg. a. Whle Fg. b demonstrates the performance comparson between conventonal TR networ and TRbased DD networ. The DD networ adopts the proposed optmal prcng algorthm for power control. Indcated by the gap between sold curves and dashed curves n Fg. a, the rate performace for both PUs and s wth TR sgnature mproves a lot compared wth the case wthout TR sgnature. The reason s that TR sgnature naturally boosts the sgnal strength and suppresses the nterference due to ts spatal-temporal resonance. Moreover, n Fg. b the two PU curves denoted as overlap due to the same QoS prorty and constrants for PUs. However, as DD communcaton taes advantages of proxmty, by applyng DD mode to lower prorty users, they can acheve a much hgher sum rate wth a lower power consumpton as shown by the curves denoted as. Consequently, the overall system performance of the TR-based DD hybrd networ outperforms others. B. Prce of Anarchy The PoA s a measure of the neffcency of equlbra n game theory, whch s defned as the rato between the worst objectve functon value of an equlbrum of the game and that of an optmal outcome. To evaluate the effcency of the proposed optmal prcng algorthm, { we evaluate the PoA n terms of throughput as η Throughputexhaustve E }. Throughput optmalprce The results are shown n Fg. 3, and we can see that n both equal weght and random weght cases, the proposed optmal prcng 5 VI. CONCLUSIONS In ths wor, we propose a TR-based DD hybrd networ where PUs communcate through AP and DD users communcate drectly to ther partners. Thans to the sspatal-temporal resonances generated by the TR sgnal transmsson n a rch-scatterng envronment, ITI and IUI wll be allevated n ths sngle-carrer system. We desgn an effcent optmal prcng algorthm that wors to mprove the DD throughput and guarantee a QoS protecton for PUs. Smulatons valdate the performance mprovement of the proposed scheme. REFERENCES [] M. N. Tehran, M. Uysal, and H. Yanomeroglu, Devce-to-devce communcaton n 5g cellular networs: challenges, solutons, and future drectons, Communcatons Magazne, IEEE, vol. 5, no. 5, pp. 86 9, 04. [] A. Asad, Q. Wang, and V. Mancuso, A survey on devce-to-devce communcaton n cellular networs, 04. [3] K. Doppler, M. Rnne, C. Wjtng, C. B. Rbero, and K. Hugl, Devceto-devce communcaton as an underlay to lte-advanced networs, Communcatons Magazne, IEEE, vol. 47, no., pp. 4 49, 009. [4] J. 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5 05 IEEE Global Conference on Sgnal and Informaton Processng (GlobalSIP) [4] T. Roughgarden and E. Tardos, Introducton to the neffcency of equlbra,