Energy-Efficient Link Adaptation on Parallel Channels

Transcription

1 Energy-Efficien Link Adapaion on Parallel Channels Chrisian Isheden and Gerhard P. Feweis Technische Universiä Dresden Vodafone Chair Mobile Communicaions Sysems D-0069 Dresden Germany (see hp:// Absrac Energy-efficien link adapaion is sudied for ransmission on a parallel channel. The oal power dissipaion model includes circui power and a power amplifier inefficiency parameer. Earlier resuls are derived in various ways based on convex minimizaion problems and concave maximizaion problems respecively. I is shown ha he fixed-poin algorihm proposed earlier by he auhors is equivalen o he Dinkelbach mehod for solving nonlinear fracional programs. I. INTRODUCTION Energy efficiency in mobile communicaion devices is becoming more and more imporan because of he increasing gap beween power consumpion of signal processing circuis and baery capaciy. Improved energy efficiency involves minimizing he energy consumpion per bi ] or equivalenly maximizing a hroughpu per Joule meric 2]. Previous work on opimizaion of energy efficiency during ransmission has modeled he power dissipaed in a mobile erminal during ransmission as he sum of a consan power dissipaion in he processing circui and he ransmission power divided by he power amplifier drain efficiency. Energy-efficien link adapaion for parallel channels is an imporan problem since boh frequency-selecive block fading channels 3] and MIMO channels (via singular value decomposiion 4] can be described wih his model. In his paper we develop our previous resuls on energyefficien link adapaion for parallel channels in 5]. The main resuls from ha paper are derived wih he aid of ransformed problems ha are concave in he case of maximizaion and convex in he case of minimizaion respecively. Thus he opimum is guaraneed o be global. Furhermore i is shown ha he algorihm proposed in 5] is equivalen o he Dinkelbach mehod for solving nonlinear fracional programs. The paper is organized as follows. In Secion II he energy efficiency is d over power based on he ransformaion o a concave program. In Secion III he inverse problem of minimizing energy consumpion per bi over rae is considered. In Secion IV he equivalen resuls of hese wo opimizaion problems form he basis of a fixed-poin algorihm ha is shown o be equivalen o he Dinkelbach mehod in Secion V. Secion VI concludes he paper. II. MAXIMIZING THE ENERGY EFFICIENCY Consider a parallel AGN channel consising of a se of K non-inerfering subcarriers he noise is independen across subcarriers. Assuming ha perfec channel sae informaion is available a boh ransmier and receiver he maximum rae (in bis/s ha can be reliably ransmied over each subcarrier is r i = log 2 ( + p i i =... K ( denoes he subcarrier spacing p i is he ransmi power specral densiy of subcarrier i and is he channel o noise raio (CNR of subcarrier i given by = h i 2 N 0 h i 2 denoes he subcarrier power gain and N 0 is he noise power specral densiy. Efficiency in general is uiliy divided by cos. In he case of energy efficiency he uiliy is he amoun of daa ransferred and he cos is he amoun of energy needed for he ransmission. Since he energy efficiency (EE can be calculaed as sum rae over oal power dissipaion we have he opimizaion problem EE(p = K ri(pi P C +ε K pi (2 P C is he circui power dissipaion and ε is a parameer ha expresses power amplifier inefficiency. For simpliciy P C is assumed o be consan. Since he numeraor is concave and he denominaor affine (hence convex in p his is a sricly quasiconcave maximizaion problem. To simplify he mahemaical analysis we express he sum rae in nas/s and inroduce he parameer µ = P C ε ha expresses he relaive weigh of he erms in he cos funcion. Thus he problem o be solved is q(p = T ρ(p µ+ T p (3 ρ i = ln( + p i Noe ha EE in (2 is relaed o q by EE = q ε ln 2 and ha r i = ln 2 ρ i.

2 A. Transformaion o a concave problem By he ransformaion = /(µ + T p y = p/(µ + T p y R K + > 0 we obain he concave maximizaion problem 6] q(y/ = T ρ(y/ y R K + >0 subjec o µ + T y = ρ i = ln ( + yi and he parameer corresponds o he inverse of he oal power dissipaion. The problem above is concave since he perspecive funcion in he objecive preserves concaviy and he equaliy consrain is affine. By fixing he value of we see ha he problem of maximizing he energy efficiency in a muli-carrier sysem is closely relaed o ha of maximizing he sum rae for a given oal ransmission power allocaed o a se of communicaion channels. I is well known ha his problem has an analyical soluion given by waer-filling of ransmission power see Appendix for deails. B. Mahemaical analysis Since he objecive funcion in (4 is concave and coninuously differeniable and he equaliy consrain funcion is affine he KKT condiions are boh necessary and sufficien for opimaliy. Afer he inroducion of a Lagrange muliplier ν R for he equaliy consrain he Lagrangian is L(y ν = T ρ(y/ ν(µ + T y hence he KKT condiions are T ρ(y / + K wih (4 µ + T y = (y / ν = 0 i =... K (y / = ν µ = 0 + y i = y i. From he second KKT condiion ν = + y i Solving for yi yields ( yi = ν If > ν ε ln 2 he resuling y i would be negaive so in his case yi = 0. Thus we have he opimal power allocaion p i = y i = ν ] + which is waer-filling wih a cuoff CNR equal o ν. The corresponding opimal rae allocaion can be calculaed from ρ i = ln ν ln ] + Since he unknowns can be calculaed as funcions of ν he problem reduces o finding ν from he las KKT condiion ν µ = T ρ(y / K yi y i. Combining his wih he second KKT condiion we have or ν µ = T ρ(y / y i ν ν = T ρ(y / µ + T y / = T ρ(y / = q(y / since µ + T y / = /. Thus a he opimum poin he Lagrange muliplier ν is equal o q. III. MINIMIZING THE ENERGY CONSUMPTION PER BIT Since he numeraor and denominaor in problem (3 are boh posiive equivalenly he inverse can be d. Solving ( for p i we ge p i = 2r i/ i =... K (5 which is convex and coninuously differeniable in r i. For energy-efficien communicaion we wish o he cos funcion energy consumpion per bi (denoed E a corresponding o he dissipaed power divided by he hroughpu hus he problem o be solved can be saed as r R K + E a (r = P C+ε T p(r T r (6 p i (r i is given by (5. ih he same simplificaions as in he maximizaion case we ge ρ R K + q(ρ = µ+t p(ρ T ρ (7 p i = eρi Noe ha he energy consumpion per bi E a can be calculaed from E a = EE = ε ln 2. q

3 A. Perspecive ransformaion The opimizaion problem (7 is a convex-concave fracional problem wih an affine denominaor 7] f 0 (x/(c T x + d x subjec o f i (x 0 i =... m Ax = b f 0 f... f m are convex and he domain of he objecive funcion is defined as {x domf 0 c T x + d > 0}. I can be shown ha his problem is sricly quasiconvex 8]. By he one-o-one variable ransformaion y = x/(c T x + d = /(c T x + d y R K + > 0 he problem above becomes y f 0 (y/ subjec o f i (y/ 0 i =... m Ay = b c T y + d = f 0 (y/ is called he perspecive of f 0. This problem is convex since he perspecive operaion conserves convexiy and he equaliy consrain is affine. Applying he variable ransformaion above o problem (7 we obain µ + T p(y/ y R K + >0 (8 subjec o T y = p i (y i / = eyi/ For his paricular problem corresponds o he inverse sum rae (which can be inerpreed as he average na ransmission ime and y corresponds o a normalized rae vecor wih he fracional disribuion of he raes. B. Mahemaical Analysis Since he objecive funcion in problem (8 is a coninuously differeniable convex funcion and he equaliy consrain is an affine funcion he Karush-Kuhn-Tucker (KKT condiions are boh necessary and sufficien for opimaliy. Afer inroducion of a Lagrange muliplier ν r R for he equaliy consrain he Lagrangian is L(y ν r = µ + T p(y/ + ν r ( T y hence he KKT condiions are T y = 0 p i yi νr = 0 i =... K K µ + T p(y / + p i = 0 wih p i = eyi/ p i = y i p i. The second KKT condiion yields ν r = p i y i = ey i / Solving for yi we ge ( yi = ln νr ln If > νr he resuling yi would be negaive so in his case yi = 0. Thus we have he opimal rae allocaion ρ i = y i = ln νr ln ] + wih he corresponding subcarrier power allocaions p i = νr ] + These expressions are idenical o he ones obained hrough maximizing he energy efficiency wih ν = /νr. Since all unknowns have been expressed as explici funcions of νr he problem reduces o finding νr from he las KKT condiion µ + T p(y / + K ( y i p i yi = 0. Combining his wih he second KKT condiion we have which leads o he resul or since T y = µ + T p(y / ν r T y = 0 ν r = (µ + T p(y / T y ν r = (µ + T p(y / = E a. Tha is a he opimum poin we have ν r = E a. IV. NUMERICAL SOLUTION ALGORITHM Summarizing he resuls from he mahemaical analysis of he wo equivalen problem formulaions he necessary and sufficien KKT condiions resul in he following se of equaions a an opimal poin (remember ha ν = /ν r : ρ i = ln ν ln ] + i =... K (9 p i = ν ] + i =... K (0 T ρ ν = µ + T ( p

4 Require: q 0 saisfying F (q 0 0 olerance n 0 repea Solve problem (2 wih q = q n o obain x n q n+ f(x n g(x n n n + unil F (q n Fig.. The Dinkelbach mehod. This se of nonlinear equaions is difficul o solve explicily. Therefore in 5] we proposed he following fixed-poin algorihm exhibiing superlinear convergence rae: given iniial value ν 0 olerance repea Use ν n o calculae new values for ρ i and p i i =... K 2 Use hese values o compue ν n+ 3 n := n + unil ν n ν n V. COMPARISON ITH THE DINKELBACH METHOD A concave-convex fracional program x S q(x = f(x g(x can also be associaed wih a parameric concave program 0] ] f(x qg(x (2 x S q R is reaed as a parameer. The problem above migh be mahemaically more racable han a concave-convex fracional program since i is concave. Le x be an opimal poin in problem (2. The opimal value of he objecive funcion F (q = f(x qg(x is a convex coninuous and sricly decreasing funcion of q 0]. Moreover le he opimal value of he objecive funcion in he concave-convex fracional program be denoed by q. Then he following saemens are equivalen: F (q > 0 q < q F (q = 0 q = q F (q < 0 q > q Thus solving he concave-convex fracional program is equivalen o finding he roo of he nonlinear equaion F (q = 0. The algorihm described in Fig. known as he Dinkelbach mehod 0] can be used o find his roo. The algorihm is in fac he applicaion of Newon s mehod o a nonlinear fracional program ]. Therefore he sequence converges o he opimal poin wih a superlinear convergence rae. A deailed convergence analysis can be found in 2]. The iniial poin can be any q 0 = f( x g( x wih a feasible x ha saisfies F (q 0 0. For problem (3 he corresponding parameric concave opimizaion problem is ρ i = ln( + p i T ρ(p q(µ + T p (3 i =... K and q R is a given parameer. Problem (3 needs o be solved in each sep of Dinkelbach s algorihm. Since i is obvious ha a sricly feasible poin exiss for problem (3 srong dualiy holds according o Slaer s condiion 9] and he Karush-Kuhn-Tucker (KKT condiions are necessary and sufficien for opimaliy. The saionariy condiion (obained by seing he derivaive wih respec o p i equal o zero is + p i q = 0 Solving his equaion for p i we ge p i = q Since he ransmi power mus be nonnegaive we have p i = q ] + The corresponding opimal rae adapaion is given by ρ i = ln q ln ] + Comparing hese resuls wih he algorihm proposed in 5] we see ha he wo numerical mehods are equivalen wih he parameer q corresponding o ν. VI. CONCLUSION e have seen ha he resuls in 5] can be derived in a number of ways based on concave maximizaion problems or convex minimizaion problems respecively. The fixed-poin algorihm urns ou o be equivalen o he Dinkelbach mehod for solving concave-convex fracional programs. For deailed convergence properies of he algorihm he reader is referred o 2]. APPENDIX A ATER-FILLING POER ALLOCATION Find he subcarrier power allocaion p i ha s he capaciy under a oal power consrain K p i ˆP : subjec o K r i(p i K p i ˆP Since he objecive and he inequaliy consrain funcions are convex and coninuously differeniable he KKT condiions are boh necessary and sufficien for opimaliy.

5 Inroducing a Lagrange muliplier λ R for he inequaliy consrain he Lagrangian is ( K L(p λ = r i (p i + λ p i ˆP hence he KKT condiions (primal feasibiliy dual feasibiliy complemenary slackness and saionariy are p i ˆP 0 λ 0 ( K λ p i ˆP = 0 dr i dp i + λ = 0 i =... K respecively. Since r i (p i is monoonically increasing he las row has no soluion if λ = 0 so λ has o be sricly greaer han 0. Thus he complemenary slackness condiion implies K p i ˆP = 0 so he oal power consrain is always acive. Insering he expression for r i (p i from ( we obain ln 2 + p i + λ = 0 Rearranging yields ( p i = λ ln 2 REFERENCES ] S. Cui A. J. Goldsmih and A. Bahai Energy-Consrained Modulaion Opimizaion IEEE Transacions on ireless Communicaions vol. 4 No. 5 pp ] G. Miao N. Himaya and G. Y. Li Energy-Efficien Link Adapaion in Frequency-Selecive Channels IEEE Transacions on Communicaions vol. 58 No. 2 pp ] R. S. Prabhu and B. Daneshrad An Energy-efficien aer-filling Algorihm for OFDM Sysems IEEE ICC ] R. S. Prabhu and B. Daneshrad Energy-efficien power loading for a MIMO-SVD sysem and is performance in fla fading IEEE GLOBE- COM ] C. Isheden and G. Feweis Energy-Efficien Muli-Carrier Link Adapaion wih Sum Rae-Dependen Circui Power IEEE Global Communciaions Conference (GLOBECOM ] S. Schaible Fracional Programming Zeischrif für Operaions Research Vol. 27 pp ] S. Boyd and L. Vandenberghe Convex Opimizaion Exercise 4.7 pp ] S. Schaible Minimizaion of Raios Journal of Opimizaion Theory and Applicaions vol. 9 No. 2 pp ] S. Boyd and L. Vandenberghe Convex Opimizaion Cambridge Universiy Press ]. Dinkelbach On Nonlinear Fracional Programming Managemen Science vol. 3 No. 7 pp ] S. Schaible and T. Ibaraki Fracional programming European Journal of Operaional Research vol. 2 pp ] S. Schaible Fracional programming. II On Dinkelbach s algorihm Managemen Science vol. 22 No. 8 pp If > λ ln 2 he resuling p i would be negaive so in his case p i = 0. Thus we have he opimal power allocaion p i = λ ln 2 ] + i =... K wih λ chosen such ha he oal power consrain is me wih equaliy λ ln 2 ] + = γ ˆP. i The sum on he lefhand side is a piecewise-linear increasing funcion of λ ln 2 wih breakpoins a so he equaion has a unique soluion. The values ploed as a funcion of he subcarrier index i can be hough of as racing ou he boom of a vessel. If ˆP unis of waer are filled ino he vessel he amoun of waer in subcarrier i is he power allocaed o ha subcarrier and is he waer level. λ ln 2 ACKNOLEDGMENT This work was financed by BMBF (German Minisry of Educaion and Research hrough he Cool Silicon Cluser of Excellence.