ENERGY CONSERVING MODEL FOR THE CHANNEL POWER-GAIN IN WIRELESS MULTI-USER DOWNLINK SCENARIOS

Transcription

1 ENERGY CONSERVING MOEL FOR THE CHANNEL POWER-GAIN IN WIRELESS MULTI-USER OWNLIN SCENARIOS Michel T. Ivrlač and Joef A. Noek Intitute for Circuit Theory and Signal Proceing Techniche Univerität München {ivrlac, ABSTRACT Etablihed model for the prediction of channel power gain in radio communication ytem have got a light drawback which grow into a real problem in a theoretical large ytem analyi of downlink cenario, where the number of uer terminal i allowed to grow unboundedly. The problem i that the model prediction of channel power gain do not take into account the exitence of additional receiver. A the number of terminal grow unboundedly, the um of the predicted receive power increae unboundedly, and eventually become larger than the total tranmit power. In a paive propagation medium like air uch model error conflict with the principle of conervation of energy, when applied to aymptotic large ytem analyi. We propoe a imple generic modification which while applicable to any of the exitent channel power gain model correct thi model error and i intended a a firt tep toward the proviion of a channel power gain model that allow for a meaningful large ytem analyi. We preent a fully worked out example which, beide providing intereting inight, i alo intended to demontrate that mathematical tractability i not eriouly affected by the propoed modification. 1. INTROUCTION In radio communication, the channel power gain (CPG) ia function of the ditance between the tranmitting and the receiving antenna. In a free-pace cenario, for example, the CPG decreae with the quare of thi ditance [1]. In application, like network planning, it i obviou that model for the accurate prediction of CPG are required []. But even in a purely theoretical analyi thoe model are of coniderable interet, epecially in the context of the multi-uer downlink (broadcat channel). The reaon i that the uer pecific channel power gain impact performance of cheduling and ignal proceing algorithm [3, 4]. There are many CPG model publihed and in ue today. See, for intance, [5 10], and the reference therein. What all thee model have in common i that they actually aume a ingle-uer ytem, ince the predicted average channel gain do not depend on whether there exit other terminal. In practical multi-uer downlink cenario, uch kind of model error i of little importance, though. The number of uer i uually low enough uch that it can afely be aumed that the exitence of one uer receiving antenna ha negligible effect on the electro-magnetic field, and therefore, doe not change the receive power of another uer receive antenna in any noticeable way. However, in an aymptotic ytem analyi, where the number of terminal i allowed to grow unboundedly, thi behavior of the etablihed model turn into a real problem. That thi i o, can be mot clearly een from the fact that for a large enough number of terminal, the um of the predicted receive power exceed the tranmit power. Of coure, thi i impoible in a paive medium, like air. A a conequence, the conventional CPG model hould not be ued»out of the box«in an aymptotic ytem analyi becaue of the inherent conflict with the principle of conervation of energy. In thi paper, we propoe a imple, generic modification which can be applied to any CPG model making it compliant with the principle of conervation of energy. When the number of terminal i mall enough, uch that conflict with thi principle i avoided well already by the conventional model, the propoed modification will have only a negligible effect. The original model behavior for mall ytem i preerved while making it applicable for a large ytem analyi. After providing it definition and baic propertie, we demontrate it application to aymptotic ytem analyi of a multi-uer downlink with an opportunitic proportional fair cheduling. Beide providing intereting inight, thi example erve to demontrate that the mathematical tractability i not affected eriouly by the propoed modification.. ENERGY CONSERVING CHANNEL POWER GAIN MOEL In the following, we conider a multi-uer downlink cenario, where a baetation communicate with a number of (at leat two) uer. efinition 1 (Channel power-gain in a multi-uer downlink). The channel power-gain of a uer j of the multi-uer down /08/$5.00 c 008 IEEE Authoried licened ue limited to: T U MUENCHEN. ownloaded on April,010 at 09:41:49 UTC from IEEE plore. Retriction apply.

2 link i defined a the ratio j.d j / P R;j ; (1) P T of the average received power P R;j and the tranmit power P T. The channel power-gain eentially depend on the ditance d j between the terminal and the baetation. Conventional model for the prediction of the channel power gain eentially aume a ingle-uer context, with one tranmitter and one receiver. When they are neverthele applied to a multi-uer context, we ue the term ingle-uer channel power gain to tell it apart from the true channel power gain. efinition (Single-uer channel power gain). We define the ingle-uer channel power gain of uer j a: j.d j / P R;j () P T ˇ Only uerj exit. Becaue the ingle uer channel power gain may conflict with the principle of conervation of energy, we propoe the imple, generic modification: efinition 3 (Energy conerving channel power gain). Let u denote with j.d j /,where0< j.d j /<1,theingle-uer channelpower gainof uerj in a multi-uer downlink, where d j, with d j >0, i the ditance between the tranmitter (the baetation) and the receiver of uer j.the j.d j / are computed according to any tandard channel power gain model. Furthermore, let all d j be pairwie different, i.e., d j d i j. The energy conerving channel power gain i defined by: j.d j / j.d j / Y 1 i.d i / : (3) The idea behind thi definition can eaily be undertood from the following Theorem. Theorem 1. The energy conerving channel power gain from efinition 3 can equivalently be written a: j.d j / Proof. See Appendix A. 1 i.d i / j.d j /: (4) We can oberve from (4) that the part of the tranmit power which i»eaten up«by receiver which are poitioned cloer to the baetation cannot be conumed anymore by receiver which happen to be located farther away. Theorem (Energy conervation). For any d j < 1, it hold true that: i.d i / 1; (5) where equality hold only if the number of uer i not finite. Proof. See Appendix B. Thi how that in no circumtance the um of the received power of all finitely or infinitely many terminal can exceed the tranmit power, which make the model (3) obey the principle of conervation of energy, even when applied within an aymptotic large ytem analyi. 3. APPLICATION EAMPLE: PROPORTIONAL FAIR OPPORTUNISTIC SCHEULING Let u have a look at how the propoed channel power gain model can be applied in an aymptotic large ytem analyi of a multi-uer downlink ytem. Suppoe there are a number of equal terminal located in a finite pace with d 1 <d <d 3 < <d ; (6) located far enough from the tranmitter uch that d =d 1 1. The ingle-uer channel power gain i therefore almot the ame for each terminal. Applying the propoed model from (3) lead to the channel power gain: for j f1;;:::;g. Notethat j 1 j 1 ; (7) 1 j 1; and 8 <1W j <1; (8) independent of. Therefore, a the number of terminal approache infinity, the um of all received power approache the tranmit power from below. On top of thi average channel power gain let there be an independent and identically ditributed random fading power gain j for each terminal. Take note that the all too common choice of an exponential probability denity function (pdf) for the j, i.e., Rayleigh Fading, i not an option in thi context. Becaue it allow for arbitrarily large fading power gain, it ak for trouble with the principle of conervation of energy. Since trouble with thi principle i exactly what we wanted to avoid in the firt place, the pdf of the j ha to have a finite upport. In thi paper, we chooe the implet pdf with finite upport, the uniform ditribution: ( 1 for 0 x 1;.x/ pdf.x/ (9) 0 ele. The tranmitter decide to communicate with only one terminal at a time. Proportional fair cheduling [11] elect the terminal j among the terminal which momentarily ha the larget fading power gain: j arg max j f1;;:::g j : (10) 008 International ITG Workhop on Smart Antenna (WSA 008) 11 Authoried licened ue limited to: T U MUENCHEN. ownloaded on April,010 at 09:41:49 UTC from IEEE plore. Retriction apply.

3 Becaue of the»proportional fair«nature of the cheduling, each of the terminal ha the ame chance to be cheduled at a given time. Hence, the average channel power gain of the cheduled terminal i given by ched E j ; (11) where EŒ: i the expectation operation, and 1 j 1 1 j 1 1 : (1) Note that! 0, a!1, ince more and more terminal compete for a finite tranmit power reource. Becaue 1, it alo follow that ched! 0, a!1.in thi way, it i not good for the average cheduled power gain to have too many terminal around.»too much competition kill the port«, o to peak. Lemma 1. The average channel power-gain of a cheduled uer i given by: Proof. See Appendix C. ched./ 1 1 : (13) C 1 The optimum number opt of uer can be defined a: opt arg max >1 ched./ ; (14) ince thi number of uer maximie throughput. Theorem 3. The optimum number opt of uer ha the following aymptotic behavior: opt! Proof. See Appendix. a! 0: (15) Setting 10 6, the maximum throughput i achieved with approximately 1: terminal. Thi particular number of terminal enure the bet compromie between a high multiuer diverity (large) and not too much competition for the finite tranmit power reource (mall ). For 10 1,the optimum number of terminal i more than a million already. 4. CONCLUSION In the aymptotic analyi of the downlink of multi-uer radio communication ytem, where the number of uer terminal i allowed to grow unboundedly, conventional model for the calculation of channel power gain cannot be applied directly, ince they are in conflict with the principle of conervation of energy. We propoe a generic modification applicable to any conventional channel power gain model which aure compliance with the principle of conervation of energy. With an exemplary, fully worked out aymptotic ytem analyi, it i demontrated that there i an optimum compromie between multi-uer diverity on the one hand, and multi-uer competition for finite tranmit power reource, on the other hand. It hould be pointed out here that beide the dicued channel power gain model conventional model for hadowing and mall-cale fading (like log-normal hadowing or the popular Rayleigh fading) have to be adapted properly, a they, too, can conflict with the principle of conervation of energy. 5. REFERENCES [1] J.. Paron, The Mobile Radio Propagation Channel, John Wiley & Son Ltd., 000. [] Cotare Ltd., Cambridge Broadband Ltd., and Cambridge Univerity, A Study on Efficient imenioning of Broadband Wirele Acce Network, Tech. Rep., Ofcom, U, 003. [3] P. Gupta and P.R. umar, The Capacity of Wirele Network, IEEE Tranaction on Information Theory, vol. 46, pp , March 000. [4] T. ElBatt and A. Ephremide, Joint Scheduling and Power Control for Wirele ad hoc Network, IEEE Tranaction on Wirele Communication, vol.3,no. 1, pp , January 004. [5] COST Action 31, igital Mobile Radio toward Future Generation Sytem, Final Report, Tech. Rep., European Communitie,EUR18957,1999. [6] Y. Okumura, Field Strength and it Variability in VHF and UHF Land-Mobile Radio-Service, Review of the Electrical Communication Laboratory, vol. 16, September-October [7] M. Hatta, Empirical Formula for Propagation Lo in Land Mobile Radio Service, IEEE Tranaction on Vehicular Technology,, no. VT-9, pp , September [8] Recommendation ITU-R P.1546, Method for Pointto-Area Prediction for Terretrial Service in the Frequency Range 30 MH to 3000 MH, Tech. Rep., International Telecommunication Union, International ITG Workhop on Smart Antenna (WSA 008) Authoried licened ue limited to: T U MUENCHEN. ownloaded on April,010 at 09:41:49 UTC from IEEE plore. Retriction apply.

4 [9] V. Erceg, L.J. Greentein, et al., An Empirically Baed Path Lo Model for Wirele Channel in Suburban Environment, IEEE Journal on Selected Area in Communication,vol.17, pp , July C. PROOF OF LEMMA 1 The probability denity function of j can be found from it cumulative ditribution function (cdf ): [10] Electronic Communication Committee (ECC) within the European Conference of Potal and Telecommunication Adminitration (CEPT), The Analyi of the Coexitence of FWA Cell in the GH Band, Tech. Rep., ECC, vol. 16, no. 33, May 003. [11] P. Viwanath,.N.C. Te, and R. Laroia, Opportunitic Beamforming uing umb Antenna, IEEE Tran. Information Theory,vol.48, pp , June 00. A. PROOF OF THEOREM 1 cdf j.x/ PrΠj <x Z x Z x Y pdf. j / d j ; from which follow immediately: 1 pdf. 1 / cdf. 1 / d1 ; (18) Without lo of generality, let d 1 <d < <d j.wehave to prove that i.d i / j 1 1 i1 Y k1 1 k.d k / jy 1 1 i.d i / ; (16) for j N nf1g. Obviouly, for j, the equality hold. We jut have to how that equality till hold for j j C 1, provided that it hold for j. 1 j i1 Y i.d i / 1 k.d k / j 1 k1 j 1 i1 Y 1 i.d i / 1 k.d k / jy 1 j.d j / 1 i.d i / k1 Y 1 i.d i / 1.x/ pdf.x/ cdf.x/ : (19) From (9)and(19) we then obtain:.x/ ( x 1 for 0 x 1; 0 ele. (0) The average fading power-gain of the cheduled uer j then compute to: Z 1 x0 x dx; (1) C 1 : () With (11) and (1) the claim of the lemma follow. jy 1 i.d i / : B. PROOF OF THEOREM. PROOF OF THEOREM 3 The optimiation problem (14) can be olved by treating the number of terminal a a real-valued variable, and by computing the firt derivative of ched./ with repect to : From (4) it follow that i.d i / 1 j.d j / j.d j / 1: (17) 0 Take note that j.d j /= j.d j / 0 i a direct conequence of 0< j.d j /<1(by definition) and the relationhip from (3) which make j.d j / 0, where equality only hold for infinitely many uer. F.; def 1 C 1 1 C 1 C log e 1 C 1 : (3) While olving for it root in cloed form appear to be difficult, we can neverthele find the olution for mall value of in the following way. Let u firt obtain a Taylor erie 008 International ITG Workhop on Smart Antenna (WSA 008) 13 Authoried licened ue limited to: T U MUENCHEN. ownloaded on April,010 at 09:41:49 UTC from IEEE plore. Retriction apply.

5 expanion of F.; / around the point.0; /: F.; / F.0; / C ˇˇˇˇˇ ˇ 0 F ˇ 0 (4) which compute to C 4 6 C 3 C : (5) 1 C The root of (5) with repect to ha to equal, inorderto make the Taylor expanion (4) centered around it root: C C 4 C : (6) Solving (6) for we find 1; 1 C : (7) Since we are looking for a poitive olution, it follow: 1 C C : (8) Since ched./ i continuou in, and the Taylor erie expanion i centered around it root with repect to, itfollow that opt! a! 0: (9) From thi and (8) then follow the claim of the Theorem: opt! a! 0: (30) International ITG Workhop on Smart Antenna (WSA 008) Authoried licened ue limited to: T U MUENCHEN. ownloaded on April,010 at 09:41:49 UTC from IEEE plore. Retriction apply.