Interference Strength Alignment and Uplink Channel Allocation in Linear Cellular Networks

Transcription

1 Iterferece Stregth Aligmet ad Uplik Chael Allocatio i Liear Cellular Networks Yue Zhao ad Gregory Pottie Departmet of Electrical Egieerig Uiversity of Califoria, Los Ageles Los Ageles, CA, yuezhao@ucla.edu, pottie@ee.ucla.edu Abstract We fid the uplik chael allocatio that maximizes the total throughput i liear (oe-dimesioal) cellular systems, uder arbitrary fairess costraits defied by resource distributio fuctios. We exploit a reveue-cost separatio priciple i this multi-cell iterferece chael problem, ad equivaletly trasform maximizig the total throughput ito miimizig the total iterferece cost of all cells. The key idea i obtaiig the optimal chael allocatio is the aligmet of iterferece sigal stregths. We show that with very low complexity, a complete optimal chael allocatio ca be costructed by ripplig alog all the cells the proposed iterferece stregth aligmet procedure. From aalyzig the cosequece of the iterferece stregth aligmet, the superiority of the optimal chael allocatio over CDMA is justified. Numerical results are provided to demostrate the performace gap betwee these two schemes. I. INTODUCTION We study the optimal uplik chael allocatio (CA) of liear (i.e., oe-dimesioal) cellular commuicatio etworks. The sigle cell cases take the form of multiple access chaels [8], for which the optimal power ad chael allocatio schemes i Gaussia frequecy selective chaels are well uderstood [5], [13]. The multi-cell cases have a added iterferece chael ature, for which (weighted) throughput maximizatio by power ad chael allocatio is a ocovex optimizatio, ad is NP-hard to solve [11], [12]. For geeral (icludig cellular) iterferece etworks, much work has bee doe i reducig the complexity of solvig this o-covex problem (although still NP) [4], [14], [15]. With covexified utility fuctios or other covex approximatios, (polyomial time) low complexity methods have also bee developed [6], [7], [10]. ecetly, progress i iterferece aligmet [2] ad the determiistic chael model [1] has provided ew perspectives ito maagig iterferece. Combiig these two leads to the idea of aligig the iterferece i bit-levels, or sigal stregths [3], [15]. I this paper, we exploit the idea of iterferece sigal stregth aligmet i multi-cell uplik chael allocatio. We first separate optimizig fairess ad throughput by itroducig the resource distributio fuctio (rdf), which ca be desiged i a atural way to match the fairess requiremet. We the assume rdf is give as a fairess costrait o CA, ad focus o fidig the optimal CA that maximizes the total throughput. With a reveue-cost separatio priciple, i our problem settig, throughput maximizatio is equivaletly trasformed ito iterferece cost miimizatio. For every cell, we sort the iterferece stregths (received at its base statio) geerated by users i each of its eighborig cells. We the alig the i th (i = 1,2,) strogest iterferece i each of its eighborig cells to co-exist i the same chael. We prove that oce the aligmet is doe for every cell, the resultig CA miimizes the total iterferece cost, ad hece maximizes the total throughput. For liear cellular etworks, we show that aligmet of all the cells ca be doe i a ripplig maer. As a result, this procedure of obtaiig the complete optimal CA has a low complexity of O( cell mlog(m)), (where cell is the umber of cells, ad m is the umber of chaels.) As will be show, our methods eable both aalytical ad umerical comparisos of optimal CA with CDMA schemes. The rest of the paper is orgaized as follows. The system model is established i Sectio II, where several key steps are take to clarify ad simplify the problem structure. I Sectio III, the complete uplik optimal chael allocatio usig iterferece stregth aligmet is derived. I Sectio IV, the performace of the optimal CA is compared with typical CDMA schemes through two examples with differet fairess requiremets. Coclusios are draw i Sectio V. II. SYSTEM MODEL We cosider the uplik chael allocatio of a liear cellular system with base statios (BS) positioed o a straight lie. We make the followig assumptios: i) Users withi the same cell use orthogoal frequecy chaels. ii) A frequecy reuse factor of 1 is applied amog all the cells. iii) Iterferece is treated as oise, (ad hece o iterferece cacellatio or joit decodig is used.) iv) Iterferece from users ot from the immediate eighborig cells is igored. Thus, each cell has two iterferig cells (Figure 1). v) There are m parallel chaels with uit badwidth. I every cell, all the m chaels are utilized, (i.e., there is o vacat chael i ay cell.) vi) Chaels experiece frequecy flat fadig, ad the oise spectral desity is flat at all receivers. vii) Each user trasmits at its ow uiform power spectral desity (PSD) over all the chaels it occupies. Accordigly, we focus o the problem of chael allocatio, assumig that users do ot vary their selected power levels.

2 A. Approximate ate Fuctio Employig the Shao capacity formula for Gaussia chaels [8], we have = log(1 + SIN) bits/sec/hz. We approximate this formula i the followig way: log(1 + SIN) max(0, log(sin)) (1) It ca be verified that the maximum deviatio from the.h.s. of (1) to the L.H.S. is 1 bit/sec/hz, which occurs whesin = 0dB. Furthermore, ote that (1) = log(sin) provided that SIN 0dB, ad this leads to the decompositio of the problem ito two steps: 1) Active user selectio: i each chael, we select the users that are expected to have SIN 0dB, so that the zero floor becomes redudat. (The iactive users are the dropped from the objective fuctio i this chael.) 2) Optimizatio with respect to the selected active users, with log(sin). Note that without SIN 0dB, log(sin) ca be arbitrarily egative. Thus, choosig the correct active users i step 1) is crucial for the approximatio to be close. B. eveue-cost Separatio Priciple From ow o, we will assume that the active user selectio is properly doe i each chael so that the zero floor i (1) ca be equivaletly dropped. I cellular scearios with i-cell orthogoalizatio, (i.e. iterferece oly comes from other cells,) this assumptio is typically easy to satisfy. Thus, we have the followig separatio priciple for a active user i s rate i ay oe chael: g ii P i i = log(sin i ) = log( k i g ) kip k +N i = reveue i cost i (2) wherereveue i log(g ii P i ),cost i log( k i g kip k +N i ), ad k rus over all co-chael users of user i. P k is the trasmit PSD of user k, N i is the oise spectral desity at receiver i, ad g ki is the chael gai from trasmitter k to receiver i. As will be show i Sectio III, this reveue-cost separatio priciple has a sigificat impact i both aalysis ad desig it greatly reduces the complexity of fidig the optimal CA i multi-cell cases. C. esource Distributio Fuctio (rdf) Defiitio 1: The resource distributio fuctio rdf(i) (i = 1,,, where is the total umber of users,) is defied to be the umber of chaels allocated to user i, or equivaletly, the amout of badwidth allocated to user i. Note that there are may differet CAs that correspod to the same rdf, which may result i very differet total throughput due to the differeces i multi-cell co-chael user selectios. With the cocept of rdf, the chael allocatio problem is decomposed ito two steps: i) decide rdf, ad ii) decide CA give rdf. For step i), the desig of rdf should match users demads ad chael coditios: users with higher target rates ad/or worse chael coditios should have more resources (chaels) distributed to them. I this paper, we assume that step i) has bee properly doe, i.e. rdf is give accordig to the fairess requiremet. (Differet cells ca have differet rdf.) We the ivestigate the optimal chael assigmet i all cells that maximizes the total throughput. (A future directio to ivestigate is to iteratively optimize step i) ad step ii) i a alteratig maer, which remais a ope problem.) III. OPTIMAL UPLINK CHANNEL ALLOCATION A. Facts from the Sigle Cell ad the Two-Cell Cases For each sigle cell, we have the followig lemma: Lemma 1 (eveue Ivariace): Give rdf, the reveue of ay oe user withi a cell is ivariat to differet CAs that coform to this rdf. This is a immediate implicatio of the flat fadig ad the i-cell orthogoalizatio assumptios. For the multi-cell cases, the reveue-cost separatio priciple (Sectio II-B) ad the reveue ivariace property (Lemma 1) imply the followig: Corollary 1: To maximize the multi-cell total throughput give rdf of all cells, it is equivalet to fid the optimal CA that has the miimum total cost icurred. For otatioal simplicity, wheever cosiderig three cosecutive cells o a lie, we deote the ceter cell by C, the left cell by L, ad the right cell by (Figure 1). We deote the user idices occupyig chael j(= 1,,m) i L, C, ad by L(j), C(j), ad (j). These user idex fuctios ca also be viewed as chael allocatio fuctios because they fully specify the CAs i L,C ad. Now, cosider two adjacet cells L ad C. We have the followig lemma o the iterferece received by C from L. Lemma 2: Give rdf of L, the set of (a total of m) iterferece stregths geerated from the users i L to the BS of C is ivariat to the actual CA of L. This is agai a immediate implicatio of the flat chael assumptio. I fact, for ay oe user i i L, its (a total of rdf(i)) iterferece stregths see at the BS of C do ot deped o which rdf(i) chaels it occupies. B. Iterferece Stregth Aligmet of the Two Neighborig Cells For ay three cosecutive cells L, C ad, we ow cosider miimizig the total cost that C sustais, geerated (BSs) (Users) L C l 4 l 3 l 2 l 1 c 1 c 2 c 3 c 4 r 1 r 2 r 3 r 4 Fig. 1. Three cosecutive cells i a liear etwork. Users coected by lies are suggested co-chael users.

3 from L ad. Clearly, the total cost that C sustais does ot deped o its ow CA, but depeds o the CAs of L ad. Accordig to Lemma 2, deote the set of iterferece stregths from L to C by {IL 1,,Im L },I1 L I2 L Im L. Due to the flat chael assumptio, WLOG, we idex the m chaels such that I j L is the iterferece from L to C i chael j. Cosequetly, the user idex fuctio L(j)(j = 1,,m) implied by such idexig satisfies that I j L = g L(j)C(j)P L(j). (Note that receiver C(j) is just the BS of C regardless of j.) Next, we would like to specify the CA i (i.e. {(j),j = 1,, m}) uder the give rdf, such that the total cost geerated from {L ad } to C is miimized. We start with a arbitrary iitial CA (j), ad compute {I j = g (j)c(j)p (j), j = 1m}. With the chael allocatio fuctiosl(j) ad(j), the total cost that the users i C sustai is m cost j = m log(i j L +Ij +N), (3) where cost j is the iterferece cost that C sustais i chael j, ad I j L ad Ij (j = 1,,m) are co-chael iterfereces. For ay other CA (j) uder the same rdf, it ca be represeted by a permutatio fuctio P(j)(j = 1,,m) applied to the iitial CA (j), such that (j) = (P(j)). I other words, chael j is assiged to the user who iitially has chael P(j) assiged to it. With L(j) ad (j), the total cost that the users i C sustai chages to m m cost j = log(i j L +IP(j) +N), (4) where I j L ad IP(j) (j = 1,,m) are the ew co-chael iterfereces. The key i fidig the optimal P(j) (ad hece the optimal CA i) amog them! permutatios is a idea of iterferece stregth aligmet. We sort {I j,j = 1,,m} i descedig order: I j1 Ij2 Ijm, ad defie P (k) j k,k = 1,,m. The followig theorem ca be show: Theorem 1: Amog all permutatio fuctios P(j), {P (j),j = 1,,m} yields the miimum total cost that the users i C sustai. emark 1: From the defiitio of P (j), we have IL 1 IL 2 Im L ad (1) IP I P (2) I P (m). What has bee doe i the optimal CA is that we alig the strogest iterferece each from L ad to co-exist i the same chael, ad so o for the 2 d strogest,..., all the way to aligig the weakest iterferece each from L ad to co-exist i the same chael. Proof of Theorem 1: We prove a equivalet form of the theorem: If a 1 a 2 a m 0,b 1 b 2 b m 0, the m log(a j +b j +N) m log(a j +b f(j) +N), for all permutatio fuctios f(j),j = 1,,m. We use iductio o m as follows. a 1 b 1 a 2 b 2 a f -1 (1) a m b f(1) b m Fig. 2. After replacig (a 1,b f(1) ) ad (a f 1 (1),b 1) with (a 1,b 1 ) ad (a f 1 (1),b f(1)), the total cost decreases, (or remais uchaged.) m = 1 is trivial. For m = 2, by Jese s iequality, a 1 a 2 log(a 1 +b 1 +N) b 1 b 2 + log(a 2 +b 2 +N) log(a 1 +b 2 +N) b 1 b 2 log(a 1 +b 1 +N) a 1 a 2 + log(a 2 +b 2 +N) log(a 2 +b 1 +N) (5)+(6) implies the theorem for m = 2. Suppose the theorem holds for all m k. For m = k + 1, we first represet the problem by a bipartite graph (Figure 2): it cosists of upper m poits a 1,,a m ad lower m poits b 1,,b m, ad there is a (udirected) edge betwee every a j ad b f(j). The theorem is the equivalet to claimig that the graph with all vertical edges (a j,b j ),j = 1,,m yields the miimum total cost. Proof follows: Give a graph geerated from a permutatio fuctio f(j): i) If edge (a 1,b 1 ) is i the graph, the after removig (a 1,b 1 ), the rest of the graph degrades to a m = k case, ad applyig the iductio assumptio proves this case. ii) If edge (a 1,b 1 ) is ot i the graph, Applyig the iductio assumptio with m = 2 yields, log(a 1 +b f(1) +N)+log(a f 1 (1) +b 1 +N) (5) (6) log(a 1 +b 1 +N)+log(a f 1 (1) +b f(1) +N) (7) I other words, after replacig the two edges (a 1,b f(1) ) ad (a f 1 (1),b 1 ) with (a 1,b 1 ) ad (a f 1 (1),b f(1) ), the total cost decreases (or remais uchaged), ad the ew bipartite graph falls ito the case of i). C. Optimal Chael Allocatio i Ifiite Liear Cellular Networks Now, cosider a two-sided ifiite liear cellular etwork, with rdf of users i all cells give. The above aligmet procedure ca the be used i a ripplig maer alog the liear etwork to obtai the optimal CA of all cells idexed as, 3, 2, 1,0,1,2,3,. We first miimize the total cost sustaied by cells {, 3, 1,1,3,}: this leads to the optimal CA i cells {, 2,0,2,}, specified by the followig procedure.

4 Average cost of CDMA Average cost of optimal CA Bits/sec/Hz Step 0. Assig a arbitrary CA to cell 0; Step 1a. Based o the CA i cell 0, assig the CA i cell 2 accordig to the the aligmet rules i Theorem 1, ad the total cost that cell 1 sustais is miimized; Similarly, Step 1b. CA i cell 0 CA i cell -2: the cost i cell -1 is miimized; Step 2a. CA i cell 2 CA i cell 4: the cost i cell 3 is miimized; Step 2b. CA i cell -2 CA i cell -4: the cost i cell -3 is miimized; Ad so o. Note that i Step 0, the arbitrary CA assigmet of cell 0 does ot lose ay geerality (ad hece optimality) due to the flat chael assumptios. Next, we miimize the total cost sustaied by cells {, 2,0,2,} by applyig the ripplig procedure to cells {, 3, 1,1,3,}. Fially, we obtai a complete optimal CA that yields the miimum total cost of all the cells. Because of the reveue ivariace property (Lemma 1), it achieves the maximum total throughput. emark 2: Fidig a complete optimal CA has a complexity of O( cell mlog(m)), where cell is the umber of cells: The term cell comes from the ripplig procedure as above, ad the term m log(m) comes from sortig the iterferece stregths for each cell (usig e.g. Heapsort.) emark 3: For all the users i ay oe cell, the orderig of their iterferece stregths at the left eighborig BS could be differet from that at the right eighborig BS. (E.g., i Figure 1, user c 1 (amog all the users i C) creates the strogest iterferece to the BS of L, but the weakest iterferece to the BS of.) To guaratee that a CA is global optimal, users withi each cell must alig their iterferece stregths with those from both their 2 d left ad 2 d right eighborig cells, (as achieved by the proposed ripplig procedure.) emark 4: We have cosidered liear cellular etworks. Geeralizatios to fidig the optimal CA i two dimesioal cellular etworks also follow the iterferece stregth aligmet idea. Each cell, however, eeds to alig with more tha two other cells (istead of aligig with just the 2 d left ad the2 d right eighborig cells as i the liear case.) This ofte leads to the impossibility of perfect aligmet of all the cells, ad sigificatly raises the complexity of fidig the global optimal CA. IV. PEFOMANCE EVALUATION: OPTIMAL CA VS. CDMA I this sectio, we give two typical examples that umerically compare the average spectral efficiecy of the optimal CA with that of CDMA schemes. We assume the simplified path loss model [9] P r = P t K( d0 d )γ with d 0 = 50m (outdoor eviromet), ad o multipath or shadow fadig. The parameter K is irrelevat to the compariso betwee optimal CA ad CDMA, ad is assumed to be 1. γ betwee 2.5 ad 4 will be tested below. We assume a liear cellular etwork with cell radius = 500m, ad that there are users equally spaced i every cell. Example 1. Users with Equal esource Share Uiform rdf I this example, we assume that every user gets the same uit chael resource (uiform rdf,) ad all users trasmit at the same uiform PSD. We compute a completely iterferece limited case, i.e., oise power is igored. I this case, the value of the PSD is irrelevat to the compariso betwee optimal CA ad CDMA, ad is assumed to be 1. For ay three cosecutive cells L, C,, the optimal CA derived i Sectio III yields a fully symmetric CA i L ad such that users i L ad with the same distace to the BS of C co-exist i the same chael (Figure 1). With the uiform rdf, the average cost per user is cost opt.ca = 1 log(2i i ) bits/sec/hz (8) where I i (i = 1,,) traverses the iterferece from all positios of the users i L (ad symmetrically.) With CDMA, every user i a cell sustais the same amout of cost from other cells (sice iterferece is averaged:) cost CDMA = log( 1 2I i ) bits/sec/hz (9) From Jese s iequality, the superiority of the optimal CA over CDMA becomes evidet: cost CDMA cost opt.ca always. We plot i Figure 3 the differece cost CDMA cost opt.ca as a fuctio of, parameterized by γ = 2.5,3,3.5,4. Note that from reveue ivariace, (cost CDMA cost opt.ca ) equals the throughput differece i terms of the average spectral efficiecy. We observe the followig: 1) As the umber of users icreases ad/or as γ icreases, the variatio i the set of iterferece stregths {I i } icreases. Thus the gap from Jese s iequality, ad hece the cost differece, icreases. Fig Number of users per cell = 2.5 = 3.0 = 3.5 = 4.0 Superiority of optimal CA over CDMA Equal resource case

5 2) As oe umerical rule of thumb i this particular example, with γ = 4, the superiority i throughput of optimal CA over CDMA reaches above 1 bits/sec/hz whe the umber of users reaches 25 per cell. Example 2. Users with Equal ate Hard Fairess I this example, we guaratee that users at all positios i a cell achieve the same rate i bits/sec. Thus, rdf eeds to be desiged such that the edge users have more badwidth. We achieve this by desigig rdf such that every user i a cell has the same total reveue i bits/sec. This requires rdf(i) beig iversely proportioal to reveue i i bits/sec/hz (we assume that the uiform trasmit PSD of all users is set such that a user at the cell edge has a reveue of 1 bit/sec/hz.) Next, equal total cost that each user sustais ca be achieved by properly time sharig differet optimal CAs. At the ed, each user achieves a equal rate i bits/sec. We first ormalize rdf withi a cell: rdf(i) such that rdf(i) rdf(j), rdf(i) = 1. The, similarly to Example 1, the average cost with optimal CA is cost opt.ca = log(2i i +N)rdf(i) bits/sec/hz (10) With CDMA, the cost every user sustais is agai equal to each other: cost CDMA = log( 2I i rdf(i)+n) bits/sec/hz (11) Clearly, (10) (11) are geeralizatios of (8) (9): whe rdf is a uiform fuctio, (with N igored,) (10) (11) degrade ito (8) (9). Agai, from Jese s iequality, cost CDMA cost opt.ca always. I additio to the previous observatios i Example 1 which appear similarly here, we have computed ad observed that requirig hard fairess makes the throughput differece of optimal CA over CDMA eve larger The asymptotic differeces icrease almost 50% compared to the equal resource share case. The ituitio is that allocatig more badwidth to edge users icreases the total variatio i the set of iterferece stregths, ad hece the gap from Jese s iequality. emark 5: We have used rdf to achieve fairess betwee users. We ote that while usig power cotrol (PC) is also a optio, prelimiary ivestigatio has show that with CDMA, PC is ofte less efficiet tha adjustig rdf. The ituitio is that to compesate cell-edge users path loss disadvatage, raisig their power (istead of allocatig more chaels to them) ofte creates more aggregate iterferece to eighborig cells. For the optimal CA problem, the compariso betwee adjustig rdf ad PC remais a ope questio. V. CONCLUSION We defied the resource distributio fuctio (rdf) which takes care of the fairess requiremet i cellular etworks. Uder ay give rdf as a fairess costrait, we have foud the uplik chael allocatio that maximizes the total throughput i liear cellular etworks, assumig flat fadig ad that users have fixed power levels (but ot ecessarily equal to each other.) The optimal chael allocatio is characterized by a iterferece stregth aligmet property for every cell, its eighborig cells iterferece i all m chaels must be aliged such that each eighborig cell s i th strogest iterferece (i = 1,,m) co-exists i the same chael. The complete optimal chael allocatio is foud with O( cell mlog(m)) complexity. From Jese s iequality, our results provide a clear justificatio of the superiority of optimal chael allocatio over CDMA i terms of throughput. Numerical comparisos uder various fairess requiremets i liear cellular etworks have show evidet gaps i bits/sec/hz betwee the throughput of optimal CA ad that of CDMA. For two (or higher) dimesioal cellular etworks i frequecy selective chaels, fidig the global optimal uplik CA remais ope due to its high complexity. Future research will also be pursued o the combiatio of power allocatio ad rdf desig with the proposed chael allocatio algorithm. EFEENCES [1] A. Avestimehr, S. Diggavi, ad D. Tse. 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