CROSSFIRE. FP7 Contract Number:

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1 CROSSFIRE Uncoordnated network strateges for enhanced nterference, moblty, rado resource, and energy savng management n LTE-Advanced networks FP7 Contract Number: 376 WP Interference-Aware LTE-A Heterogeneous Wreless Networkng D. Interference Modelng and Performance Assessment Contractual Date of Delvery: February 5 Actual Date of Delvery: February 5 Responsble Benefcary: Contrbutng Benefcares: Securty: Nature Verson: CNRS VODAFONE, STEIN Publc Report I PROPRIETARY RIGHTS STATEMENT Ths document contans nformaton, whch s propretary to the CROSSFIRE Consortum. Nether ths document nor the nformaton contaned heren shall be used, duplcated or communcated by any means to any thrd party, n whole or n parts, except wth pror consent of the CROSSFIRE consortum.

2 Document Informaton Verson Date: February 5 Total Number of Pages: 66 Authors Name Organsaton Emal Marco D Renzo (edtor) CNRS marco.drenzo@lss.supelec.fr Peng Guan CNRS peng.guan@lss.supelec.fr We Lu CNRS we.lu@lss.supelec.fr Hsham Elshaer VODAFONE hsham.elshaer@vodafone.com Nestor Hernandez STEIN nestor@stenwurf.com Document Hstory Revson Date Modfcaton Contact Person. Jan. 5 ToC released Marco D Renzo. Feb. 5 Fnal verson released Marco D Renzo Securty: Publc Page

3 . Executve Summary Ths delverable s amed at provdng a comprehensve modelng and performance assessment of heterogeneous cellular networks for applcaton to the LTE-A standard. Ths delverable n dvded n three man parts. Part I encompasses Sectons -4 and t s of methodologcal nature. It s amed at developng new methodologes for modelng, analyzng and optmzng heterogeneous cellular networks. To ths end, the mathematcal tool of stochastc geometry s used. Ths part summarzes the research conducted by CNRS researchers, whch was publshed n several journal papers. In partcular, dfferent methodologes are proposed for the analyss of the coverage probablty, the average rate and the error performance of heterogeneous cellular networks, whch use advanced transmsson schemes, ncludng multple-antenna transmsson, relay-aded communcaton and consder uplnk and downlnk scenaros. The proposed frameworks are shown to be hghly accurate and sutable for system-level optmzaton. Based on them, several fundamental performance trends of heterogeneous cellular networks are dentfed. Part II encompasses Secton 5 and t s amed at proposng a practcal scheme for enhancng the throughput of heterogeneous cellular networks by ntroducng the dea of dual connectvty,.e., the possblty of allowng dfferent base statons to serve the moble termnals n the downlnk and n uplnk. Ths part s based on a patented technology from VODAFONE, whch s called DUDE. In partcular, t s shown that DUDE s capable of provdng tenfold gans n the achevable throughput compared to the state-of-the-art LTE-A standard. It s consdered, as a consequence, a sutable canddate for beng consdered n the next releases of the 5G standard. Part III encompasses Secton 6 and t s amed at mprovng relablty, throughput and energy effcency of the LTE-A standard by captalzng on the prncple of devce-to-devce communcatons. The proposed protocols are based on the concept of network codng and have been developed and mplemented n the Stenwurf Kodo lbrary and n the ns-3 smulator. The developed code and algorthms are avalable for free for academc/research use and can be downloaded from the Stenwurf webste wth a comprehensve tutoral gude. Ths s a fundamental step forward n order to stmulate and harmonze research n ths feld, whch could eventually lead to the ntegraton of network codng enhanced devce-to-devce technology nto the next releases of the 5G standard. Securty: Publc Page 3

4 Table of Contents. Executve Summary Performance Analyss of Downlnk Heterogeneous Cellular Networks Stochastc geometry modelng and analyss of the error probablty of sngleter SISO cellular networks: the Gl-Peleaz nverson approach Introducton System Model Characterstc Functon of Interference Error Performance Numercal Results Concluson Stochastc geometry modelng and analyss of the coverage probablty and average rate of sngle-ter SISO cellular networks: the Gl-Peleaz nverson approach 8.. Introducton System Model and Problem Formulaton Coverage Probablty and Average Rate Gamma Dstrbuted Per-Lnk Power Gans Numercal Results Concluson Extenson of the Gl-Peleaz nverson approach to sngle-ter MIMO cellular networks Introducton System Model Characterstc Functon of Interference Error Performance Numercal Results Concluson Stochastc geometry modelng and analyss of the error probablty of sngleter SISO cellular networks: the Equvalent-n-Dstrbuton (ED) based approach Introducton System model Characterstc functon of the nterference The ED-Based Approach... 4 Securty: Publc Page 4

5 .4.5 Performance Analyss Numercal Results Extenson of the Equvalent-n-Dstrbuton (ED) based approach to sngleter MIMO cellular networks Introducton System model Problem Formulaton: Prelmnares Man Results Applcaton to MIMO Cellular Networks Performance Analyss of Uplnk Heterogeneous Cellular Networks Introducton Stochastc geometry modelng and analyss of the coverage probablty and average rate of sngle-ter uplnk cellular networks System Model and Problem Formulaton Gl-Pelaez Based Mathematcal Modelng Extenson to mult-antenna uplnk cellular networks wth MRC recever System Model Gl-Pelaez Based Mathematcal Modelng Extenson to mult-ter heterogeneous uplnk cellular networks System Model and Problem Formulaton Gl-Pelaez Based Mathematcal Modelng Large Scale Receve Antenna System Numercal Results Performance Evaluaton of Relay-Aded Downlnk Cellular Networks Performance evaluaton of cellular networks wth fxed relays Introducton System Model Network Interference Model Problem Statement Dversty Order n Wreless Networks wth Nose and Interference End-to-end error probablty of dual-hop cooperatve relayng usng a maxmum rato combnng recever End-to-end error probablty of dual-hop cooperatve relayng usng a selecton combnng recever Numercal and Smulaton Results Securty: Publc Page 5

6 4. Performance evaluaton of cellular networks wth randomly dstrbuted relays 4.. Introducton System Model Assocaton Polces Smulaton Results Decoupled Uplnk and Downlnk Access n Future Cellular Networks (DUDe) Introducton Motvaton Toy example showng the concept Evaluaton of the basc concept of DUDe System model Results Load and backhaul aware decoupled access (DUDe.) Motvaton System model Cell assocaton algorthm Results RAN Archtecture consderatons for DUDe Underlay Devce-to-Devce Communcatons for LTE-A Wreless Networks Introducton RLNC transmsson scenaros and schemes analyss Network and System Model Transmsson Scenaros Dstrbuton Prelmnares Sngle Feld Schemes Multple Felds Schemes Performance Metrcs Comparson Between Cooperatve and Non-Cooperatve Schemes RLNC Models for the ns-3 Smulator and the Stenwurf Kodo Lbrary Conclusons References Securty: Publc Page 6

7 . Performance Analyss of Downlnk Heterogeneous Cellular Networks. Stochastc geometry modelng and analyss of the error probablty of sngle-ter SISO cellular networks: the Gl-Peleaz nverson approach In ths secton, we ntroduce a mathematcal framework for computng the average error probablty of downlnk cellular networks n the presence of other cell nterference, arbtrary fadng, and thermal nose. A stochastc geometry based abstracton model for the locatons of the Base Statons (BSs) s used, hence the BSs are modeled as ponts of a homogeneous spatal Posson Pont Process (PPP). The Moble Termnal (MT) s assumed to be served by the BS that s closest to t. The techncal contrbuton s twofold: ) we provde an exact closed form expresson of the Characterstc Functon (CF) of the aggregate other cell nterference at the MT, whch takes nto account the shortest dstance based cell assocaton mechansm; and ) by relyng on the Gl Pelaez nverson theorem, we provde an exact closed form expresson of the Average Parwse Error Probablty (APEP), whch accounts for fadng and for the spatal dstrbuton of the BSs. From the APEP, the Average Symbol Error Probablty (ASEP) s obtaned by usng the Nearest Neghbor (NN) approxmaton, whch s shown to provde accurate estmates. Fnally, the mathematcal framework s substantated through extensve Monte Carlo smulatons and nsghts on the achevable performance are dscussed. More detals ncludng mathematcal proofs, bounds and approxmatons can be found n [.7]... Introducton The mathematcal modelng of cellular networks s usually conducted through abstracton models, whch rely upon smplfed spatal models for the locatons of the Base Statons (BSs). Common approaches nclude the Wyner model, the sngle-cell nterferng model and the hexagonal grd model [.][.3]. However, these abstracton models are ether naccurate for many operatng condtons or they stll requre extensve numercal computatons. As a result, the analyss and desgn of cellular networks s often conducted by resortng to network smulatons for selected scenaros, whch represent specfc arrangements of BSs. To crcumvent these problems, a new abstracton model for the mathematcal analyss of cellular networks s emergng, whch s referred to as Posson Pont Process (PPP)-based abstracton[.][.3][.9][.37]. Wth the ad of ths abstracton model, the locatons of the BSs are modeled as ponts of a homogeneous PPP. Recent results have confrmed that the PPP-based abstracton model s capable of accurately reproducng the man structural characterstcs of operatonal cellular networks [.38]. The usefulness of the PPP-based Securty: Publc Page 7

8 abstracton model orgnates from ts analytcal tractablty and from the possblty of leveragng mathematcal tools of appled probablty, such as stochastc geometry, for mathematcal performance analyss [.47][.49]. Owng to ts mathematcal tractablty, the PPP-based abstracton model s now routnely used to the analyss and desgn of wreless networks n general and cellular networks n partcular. Notable examples nclude[.][.3], [.4]-[.7]. For a comprehensve lterature survey, the nterested reader s referred to [.3][.]. More specfcally, n [.]the coverage probablty and the average rate of cellular networks are computed n closed-form for transmsson over Raylegh fadng channels. In [.4], the framework of [.]s extended to heterogeneous cellular networks, whch are modeled as the superposton of many PPPs. In [.5], the PPP-based abstracton model s exploted to study heterogeneous cellular networks wth a based cell assocaton mechansm. In [.6], the authors ncorporate the load characterstcs of the BSs nto the mathematcal framework, by usng a condtonally thnnng approach. In [.7], a mathematcal framework to the analyss of uplnk of cellular networks s ntroduced. In [.3], the authors ntroduce a framework for computng the average rate of heterogeneous cellular networks wth based cell assocaton and for transmsson over general fadng channels. The papers mentoned above focus ther attenton on the computaton of the coverage/outage probablty and of the average rate of a typcal Moble Termnal (MT). On the other hand, to the best of the authors knowledge, there are no mathematcal frameworks to the analyss of the average error probablty of PPP-based cellular networks. Indeed, the error probablty and the outage probablty n the presence of a Posson feld of nterferers have been studed n the lterature [.8]-[.]. For example, n [.] a comprehensve framework s ntroduced for computng the error probablty of a mult-antenna recever n the presence of dfferent models of network nterference. By consderng a smlar nterference model, the outage probablty s studed n [.9]. In [.], the effect of spatal nterference correlaton on the performance of maxmum rato combnng s nvestgated. These frameworks, however, are not applcable to cellular networks, snce the BS-to-MT cell assocaton s neglected. The nterferers are, n fact, assumed to be arbtrarly close to the typcal MT, even closer than the servng BS. Thus, they are applcable to, e.g., ad hoc, cogntve and underlay devce-to-devce wreless networks, where the dstance from the transmtter to the recever s fxed and the nterferers can be closer to the recever than the ntended transmtter. Securty: Publc Page 8

9 Motvated by these consderatons, n ths secton we ntroduce a mathematcal framework for computng the error probablty of SISO cellular networks, by explctly takng nto account the cell assocaton mechansm based on the shortest BS-to-MT dstance. The framework s applcable to cellular networks where the locatons of the BSs are modeled accordng to a homogeneous PPP, the downlnk channels experence ndependent and dentcally dstrbuted fadng.the mathematcal approach s applcable to arbtrary fadng dstrbutons on both useful and nterferng lnks. The proposed frameworks are useful for better understandng and for smplfyng the analyss of cellular networks, snce they do not requre the explct generaton and smulaton of the BSs locatons, by usng Monte Carlo smulatons... System Model In cellular downlnk, a probe MT s located at the orgn of a b-dmensonal plane and BSs are modeled as ponts of a homogeneous PPP ( ) of densty. The dstance from the th BS to the MT s denoted by r for. The MT s assumed to be tagged to the nearest BS. The servng BS s denoted by BS, and ts dstance from the MT s denoted by r, whch s a RV wth PDF fr exp [.]. The set of nterferng BSs \ BS s stll a homogeneous PPP [.9] whch s denoted by \. \ has densty p, where \ p s the actvty factor denotng the probablty that a nterferng BS transmts n the same frequency band as BS. The setup wth p corresponds to the full frequency reuse case [.], [.3]. In the depcted downlnk SISO cellular network model, the sgnal receved at the MT s as follows: b b y Er hs E \ r hs n x agg ( r ) (..) where x s the useful sgnal transmtted by BS, agg (r ) s the aggregate other-cell nterference and n s the Addtve Whte Gaussan Nose (AWGN) wth nose powern. More specfcally: E s the BSs transmt-energy per transmsson; s s the nformaton symbol emtted by BS, wheres M and the set M has M modulated symbols denoted by μ χ C;h s the channel matrx of the BS -to-mt lnk; b > denotes the ampltude path-loss exponent. A smlar notaton s adopted for all the nterferng channels of agg ( ).The fadng envelopes h and h \ for are assumed to be..d. and to follow a generc dstrbuton wth Securty: Publc Page 9

10 h h E E. As an example, Gamma [.4]and composte Gamma/Log-Normal [.4] fadng channels are explctly analyzed. At the MT, an nterference-unaware Maxmum-Lkelhood (ML)-optmum demodulator s consdered. It s assumed to have perfect Channel State Informaton (CSI) of the BS -to-mt lnk, whle gnorng the other-cell nterference. The decson metrc [.4]: where b b Re * * can be formulated as s r E h r E v r h (..) s s, \ r E r h s agg and b agg v r r n. In ths secton, we provde a mathematcal framework for computng the Average Symbol Error Probablty (ASEP) of the demodulator n (..). The ASEP s defned as the symbol error probablty averaged over the dstrbuton of the fadng channels and of the spatal locatons of the BSs. The proposed approach for computng the ASEP conssts of four steps: ) frst, the statstcal dstrbuton of condtoned upon r s derved n closed-form; agg s studed and a closed-form expresson of ts CF ) then, the Parwse Error Probablty (PEP) condtoned upon the fadng envelope ( h ) and the transmsson dstance ( r ) of BS s computed; 3) subsequently, the Average PEP (APEP) s obtaned by removng the condtonng upon h and r ; 4) fnally, the ASEP s computed from the APEP by usng the NN approxmaton...3 Characterstc Functon of Interference In ths secton, the CF of fadng models. agg condtoned upon r s computed n closed-form for arbtrary Theorem.. r E r h s. By condtonng upon r, ts CF s as follows: Let \ agg b Securty: Publc Page

11 agg where T x p x p and: where p Fq a,, ap; b,, bq ; agg b ; r ; r exp r T r E ω ω ω (..3) M q q q q / b q q q E b x x h M 4 q! / M x E F ;, ; h M b b 4 h s the generalzed hypergeometrc functon. (..4) Theorem.. s general, as t s applcable to arbtrary fadng dstrbutons. However, the expectaton over the fadng square envelope h of the generc nterferer channel needs to be computed n (..4). As an example, Gamma [.4]and composte Gamma/Log-Normal [.4] fadng channels are explctly analyzed. Gamma Channels Let g h follow a Gamma dstrbuton wth parameters, g ~ Gamma m, [.4]. Then, m, whch we denote as n (..4) has closed-form expresson as follows: x M F, m;, ; x b b 4 m M (..5) Gamma/LogNormal Channels Let g h follow a Gamma dstrbuton by condtonng on ts mean power, whch follows a g m [.4]. Log-Normal dstrbuton. We denote ths dstrbuton as ~ Gamma/LogN,, Then, n (..4) has closed-form expresson as follows: M NGHQ k x wk F, m;, ; x M k b b 4 m where k /, and w k and k for k the Gauss-Hermte quadrature rule[.35], respectvely. GHQ (..6) k,,, N are weghts and abscssas of Securty: Publc Page

12 Arbtrary Channels Closed-form expressons of the CF of agg can be computed for a general class of fadng dstrbutons wth the ad of the Mejer G-functon. For a wde class of fadng models, the PDF of G mn, pq, g h can be formulated as fg a b p q a mn, / p C G p, q D [.36], where s the Mejer G-functon. Wth the ad of [.]the expectaton n (..4) and can be re-wrtten as follows: E g x F ;, ; g b b 4, / b C x a mn, / p G,3 G pq, d b D 4 / b b q whch can be solved n closed-form wth the ad of the Melln-Barnes theorem n[.]. b q (..7)..4 Error Performance APEP By defnton, the APEP s the probablty that the actual transmtted symbol s s decoded as ŝ s s, by assumng that s and s are the only two symbols of the constellaton dagram. From (..), ths occurs when s s s s. Snce s s (..), the APEP reduces to the computaton of s s by computng the APEP condtoned upon h and Hence, the APEP s APEP E PEP, ; h, r h r formulated as summarzed n the followng theorem. from APEP Pr. We start r, whch s denoted by PEP ; h, r.. From (..), the APEP can be Theorem.. SNR E / N Let the demodulator n (..5). Let. The APEP can be formulated as s N NI APEP Pr APEP APEP, where: Securty: Publc Page

13 b x APEPN b x e SNR Q N x;,, bdx SNR b y 4 SNR yt x APEPNI e e e Q NI x; dxdy x b xy SNR E Q ;,, exp 4 h Q x; E sn / h b N x b h SNR h x NI h x / (..8) (..9) Theorem.. provdes an exact mathematcal formulaton of the APEP, whch s applcable to arbtrary dstrbutons of the fadng envelope h. The expectatons n (..9) can be computed n closed-form for varous fadng models. As examples, closed-form expressons of N and Q ;, Q ;,, NI are provded for Gamma and Gamma/Log-Normal channels. Gamma Channels Let g h m ~ Gamma,. Let expectatons: K and K two non-negatve constants. Let the m / m m m I K E g g exp K g m m K m 3 K KF m ;, 4m I K E g sn K g m m m b N x b x Then, Q ;,, I / 4 SNR SNR and Q x; / (..) / NI I x. Gamma/LogNormal Channels Let g h ~ Gamma/LogN m,,. Let Let the expectatons: K and K two non-negatve constants. Securty: Publc Page 3

14 E g exp g I K K m m g m m NGHQ m m w k k k m K k I K E g sn K g m NGHQ k 3 kk K w k F m ;, k m m 4m b N x b x Then, Q ;,, I / 4 SNR SNR and Q x; / (..) / NI I x. Asymptotc Analyss We then provde smplfed mathematcal frameworks n two lmtng operatng condtons: ) nose-lmted,.e., SNR, and ) nterference-lmted,.e., SNR, cellular networks, where SNR E / N denotes the Sgnal-to-Nose-Rato (SNR). As dscussed n[.3], typcal cellular networks operate n the nterference-lmted regme. Corollary.. Let a nose-lmted cellular network. Then, SNR N N lm APEP APEP APEP Then, lm APEP APEP NI SNR NI NI APEP lm APEP : SNR. Let an nterference-lmted network. NI / x T x T x Q x;, b dx (..) ASEP From the APEP ntheorem.., the ASEP can be obtaned by usng the NN approxmaton [.3]. In ths secton, we propose the NN approxmaton for computng the ASEP from the APEP because of ts smplcty and accuracy. The advantage s, n fact, that the ASEP s obtaned by computng a sngle APEP for every symbol ( for,,, M ) of the constellaton dagram. By assumng equprobable transmtted symbols, the NN approxmaton of the ASEP can be formulated as follows: Securty: Publc Page 4

15 M M N (..3) ASEP / APEP mn where: ) mn,, mn mn M M s the mnmum Eucldean dstance among all pars, of the constellaton dagram for M and ) N s the number of nearest neghbors of,.e., the number of ponts of the constellaton dagram mn whose Eucldean dstance from s equal to. mn For some constellaton dagrams, we may have mn, whch s ndependent of mn. In ths case, (..3) reduces to N avg mn M N mn avg APEP mn ASEP, where N mn / M s the average number of nearest neghbors of the constellaton mn sn / M wth dagram. For example, mn avg N f M and mn avg N f M for PSK modulaton. If the standard square QAM wth M 6 s consdered, we have mn / and mn avg N. Smlar results apply for dfferent values of M. 3 Trends and Insghts Let us consder a SISO cellular network operatng n the nose-lmted regme. It follows that APEP APEPN. Thus, the followng conclusons can be drawn. The APEP decreases by ncreasng the SNR. The dversty order, D, depends on N N Q Q ; SNR. The APEP decreases by ncreasng the BSs densty. The APEP ncreases by ncreasng the shadowng standard devaton. The APEP ncreases by ncreasng the path-loss exponent. Let us consder a SISO cellular network operatng n the nterference-lmted regme. It follows that APEP APEP NI. Thus, the followng conclusons can be drawn. The APEP s ndependent of the SNR. The APEP s ndependent of the BSs densty. We expect that the APEP decreases by ncreasng m. Securty: Publc Page 5

16 We expect that the APEP degrades by ncreasng.however, t s very dffcult to study the monotoncty of the hypergeometrc functons as a functon of ther many parameters The APEP decreases by ncreasng the path-loss exponent. The APEP decreases by decreasng the actvty factor. All the trends nferred from mathematcal frameworks can be supported by Monte Carlo Smulatons, whch are shown n the next secton...5 Numercal Results We valdate n ths secton the mathematcal frameworks aganst Monte Carlo smulatons, whch are obtaned by usng the procedure descrbed n [.3], and study the mpact of dfferent parameters. Before that, we frst valdate the PPP-based abstracton model by comparng t wth modelng the BS locatons va grd-based model. A smlar comparson s avalable n [.]for the coverage probablty. We provde n ths secton the comparson for ASEP. Fg... shows that the PPP-based abstracton model ( sold lnes ) provdes a worst estmate of the error probablty compared to the grd-based abstracton model ( dash curves ), snce nterferng BSs may be arbtrarly close to each other. On the other hand, the PPP-based abstracton model provdes tractablty and performance trends as a functon of system parameters can be nferred from the resultng mathematcal frameworks. We observe a good accuracy of the obtaned frameworks (depcted n curves wth colored markers) compared wth the Monte Carlo smulatons (depcted n black curves). Fgs all support the nferred trends and nsghts from mathematcal frameworks. More specally, Fg... confrms that ASEP decreases by decreasng the actvty factor. Fg... shows that by ncreasng the path-loss exponent, ASEP decreases n the nterference-lmted regme whle ncreases n the nose-lmted regme. Fg..3.3 confrms that the BS denstyaffects the ASEP only n the nose-lmted regme. Securty: Publc Page 6

17 - - ASEP -3-4 p= p= - p= - p= -3 p= E/N [db] Fgure.. ASEP of a SISO system aganst the transmt SNR. Setup: 6QAM, 5 Gamma/Log-Normal channel model wth 6dB, m, and b 3. Fgure.. ASEP of a Nt Nr system aganst the transmt SNR E / N. Setup: (a) 5 QAM wth M 6, Gamma/Log-Normal channel model wth 6dB, b 3, and p 3. (b) QAM wth 6 M, Gamma/Log-Normal channel model wth m, b 3, 5 and p 3. Securty: Publc Page 7

18 Fgure..3 ASEP of a SISO system aganst the transmt SNR E / N. Setup: (a) QAM wth M 6, Gamma/Log-Normal channel model wth m and 6dB, p 3. (b) QAM wth 6 5 and M, Gamma/Log-Normal channel model wth m and 6dB, b 3 and p Concluson In ths secton, we have proposed a new mathematcal framework for computng the average error probablty of downlnk cellular networks by relyng on a PPP based abstracton model for the locatons of the BSs. A new closed form expresson of the CF of the aggregate other cell nterference has been proposed, and an easy to compute ntegral expresson of the ASEP has been provded. The mathematcal framework s applcable for general fadng models and has been substantated wth the ad of Monte Carlo smulatons and a good accuracy has been observed. From the mathematcal framework, varous performance trends have been dentfed, whch have been confrmed by Monte Carlo smulatons.. Stochastc geometry modelng and analyss of the coverage probablty and average rate of sngle-ter SISO cellular networks: the Gl-Peleaz nverson approach In ths secton, we ntroduce new mathematcal frameworks to the computaton of coverage probablty and average rate of cellular networks, by relyng on a stochastc geometry Securty: Publc Page 8

19 abstracton modelng approach. Wth the ad of the Gl-Pelaez nverson formula, we prove that coverage and rate can be compactly formulated as a twofold ntegral for arbtrary per-lnk power gans. In the nterference-lmted regme, sngle-ntegral expressons are obtaned. As a case study, Gamma-dstrbuted per-lnk power gans are nvestgated further, and approxmated closed-form expressons for coverage and rate n the nterference- lmted regme are obtaned, whch shed lght on the mpact of channel parameters and physcallayer transmsson schemes. More detals ncludng mathematcal proofs can be found n [.8]... Introducton Recently, dfferent technques to the mathematcal modelng and performance evaluaton of cellular networks based on stochastc geometry have been reported[.]. To the best of the authors knowledge, several technques are commonly used to the computaton of mportant performance metrcs, whch nclude coverage probablty, average rate and error probablty. They offer a dfferent trade-off n terms of modelng accuracy, mathematcal tractablty, numercal complexty, etc. [.]. These technques can be classfed as based on: ) the Raylegh fadng assumpton [.], [.]; ) the domnant or nearest nterferers approxmaton [.]; 3) approxmatons of the dstrbuton of the other-cell nterference [.]; 4) the Plancherel-Parseval theorem that s applcable to arbtrary fadng for the desred lnk [.], [.]; 5) numercally nvertng the Moment Generatng Functon (MGF) of the other-cell nterference; 6) MGF-based equvalent representatons of the performance metrcs of nterest[.3]; 7) equvalent n dstrbuton representatons of the other-cell nterference[.]; and 8) the drect computaton of spatal averages wthout usng the MGF [.3]. In ths secton, we ntroduce another technque to the computaton of coverage and rate of cellular networks. The proposed approach s based on the Gl-Pelaez nverson formula [.4]. The applcaton of the Gl-Pelaez theorem to the analyss of wreless networks n the presence of nterference s not new and varous papers are avalable, e.g., [.5], [.6]. These mathematcal frameworks, however, are not based on a stochastc geometry abstracton modelng. In ths context, to the best of the author s knowledge, the Gl-Pelaez theorem has been employed only n [.7] and [.8], where the error probablty of cogntve rado and cellular networks s nvestgated, respectvely. In ths secton, on the other hand, we are nterested n the analyss of coverage and rate of cellular networks, whch lead to a dfferent and novel mathematcal formulaton. More specfcally, we provde novel two-fold Securty: Publc Page 9

20 ntegral expressons for coverage and rate, whch have a compact mathematcal formulaton and are applcable to arbtrary per-lnk power gans and path-loss exponents. In the nterference-lmted regme, exact sngle-ntegral expressons formulated n terms of generalzed hypergeometrc functons are provded. As a case study, we focus our attenton on fadng channels and transmsson schemes whose equvalent per-lnk power gans follow a Gamma dstrbuton wth arbtrary parameters. In ths scenaro, we provde approxmated closed-form expressons for coverage and rate, whose accuracy s assessed wth the ad of Monte Carlo smulatons. The ratonale of ths choce orgnates from [.9] and [.], where t s shown that the per-lnk power gans of a large class of multple-antenna transmsson schemes for transmsson over Raylegh fadng channels can be approxmated by a Gamma dstrbuton wth adequate parameters. In [.9], the mpact of the parameters of the Gamma dstrbuton s nvestgated by relyng on approxmated expressons of the other-cell nterference obtaned through moment-matchng methods. In [.], the same problem s solved wth the ad of stochastc orderng. In [.], t s shown that, n general, the analyss of multple-antenna transmsson schemes requres the computaton of the dervatves of the MGF of the other-cell nterference. In ths secton, approxmated but smple closed-form expressons for coverage and rate are provded, whch provde nsght on the achevable performance of cellular networks as a functon of the parameters of the per-lnk power gans, e.g., the multple-antenna transmsson scheme f Raylegh fadng s assumed... System Model and Problem Formulaton Smlar to the case study of the error performance n Secton., we assume a probe MT s located at the orgn of b-dmensonal plane and the BSs are modeled as ponts of a homogeneous PPP ( ) of densty. The dstance from the th BS to the MT s denoted by r for. The MT s assumed to be tagged to the nearest BS. The servng BS s denoted by BS, and ts dstance from the MT s denoted by r, whch s a RV wth PDF r exp. The set of nterferng BSs \ BS f s stll a homogeneous PPP whch s denoted by \. \ has densty. The SINR of ths downlnk cellular network can be formulated as follows: P r SINR ; N PI agg r I agg r \ r (..) Securty: Publc Page

21 where P s the BSs transmt-energy per transmsson; s the nose power, α > denotes the path-loss exponent. I agg s the aggregate other-cell nterference, and for \ are the per-lnk power gans of ntended and nterferng lnks, whch have an arbtrary dstrbuton that usually depends on fadng channel and transmsson scheme [.9][.]. N The coverage probablty ( as follows: P cov ) and average rate ( R ) are studed. They can be formulated Pcov T PrSINR T cov a (..) b R E ln SINR P exp t dt ln y P y dy (..3) where T s a relablty threshold, (a) follows from [.] and (b) follows by applyng ntegraton by parts, snce P T and cov cov P T cov..3 Coverage Probablty and Average Rate New mathematcal frameworks to the computaton of (..) and (..) are provded, by assumng that and \ for have an arbtrary dstrbuton. For generalty, and accordng to, e.g., [.9][.], the dstrbutons of and are dfferent and ndependent. The nterferers power gans ndependent but dentcally dstrbuted. \ for on the other hand, are assumed to be Theorem.. P cov n (..) can be formulated as: where E exp x s sx x dx Pcov T Im j x T x M F NI (..4) M s the MGF of RV x and the followng functons are ntroduced: where p Fq a,, ap; b,, bq ; N F NI x y exp jy x exp y I jx dy (..5) P I z E F ; ; z (..6) s the generalzed hypergeometrc functon. Securty: Publc Page

22 Proof: See [.8]. Corollary.. P cov n (..) when can be formulated as: N Proof: See [.8]. P T cov x j M T dx Im I jx x (..7) The computaton of (..4) and (..7) requres a closed-form expresson of the expectaton n (6). Remark.. provdes a general approach to solve ths problem. Remark.. k k Let p / q E for,,... k k p and k, where,,...,, p p p N q q, q,..., qm are vectors wth N and M real-valued entres, and p k and k q are shorthands for p p p... pn and q q k k k k q... q k k k M respectvely where k k denotes the Pochhammer symbol. Then the followng equaltes hold: a p k k q k b z / I z N FM, p;, q; z / k! k k k (..8) It follows from the seres representaton of the hypergeometrc functon[.]. The formulaton holds for a large number of fadng dstrbutons and transmsson schemes. A general class of dstrbutons s avalable n[.3], whose moments can be computed usng [.]. Theorem.. R n (..3) can be formulated as follows: dx R Im jf jxf NI x (..9) x Where the followng functons are ntroduced: Proof: See [.8]. Corollary.. R n (..3) when N ln y z F jx M dy (..) y y can be formulated as follows: Securty: Publc Page

23 Proof: See [.8]. F jx Im j jx dx I x R (..) Remark.. The computaton of (..) and (..) requres closed-form expressons for the frst dervatve M be found n[.3].. Luckly, they are avalable for many fadng dstrbutons. A summary can..4 Gamma Dstrbuted Per-Lnk Power Gans In ths secton, we focus our attenton on the case study where the power gans of ntended and nterferng lnks follow a Gamma dstrbuton wth arbtrary parameters,.e., ~ Gamma m, and ~ Gamma m,. Ths case study s meanngful because t I I I fnds applcaton to the analyss of cellular networks for propagaton over Raylegh fadng, whch rely on multple antenna transmsson schemes at the physcal layer. The readers referred to [.9][.]for further detals. In order to get nsght on the mpact of the multpleantenna transmsson scheme, the authors of [.9][.] resort to approxmated representatons of the aggregate other-cell nterference and to stochastc orderng analyss. In [.], t s shown thatan accurate analyss of ths scenaro would requre the computatonof the hgher-order dervatves of the MGF of the aggregate other-cell nterference. Unlke these papers, wth the ad of the mathematcal formulatons n Corollary and Corollary, we propose approxmated but closed-form expressons for coverage and rate. The obtaned mathematcal expressons are shown to provde relevant nformaton on the mpact of system parameters, whch may offer a smple approach for comparng varous multple-antenna transmsson schemes at the physcal layer [.9][.]. In Secton V, we show that the performance trends obtaned from the proposed mathematcal frameworks are confrmed wth the ad of Monte Carlo smulatons. By assumng ~ Gamma m, and ~ Gamma m, can be smplfed as follows: P T cov, Corollary and Corollary I I I m j x dx (..) x F, mi; ; j Ix Im Securty: Publc Page 3

24 where,3 m G3,3 j x dx m x F, mi; ; j Ix R Im (..3) / m, / T and / m Proof: See [.8]. I I I. a mn, p G pq, bq s the Mejer G-functon. Approxmated closed-form expressons of (..) and (..3) are provded n Proposton and Proposton, respectvely. Proposton.. P cov n (..) can be approxmated as: Proof: See [.8]. Pcov T m I T m (..4) Remark..3: By drect nspecton of (..4), the coverage probablty has the followng performance trends: ) t ncreases as T ) t s ndependent of m I. / I ncreases; ) t ncreases as m ncreases; and Remark..4: LetK I mt P / / cov K ncreases. P cov Proof: See [.8].. The accuracy of the approxmaton n (..4) ncreases as Proposton.. R n (..3) can be approxmated as: m 3, R G (..5) 3,3 m I Proof: See [.8]. Securty: Publc Page 4 m

25 Remark..5: By plottng (..5) as a functon of probablty (Remark 3) hold. /, the same trends as n coverage I Remark..6: Let K / I / m Rate K Rate ncreases.. The accuracy of the approxmaton n (..5) ncreases as..5 Numercal Results In Fgs... and.., numercal examples are shown to substantate the proposed mathematcal frameworks ( sold lnes ), approxmatons ( markers ) aganst Monte Carlo smulatons ( black dots ), whch are obtaned as descrbed n [.3]. For smplcty, the nterference-lmted regme s analyzed,.e., and the per-lnk power gans are Gamma N dstrbuted. The llustratons confrm that (..) and (..3) are exact. They also show that (..4) and (..5) are farly accurate. In partcular, the expected accuracy dscussed n Remark..4 and Remark..6 s confrmed. More mportantly, (..4) and (..5) well reproduce the behavor of coverage and rate as a functon of the system parameters, as dscussed n Remark..3 and Remark..5. Ths s, n fact, the man usefulness of these approxmatons. Furthermore, the fgures show that the accuracy of (..4) and (..5) ncreases as α decreases. Ths s an mportant outcome, snce Monte Carlo smulatons are ether less accurate or requre more smulaton tme for small values...6 Concluson In secton., new mathematcal expressons for coverage and rate of cellular networks are provded wth the ad of the Gl-Pelaez nverson formula. The frameworks are shown to be general enough for the analyss of dfferent fadng channels and transmsson schemes. Furthermore, closed-form approxmated expressons are proposed when the per-lnk power gans are dstrbuted accordng to a Gamma dstrbuton Securty: Publc Page 5

26 Fgure... P as a functon of T and K / I / m cov Fgure... R as a functon of α and K / I / m Securty: Publc Page 6

27 .3 Extenson of the Gl-Peleaz nverson approach to sngle-ter MIMO cellular networks In ths secton, a mathematcal framework to evaluate the error performance of downlnk Multple-Input-Multple-Output (MIMO) cellular networks s ntroduced. It s based on the Posson Pont Process (PPP)-based abstracton for modelng the spatal locatons of the Base Statons (BSs) and t explots results from stochastc geometry to characterze the dstrbuton of the other-cell nterference. The framework s applcable to spatal multplexng MIMO systems wth an arbtrary number of antennas at the transmtter ( N t ) and at the recever ( N ). It s shown that the proposed framework leads to easy-to-compute ntegral r expressons, whch provde nsghts for network desgn and optmzaton. The accuracy of the mathematcal analyss s substantated through extensve Monte Carlo smulatons for varous MIMO cellular networks setups..3. Introducton As dscussed n Secton., the PPP-based abstracton model s now commonly used to the analyss and desgn of wreless networks due to ts mathematcal tractablty. On the other hand, wth the excepton of a few papers, e.g., [.], the mathematcal analyss s lmted to sngle-antenna BSs and to sngle-antenna MTs. By extendng analyss n secton. and [.7]to mult-antenna networks, the techncal contrbuton of ths secton s threefold: ) we provde an exact closed-form expresson of the Characterstc Functon (CF) of the aggregate other-cell nterference at the Moble Termnal (MT); ) by usng the Gl-Pelaez nverson theorem[.4], we provde an exact expresson of the average parwse frame error probablty ( APEP dstrbuton and all BSs deployments. From (F) (F) APEP ), whch s averaged over both the fadng, the average frame error probablty (AFEP) s obtaned by usng the Nearest Neghbor (NN) approxmaton[.3]; 3) asymptotc frameworks are proposed to provde nsghts on the achevable error performance as a functon of mportant system parameters..3. System Model Smlar to Secton., n downlnk cellular networks, a probe Nr -antenna MT s located at the orgn of a b-dmensonal plane and the Nt -antenna BSs are modeled as ponts of a homogeneous PPP ( ) of densty. The dstance from the th BS to the MT s denoted by Securty: Publc Page 7

28 r for. The MT s assumed to be tagged to the nearest BS. The servng BS s denoted by BS, and ts dstance from the MT s denoted by r, whch s a RV wth PDF r exp [.]. The set of nterferng BSs \ BS f s stll a homogeneous PPP [.9], whch s denoted by \. \ has densty p, where p \ s the actvty factor denotng the probablty that a nterferng BS transmts n the same frequency band as BS. The setup wth p corresponds to the full frequency reuse case [.], [.3]. In the depcted downlnk MIMO cellular network model, the sgnal receved at the MT s as follows: y E / N r H s E / N r H s n (.3.) b b t t \ x agg ( r ) where x C N r s the useful sgnal transmtted by BS, agg (r ) C N r s the aggregate other-cell nterference and n C N r s the Addtve Whte Gaussan Nose (AWGN) wth..d. n (r) ~CN(, N ). More specfcally: E s the BSs transmt-energy per transmsson; s s the (t) vector of nformaton symbols emtted by BS, where s M and the set M has M modulated symbols denoted by μ χ C; H C N r N t s the channel matrx of the BS -to-mt lnk; b > denotes the ampltude path-loss exponent. A smlar notaton s adopted for all the nterferng channels of agg ( ). At the MT, an nterference-unaware Maxmum-Lkelhood (ML)-optmum demodulator s consdered. It s assumed to have perfect Channel State Informaton (CSI) of the BS -to-mt lnk, whle gnorng the other-cell nterference. The decson metrc [.4]: can be formulated as E E E E s y H s (.3.) where b b b b r r U r ReI( r ) r ReN Nt Nt Nt Nt Δ s s and: N r Nt * Nr Nt Nr Nt * r r, t t r, t t r r, t t agg r H Δ H Δ n H Δ r t r t r t I( r ), U, N Securty: Publc Page 8

29 Assume Raylegh channel model, rt, H ~ Gamma, and the channel phases are unformly dstrbuted, then condtonng upon Δ, we have U ~ Gamma N, N Δ. r r, H rt ~ CN, for,,, t t t N and r,,, Nr. Thus, by, ~ CN, Δ Nt r t t H Δ Δ. Ths mples that.3.3 Characterstc Functon of Interference In ths secton, we provde a closed-form expresson of the CF of I( ) n (.3.), whch s the other-cell nterference at the output of the demodulator. Theorem.3. Let the downlnk channels H \ for be..d. Raylegh dstrbuted, wth rt, E H. By condtonng upon r and upon U gven n (.3.), a closed-form expresson of the CF of I( ) n (.3.), whch s the other-cell nterference at the output of the demodulator, s provded as follows: T x p x p and: wth ;, U ;, U exp T b r r r r EU I I Securty: Publc Page 9 ω ω ω (.3.3) x where p Fq a,, ap; b,, bq ; expectaton x s s F ; ; Ω b b 4 Nt E (.3.4) s the generalzed hypergeometrc functon and the E s computed over the t N -tuple of nformaton symbols N t s C can be formulated wth g() beng a generc functon and K beng a postve constant as follows: E s g s N M N M M M Nt K K g N t t t N t t t Proof: The proof s smlar as [.7]. The only dfference s that to be replaced by z n I( ). In addton, and smlar to Z n Iagg (.3.5) n [.7] needs Z, the RVs z are stll crcularly \ symmetrc and..d. for. As a consequence, the CF of I( ) can be formulated as shown n [.7], by smply replacng Re Z wth z re Re z. Accordngly, the moments of

30 re z condtoned upon H and Δ need to be computed. Let us denote these condtoned moments by, rt, q z H, Δ Re H ~ CN, /. Thus, H, Δ and q re H Δ E z. Snce s, whch we denote by z H ~ CN,, re z turns out to be a Gaussan RV by condtonng upon rt, re Nt t ~ N,,, / U / Nt s H Δ s t. Then the proof can be proceeded as [.7]by followng from the moments of Gaussan RVs [.3] q Nt q t q q q H, Δ q U E s U s / t q N t and from the seres representaton of the generalzed hypergeometrc functon [.3], and fnally from the denttes N t t s s and F /, ;/, ; x t ; ;. F x.3.4 Error Performance (F) APEP By captalzng on the CF, transmtted vector s s decoded as two nformaton vectors possbly beng transmtted. (F) APEP s computed, whch s defned as the probablty that s s s, by assumng that s and s are the only Theorem.3. SNR E / Let N, Then, whch are defned as: (F) APEP can be formulated as APEP Δ Pr s APEP Δ APEP Δ F F F N NI where: x F b SNR / N t e F APEPN Δ Q N x;, SNR / N t, b dx b Δ x b xy Nt F y b b 4 SNR ytnt x F APEP NI Δ x e e e Q NI x; dxdy Δ F SNRU Q N x; Δ, SNR / Nt, b E U U exp b 4Nx t F Q NI x; Δ E U sn Ux (.3.6) (.3.7) Securty: Publc Page 3

31 Proof: The proof s smlar to [.7]. By applyng the Gl-Pelaez nverson theorem [.4]to (.3.), where: ) (.3.); ) (F) APEP condtoned upon H and r can be formulated as follows: F (F) Δ H r r APEP ;, APEP U, r E b sn U ; U ;, U ω N I N ω ω t re / re re r d ω N Re N ~ N, N U follows from the AWGN assumpton and from I re re ; r, U I ; r, U I ReI s defned n (.3.); ) ;U ;U ω ω ω and N re N ω ω, snce both N and I are crcularly symmetrc RVs; v) N ω ;U exp / 4 ω N U, snce t s a Gaussan RV [.33]; and v) gven n (.3.3). The rest of the proof s the same as n [.7]. I ;, s Wth the ad of [.34] and of the Kummer's transformaton ;3 /, exp / ;3 /, F m x x F m x, for arbtrary K and constants and a Gamma RV U ~ Gamma Nr, Nr for (.3.7) as follows: K postve Δ, we have close-from expressons E E U exp N r Nr Nr KU K Δ N Nr Nr Δ Nr U Nr Δ N 3 r U sn K U KF Nr ;, Δ K r 4 (.3.8) We then provde smplfed mathematcal frameworks n two lmtng operatng condtons: ) nose-lmted,.e., SNR, and ) nterference-lmted,.e., SNR, cellular networks, where SNR E / N denotes the Sgnal-to-Nose-Rato (SNR). As dscussed n[.3], typcal cellular networks operate n the nterference-lmted regme. Corollary.3. For nose-lmted ( SNR ) and nterference-lmted ( SNR frameworks can be drawn from (.3.6): ) networks, asymptotc Securty: Publc Page 3

32 F F Δ Δ N lm APEP APEP SNR F lm APEP Δ SNR T N x t T Q F NI Nx t x dx (.3.) These asymptotc frameworks can better provde nsghts on the error performance as a functon of mportant system parameters. For example, t can be noted that error probablty s ndependent of both SNR and BS densty n nterference-lmted networks. On the other hand, n nose-lmted networks, the error probablty decreases by ncreasng SNR and BS densty respectvely.also n nterference-lmted regme (F) APEP decreases by ncreasng the path-loss exponent b and (F) APEP decreases by decreasng the actvty factor p.all the trends and nsghts nferred from mathematcal frameworks can be substantated by Monte Carlo smulatons, whch we wll show n the next secton. AFEP Fnally, based on the approxmated as follows: (F) APEP n (.3.6) and on the NN approxmaton[.3], AFEP can be N t M N F / t N PE mn AFEP M Δ A P Δ (.3.9) Nt where: ) Δ mn μ μ, μ M s the mnmum Eucldean dstance mn μ mn among all pars μ, μ of the Nt -dmensonal constellaton dagram and ) number of nearest neghbors of M symbols, we have N M 4,, N t N Δ mn s the μ. For example, consderng the standard square QAM wth Δ f, mn, 4, Δ mn avg mn, N.4,6 Δmn Δ f 6, M, N t 4 and mn, N,8 4,5,6,7,8 N Δmn Δ mn M, N N t Δ f M 4, 4 f M 6,. N t N t Δ mn,, 4 f M, N N t Δ mn Δ mn, 4, 4 f Δ mn. It s worth notng that Securty: Publc Page 3

33 .3.5 Numercal Results We valdate n ths secton the mathematcal frameworks aganst Monte Carlo smulatons, whch are obtaned by usng the procedure descrbed n [.3], and study the mpact of dfferent downlnk MIMO cellular setups. Before that, we frst valdate the PPP-based abstracton model by comparng t wth modelng the BS locatons va grd-based model. A smlar comparson s avalable n [.]for the coverage probablty. We provde n ths secton the comparson for AFEP. Fg. shows that the PPP-based abstracton model ( sold lnes ) provdes a worst estmate of the error probablty compared to the grd-based abstracton model ( dash curves ), snce nterferng BSs may be arbtrarly close to each other. On the other hand, the PPP-based abstracton model provdes tractablty and performance trends as a functon of system parameters can be nferred from the resultng mathematcal frameworks. We observe a good accuracy of the obtaned frameworks (depcted n curves wth colored markers) compared wth the Monte Carlo smulatons (depcted n black curves). Fgs all support the clam that AFEP s ndependent of SNR n the nterferencelmted regme.fg..3. confrms that AFEP decreases by decreasng the actvty factor p. Fg..3. shows that by ncreasng the path-loss exponent b, AFEP decreases n the nterference-lmted regme whle ncreases n the nose-lmted regme.fg..3.3 confrms that the BS densty affects the AFEP only n the nose-lmted regme.fg..3.4 confrms that havng multple-antenna at the recever leads to an mprovement of the ASEP/AFEP. However, no receve dversty gan s obtaned n the nterference-lmted regme.the rate of Rate t log the MIMO system s defned as N M bts per channel use (bpcu). Condtoned on a consstent rate per channel use, Fg..3.5 shows that ncreasng AFEP but the performance dfference as a functon of N N N t provdes a better N t s smaller compared to N r.wth symmetrc antenna setup t r, Fg..3.6(a) shows the AFEP by assumng that the densty of BSs s kept the same but the number of BSs antennas N t s dfferent and Fg..3.6(b) shows the AFEP by assumng that the densty of BSs antennas Nt s kept the same and the BSs densty depends on N t. The trends shown n Fg..3.6 allows one to reduce the densty of BSs by ncreasng N t wthout performance degradaton n the Securty: Publc Page 33

34 nterference-lmted regme and wth a small performance degradaton n the nose-lmted regme. Fgure.3. AFEP of a N N system aganst the transmt SNR. Setup: 6QAM, t r 5 and 3 b. Fgure.3. AFEP of a N N system aganst the transmt SNR. Setup: 6QAM, t r 5 and p 3. Securty: Publc Page 34

35 Fgure.3.3 AFEP of a N N system aganst the transmt SNR. Setup: 6QAM, t r b 3 and p 3. Fgure.3.4 AFEP of a N N N system aganst the transmt SNR. Setup: 4QAM, t r r b and 5 /, 3 p 3. Securty: Publc Page 35

36 Fgure.3.5 AFEP of a N N N system wth Rate 4 t r t bpcu aganst the transmt 5 SNR. Setup: b 3,, b 3 and p 3. Fgure.3.6 AFEP of a system wth Rate 4 bpcu aganst the transmt SNR. 5 Setup: (a) b 3,, p 3. (b) 3 5 b,, p 3. Securty: Publc Page 36

37 .3.6 Concluson In ths secton, a new mathematcal framework to the computaton of the average error probablty of downlnk MIMO cellular networks have been proposed and have been substantated wth the ad of Monte Carlo smulatons. Ther analyss has revealed mportant performance trade-offs that may emerge dependng on the SNR operatng regme, the channel attenuaton parameters and the number of antennas avalable at the BSs and MT..4 Stochastc geometry modelng and analyss of the error probablty of sngle-ter SISO cellular networks: the Equvalent-n-Dstrbuton (ED) based approach.4. Introducton As dscussed n the prevous secton, the PPP-based approach provdes a tractable mathematcal analyss and s as accurate as other abstracton models [.]. In general, two man performance metrcs have been studed to date,.e., the outage probablty and the average rate [.], [.], [.3]. Less attenton has been gven, on the other hand, to the computaton of the ASEP, whch s, however, a relevant fgure of mert to wreless systems analyss and desgn. In fact, t s drectly related to the bt, packet, block and frame error probabltes, whch are mportant performance metrcs to the desgn of cellular networks [.5]. Indeed, the framework ntroduced n Secton. and.3, based on the Gl-Peleaz nverson approach as well as the APEP, provdng the ASEP usng the nearest neghbor approxmaton, s, however, not exact. In the followng three sectons, we ntroduce a new mathematcal methodology for the computaton of the error probablty of downlnk MIMO cellular networks. The proposed approach captalzes on the so-called Equvalent-n- Dstrbuton (ED)-based method, by fndng ED representatons of the aggregate other-cell nterference, whch s formulated as a lnear combnaton of condtonally Guassan Random Varables (RVs). Wth the ad of ths mathematcal formulaton, the error probablty s computed by frst condtonng upon the non-guassan RVs and by then removng the condtonng. The usefulness of ths approach les n the possblty of obtanng exact mathematcal expressons n the presence of non-gaussan dstrbuted nterference. In Secton.4, we fst ntroduce the ED-based framework by only consderng a sngle-ter SISO cellular network, whle the framework s extended to studyng a varety of MIMO setups n Secton.5. In Secton.6, we further extend the applcaton of the ED-based approach Securty: Publc Page 37

38 from the conventonal sngle-ter cellular network to mult-ter heterogeneous cellular networks. Compared to Secton. and.3, the new approach: ) s not based on the Gl-Pelaez nverson theorem; ) s applcable to many MIMO schemes (not just to spatal multplexng); and ) provdes, n many cases, exact ntegral expressons of the error probablty. In the presence of other-cell nterference and nose, the error probablty s formulated n terms of a two-fold ntegral. The framework s shown to reduce to the computaton of a sngle ntegral n nterference-lmted cellular networks. The followng notatons are used n ths secton: denotes the set of postve ntegers. E s the expectaton operator. j s the magnary unt., Gaussan Random Varable (RV) wth mean and varance. CN s a complex Re and and magnary part operators. exp Re Im Im are real z ω E j z z s the Characterstc Functon (CF) of the complex RV z Rez jimz, where ω, s the absolute value of vectors and complex RVs. x! s the factoral of x. x x t exp tdt s the Gamma functon. x x k / x k symbol.. s the Pochhammer d x y denotes that x and y are ED. M x s E expsx s the Moment Generatng Functon (MGF) of x. ;; x Eq. ()]. erf / exp x t dt p q F s the generalzed hypergeometrc functon [Ch. 5, s the error functon..4. System model A b-dmensonal downlnk cellular network deployment as depcted n [.] s studed, where a probe sngle-antenna Moble Termnal (MT) s located at the orgn and the sngle-antenna Base Statons (BSs) are modeled as ponts of a homogeneous PPP ( ) of densty. The MT s served by the nearest BS ( BS ). Ther dstance s denoted by r, whch s a RV havng Probablty Densty Functon (PDF) equal to fr exp [.]. Smlar to [.], a per-cell random fractonal frequency reuse s consdered, where p denotes the probablty that the generc nterferng BS transmts n the same frequency band as BS. Securty: Publc Page 38

39 \ Accordng to [.47], the set of nterferng BSs ( ) s a homogeneous PPP of densty p. p The dstance from the th nterferng BS to the MT s denoted by r for r. \ p In the depcted downlnk cellular network model, the (complex) sgnal receved at the MT can be formulated as follows: y E s h E s h n x r n (.4.) \ I agg where x E sh s the useful sgnal from p BS, \ r r E s h s the other-cell agg nterference, whch depends on r snce r n~ CN, N s the Addtve Whte Gaussan Nose (AWGN), E s the average symbol energy of BS, s a exp j s the symbol transmtted by b BS and h / r exp j, p I s the channel mpulse response of the BS -to-mt lnk, wth b denotng the path--loss exponent, denotng the fadng beng a unformly dstrbuted RV n,. For ease of llustraton, envelope, and Raylegh fadng s assumed. Hence, E [.4]. A smlar notaton s used for agg s an exponental RV wth parameter. In partcular, we assume that all E E E ) and that all the the BSs transmt the same average energy per symbol ( I channels are ndependent and dentcally dstrbuted havng the same path-loss exponents b b b) and the same fadng parameters ( I ). A general b-dmensonal ( I modulaton scheme s consdered, whose M equprobable symbols are ( m ) ( m ) exp ( m s a j ) ( m) ( ) for m,,, M s s s s m \ for. The. Then, and, M m constellaton dagram s assumed to have unt average energy,.e., / M s thus s s E E. At the MT, the optmal demodulator n AWGN s used [.4]: arg mn m s s, m,,, M m sˆ y E h s (.4.) where ŝ s the estmate of the actual transmtted symbol s. In ths secton, we are nterested n computng the ASEP of the demodulator n (.4.), whch s the probablty that s. ŝ p Securty: Publc Page 39

40 .4.3 Characterstc functon of the nterference In ths secton, the CF of agg s presented agan to ncrease the readablty. For smplcty, we use the notaton z a exp j exp j. Hence, b \ r E z r agg / as follows: p. Then the CF of agg r gven smplfes to agg r s ; r agg agg ; r ω ω, q M q m E pr ω agg ω ; r exp q s b (.4.3) M m q r q where 4 q! / b / b. q q q.4.4 The ED-Based Approach From (.4.3), t s apparent that the aggregate other-cell nterference common probablty dstrbuton. In partcular, agg agg does not follow a does not follow ether a Gaussan dstrbuton, as n AWGN channels [.4], or a symmetrc alpha stable dstrbuton, as n ad hoc networks [.], for whch tractable frameworks to the computaton of the ASEP exst. Ths makes the mathematcal computaton of the ASEP of cellular networks dffcult. If a bnary modulaton scheme s consdered,.e., M, the ASEP can be obtaned by explotng the Gl-Pelaez nverson theorem. It provdes, however, ether approxmatons or bounds for M. In ths subsecton, we ntroduce a new mathematcal approach that leads to exact expressons of the ASEP of cellular networks for arbtrary b-dmensonal modulatons wth M. The proposed approach fnds nspraton from the methodology recently ntroduced n [.], whch s applcable to decentralzed wreless networks. To better ntroduce our approach, we brefly summarze the methodology adopted n [.]. The system model n (.4.) reduces to that of [.] by lettng r,.e., by replacng r wth: agg SS / agg agg d agg agg r B G (.4.4) where SS follows a symmetrc alpha stable dstrbuton, B agg s a real stable RV totally agg skewed to the rght and avalable n [.]. G agg s a complex Gaussan RV wth zero mean and varance Securty: Publc Page 4

41 By captalzng on the ED-based representaton n (.4.4), the authors of [.] propose a margnalzaton-based approach to the computaton of the ASEP: ) frst, the ASEP condtoned upon h and [.4] and ) then, the condtonng upon h and ntegrals. B agg s computed by usng formulas applcable to AWGN channels B agg s removed by computng the related Inspred by ths approach, n what follows we seek to answer the followng queston: Is t possble to develop an ED-based representaton of agg r for arbtrary values of r n order to leverage a margnalzaton-based approach to the performance analyss of cellular networks? A postve answer to ths queston s provded n the followng theorem: Theorem.4.: Let agg r havng CF ; r n (.4.3). Let agg q B for q be ndependent real RVs agg q whose MGF s q s exps M B agg agg RVs G q ~ CN, q ( r ) wth:. Let q G for q be ndependent complex Gaussan agg The RVs q B and agg / q q M E q( r) 4 p rq s. b r M m q G are ndependent for q agg. Then: d d q q agg agg agg agg q r r q m (.4.5) B G (.4.6).4.5 Performance Analyss From Theorem.4., an ED-based formulaton of (.4.) s: j q q d exp y Es B G n (.4.7) b agg agg r q Condtonng upon s, h and q B for q agg, y n (.4.7) s condtonally-gaussan. So, the ASEP of (.4.) can be computed from the classcal defnton of SINR n AWGN [.4]: Securty: Publc Page 4

42 s q n, G agg q q E B agg q( r ) B (.4.8) E y SIN R, r, where X agg b N r q N E s E q y E, q n y Gagg n, Gagg E E denotes the expectaton computed only over X. By defnton from (.4.), ASEP Prs s. Based on the ED-based representaton n ˆ (.4.7), the ASEP of b-dmensonal modulatons can be formulated as the lnear combnaton of ntegrals lke the followng [.]: M where SINR q, r, Bagg s E sn J SINR,, SINR d M sn (.4.9) exp ssinr s a trplet of modulaton-dependent parameters. s the MGF of the SINR n (.4.8) and,, For example, the ASEP of square Quadrature Ampltude Modulaton (QAM) s: ASEP 4 / J,, 4 / J,, (.4.) QAM SINR SINR where /, /4, /, M / M. J s: Let the SINR n (.4.8). Then, where and y SINR SINR,,,, / sn, 3 / M and J x y dxdy (.4.) E, exp b x y x yexp xq yy sn y (.4.) N M m Q y p / M F / b; / b; s y p (.4.3) y m Y Y f /, y exp / / erf cot Y y y y y. y Y Y f / wth Securty: Publc Page 4

43 Typcal cellular networks are nterference-lmted [.3]. Let J n (.4.). If N, SINR,, then: J SINR sn,, Y sn y Q y dy (.4.4).4.6 Numercal Results In Fg..4., some numercal examples are shown n order to substantate the accuracy of the ED-based approach. Monte Carlo smulatons are obtaned as descrbed n [.3]. At the MT, the demodulator n (.4.) s used. A good agreement between mathematcal framework and smulatons s observed. As suggested by (.4.4), the ASEP s ndependent of n the nterference-lmted regme (hgh E/ N ). Decreasng the frequency reuse factor p leads to a better ASEP. However, the average rate decreases by decreasng p [.3]. Hence, a tradeoff emerges. The mpact of b depends on E/ N. In the nterference-lmted regme, the hgher b the better the ASEP. Fg..4.: Vadaton of the ED-based Approach Securty: Publc Page 43

44 .5 Extenson of the Equvalent-n-Dstrbuton (ED) based approach to sngle-ter MIMO cellular networks.5. Introducton In the present secton, we ntroduce a new mathematcal methodology for the computaton of the error probablty of downlnk MIMO cellular networks based on the ED representaton of the other-cell nterference smlar as that proposed n Secton.4. The mathematcal framework ntroduced n Secton.4, however, s applcable only to SISO cellular networks for transmsson over Raylegh fadng channels. Its generalzaton to MIMO cellular networks and to other fadng dstrbutons s, however, not straghtforward. In Secton.4, n fact, the error probablty s obtaned by frst computng the Cumulatve Dstrbuton Functon (CDF) of the Sgnal-to-Interference-plus-Nose-Rato (SINR) and by then applyng the so-called CDFbased approach. Ths methodology s effectve n Raylegh fadng, as the CDF of the power gan of the ntended lnk s an exponental functon, whch s convenently formulated for further analyss. It s known, on the other hand, that the generalzaton of CDF-based methods to other fadng dstrbutons s problematc [.], [.3], [.8]. Unfortunately, the equvalent power gan of the ntended lnk of MIMO transmsson schemes s not exponentally dstrbuted even n Raylegh fadng channels [.]. As a consequence, a new mathematcally tractable approach s necessary n ths context. In the present secton, the lmtatons are overcome by ntroducng a new mathematcal framework that s based on the computaton of the Moment Generatng Functon (MGF) of the equvalent power gan of the ntended lnk, whch makes the ED-based approach applcable to a number of MIMO arrangements for transmsson over Raylegh fadng channels. As a byproduct, we show that the proposed approach s applcable to SISO cellular networks for transmsson over Nakagam-m fadng channels. Compared to [.], the approach n ths secton s dfferent snce t does not explot stochastc orderng. Our approach: ) s not based on the Gl-Pelaez nverson theorem; ) s applcable to many MIMO schemes (not just to spatal multplexng); and ) provdes, n many cases, exact ntegral expressons of the error probablty. In the presence of other-cell nterference and nose, the error probablty s formulated n terms of a two-fold ntegral. The framework s shown to reduce to the computaton of a sngle ntegral n nterference-lmted cellular networks. Also, a smple closed-form expresson s ntroduced, whch provdes meanngful nsghts on the mpact of varous system parameters that determne the achevable performance of MIMO cellular networks. Securty: Publc Page 44

45 In ths secton, the followng notaton s used. * z, z and argz denote conjugate, modulus and phase operators of a complex number z. z denotes that z s drectly proportonal z to z. denotes the feld of complex numbers. K x S denotes a K column-vector wth entres belongng to the set S. The k th entry s denoted by K ( k ) x. L matrx wth entres belongng to the set S. The ( kl, ) th entry s denoted by denotes the Hermtan of X. K L X S denotes a ( kl, ) X. H X cards denotes the cardnalty of the set S. x denotes the norm of vector x. X denotes the Frobenus norm of matrx X. j denotes the magnary unt. CN, and, dstrbuton wth mean and varance N denote a complex and a real Gaussan. The notaton, X used for Gaussan RVs condtoned upon the RV X. ab, ab. m, n (, ) CN and, X N s U denotes a unform dstrbuton G denotes a Gamma dstrbuton wth fadng parameter m and mean square value [.4]. d denotes a Ch-Square dstrbuton wth d degrees of freedom [.54]. Y X / X ~ d, d where X F denotes a F-dstrbuton [.54] wth parameters ~ d / d and X ~ d / d d and d,. q denotes the Pochhammer symbol, where q s a non-negatve nteger.! and!! denote factoral and double factoral operators. Re and X Im denote real and magnary part operators. E denotes the expectaton operator. f denotes the Probablty Densty Functon (PDF) of RV X. re m exp X ω E j X X denotes the Characterstc Functon (CF) of a re m complex RV X ReX jimx X jx, where ω, short-hand ω E exp jωx s used. E exp X a real RV X. Pr denotes probablty. denotes that the RVs X and Y are ED.. For smplcty, the M s sx X denotes the MGF of denotes the bnomal coeffcent. s the Gamma functon [.35]., upper-ncomplete Gamma functon [.35]. p Fq a,, ap; b,, bq ; X d Y s the s the generalzed Securty: Publc Page 45

46 hypergeometrc functon [.]. G mn, pq, a b p q s the Mejer G-functon [.]. Jv s the Bessel functon of the frst knd [.35]..5. System model We consder a b-dmensonal downlnk cellular network deployment, where a typcal multantenna Moble Termnal (MT) s located at the orgn and the mult-antenna BSs are modeled as ponts of a homogeneous PPP ( ) of densty. The number of antennas at each BS and at the MT s denoted by as those depcted n Secton.4. N t and N r, respectvely. The other setups are smlar In ths MIMO cellular network model, data transmsson occurs n frames of N s tme--slots each. The sgnal receved at the MT n the th tme-slot can be formulated as follows (,,, Ns ): y( ) E / N r H s ( ) E / N r H s ( ) n ( ) (.5.) b b t t \ x( τ) agg ( ; r ) where r y () N, BS, r x () N s the ntended sgnal from N r s the agg (, ) aggregate other-cell nterference and N n s the Addtve Whte Gaussan Nose r () (AWGN). The aggregate other-cell nterference depends on r, snce the nterferng BSs must le outsde the ball of radus r and centered at the orgn. Ths orgnates from the shortest dstance cell assocaton crteron. More specfcally: ) E s the BSs transmt-energy per transmsson, whch s equally splt among the N t antennas; ) t, t ; s Θ η S I, for,,, t encoded symbols emtted by t N and,,, Ns, s the vector of space-tme BS, where ( ; ) η s the vector of modulated nformaton symbols and N Θ s the s Nt space-tme encodng matrx, S s the sde nformaton avalable at BS. In partcular, M ndependent nformaton symbols are transmtted n.e., η s a M column-vector and modulated symbols. The generc cardm η M for m,,, m I N s tme-slots, M wth M denotng the set of M symbols of M are denoted by for Securty: Publc Page 46

47 ,, M,. A zero-mean and an average unt-energy constrants are assumed,.e., M (/ M ) and M (/ M ), respectvely. For example, they can be the M symbols of ether a Phase Shft Keyng (PSK) or a Quadrature Ampltude Modulaton (QAM) constellaton dagram. The rate provded by (.5.) s R M / N s log Nr Nt use (bpcu); ) H s the channel matrx of the r, t r, t r, t BS M bts per channel -to-mt lnk, where H H exp j arg H and rt, arg H ~ U, for t,,, Nt and r,,, Nr. Quas-statc fadng s assumed n (.5.), whch mples that H s constant n the N tmes-slots of a frame,.e., () s H H for,,, Ns, whle t changes ndependently from one frame to another. The channel envelope rt, H s assumed to follow a Raylegh dstrbuton[.4]. The only excepton s the SISO setup n Secton.5.4.A, where H follows a Nakagam--m dstrbuton wth fadng parameter m [.4]; v) b s the rt, ampltude path-loss exponent; v) r n ( ) ~ CN, N are ndependent and dentcally dstrbuted (..d.) RVs for r,,, Nr,,,, Ns. Smlar notaton and assumptons are adopted for the nterferng channels of ( ; ) agg. As for the other-cell nterference model, (.5.) assumes the so-called sotropc scenaro [.], where the N r antennas at the MT are omndrectonal and are subject to the nterference generated by all nterferng BSs. Transmt- and receve-antennas are assumed to be co-located, hence the transmsson dstances r and r \ for are ndependent of the antennas nter-dstances. Smlar to the ntended lnk, locatons and channels of all nterferng BSs are assumed not to change n the, r, t H N s tmesslots of a frame. All channels are..d. wth mean square value r t E H E for t,,, Nt, r,,, N \ r,. The sgnal model n (.5.) s suffcently general to account for a large number of MIMO schemes, whch are studed n Secton Problem Formulaton: Prelmnares In ths secton, some defntons and enablng results for further analyss are ntroduced. The problem s formulated n a general manner, so that the MIMO arrangements analyzed n Secton.5.4 turn out to be specal cases of t. Ths provdes a unfed mathematcal framework to the performance evaluaton of MIMO cellular networks. As a consequence, Securty: Publc Page 47

48 defntons and results presented n ths and n the next secton are formulated n terms of fundamental propertes of the nvolved RVs, wthout lnkng them to a specfc MIMO setup. The connecton wth MIMO cellular networks wll become apparent n Secton.5.4. In order to facltate the readng, however, some remarks that lnk the general mathematcal formulaton used n ths and n the next secton to Secton.5.4 are provded. Defnton.5.: Let a complex RV X. It s sad to be sphercally symmetrc (or crcularly symmetrc or rotatonally nvarant) f ts PDF, X X f x f x [.53]. X f, depends only on X,.e., Remark.5.: Let followng propertes hold [.8]: d re m ) X X exp j, where, ) X X ω ω, ) re m X X v) re X X X jx be a complex sphercally symmetrc RV. Then, the ω ω ω, s an arbtrary constant, re cos X m ω ω ω m X E E cos X, X X v) a lnear combnaton of sphercally symmetrc RVs s stll a sphercally symmetrc RV. Defnton.5.: Let a complex RV X. The RV (GCG) X s sad to be a Generalzed Compound Gaussan (GCG) representaton of X f the followng equalty n dstrbuton holds: d GCG X X BqGq q (.5.) B and: ) q q are ndependent real RVs wth M q B s exps, ) G q q q ndependent complex Gaussan RVs wth dstrbuton Gq ~, G q CN, ) B q q are and Gq q are ndependent RVs. Defnton.5.3: Let S I be the a pror nformaton avalable at the MT about the receved sgnal n (.5.),.e., the sde nformaton. The Maxmum-Lkelhood (ML)-optmum [.4] Securty: Publc Page 48

49 demodulator based on (.5.) and S I s sad to be SI -optmum. If S I s ndependent of the other-cell nterference, t s sad to be nterference-oblvous SI -optmum. Remark.5.: As for the MIMO setups n Secton.5.4, the a pror sde nformaton usually a functon of the channel matrx of the ntended lnk,.e., S I depends on H. S I s Defnton.5.4: Let the receved sgnal n (.5.). Let vectors z for,,, Ns. Let Nt ; I ; I z τ be a short-hand for the N s η be the hypothess of η and b Δ τ s τ s τ Θ η S Θ η S. Let S E / Nt r I S be the sde nformaton at the MT, where S I s ndependent of r and of the other-cell nterference. Let be the decson metrc of an nterference-oblvous SI -optmum demodulator based on (.5.). Let be wrtten as: N s Δ τ y y b / t b I, Δ τ / t I, Δ τre b D SI / t D SI E N r D S E N r D S IAI IAI Δ τ n τ Δ τ τ E / N r Re,, E N r Re,, ; r b t 3 agg I (.5.3) where () and () are demodulator- and modulator-dependent functons, respectvely, y τ N I, yτ S s the post-processed receved sgnal, N I, y τ S s τ s the hypothess at the recever, N s the vectors sze that depends on the transmsson scheme beng consdered, D and real-valued functons, wth zero mean, follows: 3,, D,, s, condtonng upon I S and, D are postve IAI, D s a complex RV whose GCG representaton s d GCG I r b/ q / q Δ, a complex Gaussan RV r D S, Δ τ, τ; D S, Δ τ, τ; 3 agg 3 I agg E / N r p BG t q q q (.5.4) GCG D 3,, as Securty: Publc Page 49

50 where q CN IAI wth IAI IAI s τ, s exps q IAI s τ ~, G. If D ~ CN, S I, Δ τ q IAI M B, as well as q, the nterference-oblvous SI -optmum demodulator n (.5.4), s sad to be n a desred form f the followng equaltes hold: E Gq q D where, E D SI, Δτ, nτ n τ D D I IAI I S, Δ S, Δ τ τ D I S, Δ τ GCG b/ q / q 3 SI, Δ τ, ag g τ; r E / Nt r p Bq Gq q D SI, Δ τ D SI, Δ τ N D SI, Δ τ D SI, Δ τ (.5.5) D s a postve real-valued functon and G s a non-negatve constant related to G through the relaton: If D q IAI desred form f (.5.5) Gq τ I, Δ Gq q D SI, Δ τ D SI, Δ τ S G (.5.6), the nterference-oblvous SI -optmum demodulator n (.5.4) s sad to be n a Remark.5.3: As for the MIMO setups n Secton.5.5, the physcal meanng of the addends n (.5.4) s as follows: ) D s related to the servng BS; ), to the AWGN at the recever; ) D IAI, and IAI v) 3,, D s related,, D s related to the aggregate other-cell nterference; are related to the Inter-Antenna Interference (IAI) that may orgnate from couplng nformaton symbols n space and tme f removng t at the recever. S I s not suffcent enough for Remark.5.4: In (.5.4), the proportonalty symbol ( ) s used because all terms ndependent of the hypothess at the demodulator,.e., Δ, are neglected. Ths s known not to ntroduce any sub-optmalty n the defnton of the demodulator and any approxmatons for ts performance evaluaton [.4]. Securty: Publc Page 5

51 Remark.5.5: Let an nterference-oblvous SI -optmum demodulator formulated n the desred form n (.5.4) and (.5.5). It s sad to be n a sngle-stream desred form f the decson metrc n (.5.4) can be wrtten as follows: where m M m m Δ τ Δ τ (.5.7) has the same structure as (.5.3) except that t depends only on the m th nformaton symbol,.e., D D for,,,3,iai notaton holds for other symbols.,m and IAI IAI,m. A smlar Remark.5.6: Demodulators formulated n the sngle-stream desred form n (.5.7) allow the MT to demodulate the nformaton vector η n a symbol-by-symbol fashon wthout loosng optmalty. Demodulators formulated as n (.5.3) need mult-stream algorthms because of the couplng of the M nformaton symbols. In Secton.5.6, performance metrcs related to SI -optmum demodulators are computed. In Secton.5.5, t s shown that many MIMO detectors can be formulated as demodulators. SI -optmum.5.4 Man Results In ths secton, the man results of the paper are summarzed. As antcpated n Secton.5.3, the results are formulated n general terms based on the system model n Secton.5. and subsequently they are lnked to MIMO cellular networks, whch are further nvestgated n Secton.5.5. Proposton.5.: Let and \ be the PPPs of densty of avalable and nterferng BSs, respectvely. Let p be the actvty factors of the BSs. Let b be the ampltude pathloss exponent and r r \ for be the dstances from the nterferng BSs to the MT. Z \ be..d. sphercally symmetrc complex RVs for and let Z Z Let, have zero mean and fnte raw nteger moments of any even order. Let 3,, re, Re, D n (.5.3) as follows: I Δ τ τ \ D S,, ; r E / N r Z (.5.8) b 3 agg t, Securty: Publc Page 5

52 The GCG representaton of D can be formulated as: 3,, d GCG I r b/ q / q r D S, Δ τ, τ; D S, Δ τ, τ; 3 agg 3 I agg E / N r p BG t q q q q where M B s exps and q ~ CN, G S I, Δ τ q The proof s avalable n [.55]. G wth: q / bq / / b (.5.9) q q re S, ( ), q I Δ τ E Z (.5.9) G q! q q Remark.5.7: As for the MIMO setups n Secton.5.5, 3,, / q D n (.5.8) represents the aggregate other-cell nterference at the output of the demodulator and Z \, for s the contrbuton orgnatng from each nterferng BS. Lemma.5.: Let Z, be a complex RV defned as follows: where t N r Nt Ns r, t t r Z H s U (.5.), r t rt, H are..d. complex Gaussan RVs,.e.,,,, Nt, s and for,,, Ns. Let wth respect to H and r N, rt, H ~ CN, for,,, r U are N and N complex random vectors, respectvely, re, Re, t r re Z Z. The raw nteger moments of any even order of Z s τ are: re re q Z H Z q Nr Nt N q q s q t r E, E E, E / k s U s τ s τ r t (.5.), Remark.5.8: As for the MIMO setups n Secton.5.5, Z, n (.5.) depends on the channel gans, on the nformaton symbols, rt, H, and on the nformaton symbols, s, of the nterferng BSs, as well as Securty: Publc Page 5 U, of the servng BS. Snce rt, H are complex Gaussan RVs, ther phase s unformly dstrbuted and, thus, from Defnton.5., they are sphercally

53 symmetrc RVs. Snce Z, s the lnear combnaton of s sphercally symmetrc as well. Lemma.5.: Let, functon Let such that: q H q rt,, accordng to Remark.5., t G n (.5.9) wth re E Z formulated as n (.5.). Let a, N r Nt Ns N Ns t r r r t r r s U s τ U (.5.) G satsfy the equalty n (.5.6) wth q D S N r Ns r, Δ τ / D S I I, Δ τ r U. Let X X q q X, the followng dentty holds: Q G. For every Q X E s F ; ; τ s τ X (.5.3) b b q Remark.5.9: If Ns, the equalty n (.5.) s always satsfed wth N t t s U s U. s s,.e., t N r Nt t r Nt t Nr r r t t r APEPF η η E r P IAI EF IAI, r, q ; /,/, / E E B q q (.5.) B q Lemma.5.: Let a complex RV, whle Z h s h, where h ~ m, s, h and are complex random numbers. Let moments of any even order of re re Z Z G, arg h ~, re, Re, U, Z Z. The raw nteger re Z, wth respect to h and s are as follows: h q / q s q q q q m q q q E, E s E h, E s (.5.4) m m q Remark.5.: The formulaton of Z, n Lemma.5. s used to study the SISO setup of Secton.5.5 I. Snce the phase of accordng to Remark.5., as well. h s unformly dstrbuted, Z, s sphercally symmetrc, Securty: Publc Page 53

54 Lemma.5.3: Let, satsfy the equalty n (.5.6) wth X q X q G n (.5.9) wth re E Z formulated as n (.5.3). Let q q, q I, Δ τ I, Δ τ / h G D S D S. Let Q G. For every X, the followng dentty holds: s Q X E F, m; ;; s X (.5.5) b b m q Lemma.5.4: Let a b-dmensonal modulaton scheme wth equ-probable symbols, whch s dentfed by the quadruplets of parameters and,,,,,, [.] Let an nterference-oblvous SI -optmum demodulator formulated n the sngle-stream desred form of (.5.7), as: Let I, Δ τ I m Δ τ, m, m arg mn m m η M ηˆ Δ τ D S D S D wth E D Δ τ Δ τ, m (.5.6). The Average Symbol Error Probablty (ASEP) of (.5.).e., m m ASEP Pr η η can be formulated as n (.5.5), where: m ˆ sn P E IAI, r, q ;,, IAI,, q q sn r B d M SINR B D q (.5.7) wth M s exp sd S D S I I E and,, M SINR s E r IAI s IAI, r, E E M SINR Bq q q B q m E / N s t SINR s defned n (.5.8) and (.5.9) b F m ;; E / N sy P E / N y exp N x y exp Q E / N y x dxdy t IAI t t E b E b E b/ q IAI, r, q r N r IAI r p q / q SINR B B q G (.5.9) q Nt Nt Nt q (.5.8) The mathematcal formulaton of the ASEP s possble thanks to the ED-based representaton of the aggregate other-cell nterference,.e., D upon IAI, r, and B q q Securty: Publc Page 54 3,,. In fact, by condtonng, the decson metrc of the demodulator n (.5.5) bols down to that of an equvalent demodulator n AWGN. As a consequence, the wdely adopted

55 mathematcal formulaton of the ASEP of b-dmensonal modulatons can be used. As a frst G s step, n fact, only the randomness of the AWGN and of the complex Gaussan RVs q taken nto account. The condtonng wth respect to IAI, subsequently. r, and q q q B s removed The constrant E Δ τ represents a normalzaton factor that can be Δ τ D, m understood, for specfc MIMO setups, by drect nspecton of Secton.5.5. From Secton.5.5, n partcular, D Δ τ Δ τ, m, whch mples E D Δ τ as a Δ τ, m result of the zero-mean and average unt-energy constrants assumed for the constellaton dagram. Lemma.5.5: Let an nterference-oblvous desred form of (.5.3), as: SI -optmum demodulator formulated n the M η M ηˆ arg mn Δ τ The Average Parwse Frame Error Probablty ( F APEP F APEP η η Pr Δ τ, can be formulated as (.5.) ), whch s defned as sn P EF IAI, r, q ;,, IAI,, q q sn r B d M D SINR B q (.5.) wth M s exp sd S, Δ τ D S I E. I Remark.5.: From APEP F, the Average Frame Error Probablty (AFEP) of b-dmensonal modulatons can be obtaned by usng the Nearest Neghbor (NN) approxmaton. The AFEP can be calculated by consderng only the pars of transmtted ( η ) and hypothess ( η ) vectors that dffer n a sngle entry. Assumng that the M possbltes when ths condton holds are equprobable, the ASEP follows from the AFEP and can be formulated as ASEP AFEP / M. Securty: Publc Page 55

56 Theorem.5.: Let s s q q s exps m M, fr exp and M B for every nteger q. Let the SINR (, ) defned n (.5.9). Its MGF can be formulated as shown n (.5.8), where exp Q p Q and q q Q G. q PIAI E IAI s τ s τ, E / N Corollary.5.: Let the SINR (, ) n (.5.8) wth regme s consdered. Then, M M SINR n (.5.9) smplfes as follows:,.e., an nterference-lmted ;; P Q z F m sz z IAI s M s m E / s dz N (.5.) SINR SINR Theorem.5.: Let s s m M, fr exp q, M B s exps for every nteger q. Let the SINR (, ) and the error probablty ntegral n (.5.9) and (.5.7), respectvely. The dentty n (.5.8) holds, where P E exp, p IAI IAI s τ s τ T ;, s defned n [.55]. q sn /, Q Q, q q Q G and q.5.5 Applcaton to MIMO Cellular Networks In ths secton, varous MIMO arrangements are studed and t s proved that ther error probablty performance can be formulated as n Secton.5.4. By drect nspecton of Theorem.5. and Theorem.5., t s apparent that the error probablty ntegrals n depend only on three parameters,.e., m, and Q. In the followng sub-sectons, such a trplet of parameters s computed for relevant MIMO schemes. It s worth mentonng that for some MIMO setups the proposed mathematcal framework provdes only approxmated expressons of the error probablty, as explaned n the sequel. A summary of the trplet m,, Q s provded n [.55] for all MIMO schemes analyzed n the present report. Also, [.55] hghlghts when the framework s exact or approxmated. For all analyzed MIMO transmsson schemes, the followng procedure s appled: Securty: Publc Page 56

57 D, ) From the sgnal model n (.5.) and the demodulator n (.5.3), the functons D and,, D are computed. If 3,, the MIMO scheme s IAI-free. ) The functon,, IAI IAI,,, D s computed as well. Otherwse, D s computed such that the constrants n (.5.4) and (.5.6) are all satsfed. The dfference between exact and approxmated results may emerge at ths step: f all constrants are satsfed wth equalty, the mathematcal formulaton s exact. Otherwse, t s an approxmaton. 3) By lettng M M for the sngle-stream demodulator and s D s s s D the mult-stream demodulator, the parameters m and m M holds. that s s M M for are computed from D such,.5.5 Sngle-Input-Sngle-Output Transmsson over Nakagamm Fadng Let a SISO transmsson scheme and a Nakagam-m fadng channel model. Thus, Nt Nr Ns, M,, ~ m, s η, s η \ for,, m H ~ G, and H G \ for. Let the nterference-oblvous demodulator n (.5.7) wth, S I H, I y, y y S,, b, I Er y S s H η and N. Snce a sngle symbol s transmtted, the demodulator s IAI-free and IAI. By nsertng (.5.) n (.5.3), we obtan Δ η η, I, Δ τ H Δ, D S, (), *, D, S I, Δ, nτ H Δ () n and 3,,,, *,, D formulated as n (.5.8) wth I, Z H Δ () H s (). From (.5.4), D S, Δ τ H Δ (). Thus,, I H m D S G and m, m, / m. ~,, Q follows from (.5.4)..5.5 Spatal Multplexng MIMO Transmsson over Raylegh Fadng -- Optmal Demodulaton Let a spatal multplexng MIMO transmsson scheme and a Raylegh fadng channel model. N, M Nt Thus, s, s η, s η \ for, rt, H ~ G, and Securty: Publc Page 57

58 rt, H ~ G, for t,,, Nt, r,,, N \ r and. Let the nterferenceoblvous demodulator n (.5.3) wth I y, s / H η N N b S I E Nt r and r demodulator s IAI-free and IAI I S H, y S, y y,. Wth ths choce, the mult-stream, as well as Nr Nt, Nr Nt, D SI r D S t I r t r t, and H s U Δ η η, r t t r t t * r Δ τ H Δ Δ τ n τ H Δ n, (),,, () N N D r u u r as n (.5.8) wth Z () () and 3,,, r u * r N t r, t t U H Δ. From (.5.4),,Δ τ () t () D S I, Nr Nt r t t r t H Δ. Snce () () ~, () Nt r, t t Nt t H Δ t Δ t and CN, then N t t,, () Δ. Fnally, m N r t N t t s s. t Nt t I Δ τ N N Δ D S, ~ G r, r () t Q follows from (.5.) wth.5.5 Sngle-Input-Multple-Output (SIMO) Transmsson over Raylegh Fadng Let a SIMO transmsson scheme and a Raylegh fadng channel model. Thus, N t N, s M, s η, s η \ for, r, H ~ G, and r, ~, H G for r,,, N \ r and. Let the nterference-oblvous demodulator n (.5.7) wth, S H r I for,,, r r I r N, b r, S I Er for r,,, Nr and N Nr y, s H η s transmtted, the demodulator s IAI-free and y S, y y,. Snce a sngle symbol IAI. By nsertng (.5.) n (.5.3), we obtan Δ η η, I Nr r,, Δ τ Δ () H D, S r, * * N,,,, () r r r D S I Δ τ n τ Δ H n and 3,,, wth r, Nr r r, H r s U Z () () and * r r, D formulated as n (.5.8) U () H Δ (). From (.5.4), D, S I,Δ τ Securty: Publc Page 58

59 Nr r, Δ () H. Thus, N r r, H ~ N, N r Fnally, Q follows from (.5.) wth D SI r G r r and m, N r, s s Orthogonal Space-Tme Block Codng (OSTBC) Transmsson over Raylegh Fadng Let an OSTBC MIMO transmsson scheme and a Raylegh fadng channel model. Based on [.56], generalzed complex orthogonal desgns of sze t N N are consdered. Thus, M / N I and the s Nt space-tme encodng matrx Θ η ; S Θ η satsfes the property D H Θ η Θ η Θ η, where t, t M m m D Θ η pt η for,,, t m t t t N, s t, t η t t N and D Θ for,,, t t N m p t are strctly postve numbers for,,, t and m,,, M. For example, t N and m,,, m p p t for,,, t M wth p and p f the matrces n [.56] and n [.56] are consdered, respectvely. As for the channels, rt, H ~ G, and rt, H ~ G, for t,,, Nt, r,,, Nr and \. Let the nterference-oblvous demodulator n (.5.7) wth S I H, S I,, I y τ y τ y τ b, E / Nt r N N y τ S s τ Θ η and r. Wth ths choce, the mult-stream demodulator s IAI-free and the mult-stream demodulator can be rewrtten n terms of the sngle-stream demodulator n (.5.) by explotng the propertes of Θ. In partcular, let where N ): Δ for be formulated as follows (,,, s M * M t, t, t t m m t m m m m Δ Θ η Θ η α η η β η η (.5.3) α and space-tme encodng matrx for ): we obtan, for m,,, M the defntons gven n (.5.6) β are Nt complex vectors for,,, Ns, whch depend on the Θ. Then, wth the ad of the property [.56] as follows (t holds N N N N N M r t t r t m m m t r t r t m H Δ H p η η (.5.4) r s t r t,,, the denttes n (.5.5) and D can be formulated wth 3, m,, Securty: Publc Page 59

60 r t m m m r, t I Nt, N N D S, Δ τ η η p H D S, Δ τ, n τ, m t, m I r t Nr Ns * * r r t t m m t m m n H α η η β η η r t Z Nr Ns Nt Nr Nt Ns r, u u r r, u u r, H s U H s U r u r u t, N r r t t m m t m m t * U H α η η β η η (.5.5) (.5.6) It s apparent that Z, n (.5.6) s formulated as shown n (.5.) of Lemma. To proceed wth the analyss, t s mportant to understand whether the equalty n (.5.3) s satsfed for arbtrary generalzed complex orthogonal desgns. Ths ssue s addressed n the followng. Defnton.5.5: Let a generalzed complex orthogonal desgn Θ, accordng to [.56]. Let Z, n (.5.6). Θ s sad to be nterference-orthogonal f the equalty n (.5.3) s satsfed for M m s τ η. m Proposton.5.: The generalzed complex orthogonal desgns n [.56] are nterferenceorthogonal, whle those n [.56] are not nterference-orthogonal. The reason why some generalzed complex orthogonal desgns are not nterferenceorthogonal s due to the quas-statc assumpton for the other-cell nterference. Ths mples, n fact, that the terms r N t r, u u H s n (.5.6) are not ndependent, snce they u orgnate from nterferng BSs belongng to the same PPP. Comparng wth, r Z n (.5.6), we note that ths does not occur for D, m,, D n (.5.5), m,,, snce the nose terms n are ndependent for r,,, Nr,,,, Ns. The generalzed complex orthogonal desgns n [.56] are desgned based on the ndependence property of the AWGN. Hence, some code constructons may not satsfy the nterference-orthogonal property that orgnates from the partal correlaton of the nterference across the receve-antennas and the tme-slots. Securty: Publc Page 6

61 The proposed mathematcal approach s applcable to nterference-orthogonal generalzed complex orthogonal desgns. It can be appled to generalzed complex orthogonal desgns that are not nterference-orthogonal, by assumng that the equalty n (.5.3) holds true. In ths latter case, the framework s no longer exact, but t s an approxmaton. In Secton.5.6, t s shown that t s accurate enough for typcal MIMO setups though. If Θ s nterference-orthogonal, the equaltes n (.5.4) are satsfed and we obtan m m Nr Nt m r, t I, Δ τ η η pt H D S and N, r Nt p m r t, m I t H, m r t Snce, for typcal OSTBCs [.56], m p p t for,,, t D S. r t t N and m,,, D, m SI ~ G NrNt, NrNtp. Ths mples m, NrNt, p formulated as n Lemma wth Ns, we conclude that M m s τ η. m M, then. Snce, Z s Q follows from (.5.) wth.5.5 Spatal Multplexng MIMO Transmsson over Raylegh Fadng - Worst-Case Let a spatal multplexng MIMO transmsson scheme and a Raylegh fadng channel model. N, M Nt Thus, s, s η, s η \ for, rt, H ~ G, and rt, H ~ G, for t,,, Nt, r,,, N \ r and. Let the nterferenceoblvous demodulator n (.5.7) wth (, ) S H rm I for,,, r r N, where m denotes the sngle stream of the ntended lnk that MT s nterested n demodulatng. Also, let I y S, y y, and r b, S I E / Nt r for r,,, Nr y, s H η r r m m N N. Thus, unlke the spatal multplexng MIMO scheme of Secton.5.5 B, N symbols of the ntended lnk are treated as nterference. Accordngly, ths MIMO scheme s affected by other-cell nterference and by IAI,.e., (.5.7), we obtan Δ ( m) η ( m) η ( m), r * * m N,,,, () r r m r D m S I Δ τ n τ Δ H n, 3, m,,, Nr Nt H r t s t U r, r t Z and IAI n (.5.3). By nsertng (.5.) n m N,, () r r m I Δ τ Δ H D S,, m r r r, m m D as shown n (.5.8) wth * U () H Δ (), as well as t Securty: Publc Page 6

62 m N,, r r m I Δ τ Δ H D S r and IAI IAI s τ ~ CN, wth IAI N t u M m IAI IAI u, u m m, m m s τ s η. From (.5.4), we obtan m N,, r r m I Δ τ Δ H D S and N, r rm, m I H, m r D, m SI ~ G Nr, Nr and m, N r, wth Ns, we conclude that. Snce, Q follows from (.5.) wth D S. Thus, r Z s formulated as n Lemma N t t s s. t.5.5 Zero-Forcng (ZF) MIMO Recever over Raylegh Fadng Let a MIMO transmsson scheme wth ZF-based recepton and a Raylegh fadng channel N model [.57]. Thus, r t N, Ns, M Nt, s η, s η \ for, rt, H ~ G, and rt, H ~ G, for t,,, Nt, r,,, N \ r and. Let the nterference-oblvous demodulator n (.5.3) wth S I H, H H S I, I y, y H H H y b E Nt r y S, s / s and N M N t. Accordngly, the mult-stream demodulator n (.5.3) s IAI-free wth IAI and t can be re-wrtten n the sngle-stream formulaton of (.5.7), where W H H H denotes the ( M N ) N H H m D, m S I, Δ τ Δ, m I ZF matrx at the recever, t r Δ η η, * m N r m, r r r D S, Δ τ, n τ Δ W n and, D 3, m,, as shown n (.5.8) wth Nr Nt H r, t s t, U r r t *, U r Δ m W m r. From (.5.4), a m m, m H Δ W, where mm, Thus, a Z and D S I m N,, r m r Δ τ Δ W, m r W H H and (a) follows from drect nspecton of W. D, m SI W ~ G Nr Nt, Nr Nt, where (a) follows from [.57] and [.58]. Ths mples m, N N, wth Ns, we conclude that r t. Snce, Q follows from (.5.) wth.5.5 Zero-Forcng MIMO Precodng over Raylegh Fadng Z s formulated as n Lemma N t t s s. t Securty: Publc Page 6

63 Let a MIMO transmsson scheme wth ZF-based precodng and a Raylegh fadng channel model [.59]. Let N u sngle-antenna MTs be served by ntended and nterferng BSs n the same channel use wthn ther respectve cells. The sgnal model n (.5.) s stll applcable wth mnor changes. Wth a slght abuse of notaton, let H denote the u Nt downlnk channel matrx of the lnks from BS to ts N N u ntended MTs. Lkewse, let denote by N H for \ the u Nt downlnk channel matrces of the nterferng BSs towards the same u MTs as BS. Also, let ˆ N H \ for denote the u Nt downlnk channel matrx of the lnks from the th nterferng BS ( BS ) towards ts ntended general, ˆ H H. Then, (.5.) stll holds by replacng r N wth N u sngle-antenna MTs. In N and by lettng Nt Nu, u N N r Ns, M Nu, ut, H ~ G, and ut, H ~ G, for t,,, Nt, u,,, N \ u and s defned as N. Let V denote the t Nu precodng matrx used at BS, whch V V V η H H wth / V H H H. Lkewse, let V ˆ denote the precodng matrx used at BS, whch s ˆ ˆ ˆ V V V η wth ˆ ˆ H ˆ ˆ H / V H H H \ for. Based on these precodng matrces, whch assume that the sde nformaton avalable at BS and BS s S and I V ; I s Θ η V η S and Vˆ S, respectvely, the transmtted vectors are I ; ˆ I \ s Θ η S V η for. Let the nterferenceoblvous demodulator n (.5.3) wth I b I E Nt r y, s / η / V η S and u N M N. Accordngly, the multstream demodulator n (.5.3) s IAI-free wth I S V η, y S, y y, IAI and t can be re-wrtten n the snglestream formulaton of (.5.7) for each ntended user u m,,, Nu. In partcular, we have Δ η η, m I, Δ τ Δ V η D S,, m * m D, m S I, Δ τ, n τ Δ Vη n and 3, m,, D as shown n (.5.8) wth Nt m, t t m Z H s U, ˆ, t s Vη s and * U Δ V η. From m m m (.5.4), D, m S I, Δ τ Δ V η and m I D S V η. Snce Z, s, Securty: Publc Page 63

64 formulated as n Lemma wth N s N, we conclude that s N ˆ t s t V η Nt s t, snce ˆ t t r Q follows from (.5.) wth s Vη \ for. So far, the analyss for ZF precodng s exact and no approxmatons have been used. To complete * H the analyss, however, the dstrbuton of D S V η η V Vη needs to be, m I computed. To the best of our knowledge, however, t s unknown for dscrete modulaton schemes. To get a tractable yet accurate mathematcal framework, we explot two approxmatons for the computaton of the dstrbuton of η follows a unt-energy complex Gaussan dstrbuton,.e., D,m. Frst of all, we assume that m η ~ CN, for m,,, M Nu. From [.54], we obtan * H ηv ~, Vη Nu Nt Nu F Nu Nt N, whch mples u I D, S ~ N N / N F N N,N. Second of all, we approxmate ths m t u u t u u resultng scaled F-dstrbuton wth a scaled Ch-Square dstrbuton,.e., D S, whch s known to be accurate for N and, n turn, can be I ~ N, m u N t N u re-wrtten n terms of a Gamma dstrbuton,.e., D, m S ~ Nt Nu, N I G u Nt Nu. Ths mples m, Nt Nu, / Nu u. In Secton.5.6, these approxmatons are shown to be accurate enough for typcal MIMO setups..5.5 Numercal and Smulaton Results In ths secton, numercal examples are shown to substantate the accuracy of the mathematcal frameworks and to confrm the performance trends.the frameworks are compared aganst Monte Carlo smulatons, whch are obtaned by usng the procedure descrbed n [.], [.55]. The accuracy of the PPP-based abstracton for modelng the error performance of cellular networks s nvestgated by comparng t wth grd-based abstracton models. Hence, smlar curves are not reported n ths secton. The smulaton setup s summarzed n the capton of each fgure, where markers show Monte Carlo smulatons, sold lnes the analytcal framework and dashed lnes the asymptotc framework. As for the mplementaton of the mathematcal frameworks for QAM wth M 4 follows are used: /, 3 / M, /,, the parameters as 4 M / M and /4, Securty: Publc Page 64

65 6 / M, /4,,,, /,, /, s used. 4 M / M. If M, the quadruplet Selected numercal examples are llustrated n Fgs , where the ASEP s depcted as a functon of E / N, whch s a reference sgnal-to-nose-rato that s computed at a fxed reference dstance of one meter from the transmtter. These fgures confrm the accuracy of the proposed mathematcal frameworks. The approxmatons proposed n Secton.5.5 are confrmed to be suffcently accurate n the consdered setup. Smlar accuraces are obtaned for dfferent parameters. Fg.5.: ASEP as a functon of m and b Ths confrms that the mpact of the fadng severty s neglgble n the presence of other-cell nterference. The path-loss exponent has a dfferent mpact n nose- and nterference-lmted regmes. In partcular, a bgger path-loss s benefcal n nterference-lmted cellular networks, snce the other-cell nterference s reduced. Fgure.5. (b) shows a smlar behavor n the presence of receve-dversty. Fgure.5. (a) confrms that receve-dversty s stll benefcal, but the gan n the presence of other-cell nterference s reduced compared to the noselmted scenaro. Securty: Publc Page 65

66 Fg..5.: ASEP as a functon of Nr and b Fgure.5.3 shows that the performance gan offered by transmt-dversty compared to receve-dversty n nose-lmted networks s not observable n the presence of other-cell nterference. In fact, the ASEP of Fg..5.(a) and Fg..5.3(a) s almost the same. Fg.5.3: ASEP as a functon of Nr Fgure.5.4 shows the ASEP of spatal multplexng MIMO and t brngs to our attenton the detrmental mpact of IAI, even n the presence of strong other-cell nterference. In general, we observe that ncreasng the number of antennas s benefcal f a mult-stream Securty: Publc Page 66

67 demodulator s used (see Fg..5.4 (a)). By comparng Fg..5.4 (a) wth Fg..5.3 (a) for Nr, whch provde the same rate, we observe that spatal multplexng provdes better performance than the Alamout code, but at the cost of a hgher demodulaton complexty. Fg..5.4: ASEP as a functon of Nt, Nr and modulaton order Fgure.5.5 compares ZF recepton and ZF precodng under smlar operatng condtons, and under the assumpton that M s ndependent of confrms that the ASEP gets worse by ncreasng N t and N t and N u, respectvely. The fgure N u. A close nspecton of Fgs..5.5(a) and.5.5(b) reveals that ZF recepton and precodng provde almost the same performance n the nterference-lmted regme. Archtectural desgn and mo/demodulaton complexty are, however, qute dfferent between them. Fgure.5.6 provdes a sound confrmaton of some non-trval trends hghlghted n [.55]. Fgure.5.6(a) shows that the ASEP may get worse by ncreasng shows that the ASEP gets better by ncreasng N t, whle Fg..5.6(b) N t. The trend n Fg..5.6 (a) orgnates from the fact that K PSK decreases by ncreasng N t and that ths effect s not counterbalanced by the reducton of the modulaton order M. On the other hand, the trend n Fg..5.6 (b) follows because K PSK s kept fxed by ncreasng N r. As a result, reducng the modulaton order M s benefcal ( K decreases). By comparng Fg..5.6 (a) wth Fg..5.4 (a) for PSK Securty: Publc Page 67

68 N 4 (same MIMO setup and rate), the performance vs. complexty trade-off between t spatal multplexng wth ML-optmum demodulaton and ZF-based recepton clearly emerges n the nterference-lmted regme. In concluson, the proposed mathematcal frameworks are suffcently accurate and nsghtful to the analyss, desgn and optmzaton of MIMO-aded cellular networks. Fg.5.5: ASEP as a functon of Nt and Nu Fg..5.6: ASEP as a functon of Nt, Nr and modulaton order Securty: Publc Page 68

69 3. Performance Analyss of Uplnk Heterogeneous Cellular Networks 3. Introducton The use of PPP-based abstracton model for modelng heterogeneous networks and dervaton of the correspondng downlnk coverage and rate under varous assocaton and nterference coordnaton strateges has been extensvely explored, several notable examples can be found n Secton and reference theren. However, the analyss of the uplnk by PPP-based abstracton model s lmted, because the nterferng MTs n uplnk are not strctly Posson dstrbuted nodes and the transmt power of an nterferng MT s correlated wth ts path loss to the BSdue to the uplnk power control. Several examples on uplnk cellular analyss by utlzng stochastc geometry can be found n [3.][3.][3.3]. However, there s no studes on performance n the uplnk wth mult-antenna recever to the best of authors knowledge. In ths secton we provde a mathematcal framework to study the coverage and rate performance cellular uplnks wth fractonal power control and then extend the analyss to mult-antenna recever and heterogeneous networks. 3. Stochastc geometry modelng and analyss of the coverage probablty and average rate of sngle-ter uplnk cellular networks 3.. System Model and Problem Formulaton In ths secton, both the MTs and BSs are equpped wth sngle transmt/receve antenna. We assume that the BSs are modeled as ponts of a homogeneous PPP ( ) of densty and that the MTs are modeled as ponts of a homogeneous PPP of densty MT. Each MT s assumed to be connected to ts nearest BS. The probe BS s denoted by BS. Some mportant assumptons are made to study the uplnk performance accordng to [3.]: )Full load assumpton, whch means that MT and all BSs are actve; ) Each BS selects only one MT to serve at one resource block; ) Correlatons between MTs are neglected and actve MTs approxmately form a PPP wth same densty as BS densty. Securty: Publc Page 69

70 The PDF f and CCDF F R R of the closest dstance between a MT and BSs are as follows: R F R exp exp f (3..) The MTs apply truncated fractonal power control [3.][3.][3.3], mathematcally the transmt power of MTs can be formulated as follows: P TX,, f, otherwse p R p R PMT (3..) where p s the pre-defned reference power, s the path-loss exponent, s the fractonal factor, P MT s the maxmum allowed transmt power, and R denotes the closest dstance condtoned on connecton, whose dstrbuton s as follows: Let the event f R f R P MT,, (3..3) Pr X p X denote connecton, the probablty of connecton p Pr X f exp p Pr X s: P MT P MT d R (3..4) Wth fractonal power control, the receved sgnal at the probe BS n ths uplnk cellular network s as follows: p R p R D D R n (3..5) y s h s,,,,,,,,,, where D, denotes the dstance to probe BS from th nterferng MTs, \ \ denotes the set of nterferng MTs, whch s stll approxmately PPP wth densty. The ndcator functon,, D R s the constrant due to shortest-dstance cell assocaton n the uplnk. The nose n, s AWGN wth nose power. The SINR of ths uplnk cellular network can be formulated as follows: U SINR, \ I, (3..6) Securty: Publc Page 7

71 where: where U, s useful sgnal power and U, pr, h, \, pr, D, h, D, R, \ I \ I, s the aggregate other-cell nterference,, (3..7) h and, h for \ are the per-lnk power gans of ntended and nterferng lnks, whch are assumed to be Gamma dstrbuted Gamma,,.e., Raylegh fadng channel model s used. The coverage probablty ( as follows: P cov ) and average rate ( R ) are studed. They can be formulated Pcov PrSINR cov a (3..8) b R E ln SINR P exp t dt ln y P y dy (3..9) where s a relablty threshold, (a) follows from [.] and (b) follows by applyng ntegraton by parts, snce P and cov cov P. cov 3.. Gl-Pelaez Based Mathematcal Modelng Mathematcal frameworks to the computaton of (3..8) and (3..9) are provded, by assumng that RVs., h and, h for \ as ndependent but dentcally dstrbuted Gamma Theorem 3.. If sngle ter SISO cellular uplnk, the SINR coverage probablty s as follows: P p d I, (3..) r j P X m Q j e CF \ I cov where the functons are computed as: Q PMT p x CF \ I, x exp d exp F,, PMT p, j px 3 x exp x Proof: Theorem 3.. can be derved by applyng Gl-Peleaz nverson as n Secton.3. Pr X (3..) dx Securty: Publc Page 7

72 Theorem 3.. If sngle ter SISO cellular uplnk, the average rate can be computed as: where: Im d R Z ; d j \ e CF f I (3..), R R fr Pr ln X jp p Z ; ln y j dy y y jp f jp jp Proof: Applyng ntegral by parts we have ' R Eln SINR Pcov y dy ln y Pcov y dy y ' j Pcov y Im e CF \ Q ; y I, jp N R p Q ; y j f d R y y d And the rest closed-form s wth the ad of the Melln-Barnes theorem n[3.8]. (3..3) (3..4) 3.3 Extenson to mult-antenna uplnk cellular networks wth MRC recever 3.3. System Model The same system models and assumptons as n Secton 3.. hold. Smlar to (3..5),but assumng that all BSs are all equpped wth probe BS s as: N r receve antennas, the receve sgnal at the s s D R y p R h p R D h n,,,,,,,,,,, \ (3.3.) where h, and h, are channel vectors whose entres are assumed to be..d. complex Gaussan RVs,.e., Raylegh fadng channel models are appled. n, s AWGN nose vector wth per dmenson nose power. If MRC recever appled, the weght vector s * h, [3.4] and the post-processng sgnal s as: Securty: Publc Page 7

73 z h y *,,,,, s s p R p R D D R h h h h h n * * *,,,,,,,,,,, The SINR can be formulated as follows [3.4]: \ U SINR, \ I, (3.3.) (3.3.3) where: U, pr, h, * \ h,, pr, D, h, D, R, \ h, I (3.3.4), h and h h *,, h, are ndependent RVs: h h h, *,, ~ Gamma Nr, N r h, ~ Gamma, (3.3.5) Proof: Nr r r h h ~ Gamma, ~ Gamma Nr, Nr.Condtoned on h,,,, r r N h h *,, h, s a lnear combnaton of complex Gaussan random varables because h, and h, are ndependent, so that h h *,, h, condtoned on h, s complex Gaussan. Compute the mean and varance of h h *,, h, condton on, * * h h E E and h, h,,, h : h, h, h, * * * h, h,e h, h, h, E h, h, respectvely, so that the PDF of h, h, condtoned on, h can be shown as f * x h h, h, h, *,, exp h h, h, h h *,, h,, whch s Securty: Publc Page 73

74 ndependent of h,. So that h h *,, h,, s complex Gaussan and h ~ Gamma, h h *,,, whch s ndependent wth h, Gl-Pelaez Based Mathematcal Modelng Smlar to Theorem 3.. and 3.., mathematcal frameworks to the computaton of (3..8) and (3..9) are provded, by havng, h and h h *,, h, for \ as ndependent but dentcally dstrbuted Gamma RVs. Theorem 3.3. In sngle ter uplnk cellular networks, f a MRC recever s used, the SINR coverage probablty s as follows: P p d I, (3.3.6) r j P X m Q j e CF \ I cov where the functons are computed as: PMT p \ I, N R exp Q x x d CF PM T p exp F,,, j p x 3 x exp x Proof: Theorem 3.3. can be derved by applyng Gl-Peleaz nverson as n Secton.3. Pr X (3.3.7) dx Theorem 3.3. In sngle ter uplnk cellular networks, f a MRC recever s used, the average rate can be computed as: where: Im d R Z ; d j \ e CF f I (3.3.8), R Z f R Wth nteger-valued ; ln y X jp N r p j y y fr Pr N, Z r a has closed-form exsts, as follows: ; N r dy (3.3.9) Securty: Publc Page 74

75 N N N r r r, Z 3, Z 4, Z ; where we use short-hands ln A A A A A A ; ln A A A A A A ; ln A A A 6 A 3A 6 A A A 6 A for smplcty. A jp Proof: See proof of Theorem (3.3.) 3.4 Extenson to mult-ter heterogeneous uplnk cellular networks 3.4. System Model and Problem Formulaton Assume ak ters uplnk cellular network, where the MTs are equpped wth sngle transmt antenna and that BSs are equpped wth N r receve antennas and apply MRC recever. BSs locatons n each ter can be modeled by PPPs wth denstes j, j,..., K. MTs are modeled by a PPP wth densty MT. Same assumptons as n Secton 3.. and 3.3. hold. PDF and CCDF of the closest dstance between MT and ter j BSs are as follows: Rj F R j j exp j exp j f (3.4.) Cell Assocaton The assocaton beng n ter j s based on [3.5]: k j : arg max T R k,..., K k k, (3.4.) Where s the path-loss exponentper ter, Rk, s the closest dstance between MT and th k BS n ter k and PDF and CCDF of R k, are the same as defned n. T k denotes the assocaton weght and three dfferent assocaton schemes can be studed and compared. Assocaton : based on the receved power n the downlnk. where P k denotes ter k BS transmt power. T k P k Securty: Publc Page 75

76 Assocaton : based on best based receved power n the downlnk. where B k denotes ter k bas factor. T P B k k k Assocaton 3: smallest path loss assocaton. Let T k X j denote the event that cell assocaton beng wth ter j. The probablty Pr X j denotes the assocaton probablty, whch can be computed as follows: The CDF and PDF of be formulated as follows. j k X j TjRj, Tk Rk, k j Pr Pr, (3.4.3) R j, whch s the shortest dstance condtoned on ter j assocaton, can F f Rj Rj R j / k K Tk j f R j F R R j R j dr j k k j T j j j P MT,, / P j j k MT K p j T p k j j f R R j F R j dr j j R k k j T j f Rj P MT j j p j Proof: t follows the defnton of condtonal probablty: F Rj / k K Tk j Rj F R R j k k T j j j j P MT, Rj, / k K p j Tk j fr Rj F R j dr j j R k k j T j Pr R, j j j X j P MT,, p j j j P MT Pr Rj, X j p j / k Tk j Pr Rj, Rk Rj, k j T j P,, / k p j j P MT Tk j Pr Rj, Rk R j, k j p j T j Ths conclude the proof MT j j j (3.4.4) (3.4.5) Securty: Publc Page 76

77 As n (3..), fractonal power control scheme defnes the MT transmt power condtoned on a MT beng assocated wth ter j of BS: P TX, j, f j j j j p R p R P, otherwse j j j j MT (3.4.6) where p j s the pre-defned reference power per ter, j s the path-loss exponentper ter, j s the fractonal factorper ter. Sgnal and System Model Smlar to (3.3.),but assumng K ter cellular uplnk, the receve sgnal at the probe BS s as: where j j y pr h s n j, j j, j, j, j, \ j K k j k j j j j j, j, h j, s j, j, j, p R D D R p R D D T R k k j j k k k, k, gk, sk, k, k, Tk k denotes the set of actve MTs n ter k, whch s PPP wth densty / j (3.4.7) k. D denotes the dstance to probe BS from nterferng MT. The ndcator functons Dj, Rj, and / j Tj k Dk, R k, are the constrants due to cell assocaton n the uplnk. Raylegh T k d fadng model s assumed,.e., h g ~ CN, I N dmenson nose power. R. The nose vector s AWGN wth per The MRC recever multples a weght vector N h *, j C r for the receved sgnal j, y. Thus, the post-processed sgnal s: z * j, j, j, \ j * * h h s h n j j j j, j, j, j, j, j, pr K h k j k j j j * j j, j, h j, h j, s j, j, j, p R D D R The output SINR can then be formulated as: y p R D D T R k k j * j k k k, k, h j, gk, sk, k, k, Tk / j (3.4.8) Securty: Publc Page 77

78 SINR j U j, K \ I j, j Ik, j k j (3.4.9) where the short-hands a follows as used: U I j j, pr j j, h j, * \ j j h j j, j, j p jrj, Dj, h j, Dj, Rj, \ h j j, (3.4.) I * k k j j, T h j k k, j pk Rk, Dk, gk, Dk, Rk, h T k j, k / j j, h N N, h ~ Gamma, j r r, ~ Gamma, ndependent Gamma RVs. Proof: See proof of (3.3.5). h h * j, j, h * j, and g ~ Gamma, h j, k, are Problem Statement We focus on the coverage probablty: cov where the per ter coverage s defned as: K cov jpr j P P X X (3.4.) j Pr SINR Pcov X j j (3.4.) 3.4. Gl-Pelaez Based Mathematcal Modelng Smlar to Theorem 3.. and 3.3., mathematcal frameworks to the computaton of (3.4.) are provded, by havng j, h, h h * j, j, h j, for \ j and h h * j, j, g k, for k as ndependent but dentcally dstrbuted Gamma RVs. Theorem 3.4. Let SINR n (3.4.), the general framework to compute SINR coverage probablty n the uplnk SIMO networks wth MRC recever can be summarzed as follows: K K K p d Pco v X CF j j T k j j j Pr j Im \ Qj j e CF,, I j j I (3.4.3) k j Securty: Publc Page 78

79 where: Nr j j Qj x x f d Rj j j CF \, exp j x F,,, j pjx f xdx I j j R j j (3.4.4) j / j Tj k k T j CF expk x F,,, jpk x f xdx I k, j R k Tk j j T k And the assocaton probabalty: wth f Rj f Rj Pr X j Pr / k T (3.4.5) k f j X x F x dx k j Rj Rk k j Tj Proof: Theorem 3.4. can be derved by applyng Gl-Peleaz nverson as n Secton Large Scale Receve Antenna System Wth many recever antennas, we have SINR lnearly scaled wth the number of receve antennas N r : SINR j Nr N j j pr j j, r K \ I j, j Ik, j k j (3.4.6) Proof: By usng law of large number,let M xy, C be two ndependent vectors wth dstrbuton CN,cI. Then *.. * x y a s x x a. s. lm and lm c M M M M. We can have j, h N. N r r The framework to compute SINR coverage stays the same as n (3.4.4) but replaces Q x wth: j j Q exp j x N R f d (3.4.7) R j 3.5 Numercal Results We valdate n ths secton the mathematcal frameworks aganst Monte Carlo smulatons, and study the mpact of dfferent uplnk cellular setups. The smulatons are done by Securty: Publc Page 79

80 generatng PPP located BSs per ter and MTs n a D plane and then dong assocaton wth specfc scheme dscussed n (3.4.). To guarantee saturaton, a MT near orgn s selected and ts assocated BS s defned as the probe BS. The SINR coverage probablty and average rate are smulated for the probe lnk. Fg 3. and 3. valdate the coverage probablty framework for varous cellular uplnk setups and a good accuracy can be found. More specally, Fg 3. studes coverage performance n sngle ter cellular uplnk and t confrms the ntuton that more receve antenna leads to better coverage. Fg 3. sustans the general framework n Theorem Fg 3.3 valdates the average rate frameworks for varous cellular uplnk setups and llustrates the trends of average rate as a functon of path loss exponent and fractonal factor. It supports the ntuton that average ncreases by ncreasng path loss exponent. The effect of fractonal factor, on the other hand, s complcated. A detaled dscusson can be found n [3.][3.3]. as the fractonal factor ncreases, the rate decreases due to the loss n rate for some users whose transmt power s reduced, whch s not overcome on average by the reducton n nterference and ncreased rate for other users, especally those near the celledge. Fg 3.4 plots the coverage probablty for the scaled SINR SINR N r and confrms the analyss n Secton that output SINR at the MRC recever s lnearly scaled wth the number of receve antennas N r..9.8 Nr= Nr= Nr=3 Nr=4.7 Coverage Probablty Fgure 3. Pcov of a sngle ter cellular uplnk aganst threshold. Setup:, P MT Threshold [db] Watt, p 6dBm,.9, db. 5 and 6 Securty: Publc Page 8

81 Coverage Probablty Fgure 3. Pcov 5 and Threshold [db] of a two ter cellular uplnk aganst threshold. Setup: 6,, 5, P.4Watt, p p 6dBm,.9, T/ T 5, MT. db =3 =4 =5 =6 3 Average Rate fractonal factor Fgure 3.3 Average rate n nterference lmted regme of a sngle ter cellular uplnk aganst fractonal factor. Setup: 6, MT P, p 6dBm. Securty: Publc Page 8

82 Coverage Probablty N r = N r = N r =3 N r =4 approxmaton Threshold Fgure 3.4 Scaled SINR coverage of a sngle ter cellular uplnk wth massve receve antennas aganst threshold. Setup:.9, 6 and 4 Watt., PMT, p 6dBm, Securty: Publc Page 8

83 4. Performance Evaluaton of Relay-Aded Downlnk Cellular Networks 4. Performance evaluaton of cellular networks wth fxed relays 4.. Introducton Relay-aded wreless networks have been an actve feld of research for the last few years n both academa and ndustry[4.]. Furthermore, the Thrd Generaton Partnershp Project's Long Term Evoluton-Advanced (3GPP LTE-A) s consderng relay-aded archtectures for cost-ffectve throughput enhancement, coverage extenson and energy consumpton reducton[4.], [4.3]. Among the many cooperatve transmsson protocols, Amplfy-and- Forward (AF-) based relayng s consdered to provde a good trade-off between mplementaton complexty, cost and achevable performance [4.4]. As such, ts end-to-end error probablty and achevable dversty have been studed extensvely n the last years[4.5]. The avalable lterature on the mathematcal performance evaluaton of AF-based relay-aded wreless networks s vast. In partcular, the end-to-end error probablty and achevable dversty over dfferent fadng channels have been studed n nose-lmted wreless networks [4.5]-[4.] as well as n the presence of addtve nose and nterference [4.]-[4.3]. As far as the latter scenaro s concerned, the avalable frameworks usually rely on the assumptons that ether a fnte number of nterferers or a sngle domnant nterferer [.], [.6] s avalable n the network, that the locatons of the nterferers are fxed and known t a pror, as well as that the nterference follows a Gaussan dstrbuton [.]. Furthermore, the dversty combners at the destnaton are usually assumed to be oblvous to the nterference statstcs and dstrbuton, whch results n a low-complexty mplementaton but n sub-optmal performance. Examples of papers studyng more advanced demodulators are [.3]-[.36]. However, they do not consder relay-aded transmssons. These assumptons may be approprate for modelng and studyng classcal communcatons networks, e.g., carefully planned macro cellular systems. However, they may be less approprate to the analyss of emergng communcatons systems, such as heterogeneous cellular networks [4.38]-[4.4], where, e.g., cogntve rados [4.4] and closed-access femto base statons [4.] are randomly overlad wthn the macro cells. In such a scenaro, both cogntve rados and closed-access femto base statons act as nterferers to the macro cell users, whose number and locatons are random and hence unknown t a pror. Accordngly, none of the nterferers can be consdered domnant compared to the others and the aggregate nterference largely devates from a Gaussan dstrbuton [4.43]-[4.47]. Motvated by these consderatons, a few papers have recently studed the performance of relay-aded Securty: Publc Page 83

84 wreless networks n the presence of random nterference [4.48]-[4.54]. In [4.49], the outage of optmal, maxmal rato and selecton combnng s nvestgated. In [4.5], AF-based dualhop relayng s studed wthout dversty combnng at the destnaton. In[4.5], the achevable spatal-contenton dversty order of cooperatve relayng s computed. In [4.53] and [4.54], error probablty and dversty order of mult-hop relayng are studed n nterference-lmted networks wthout dversty combnng at the destnaton. In ths secton, we study AF-based dual-hop cooperatve protocols n the presence of Nakagam-m fadng, addtve nose at the relay, as well as addtve nose and symmetrc alpha-stable nterference at the destnaton. Compared to [4.48]-[4.54], the study n the secton s dfferent n many aspects: ) two nterference scenaros are nvestgated, whch arse, e.g., when ether the same or dfferent nterferers are actve durng the broadcast and relayng phases; ) a Maxmal Rato Combnng (MRC) and Selecton Combnng (SC) demodulators at the destnaton are analyzed. We provde closed-form expressons of the end-to-end Moment Generatng Functon (MGF) and study the achevable dversty for all consdered setups. Four man takeaway messages emerge from the mathematcal frameworks: ) the dversty order depends on the ampltude path-loss exponent ( bi ) of the nterferng network; ) under the assumpton that the transmt-powers of cooperatve and nterferng networks are ndependent, the dversty order s equal to / b I. The followng notaton s used throughout ths secton: Securty: Publc Page 84 s the set of postve natural numbers. GFM denotes a Galos Feld (GF) of sze M. j s the magnary unt. * s the complex conjugate operator. dstrbuton., varance. Re s the real part operator. d denotes an equalty n CN s a complex Gaussan Random Varable (RV) wth mean and A CN s a complex Gaussan RV condtoned upon the RV A. expectaton operator. E exp E s the M s X sx s the MGF of RV X. Pr X X F s the Cumulatve Dstrbuton Functon (CDF) of RV X. Complementary Cumulatve Dstrbuton Functon (CCDF) of RV X. Densty Functon (PDF) of RV X. the Gamma functon., z exp z x t t dt x c F F s the X f X X s the Probablty s the bnomal coeffcent. x exp s x t t dt s the upper-ncomplete Gamma functon.

85 x, z exp s the lower-ncomplete Gamma functon. z x t t dt, / B x y x y x y s the Beta functon. x x y / x Pochhammer symbol. x! x y s the s the factoral operator wth x be a natural number. n exp s the generalzed exponental ntegral functon.,, En x t xt dt Kummer confluent hypergeometrc functon [4.55]. a mn, p G pq, bq functon[4.56]. F s the s the Mejer G- x, y y / x, y / x,, y x / x wth x and y. g and g bi f for g f. 4.. System Model The dual-hop network topology sketched n Fg. 4. s studed, whch corresponds to a typcal scenaro consdered by the IEEE 8.6 workng group for relay-aded communcatons [4.]. A three-node network s consdered, where a tower-mounted source ( S ) communcates wth the ntended destnaton ( D ) located at the street level, wth the ad of a fxed relay staton ( R ) deployed on the rooftop of a buldng. The destnaton receves two copes of the same nformaton-bearng sgnal from the source and the relay. The destnaton s surrounded by randomly dstrbuted co-channel nterferers, whose spatal locatons are unknown t a pror and can vary over the whole b-dmensonal plane. The locatons of source, relay and destnaton, on the other hand, are assumed to be fxed and known. Snce the relay s located on the rooftop and the randomly-dstrbuted nterferers are located at the street level (see Fg. 4.), the relay s assumed to be nose-lmted and the destnaton s subject to both nose and co-channel nterference. Ths scenaro s smlar to [4.]. In ourwork, however, the nterferers have random spatal locatons. Possble applcaton scenaros of ths network deployment to heterogeneous cellular networks are descrbed n [4.47]. Securty: Publc Page 85

86 Fg.4.: System model The transmsson of S and R occur n two orthogonal tme-slots. In the frst tme-slot, S broadcasts ts data to R and D. Let GFM sgnals receved at R and D can be wrtten as follows: be the symbol transmtted by S. The S y E h x n SR S SR S SR, y E h x n SD S SD S SD T where: ) x s the M -ary modulated symbol emtted by S ; ) S M S M (4.) s the modulaton mappng functon; ) E S s the average symbol energy of S ; v) n XY s the complex Addtve Whte Gaussan Nose (AWGN) at the nput of node Y and related to the transmsson from node X. The AWGN s ndependent and dentcally dstrbuted (..d.) wth zero mean and varance / N per real dmenson, t{.e.}, n ~, N XY CN ; v) aggregate nterference at D n the frst tme-slot, whch s descrbed n Secton 4..; v) T s the s the fadng gan from node X to node Y. In partcular, h / d b XY exp j where: h XY, XY XY XY XY XY XY s a propagaton-dependent constant; d XY s the nodes' dstance; b XY s the ampltude path-loss exponent; XY s the fadng envelope, whch s assumed to follow a Nakagam-m dstrbuton havng parameters s assumed to be an nteger. It s known that, n general, mxy, XY. For mathematcal tractablty, m XY m XY may take non-nteger values [4.],[4.58]. The analyss of these scenaros, however, s postponed to future research. The Nakagam-m fadng model s consdered snce t s useful for modelng both lne-of-sght and Securty: Publc Page 86

87 non-lne-of-sght propagaton condtons [4.57]; and XY s the channel phase, whch s assumed to follow a unform dstrbuton n,. In the second tme-slot, R uses the AF protocol for relayng y SR to D. The sgnal receved y E h G y n, where: ) E R s the average at D can be formulated as RD R RD AF SR RD T symbol energy of R ; ) GAF / ES hsr s the AF relayng gan, by assumng an deal Channel State Informaton (CSI-) asssted protocol[4.9]. More accurate harmonc mean approxmatons for AF-based relayng have recently been proposed n the lterature [4.59]. For mathematcal tractablty, however, they are not consdered n ths paper; and ) the aggregate nterference at D n the second tme-slot, whch s descrbed n Secton 4... T s 4..3 Network Interference Model For ease of notaton but wthout loss of generalty, D s located at the orgn of the bdmensonal plane where the nterferers are randomly dstrbuted e.g., the street level of Fg. 4.. The nterferers are modeled as ponts of a Posson Pont Process (PPP) of fxed densty, smlar to [4.43]-[4.47]. s the average number of nterferers per square meter and t s measured n nterferers /m ''. By relyng on a statstcal-physcal generaton and propagaton mechansm of the network nterference, [4.43]-[4.47] have proved that T and T follow a Symmetrc Alpha-Stable ( S S) dstrbuton [4.6]. Wth the ad of the compound Gaussan representaton of [4.47], they can be formulated as T I Z Z Z E BG, where: ) E I s the average symbol energy of the nterferers; ) Z, dentfes the tme-slot; ) B Z s a Stable b RV totally skewed to the rght, whch s denoted by ~ S / b,,cos I / b Securty: Publc Page 87 B wth Z sk I I b beng the ampltude path-loss exponent of the nterferers-to- D lnks [4.6]. The RV I / bi B Z s characterzed by ts MGF, whch s B s exps The RVs B Z and G Z are ndependent [4.6]. The varance parameters of the nterferers-to- D lnks (.e., fadng dstrbuton, M ; and v) ~, Z Z GZ G CN. G depends on the propagaton Z b I ), and the modulaton scheme of the nterferers. Closed-form expressons of G are avalable n [4.47], to whch the reader s referred for further nformaton. As an example, let ( m, ) be the parameters of the Nakagam-m fadng of the..d. nterferng channels and let the modulaton used by the Z ID ID

88 nterferers be the Multlevel Phase Shft Keyng (MPSK). In ths case, 4KK GZ bi [4.47], where: / bi K / f bi and K f bi / bicos / b I mid / b I / b / b / / / / I I bi bi K / b ID I mid m ID / bi / bi / bi (4.) Snce the communcaton occurs n two tme-slots, the temporal correlaton propertes of and T have to be characterzed. Two practcal scenaros are consdered, whch take nto account the dfferent sesson lfetmes of the dual-hop relayng protocol and of the nterferers: ) the quas-statc scenaro and ) the fast-varyng scenaro. In the former case, the nterferers are assumed to have a longer sesson lfetme than the duraton of the cooperaton phase, whch accounts for broadcast and relayng. Thus, t s reasonable to assume that the same nterferers are actve n both tme-slots and that they do not change ther locatons. Also, the fadng envelopes and phases are assumed not to change n the two tme-slots. Hence, T and T are correlated and T T T. In the latter case, the nterferers are assumed to have a shorter sesson lfetme than the duraton of the cooperaton phase. More precsely, dsjont sets of nterferers are actve durng broadcast and relayng phases. Owng to the dfferent actve nterferers and to ther dfferent locatons, the assocated fadng envelopes and phases are assumed to be ndependent n the two tme-slots. Hence, T and T can be assumed to be ndependent. The quas-statc scenaro may occur when the relayng phase s scheduled rght after the broadcast phase. The fast-varyng scenaro may occur when broadcast and relayng phases are delayed. It s worth mentonng that the mpact of nterference correlaton on the performance of wreless networks s recevng an upsurge of research nterest [4.46],[4.47],[4.6]-[4.64]. The analyss and comparson of quas-statc and fast-varyng scenaros provde a contrbuton to ths feld of research Problem Statement Let the system model of Secton 4.., the sgnals receved at D can be formulated as follows: a y E h x n E BG E h x SD S SD S SD I S SD S SD b y E h x n E h E h n E BG E h x RD R RD S RD R RD S SR SR I R RD S SRD Securty: Publc Page 88 (4.3)

89 b where / XY E E d ( X S, R, Y R, D XY X XY XY n E B G ~ CN, N E B ) and (a), (b) follow by ntroducng: SD SD I B I G SRD nrd ERhRD ES hsr nsr EI B G CN N NERD RD ESR SR EIB G ~, B, XY B B B, where G G G n the quas-statc scenaro, whle B, B,, n the fast-varyng scenaro. Ther dstrbutons are determned by b I, (4.4) GG are..d. G G G. The objectve s to compute the end-to-end error probablty and to study the dversty order of the dual-branch dstrbuted network of (4.). These performance metrcs are computed by averagng over all possble network deployments of the nterferers, accordng to the defnton of ''spatal average'' gven n [4.47]. From (4.), ths mples the need of computng the expectatons over the RVs B Z and G Z. Snce condtonng upon y SD and y RD are complex Gaussan RVs by h XY, B, B, a two-step methodology s proposed: ) frst, the performance metrcs are computed by averagng over the dstrbuton of the AWGNs and the RVs G Z ; and ) then, the expectaton over the other RVs s computed. Ths approach s convenent because the error probablty of dversty systems n AWGN (frst step) has been wdely studed. By usng ths two-step approach, the end-to-end error probablty of general b-dmensonal modulatons can be formulated as the lnear combnaton of ntegrals lke [4.47]: a sn Eh B sn J,,, exp SINR h, Bd M d SINR (4.5) where: ) h and B are short-hands collectng the RVs b sn sn h, h, h and B, B SR SD RD SINR h, B s the end-to-end Sgnal-to-Interference-plus-Nose-Rato (SINR), whch depends on the demodulator used at D and s condtoned upon h, B ; ) M SINR s ssinr h B ; and v),, are modulaton-dependent Eh B, exp, parameters [.]. For example, ; ),, M / M, / M, for MPSK modulaton [4.57]. The equaltes n (a) and (b) correspond to frst and second step of the two-step Securty: Publc Page 89

90 approach descrbed above, respectvely. The computaton of (4.) may be avoded usng approxmatons [4.65]. The applcaton of (4.) requres a formal defnton of SINR,, whch depends on the demodulator used at D. The decson statstc of a general demodulator can be formulated as: * * S ES, ER, h S Re ES, ER, h, n S Re 3 ES, ER, EI, h,, S D D D G B (4.6) where S s the hypothess of S collectng the RVs n, n, n and, SR SD RD,, n and G are short-hands S M S M S GG, and D (), D (), () 3 D are related to useful, nose and nterference terms, respectvely. The general formulaton n (4.6) apples to all the demodulators of ths paper. From (4.6), the, SINR can be formulated as: SINR h, B D ES, ER, h D ES ER h n E G D3 ES ER EI h G B E Re,,, Re,,,,, n (4.7) 4..5 Dversty Order n Wreless Networks wth Nose and Interference Let ET ES ER. In nose-lmted networks, the dversty order s defned as the slope of the end-to-end error probablty n (4.5) as a functon of T E / N n a log-log scale [4.66]. In nterference-lmted networks, on the other hand, the dversty order s defned as the slope of the end-to-end error probablty n (4.5) as a functon of E / E n a log-log scale [4.6]. In wreless networks wth nose and nterference, t s relevant to study the asymptotc behavor of the end-to-end error probablty n (4.5) as a functon of both T T I E / N and E / E. In ths paper, two case studes are of nterest [4.44]: ) the homogeneous T I E / N ( E / N ) wth and ) the heterogeneous scenaro,.e., scenaro,.e., I I T I E / N and EI / N are ndependent of each other. The nterested reader s referred to [4.44] T for further detals. As for the heterogeneous scenaro, two stuatons are worth beng studed: a) the dversty order as a functon of T E / N when EI / dversty order as a functon of ET / E I when ether EI / N or ET / ET E I can be ncreased by ether ncreasng T as / N s kept constant and b) the E and keepng N are kept constant, E I constant or by Securty: Publc Page 9

91 decreasng E I and keepng E T constant, respectvely. Throughout ths paper, N s assumed to be constant for all the analyzed case studes. In practce, scenaros a) and b) are useful for comparng the system setup under analyss aganst nose- and nterference-lmted networks, where the dversty order s computed as a functon of T E / N and E / E, respectvely. For example, scenaro a) allow us to readly assess the dversty order loss due to the aggregate nterference compared to nterference-free wreless networks, e.g., [4.], [4.3], [4.], [4.] and [4.44]. A smlar comment apples to scenaro b) as far as nterference-lmted networks are concerned, e.g., [4.6]. However, scenaro b) reduces ether to the nose-lmted setup or to scenaro a). In partcular: ) f E T s kept constant but E I decreases, the system reduces to the nose-lmted case as the nterference power s neglgble compared to the nose power. Ths scenaro s of no nterest n ths paper, as t has been studed extensvely n the lterature already [4.5]-[4.]; and ) f E I s kept constant but E T ncreases, the dversty analyss s the same as a). Ths mples that the dversty order of case study b) can be obtaned from the analyss of case study a) by smply replacng ET / N wth ET / E I. A smlar comment apples f ET / N and ET / E I,.e., E T ncreases and analyzed n ths paper. E I decreases at the same tme. Thus, only case studes ) and a) are T I Smlar to [4.66], the dversty order can be obtaned from the asymptotc behavor of M SINR n (5). In fact, of nterest, e.g., T M SINR s a functon of T E / N or E / E T I E / N and E / E. Let be the rato T. Then, the followng defnton of dversty holds. I Defnton : Let d M s o SINR K () s, where K and () s ndependent of. From (5), the asymptotc error probablty can be formulated as the lnear combnaton of: sn d sn o d K o J,, d d K sn (4.8) sn Then, the dversty order as a functon of s defned to be d o. Securty: Publc Page 9

92 From (5) and Defnton, error probablty and dversty order can be obtaned from M SINR. In the next sectons, demodulators at D. M SINR s computed n closed-form for dfferent 4..6 End-to-end error probablty of dual-hop cooperatve relayng usng a maxmum rato combnng recever Consder a MRC demodulator, whch has perfect knowledge of the CSI,.e., h XY, of the three-node cooperatve network but s oblvous to the network nterference. Ths corresponds to the optmal dversty combner wthout network nterference [4.57] ( x ): S M S ˆ arg mn N y E h x N NE E y E h x (4.9) S S SD S SD S RD RD SR SR RD R RD S S GFM Quas-Statc Interference Scenaro Consderng the condtonal-awgn representaton, the, SINR SINR n (4.7) s:, cos (4.) SR RD SR RD SR RD SR SD RD h B SD SD SD B B B SR RD SR RD SR RD SR RD where, E / N XY XY XY B E I / N B G, RD SD. The SINR n (4.) s exact but mathematcally ntractable. Lemma overcomes ths ssue wth the ad of equvalent channel. Lemma 4.. Let the equvalent channel model, the optmal nterference-oblvous demodulator and ts equvalent output SINR are: ˆ S GFM eq SINR h B eq eq S arg mn S ysd ES hsd xs yrd Eeq heq xs, mn, B SD SR RD (4.) From (4.), the MGF of the SINR n (4.) s computed n Proposton 4.. Accordng to Defnton, the asymptotc MGF s computed n Propostons 4., 4.3 for case studes ) and a) dscussed n Secton II.D. Fnally, the dversty order of both case studes s analyzed n Proposton 4.4. Securty: Publc Page 9

93 Proposton 4.:Let the system model of Secton II. The MGF of (4.) s M SINR M eq SINR, where m RD msd k mrd msr k msd msd msr mrd msr mrd, k k ;, k k ; k SD SD k SR RD SR RD M eq s Z Y s Z Y s SINR m SR msd k m SR mrd k 3 m SD m SD 4 ms R mrd msr mrd Z, k Yk s; Z, k Yk s; k SD SD k SR RD SR RD (4.) Z T P Z T P msd mrd 3 msd m RD m SR Zk,,,,,, msd m T m m P k m m RD SD msd RD mrd! SR msd mrd 4 msd mrd m SR Z, m, m, k, m, m, msd m T P RD SD msd RD mrd! SR (4.3) Y k msd m SR msd m SR m RD, k m SD SR SRD SD SR SRD SD m SR SD msd SR msr! RD m, m, k, m, m, msd m SR msd m SR m RD, k m SD SR SRD SD SR SRD SD m SR SD msd SR msr! RD m, m, k, m, m, SD RD SRD SD RD SRD k SD RD SRD SD RD SRD l ; I ri / / s k k l r, r, f g l, r, f ri k g g li g f f g / s li,r E I I li I I ri G G g f ri, li r I ri r! I g g f f s N s r I I I T I x y x, y, z B x, yx y! y! x y z P x, y, z, w where: ) / b r / l x x z w x! y z y x (4.4) y w x! y z P x, y, z, w (4.5) x x I I I m / m / m / SRD SD SD SR SR RD RD x wth r I, l I beng postve ntegers; ) ; and ) E / N. XY XY XY Proposton 4.:Let the same assumptons as n Proposton 4.. Let S KT ET, ER KT ET and ET ES ER wth K T. Let EI / N I ( ET / N) E E /. If T N, the MGF of the SINR n (4.) s approxmated by (4.) obtaned by replacng ; (4.4) wth ; Y : k Y of k Securty: Publc Page 93

94 r, r, l, r, g r l k g li li ri / g, r I l I I ri I I I k s; k k ri li G r, I li r I I ri ri g g! G s r I I I Y and XY wth XY b, where b and K SY SY T SY / dsy SY RD K / d. RD T RD RD RD (4.6) Proposton 4.3:Let the same assumptons as n Proposton 4.. Let S KT ET, ER KT ET and ET ES ER wth K T. Let EI / N I dscussed above (heterogeneous case). If T E be constant, as E / N, the MGF of the SINR n (4.) s approxmated by (4.) obtaned by replacng Y k ; of (4.4) wth ˆ ; / b T I Y : Y ˆ s; k g s (4.7) k / bi g k E / b I / bi I G N b I g g! XY wth XY defned n Proposton. k Proposton 4.4:The dversty order of (4.9) n a quas-statc nterference scenaro s equal to under the assumptons of Proposton 4. and to / b I under the assumptons of Proposton 4.3. Remark 4.:From Propostons -4, we conclude that the dversty order s ndependent of the fadng parameters m XY. Under the assumptons of Proposton,.e., the homogeneous scenaro of Secton II.D[4.44], the error probablty reaches an horzontal asymptote for hgh- SNR, whch mples that an optmal transmt-power exsts beyond whch the error probablty cannot be further mproved. Under the assumptons of Proposton 3,.e., the heterogeneous scenaro of Secton II-D[4.44], the dversty order s the same as n sngle-hop networks [4.47], whch mples that there s no dstrbuted dversty gan by usng the nterferenceoblvous MRC demodulator n (4.9). A comprehensve physcal justfcaton for the fractonal dversty order equal to / b I s avalable n[4.47]. Fast-Varyng Interference Scenaro Smlar to Secton 4.3., the SINR, n (4.7) can be formulated as ( E / N BZ I Z B ): G Securty: Publc Page 94

95 SR RD SR RD SR RD SINR h, B SD SD SD B B (4.8) SR RD SR RD SR RD The SINR n (4.8) s mathematcally ntractable and Lemma 4. s used nstead. Lemma 4.: The optmal dversty combner n the absence of nterference s stll eq n (4.). A Lower-Bound (LB) and a Upper-Bound (UB) for the output SINR are as follows: where eq eq eq LB h, B h, B UB h, B eq Z h, B KZ B SD mn SR, RD SINR SINR SINR SINR K LB bi for the LB, UB K for the UB and LB,UB Z. (4.9) Remark 4.: By comparng (4.) and (4.9), the MGF of the SINR n (4.9) can be obtaned from Propostons by replacng G wth K Z G. Hence, the dversty order n the fastvaryng scenaro s the same as n the quas-statc scenaro, as summarzed n Proposton 4.4. Remark 4.3: Snce eq SINR n (4.9) s equal to UB, eq, SINR n (4.), the error probablty n the fast-varyng scenaro s never better than n the quas-statc scenaro End-to-end error probablty of dual-hop cooperatve relayng usng a selecton combnng recever Consder a SC demodulator, whch has perfect knowledge of the CSI,.e., h, of the threenode cooperatve network but s oblvous to the network nterference. Under these assumptons, t can only choose the branch provdng the hghest Sgnal-to-Nose Rato (SNR), whch s computed by neglectng the aggregate nterference. From (4.3), t can be formulated as [4.57]: ˆ arg mn S y SD ES hsdm S f SD eq mn SR, RD eq f mn, (4.) S GF M S eq ysrd Eeq heq M S SD eq SR RD where ˆS s the estmate of and S E / N. XY XY XY Securty: Publc Page 95

96 T T I Quas-Statc Interference Scenaro Consder the quas-statc scenaro wth E BG. Lemma 4.3 provdes the end-toend error probablty of (4.),.e.}, ˆ Pr S S. Lemma 4.3:Let S belong to a generc b-dmensonal constellaton dagram characterzed by the trplet of parameters,, [4.47]. The end-to-end error probablty of (4.) can be formulated as the lnear combnaton of ntegrals lke (4.5) wth eq h, B max,mn, SINR. B SD SR RD B E I / N B and G M SINR From Lemma 4.3, error probablty and dversty order can be obtaned from () n Defnton 4.. A closed-form expresson of M SINR () s gven n Proposton 4.5. eq eq Proposton 4.5:The MGF of eq h, B max,mn, SINR s: B SD SR RD m / 3 SD m ; SR mrd m, ; SD m SR mrd li r I M eq s liri Qk s Q t s Qk,, t s; 3 SINR k t k t (4.) where Q ;, k Q t, ;, and msr / SR mrd / RD, 3 msd / SD msr / SR mrd / RD. 3 Q k,, t ; are defned n [4.83], and m SD / SD, From the MGF n (4.), the dversty order can be obtaned followng the same lne of sght as Proposton 4. and 4.3 by consderng the homogeneous and heterogeneous scenaros. E Proposton 4.6:Let S T T E / N ( E / N ). If T I I T (4.) obtaned by replacng K E, ER KT ET and ET ES ER wth K T. Let E / N, the MGF of the equvalent SINR s approxmated by Q ;, k Q, t, ; 3 Q wth k,, t ; k Q ;, t, ; Q and 3 Q k,, t ; defned n [4.83], respectvely, as well as XY wth XY, where / b SY b K d and / RD K d SY T SY SY SY. RD T RD RD RD Securty: Publc Page 96

97 E Proposton 4.7:Let S T T K E, ER KT ET and ET ES ER wth K T E / N heterogeneous scenaro,.e., I s kept constant. If T (4.) s approxmated by: M SINR / b I T eq s I G I. Let the E / N, the MGF n E / bi / bi s N bi (4.) msd msr mrd msd msr mrd / bi / bi / bi 3 Qk Q, t 3 Qk,, t 3 k t k t Proposton 4.8:The dversty order n a quas-statc nterference scenaro s equal to under the assumptons of Proposton 4.6 and to / b I under the assumptons of Proposton 4.7. Fast-Varyng Interference Scenaro Consder the fast-varyng scenaro, where T I Z Z Z E BG are..d. for Z,. Lemma 4.4:The end-to-end error probablty of the SC demodulator n (4.) n quas-statc and fast-varyng nterference scenaros s the same. Based on Lemma 4.4, the error probablty of SC n the fast-varyng scenaro can be studed from Propostons Also, the same performance n both nterference scenaros s expected Numercal and Smulaton Results In ths secton, we show some numercal results to verfy the accuracy of the proposed mathematcal methodology aganst Monte Carlo smulatons and to valdate our fndngs. As an example, the followng setup s consdered. Dual-hop network}: ) d SR d 5m, d 5m ; ) ; ) b 3, b.5, b ; v) RD SD SR RD ; v) m m m ; v) MPSK modulaton wth M 6 SD SR RD SD SR RD ES ER ET /. Interferng network}: ) ; ) bi 3.5 ; ) m I SD SR I RD SD ; and v) ; v) MPSK modulaton wth M 6 ; v) synchronous transmsson[4.47]; and v) {,,,, } nterferers /m. As for the AWGN, B N k T where Securty: Publc Page 97

98 k B T s the Joule/Kelvn s the Boltzmann's constant and 9Kelvn nose temperature. As for E I, two scenaros are consdered: ) I ET to the homogeneous case of, e.g., Proposton 4.; and ) I corresponds to the heterogeneous case of, e.g., Proposton 4.3. E, whch corresponds E / N 9dB, whch For assessng the accuracy of the mathematcal analyss, the error probablty s computed n four ways. ) Monte Carlo Smulatons. These are obtaned by smulatng the whole communcaton system, ncludng modulator, channel and demodulator, wthout any a pror assumptons about the dstrbuton of the network nterference. The followng procedure s used[4.47]: ) a fnte crcular area of (normalzed) radus s located, s consdered. The radus s chosen such that R A around the orgn,.e., where D, n order to mnmze the error commtted n smulatng an nfnte b-dmensonal plane; ) the number of nterferers s generated accordng to a Posson dstrbuton wth densty and area R A R A ; ) the locatons of the nterferers are dstrbuted followng a unform dstrbuton over the crcular regon of area R A ; v) ndependent channel gans are generated for each nterferer; and v) the exact dversty demodulators are used at D ; ) Sem-Analytcal Framework. Consder the MRC demodulator of Secton 4.3 as an example. These results are obtaned by nsertng the SINR of (4.) n (4.5) and by numercally computng the expectatons wth respect to h and B. The RVs B are generated wth the ad of [4.8]; 3) Analytcal Framework. These results are obtaned by nsertng the MGF of the equvalent SINR of, e.g., (4.3) n (4.5) and by numercally computng the ntegral; and 4) Asymptotc Framework. These results are obtaned by nsertng the MGF of the asymptotc equvalent SINR of, e.g., Propostons 4., 4.3 n (4.5) and by numercally computng the ntegral. As for sem-analytcal, analytcal and asymptotc frameworks, G s computed usng (4.). Selected numercal examples llustratng the end-to-end Average Symbol Error Probablty (ASEP) at D are shown n Fgs For ease of llustraton, the results obtaned wth Monte Carlo Smulatons are reported only for {,, }, whle the results obtaned wth the Sem-Analytcal Framework are reported only for 8 6 {,,}. As for 3, t s mportant to remark that t corresponds to a very sparse nterferng network, whch nearly corresponds to an nterference-free scenaro n the analyzed range of SNRs. Ths setup s studed to substantate the frameworks for all possble values of and to show Securty: Publc Page 98

99 the achevable lower-bound wthout nterference n the SNR range of nterest. In Fgs , n partcular, the sold lnes show the Analytcal Framework, the dashed lnes show the Asymptotc Framework and the markers show ether Monte Carlo Smulatons or the Sem- Analytcal Framework dependng on. Overall, we observe a good agreement between Monte Carlo smulatons and the proposed mathematcal frameworks. The numercal examples confrm the fndngs about the achevable dversty, as a functon of the demodulator used at the destnaton and of the nterference scenaro. Fg 4. Interference oblvous MRC demodulator wth quas statc nterference n (a) homogeneous, (b) heterogeneous scenaro Fg 4.3 Interference oblvous MRC demodulator wth fast varyngnterference n (a) homogeneous, (b) heterogeneous scenaro Securty: Publc Page 99

100 Fg 4.4 Interference oblvous SC demodulator of quas statc nterference and heterogeneous scenaro Fg 4.5Interference oblvous SC demodulator of fast varyng nterference and homogeneous scenaro 4. Performance evaluaton of cellular networks wth randomly dstrbuted relays 4.. Introducton The deployment of relays, as nfrastructures wthout a wred backhaul connecton, have been consdered by IEEE 8.6j workng group [4.] and the Thrd Generaton Partnershp Project's Long Term Evoluton-Advanced (3GPP LTE-A) [4.3] to enhance the throughput and Securty: Publc Page

101 coverage as a cost-effectve soluton n future cellular networks. Currently, practcal systems usually consder half-duplex relays (the relays can ether receve or transmt but not at the same tme) to relay the message from the base statons (BSs) and the moble termnals (MTs) usng amplfy-and-forward, decode-and-forward or demodulate-and-forward protocols [4.], [4.5], [4.83], [4.84]. However, the relays deployed n the commercal multhop wreless network are not desgned to specfcally mtgate nterference. Consequently, nterference s one of the man performance lmtng factors n the relay-aded wreless networks. In ths context, the performance of the relay-aded transmsson n nterference lmted schemes have been an actve feld of research n the last few years. For example, n [4.5], the outage probablty of the dual hop transmsson s computed n the presence of randomly dstrbuted nterferng nodes around the destnaton.in Secton , the achevable dversty order of the dual-hop cooperatve relays are studed by assumng the nterference follows symmetrc alpha stable dstrbuton at the destnaton and the performance of dfferent recevers n nterference-lmted cooperatve wreless networks are compared. Even though the above cases studed gve a clear understandng of the performance of relay-aded wreless networks n the presence of nterference, all these studes assume that ether the poston of the BS or the poston of the relay s fxed. Furthermore, the extra nterference generated by the actve relays are not consdered n the system model of the cted papers, whch s one of the man challenges faced by the relay deployment n the cellular networks[4.]. Motvated by these consderatons, n the present paper we use stochastc geometry to model the postons of the BSs, the MTs, and the relays to study the coverage probablty of the relay-aded downlnk transmsson by consderng the addtve nose as well as the nterferences generated by the BSs and the relays. In partcular, the Posson Pont Process (PPP)-based approach [.] has been used, whch s consdered as accurate as other abstracton models [.]and leads to a tractable mathematcal analyss for further study. More precsely, both cooperatve and non-cooperatve transmsson are consdered. The MT can choose a BS or a relay as ts servng node accordng to the assocaton polcy, whle the servng nodes can serve multple users usng orthogonal resource blocks, e.g., tmefrequency resource block n LTE [.5]. Thus, the ntra-cell nterference s not consdered n ths paper. All the nterferences are generated from the other-cell BSs or other relays usng the same resource block as the servng node. Further, we assume all the BSs and relays operate n open access model but not wth full load,.e., all the MTs can connect to them Securty: Publc Page

102 wthout any restrctons. Partcularly, three assocaton polces are studed. The coverage probabltes of the relay-aded wreless network wth dfferent assocaton polces are nvestgated through extensve Monte Carlo smulatons. The numercal results also show the beneft or loss caused by the deployment of relays compared to the non-cooperatve cellular network. 4.. System Model As shown n Fg. 4.6., we consder a b-dmensonal network consstng of BSs and relays spatally dstrbuted accordng to two ndependent PPPs BS and R wth densty R, respectvely. The locatons of the MTs are also modeled an an ndependent PPP wth densty MT. BS and MT Fg A relay-aded cellular network wth PPP dstrbuted BSs, relays, MTs, and cell boundary of BSs (lne). Each MT can be served ether by the BS or by the relay accordng to the assocaton polces ntroduced n Secton Let a typcal MT be denoted as MT. If MT s assocated to a BS, denoted as BS, then BS would allocate a free resource block to serve MT drectly. In ths case, the nterferng sgnals are generated from other the BSs usng the same resource block as BS. On the other hand, f MT s tagged to a relay, denoted as R, the transmsson s dvded nto two orthogonal tme-slots [4.83], [4.84]. In the frst tme-slot, R Securty: Publc Page

103 receves useful sgnals from ts servng BS ( BS ( R ) ) occupyng a free resource block of ( R ) BS. The nterferences receved by R are from other BSs usng the same resource block as ( R ) BS. In the second tme-slot, R uses the decode-and-forward protocol for relayng the detected message to MT usng a free resource block of R. The nterferng nodes n the second tme-slot are other actve relays usng the same resource blocks as R. For smplcty, we assume that the relays are always served by the nearest BSs of the relays. It s worth mentonng that BS and ( R ) BS may not be the same BS. SINR If MT s served by BS drectly usng resource block m, the SINR at MT s: SINR E h r BS, D () T MT N BS, D (4.6.) where ) the BS -to- E T s the transmt power of BS ;), hr BS D s the channel propagaton coeffcent of MT lnk wth path loss exponent BS, D, dstance r and the standard Raylegh fadng coeffcent h ~ () exp [.4].For smplcty, ndependent and dentcally dstrbuted (..d.) Raylegh fadng channels are assumed throughout ths paper;) N s the varance of the Addtve Whte Gaussan Nose (AWGN) at MT ;v) BS, D s the cumulatve nterference at MT, whch can be formulated as follows: (4.6.) BS, D BS, D BS, D BS \ T BS BS j j jbs E h r E h r where represents the set of actve BSs servng MTs drectly usng resource BS BS block m. All the BSs n BS have transmt power E T ; servng relays wth resource block m ; EBS KT ET denotes the set of BSs BS, K T BS, s the transmt power of the BSs n. BS When MT s served by R usng the dual hop lnk, R receves the message from the frst tme slot, and the SINR of R can be expressed as: ( R ) BS n Securty: Publc Page 3

104 SINR R E ˆ ˆ BSh r N BS, R BS, R (4.6.3) where ) hr ˆ ˆ BS, R s the channel gan of the ( R ) BS -to- R lnk wth Raylegh fadng envelop ĥ, dstance ˆr and path loss exponent BS, R ;), BS R s the nterference at R smlar to BS, D n (4.6.): ˆ ˆ ˆ E h r E h rˆ (4.6.4) BS, R BS, R BS, R T ( R ) j \ BS j j BS BS BS In the second tme-slot, MT receves the relayng message from R, and the SINR n ths case s: where ) SINR E h r R () R MT N R ER KT ET s the transmt power of the relays;) hr R (4.6.5) s the channel gan of the R -to- MT lnk wth fadng envelop h, dstance r and the path loss exponent R ; ) R s the aggregate nterference at MT whch can be expressed as: E h r (4.6.6) R R kr \ R R k k where R R represents the set of actve relays usng the same resource block as R. It s worth mentonng that the transmsson from the relay to the MT s not affected by nterferences from BSs because of the orthogonal two tme-slots transmsson. Coverage Probablty The BSs and relays are assumed to have M BS and M R orthogonal resource blocks whch are chosen wth equal probablty. Snce the MTs and the BSs are dstrbuted accordng to two ndependent PPPs, the average number of MTs located n each cell s / [.6]. In order to avod spatal blockng [4.85],.e., the MTs cannot be served by the assocated BSs because all the resource blocks of the servng BSs have been occuped, we assume / M. Thus, the probablty of spatal blockng when the MTs are tagged to the MT BS BS BSs s very small and can be neglected. Smlarly, we assume / MT MT R R BS M to avod the spatal blockng of the relays. Wth these assumptons, all the MTs can be served by ther servng nodes and the BSs and relays are not operated wth full load. Securty: Publc Page 4

105 Usng the assocaton polces n Secton 4.6.3, all the MTs are served by a BS or a relay, and the coverage probablty of the system can be defned as: P c E MT SINR t MT MT (4.6.7) t where SINR s defned n (4.6.8); s the target SINR threshold. It s notceable MT t that f a MT s tagged to the dual hop lnk, t s n coverage when the SINRs of both lnks are above the threshold. In other words, the performance of the dual hop lnk depends on the weakest hop [4.8]. SINR MT t () SINRMT t () MTt s served by BS (4.6.8) SINRR SINR s served by relay t MT MT t t 4..3 Assocaton Polces Assocaton Smlar to Secton 4.6., let BS and R denote the closest BS and closest relay to the MT ( MT ), respectvely. Let r be the dstance between BS and MT. Let r be the dstance from R to MT. Then, usng assocaton, MT s tagged to the node t defned as follows: BS f r r t R f r r (4.6.9) In other words, MT s served by the closest servng node n the network. Wth a low mplementaton cost, the nearest dstance assocaton s wdely consdered n relay-aded wreless networks and mult-ter cellular networks. Assocaton Let the same notatons as Assocaton. Accordng toassocaton, MT s tagged to the node t defned as follows: where BS BS B E r B E r t R B E r B E r BS, D R f BS T R R BS, D R f BS T R R (4.6.) B and B are the basng factors of the BSs and relays. The MTs usng R Assocaton chooses the servng nodes based on the based average receved power by takng nto account the channel propagaton as well as the transmt power of the servng nodes. Ths assocaton fnds ts ratonale from the long-term averaged maxmum basedreceved-power assocaton polcy used for mult-ter cellular networks n [.3]. Securty: Publc Page 5

106 Assocaton 3 Usng the same notatons as n Assocaton, and let ˆr represent the dstance from R BS to R, where defned as follows: ( R ) BS s the closest BS of R. From Assocaton 3, ( ) MT s tagged to the node t BBS E T EBS E R BS f B mn, BS, D R BS, R R r ˆ r r t BBS E T EBS E R R f B mn, BS, D R BS, R R r ˆ r r (4.6.) Snce the performance of the dual hop lnk depends on the weakest lnk, Assocaton 3compares the overall averaged receved power of the drect lnk and the dual hop lnk at the cost of extra complexty. Compared toassocaton, Assocatons and 3 provde more flexblty by ntroducng the basng factors. It s apparent that when BBS, the MTs wll always choose the relay-aded transmsson. When BR, the system reduces to the non-cooperatve cellular networks. These three assocaton polces are based on the concept of ncreasng the average receved power wth a dfferent mplementaton cost. However, choosng the lnk wth a strong useful receved power may also ncrease the nterference receved by other MTs n the network. Indeed, nterference s one of the man lmtng factors n future cellular networks [.]. In the next secton, Monte Carlo smulatons are used to study the coverage probablty of the relay-aded wreless network usng these three assocaton polces Smulaton Results In ths secton, selected numercal examples are dsplayed to show the coverage probablty of the relay-aded cellular network as descrbed n Secton More specfcally, as far as Monte Carlo smulatons are concerned, the followng seven-step methodology has been used: Step : A real b-dmensonal network n a fnte crcular area wth radus R s consdered. The radus s chosen such that R 3 BS,.e. around 3 BSs are smulated n ths area, n order to mnmze the error commtted by the truncaton problem. Securty: Publc Page 6

107 Step : The BSs, relays, and MTs are generated n the smulated area accordng to three ndependent PPPs wth densty BS, R and MT, respectvely. Step 3: For each MT, the assocaton polces descrbed n Secton are appled to determne the lnks and the resource blocks used to serve the MT,.e., the transmsson status of all BSs and relays are dentfed. Step 4: Independent channel gans are generated for all useful and nterferng lnks between BSs and MTs, BSs and relays, and relays and MTs. Step 5: The SINRs of each MT and each actve relay are computed as shown n (4.6.), (4.6.3) and (4.6.5). Step 6: The coverage rate of ths partcular network deployment s obtaned accordng to (4.6.7) and the SINRs computed nstep 5. Step 7: Fnally, the coverage probablty of the system s computed by repeatng Step - Step 6 for at least 6 tmes. For comparson, the coverage probablty of a non-cooperatve cellular network s also presented to show the beneft or loss of the deployment of relays. The MTs and the BSs are dstrbuted n the smulated area accordng to two ndependent PPPs wth the same densty as n Step. The MTs n the non-cooperatve cellular network are assocated to ther nearest BSs. All the BSs are transmttng wth full power E T. In Fg , we llustrate the coverage rate of the relay-aded archtecture wth a set of values of path loss exponents BS, D, BS, R, R and wth a wde range of densty of relays R. In general, we consder the densty of MTs MT 4 whle the densty of BSs 5 BS and the number of resource blocks per BS BS M throughout the smulatons to avod the spatal blockng descrbed n Secton Snce the study focuses on the nterference lmted network, we assume the total power T E / N 3 db. In addton, as far as the basng factors nassocaton and Assocaton 3 are concerned, we consder BR to be fxed and only B BS s changng. Securty: Publc Page 7

108 Fg Comparson of the coverage probabltes as a functon of the threshold wth dfferent assocaton polces. Setup:,, 6, 6, K.5. BS, D 3 BS R R T M, 4 R, R In partcular, Fg provdes the coverage probablty of three assocaton polces when 5 R BS,.e., a dense relay deployment compared to the BSs. It s shown thatassocaton wth the lowest mplementaton cost gves the best performance, n terms of the coverage rate, among three assocaton polces. On the other hand, when BS R, usng Assocaton and 3 almost provdes the same performance as the cellular network wthout relays. The coverage rate ncreases, n general, for smaller basng factor B B BS B. In other words, servng more MTs waa dual hop lnk n the network, whch decreases the nterference, wll mprove the overall performance, though the average receved sgnal from the dual hop lnk may be weaker than the drect lnk. Ths agrees wth the observaton that the path loss exponents n the dual hop lnk are much hgher than that n the drect lnk. So the nterferng nodes that use cooperatve transmsson and are far from the probe MT can be neglected. Securty: Publc Page 8

109 Fg Comparson of the coverage probabltes as a functon of the threshold wth dfferent assocaton polces. Setup:,, 6, 6, K.5,, 3 BS D BS R R T M 5 R, R In Fg , the coverage probabltes when the BSs and relays are dstrbuted wth the same densty,.e., R BS, are dsplayed. In ths network, changng the basng factors, Assocaton and Assocaton 3 may outperform Assocaton. Furthermore, compared to Fg 4.6., the performance of Assocaton gets worse wth the decreasng of R. Fg provdes the coverage probablty when R BS,.e., the relays are dstrbuted sparer than the BSs. We notce that the performance of applyng Assocaton drops dramatcally compared to the results n Fg. 4.6., Also n ths case, the performance of usng Assocaton 3 remans as n denser relay deployment networks by carefully choosng the basng factors. Securty: Publc Page 9

110 Fg Comparson of the coverage probabltes as a functon of the threshold wth dfferent assocaton polces. Setup: M,, 3,, 6, 6, K.5. R BS D BS R R T 6 R 4, Fg Comparson of the coverage probabltes as a functon of the threshold wth dfferent assocaton polces. Setup: BS, R 3 4 R, R, 3, (b):, 5,, 3, 3 R BS D BS R M,(a):, 3, R BS D Securty: Publc Page

111 Fnally, the mpact of the path loss exponents s studed n Fg We observe that the beneft of the deployment of relays s lmted when the path loss exponent of the drect lnk ncreases compared to the path loss exponents n dual hop lnks. Specfcally, the cooperatve transmsson provdes worse coverage probablty compared to the noncooperatve cellular network n the nterference lmted case f,,, BS D BS R R. Ths observaton s nterestng because t shows that many conclusons about the relay-aded archtecture n nose lmted envronment are not applcable to the network wth randomly dstrbuted nterferers. From Monte Carlo smulatons, t s observed that the performance of relay-aded transmsson depends heavly on the path loss exponents of the channels, the densty of the relays, as well as the assocaton polces and the basng factors. However, the smulaton cannot provde nsghtful nformaton on the system desgn and on the dependency of the system parameters to optmze. Thus, provdng a mathematcally tractable analyss on the performance of the relay-aded cellular network s of great value, whch s our future research nterest. Securty: Publc Page

112 5. Decoupled Uplnk and Downlnk Access n Future Cellular Networks (DUDe) 5. Introducton In order to keep up wth the ever ncreasng network traffc, cellular networks are shftng from a sngle-ter homogeneous network approach to mult-ter heterogeneous networks (HetNets). HetNets, composed of dfferent types of small cells (mcro, pco and femto) and macro cells, have been a popular approach n the past few years as an effcent and scalable way to mprove the network capacty n hotspots. However, most network technologes such as 3G or 4G were desgned wth Macro cells n mnd and heterogenety was just an afterthought. Ths dramatc change n cellular networks requres a fresh look on how present networks are deployed and what fundamental changes and mprovements need to be done for future networks to operate effcently. Cellular networks have often been desgned based on the downlnk (DL); ths s due to the fact that network traffc s mostly asymmetrc n a way that the throughput requred n the downlnk s hgher than the one requred n the uplnk. However, uplnk s becomng more and more mportant wth the growth of sensor networks and machne type communcatons (MTC) where the traffc s often uplnk centrc and also the ncreasng popularty of symmetrc traffc applcatons, such as socal networkng, vdeo calls, real-tme vdeo gamng, etc. As a consequence, the optmzaton of the uplnk has become ncreasngly mportant and the queston that we try to tackle n[5.] s what mprovements are possble to optmze the uplnk of a hghly densfed HetNet? 5. Motvaton Cell assocaton n cellular networks s normally based on the downlnk receved sgnal power only [5.]. Despte dfferng UL and DL transmsson powers and nterference levels, ths approach was suffcent n a homogeneous network where all the base statons (BS) are transmttng wth the same or smlar average power level. However, n HetNets where we have a large dsparty n the transmt power of the dfferent layers ths approach s hghly neffcent n terms of the uplnk. To understand ths asserton we consder a typcal HetNet scenaro wth a macro cell (Mcell) and a small cell (Scell), where n ths paper we consder outdoor Scells. The DL coverage of the Mcell s much larger than the Scell due to the large dfference n the transmt powers of both. However, n the UL all the transmtters, whch are battery powered moble devces, have Securty: Publc Page

113 about the same transmt power and thus the same range. Therefore, a user equpment (UE) that s connected to a Mcell n the DL from whch t receves the hghest sgnal level mght want to connect to a Scell n the UL where the pathloss s lower to that cell. As HetNets become denser and small cells smaller, the transmt power dsparty between macro and small cells s ncreasng and, as a consequence, the gap between the optmal DL and UL cell boundares ncreases. For the sake of optmal network operaton, ths necesstates a new desgn approach whch s the Downlnk and Uplnk Decouplng (DUDe) where the UL and DL are bascally treated as separate network enttes and a UE can connect to dfferent servng nodes n the UL and DL. The concept of DUDe has been dscussed as a major component n future cellular networks n [5.3]-[5.5]. In [5.5], n partcular, DUDe s consdered as a part of a broader devce-centrc archtectural vson, where the set of network nodes provdng connectvty to a gven devce and the functons of these nodes n a partcular communcaton sesson are talored to that specfc devce and sesson. A study n [5.6] tackles the problem from an energy effcency perspectve where the UL/DL decouplng allows for more flexblty n swtchng-off some BSs and also for savng energy at the termnal sde. In [5.7] Mult-Rado HetNets are dscussed where all rado access technologes (RAT) lke WF and LTE are managed under a sngle network and ths can be consdered as an extenson to DUDe n future work where UL and DL can be scheduled on dfferent RATs. One technque that brngs some farness to the UL s Range Extenson (RE) where the dea s to add a cell selecton offset to the reference sgnals of the Scells to ncrease ther coverage n order to offload some traffc from the Mcells [5.8]. However, usng offsets greater than 3-6 db may lead to hgh nterference levels n the DL whch s why technques lke enhanced Inter-Cell Interference Coordnaton (eicic) have been developed to try to combat ths type of nterference [5.9]. Nevertheless, the RE technque s lmted to moderate offset values due to the harsh nterference n the DL. So DUDe would brng n the benefts of havng very hgh RE offsets n the UL wthout the nterference effects n the DL. The man contrbuton of ths work s to study the gans that can be acheved by the DUDe technque n terms of UL capacty and throughput and also to study the effects that ths approach has on nterference. We use a realstc scenaro of a cellular network based on realworld plannng/optmsaton tools whch, we beleve, adds a lot of value and credblty to ths work. In our best knowledge, ths s the frst work that assesses the benefts of decouplng UL and DL n a real world deployment. Securty: Publc Page 3

114 5.3 Toy example showng the concept In ths study we drop the tradtonal UL/DL cell assocaton based on DL receved power (RP). We assume that whle the downlnk assocaton s stll based on DL RP, the uplnk assocaton s n fact based on pathloss. Ths apparently smple assumpton n realty leads to radcal changes n system desgn and archtecture. One ssue wth ths approach s when a UE has a lnk n drecton to a node (UL or DL) t needs a mechansm to allow the Acknowledgment process, channel estmaton, etc. Ths would requre major desgn changes. Theren we am at studyng whether the gans of DUDe justfy such major changes. DUDe results n dfferent cell boundares n the UL and DL n a HetNet scenaro where a UE n the regon between the UL and DL cell boundares wll be connected to the Scell and Mcell n the UL and DL respectvely as shown n Fgure 5.. We wll focus on the gans n the UL as ths s the man motve for applyng ths technque. Note that DL capactes are not affected snce the assocaton remans unchanged. In ths secton, we consder a two cell network model composed of a Mcell and a Scell to present the advantages of DUDe n a smplfed way. The model s used to study two cases; the frst case s a nose lmted scenaro wth only one UE, to show the benefts n terms of uplnk UE capacty. The second case s an nterference lmted scenaro where there are three UEs n the network to show the benefts n reducng the nterference. The two cases are explaned n detals below. Fgure 5. System model for UL/DL decouplng. Case (nose lmted) In ths case we have one UE movng from the Scell vcnty towards the Mcell and the UE UL rate s calculated for two cases; the frst s the conventonal case where cell selecton s based on the DL receved power so the UE performs a Handover (UL & DL) from the Scell to the Securty: Publc Page 4

115 Mcell when passng the DL cell border (shown n Fgure 5.) and the second case s where the UL cell selecton s based on the PL where the UE s stll connected to the Scell untl passng the UL cell border whch represents the DUDe technque. Neglectng, for smplcty, fadng and shadowng and normalzng varous quanttes, the UL rate calculaton s based on the below equatons: R = BW log ( + SNR) SNR = P ue N d α (5.) Here, R s the rate; SNR s the sgnal to nose rato, Pue s the UE transmt power and N s the nose power whch s consdered to be dbm. BW s the bandwdth and s consdered to be unty for smplcty. The dstance based PL s dependent on the dstance d and the pathloss exponent α. We now calculate the UL rate for a UE movng from the Scell towards the Mcell for the two cell assocaton methods, assumng, Pue to be dbm and the Scell and Mcell to have a PL exponent of 3.6 and 4 respectvely. Fnally, the Mcell and Scell have a transmt power of 46 and 3 dbm respectvely. Fgure 5. llustrates the UL normalzed rate for the PL and RP cell assocaton cases. And t shows that the PL case has a hgher performance n the area between the DL cell border and the UL cell border snce n that area the UE has a lower pathloss to the Scell, thus obtanng a hgher rate when connected to the Scell. The two curves are the same outsde that area snce the PL and RP cell assocaton result n selectng the same cell..9.8 DL RP assocaton Pathloss assocaton (DUDe) Normalzed Rate DL cell border UL cell border dstance (m) Fgure 5. UE rate comparson between the DL Receved Power (RP) case and the Pathloss (PL) case. Securty: Publc Page 5

116 Case (nterference lmted) In ths case, we have the same setup as the prevous one but wth three UEs nstead of only one UE as shown n Fgure 5.. We calculate the overall UL rate of the network usng the PL based cell assocaton where UE s connected to the Scell n the UL and then usng the RP based cell assocaton where UE s connected to the Mcell n the UL. UE s always connected to the Scell n the UL and UE3 s always connected to the Mcell n the UL. R = BW log ( + SIR) SNR = P ue N d α (5.) The UL rate s calculated based on the above equaton, where SIR s the Sgnal to Interference Rato (we neglect the nose for smplcty). The total normalzed UL rate (RT) s the sum of the normalzed UL rate at the Mcell (RM) and the Scell (RS) whch means the UL rate of the whole system (RT = RM + RS). We use the same parameters as case and settng d, d, d3, and d4 n Fgure to, 5, 8, and respectvely. So calculatng RT n the PL case yelds RT = = and n the RP case RT = =.67. We can see that RT s almost 5% hgher n the PL case for the followng reasons. - UE n the PL case has a lower PL to the Scell whch means that UE has a better channel to the Scell and n turn gets a better rate when connected to t. - UE causes less nterference to the Mcell n the PL case than the nterference t causes to the Scell n the RP case for the same reason as above, so the nterference level n the network s lower and n turn the rate s hgher. 5.4 Evaluaton of the basc concept of DUDe In ths secton we present our realstc smulaton setup whch s based on an exstng cellular network and we use ths setup to valdate our fndngs and llustrate the gans from the studed concept System model In our smulatons we use the Mult-technology rado plannng tool Atoll [5.] n conjuncton wth a hgh resoluton 3D ray tracng pathloss predcton model [5.]. The model takes nto account clutter, terran and buldng data. Ths guarantees a realstc and accurate propagaton model. Securty: Publc Page 6

117 Atoll has the capablty of performng system level smulatons where a smulaton s a snapshot of the LTE network. For each smulaton, t generates a user dstrbuton usng a Monte Carlo algorthm. The user dstrbuton s based on traffc data extracted from the real network. Resource allocaton n each smulaton s carred out over a duraton of second ( frames). As deployment setup, we use a Vodafone LTE small cell test bed network that s up and runnng n the London area. The test network covers an area of approxmately one square klometer. We use ths exstng test bed to smulate a relatvely dense HetNet scenaro.the consdered network s shown n Fgure 3 where the black shapes are macro stes and the red crcles are small cells whch are consdered to be pco cells.we consder a realstc user dstrbuton based on traffc data from the feld tral network n peak tmes. The dstrbuton s up-scaled to smulate a hgh user densty. We use an uplnk power control algorthm where each cell has a predefned nterference upper lmt. If the UL receved nterference at a cell s hgher than ths lmt the cell sgnals the neghborng cells to lower the UL transmt power of ther UEs n order to lower the nterference level at that cell. We smulate two cases; the frst case s where the UL cell assocaton s based on PL whch represents the DUDe technque. The other case s where the UL cell assocaton s based on the DL Reference Sgnal Receved Power (RSRP) whch s the conventonal LTE procedure [5.]. In the DL RSRP case we smulate low and hgh power Scell cases to understand the gans of the PL approach compared to the DL RSRP approach wth dfferent Scell szes.as ponted out before, all the results n the next secton wll focus on the UL performance. The smulaton parameters are lsted n Table 5.where we consder an LTE deployment. One deployment ssue s that a UE connected to dfferent nodes n the UL and DL needs a way to send Acknowledgment, plot and relevant control sgnalng to ts DL node wth whch t has no UL establshed. A possble way s to route the data to the UL node and through the backhaul to the DL node and vce versa wth recevng control sgnals from the UL node.we assume an deal backhaul where control sgnals are delvered wth no notable delay. Nondeal backhaul operaton and alternatve control sgnalng delvery mechansms are left for future work. Securty: Publc Page 7

118 Fgure 5.3 Vodafone s LTE small cell test network n London. Table 5. Smulaton parameters Operatng frequency Bandwdth Network deployment User dstrbuton Scheduler Smulaton tme Traffc model Propagaton model Max. transmt power Antenna system UEs moblty Supported UL modulaton schemes.6 GHz (co-channel deployment) MHz ( frequency blocks) 5 Mcells and 64 Scells dstrbuted n the test area as shown n Fgure UEs dstrbuted accordng to traffc maps read from a lve network Proportonal far 5 smulaton runs wth second each. Full buffer 3D ray-tracng model Macro=46 dbm, Hgh power Pco = 3dBm, Low power Pco = dbm, UE= dbm. Macro: Tx, Rx, 7.8 db gan Pco: Tx, Rx, 4 db gan UE: Tx, Rx, db gan Pedestran (3km/h) QPSK, 6 QAM, 64 QAM Securty: Publc Page 8

119 5.4. Results In ths secton, we present results comparng three cases: - DL RP based cell assocaton where Scells are Pco cells (Pcells) wth low transmt power (LP) of dbm. Ths case s referred to as DL-LP. - DL RP based cell assocaton where Scells are Pcells wth hgh transmt power (HP) of 3 dbm. Ths case s referred to as DL-HP. - Pathloss based cell assocaton whch represents the Downlnk and Uplnk Decouplng (DUDe) (Pco transmt power s rrelevant as cell assocaton s not based on DL RP). Fgure 5.4 Uplnk coverage of the DL_LP (left), DL_HP (mddle) and DUDe (rght) cases where green and red represent the Macro and Pco cells coverage respectvely. Fgure 5.4 llustrates the UL coverage of the Pcell layer (red) and Mcell layer (green) for the above three cases; t shows a much larger coverage for the Pcells n the DUDe case whch ensures a more homogeneous dstrbuton of UEs between the nodes whch, n turn, results n a much more effcent use of the resources as wll be shown n the followng results. In our smulatons we defne a UE mnmum and maxmum throughput demand where bascally a UE has to reach the mnmum throughput requrement to be able to transmt ts data otherwse t s consdered n outage. On the other hand the maxmum throughput demand puts a lmt to the amount of throughput that each UE can get, so settng a hgh value for t helps n smulatng a hghly loaded network. The used scheduler tres frst to satsfy the mnmum throughput requrements for all the UEs and then dstrbutes the remanng resources among the UEs to satsfy the maxmum throughput demand of each UE accordng to the proportonal far crteron. Securty: Publc Page 9

120 UL UE throughput (Kb/s) DUDe DL-HP DL-LP Number of Small Cells Fgure 5.5 The 5 th percentle UL UE throughput, comparng the DUDe, DL-HP and DL-LP cases wth ncreasng the number of Pco cells. Fgure 5.5 shows the effect of addng Pco cells on the 5th percentle UL throughput for the dfferent cases. Pcells are all placed n ther respectve locaton as shown n Fgure 5.3 but they are all swtched off at the begnnng and are actvated one by one to understand the effect of ncreasng the number of Pcells n each case. In these results, we set the mnmum throughput requrement to a relatvely low value (Kb/s) to show how the 5th percentle throughput evolves n the dfferent cases wthout the constrant of a hgh mnmum throughput requrement. In the DUDe case, we see that the 5th percentle throughput s ncreasng wth the number of Pcells. Ths s due to the fact that Pcells have a large coverage n the UL so they serve a large number of UEs and n turn have a bg effect on the 5th percentle throughput. As the number of Pcells ncreases we notce that the 5th percentle UEs throughput starts to saturate as they are more lmted by the channel qualty and transmt power. So the extra capacty offered by addng more Pcells s used to serve the UEs wth better channel condtons. Lookng at the case of DL-LP and DL-HP, we see that addng Pcells has lttle effect on the 5th percentle throughput as Pcells have very lmted coverage so ther effect s more n the 9th percentle throughput rather than the 5th percentle. Moreover; we see that the 5th percentle throughput s fluctuatng as we ncrease the number of Pcells. Ths s bascally due to the hgh nterference that the Pcells UEs create to the Mcell cell edge UEs snce these UEs are closer to the Pcells so they suffer from a hgh level of nterference. We see ths effect more clearly n the DL-LP case where the throughput starts to decrease after a certan pont. On the contrary, n the DUDe case we see that the throughput s ncreasng more stably snce the UEs always connect to the node to whch they have the Securty: Publc Page

121 lowest PL whch guarantees a lower nterference level as explaned n Secton II. In the next results all the Pcells n the test network are actvated. Table 5. Average number of UEs per Node (Macro and Pco cells) for the three cases. DL-LP DL-HP DUDe Macro cell Pco cell 4 8 Table 5. shows the average number of UEs per cell where we calculate the average for the Mcells and Pcells separately havng a constant total number of UEs (56) for all cases. The table shows how most of the UEs are connected to the Mcell n the DL-LP and DL-HP cases and the Pcells are under-utlzed. On the other hand n the DUDe case the UEs are dstrbuted n a more homogeneous way among the Mcells and Pcells whch ensures much more effcent resource utlzaton. Fgure th, 5 th and 9 th percentle comparson of DL-LP, DL-HP and DUDe cases. For the results n Fgure 5.6 we set a mnmum and maxmum throughput demand of Kb/s and Mb/s respectvely. The fgure shows the 5th, 5th, and 9th percentle UE throughput for the three cases n comparson. The 5th percentle UL throughput n the DUDe case s ncreased by more than % compared to the DL-LP case and by % compared to the DL-HP.As for the 5th percentle UL throughput, the DUDe case has a gan of more than 6% compared to the DL-LP case and more than a % compared to the DL-HP case.the gans n the 5th and 5th percentle are resultng from the hgher coverage of the Pcells n the DUDe case whch results n a better dstrbuton of the UEs among the nodes and a much more effcent usage of the resources. Also the fact that the UEs connect to the node to whch Securty: Publc Page

122 they have the lowest PL helps n reducng the UL nterference as shown before. Ths results n a hgher UE Sgnal to Nose and Interference Rato (SINR) that allows the UEs to use a hgher modulaton scheme and n turn acheve a better utlzaton of the resources and a hgher throughput. Lookng at the 9th percentle UL throughput we see that the DL-HP case acheves the hghest throughput whch can be explaned by the fact that Pcells serve less UEs than the DUDe case then these UEs get a hgh throughput but on the expense of the 5th and 5th percentle UEs. Interestngly the DL-HP case acheves a hgher 9th percentle throughput than the DL-LP case whch seems counter ntutve. Ths can be explaned by the fact that Pcells n the DL-LP case serve even less UEs than the Pcells n the DL-HP case so the effect of the Pcells n the DL-LP case s notceable even after the 9th percentle. So f we look at the 98th percentle throughput n the DL-LP case t s 5 Mb/s whereas n the DL-HP case t s Mb/s whch shows that the effect of the Pcells n the DL-LP case s on a very lmted number of UEs. The gans of the 5th and 5th percentles are comparable to the results shown n [5.] where the authors apply a hgh Range Extenson (RE) value to the Pcells and they get a two tmes gan n the 5th and 5th percentle throughput. The RE technque bascally works n the same drecton as decouplng the UL and DL n the sense that t results n an ncreased coverage n the UL. The dsadvantage of RE s that the nterference level n the DL ncreases aggressvely as the RE bas ncreases whch requres the usage of nterference management technques as mentoned before whch s not requred n the UL/DL decouplng snce the UL and DL are treated as two dfferent networks n ths technque. Fgure 5.7 Average outage rate of Macro and Small cell layers comparson between the DL- LP, DL-HP and DUDe cases. Securty: Publc Page

123 Fgure 5.7 represents the average outage rate for the Mcell layer and the Pcell layer for the three cases or n other words the percentage of the UEs that fal to acheve the mnmum throughput demand ( Mb/s) out of the total number of connected UEs to a certan node. Snce the smulated scenaro s consdered to be a hghly dense scenaro t requres a very effcent use of resources n order to satsfy the hgh requrements of the UEs. As seen n the fgure the Macro layer has a very hgh outage rate (more than 9%) n the DL-HP and DL-LP cases whch s bascally explaned by the fact that the Macro layer s very congested n the UL as seen n Table so Mcells do not have enough resources to serve all ther UEs wth a hgh throughput level. However, n the DUDe case, UEs are dstrbuted more evenly between the nodes so the outage rate n that case s low (less than %) for both Macro and Pco cell layers. These results clearly show that decouplng UL and DL where UL s based on PL s a promsng canddate for future networks where the network load s expected to ncrease n the UL and where provdng a consstent and ubqutous servce to all UEs n dfferent network deployments and UE denstes s a prorty. Ths technque would also allow freeng up spectrum resources n the UL whch could be used for DL purposes. So far n ths secton, we presented an assessment of the UL/DL decouplng concept (DUDe) n a dense HetNet deployment. We started by a smplfed model to hghlght the motves and gans of ths concept and then we presented smulaton results based on a lve Vodafone LTE test network deployment n London. The smulatons used a hgh resoluton ray tracng propagaton model and user dstrbutons based on network measurements whch make ths model hghly realstc and provdng a much better vew on the effects of deployng ths technque n the real world than normal system level smulatons. The gans are very hgh n a dense HetNet deployment where ths technque can acheve between % and % mprovement n the 5th percentle UL throughput and even more than that n the 5th percentle throughput. Also, we have shown that the outage rate s decreased dramatcally n networks wth hgh mnmum throughput requrements where the outage rate s decreased from 9% to below % on the Macro layer. We beleve that the DUDe technque s a strong canddate for 5G archtecture desgns and t can be very useful n many applcatons lke Machne Type Communcatons (MTC) where Uplnk optmzaton s very crtcal. The next secton wll nclude an evoluton of the DUDe concept ncludng cell load and backhaul awareness. Securty: Publc Page 3

124 5.5 Load and backhaul aware decoupled access (DUDe.) 5.5. Motvaton Drven by an ncreasng densty of small cells n heterogeneous 4G systems, t was recently shown that the tradtonal strategy of handng over up (UL) and downlnk (DL) smultaneously based on downlnk receved power s sgnfcantly capacty suboptmal. Indeed, the UL performance gans for cell-edge users due to decouplng the DL and UL cell assocaton were consstently shown to be n the order of -3% [5.]. These capacty mprovements n the UL are very tmely snce the UL traffc has been growng over past years wth an unprecedented rate. Ths trend s drven by new applcatons whch generate symmetrc traffc, such as real-tme gamng and vdeo calls. In addton, the emergng array of socal networkng applcatons as well as machne-to-machne technologes generates more UL traffc than DL n an uncorrelated fashon. The optmzaton of the UL, partcularly for dsadvantaged users at the cell edge, s thus of hghest mportant to a consstent qualty of experence n emergng 5G systems. The decouplng s facltated by the fact that the degree of heterogenety has ncreased dramatcally over past years and s expected to grow further as part of 4G and 5G rollouts. Ths shft from a sngle-ter homogeneous network towards mult-ter heterogeneous networks (HetNets) composed of dfferent types of small cells (Mcro, Pco and Femto) comes along wth the unque opportunty to have ample connectons avalable at any pont and tme. Ths, n turn, facltates our purpose of decouplng the UL from the DL and the thereby acheved performance gans. The concept s shown n Fgure 5. where the Small Cell (Scell) has DL and UL cell borders whch are defned by the DL receved power and pathloss respectvely; a UE between these two borders wll tend to connect to the Scell n the UL and to the Macro cell (Mcell) n the DL as shown n the fgure. Some pror art s emergng n ths feld[5.] and [5.3], but have so far assumed that the cell assocaton strategy s based on the lnk qualty n each drecton. That s, the decson to handover the DL s (and has been) based solely on the DL receved power; whereas the decson to handover the UL s based solely on the UL pathloss. The system assumptons were to some extent deal n that nether the cell load nor the backhaulng capabltes have been taken nto account both of whch have an mpact onto the actual performance gans under more realstc operatng condtons. Ths shortcomng s addressed n ths paper at Securty: Publc Page 4

125 hand, where we proceed to outlne pror related art as well as a summary of our techncal contrbutons. The concept of downlnk/uplnk decouplng (DUDe) has been dscussed as a major component n future cellular networks n[5.3]-[5.5]. In [5.5], n partcular, DUDe s consdered as a part of a broader devce-centrc archtectural vson, where the set of network nodes provdng connectvty to a gven devce and the functons of these nodes n a partcular communcaton sesson are talored to that specfc devce and sesson. In [5.4] and [5.5], backhaul aware cell assocaton was consdered but only from a DL perspectve. In [5.6] and [5.7], load aware cell assocaton was studed n the DL as well. In [5.8], the authors study UL cell assocaton n a game theoretc approach to optmze the packet success rate of the UEs. In contrast to pror art, ths paper focuses on the cell assocaton algorthm where we argue that UL pathloss alone s not suffcent to effcently apply DUDe. Notably, the assocaton algorthm ought also to consder the overall load of the cell(s). Furthermore, snce DUDe requres sgnfcant backhaulng support, we also condton assocaton wth backhaulng capacty. Therefore, nstead of takng the decson based only on lnk qualty, the system now consders the lnk qualty, the cell load and the cell backhaul capacty. We then use a realstc scenaro of a cellular network based on Vodafone s real-world plannng/optmsaton tools whch, we beleve, adds a lot of value and credblty to ths work. We gve specal attenton to UL power control where we show that the performance depends greatly on the power control settngs. We use a flow level traffc model that s more realstc than the full buffer model consdered n pror art. The results are then dscussed and evaluated n great detals, thus offerng unque nsghts nto the performance trends of the emergng decouplng concept System model We consder the UL of a HetNet where, as deployment setup, we use the Vodafone LTE small cell test bed network that s up and runnng n the London area. The test network covers an area of approxmately one square klometre. We use ths exstng test bed to smulate a relatvely dense HetNet scenaro. The consdered network s stll shown n 3 where the black shapes are Macro stes and the red crcles are Small cells where n total we have B cells.we consder a realstc user dstrbuton based on traffc data from the feld tral network n peak tmes where the total number of users s Nu.Network traffc s modelled on a flow level where flows represent ndvdual fle or data transfers e.g. vdeo, audo or generc fle uploads. Ths model reflects a much more realstc Securty: Publc Page 5

126 traffc model than the full buffer model consdered n [5.]. We assume that a flow of sze ρ arrves to a UE s queue after a certan perod wat tme TW. TW and ρ are exponentally dstrbuted wth certan mean values. UEs experence a dfferent TW each tme a flow transmsson s fnshed.the rado lnk qualty s determned by many factors ncludng pathloss, fadng, nterference, and transmt power of the UEs. The UL SINR of UE connected to BS j s gven by: SINR j = h jp N+I where P s the th UE transmt power, h j ncorporates pathloss, shadowng and fast fadng (5.3) between UE and BS j, N s the nose power and I s the UL ntercell nterference. We characterze the achevable data rate usng the Shannon formula where BW represents the system bandwdth: C j Access = BW log ( + SINR j ). (5.4) Uplnk power control for the UEs follows the 3GPP specfcatons [5.9], where we consder open loop power control whch s gven by: P UE = mn{p MAX, log (M) + P + α. L }, (5.5) where P MAX s the maxmum permttable transmt power of the UE, M s the number of physcal resource blocks (PRBs) assgned to the UE, P s a normalzed power, α s the pathloss compensaton factor and L s the pathloss towards the servng cell. However, the power control algorthm does not account for nter-cell nterference whch, as we wll show n the results, affects greatly the UL performance. The effect s more pronounced when load balancng takes place snce UEs connect to a suboptmal cell so they are more vulnerable to nterference. Therefore we wll use an nterference aware power control algorthm whch sets a lmt to the transmt power of the UEs dependng on the nterference level that the UE causes to the closest neghborng cell. Smlar algorthms have been proposed n the lterature such as [5.]. The algorthm s as follows: P UE = mn{p MAX, log (M) + P + α. L, Ι + L s + log (M)} (5.6) where Ι represents the UL nterference power spectral densty (PSD) target for the UE and L s s the pathloss towards the most nterfered cell by the UE. Ths allows us to control the nterference level n the system by changng Ι. In a real world deployment, the Scell backhaul s always an ssue snce outdoor Scells are usually mounted on street furnture where there s no guaranteed wred connecton or lne-ofsght to the Mcell. Furthermore wth the ncreasng bt rates provded by access technologes the bottleneck s movng slowly from the access network to the backhaul. We consder that all Securty: Publc Page 6

127 cells n the test network have a lmted backhaul capacty C j bk where, naturally, Scells would have tghter backhaul constrant than Mcells Cell assocaton algorthm In our prevous study [5.]we have consdered the UL cell assocaton to be based on pathloss (PL) whch showed very hgh performance mprovements that were manly due to the load balancng effect and the mproved lnk qualty of the UEs.We extend ths approach to nclude the cell load and backhaul capacty n the decson crteron; consequently nstead of takng the decson based only on lnk qualty the UE consders the lnk qualty, the cell load and the cell backhaul capacty. Ths approach makes sense snce n real networks users are dstrbuted n a non-unform way where a UE that s close to a congested cell mght be better off connectng to a cell that s further but less congested. We consder a cell assocaton crteron that was consdered n [5.6] n the DL. We extend ths by applyng t to the UL and ncludng the backhaul capacty so that the optmal BS chosen by UE s gven by s(): s() = arg max j B ( η j )C j Max where C j Max = mn {C j Access, C j bk }, η j s the jth BS load whch s reflected n [5.6] as beng the (5.7) average resource utlzaton per cell. We found that ths approach for η j works fne n the DL whereas n the UL the stuaton s dfferent snce the UEs are power lmted whch means that a UE wth bad channel condtons would not be able to transmt on a large number of resource blocks. Ths would result n a low utlzaton of the resources of the cell even though ths cell could be servng many UEs. Therefore the cell utlzaton s a poor metrc to characterze the cell load n the UL and we resort to a dfferent way of estmatng the load. Notably, snce the flow arrval s exponentally dstrbuted and assumng the s`ystem to be statonary, the statonary dstrbuton of the number of flows N j s dentcal to that of an M/GI/ mult-class processor sharng system [5.]. The average number of flows s then gven by [N j ] = whch yelds η j = E[N j] E[N j ]+. Insertng η j nto (5.5) yelds: η j η j, s() = argmax j B C j Max E[N j ] + (5.8) The cell assocaton crteron n (5.8) wll be used for the rest of ths secton. We consder a fully dstrbuted algorthm where the man dea s that a UE does not need to stay connected to one BS n the UL all the tme. Instead a UE can keep ts anchor DL cell and every tme the UE Securty: Publc Page 7

128 has data (flow) to transmt n the UL, the UE connects to the cell wth the hghest crteron accordng to (5.8). The algorthm thus functons as follows: The BSs broadcast ther load E[N j ] and backhaul capactyc bk j. All UEs n the system start wth an exponentally dstrbuted wat tme (Tw) after whch a UE has a flow of sze ρ to transmt. The UE uses the crteron n (5.6) to fnd the best cell to connect to and after fnshng ts transmsson the UE dsconnects from the cell and goes dle for another random perod Tw; thereupon the operaton s repeated. The steps are detaled n Algorthm 5.. Algorthm 5.: Load/backhaul aware UL cell assocaton. BSs broadcast E[N j ] and C bk perodcally.. UEs ( N u ) are dle for a random T w ( N u ). 3. for Number of subframes 4. for each dle UE 5. f T w = 6. UE_queue = ρ. 7. UE connects to BS () accordng to (5.6) 8. UE s scheduled n BS () untl UE_queue =. 9. UE goes dle for a random T W.. else. T w = T w -. end for 3. end for Results In our smulatons we use the deployment setup of the Vodafone LTE test network n the London area. The setup conssts of 5 Mcells and outdoor Scells. Our propagaton model s based on a hgh resoluton 3D ray tracng pathloss predcton model. The model takes nto account clutter, terran and buldng data. Ths guarantees a realstc and accurate propagaton model. The user dstrbuton s based on traffc data extracted from the real network. We consder three power control settngs: - Loose power control wth full pathloss compensaton. We use (5.3) where (α, P ) are set to (, -8 dbm). Ths s referred to as Settng. - Conservatve power control wth partal pathloss compensaton. We use (5.3) where (α, P ) are set to (.6, -7 dbm). Ths s referred to as Settng. Securty: Publc Page 8

129 - Interference aware power control where we use (5.4) and set (α, P ) to (, -8 dbm) andι to - dbm. Table 5.3 Smulaton parameters Operatng frequency Bandwdth Network deployment User dstrbuton Scheduler Smulaton tme Traffc model Propagaton model Max. power transmt Antenna system UEs moblty Supported UL modulaton schemes.6 GHz (co-channel deployment) MHz ( frequency blocks) 5 Mcells and Scells dstrbuted n the test area. 33 UEs dstrbuted accordng to traffc maps read from a lve network Proportonal far seconds (, subframes) Flow level traffc. Mean flow sze = Mbt. Mean wat tme = ms. 3D ray-tracng model Mcell = 46 dbm, Scell power = 3dBm, UE = dbm. Macro: Tx, Rx, 7.8 db gan Pco: Tx, Rx, 4 db gan UE: Tx, Rx, db gan Pedestran (3km/h) QPSK, 6 QAM, 64 QAM We compare 3 UL cell assocaton cases: - Cell assocaton based on the DL Reference Sgnal Receved Power (RSRP) whch s the conventonal LTE procedure [5.9]. Ths case s termed DL-RSRP. - Cell assocaton based on the pathloss whch represents the DUDe algorthm as consdered n [5.] and s termed as DUDe. - Cell assocaton based on Algorthm 5. whch consders the cell load and backhaul capacty on top of the conventonal DUDe. Ths case s termed DUDe-Load. As ponted out before, all the results n the next secton wll focus on the UL performance. The smulaton parameters are lsted n Table 5.3 where we consder an outdoor LTE deployment. Securty: Publc Page 9

130 Intally we assume havng an deal backhaul (.e. no lmt on the backhaul capacty) on all the cells n order to study the load balancng effect. We start by comparng the throughput results wth dfferent power control settngs accordng to Settng and Settng. Fgure 5.8 Throughput percentles for the three cases wth power control Settng and. The throughput results for the three cases n comparson are shown n Fgure 5.8. Comparng DUDe to DL-RSRP, we see smlar gans as n [5.]where the 5th and 5th percentles are ncreased by more than % and 5% respectvely for both power settngs. The gans are due to the load balancng effect of DUDe and the better lnk qualty as UEs connect to the cells to whch they have the lowest PL. The 9th percentle throughput s less n DUDe than DL-RSRP as n the latter case only a few UEs are served by the Scells; therefore these UEs acheve a hgh throughput. We notce also that DUDe-Load s more affected, n terms of 5th and 5th percentles, by the change n the power settngs than DUDe. Ths s due manly to the fact that UEs connect to suboptmal cell n terms of PL due to the load balancng effect whch makes these UEs more vulnerable to nterference and more affected by the other UEs transmt power. We then compare DUDe and DUDe-Load startng by Settng where we see that the 5th percentle throughput s reduced by about % n the DUDe-Load case whle the 5th percentle s ncreased by 4% compared to DUDe. The loose power control causes the nterference level to ncrease whch has a negatve effect on the cell edge UEs as explaned below. Securty: Publc Page 3

131 Ths result shows that cell edge UEs (5th percentle) are better connected to a loaded cell to whch they have the better lnk qualty than connectng to an unloaded cell wth a worse channel. On the other hand the 5th percentle UEs can afford a reduced channel qualty and wth the hgher power headroom they actually acheve a hgh gan by usng the extra resources provded by the load balancng effect of DUDe-Load. Fnally, the fgure also shows a loss of about % n the 9th percentle throughput whch s logcal snce load balancng s always a trade-off between peak and (cell-edge/average) throughput. Then we compare DUDe and DUDe-Load for Settng where the 5th percentle throughput n DUDe-Load s mproved by about 4% over DUDe whereas the 5th percentle throughput s almost the same. Ths result shows how power control affects the network performance greatly. The used power control scheme sets a lower lmt on the transmt power of the UEs than the one used n Settng ; ths causes the UL nterference level n the network to be lower than the prevous case whch, n turn, allows the cell edge UEs to acheve a hgher throughput when connected to a suboptmal cell n terms of pathloss. On the other hand, the 5th percentle UEs do not acheve a hgher throughput wth the load balancng effect due to the lower bound on the UEs transmt power. These UEs hence mght not be able to use all the resources avalable to them; therefore, these UEs acheve a relatvely low gan from the hgher resource avalablty whereas the lower lnk qualty to the suboptmal cell reduces the throughput. Consequently, both effects almost even out and there s no gan n terms of 5th percentle throughput. The man message n Fgure 5.8 s that cell edge UEs are mostly nterference lmted whereas 5th percentle UEs are power lmted so havng power control Settng would beneft the 5th percentle UEs but would be harmful for cell edge UEs whle power control Settng has the opposte effect. Fgure 5.9 shows a CDF of the varance of the UEs UL SINR over tme for Settng where nterference s qute hgh. DUDe shows an average reducton of varance of about db compared to DL-RSRP whereas DUDe-Load shows an even lower average varance of about 5dB compared to DL-RSRP. The lower varance reflects a more stable nterference scenaro n DUDe where the lower varance of DUDe-Load results from the mproved load balancng effect whch mproves the resource utlzaton and, n turn, helps n stablzng the nterference. Ths s a very mportant feature snce UL nterference s known to be very volatle and dynamc and ths result shows that rado resource management (RRM) and self-organzng network (SON) operaton n general can be facltated usng DUDe. Securty: Publc Page 3

132 DL RSRP DUDe DUDe-Load CDF Users SINR varance (db) Fgure 5.9 CDF of the SINR varance where DUDe clearly shows nterference calmng propertes. Fgure 5. 5th, 5th and 9th percentle throughput of the three cases wth nterference aware power control In Fgure 5., we show throughput results for the nterference aware power control n (5.6). The am here s to try to fnd a trade-off between 5th and 5th percentle performance. We see, ndeed, that usng ths power control setup we acheve a smlar or even hgher 5th percentle throughput as n Settng n Fgure 5.8 where DUDe-Load outperforms DUDe by 5%. Also, n the 5th percentle the performance s smlar to Settng n Fgure 3 where DUDe-Load outperforms DUDe by %. The better performance of DUDe-Load n the 5th and 5th percentle throughputs results from the fact that the nterference aware power control affects more the UEs that cause hgher nterference, mostly cell edge UEs, to neghborng cells whle allowng the other UEs, 5th and 9th percentle UEs, to transmt wth a hgher power. Ths results n a lower nterfernce scenaro whch benefts the cell edge UEs that are nterference lmted and also allows the hgher achevng UEs to transmt wth a hgher power and, n turn, explot the extra resources resultng from load balancng. Securty: Publc Page 3

133 In the results n Fgure 5. we study the throughput behavour n the 3 cases whle changng the backhaul capacty of Scells from to Mbps. The Mcells backhaul capacty s assumed to be Mbps n all cases. We present the results for the nterference aware power control setup used n Fgure 5.. Fgure 5. Throughput percentles aganst backhaul capacty In the 5th percentle result the DUDe-load case shows the hghest throughput snce the UEs know of the backhaul and load capabltes of the cells. The DL-RSRP case performs better than the DUDe case up to a backhaul capacty of Mbps after whch DL-RSRP saturates and DUDe keeps on ncreasng. Smlarly, n the 5th percentle the DL-RSRP case s performng almost the same as DUDe- Load for very low Scell backhaul capactes snce n the former case the UEs are mostly connected to the Mcells but as the Scell backhaul capacty ncreases DL-RSRP starts saturatng and DUDe-Load surpasses t. Also the DUDe-load case s outperformng DUDe for the dfferent capactes where the gan ncreases as the backhaul capacty of Scells ncreases as wth the ncrease of Scell capacty DUDe-Load can have more optons to assgn UEs to Scells n a more effcent way. Fnally for the 9th percentle throughput, DUDe outperforms both DUDe-load and DL-RSRP snce t has the lowest number of UEs connected to the Mcells. These UEs can get very hgh throughputs, up to a certan pont where DL-RSRP surpasses DUDe. The reason s that Scells n DL-RSRP serve fewer UEs than the other cases. Therefore after a certan backhaul capacty Scells can provde very hgh data rates to these UEs. Lookng at the DUDe-load case, wth lower Scell backhaul capactes the UEs are pushed more towards the Mcells but Securty: Publc Page 33

134 stll DUDe-load has less UEs connected to Mcells than DL-RSRP whch explans why DUDeload outperforms DL- RSRP at the begnnng but as the Scells backhaul capacty ncreases the load balancng role s stronger whch stops the 9th percentle throughput of DUDe-load from ncreasng as explaned before. Fnally, n order to have some nsght on the load balancng effect of DUDe we compare the varance of the number of UEs per cell n the 3 cases. Ths measure gves an ndcaton of how UEs are dstrbuted among the cells. A hgh varance ndcates low load balancng effect and vce-versa. The varance s 47, 83 and for DL-RSRP, DUDe and DUDe-load respectvely. The DUDe case shows a clear mprovement of load balancng over DL-RSRP whch s shown by a dramatcally reduced varance whch, n turn, shows that the varaton n the number of UEs/cell s small. The DUDe-load case shows an even lower varance (.e. better load balancng) than DUDe as t s not only restrcted on balancng the UEs between Mcells and Scells but t also mproves the load balancng among Scells whch s a very mportant feature n future ultra-dense Scell networks. The decouplng of the downlnk and uplnk, referred to as DUDe, s an emergng paradgm shown to mprove capacty sgnfcantly for cell edge users. The underlyng prncples of DUDe relate to a proper and ndependent assocaton of the uplnk and downlnk. The focus of ths paper has thus been to extend the pror smple assocaton algorthms, based on the lnk qualty n the respectve lnks only, to a more advanced approach whch consders the load n the cells as well as any backhaulng constrans. Havng frst ntroduced the general system archtecture, the assocaton algorthms as well as the smulaton framework, we then presented and dscussed an ample amount of performance results. The fndngs confrm that the enhanced DUDe acheves a reduced UL SINR varance over baselne LTE, n the order of -5 db, whch facltates RRM and SON operatons. Results for our load-aware DUDe show that the system throughput mproves even further compared to the pror ntroduced baselne DUDe approach. The performance mprovement depends very much on the power control mechansm used. We have shown performance results for dfferent power control settngs and we used an nterference aware power control algorthm where throughput gans of the load aware DUDe over baselne DUDe are 5% and % n the 5th and 5th percentle throughput respectvely. We beleve that the DUDe technque s a strong canddate for 5G archtecture desgns and t can be very useful n many applcatons lke real-tme vdeo gamng, Machne Type Communcatons (MTC), among others, where uplnk optmzaton s very crtcal. In the next Securty: Publc Page 34

135 secton we dscuss some archtecture consderatons for 5G networks based on today s LTE archtecture that could facltate the mplementaton of DUDe n future networks. 5.6 RAN Archtecture consderatons for DUDe Decouplng the downlnk from the uplnk naturally requres some changes n the overall archtecture, spannng from the rado access network (RAN) to the core network (CN).Decouplng flow notably requres some tght control to enable a smooth flow splttng and flow reassembly. From a physcal channel perspectve, two possble solutons have been proposed n [5.] durng the Rel. works. Heren we study DUDe-enablng archtectures from the perspectve of Access-Stratum (AS) and Non-Access Stratum (NAS) sgnalng. AS sgnallng refers to Layer, Layer and RRC control messages exchanged between UE and BS.NAS sgnalng refers to control messages exchanged between UE and core-network. It ncludes e.g. establshng and managng bearers, authentcaton and dentfcaton messages, moblty management and trackng area update [5.3]. We note that delay, relablty and throughput requrements for each of these are very dfferent, whch naturally leads to a plethora of possble archtectures. To ths end, we propose a few optons whch rely on the followng assumptons: - a baselne 3GPP archtecture; - avalablty of Mult-Flow TCP, whch s able to handle dfferent data flows at networkng layer; - CoMP, whch defnes RAN traffc anchor ponts; - support of mult-homng whch allows dfferent RATs to be connected at the same tme to the same UE. The optons dscussed below are manly dfferentated n the layer where the separaton occurs, as well as the anchor pont of choce where the traffc s reunted agan. The dfferent choces are depcted n Fg. 5.3 and dscussed below. Securty: Publc Page 35

136 Fgure 5.. Archtecture consderatons NAS-Decouplng wth RAN Anchor Pont. In ths proposal a dedcated AS bdrectonalconnecton s kept for both the Mcell and the Scell. The strength of ths archtectures that all the delay-senstve sgnalng (such as H-ARQ sgnalng) s handled by each cell. However, ths requres the use of bdrectonal physcal control channels. Moreover, we note that havng the traffc-mergng anchor pont n the RAN requres the traffc to be handled va an establshed X nterface, smlar to the CoMP procedures currently outlned n 3GPP. NAS sgnalng s also handled va the X nterface. NAS-Decouplng wth CN Anchor Pont. Ths proposal dffers from the prevous one for the fact that data and NAS sgnalng are drectly routed from the Scell to the core. The advantage here s that moblty s handled n a more effcent way, at the expense of delays due to the MME, whch often resdes physcally far from the RAN. Also n ths case, delay-senstve sgnalng s handled va bdrectonal exchanges at each cell. AS-Decouplng wth RAN Anchor Pont. In ths embodment, the most aggressve of all, there s a complete separaton of the traffc;.e. f the UE communcates n the UL to the Scell, no DL s mantaned. Ths requres AS and NAS nformaton to be sent wth mnmal delay va the DL of the Mcell. The dsadvantage here s that the X needs to facltate close-to-zero delay communcatons; the advantage s that rado capacty s completely freed n the decoupled lnk. Securty: Publc Page 36

137 We note that we ddn t consder the case of AS-Decouplng wth CN Anchor Pont, due to the fact that for delay-senstve control sgnalng s not possble to tolerate delays due to the anchor n the CN often physcally resdng far from the RAN. Securty: Publc Page 37

138 6. Underlay Devce-to-Devce Communcatons for LTE-A Wreless Networks 6. Introducton Devce-to-devce (DD) communcaton enables nearby moble devces to establsh drect lnks n cellular networks, unlke tradtonal cellular communcaton where all traffc s routed va base statons (BSs). DD has the potental to mprove spectrum utlzaton, shorten packet delay, and reduce energy consumpton, whle enablng new peer-to-peer and locatonbased applcatons and servces and beng a requred feature n publc safety networks. Introducng DD poses many challenges and rsks to the exstng cellular archtecture. In partcular, n a DD underlad cellular network where the spectrum s reused DD transmsson may cause nterference to cellular transmsson and vce versa. Exstng operator servces may be severely affected f the newly ntroduced DD nterference s not approprately controlled. In ths secton, we focus our attenton on the applcaton of network codng (NC) to cooperatve wreless networks and devce-to-devce (DD) communcatons. In partcular, we have carred a set of analytcal studes wth numercal smulatons to fnd a transmsson scheme for relably conveyng nformaton n cooperatve networks usng NC [6.]. The key dea behnd these studes s to study and analyze a set of Random Lnear Network Codng (RLNC) [6.] based schemes n the State-Of-The-Art (SoA), that allows a moble network, for example LTE-A, to decrease the number of transmssons for a set of devces sharng common nformaton content even n the presence of hgh packet losses. The reason for ths choce s two fold: () RLNC based schemes have proven to provde sgnfcant gans for sngle hop and wreless mesh heterogeneous topologes [6.4] and () for cooperaton based RLNC schemes, nvolved devces typcally wll need less transmssons from the system or among themselves compared to other classcal strateges [6.3]. Ths knd of approach reduces the network load n terms of total transmtted power meanng a reducton n energy consumpton and hence, the overall nterference. Also, by reducng the amount of transmssons, network throughput s ncreased due to the completon tme for the same nformaton content sharng also beng reduced [4-7]. For our analytcal studes, we revew dfferent transmsson scenaros and schemes n typcal moble network topologes. The key pont s to evaluate whch are the benefts compared to SoA network codng schemes and when are they preferable than other conventonal technques under a set of performance metrcs for the gven scenaros [6.]. Also, a new scheme for low overhead n cooperatve networks s currently beng proposed as part of these studes [6.]. Frst, we provde an ntroductory motvaton and defnton for the 38

139 general problem of relably transmttng nformaton to a set of devces n a wreless network wth ntra-sesson based NC cooperaton mechansms. Then, we provde a revew of RLNC based technques to observe the behavour of the proposed schemes n terms of number of transmssons and metrcs gven scenaros. 6. RLNC transmsson scenaros and schemes analyss In ths secton we revew and provde a set of avalable technques that may be appled for our scenaro of nterest. To ths end, we frst defne the problem of relably transmttng a common nformaton content among a set of recevers. Then, we revew dfferent transmsson schemes n order to dentfy whch are sutable strateges accordng to a set of metrcs. 6.. Network and System Model We consder the problem of relably transmttng a batch of g packets from a source to N recevers n a sngle hop network as shown n Fg. 6.. Each packet has the same length of B bts. We assume a fxed tme slotted system to whch all nodes are synchronzed whch s a sutable case for LTE-A. We consder ndependent heterogeneous packet erasure rates on the connectvty lnks from the source to the recevers, ε j, j [, N], e.g. the packet recepton dstrbuton of recever j s Bernoull( ε j ) and s ndependent from all others n general. Fgure 6.. Sngle source - multple snk topology of N recevers wth erasures Besdes the sngle hop connectons, recevers mght be n a fully nterconnected fashon whch we call a moble cloud [6.]. Hence, the connectons can be regarded as bdrectonal symmetrc channels n a local area technology wth a homogeneous erasure rate ε. We also assume that n a tme slot a devce s ether transmttng, recevng or watng for recepton and that a successful transmsson s receved n the same slot t was sent. Lastly, we consder that an deal error free nstantaneous orthogonal feedback channel exsts n order to acknowledge packet recepton when requred. 39

140 We revew two transmsson scenaros, e.g. broadcast wth RLNC and cloud cooperaton wth RLNC snce these schemes outperform ther uncoded counterparts as descrbed n [3, 3-4]. We wll gve a bref descrpton of them to ndcate the behavour of the encodng, decodng and (f necessary) recodng process. In our work, we dfferentate our schemes accordng the Galos felds choces n a gven generaton. We separate them by () the sngle feld schemes, n whch the same Galos feld q s used for all codng coeffcents and () the multple felds schemes where a Galos feld q, [, g], s pcked for each codng coeffcent c, [, g] n the generaton. We wll specfy the felds choce and fnte felds arthmetcs and operatons for the multple felds schemes when we address them n the next sectons. We provde an n-depth study of the dstrbutons for the number of transmssons for each of the gven scenaros and schemes. We separate the analyss on a feld scheme bass snce the computaton of the basc probablty mass functon (pmf) heavly depends on t. In ths way, we model the number of transmsson as random varables usng the geometrc dstrbuton as a buldng block to derve the pmf n order to obtan a complete descrpton of the transmsson process. We frst gve a new expresson of the probablty mass functon for RLNC wth no erasures and then compute the pmf for the transmsson scenaros and schemes. 6.. Transmsson Scenaros Broadcast wth RLNC For broadcast, the source generates encoded packets from a generaton of sze g, where each packet has B bts. For the encodng process, coded packets are typcally generated by selectng random coeffcents from some partcular choce of Galos felds of sze q, namely GF(q ) for each codng coeffcent n general, n order to nclude both sngle and multple feld schemes. Then, the source attaches the codng coeffcents values n bts as overhead requred for the decodng process. Followng, the source broadcasts each coded packet to all recevers through the packet erasure channel descrbed n the prevous secton. When a packet successfully arrves at a recever, t checks f the packet s lnearly ndependent from all prevous. If not, t dscards t and wats for a new one. In case of beng lnearly ndependent, the recevers adds t to the codng matrx n order to effcently perform Gaussan elmnaton for decodng. In ths way, an acknowledgement s sent when all recevers have collected ther requred combnatons through an deal feedback channel of the model. If ths s not the case, the source keeps sendng coded packets untl ths occurs. For ths scenaro, recodng s unnecessary. 4

141 Cloud Cooperaton wth RLNC In a cloud cooperatve scenaro, packet transmssons takes place n two stages: () between source and recevers as group and () nternally n the cloud between recevers whch have mssng packets, whch we label the remote and local stage respectvely. For the remote stage, the source creates a telescopc coded packet, as descrbed above, and broadcasts t for the cloud to get t collectvely, meanng that t s enough that at least one recever gets a coded packet to consder the cloud has t. Ths stage fnshes once the cloud gets the generaton as a group. After the remote stage has ended, there wll be recevers whch do not have g lnearly ndependent equatons to decode. However, to manage ths stuaton, n the local stage each recever broadcasts recoded packets from the same feld n the sngle feld schemes. For the multple felds scheme, a packet s recoded wth all the coeffcents of the recodng vector pcked from lowest feld n the generaton, e.g. mn(q ). Ths recodng method provdes a smple way to recode but has an mpact n the decodng probablty [6.]. Broadcastng n ths stage occurs n a sequental fashon n order to guarantee that each recever dstrbutes all ts knowledge to the set. At the end, the local stage fnshes once all recevers have decoded the generaton and any node sends an acknowledge to the source Dstrbuton Prelmnares We provde some defntons for the geometrc dstrbuton gven that t wll be a basc tool for our dervatons. We focus on ths dstrbuton gven that ts a reasonable standard model for packet dstrbutons and allows us to easly compute more complcated dstrbutons for other scenaros and schemes. To avod ambguty, we employ the defnton of the geometrc dstrbuton that stands for the number of Bernoull trals (each of success probablty p) requred to get one successful event whch s defned as follows: (6.) From ths dstrbuton, we wll often use ts moments whch can be obtaned from (6.) and defned as follows: (6.) Another relevant property of the geometrc dstrbuton s ts probablty generatng functon (pgf). Ths property allows us to represent any dstrbuton as a functon G(z), n a complex z doman, retanng all ts propertes. The pgf for the geometrc dstrbuton s defned as: 4

142 (6.3) The pgf n (6.3) s defned n the complex doman wth a regon of convergence for whch the pgf seres converges. Fnally, we notce that the pgf transformaton over the orgnal pmf holds a relaton wth the Z-transform of a dscrete sequence. Let P(z) be the Z-transform of a pmf. Then, the relatonshp between both transformatons s defned n equaton (6.4). We see from t that the relatonshp s just evaluate analytcally n the nverse of the argument. Ths relatonshp wll play a role later when dervng the pmfs for our scenaros and schemes. (6.4) 6..4 Sngle Feld Schemes RLNC Probablty Mass Functon Wthout Erasures For the sngle feld cases, all the codng coeffcents are drawn unformly at random from GF(q). As already mentoned, we frst calculate the pmf for the number of transmssons requred for decodng for a sngle lnk usng RLNC wth no erasures as shown n Fg. 6.. Fgure 6.. Source - Snk RLNC lnk Although, prevous work exsts n ths area [5-6], the prevous defntons wll be helpful for obtanng other dstrbutons for whch the current analyss tools becomes cumbersome. A model for the requred number of transmssons for decodng RLNC packets employs an Absorbng Markov Chan as shown n Fg Fgure 6.3. Absorbng Markov Chan for RLNC 4

143 Ths chan comprses g+ states. Frst, state, wth [, g], s the case where the -th lnearly ndependent (l..) coded packet has not been receved by the destnaton. Then, D s the absorbng state where decodng s performed. The transton probabltes of each state depend only on the amount of prevously receved l.. combnatons. Usng the tools from secton 6..3, we model the amount of transmssons to leave any stage of the chan n Fg. 6.3 as a geometrc dstrbuton wth a success probablty dependng on the state as follows: (6.5) Wth the prevous defnton, the total number of transmssons to decode s just the sum of each of the random varables n (6.5) as follows: (6.6) However, analytcally computng the pmf for the sum of several random varables becomes very demandng snce t nvolves g- dscrete convolutons. Instead, we use a pgf approach to get the analytcal pmf. To acheve ths, we use (6.3) to get the pgf of each of the stages and use the fact the pgf for the sum of ndependent random varables s the product of ther pgfs. Also, we use the lnear dependence probabltes, defned as γ = p, [, g], n order to smplfy our calculus. Then, the pgf becomes as follows: (6.7) In (6.7), for the thrd equalty, P g stands for the product of all the lnearly ndependent probabltes from (6.5). We recognze ths scalar term as the probablty of decodng n exactly n g transmssons. We use the relatonshp descrbed n (6.4) to further smplfy our computatons. In ths way, and rearrangng terms, the Z-transform of the RLNC pmf wthout erasures s as follows: Wth the Z-transform of the pmf n (6.8), we perform a partal fracton expanson snce the product term of the second equalty s ratonal. Ths turns nto the followng: (6.8) 43

144 (6.9) In (6.9), we have spl the product as a sum of g terms snce all the poles of the Z-transform n (6.8) are smple due to the fact the lnearly dependent probabltes are unque. The a term n the summaton s the resdue for the -th term whch s easy to calculate as follows. (6.) Afterwards, wth the resdues values, we use smple nverse z-transforms pars to obtan the pmf for RLNC wthout erasures, as follows: (6.) Uncast RLNC wth Erasures Gven that we have computed the RLNC pmf n (6.), for a uncast pmf wth erasures we now account for the erasure process for a sngle lnk as descrbed n Fg Fgure 6.4. Source - Snk Uncast (RLNC) lnk For a uncast sesson, we need to have g lnearly ndependent receved packets n t transmssons. Therefore, we need to consder all the cases where. packets are receved (wth the fnal success n t, whch [ ] do not consder) and t- packets were lost or lnearly dependent. For ths, we revew two man probabltes n the same way as descrbed n [6.7,6.]. Frst, the probablty for successfully recevng lnearly ndependent coded packets n t transmssons by just consderng erasures, s NB(, ε) whch s the negatve bnomal probablty for trals and success probablty ε, defned as follows: (6.) Second, the probablty that g coded packets are lnearly ndependent n slots, s just (). Then by the total law of probablty, the probablty of decodng n exactly t slots for a sngle user wth RLNC based uncast wth erasures s: 44

145 (6.3) Broadcast RLNC wth Erasures The same broadcast RLNC scenaro as above s consderedwhere all the codng coeffcents are generated from the same feld. In ths stuaton, each recever needs to collect dfferent lnear combnatons to decode the packets. Then, we can regard the dstrbuton of ths scenaro as the case of fndng the dstrbuton for the maxmum ofn uncast sessons. In ths manner, the number of transmssons wll be determned by the recever that performs the worst n terms of retransmssons, the random varable for the number of transmsson n a broadcast scenaro s modelled as: (6.4) We calculate (6.4) by usng a cummulatve dstrbuton functon (cdf) approach. For the probablty of the maxmum beng less than or equal to t transmssons, ths must occur for all recevers. Next, under the ndependence assumpton, we compute the cdf for broadcast RLNC as shown as follows: Wth the cdf n (6.5), the pmf from broadcast can be obtaned from the cdf as follows: (6.5) (6.6) Wth (6.6), we compute all the values of the pmf. For the partcular case of t = g, the subtracted term n the thrd equalty s zero by defnton. Cloud Cooperaton RLNC wth Erasures We conduct the analyss consderng the possblty of users beng only connected to the remote moble network and users beng connected to the local network to observe the effect n the performance metrcs that we wll be detaled n the next sectons. We refer to the users 45