Performance Enhancement of Adaptive Orthogonal Modulation in Wireless CDMA Systems

Transcription

1 Performance Enhancemen of Adapive Orhogonal Modulaion in Wireless CDMA Sysems 1 Alaa Muqaash, Marwan Krunz, and Tao Shu Deparmen of Elecrical and Compuer Engineering The Universiy of Arizona Tucson, AZ {alaa,krunz,shu}@ece.arizona.edu Absrac Recen research in wireless CDMA sysems has shown ha adapive rae/power conrol can considerably increase nework hroughpu relaive o sysems ha use only power or rae conrol. In his paper, we consider join power/rae opimizaion in he conex of orhogonal modulaion (OM) and invesigae he addiional performance gains achieved hrough adapaion of he OM order. We show ha such adapaion can significanly increase nework hroughpu while simulaneously reducing he per-bi energy consumpion relaive o fixed-order modulaion sysems. The opimizaion is carried ou under wo differen objecive funcions: minimizing he maximum service ime and maximizing he sum of user raes. For he firs objecive funcion, we prove ha he opimizaion problem can be formulaed as a generalized geomeric program (GGP). We hen show how his GGP can be ransformed ino a nonlinear convex program, which can be solved opimally and efficienly. For he second objecive funcion, we obain a lower bound on he performance gain of adapive OM (AOM) over fixed-modulaion sysems. Numerical resuls indicae ha relaive o an opimal join rae/power conrol fixed-order modulaion scheme, he proposed AOM scheme achieves significan hroughpu and energy gains. I. INTRODUCTION Efficien uilizaion of he limied wireless specrum while saisfying applicaions qualiy of service (QoS) requiremens is an essenial design goal of fourh-generaion (4G) wireless neworks and a key o heir successful deploymen [46]. Despie heir appealing simpliciy, resource allocaion policies in currenly deployed wireless neworks, such as he IEEE , are inefficien, perform poorly under moderae loads [10], and are unable o mach he growing demand for high daa raes. The need for specrally efficien sysems has moivaed he developmen of adapive ransmission echniques, several of which are in he process of being sandardized. These echniques adap users parameers according o he ime-varying channel condiions, inerference levels, rae requiremens, bi error rae (BER) needs, and energy consrains [29]. This work was suppored by he Naional Science Foundaion under grans ANI , ANI , and ANI , and by he Cener for Low Power Elecronics (CLPE) a he Universiy of Arizona. CLPE is suppored by NSF (gran # EEC ), he Sae of Arizona, and a consorium of indusrial parners.

2 2 In narrow-band (i.e., non-spread specrum) sysems, adapaion includes varying he ransmission power [16], modulaion order [14], symbol rae [9], coding rae [41], or any combinaion of hese parameers [3], [13], [15], [28]. In paricular, i is well known ha adapive modulaion is a promising echnique for increasing he user daa rae in narrowband sysems. This was demonsraed in [14] for he single-user case, where i was shown ha adapive modulaion can provide up o 10 db gain over a fixed-rae sysem ha uses only power conrol. In [33], he auhors sudied he muliuser case and showed ha even wihou power conrol, adapive modulaion has a significan hroughpu advanage over fixed-rae power conrol schemes. Much of he work on adapive modulaion in narrow-band sysems (e.g., [4], [14], [23], [24], [43]) has been moivaed by recen advances in designing low-complexiy adapive modulaion circuiry and channel esimaion echniques [14]. In he conex of (wide-band) direc-sequence code division muliple access (CDMA) neworks, power conrol has radiionally been he single mos imporan adapaion parameer [12], and has been horoughly sudied (see [37] and he references herein). Recen effors on adapaion in CDMA neworks have also focused on adaping he ransmission rae using muliple codes [18], [36], parallel combinaory spread specrum [48], muliple chip-rae [44], adapive modulaion and coding (AMC) [1], [6], [17], and classical variable processing gain (VPG) echniques [10], [11], [19], [22], [25], [27], [30], [35], [42], [45] in which boh he ransmission power and daa rae are adaped, bu he modulaion and coding are kep fixed. For CDMA sysems ha require coheren recepion, a pilo signal mus usually be ransmied for each user. This is he case, for example, in WCDMA sysems [1], where a high-rae coheren wo-dimensional modulaion 1 such as 16QAM [1], [17] is used. Alernaively, o reduce he implemenaion complexiy associaed wih coheren recepion (e.g., recovering he pilo signals from users) and o poenially improve energy efficiency (a pilo signal consumes a considerable amoun of he mobile user s energy), noncoheren recepion can be used [21]. M-ary orhogonal modulaion (OM) is a specrally-efficien modulaion echnique ha is well suied for his applicaion [12]. Alhough differenial phase shif keying (DPSK) can also be used for noncoheren recepion, i has been shown ha OM ouperforms DPSK for M > 8 in addiive whie Gaussian noise (AWGN) and in Rayleigh fading channels [32]. OM has been used successfully in he uplink of IS-95 and is also par of he radio configuraions of he cdma2000 sandard [18]. This paper focuses on CDMA sysems for which coheren recepion is no possible and where OM is used (e.g., uplink IS-95). For such sysems, classical (i.e., fixed OM order) VPG has been he focus of research because of is performance benefis, flexibiliy, and pracicaliy (e.g., low peak-o-mean envelope power, fixed chip rae, ec. [19]). The exensive work on VPG has clearly quanified he performance 2 advanages of combined rae/power conrol over power conrol alone (e.g., see [35], [19]). However, o he bes of our knowledge, adaping he modulaion order for variable-rae OM-based sysems remains an unexplored area of research, and one for which join rae/power conrol has no ye been invesigaed. Our firs conribuion (Secion II) is o show ha when OM is used, he performance of variable-rae CDMA 1 By wo-dimensional modulaion, we mean modulaion schemes for which he modulaion symbol can be represened by a 2-dimensional vecor, i.e., by a poin in he 2-dimensional signal space (or consellaion). 2 Throughou he paper, he erm performance is used o refer o nework hroughpu and/or per-bi energy consumpion.

3 3 neworks can be improved by using higher OM orders a lower daa raes. We hen use hese resuls o show ha, in he single link case, variable-rae sysems wih adapive orhogonal modulaion (AOM) significanly ouperforms VPG sysems wih a fixed OM order 3. Thus, similar o adapive modulaion in narrow-band sysems, AOM in CDMA sysems is shown o be a promising echnique for increasing he user daa rae. Noe ha he processing gain and ransmission power are varied in boh AOM and VPG. However, in AOM he OM order is also varied depending on he daa rae, whereas VPG uses he same OM order for all daa raes. The main goal of our sudy is o invesigae he heoreical performance limis of join rae/power conrol for AOMbased CDMA neworks and o gain insighs ino he echnique iself. We consider boh poin-o-poin (PTP) as well as mulipoin-o-poin (MuliPTP) neworks (see Figure 1). PTP neworks is he more general communicaion paradigm. I can represen a compleely disribued mobile ad hoc nework, or a microcellular nework in which mobile-base saion pairs compee for he same frequency specrum. In MuliPTP neworks, muliple nodes ransmi o one node, as in he case of a cluser-based ad hoc or sensor nework [31] or in he case of he uplink of a single cell in a CDMA-based cellular nework (e.g., IS-95 [32]). Wih very few excepions, previous work has mainly considered MuliPTP neworks. (a) PTP Neworks. (b) MuliPTP Neworks. Fig. 1. Nework opologies considered in he paper. To joinly opimize he powers and raes, we consider wo hroughpu-relaed objecive funcions: (1) minimizing he maximum service ime, and (2) maximizing he sum of users ransmission raes. Boh funcions are opimized subjec o consrains on he maximum ransmission power, on he minimum and maximum ransmission raes, and on he BER. The firs funcion is novel in our conex and has no received much aenion; previous research has primarily focused on he second objecive funcion. However, as we argue in Secion III, here are imporan pracical advanages of he firs objecive funcion. We obain he opimum soluion o he problem of minimizing he maximum service ime in boh PTP and MuliPTP neworks by formulaing he problem as a generalized geomeric program (GGP) [8]. We hen ransform his GGP ino a geomeric program (GP), which iself can be ransformed ino a nonlinear convex program. The advanage of hese ransformaions is ha a convex program has a global opimum ha can be found very efficienly [8]. Furhermore, in he case of MuliPTP neworks, we derive a simple expression for compuing he opimal powers and raes ha minimize he maximum service ime. Our soluions are compuaionally efficien. They can also be used o deermine he feasibiliy of 3 For breviy, we use he acronym AOM o refer o a variable-rae sysem wih adapive OM, while he acronym VPG refers o a variable-rae sysem wih a fixed OM order.

4 4 a se of rae and BER requiremens under cerain consrains, hus, allowing for he use of admission conrol policies. Alhough he second objecive funcion (i.e., maximizing he sum of raes) has he advanage of being in he exac form of hroughpu, i has he limiaion of having several local maxima. As a resul, here are no compuaionally efficien algorihms o solve his problem 4. Hence, for PTP neworks, alhough we do no know he opimal rae/power soluion for VPG and AOM, we provide some numerical resuls ha demonsrae he performance advanages of AOM over VPG. For MuliPTP neworks, we sar from heorems proved in [19], and we analyically derive a simple procedure for maximizing he sum of raes for VPG sysems. Then, we show how his soluion, which is opimal in VPG sysems, can be used heurisically in AOM MuliPTP neworks. Using hese resuls, we derive a lower bound on he achievable gain of AOM over VPG schemes. As shown in Secion IV, his gain is subsanial. Noe ha our goal in his paper is no o promoe OM as a modulaion scheme, bu raher o advocae adaping he order of OM for CDMA sysems ha already use OM (e.g., he uplink of IS-95). The res of he paper is organized as follows. In he nex secion, we ake a sysem-level approach o he analysis of AOM in CDMA mulimedia neworks and show is performance advanages over VPG. In secion III, we presen he objecive funcions, formulae he opimizaion problems, and presen heir soluions. The performance of AOM is presened and conrased wih VPG in Secion IV. Finally, our main conclusions and several open issues are drawn in Secion V. II. ORTHOGONAL MODULATION IN CDMA NETWORKS A. Moivaion for Higher Orhogonal Modulaion Orders The main goal of his secion is o show ha for any daa rae, increasing he OM order improves he performance of a CDMA sysem. The maximum OM order ha can be used, however, is consrained by he chip rae. We firs sar wih a sysem-level analysis of CDMA sysems. The benefis of a higher OM order is hen esablished using his analysis and hrough an analogy beween OM and FEC. The message we will ry o convey is ha, in CDMA sysems, i is always advanageous o use an FEC or an OM order ha reduces he bi-energy-o-noise specral densiy raio (E b /N 0 ) required for a given BER. Digial Processor R (bps) FEC R c (bps) Modulaor R (bps) m Spreader W (Hz) Transmier Fig. 2. Simplified block diagram of he ransmier circui. The ransmier circui of he sysem under sudy is shown in Figure 2. I consiss of (digial) FEC encoder, modulaor, direc-sequence spreader, and (analog) amplifier and ransmier [12]. Consider packe recepion for link i. Le I be he se of acive links in he nework, P (i) be he ransmission power of link i, and h ji be he channel gain beween he 4 This may be one reason why previous sudies ha pursued an algorihmic approach o his problem considered oher objecives, such as minimizing he power or even minimizing he sum of raes [25].

5 5 receiver of link i and he ransmier of link j. Then he signal-o-noise (and inerference) raio a i is: SNR (i) = j I {i} h ii P (i) h ji P (j) (1) + P hermal where P hermal is he hermal noise, which is modeled as a whie Gaussian noise process. The inerference from oher users is also assumed o be Gaussian. This assumpion has been shown o produce hroughpu resuls ha are reasonably accurae [34]. For reliable communicaion, a more relevan meric han SNR (i) is he effecive bi energy-o-noise specral densiy raio a he deecor, denoed by µ (i) and given by [12]: (i) def µ = E b = W N 0 R (i) j I {i} h ii P (i) h ji P (j) (2) + P hermal where W is he Fourier bandwidh occupied by he signal (i.e., chip rae) and R (i) is he daa rae of i s inended signal. Le µ req be he required µ (i) for a cerain BER. Then, he maximum achievable daa rae a i is: R (i) = W SNR(i) µ req. (3) Boh (2) and (3), which hold for any CDMA sysem, do no explicily indicae he effecs of FEC and modulaion on he achievable daa rae. However, hese effec appear indirecly hrough he value of µ req. For example, he sronger he FEC code (i.e., he lower he code rae), he lesser is µ req and he higher is he achievable daa rae. This analysis is inline wih he findings of Vierbi [40], in which he showed ha he jamming margin is acually increased by coding; he idea is ha wih coding, µ req is lower, and so more inerference is allowed for he same rae (i.e., SNR (i) in (3) can be decreased). In oher words, for CDMA sysems i is always preferable o use schemes ha enable operaion a a lower µ req. In he case of M-ary OM, he modulaor akes k = log 2 M FEC-coded bis and maps hem ino one of he M Walsh (or Hadamard) orhogonal sequences [32] of lengh M bis. So he resuling modulaed bi rae R m is equal o R c M/k, where R c is he coded bi rae (see Figure 2). A he receiver, he signal is firs despread and hen noncoherenly deeced, generaing k sof oupu bis for each ransmied Walsh symbol, which are fed o he Vierbi decoder (see [39] for furher deails). A igh upper bound on he probabiliy of bi error in OM is given by [32]: P b < 1 2 e k(µ(i) 2ln2)/2. (4) I is clear from (4) ha he higher he value of k, he lower is he BER. Therefore, he higher he OM order M, he beer is he BER performance for he same E b /N 0 value. OM in his sense works as an FEC code; he higher he value of M, he lower is he modulaion rae k/m, bu he beer is he BER performance. Noe ha he higher he OM order, he higher is R m ; however, his has no impac on he sysem bandwidh as long as R m W, since he signal is spread

6 6 by a high-rae CDMA code. B. Performance Advanages of Adapive Orhogonal Modulaion In he previous secion, we showed ha increasing he OM order is beneficial for he performance of a CDMA nework. However, he higher he user daa rae R, he lower mus be he maximum allowable M o ensure ha R m W. Thus, in AOM, M mus be adaped according o R. Our goal in his secion is o quanify he performance gains of adaping M according o R. To do his, we derive he relaionship beween he user s SNR and he achievable daa rae for AOM and for non-adapive OM (i.e., VPG). Firs, we claim ha i is sufficienly accurae o use (2) and he upper bound in (4) o analyze OM in CDMA sysems. To subsaniae our claim, we compare he performance obained from hese wo simple equaions wih he resuls repored in [26], which were obained using rigorous analysis. We simulae he same seup of [26]: a MuliPTP nework ha uses 64-ary OM wih equal received powers a he common receiver. The number of ransmiers is varied o obain differen E b /N 0. Par (a) of Figure 3 shows he probabiliy of bi error versus E b /N 0. The exac plo is he same one ha was obained in [26], while he upper-bound curve is he one obained using (2) and (4). This figure demonsraes ha he bound is sufficienly igh for all pracical purposes. To verify he ighness of he bound for oher values of M, we show in Par (b) of Figure 3 he probabiliy of bi error versus M for E b /N 0 = 8 db and E b /N 0 = 10 db. As can be seen, he bound is igh, and hence will be used in our subsequen analysis Uppeound Exac 10 2 Uppeound Exac Probabiliy of Bi Error Probabiliy of Bi Error Eb/No = 8 db Eb/No = 10 db Eb/No (db) (a) Orhogonal Modulaion Order (b) Fig. 3. Probabiliy of bi error in an OM-based CDMA sysem. Nex, we use (2) and (4) o derive he relaionship beween he user s SNR and he achievable rae wih and wihou adaping M. From his relaionship, we demonsrae he performance advanages of AOM over VPG for he single-link case. Wihou loss of generaliy, we assume ha he sysem under sudy does no use any FEC (i.e., R c = R). VPG uses he same modulaion order M for all daa raes. This M is chosen such ha for a given R, R m Z W, where Z is a hreshold ha is ofen deermined by regulaory laws. For example, he Federal Commission Commission (FCC) calls for a leas a raio of 10 (i.e., 10 db) of spreading rae o modulaion bi rae in he 2.4 GHz ISM band [5], so in his case

7 7 Z = W/10. Accordingly, he modulaion order for VPG is decided based on W, Z, and he maximum desired daa rae (R max ). If Z = W and R max = W/2, hen he (fixed) modulaion order M = 2. If he required BER is 10 6, hen for his VPG sysem, µ req is abou 14.8, and so using (3), he required SNR a R max is 7.4. Noe ha whereas µ req is fixed, he required SNR is a funcion of R. AOM, on he oher hand, uses a variable M ha depends on R. The higher he value of M, he smaller is he value of µ req, bu also he higher is R m. For Z = W and R max = W/2, he value of M a R max canno exceed 2 (o ensure ha R m Z), implying ha here is no performance advanage of AOM over VPG a R max. However, for R < R max, AOM uses a higher value for M, enabling operaion a a lower µ req, or equivalenly, resuling in a higher daa rae (see (3)). For each daa rae R, he corresponding value of M is he larges value such ha R m, which in he absence of FEC is equal o R M/k, does no exceed Z. Assuming M is coninuous (more on his assumpion shorly), R can be expressed as: R = Z k 2 k. (5) For a given arge P b, we use (4) as an equaliy, replace µ (i) wih µ req, and derive µ req as a funcion of k. This funcion along wih (5) is used o approximae µ req as a funcion of R, say g(r). The approximaion can be done by simple curve fiing. Finally, using µ req = g(r) and (3), one can express he required SNR as a funcion of R: SNR req = R W def g(r) = f(r) W. (6) In he case of AOM, f(r) can be well-approximaed (less han 1% fiing error) by he posynomial funcion 5 ari b, for some real-valued coefficiens a > 0 and b > 1. On he oher hand, in he case of VPG, g(r) is a consan ha is equal o µ req (e.g., g(r) = 14.8 for M = 2), and herefore, SNR req is simply a linear funcion of R. This lineariy beween R and SNR req has been he underlying assumpion in all previous adapive rae/power conrol schemes for OM-based CDMA neworks. We now know ha his assumpion does no hold for AOM Required SNR (db) VPG AOM Rae Enhancemen of AOM Relaive Energy Consumpion of AOM Rae (normalized) (a) Required SNR versus daa rae. Maximum Rae SNR (db) Maximum Rae (b) Rae enhancemen of AOM over VPG SNR (db) (c) Energy consumpion of AOM relaive o VPG. Fig. 4. Performance of AOM and VPG for a single link. 5 The definiion of a posynomial can be found in Appendix A.

8 8 Using he relaionships beween R and SNR req, we are now in a posiion o compare he performance of AOM wih VPG for he single-link case. Figure 4 demonsraes several performance merics obained using Z = W and R max = W/2. Par (a) of he figure depics SNR req versus he normalized rae R/W. I is clear ha for all R < R max, AOM requires a significanly less SNR han VPG o achieve a cerain daa rae. Such an improvemen essenially reflecs a power gain. Equivalenly, AOM achieves a much higher rae han VPG for he same SNR req (i.e., rae gain). Par (b) of he figure shows he relaive rae enhancemen of AOM over VPG versus he SNR. I is shown ha he rae advanage of AOM over VPG increases as he SNR decreases, and is very significan in he low SNR regime. Noe ha when SNR 8.7 db, he link operaes a R max, and AOM uses he same modulaion order as VPG, i.e., here is no rae improvemen. Par (c) of he figure shows he energy-per-daa bi (E b ) consumpion of AOM relaive o ha of VPG versus he SNR. E b is defined as he ransmission power divided by R. The figure shows ha AOM consumes much less E b han VPG in he low SNR regime. The E b consumpion of AOM increases as he SNR increases unil he maximum rae is reached, a which AOM consumes he same E b as VPG. In he above discussion, we permied he modulaion order M o ake any real posiive value; however, in real life, M is resriced o a finie se 6. Noneheless, we evaluae he poenial gains wihou his addiional consrain o serve as an upper bound on he performance of AOM in pracice. III. JOINT RATE/POWER OPTIMIZATION FOR AOM SYSTEMS The analysis presened in he previous secion focused on he single-user case. For a nework of users, increasing one user s power increases ha user s SNR, and consequenly is rae. However, his comes a he expense of he SNR for oher users, whose daa raes mus now be reduced o comba he added inerference. Deermining he bes powers and raes ha opimize a given objecive funcion (e.g., nework hroughpu) is no sraighforward. The goal of his secion is o define objecive funcions and derive policies ha opimize hem for he case of a nework of users (i.e., muliuser case). We sudy wo hroughpu-oriened objecive funcions: (1) minimizing he maximum service ime, and (2) maximizing he sum of users ransmissions raes. The wo funcions differ in wo aspecs: he ime scale a which rae adapion is carried ou and he required hardware. A. Minimizing he Maximum Service Time Le L i be he load (in bis) o be ransmied over link i, i I, where I is he se of acive links in he nework. Recall ha R i is he daa rae (in bis/sec) for link i. The service ime for link i, denoed by S i, is L i /R i. A scheme ha minimizes he maximum service ime S max = max {S i, i I} has he advanage of being easy o inegrae in many curren wireless nework sandards. For example, he access poin (AP) of an IEEE WLAN (or he Picone conroller of an IEEE 6 The burden of demodulaion for high values of M can be alleviaed by using he Fas Walsh Transform mehod [2], which requires only Mlog 2M real addiions and subracions.

9 WPAN) can uilize is polling medium access mechanism o measure he channel gains beween he AP and each mobile node, and o probe nodes abou heir loads. Using channel gains and load values, he AP can compue he opimum powers and raes ha minimize S max. A scheme ha minimizes S max does no require users o receive any feedback from he AP while ransmiing, i.e., only one ransceiver is required a a node. Furhermore, rae adapaion is carried ou on a per-packe basis (i.e., he whole packe is ransmied a one rae), which is pracical for curren wireless neworks sandards [29]. Given he channel gains and he loads L i i I, he goal is o find he ransmission powers and raes (i.e., P (i) R i, i I) so as o minimize S max. Formally, his problem is saed as follows: and minimize {R i,p (i), i I} subjec o: { max i I } L i R i h ii P (i) P f(r i) h ji P (j) +P W hermal j I {i}, i I 0 P (i) P max, i I R min R i R max, i I (7) The firs consrain reflecs he BER requiremen of link i, since i mandaes ha i s SNR be greaer han or equal o f(r i ) W = SNR req (see (6)). f(r i ) W is equal o R i W µ req for VPG and is approximaed by a(r i /Z) b (Z/W) for AOM, where a and b are wo consans whose values are obained from he fiing of f(r). In our simulaions, a 9.8 and b 1.2, wih less han 1% fiing error. Alhough he formulaion in (7) assumes he same minimum rae, maximum rae, and maximum power consrains for all nodes, his can be easily exended o handle he case of node-specific consrains. Noe ha his formulaion is applicable o boh PTP and MuliPTP neworks. Proposiion 1: The opimizaion problem in (7) is a generalized geomeric program (GGP). This GGP can be ransformed ino a geomeric program (GP), which iself can be ransformed ino a nonlinear convex program 7. Proof: Wih simple algebraic manipulaions, (7) can be expressed as: minimize {R i,p (i), i I} { } max {L iri 1 } i I subjec o: [ h ji P (j) j I {i} P (i) Pmax 1 1 R i R 1 max 1 R 1 i R min 1 ] [ ] + P hermal h ii P (i) 1 f(ri ) W 1 (8) 7 See Appendix A for a brief descripion of GGP and GP.

10 10 where he consrains in (8) are o be saisfied for all i I. If f(r) is a posynomial (see Appendix A), which is he case for boh VPG and AOM, (8) is a GGP. In is curren form, his GPP canno be solve opimally and efficienly. Therefore, we make wo ransformaions. The firs one ransforms he above GGP ino a GP. To his end, we inroduce a new auxiliary variable such ha: L i R i, i I. (9) Wih he inroducion of, (8) becomes: minimize {,R i,p (i), i I} subjec o: L i R 1 i 1 1 [ h ji P (j) j I {i} P (i) Pmax 1 1 R i R 1 max 1 R 1 i R min 1 ] [ ] + P hermal h ii P (i) 1 f(ri ) W 1 (10) I is obvious ha (8) and (10) are equivalen forms, meaning ha he powers and raes ha minimize also minimize he objecive funcion in (8). Formulaion (10) is an example of a GP, which can be easily ransformed ino a nonlinear convex program using a logarihmic change of variables [8]. Formally, le z = def def log, x i = log P (i) def, and y i = log R i i I (so ha = e z, P (i) = e x i, and R i = e y i ). Insead of minimizing he objecive funcion, we now minimize log. Also, each consrain of he form f 1 is changed o log f < 0. This resuls in he following (equivalen) opimizaion problem: minimize z {z,x i,y i, i I} subjec o: log L i e y i e z 0 [ ] log h ji e x j + P hermal h 1 ii e x i f(ey i) W 0 j I {i} log e x i P 1 max 0 log e y i R 1 max 0 log e y i R min 0 (11) A firs, he above formulaion may look more complicaed han (10). However, unlike (10), (11) is a convex opimizaion problem ha can be solved efficienly (see [8] for more deails). Once (11) is solved for x i and y i, i I, he opimal power and rae allocaion is simply given by P (i) = e x i and R i = e y i i I. Proposiion 1 applies o boh PTP and MuliPTP neworks, and also for VPG as well as AOM schemes. In he case of

11 11 MuliPTP neworks, he srucure of he problem can be furher simplified o allow for even a faser compuaion of he opimal soluion. The following proposiion enables he subsequen derivaion of his soluion. Proposiion 2: The powers and raes ha opimize (7) are such ha he firs consrain is saisfied wih equaliy. Proof: See Appendix B. In MuliPTP neworks, he receiver is common o all ransmiers, and so he channel gains h ji and h ii can be simply wrien as h j and h i, respecively. Hence, uilizing Proposiion 2, he opimal power and rae allocaion in he case of MuliPTP neworks mus saisfy he following se of linear equaions: j I {i} Using he same derivaion mehodology as in [35], (12) can be reduced o: h i P (i) h j P (j) = f(r i), i I. (12) + P hermal W 1 P ( ) = 1 ( hermal ), i I. (13) W j I f(r + 1 j) P (i) W h i f(r + 1 i) By imposing he consrain P (i) < P max and noing ha (13) is valid i I, he following inequaliy can be obained: ( j I 1 ) 1 W f(r + 1 j) min i I P hermal ( )]. (14) W [P max h i f(r + 1 i) This equaion deermines he feasibiliy of a se of raes, BER requiremens, and maximum power consrains. Nex, we use (14) o derive he opimal soluion for (7). Consider he following proposiion: Proposiion 3: The powers and raes ha opimize (7) are such ha L i R i = L j R j i,j I. Proof: See Appendix C. This proposiion says ha, a he opimal soluion o (7), all users have he same service ime (S). Hence, R i = L i /S i I. Accordingly, (14) can be wrien as: ( j I 1 ) 1 W f(l + 1 j/s) min i I P hermal ( )]. (15) W [P max h i f(l + 1 i/s) The only unknown in his equaion is S, and so i can be easily solved for he minimum S. Noe ha a unique soluion always exis, since he lef-hand side (LHS) of (15) is 0 a S =, and i increases as S decreases, while he RHS is 1 a S =, and i decreases as S decreases. In Secion IV, we use (15) o show he significan performance improvemen of AOM over VPG. B. Maximizing he Sum of Users Raes The goal of his objecive funcion is o maximize nework hroughpu, subjec o consrains on he BER, he maximum ransmission power, and he minimum and maximum ransmission raes. This funcion, which has been he focus of much

12 12 previous research, requires fas rae adapaion; for he nework o operae a he opimal poin, whenever a user complees he ransmission of a packe, all oher ransmiers mus updae heir raes in he mids of ransmiing heir packes. This means ha users mus use inra-packe rae adapaion (i.e., differen porions of he same packe mus be ransmied a differen raes). Furhermore, maximizing he sum of raes requires users o be able o receive feedback abou heir new raes while ransmiing, which may necessiae he use of a muliple-channel muliple-ransceiver archiecure. Noe ha he minimum-rae consrain, which has been overlooked in mos previous sudies, is crucial for mulimedia neworks; wihou his consrain, some users may never be allowed o ransmi, paricularly if hey experience a bad channel relaive o oher users (i.e., heir channel gains are relaively small). The power/rae opimizaion problem for boh AOM and VPG can be formulaed as follows: maximize {R i,p (i), i I} R i i I subjec o: h ii P (i) P f(r i) h ji P (j) +P hermal j I {i} W, i I 0 P (i) P max, i I R min R i R max, i I. (16) Unforunaely, his objecive funcion canno be ransformed ino he minimizaion of a posynomial as was done in he previous secion. So i is no possible o formulae his problem as a GGP, a GP, or a nonlinear convex program. In fac, he problem exhibis an unknown number of local maxima, and here are no efficien algorihms o solve i opimally for he general case (i.e., PTP neworks). However, in order o ge a feeling of how much improvemen AOM can provide over VPG, we fix one dimension of he problem, namely, he ransmission powers, and limi our aenion o rae opimizaion. Specifically, for PTP neworks, we examine he case when nodes use he maximum power (P max ). Firs, consider he following resul. Proposiion 4: The powers and raes ha opimize (16) are such ha he firs consrain is saisfied wih equaliy. Proof: The proof is similar o he one for Proposiion 2, and is omied for breviy. If all users operae a P max, hen from Proposiion 4, i is easy o compue he users raes for boh AOM and VPG by solving he following se of equaions: R i = f 1 W h ii P max, i I. (17) h ji P max + P hermal j I {i} For MuliPTP neworks, we follow a differen approach ha allows us o obain a lower bound on he achievable gain of AOM over VPG schemes. Wihou loss of generaliy, le he users in he se I be ordered according o heir link-channel

13 13 gains, i.e., i < j h i h j. I has been shown in [19] ha in he case of VPG 8, he opimal soluion for (16) has he following srucure: The se of bes v 1 users (I v1 ) operae a rae R max (i.e., a he maximum-rae boundary) and heir powers saisfy h i P (i) = h j P (j) i,j I v1, i.e., hey have equal received powers. The se of nex v 2 bes users (I v2 ) operae a power P max (i.e., a he maximum-power boundary) and raes R i < R max i I v2. Noe ha h i P (i) < h j P (j) i I v2 and j I v1 (see [19] for more deails). A mos, here is one user U (whose order in I is v 1 + v 2 + 1) ha operaes a rae R U and power P (U) R min < R U < R max and P (U) < P max. Furhermore, h U P (U) < h i P (i) and R U < R i i {I v1 Iv2 }. such ha The remaining users, I v3 = I I v1 I v2 {U}, operae a rae R min (i.e., a he minimum-rae boundary) and power h i P (i) = h j P (j) and j {I v1 Iv2 U}. i,j I v3, i.e., hey have equal received powers. Furhermore, h i P (i) < h j P (j) i I v3 Using his soluion srucure, we now presen a proposiion ha will enable us o derive a novel algorihm for finding he opimal soluion for VPG neworks. We hen show how his algorihm can be used as a heurisic for AOM neworks. Proposiion 5: For VPG MuliPTP neworks, he opimal soluion o (16) is such ha here is only one elemen in he se {I v2 U}, i.e., some users operae a he maximum-rae boundary, ohers operae a he minimum-rae boundary, and only one user operaes a a rae in beween hese wo boundaries. Proof: See Appendix D. This opimal soluion is inuiive and agrees wih previously repored informaion heoreic resuls [38]; if here is no consrain on he maximum rae, he sysem hroughpu is maximized while simulaneously saisfying each user s minimum-rae consrain only when he bes-channel user is allowed o ransmi a a power larger han he one required for i o achieve R min. If here is a consrain on he maximum rae, allowing only he bes user o increase his power may no achieve he maximum nework hroughpu. The reason is ha he bes user canno uilize any exra power beyond he one required o achieve R max. Hence, he opimal policy will hen be ha some bes-channel users operae a R max (wihou using P max ), some bad-channel users operae a R min, and a mos one user operaes a a rae ha is beween R max and R min. Based on Proposiion 5, he opimal soluion for VPG neworks can be found by assigning rae R max o he maximum possible number of users such ha he feasibiliy condiion (14) is no violaed, and hen assigning o he nex bes user he maximum power a which (14) is saisfied wih equaliy. The deails of he algorihm are as follows: 1) Assign rae R min i I and check he feasibiliy condiion in (14); if his condiion is no saisfied, hen here is no soluion o his problem; oherwise go o he nex sep. 2) Assign rae R max o he bes user in I, say user j, and check he feasibiliy condiion in (14); if saisfied, hen se I = I {j} and repea his sep; oherwise, go o he nex sep. 8 The auhors in [19] did no consider a minimum-rae consrain; however, heir resuls exend o he case when R min > 0.

14 14 3) Find he maximum power (P allowed ) ha j can use such ha (14) is saisfied wih equaliy; he ransmission power of j is hen given by P j = min{p allowed,p max }. This rae/power assignmen (RPA) algorihm gives an opimal soluion for VPG. We now explain he inuiion behind using he same algorihm as a heurisic for AOM. The main idea is o replace he objecive funcion in (16) by a slighly differen bu relaed objecive funcion, and hen measure he acual hroughpu under his new funcion. Firs, noe ha in he case of AOM, f(r i ), which was shown in Par (a) of Figure 4, can be well-approximaed by a second-degree polynomial in R i, say a 1 R 2 i +b 1R i +c 1, which can be wrien as ( a 1 (R i + b 2 ) 2 + c 2 ) for some coefficiens a1, b 1, c 1, b 2, and c 2. Of course such an approximaion has a higher fiing error han he posynomial fiing chosen earlier ( i.e., ar b i). def Le O i = (R i + b 2 ) 2 def, O min = (R min + b 2 ) 2 def, and O max = (R max + b 2 ) 2, and replace he objecive funcion in (16) by i I O i. Then, he opimizaion problem becomes: maximize {O i,p (i), i I} O i i I subjec o: h ii P (i) P a 1O i +c 2 h ji P (j) +P W hermal j I {i}, i I 0 P (i) P max, i I O min O i O max, i I (18) This formulaion has a similar srucure o he one of VPG. Since RPA finds he opimal soluion o VPG, i can also find he opimal soluion o (18) (a 1 and c 2 are consans ha do no affec he opimizaion algorihm). This means ha RPA can be used o maximize i I (R i +b 2 ) 2. The powers and raes ha maximize i I (R i +b 2 ) 2 are no necessarily equal o he ones ha maximize i I R i. However, we expec hem o be close. In his sense, RPA can be used as a heurisic mehod o maximize i I R i. The simulaion resuls in Secion IV show ha based on his heurisic, AOM provides significan performance advanages over VPG. A. Simulaion Seup IV. PERFORMANCE EVALUATION In his secion, we evaluae he performance of AOM and conras i wih ha of VPG [19]. Our resuls are based on numerical experimens conduced using MATLAB. Our performance merics include he service ime (S) 9, he sum of users raes, and he average energy consumpion per bi (E b ), defined as P N i=1 P i P N i=1 R i. Noe ha E b is a more significan measure han he average ransmission power. In fac, i is misleading o compare he average ransmission power of wo sysems ha ransmi a differen daa raes, as he cos of ransmiing a cerain number of bis depends on boh he ransmission power and he rae. In some cases, we also sudy he hroughpu and energy fairness indexes I R = (P N i=1 R i) 2 N P N i=1 R2 i 9 A he opimal soluion for he firs objecive funcion, all users have he same service ime S (see Proposiion 3).

15 15 and I E = (P N i=1 (P i/r i )) 2 N P N i=1 (P i/r i ) 2, respecively [20]. The fairer he sysem, he higher are he values of I R and I E. I R measures he equaliy of users allocaion of hroughpu. If all users ge he same amoun of hroughpu, hen he fairness index is 1, and so he sysem is 100% fair. As he discrepancy in hroughpu increases, I R decreases. A scheme ha favors only few users has a fairness index close o zero. I E measures he discrepancy in he amoun of energy each user invess in delivering one bi of informaion. Typically, a sysem designer would like his per-bi energy o be equal for all users o exend he lifeime of users baeries. To simulae he channel gains, we assume he wo-ray propagaion model wih a pah loss facor of 4. Noe, however, ha he problem formulaion does no depend on how he channel aenuaion marix is generaed, i.e., any oher fading model can be used. The oal bandwidh of he sysem (i.e., he chip rae) is W = 1 MHz. We le P max = 20 dbm. B. Poin-o-Poin Neworks In his scenario, N ransmiing nodes are randomly placed across a square area of lengh 600 meers. For each ransmier j, he receiving node i is placed randomly wihin a circle of radius 100 meers ha is cenered a j. Given he locaion of he N receivers and N ransmiers, he channel aenuaion he channel aenuaion beween any pairs of nodes i and j is compued using he he wo-ray propagaion model wih aenuaion facor equals o 4. The marix H is hen formed wih enries h ij. Whenever he soluion se is empy for he generaed H (i.e., R min canno be achieved for all users), a new se of ransmiers and receivers are randomly generaed. The maximum-rae consrain R max is chosen such ha he modulaion order M used in VPG is equal o 16, which is he minimum M used in AOM. For his experimen, we le R min = R max /100. Number of Scheme Minimize Max S i Maximize R i Node Pairs S E b Ri E b I R I E (N) (sec) (microjoules/bi) (Mbps) (microjoules/bi) VPG AOM VPG AOM TABLE I PERFORMANCE COMPARISON BETWEEN AOM AND VPG IN PTP NETWORKS. The performance of AOM and VPG is shown in Table I. The resuls are repored for N = 20 and N = 30 based on he average of 100 independen realizaions of he marix H. For he firs objecive funcion, (i.e., minimizing he maximum service ime), all nodes are assumed o have 1 Mbis of daa. Alhough a randomly generaed workload is more pracical, he choice of equal workloads is mean o faciliae he discussion. For N = 20 and N = 30, AOM achieves a reducion in S by 65.4% and 50%, respecively, while simulaneously achieving abou 75% and 42% energy savings, respecively.

16 16 The reason for his considerable improvemen can be explained as follows. From Proposiion 3, we know ha a he opimal powers and raes, all users have he same S. Since users have he same load (1 Mbis), he opimal soluion is when all users ransmi a he same rae. This rae mus be chosen o accommodae he wors-channel user (i.e., lowes SNR). AOM, as Figure 4 shows, has a significan performance advanage over VPG a low SNR values; hus providing a smaller service ime and a much lower energy consumpion han VPG. For he second objecive funcion (i.e., maximizing he sum of users raes), he opimal soluion in PTP neworks is unknown; however, o provide a feeling of wha he AOM improvemen is, we le all users ransmi a P max, and compue he corresponding opimal raes using (17). For boh N = 20 and N = 30, AOM achieves abou 15% increase in hroughpu, 13% saving in energy, and abou 16% improvemen in I R relaive o VPG. The improvemen in I E for N = 20 is pariculary significan (abou 42%). Such an improvemen is jusified by noing ha AOM achieves a significan hroughpu gain for low-rae (low-snr) links, someimes wice ha of VPG, bu provides lile gain for high-rae links. This has a negligible impac on hroughpu, bu has a significan impac on I E. C. Mulipoin-o-Poin Neworks In his secion, we consider N ransmiing nodes ha are randomly placed wihin a square area of lengh 200 meers. The common receiver is placed a he cener of he square. Given he locaion of he nodes, he channel aenuaion marix H. Similar o he PTP case, whenever he soluion se is empy for he generaed H, a new se of ransmiers is randomly generaed. The resuls are obained based on he average of 100 independen realizaions of he marix H. For he firs objecive funcion, he workload a each ransmier is seleced randomly beween 1 and 20 Mbis. As before, R max is chosen such ha he modulaion order M used in VPG is equal o 16. Figure 5 depics he performance of AOM and VPG for he firs objecive funcion. Par (a) of he figure depics he service ime S versus N. I is shown ha as N exceeds 10, AOM achieves considerably lower S han VPG. For example, when N = 50, S under AOM is only 45% of S under VPG. I is also shown ha S under boh AOM and VPG increases wih N. This is expeced since as N increases, he muliple access inerference (MAI) also increases, and users are forced o ransmi a lower raes, which increases heir service imes. Noe ha for VPG, he S-versus-N curve is approximaely linear, while for AOM, he slope of ha curve decreases slighly wih N. This can be explained by examining (15). The RHS of (15) is close o 1, as P hermal is ypically very small. In he case of VPG, f(l i /S) = µ req L i /S, WS/L i µ req 1, and so he LHS of (15) can be well-approximaed by µreq S i I L i µreq S NL ave, where L ave is he expeced value of L i. This explains why S increases almos linearly wih N. For he AOM case, f(li/s) = a(l i /S) b for some coefficiens a > 0 and b > 1. I is easy o show ha he S-versus-N curve can be approximaed by S cn 1/b, for some coefficiens c > 0. Thus, is derivaive (or slope) decreases wih N. Par (b) of Figure 5 depics E b versus N. I shows ha in addiion o reducing he service ime, AOM achieves a significan energy saving over VPG. For example, for N = 50, AOM energy expendiure is less han 40% ha of VPG.

17 VPG AOM 6 x 10 6 VPG AOM Service Time (seconds) Energy per Informaion Bi (joules) Number of Transmiing Nodes (a) Number of Transmiing Nodes (b) Fig. 5. Performance of AOM and VPG based on he minimizaion of he maximum service ime in MuliPTP neworks. Nex, we sudy he impac of increasing P hermal on he service ime S. Figure 6 shows S as a funcion of P hermal for N = 30. The workload is generaed as in Figure 5. For all values of P hermal, AOM consisenly shows a good improvemen over VPG. For boh AOM and VPG, however, S sars o increase exponenially when P hermal exceeds 60 dbm. The reason is ha a his value, P hermal becomes comparable o he maximum received powers for bad-channel users. Hence, he SNR of he users deerioraes significanly, causing a fas drop in heir raes, and a corresponding dramaic increase in S VPG AOM Service Time (seconds) Thermal Noise (dbm) Fig. 6. Service ime S of AOM and VPG in MuliPTP neworks as a funcion of P hermal. In he case of he second objecive funcion (i.e., maximizing he sum of users raes), we se he maximum modulaed bi rae Z o W/5. As before, R max is chosen such ha he modulaion order M used in VPG is equal o 16 (i.e., R m /R = 4), so R max = W/20. Par (a) of Figure 7 depics he hroughpu performance versus N for hree differen values of R min (R max /50, R max /100, and zero). Recall ha he used RPA algorihm is opimal for VPG, bu is only a heurisic for AOM, so he resuls in Figure 7 represen a lower bound on he achievable gain of AOM over VPG. Several

18 18 observaions can be made based on his figure. Toal Throughpu (Mbps) AOM VPG Rmin=Rmax/50 Rmin=Rmax/100 Rmin=0 Energy Consumpion (joules/bi) 9 x VPG AOM Number of Transmiing Nodes (a) Number of Transmiing Nodes (b) Fig. 7. Performance versus N under he hroughpu maximizaion crierion for MuliPTP neworks. Firs, AOM achieves considerably more hroughpu han VPG; e.g., for R min = R max /50 and N = 50, AOM achieves abou 30% more hroughpu han VPG. This is because for any power allocaion vecor, AOM enables higher raes han VPG. Second, in he cases of R min = R max /50 and R min = R max /100, as N increases, he hroughpu for AOM increases, while he hroughpu for VPG decreases. This can be explained as follows. For VPG, as N increases, more bad-channel users are required o operae a R min. To enable his, oher (good-channel) users mus decrease heir powers (and consequenly heir raes) o reduce he MAI. The increase in he oal hroughpu due o a higher number of badchannel users does no offse he decrease in he hroughpu of he good-channel users. Therefore, he overall effec is a sligh reducion in nework hroughpu. This is no he case, however, for AOM. Simulaion resuls indicae ha he increase in he oal hroughpu due o more bad-channel users is higher han he decrease in he hroughpu of he goodchannel users. This can be jusified as follows. Unlike VPG, AOM uses higher OM orders a low daa raes and hus requires much less SNR han VPG o achieve R min. Thus, good-channel users do no need o reduce heir powers (and heir raes) considerably o accommodae he new users, and so he reducion in he hroughpu of he good-channel users is no considerable (when compared o he VPG case). The overall effec is a sligh increase in nework hroughpu. The hroughpu of AOM increases wih N unil he RPA is unable o find a feasible soluion. Anoher observaion is ha as R min increases, he hroughpu for VPG decreases, while he hroughpu for AOM increases. So he hroughpu gain of AOM over VPG goes up wih R min. This can be explained as follows. Increasing R min ighens he consrains (i.e., reduces he soluion space), and his resuls in a lower hroughpu whenever RPA is opimal. This is exacly wha happens in he VPG case since RPA is opimal for VPG. Bu since RPA is heurisic for AOM, we conjecure ha is performance becomes closer o he opimal one as R min increases, and so he hroughpu increases. The las poin o noe abou Figure 7-(a) is ha for R min = 0, boh AOM and VPG are almos linear. The reason is

19 19 ha when R min = 0, RPA allocaes powers only o good-channel users unil he nework sauraes, i.e., v 1 users are assigned R max and only one user is assigned he res of he power such ha (14) is saisfied. Adding more users has no impac once (14) is saisfied. As in he PTP case, he hroughpu advanage of AOM over VPG comes wih energy savings. Par (b) of Figure 7 depics he energy consumpion of AOM and VPG as a funcion of N for R min = R max /50. This figure shows ha AOM achieves a significan energy saving over VPG (up o 25%). Nex, we sudy he fairness properies of AOM and VPG. Par (a) and (b) of Figure 8 depic I R and I E, respecively, as a funcion of N (recall ha he fairer he sysem, he higher are he values of I R and I E ). The resuls are for R min = R max /50. I can be observed ha relaive o VPG, AOM can improve I R and I E up o 21% and 30%, respecively. AOM VPG AOM VPG 1 1 Throughpu Fairness Index Energy Fairness Index Number of Transmiing Nodes (a) Number of Transmiing Nodes (b) Fig. 8. Fairness versus N under he hroughpu maximizaion crierion for MuliPTP neworks. Finally, we sudy he effec of varying he minimum processing gain (PG) by varying R max. We fix R min in his experimen a W/500. Figure 9 shows he performance of AOM and VPG as a funcion of he minimum PG. I can be observed ha he sum of raes decreases as he PG increases for boh AOM and VPG. This agrees wih he previous inuiion ha reducing R max ighens he soluion space, and so decreases he achieved maximum. Furhermore, i no difficul o noice ha RPA favors higher values of R max. V. CONCLUSIONS AND OPEN ISSUES In his paper, we invesigaed he poenial performance gains of using adapive orhogonal modulaion (AOM) in mulirae CDMA neworks. We showed ha, relaive o a variable processing gain (VPG) sysem ha uses fixed orhogonal modulaion (OM) order, AOM can significanly increase he nework hroughpu while simulaneously reducing energy consumpion. We sudied he problem of opimal join rae/power conrol for AOM-based sysems under wo objecive funcions: minimizing he maximum service ime and maximizing he sum of users raes. For he firs objecive funcion, we showed ha he opimizaion problem can be formulaed as a GGP, which can be ransformed ino a nonlinear convex

20 AOM VPG 0.25 Toal Throughpu (Mbps) Minimum Processing Gain Fig. 9. Throughpu in MuliPTP neworks as a funcion of he minimum processing gain (varied hrough Rmax). program, and be solved opimally and efficienly. In he case of he second objecive funcion, we obained a lower bound on he achievable gain of AOM over fixed-modulaion schemes. Unlike previous work on adapive ransmission, which have focused mainly on cellular neworks, ours is applicable o boh PTP and MuliPTP neworks. In PTP neworks, our resuls show ha, when compared wih fixed OM order VPG schemes, AOM can achieve more han 50% improvemen in he service ime and, simulaneously, more han 40% reducion in energy consumpion. In MuliPTP neworks, we derived a simple algorihm for finding he opimal powers and raes for VPG, and explained he inuiion behind using ha algorihm as a heurisic for AOM. Our resuls show ha he achievable hroughou gain can be up o 30% compared o VPG. Furhermore, AOM achieves more han 45% reducion in he service ime relaive o VPG. In our analysis, we le he modulaion order M o ake real posiive value. However, in realiy, M is resriced o a finie se. Our fuure work will focus on sudying he impac of resricing M o a finie se of values. AOM is sill a newly explored area of research. Several challenges remain o be addressed, including finding he opimal soluion for maximizing he sum of raes for AOM in MuliPTP neworks, he opimal algorihm for maximizing he sum of raes for VPG and AOM in PTP neworks, and closed-form approximaions o he opimal soluions. In addiion o solving for hese heoreical limis, our fuure work will focus on how o inegrae hese algorihms wihin curren wireless neworks proocols. APPENDIX A. Geomeric Programming Le x 1,...,x n be n variables in R +, and le x = def (x 1,...,x n ). A funcion f is called a posynomial in x if i can be wrien in he form f(x 1,...,x n ) = K k=1 c kx a 1k 1 x a 2k 2...x a nk n, where c k 0 and a ik R. If K = 1, hen f is called a monomial funcion. A GP is an opimizaion problem of he form [8]: