Research Article Robustness Maximization of Parallel Multichannel Systems

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1 Journal of Electrcal and Computer Engneerng Volume 2012, Artcle ID , 16 pages do: /2012/ Research Artcle Robustness Maxmzaton of Parallel Multchannel Systems Jean-Yves Baudas, 1 Fahad Syed Muhammad, 2 and Jean-FrançosHélard 2 1 Natonal Center for Scentfc Research (CNRS), The Insttute of Electroncs and Telecommuncatons of Rennes (IETR), UMR 6164, Rennes, France 2 UnverstéEuropéenne de Bretagne, INSA, IETR, UMR 6164, Rennes, France Correspondence should be addressed to Jean-Yves Baudas, jean-yves.baudas@nsa-rennes.fr Receved 27 February 2012; Accepted 10 May 2012 Academc Edtor: Shuo Guo Copyrght 2012 Jean-Yves Baudas et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Bt error rate (BER) mnmzaton and SNR-gap maxmzaton, two robustness optmzaton problems, are solved, under average power and btrate constrants, accordng to the waterfllng polcy. Under peak power constrant the solutons dffer and ths paper gves bt-loadng solutons of both robustness optmzaton problems over ndependent parallel channels. The study s based on analytcal approach, usng generalzed Lagrangan relaxaton tool, and on greedy-type algorthm approach. Tght BER expressons are used for square and rectangular quadrature ampltude modulatons. Integer bt soluton of analytcal contnuous btrates s performed wth a new generalzed secant method. The asymptotc convergence of both robustness optmzatons s proved for both analytcal and algorthmc approaches. We also prove that, n the conventonal margn maxmzaton problem, the equvalence between SNR-gap maxmzaton and power mnmzaton does not hold wth peak-power lmtaton. Based on a defned dssmlarty measure, bt-loadng solutons are compared over Raylegh fadng channel for multcarrer systems. Smulaton results confrm the asymptotc convergence of both resource allocaton polces. In nonasymptotc regme the resource allocaton polces can be nterchanged dependng on the robustness measure and on the operatng pont of the communcaton system. The low computatonal effort leads to a good trade-off between performance and complexty. 1. Introducton In transmtter desgn, a problem often encountered s resource allocaton among multple ndependent parallel channels. The resource can be the power, the bts or the data, and the number of channels. The resource allocaton polces are performed under constrants and assumptons, and the ndependent parallel channels can be encountered n multtone transmsson. Independent parallel channels result from orthogonal desgn appled n tme, frequency, or spatal domans [1]. They can ether be obtaned naturally or n a stuaton where the transmt and receve strateges are to orthogonalze multple waveforms. The orthogonal desgn can also be appled n many communcaton scenaros when there are multple transmt and receve dmensons. Orthogonal frequency-dvson multplexng (OFDM) and dgtal multtone (DMT) are two successful commercal applcatons for wreless and wrelne communcatons wth orthogonalty n the frequency doman. To perform resource allocaton, relatons between varous resources are needed, and one s the channel capacty. Ths capacty of n-ndependent parallel Gaussan channels s the well-known sum of the capacty of each channel C = C = log 2 (1 + snr ). (1) Ths relaton, whch holds for memoryless channels, lnks the supremum btrate C, here expressed n bt per two dmensons, to the sgnal to nose rato, snr, experenced by each channel or subchannel. Any relable and mplementable system must transmt at a btrate r below capacty C over each subchannel, and then the margn, or SNR-gap, γ s ntroduced to analyze such systems [2, 3] γ = 2C 1 2 r 1. (2)

2 2 Journal of Electrcal and Computer Engneerng Ths SNR-gap s a convenent mechansm for analyzng systems that transmt below capacty or wthout Gaussan nput, and ( r = log 2 1+ snr ), (3) γ wth r the btrate n bts per two-dmensonal symbol (bts per second per subchannel) whch s also the number of bts per constellaton symbol. Resource allocaton s performed usng loadng algorthms, and dverse crtera can be nvoked to decde whch porton of the avalable resource s allocated to each of the subchannels. From an nformaton theory pont of vew, the crteron s the mutual nformaton, and the optmal resource allocaton under average power constrant was frst devsed n [4] for Gaussan nputs and later for non- Gaussan nputs [5]. Snce the performance measure s the capacty, the SNR-gap n (3) sγ = 1forall. In other cases, γ s hgher than 1, and (3) has been exploted nto many optmal and suboptmal resource allocaton polces. In fact, resource allocaton s a constrant optmzaton problem, and generally two cases are of practcal nterest: rate maxmzaton (RM) and margn maxmzaton (MM), where the objectve s the maxmzaton of the data or btrate, and the maxmzaton of the system margn (or power mnmzaton n practce), respectvely [6]. The MM problem gathers all non RM problems ncludng power mnmzaton, margn maxmzaton (n ts strct sense), and other measures such as error probabltes or goodput. (In ths paper MM abbrevaton s related to the general famly of non-rm problems and not only to the margn maxmzaton problem n ts strct sense. The expanded form s reserved for the margn maxmzaton n ts strct sense.) It s not necessary to study all the resource allocaton strateges, and equvalence or dualty can be found. Famles of approaches are defned, and unfed processes have been used [7 10]. The loadng algorthms are also splt n to two famles. The frst s based on greedy-type approach to teratvely dstrbute the dscrete resources [11], and the second uses Lagrangan relaxaton to solve contnuous resource adaptaton [12]. Both approaches have been compared n terms of performance and complexty [7, 12 14]. All these adaptve resource allocatons are possble when channel state nformaton (CSI) s known at both transmtter and recever sdes. Ths CSI can be perfect or mperfect, and full or partal. The effects of channel estmaton error and feedback delay on the performance of adaptve modulated systems can also be consdered n the resource allocaton process [15 17]. In ths paper we shall focus henceforth on MM problems, and the man contrbutons are as follows: () the new resource allocaton algorthm and () the comparson of the dfferent resource allocaton strateges. It s assumed that the channel estmaton s perfect, and feedback CSI delay and overhead are neglgble. The consdered peak-power constrant, nstead of the conventonal average power constrant or sum power constrant, results from power mask lmtaton and has been taken nto accountnresourceallocatonproblem[13, 18 20]. Wth ths peak power constrant, each channel must satsfy a power constrant. Note that the sum power constrant s hstorcally the frst consdered constrant [4]. Btrate constrant comes from communcaton applcatons or servce requrements, where dfferent flows can exst, but one of them s chosen at the begnnng of the communcaton. In ths confguraton, the remanng parameter to optmze s then the SNR-gap γ whch s also related to the error probablty of the communcaton system. Two smlar problems of MM have the same objectve, that s, to maxmze the system robustness. What we call robustness n ths paper s the capablty of a system to mantan acceptable performance wth unforeseen dsturbances. The frst measure of robustness s the SNR gap, or system margn, and ts maxmzaton ensures protecton aganst unforeseen channel mparments or nose. The system margn maxmzaton s the maxmzaton of the mnmal SNR-gap γ n (3) over the n subchannels. In that case the conventonal equvalence between margn maxmzaton and power mnmzaton n MM problems s not generally true. In ths paper we show that ths equvalence can nevertheless be obtaned n partcular confguratons. The second robustness measure s the bt error rate (BER) and ts mnmzaton can reduce the packet error rate and the data retransmssons. In transmtter desgn, the BER mnmzaton can be realzed usng unform bt-loadng and adaptve precodng [21, 22]. Analytcal studes have been performed wth peak-ber or average BER (computed as arthmetc mean) approaches [15, 17]. Wth nonunform bt loadng, the average BER must be computed as weghted arthmetc mean, and the resource allocaton has been performed usng a greedy-type algorthm [23]. The frst man contrbuton of ths paper s the analytcal soluton of the resource allocaton problem n the case of weghted arthmetc mean BER mnmzaton. To perform the analytcal study, based on a generalzed Lagrangan relaxaton tool, we develop a new method for fndng roots of functons. Ths method generalzes the secant method to better ft the functon-dependng weght and to speed up the search of the roots. Both robustness polces are compared usng a new measure. Ths measure evaluates the dfference of the bt dstrbutons nstead of the btrates. We also prove that both robustness polces provde the same bt dstrbuton n asymptotc regme, whch s defned for hgh SNR and hgh btrate regmes, and ths s the second man contrbuton n ths paper. The proof s gven n the case of unconstraned modulatons (.e., contnuous btrates and analytcal soluton) and also for QAM constellatons and greedy-type algorthms. The convergence s exemplfed by smulaton n multcarrer communcatons systems. The organzaton of the paper s as follows. In Secton 2, the quanttes to be used throughout the paper are ntroduced, and the robustness optmzaton problem s formulated n a general way for both system margn maxmzaton and BER mnmzaton. The equvalences between margn maxmzaton and power mnmzaton are worked out.

3 Journal of Electrcal and Computer Engneerng 3 Secton3 presents the consdered expressons of accurate BER, the new measure of the bt dstrbuton dfferences, and the new search method of roots of functons. The solutons of formulated problems are gven n Secton 4 n the form of an optmum resource allocaton polcy based on greedy-type algorthms. The condtons of equvalence of both margn maxmzaton and BER mnmzaton are gven n ths secton. Secton5 presents the analytcal soluton and both greedy-type and analytcal methods are compared n Secton 6. Ths Secton 6 exemplfes the applcaton of robustness optmzaton to multcarrer communcaton systems. Fnally, the paper concludes n Secton 7 wth the proofs of several results relegated to the appendces. Notaton. Thebtrates{r } n aredefnedasanumber of bts per two dmensons and they are smply gven by a number of bts (undertone per constellaton). 2. Problem Formulaton Consder n parallel subchannels. On the -th subchannel, the nput-output relatonshp s Y = h S + W, (4) where S s the transmtted symbol, Y s the receved one, and h the complex scalar channel gan. The complex Gaussan nose W s a proper complex random varable wth zeromean and varance equal to σw 2. The conventonal average power constrant s 1 E [ S 2] P, (5) n whereas the peak-power constrant, or power spectrum densty constrant, consdered n ths paper s E [ S 2] P, = 1,..., n. (6) It s convenent to use normalzed unt-power symbol {X } n such that S = p PX, (7) whch leads to the peak-power constrant p 1, = 1,..., n. (8) It s also convenent to ntroduce two other varables. The frst one s the conventonal SNR snr = h 2 P p σw 2 (9) and the second s called power spectrum densty nose rato (PSDNR) psdnr = 1 h 2 P n σw 2, (10) whch s the mean sgnal to nose rato over the n subchannels f and only f p = 1forall. Ths PSDNR s the rato between the power mask at the recever sde (the transmtted power mask through the channel) and the power spectrum densty of the nose. The system performance wll be gven accordng to ths parameter to pont out the ablty of a system to explot the avalable power under peak-power constrant. Usng the prevous notatons, (3)becomes ( r = log 2 1+ h 2 ) p P γ σw 2. (11) Wth p /γ = 1forall, r s the subchannel capacty under power constrant P. Wth unconstraned modulatons, r s defned n R, but constraned modulatons are used n practce and r takes a fnte number of nonnegatve values. Nonnteger number of bts per symbol can also be used wth fractonal bt constellatons [24, 25]. In ths paper, modulatons defned by dscrete ponts are used wth nteger number of bts per symbol. Typcally, r {0, β,2β,..., r max }, where β s the granularty n bts and r max s the number of bts n the rchest avalable constellaton. The peak-power and btrate constrants are then p 1, r = R, r { } 0, β,2β,..., r max. (12) Obvously, the explotaton of the avalable power leads to p = 1foralland the constrant s smplfed as r = R, r { } 0, β,2β,..., r max. (13) Wth peak-power and btrate constrants, the resource allocaton strategy s then to use all avalable power and to optmze the robustness. The problem we pose s to determne the optmal btrate allocaton {r } n that maxmzes a robustness measure, or nversely mnmzes a fralness measure, under constrants gven n (13). In ts general form, ths problem can be wrtten as [ ] r 1,..., rn = arg n mn φ ( {r } n ), r =R (14) r {0,β,2β,...,r max} where φ( ) s the fralness measure. In ths paper, ths measure s gven by the SNR gap or the BER. In addton to the btrate allocaton, the recever s presumed to have knowledge of the magntude and phase of the channel gan {h } n, whereas the transmtter needs only to know the magntude { h } n. The objectve s to fnd the data vector [r1,..., rn ] whch s the fnal relevant nformaton for the transmtter. The resource allocaton can then be computed on the recever sde to reduce the feedback data rate from n real numbers to n fnte nteger numbers. Furthermore, the nteger nature of the data rates allows a full CSI at the transmtter, whch s not possble wth real numbers System Margn Maxmzaton. The SNR-gap γ of the subchannel s (3) γ = snr 2 r 1. (15)

4 4 Journal of Electrcal and Computer Engneerng Wth relable communcatons, γ s hgher than 1 for all subchannels. Let the system margn, or system SNR-gap, be the mnmal value of the SNR gap n each subchannel γ = mn γ. (16) Let γ nt be the ntal system margn of one communcaton system ensurng a gven QoS. Let γ be the optmzed system margn of ths system. Then, the system margn mprovement ensures system protecton n unforeseen channel mparment or nose, for example, mpulse nose; btrate and system performance targets are always reached for an unforeseen SNR reducton of γ/γ nt over all subchannels. Ths robustness optmzaton does not depend on constellaton and channel-codng types. The system margn γ s defned and optmzed wthout knowledge of used constellatons and codng, and the proposed robustness optmzaton works for any codng and modulaton scheme. The objectve s the maxmzaton of the system margn whch s equvalent to the mnmzaton of γ 1.Wenoteγ (r ) the functon that assocates r to γ.thefunctonφ( )n(14) s then gven by φ ( {r } n ) = max [ ] r 1,..., rn = arg n mn r =R r {0,β,2β,,r max} 1 γ (r ), (17) max γ 1. (18) Ths problem s the nverse problem of btrate maxmzaton under peak-power and SNR-gap constrants. The soluton of the btrate maxmzaton problem s obvous under the sad constrants and gven by ( 1 r = β β log 2 1+ snr ). (19) γ Followng the conventonal SNR-gap approxmaton [2], the symbol error rate (SER) of QAM dependng on the SNRgap s constellaton sze ndependent wth ser (r ) = 2erfc 3 2 γ r, (20) where the complementary error functon s usually defned as erfc(x) = 2 e t2 dt. (21) π The system margn maxmzaton s then equvalent to the peak-ser mnmzaton n hgh-snr regme. Note that, wth (16), the system margn maxmzaton can also be called a trough-snr-gap maxmzaton, and t s strongly related to the peak-power mnmzaton. Whereas the bt-loadng soluton s the same for power mnmzaton and margn maxmzaton wth sum-margn or sum-power constrants, nstead of peak constrants, the followng lemma gves suffcent condtons for equvalence n the case of peak constrants. x Lemma 1. The bt allocaton that maxmzes the system margn under peak-power constrant {p margn } n mnmzes the peak-power under SNR-gap constrant {γ power } n f p margn γ power = α for all. Proof. It s straghtforward usng (11) and (18). Both problems have the same expresson and therefore the same soluton. Ths lemma provdes a suffcent but not necessary condton for the equvalence of solutons, and t says that f the power and the SNR-gap constrants have proportonal dstrbutons for margn maxmzaton and peak-power mnmzaton problems, respectvely, then both problems have the same optmal btrate allocaton. In the general case, we cannot conclude that both problems have the same soluton BER Mnmzaton. In communcaton systems, the error rate of the transmtted bts s a conventonal robustness measure. By defnton, the BER s the rato between the number of wrong bts and the number of transmtted bts. Wth a multdmensonal system, there exsts several BER expressons [15, 23]. Let the BER be evaluated over the transmsson of m multdmensonal symbols. (We suppose that m s hgh enough to respect the ergodc condton and to make possble use of error probablty.) In our case, the multdmensonal symbols are the symbols sent over n subchannels. Let e be the number of erroneous bts receved over subchannel durng the transmsson. The BER s then gven as n e n ber = m r (e /mr ) n = r n. (22) r The BER over subchannel s e /mr, and the BER of n subchannels s then ber ( {r } n ) n r ber (r ) = (23) R wth ber (r ) the functon that assocates the BER of channel wth the btrate r. The BER of multple varable btrate r s then not the arthmetc mean of BER but s the weghted mean BER. Weghted mean BER and arthmetc mean BER are equal f r = r j for all, j or f ber = 0forall. As there exsts ber 0, then weghted mean BER and arthmetc mean BER are equal f and only f r = r j for all, j. Note that f the number m of transmtted multdmensonal symbols depends on the subchannel, (23) does not hold anymore. These obvous results on mean measures are not taken nto account, and mean BER s erroneously used nstead of mean weghted BER [15, 17]. The functon φ( )n(14) s then gven by φ ( {r } n ) 1 = r ber (r ), R (24) [ ] r 1,..., rn = arg n mn r =R ber ( {r } n ). r {0,β,2β,...,r max} (25)

5 Journal of Electrcal and Computer Engneerng 5 To smplfy the notatons, let ber(r) be the BER of the system. In hgh SNR regme wth Gray mappng, r ber (r ) = ser (r ), and then weghted mean BER can be approxmated by arthmetc mean SER dvded by the number of transmtted bts. Contrary to system margn maxmzaton, the BER mnmzaton needs the knowledge of constellaton and codng schemes, and t s based on accurate expressons of BER functons. In ths paper, the used constellatons are QAM, and the optmzaton s performed wthout a channel codng scheme. When dealng wth practcal coded systems, the ultmate measure s the coded BER and not the uncoded BER. However, the coded BER s strongly related to the uncoded BER. It s then generally suffcent to focus on the uncoded BER when optmzng the uncoded part of a communcaton system [26]. 3. Interludes Before solvng the optmzaton problem, the BER approxmaton of QAM s presented. Ths approxmaton plays a chef role n BER mnmzaton, and a good approxmaton s therefore needed. Snce ths paper deals wth btrate allocaton, a measure of dfference n the btrate dstrbuton s proposed and presented n ths secton. Ths secton also presents a new research method of roots of functons. Ths method generalzes the secant method and converges faster than the secant one BER Approxmaton. Conventonally, the BER approxmaton of square QAM has been performed by ether calculatng the symbol error probablty or by smply estmatng t usng lower and upper bounds [27]. Ths conventonal approxmaton tends to devate from the exact values when the SNR s low and cannot be appled for rectangular QAM. Exact and general closed-form expressons are developed n [28] for arbtrary one and two-dmensonal ampltude modulaton schemes. An approxmate BER expresson for QAM can be obtaned by neglectng the hgher-order terms n the exact closed-form expresson [28]. ber 1 ( ) ( ) 3 erfc r I J I 2 + J 2 2 snr (26) wth I = 2 r/2, J = 2 r/2,andr = log 2 (I J ). By symmetry, I and J can be nverted. The BER can also be expressed usng the SNR-gap γ. Usng (3)and(26), the BER s wrtten as ber 1 ( ) ( ) 3(I J erfc 1) r I J I 2 + J 2 2 γ. (27) These two approxmatons allow the extenson of the ber (r )functonfromn to R + whch s useful for analytcal studes. Fgure 1 gves the theoretcal BER curves and the approxmated ones from the bnary phase shft keyng (BPSK) to the QAM. For BER lower than , the relatve error s lower than 1% for all modulatons. BER BPSK SNR (db) Approxmaton Exact 2 15 QAM Fgure 1: Exact BER curves and approxmatons (26) Dssmlar Resource Allocaton Measure. Two resource allocatons can have the same btrate, but ths does not mean that the btrates per subchannel are the same. To measure the dfference n the bt dstrbuton between dfferent resource allocaton strateges, we need to evaluate the dssmlarty. Ths dssmlarty measure must verfy the followng propertes: (1) f two resource allocatons lead to the same bt dstrbuton, then the measure of dssmlarty must be null, whereas (2) f two resource allocatons lead to two completely dfferent bt dstrbutons n loaded subchannels, then the measure of dssmlarty must be equal to one, and (3) the measure s symmetrc; that s, the dssmlarty between the resource allocatons X and Y must be the same as the dssmlarty between the resource allocatons Y and X. We choose that the empty subchannels do not mpact the measure. Defnton 2. The dssmlarty measure between the resource allocatons X and Y s n δ(r (X) r (Y)) μ(x, Y) = max n j {X,Y} δ ( ( )), (28) r j where δ(x) = 1fx 0 else δ(x) = 0. Ths dssmlarty has the followng propertes. Property 1. μ(x, Y) = 0ff r (X) = r (Y) forall. Property 2. μ(x, Y) = 1ff r (X) r (Y) orr (X) = r (Y) = 0forall. Property 3. μ(x, Y) = μ(y, X). Property 4. If μ(x, Y) = 0, then, for all resource allocaton Z, μ(x, Z) = μ(y, Z). All these propertes are drect consequences of Defnton 2. For a null dssmlarty, μ(x, Y) = 0, all

6 6 Journal of Electrcal and Computer Engneerng the subchannels transmt the same number of bts, that s, r (X) = r (Y) forall. For a full dssmlarty, μ(x, Y) = 1, all the nonempty subchannels of both resource allocatons X and Y transmt a dfferent number of bts, that s, for all such as r (X) 0 and r (Y) 0, then r (X) r (Y). It s obvous that the measure s symmetrc μ(x, Y) = μ(y, X). If two resource allocatons have a null dssmlarty μ(x, Y) = 0, then they are dentcal and for any resource allocaton Z μ(x, Z) = μ(y, Z). The converse of ths last property s not true. Note that the dssmlarty s not defned for two empty resource allocatons. For example, let n = 4and[r 1 (X),..., r 4 (X)] = [ ]. If [r 1 (Y),..., r 4 (Y)] = [3222] or [r 1 (Y),..., r 4 (Y)] = [ ], then μ(x, Y) = 1. If [r 1 (Y),..., r 4 (Y)] = [ ], then μ(x, Y) = 1/2. The measure μ(x, Y) snullfand only f [r 1 (Y),..., r 4 (Y)] = [ ]. The dssmlarty does not evaluate the total btrate dfferences but only the bt dstrbuton dfferences; the contrbuton of two btrates r (X) andr j (Y) n the dssmlarty measure s ndependent of the btrate dfference r (X) r j (Y) Generalzed Secant Method. There are many numercal methods for fndng roots of functons. We propose a new method, called the generalzed secant method, that s, based on the secant method. Ths new method better fts the functon-dependng weght than secant method do and then mproves the speed of the convergence. Before explanng ths new method, a bref overvew of the secant method s gven. In our case, the objectve functon f (x) s monotonous, nondfferentable and computable over x [x 1, x 2 ]wth f (x 1 )/ f (x 1 ) = f (x 2 )/ f (x 2 ). The secant method s as follows for an ncreasng functon f (x): (1) = 0, y 0 = f (x 1 ); (2) x 0 = (x 2 f (x 1 ) x 1 f (x 2 ))/( f (x 1 ) f (x 2 )), y +1 = f (x 0 ); (3) { f y +1 y ɛ, then x 0 s the root of f (x), else y+1<0 } then x 1=x 0 y +1>0 then x 2=x 0, +1andgotostep2. The objectve of the secant method s to approxmate f (x) by a lnear functon g (x) = a x + b at each teraton, wth g (x 1 ) = f (x 1 )andg (x 2 ) = f (x 2 ), and to set x 0 as the root of g (x).thesearchfortherootof f (x) s completed when the desred precson ɛ s reached. The precson s gven for y, but t can also be gven for x. As the functon f (x) s computable, t can be plotted and an a posteror smple algebrac or elementary transcendental nvertble functon over [x 1, x 2 ] can be used to better ft the functon f (x). The a posteror nformaton s then used to mprove the search for the root. The functon f (x) s teratvely approxmated by a h(x) +b nstead of a x + b, where h(x) s the nvertble functon. Ths method s then gven as follows for an ncreasng functon f (x): (1) = 0, y 0 = f (x 1 ); (2) x 0 = h 1 ((x 2 f (x 1 ) x 1 f (x 2 ))/( f (x 1 ) f (x 2 ))), y +1 = f (x 0 ); (3) { f y +1 y ɛ, then x 0 s the root of f (x), else y+1<0 } then x 1=x 0 y +1>0 then x 2=x 0, +1andgotostep2. Compared to the secant method, only step 2 dffers and the computaton of x 0 s performed takng nto account the approxmated shape h(x) of the functon f (x). Ths generalzed secant method s used n Secton 5 to fnd the root of the Lagrangan and s compared to the conventonal secant method. In our case, f (x) s the sum of logarthmc functons, and the functon h(x) s then the logarthmc one. 4. Optmal Greedy-Type Resource Allocatons Thegeneralproblemstofndtheoptmalresourceallocaton [r 1,..., r n ] that mnmzes φ( ), the nverse robustness measure, or fralness. Ths s a combnatoral optmzaton problem or nteger programmng problem. The core dea n ths teratve resource allocaton s that a sequental approach can lead to a globally optmum dscrete loadng. Greedy-type methods then converge to the optmal soluton. Convexty s not requred for the convergence of the algorthm and monotoncty s suffcent [29]. Ths monotoncty ensures that the removal or addton of β bts at each teraton converges to the optmal soluton. In ths paper the used functons φ( ) are monotonc ncreasng functons. In ts general form and when the objectve functon φ( ) s not only a weghted sum functon, the teratve algorthm s as follows: (1) start wth allocaton [r (0) 1,..., r n (0) ] = 0, (2) k = 0, (3) allocate one more bt to the subchannel j for whch φ ({ r (k+1) } n ) (29) s mnmal, wth r (k+1) j = r (k) j + β and r (k+1) = r (k) for all j, (4) f r (k+1) = R, termnate; otherwse k k +1and go to step 3. The obtaned resource allocaton s then optmal [29]and solves (14). Ths algorthm needs R/β teratons. The target btrate R s supposed to be feasble; that s, R s a multple of β. Note that an equvalent formulaton can be gven startng wth r (0) = r max for all and usng bt removal nstead of bt addton wth maxmzaton nstead of mnmzaton. For btrates hgher than (n/2)r max, the number of teratons wth bt removal s lower than wth bt addton. The opposte s true wth btrate lower than (n/2)r max. Iteratve resource allocatons have been frstly appled to btrate maxmzaton under power constrant [11]. Many works have been devoted to complexty reducton of greedytype algorthms; see, for example, [6, 12, 30, 31] and references theren. In ths secton, only greedy-type algorthms are presented n order to compare the analytcal resource allocaton to the optmal teratve one. Note that the analytcal soluton can also be used as an nput of the greedytype algorthm to ntalze the algorthm and to reduce the number of teratons.

7 Journal of Electrcal and Computer Engneerng System Margn Maxmzaton. The system margn, or system SNR gap, maxmzaton under btrate and peakpower constrants s the nverse problem of the btrate maxmzaton under SNR-gap and peak-power constrants. Ths nverse problem has been solved, for example, n [18]. To comply wth the general problem formulaton, the nverse system margn mnmzaton s presented nstead of the system margn maxmzaton. Lemma 3. Under btrate and peak-power constrants, the greedy-type resource allocaton that mnmzes the nverse system margn γ 1 (16) allocates sequentally β bts to the subchannel bearng r bts and for whch s mnmum. 2 r+β 1 snr (30) Proof. It s straghtforward usng (17) and (29). See Appendx A for an orgnal proof. The man advantage of system margn maxmzaton s that the optmal resource allocaton can be reached ndependently of the SNR regme. Resource allocaton s always possble even for very low SNR, but t can lead to unrelable communcaton wth SNR gap lower than 1. Lemma 3 s gven wth unbounded modulaton orders, that s, r max = and r βn for all. Wth full constrants (13), the subchannels that reach r max are smply removed from the teratve process BER Mnmzaton. The system BER mnmzaton under btrate and peak-power constrants s the nverse problem of btrate maxmzaton under peak-power and BER constrants. Ths nverse problem has been solved, for example, n [23]. Usng (29) and(24), the soluton of BER mnmzaton s straghtforward, and the correspondng greedy-type algorthm s also known as Levn-Campello algorthm [5, 32, 33]. The man drawback of ths soluton s that t requres good approxmated BER expressons even n low-snr regme. Ths constrant can be relaxed, and the followng lemma gves the optmal greedy-type resource allocaton for the BER mnmzaton. Lemma 4. In hgh SNR regme and under btrate and peakpower constrants, the greedy-type resource allocaton that mnmzes the BER mnmzes (r + β)ber (r + β) at each step. Proof. SeeAppendx B. Lemma 4 states how to allocate bts wthout mean BER computaton at each step. It s gven wthout modulaton order lmtaton. Lke system margn maxmzaton soluton, the bounded modulaton order s smply taken nto account usng r max and subchannel removal Comparson of Resource Allocatons. To compare the two optmzaton polces, we call B the resource allocaton that maxmzes the system margn and C the resource allocaton Table 1: Example of system margn and BER wth n = 20, R = 100, psdnr = 25 db, and β = 1. System margn maxmzaton (B) BER mnmzaton (C) mn γ 6.9 db 6.6 db ber that mnmzes the BER. Table 1 gves an example of btrate allocaton over 20 subchannels where the SNR follows a Raylegh dstrbuton and wth β = 1. In ths example, the PSDNRdefnedn(10) s equal to 25 db, and the maxmum allowed btrate per subchannel s never reached. As expected, the system margn mnmzaton leads to a mnmal SNR gap, mn γ, hgher than that provded by the BER mnmzaton polcy wth a gan of 0.3 db. On the other hand, the BER mnmzaton polcy leads to BER lower than that provded by system margn mnmzaton ( versus ). In ths example, the dssmlarty s μ(b, C) = 0.1, and two subchannels convey dfferent btrates. All these results are obtaned wth r max = 10. Ths example shows that the dfference between the resource allocaton polces can be small. The queston s whether both resource allocatons converge and f they converge then n what cases. The followng theorem answers the queston. Theorem 5. In hgh-snr regme wth square QAM and under btrate and peak-power constrants, the greedy-type resource allocaton that maxmzes the system margn converges to the greedy-type resource allocaton that mnmzes the BER. Proof. SeeAppendx D. The consequence of Theorem 5 s that the dssmlarty between the resource allocaton that maxmzes the system margn and the resource allocaton that mnmzes the BER s null n hgh-snr regme and wth square QAM. Wth square QAM, β should be a multple of 2. Note that wth square modulatons, β can also be equal to 1 f the modulatons are, for example, those defned n ADSL [34]. Fgure 6 exemplfes the convergence wth β = 2aswewllseelater n Secton Optmal Analytcal Resource Allocatons The analytcal method s based on convex optmzaton theory [35]. Unconstraned modulatons lead to btrates r defned n R. Wthr R + the soluton s the waterfllng one. Wth bounded modulaton order, that s, 0 r r max, the soluton s qute dfferent from the waterfllng one. The soluton s obtaned n the framework of generalzed Lagrangan relaxaton usng Karush-Kuhn-Tucker (KKT) condtons [36].

8 8 Journal of Electrcal and Computer Engneerng As the btrates are contnuous and not only ntegers n ths analytcal analyss, the constrants (13) do not hold anymore and become r = R, 0 r r max. (31) The KKT condtons assocated to the general problem (14) wth (31) nstead of (13)wrte[36] r 0, = 1,..., n, (32) r r max 0, = 1,..., n, (33) R r = 0, (34) μ 0, = 1,..., n, (35) ν 0, = 1,..., n, (36) μ r = 0, = 1,..., n, (37) ν (r r max ) = 0, = 1,..., n, (38) ( {rj } ) n φ λ μ + ν = 0, = 1,..., n, (39) r j=1 where λ, μ and ν are the Lagrange multplers. The frst three equatons (32) (34) represent the prmal constrants, (35) and(36) represent the dual constrants, (37) and(38) represent the complementary slackness, and (39) s the cancellaton of the gradent of Lagrangan wth respect to r. When the prmal problem s convex and the constrants are lnear, the KKT condtons are suffcent for the soluton to be prmal and dual optmal. For the system margn maxmzaton problem, the functon φ( ) s convex over all nput btrates and SNR whereas ths functon s no longer convex for the BER mnmzaton problem. Appendx C gves the convex doman of the functon φ( ) n the case of BER mnmzaton problem. The propertes of the studed functon φ( ) are such that ( {rj } ) n φ = ψ (r ) (40) r j=1 s ndependent of r j for all j. The optmal soluton that solves (32) (39) s then [36] 0, f λ ψ (0), r (λ) = ψ 1 (λ), f ψ (0) <λ<ψ (r max ), r max, f λ ψ (r max ) for all = 1,..., n and wth λ verfyng the constrant (41) r (λ) = R. (42) It s worthwhle notng that the above general soluton s the waterfllng one f r max R. The waterfllng s also the soluton n the followng case. Let I be the subset ndex such that I = { r {0, r max } }, (43) and let R the target btrate over I. In ths subset, {r } I are solutons of ( {rj } ) n φ λ = 0, I r j=1 R (44) r (λ) = 0. I Ths s the soluton of (14) wth unbounded modulatons over the subchannel ndex subset I.IfI = {1,, n} and R = R, and(44) s also the soluton of (14) wth unconstraned modulatons System Margn Maxmzaton Theorem 6. Under btrate and peak-power constrants, the asymptotc bt allocaton whch mnmzes the nverse system margn s gven by r = R I + 1 I Proof. SeeAppendx E. j I log 2 snr snr j, I. (45) The soluton gven by Theorem 6 holds for hgh modulaton orders whch defnes the asymptotc regme, compare Appendx E. If the set I s known, then Theorem 6 can be used drectly to allocate the subchannel btrates. Otherwse, I should be found frst. The expresson of r n Theorem 6 s a functon of the target btrate R, the number I of subchannels, and the ratos of SNR. Ths expresson s ndependent of the mean receved SNR or PSDNR. It does not depend on the lnk budget but only on the relatve dstrbuton of subchannel coeffcents { h 2 } n BER Mnmzaton. The arthmetc mean BER mnmzaton has been analytcally solved, for example, n [22, 37]. Ths arthmetc mean measure needs to employ the same number of bts per constellaton whch lmts the system effcency. The followng theorem gves the soluton of the weghted mean BER mnmzaton that allows varable constellaton szes n the multchannel system. Theorem 7. Under btrate and peak-power constrants, the asymptotc bt allocaton whch mnmzes the BER s gven by r = R I + 1 I wth equal n-phase and quadrature btrates. Proof. SeeAppendx F. j I log 2 snr snr j I (46) The soluton gven by Theorem 7 holds for hgh modulaton orders and for subchannel BER lower than 0.1,

9 Journal of Electrcal and Computer Engneerng 9 and these parameters defne the asymptotc regme n ths case, compare Appendx F. The optmal asymptotc resource allocaton leads to square QAM wth r conveyed btrate n each n-phase and quadrature components of the sgnal of subchannel. It s mportant to note that, n asymptotc regme, BER mnmzaton and system margn maxmzaton lead to the same subchannel btrate allocaton. In that case, the asymptotc regme s defned by the more strngent context whch s the BER mnmzaton. As we wll see n Secton 6, ths asymptotc behavor can be observed when β = 2. The man drawback of the formulas n Theorems 7 and 6 s that the subset I must be known. To fnd ths subset, the negatve subchannel btrates and those hgher than r max should be clpped, and I can be found teratvely [18]. But clppng negatve btrates frst can decrease those hgher than r max, and clppng btrates hgher than r max frst can ncrease the negatve ones. It s then not possble to apply frst the waterfllng soluton and after that to clp the btrates r greater than r max to converge to the optmal soluton. Fndng the set I requres many comparsons, and we propose a fast teratve soluton based on the generalzed secant method Lagrangan Resoluton. To solve (41), numercal teratve methods are requred. It s mportant to observe that the functon defned n (41) s not dfferentable, and, thus, methods lke Newton s cannot be used [18]. We use the proposed generalzed secant method to better ft the functon-dependng weght and ncrease the speed of the convergence. An mportant pont for the teratve method s that the ntalzaton value must lead to feasble soluton and should be as close as possble to the fnal soluton. The root of the functon defned by (42) s now calculated. Let f (λ) = r (λ) R. (47) Theorems 6 and 7 show that r(λ) s the sum of log 2 ( ) functons. Ths s the reason why the functon log 2 ( ) s used n the generalzed secant method. Fgure 2 shows three functons versus the parameter λ. The frst functon s the nput functon f (λ), the second one s the functon used by the generalzed secant method, and the last one f the lnear functon used by the secant method. In ths example, the common ponts are λ = 0andλ = 2.3. As t s shown, the generalzed secant method better fts the nput functon than the secant method and therefore can mprove the speed of the convergence to fnd the root whch s around λ = 1/80 n ths example. To ensure the convergence of the secant methods, the algorthm should be ntalzed wth λ 1 and λ 2 such as f (λ 1 ) < 0and f (λ 2 ) > 0. For both optmzaton problems, system margn maxmzaton and BER mnmzaton, the parameter λ s gven by the functon ψ (r ), and t can be f (λ) λ Input functon New generalzed secant method Secant method Fgure 2: Approxmaton of the nput functon f (λ) wth the generalzed secant method and the secant method, n = 1024 and r max = 15. reduced to λ = 2 r /snr, as shown n Appendces E and F. Parameters {λ 1, λ 2 } are then chosen as 1 λ 1 =, λ 2 = 2rmax. (48) max snr mn snr Usng (41), λ λ 1 leads to r (λ) = 0forall,andλ λ 2 leads to r (λ) = r max for all. Then, t follows that f (λ 1 ) < 0and f (λ 2 ) > 0fR (0, nr max ). Fgure 3 shows the needed number of teratons for the convergence of the generalzed and conventonal secant methods versus the target btrate R. Results are gven over a Raylegh dstrbuton of the subchannel SNR wth 1024 subchannels. The possble btrates are then R [0, n r max ] and β = 2. Here, r max = 15 and then R bts per multdmensonal symbol. For comparson, the number of teratons needed by the greedy-type algorthm s also plotted. Note that the greedy-type algorthm can start by empty btrate or by full btrate lmted by r max for each subchannel. The number of teratons s then gven by mn{r, nr max R}. The teratve secant and generalzed secant methods are stopped when the btrate error s lower than 1. A better precson s not necessary snce exact btrates {r } I can be computed usng Theorems 6 and 7 when I s known. As t s shown n Fgure 3, the generalzed secant method converges faster than the secant method, except for the very low target btrates R. For very hgh target btrates, near from n r max, the number of teratons wth the generalzed secant method can be hgher than that wth the greedy-type algorthm. Except for these partcular cases, the generalzed secant method needs no more than 4-5 teratons to converge. In concluson, we can say that wth Raylegh dstrbuton of {snr } n and for target btrates R such that 3% R/nr max 97%, the generalzed secant method converges faster than the secant method or the greedy-type algorthm.

10 10 Journal of Electrcal and Computer Engneerng Number of teratons Number of teratons Target bt rate Greedy type New generalzed secant method Secant method Fgure 3: Number of teratons of the secant and generalzed secant methods, and greedy-type algorthm versus the target btrate, n = 1024, r max = Target bt rate Greedy type Bsecton method Secant method Fgure 4: Number of teratons of the bsecton and secant methods, and greedy-type algorthm for nteger-bt soluton versus target btrate, n = 1024, r max = 15. Usng the generalzed secant method, the btrates are not ntegers and for all, r [0, r max ]. These solutons have to be completed to obtan nteger btrates Integer-Bt Soluton. Startng from the contnuous btrate allocatons prevously presented, a loadng procedure s developed takng nto account the nteger nature of the btrates to be allotted. A smple soluton s to consder the nteger part of {r } I and to complete by a greedy-type algorthm to acheve the target btrate R. The nteger part of {r } I s then used as a startng pont for the greedy algorthm. Ths procedure can lead to a hgh number of teratons. Therefore, the secant or bsecton methods are sutable to reduce the number of teratons. The problem to solve s then to fnd the root of the followng functon [18]: g(α) = r + βα R, (49) I where r, I,andR are gven by the contnuous Lagrangan soluton. Ths s a suboptmal nteger btrate problem, and the optmal one needs to fnd {α } n nstead of a unque α. As the optmal soluton leads to a huge number of teratons, t s not consdered. The functon (49) s a nondecreasng and nondfferentable starcase functon such that g(0) < 0, g(1) > 0because I r = R. The teratve methods can then be ntalzed wth α 1 = 0andα 2 = 1. Two teratve methods are compared: the bsecton one and the secant one. Both methods are also compared to the greedy-type algorthm. Fgure 4 presents the number of teratons of the three methods to solve the nteger-bt problem of the Lagrangan soluton wth β = 1. Results are gven over a Raylegh dstrbuton of the subchannel SNR, wth 1024 subchannels and the target btrates are between 0andn r max = As t s shown, the convergence s faster wth bsecton method than wth greedy-type algorthm. For target btrates between 10% and 90% of the maxmal loadable btrate, the secant method outperforms the bsecton one wth a mean number of teratons around 4 whereas the number of teratons for bsecton method s hgher than 8. Fgure 4 also shows that g(0) s all the tme lower than the half of number of subchannels and around ths value for target btrate between 10% and 90% of the maxmal loadable btrate. Then, f the number of teratons nduces by the greedy-type algorthm to solve the nteger-bt problem of the Lagrangan soluton that s acceptable n a practcal communcaton system, ths greedytype completon can be used and appears to lead to the optmal resource allocaton. Ths result obtaned wthout proof means that the greedy-type procedure has enough bts to converge to the optmal soluton. If the number of teratons nduced by the greedy-type algorthm s too hgh (thsnumbersaroundn/2), the secant method can be used. The overall analytcal resoluton of (14) needs few teratons compared to the optmal greedy-type algorthm. Whereas the contnuous soluton of (14) s optmal, the analytcal nteger btrate soluton s suboptmal. 6. Greedy-Type versus Analytcal Resource Allocatons In the prevous secton, the numbers of teratons of the algorthms have been compared. In ths secton, robustness comparson s presented and the analytcal solutons obtaned n asymptotc regme are also appled n nonasymptotc regme whch means that β = 1 and modulaton orders can be low. The evaluated OFDM communcaton system s composed of 1024 subcarrers wthout nterferences between the symbols or the subcarrers. The channel s the Raylegh

11 Journal of Electrcal and Computer Engneerng 11 Bt-rate PSDNR (db) Fgure 5: Target btrate versus nput PSDNR. fadng one wth ndependent and dentcally dstrbuted elements. The rchest modulaton order s r max = 10. The robustness measures are evaluated for dfferent target btrates whch are gven wth the followng arbtrary equaton: ( R = mn (log 2 1+ snr ) ), r max. (50) 2 Ths equaton ensures relable communcatons for all the nput target btrates or PSDNR. The emprcal relatonshp between PSDNR and targetbtrate s alsogven n Fgure 5. Fgure 6 presents the output BER and the system margn of three resource allocaton polces versus the target btrate R. The frst one, A, s obtaned usng analytcal optmzaton, the second, B, s the soluton of the greedy-type algorthm whch maxmzes the system margn, and the thrd, C, s the soluton of the greedy-type algorthm whch mnmzes the BER. Two cases are presented: one wth β = 1 and the other wth β = 2.AllsubchannelBERarelowerthan to use vald BER approxmatons. Note that, wth β = 1, the system margn of allocaton B s almost equal to 8.9 db for all targetbt rates. Ths constant system margn γ s not a feature of the algorthm but s only a consequence of the relaton between the target btrate and the PSDNR. To enhance the equvalences and the dfferences between the resource allocaton polces, the dssmlarty s also gven n Fgure 6 wth β = 1andβ = 2. As expected n both cases, β = 1andβ = 2, the mnmal BER are obtaned wth allocaton C, and the maxmal system margns wth allocaton B. Wth β = 1 and when the target btrate ncreases, the Lagrangan soluton converges faster to the optmal system margn maxmzaton soluton, B, than to the optmal BER mnmzaton soluton, C. Note that Theorem 7s an asymptotc result vald for square QAM. Wth β = 1, the QAM can be rectangular, and the asymptotc result of Theorem 7 s not applcable, contrary to the result of Theorem 6 where there s not any condton on the modulaton order. The case β = 2 shows the equvalence between the optmal system margn maxmzaton allocaton and the BER System margn (db) Dssmlarty β = β = β = β = A B C A B C β = 2 β = μ(a, B) μ(a, C) μ(b, C) Target bt-rate Fgure 6: BER, system margn and dssmlarty versus target btrate for Lagrangan (A), greedy-type system margn maxmzaton (B) and greedy-type BER mnmzaton (C) algorthms, n = 1024, r max = 10, and β {1, 2}. optmal BER mnmzaton allocaton. In ths case, the asymptotc result gven by Theorems 5 and 7 can be appled because the modulatons are square QAM, and the convergence s ensured wth hgh modulaton orders, that s, hgh target btrates. Beyond a mean btrate per subchannel around 6, that corresponds to a target btrate around 6000, all the allocatons A, B and C are equvalent, and the dssmlarty s almost equal to zero. In nonasymptotc regme, the dfferences n BER and system margn are low. The system margn dfferences are lower than 1 db, and the ratos between two BER are around 3. In practcal ntegrated systems, these low dfferences wll not be sgnfcant and wll lead to smlar solutons for both optmzaton polces. Therefore, these resource allocatons can be nterchanged. 7. Concluson Two robustness optmzaton problems have been analyzed n ths paper. Weghted mean BER mnmzaton and mnmal subchannel margn maxmzaton have been solved under peak-power and btrate constrants. The asymptotc convergence of both robustness optmzatons has been proved for analytcal and algorthmc approaches. In nonasymptotc regme, the resource allocaton polces

12 12 Journal of Electrcal and Computer Engneerng can be nterchanged dependng on the robustness measure and the operatng pont of the communcaton system. We have also proved that the equvalence between SNRgap maxmzaton and power mnmzaton n conventonal MM problem does not hold wth peak-power lmtaton wthout addtonal condtons. Integer bt soluton of analytcal contnuous btrates has been obtaned wth a new generalzed secant method, and bt-loadng solutons have been compared wth a new defned dssmlarty measure. The low computatonal effort of the suboptmal resource allocaton strategy, based on the analytcal approach, leads to a good tradeoff between performance and complexty. Appendces A. Proof of Lemma 3 We prove that the optmal allocaton s reached startng from empty loadng wth the same ntermedate loadng than startng from optmal loadng to empty loadng. To smplfy the notaton and wthout loss of generalty, β = 1. Let [r1,..., rn ] be the optmal allocaton that mnmzes the nverse system margn γ(r ) 1 for the target btrate R, and then γ(r ) 1 2 = max r 1. (A.1) snr Let [r 1,..., r n ] be the optmal allocaton that mnmzes the nverse system margn γ(r+1) 1 for the target btrate R+1 R. The optmal allocaton for target btrate R s obtaned teratvely by removng one bt at a tme from the subchannel k wth the hghest nverse system margn [38] 2 k = arg max r 1 (A.2) snr or 2 rk 1 2r 1, = 1,..., n. (A.3) snr k snr The last bt removed s from the subchannel wth the lowest nverse-snr, snr 1, because the bts over the hghest nverse- SNR are frst removed. Now, let [r 1,..., r n ] be the optmal allocaton that mnmzes the nverse system margn γ(r) 1 for the target btrate R<R. Followng the algorthm strategy, the optmal allocaton for target btrate R + 1 s obtaned addng one bt on subchannel j such that 2 j = arg mn r+1 1. (A.4) snr We frst prove that γ(r +1) 1 2 rj+1 1 =. (A.5) snr j Suppose that there exsts j such that 2 r j 1 snr j > 2rj+1 1 snr j, (A.6) then one bt must be added to subchannel j to obtan r j +1 bts before addng one bt to subchannel j to obtan r j bts whch means that [r 1,..., r n ] s not optmal. As [r 1,..., r n ]s optmal by defnton, t yelds 2 r 1 snr 2rj+1 1 snr j = 1,..., n (A.7) whch proves (A.5). The frst allocated bt s from the subchannel wth the lowest nverse SNR gven by (A.4) wth r = 0forall. Comparng (A.3) wth(a.7) yelds that k = j, and the ndex subchannel of the frst added bt s the same as the last removed bt. All the ntermedate allocatons are then dentcal wth bt-addton and bt-removal methods. There exsts only one way to reach the optmal allocaton R startng from the empty loadng. Proof of Lemma 3 can also be provded n the framework of matrod algebrac theory [19, 39]. B. Proof of Lemma 4 To smplfy the notaton and wthout loss of generalty, the proof s gven wth β = 1. Let [r 1,, r n ] be the optmal allocaton for the target btrate R such that r = R.LetR+1 the new target btrate. We frst prove that Δ (r ) = (r +1)ber (r +1) r ber (r ) (B.1) s a good measure at each step of the greedy-type algorthm for the BER mnmzaton, and fnally that (r +1)ber (r +1) can be used nstead of Δ (r ). Startng from the optmal allocaton of target btrate R, the new target btrate R + 1 s obtaned by ncreasng r j by one bt ( rj +1 ) ( ber j rj +1 ) + n, j r ber (r ) ber(r +1) = 1+ n r (B.2) and, usng Δ j, ber(r +1) Δ ( ) j rj = R +1 + R R +1 ber(r). (B.3) The ber(r + 1) whch s equal to φ({r (k+1) } n ) n (29) s mnmzed only f Δ j (r j ) s mnmzed. The mnmum ber(r + 1) s then obtaned wth the ncrease of one bt n the subchannel j such that j = arg mnδ (r ). (B.4) To complete the proof by nducton, the relaton must be true for ber(1). Ths s smply done by recallng that ber (0) = 0, and then mn ber(1) = mn ber (1) = mn Δ (0). (B.5) The convergence of the algorthm to a unque soluton needs the convexty of the functon r r ber(r ). Ths convexty

13 Journal of Electrcal and Computer Engneerng 13 s verfed at hgh SNR. Appendx C provdes a more precse doman of valdty. It remans to prove that (r +1)ber (r +1)canbeused nstead of Δ (r ). In hgh SNR regme and then ber (r +1) ber (r ) lm Δ snr (r ) = (r +1)ber (r +1) + (B.6) (B.7) whch proves the lemma. In low SNR regme, the approxmaton of Δ by (r + 1)ber (r + 1) remans vald; the dssmlarty between allocaton usng Δ (B.1) and allocaton usng (r +1)ber (r +1) s null n the doman of valdty gven by Appendx C. C. Range of Convexty of r ber Let f : N R + r r ber (r,snr ) (C.1) whch equals the SER for hgh-snr regme and Gray mappng. The functon f s a strctly ncreasng functon: f (r ) <f(r +1)forallsnr,becauseber(r,snr ) ber(r + 1, snr )andr <r +1.LetΔ(r ) = f (r +1) f (r ), and then Δ(r +1) Δ(r ) = f (r +2) 2 f (r +1) + f (r ) (r +1)(ber (r +2) 2ber (r +1)). (C.2) If ber (r +2) 2ber (r + 1), then the functon f s locally convex or defnes a convex hull. Ths relaton s verfed for BER lower than and for all r 0. D. Proof of Theorem 5 WeprovethatbothmetrcsusednLemmas3 and 4 lead to the same subchannel SNR orderng. Let We then have to prove that f (r,snr ) = 2r+β 1, snr g(r,snr ) = ( r + β ) ( ber r + β ). (D.1) f (r,snr ) f ( r j,snr j ) g(r,snr ) g ( r j,snr j ). (D.2) It s straghtforward that f (r,snr ) f ( ) r j,snr j snr j 2rj+β 1 snr 2 r+β 1. (D.3) Wth square QAM, n hgh SNR regme and usng (26) ( g(r,snr ) = ) ( ) 3 erfc 2 r +β 2 ( 2 r+β 1 )snr (D.4) and t can be approxmated by the followng vald expresson ( ) 3 g(r,snr ) = 2erfc 2 ( 2 r+β 1 )snr. (D.5) Then, g(r,snr ) g ( ) r j,snr j snr j 2rj+β 1 (D.6) snr 2 r+β 1 whch s also gven by the frst nequalty. In hgh SNR regme and wth square QAM, that s, β = 2, f ( ) andg( ) leadto the same subchannel SNR orderng and then arg mn f (r,snr ) = arg mn g(r,snr ). (D.7) Ths last equaton does not hold n low SNR regme (the BER approxmaton s not vald) or when the modulatons are not square, that s, when r s odd. Note that (D.5) snot only a good approxmaton n hgh SNR regme, t can also be used wth hgh modulaton orders wth moderate SNR regme defned n Appendx C. E. ProofofTheorem 6 As the nfnte norm s not dfferentable, we use the k norm wth lm k + 1/k γ k I ( ) = max γ 1. (E.1) I In the subset I, the Lagrangan of (18)forallk s L k ({r } I, λ) = 1/k (2 r k 1) I snr k + λ R r. I (E.2) Let λ such as λ = (k 1)/k (2 r k 1) λ I snr k log 2. The optmal condton yelds (E.3) 2 r (2 r 1) k 1 = snr k λ. (E.4) In asymptotc regme, r 1 and then 2 r 1 2 r.the equaton of the optmal condton can be smplfed and r = log 2 (snr ) + 1 k log 2 λ. (E.5) The Lagrange multpler s to dentfy usng the btrate constrant, and replacng λ n the above equaton leads to the soluton r = R I + 1 I snr log 2. snr j I j (E.6)

14 14 Journal of Electrcal and Computer Engneerng Note that we do not need to calculate the convergence of the soluton wth k + to obtan the result for the nfnte norm. The result holds for all values of k n asymptotc regme. Wth k = 1, the problem s a sum SNR-gap maxmzaton problem under peak-power constrant, and t can be solved wthout asymptotc regme condton. Note that ths sum SNR-gap maxmzaton problem, or sum nverse SNRgap mnmzaton problem, under peak-power and btrate constrants s mn γ 1 σ = mn (2 r W 2 1) {r I } I I h 2 (E.7) Pp and s very smlar to power mnmzaton problem under btrate and SNR-gap constrants exchangng p wth γ 1 mn σ p = mn (2 r W 2 1) {r I } I I h 2 Pγ 1. (E.8) Both problems are dentcal f p γ = α as t s stated by Lemma 1. F. Proof of Theorem 7 To prove ths theorem, varables I and J are used nstead of r wth I = 2 r/2, J = 2 r/2, (F.1) and the btrate constrant s R = log 2 (I J ). In the subset I, the Lagrangan of (25) s then L({I, J } I, λ) = 1 (2 1 1 ) R I J I ( ) 3 erfc I 2 + J 2 2 snr + λ R log 2 (I J ). I (F.2) (F.3) Let X {I, J }, then L = X f (I, J ) + 1 X X 2 g(i, J ) 1 λ, (F.4) X wth f (I, J ) = 1 ( ) 2 3snr e 3snr/(I2 +J 2 2) R I J ( π I 2 + J 2 2 ) 3/2, g(i, J ) = 1 ( ) R erfc 3snr I 2 + J 2. 2 (F.5) The optmalty condton yelds ( I 2 J 2 ) I J f (I, J ) = (I J )g(i, J ). (F.6) AtrvalsolutonsI = J, and the other soluton must verfy (I + J )I J f (I, J ) g(i, J ) = 0. (F.7) To fnd the root of (F.7), let h ( x, y ) = x ye y erfc ( ) y wth x = 2 (I + J )I J (2 π I 2 + J ), 2 I J y = 3snr I 2 + J 2 2. (F.8) (F.9) We wll prove that ths functon s postve n a specfc doman. Consder that (1) ye y > erfc( y)fory 0.334, then for BER lower than (2) π/2x >1for{I, J } [1, + ) 2 and I 1orJ 1, and lm I,J 1 π/2x = 1 +. Then, n the doman defned by {I, J } [1, + ) 2 ber 0.1, (F.10) h(x, y) spostve,and(f.7) has no soluton. Thus, the only one soluton of (F.6)wth(F.10)sI = J.Aswewllseelater the doman of (F.10) s less restrctve than the asymptotc one. The problem s now to allocate bts wth square QAM. The followng upper bound s used: ber(r ) = 2 ( ) 3snr erfc. (F.11) r 2(2 r 1) Note that ths upper bound s a tght approxmaton wth hgh SNR and wth hgh modulaton orders. The Lagrangan s that L({r } I, λ) = 2 ( ) 3snr erfc R 2(2 I r 1) + λ R (F.12). I r And ts dervatve s L = ln 2 2 r 3snr 1) r π 2 r 1 2(2 r 1) e 3snr/2(2r λ. (F.13) Let r 1forall, then 2 r 1 2 r, and the optmalty condton yelds 3snr e 3snr/2r = 2λ2 π 2 r ln 2 2. (F.14) Wth relable communcaton over the subchannel, the Shannon s relaton states that r log 2 (1 + snr ) and 3snr /2 r 3/2 becauser 1. The relaton between r and

15 Journal of Electrcal and Computer Engneerng 15 λ s then bjectve, and the real branch W 1 of the Lambert functon [40] can be used wth no possblty for confuson ( ( )) r = log 2 (3snr ) log 2 W 1 2λ2 π ln 2. (F.15) 2 Wth the btrate constrant R = I r,wecanwrte log 2 ( W 1 ( and wth (F.15) )) 2λ2 π ln 2 2 r = R I + 1 I = R I 1 I snr log 2. snr j I j log 2 (3snr ) (F.16) (F.17) Ths result s obtaned wth square QAM n asymptotc regme (hgh modulaton orders and hgh SNR) whch s a more restrctve doman than that of (F.10). Acknowledgments The research leadng to these results has receved partal fundng from the European Communty s Seventh Framework Program FP7/ under grand agreement no also referred to as OMEGA. References [1] A. N. Akansu, P. Duhamel, X. Ln, and M. De Courvlle, Orthogonal transmultplexers n communcaton: a revew, IEEE Transactons on Sgnal Processng, vol. 46, no. 4, pp , [2] J. M. Coff, A multcarrer prmer, Tech. Rep. ANSI T1E1.4/91 157, Commttee Contrbuton, Washngton, DC, USA, [3] G. D. Forney and M. V. Eyuboglu, Combned equalzaton and codng usng precodng, IEEE Communcatons Magazne, vol. 29, no. 12, pp , [4] C. E. Shannon, Communcaton n the presence of nose, Proceedngs of the Insttute of Rado Ingneers, vol. 37, pp , [5] A. Lozano, A. M. Tulno, and S. Verdú, Optmum power allocaton for parallel gaussan channels wth arbtrary nput dstrbutons, IEEE Transactons on Informaton Theory, vol. 52, no. 7, pp , [6] T. Antonakopoulos and N. Papandreou, Bt and power allocaton n constraned multcarrer systems: the sngle-user case, Eurasp Journal on Advances n Sgnal Processng, vol. 2008, Artcle ID , 14 pages, [7] S. T. Chung and A. J. Goldsmth, Degrees of freedom n adaptve modulaton: a unfed vew, IEEE Transactons on Communcatons, vol. 49, no. 9, pp , [8] A. Fasano and G. d Blaso, The dualty between margn maxmzaton and rate maxmzaton dscrete loadng problems, n Proceedngs of the IEEE Workshop on Sgnal Processng Advances n Wreless Communcatons, pp , July [9] D. P. Palomar and J. R. Fonollosa, Practcal algorthms for a famly of waterfllng Solutons, IEEE Transactons on Sgnal Processng, vol. 53, no. 2, pp , [10] I. Km, I. S. Park, and Y. H. Lee, Use of lnear programmng for dynamc subcarrer and bt allocaton n multuser OFDM, IEEE Transactons on Vehcular Technology, vol. 55, no. 4, pp , [11] D. Hughes-Hartogs, Ensemble Modem Structure for Imperfect Transmsson Meda, Telebt Corporaton, Cupertno, Calf, USA, 1987, US Patent 4,679,227. [12] B. S. Krongold, K. Ramchandran, and D. L. Jones, Computatonally effcent optmal power allocaton algorthms for multcarrer communcaton systems, IEEE Transactons on Communcatons, vol. 48, no. 1, pp , [13] W. J. Cho, K. W. Cheong, and J. M. Coff, Adaptve modulaton wth lmted peak power for fadng channels, n Proceedngs of the 51st Vehcular Technology Conference Shapng Hstory Through Moble Technologes (VTC 00),pp , May [14] P. Uthansakul and M. E. Balkowsk, Performance comparsons between greedy and Lagrange algorthms n adaptve MIMO MC-CDMA systems, n Proceedngs of the Asa-Pacfc Conference on Communcatons, pp , Perth, Australa, October [15] A. J. Goldsmth, Varable-rate varable-power MQAM for fadng channels, IEEE Transactons on Communcatons, vol. 45, no. 10, pp , [16] S.Ye,R.S.Blum,andL.J.CmnJr, AdaptveOFDMsystems wth mperfect channel state nformaton, IEEE Transactons on Wreless Communcatons, vol. 5, no. 11, pp , [17] N. Y. Ermolova and B. Makarevtch, Practcal approaches to adaptve resource allocaton n OFDM systems, Eurasp Journal on Wreless Communcatons and Networkng, vol. 2008, Artcle ID , [18] E. Baccarell, A. Fasano, and M. Bag, Novel effcent btloadng algorthms for peak-energy-lmted ADSLtype multcarrer systems, IEEE Transactons on Sgnal Processng, vol. 50, no. 5, pp , [19] A. Fasano, On the optmal dscrete bt loadng for multcarrer systems wth constrants, n Proceedngs of the 57th IEEE Semannual Vehcular Technology Conference (VTC 03), vol. 2, pp , Aprl [20] M. A. Khojastepour and B. Aazhang, The capacty of average and peak power constraned fadng channels wth channel sde nformaton, n Proceedngs of the IEEE Wreless Communcatons and Networkng Conference, vol. 2, pp , Atlanta, Ga, USA, March [21] Y. Dng, T. N. Davdson, and K. M. Wong, On mprovng the BER performance of rate-adaptve block transcevers, wth applcatons to DMT, n Proceedngs of the IEEE Global Communcatons Conference, vol. 3, pp , San Francsco, Calf, USA, December [22] D. P. Palomar, A unfed framework for communcatons through MIMO channels [Ph.D. thess], Unverstat poltecnca de Catalunya, Barcelona, Span, [23] A. M. Wyglnsk, F. Labeau, and P. Kabal, Bt loadng wth BER-constrant for multcarrer systems, IEEE Transactons on Wreless Communcatons, vol. 4, no. 4, pp , [24] G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff,andS. U. Quresh, Effcent modulaton for bandlmted channels, IEEE Journal on Selected Areas n Communcatons, vol. 2, no. 5, pp , [25] J. M. Coff, Dgtal Communcaton, DepartmentofElectrcal Engneerng, Stanford Unversty, Stanford, Calf, USA, 2007, Course.

16 16 Journal of Electrcal and Computer Engneerng [26] D. P. Palomar, M. A. Lagunas, and J. M. Coff, Optmum lnear jont transmt-receve processng formimo channels wth QoS constrants, IEEE Transactons on Sgnal Processng, vol. 52, no. 5, pp , [27] J. G. Proaks, Dgtal Communcatons, Electrcal engneerng, McGraw-Hll, New York, NY, USA, 3rd edton, [28] K. Cho and D. Yoon, On the general BER expresson of one- and two-dmensonal ampltude modulatons, IEEE Transactons on Communcatons, vol. 50, no. 7, pp , [29] B. Fox, Dscrete optmzaton va margnal analyss, Management Scence, vol. 13, no. 3, pp , [30] P. S. Chow, J. M. Coff, and J. A. C. Bngham, Practcal dscrete multtone transcever loadng algorthm for data transmsson over spectrally shaped channels, IEEE Transactons on Communcatons, vol. 43, no. 2, pp , [31] J. Campello, Optmal dscrete bt loadng for multcarrer modulaton systems, n IEEE Internatonal Symposum on Informaton Theory, p. 193, IEEE Publshng, Cambrdge, Mass, USA, [32] H. E. Levn, A complete and optmal data allocaton method for practcal dscrete multtone systems, n Proceedngs of the IEEE Global Communcatons Conference, vol. 1, pp , San Antono, Tex, USA, November [33] J. Campello, Practcal bt loadng for DMT, n Proceedngs of the IEEE Internatonal Conference on Communcatons, vol. 2, pp , Brtsh Columba, Canada, June [34] G.992.3, Asymmetrc Dgtal Subscrber Lne Transcevers,ITU- T Recommendaton, Geneva, Swtzerland, [35] Z.-Q. Luo and W. Yu, An ntroducton to convex optmzaton for communcatons and sgnal processng, IEEE Journal on Selected Areas n Communcatons, vol. 24, no. 8, Artcle ID , pp , [36] S. Boyd and L. Vandenberghe, Convex Optmzaton, Cambrdge Unversty Press, Cambrdge, UK, [37] A. Pascual-Iserte, Channel state nformaton and jont transmtter-recever desgn n mult-antenna systems [Ph.D. thess], Unverstat Poltecnca de Catalunya, Barcelona, Span, [38] L.-P. Zhu, Y. Yao, S.-D. Zhou, and S.-W. Dong, A heurstc optmal dscrete bt allocaton algorthm for margn maxmzaton n DMT systems, Eurasp Journal on Advances n Sgnal Processng, vol. 2007, Artcle ID 12140, 7 pages, [39] R. J. Wlson, An ntroducton to matrod theory, The Amercan Mathematcal Monthly, vol. 80, no. 5, pp , [40] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, andd.e.knuth, OntheLambertWfuncton, Advances n Computatonal Mathematcs, vol. 5, no. 4, pp , 1996.

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