Joint Optimization of Bit and Power Loading for Multicarrier Systems

1 Joint Optimizati on of Bit and Po wer Loading for Multicarrie r Systems Ebrahim Bedeer , Octavia A. Dobre, Mohamed H. Ahmed, and Kareem E. B a ddour Abstract In this letter , a nov el low complexity bit and power loading algorithm is formulated for multicarrier communication systems. The proposed algorithm jointly maximizes the throughput and minimizes the transmit po wer through a weighting coef ficient α , while meeting constraints on the target bit error rate (BER) per subcarrier and on the total transmit po wer . The optimization problem is solved by the Lagrangian multiplier method if the initial α causes the transmit power not to violate the po wer constraint; otherwise, a bisection search is used to find the ap propriate α . Closed-form ex pressions are deriv ed for the close-to-optimal bit and power allocations per subcarrier , av erage throughpu t, and av erage transmit po wer . Simulation results illustrate the performance of the proposed algorithm and demonstrate its superiority with respect to existing allocation algorithms. F urthermore, the results show that the performance of the prop osed algorithm approaches that of the e xhausti ve search for the discrete optimal allocations. Index T erms Adapti ve modulation, bit loading, multicarrier systems, multiobjectiv e optimization, power loading. I . I N T R O D U C T I O N Multicarrier modulatio n is recog nized as a robust and efficient transmission techniqu e , as evidenced b y its consideratio n for d i verse co mmunicatio n systems an d adoption by several wireless standards [1]. The pe rforman ce of multicarrier com munication systems can be significantly improved by d ynamically adap ting the tr ansmission parameters, such as power , co nstellation size, sym bol rate, codin g r ate/scheme, or any co mbination of these, accordin g to the ch annel qua lity or the wir eless stand a r d specificatio ns [2]–[5]. T o date, mo st of the research literatu re has focused on the single ob jec ti ve of either maximizing the t hrou g hput or minimizing the transmit power separately (see, e.g., [2]–[5] and ref erences there in). In [2], W yglinski e t a l. propo sed an incr emental bit loading algorith m with uniform power in o rder to max im ize the through put while guaran tee ing a target BER. Liu and T ang [3] proposed a power loadin g algor ithm with uniform b it loading that aims to minimize the tr ansmit p ower while gu a ranteeing a target BER. I n [ 4], Mahmood and Belfiore p roposed an efficient greedy b it allocation algo rithm tha t minimizes the tra n smit power sub ject to fixed thr oughp ut and BER per sub carrier con straints. In emerging wireless communicatio n sy stems, v arious requir ements are need ed. For example, maximizing the throug hput is fav oured if sufficient guard bands exist to separate u ser s, while minimizing the tran sm it power is prioritized wh en o p erating in interference- prone share d spectru m environments, to prolong the battery life time of 2 battery-o perated nodes, as well as to support en v ironmen tally-friendly transmission behaviors. This motiv ates us to for m ulate a m ultiobjective op timization (MOOP) pro blem that op timizes the co nflicting and incommensur able throug hput and p ower objectives. According to th e MOOP p rinciple, th e r e is no solutio n th a t impr oves on e of the objectives without deterio rating oth ers. Th erefore, MOOP pro d uces a set of o ptimal solu tions an d it is the responsibility of the r esource allocation entity to choose the m ost p referred o p timal solu tion d epending o n its preferen ce [6]. A well kn own appr oach to solve MOOP pro b lems is to linearly combine th e co mpeting ob jectiv e function s into a single ob jectiv e fun c tio n, throu g h weigh ting co e ffi cients that reflect the req uired preferen ces [ 6 ]. These prefer ences c an be prescribe d and fixed dur in g the solution pro c e ss (as in posteriori and priori metho d s) or can be c h anged d uring the solu tion p rocess (interacti ve method s) [6]. In this pap er , we adopt an in teractiv e app roach in order to o btain a low comp lexity solu tion. W e pro pose a low complexity algor ithm that jo intly max imizes the thr o ughpu t and minimizes the total transmit power , subject to constraints o n the BER per sub c arrier and the total tr ansmit power . Limiting th e total transmit power is cr ucial for a variety of reaso ns, e.g ., to reflect the tr ansmitter’ s power amplifier limitation s, to satisfy regulatory max imum power limits, and to limit interference/ encou r age frequ ency r euse. Mor e over , including th e total su bcarrier power in the objec tive function is especially desirab le, as it minim izes the tran smit power when the power constraint is inacti ve. Closed-form expressions are deri ved for the close-to-o ptimal bit and power allocatio ns, av erage thro ughpu t, and average transmit power . Sim u lation r esults show that the prop osed algorithm o u tperfor ms existing b it and p ower lo ading schemes in the literatur e, w h ile r e q uiring similar or reduc e d comp utational effort. The results also indicate that the p roposed alg orithm’ s p erforma nce appro aches th at o f the exhausti ve search for the optimal discre te allo c ations, with significantly r educed co mputationa l effort. I I . P RO P O S E D L I N K A DA P TA T I O N S C H E M E A. Optimization P r oblem F ormulation A m ulticarrier com munication system decomp oses th e sign al ban d width into a set of N o rthogo nal n arrowband subcarriers of equ al bandwidth. Each subcarrier i transmits b i bits using power P i , i = 1 , ..., N . Follo wing the common practice in the literatur e, a delay - an d error-free feed back c h annel is assumed to exist between the transmitter and r eceiv er for repor ting the ch annel state infor mation [3]–[5]. In o rder to max im ize the thro ughpu t and minimize the transmit power subject to BER an d total tran smit p ower constraints, the o p timization pr o blem is formu lated as Maximize b i N X i =1 b i and Minimize P i N X i =1 P i , subject to BER i ≤ BER th,i , N X i =1 P i ≤ P th , i = 1 , ..., N , (1) 3 where BER i and BER th,i are the BER a nd thre sh old value of BER per sub carrier 1 i , i = 1, ..., N , respe cti vely , and P th is the to tal tran smit p ower thr eshold. An ap p roximate expression fo r the BER p er subca r rier i for M -ar y QAM is given by [ 3 ] BER i ≈ 0 . 2 exp − 1 . 6 P i 2 b i − 1 |H i | 2 σ 2 n ! , (2) where H i is the channel gain of su bcarrier i an d σ 2 n is the variance of the additiv e white Gau ssian noise (A WGN). The multi-ob jec ti ve optimization functio n in (1) can be rewritten as a lin e ar combin ation of mu ltip le ob jecti ve function s as follows Minimize P i ,b i F ( p , b ) = α N X i =1 P i − (1 − α ) N X i =1 b i , subject to g  ( P i , b i ) =      0 . 2 exp  − 1 . 6 C i P i 2 b i − 1  − BER th,i ≤ 0 ,  = 1 , ..., N , P N i =1 P i ≤ P th ,  = N + 1 , (3) where α ( 0 < α < 1 ) is a weigh ting coefficient which ind icates the rate at which the multicarrier system is willing to tr ade o ff the v alues o f th e ob jectiv e func tio ns in order to obtain a low co mplexity solutio n [6] (i.e., a hig her value of α fa vors minimiz in g the tran smit p ower , where a s a lower value of α fa vors maxim izin g the thr oughp ut). C i = |H i | 2 σ 2 n is the ch annel-to- noise ratio for sub carrier i , an d p = [ P 1 , ..., P N ] T and b = [ b 1 , ..., b N ] T are th e N -dimensional power and bit distribution vectors, re sp ecti vely , with [ . ] T denoting the tran spose op eration. B. Bit a nd P ower Allo cations The op timization p roblem in (3) can be solved numerically; howev er , this is co mputationa lly comp lex. A low complexity solution can b e ob tained by relaxin g the power co nstraint in (3), i.e.,  6 = N + 1 , and th en a pplying the method of Lag r ange multip liers. Accor d ingly , the ine quality con stra in ts are tr a nsformed to equality con straints by adding n on-negative slack variables, Y 2 i ,  = i = 1 , ..., N [7]. Henc e, the constraints are g i ven as G i ( p , b , y ) = g i ( p , b ) + Y 2 i = 0 , i = 1 , ..., N , (4) where y = [ Y 2 1 , ..., Y 2 N ] T is the vector of slack variables, and the Lag range functio n L is expressed as L ( p , b , y , λ ) = F ( p , b ) + N X i =1 λ i G i ( p , b , y ) , = α N X i =1 P i − (1 − α ) N X i =1 b i + N X i =1 λ i    0 . 2 exp  − 1 . 6 C i P i 2 b i − 1  − BER th,i + Y 2 i    , (5) 1 The constrain t on the BER per subcarrier is a suitable formulation that results in similar BER charact eristic s compared to an av erage BER constraint, especial ly at high signal-to-no ise ratios (SNRs) [5]. Further , it significantly reduces the computational complex ity by yielding closed-fo rm expressions. 4 where λ = [ λ 1 , ..., λ N ] T is the vector of Lagr a nge multiplier s. A stationary poin t is fou nd when ∇L ( p , b , y , λ ) = 0 ( ∇ d enotes th e grad ient), which yields ∂ L ∂ P i = α − 0 . 2 λ i 1 . 6 C i 2 b i − 1 exp  − 1 . 6 C i P i 2 b i − 1  = 0 , (6) ∂ L ∂ b i = − (1 − α ) + 0 . 2 ln(2) λ i 1 . 6 C i P i 2 b i (2 b i − 1) 2 exp  − 1 . 6 C i P i 2 b i − 1  = 0 , (7) ∂ L ∂ λ i = 0 . 2 exp  − 1 . 6 C i P i 2 b i − 1  − BER th,i + Y 2 i = 0 , (8) ∂ L ∂ Y i = 2 λ i Y i = 0 . (9) It can be seen that (6) to (9) represent 4 N equations in the 4 N u nknown compo nents of the vectors p , b , y , and λ . By solv in g (6) to (9), one o btains the solution p ∗ , b ∗ . Equation (9) imp lies that either λ i = 0 or Y i = 0; hen ce, two po ssible cases exist and we are going to in vestigate each case inde pendently . — Case 1 : Setting λ i = 0 in (6) to (9) resu lts in an u nderdeter mined system o f N eq uations in 3 N unkn owns, and, hence, n o un ique solution can be reached . — Case 2 : Setting Y i = 0 in (6) to (9), we ca n relate P i and b i from (6) and (7) as follows P i = 1 − α α ln(2) (1 − 2 − b i ) , (10) with P i ≥ 0 if and only if b i ≥ 0 . By substituting ( 10) in to (8), on e obtain s the so lution b ∗ i = 1 log(2) log    − 1 − α α ln(2) 1 . 6 C i ln(5 BER th,i )    . (11) Consequently , fr om (10) one gets P ∗ i = 1 − α α ln(2)    1 − − 1 − α α ln(2) 1 . 6 C i ln(5 BER th,i ) ! − 1    . (12) Since we consider M -ary QAM, b i should be greater than 2 . Fro m (11), to hav e b i ≥ 2 , C i must satisfy the condition C i ≥ C th,i = 4 1 . 6 α ln(2) 1 − α ( − ln(5 BER th,i )) , i = 1 , ..., N . (13) The re la xed optimiz a tion pr oblem is no t conve x and, hence, the Karu sh-Kuhn-T ucker (KKT) cond itio ns do not guaran tee th at ( p ∗ , b ∗ ) repr esents a global op timum [7]; the p roof of th e KKT co nditions is not p rovided du e to th e space limitations. T o cha r acterize the gap to the global optimum solutio n , we c ompare the obtained local optimu m results to the glo bal op timum results o btained thr ough th e exhaustive search in the next section . If the total transmit p ower P N i =1 P i is below P th , th en the final bit and power allocations are reach ed. On th e other han d, if the tra nsmit power exceeds P th , the algorithm ado pts th e in teractive app roach and overrides the initial value of α to mee t the p ower constra int. This is achieved by giving m ore weight to the tra n smit power minimization in (3), i.e., b y in creasing α . T he lowest α ∗ that satisfies th e co nstraint, i.e., α ∗ that results in the highest total power 5 which is lower th an P th , is fo und throu g h the bisection search 2 (please n ote that lower v alues of α pro duce lower values of the o bjectiv e function in (3)). T he pr oposed algorith m can b e f ormally stated as follows: Proposed Algorithm 1: INP UT The A W GN variance ( σ 2 n ), channel gain per subcarrier i ( H i ), target BER per subcarrier i (BER th,i ), initi al weighting parameter α , and tolerance ǫ . 2: for i = 1, ..., N do 3: if C i ≥ C th,i = − 4 1 . 6 α ln(2) 1 − α ln(5 BER th,i ) th en 4: - b ∗ i and P ∗ i are gi ven by (11) and (12), respecti vely . 5: - b ∗ i,f inal ← Round b ∗ i to the nearest integ er . 6: - P ∗ i,f inal ← Recalculate P ∗ i according to (2). 7: else 8: Null the corresponding subcarrier i . 9: end if 10: en d for 11: wh ile P N i =1 P ∗ i,f inal − P th > ǫ do 12: - Set α L = α and α U = 1. 13: - Set α ∗ = ( α L + α U ) / 2 . 14: - Repeat steps: 3 t o 9. 15: if P N i =1 P ∗ i,f inal < P th then 16: - Set α U = α ∗ , t hen α ∗ = ( α L + α U ) / 2 . 17: - Repeat steps: 3 t o 9. 18: else 19: - Set α L = α ∗ , t hen α ∗ = ( α L + α U ) / 2 . 20: - Repeat steps: 3 t o 9. 21: end if 22: en d wh ile 23: OUTP UT b ∗ i,f inal and P ∗ i,f inal , i = 1, . .., N . C. Analytical Exp r essions of A verage Thr oug hput and T ransmit P o wer When the in itial value of α re sults in an in a c ti ve power constra in t, the closed-fo rm expressions f or the average throug hput and transmit p ower can b e foun d by averaging the bit and power allocatio ns g i ven b y (11) and (12), respectively , over C i . In such a case, the average th rough put is expressed as Throu g hput av = N X i =1 E { b i ( C i ) } = N X i =1 Z ∞ C th,i b i ( C i ) " ⋋ exp ( − ⋋ C i ) # d C i , ( 14) 2 This is true as the total transmit powe r calcula ted from (12) is a decrea sing functio n of α . The proof is not provided due to the space limitat ions. 6 where ⋋ exp ( − ⋋ C i ) is th e exponential distribution of C i with m e an 1 ⋋ , given that the chann el gain H i has a Rayleigh distribution. The in tegration in ( 14) is solved by parts yield ing Throu g hput av = N X i =1 1 log (2)    log (4) exp ( − ⋋ C th,i ) − Ei ( − ⋋ C th,i ) ln (10)    , (15) where E i ( − z ) = − R ∞ z e − t t dt, z > 0 is the exponen tial integral fun ction. Similarly , the average transmit power is giv en by Power av = N X i =1 1 − α α ln (2)    exp ( − ⋋ C th,i ) + ⋋ C th,i 4 Ei ( − ⋋ C th,i )    . (16) I I I . S I M U L A T I O N R E S U L T S This section inv estigates th e perfor m ance of the prop osed algo rithm, an d compa r es its p erforma n ce with bit and power loading alg orithms presen ted in the literatu r e, as well as with the exhau sti ve search for the d iscrete glob al optimal allocations. The comp utational complexity o f the p r oposed alg orithm is also co mpared to the other schemes. A. Simulation S etup As an example of a m u lticarrier system, we consider ortho gonal fre q uency division multip lexing (OFDM) with N = 12 8 subc a rriers. Without loss of gener a lity , th e BER constraint per subcarrier, BER th,i , is assumed to be the same for all sub carriers and set to 10 − 4 . A Rayleigh fadin g en vironm ent with average channe l p ower g ain E {|H i | 2 } = 1 is considered. Repr e sentativ e results are pr e sen ted, which were obtain ed thro ugh Monte Carlo trials for 10 4 channel realiza tions w ith ǫ = 10 − 9 mW an d initial α = 0 . 5 . The tran smit p ower objective fun ction is scaled du ring simulations so that it is approx imately within the same range as the throughp u t [6 ]. For convenience, presented numerical r esults are displaye d in the origin a l scales. B. P erformance of the P r oposed Algorithm Fig. 1 depicts the average th roughp ut and tran sm it power as a fun ction o f th e average SNR 3 , with a nd without considerin g the total power co n straint. In the latter c a se, the average throug hput and tr ansmit p ower , obtain ed by av eraging (11) and (1 2), respectively , over the total nu mber of channel r e a lizations throu g h Mo nte Carlo simulations, show an excellent match to their counter p arts in (15) and (16), respectively . Further , for an average SNR ≤ 24 3 The avera ge SNR is calcul ated by av eragin g the instantaneo us SNR va lues per subcarrier ove r the total number of subcarriers and the total number of channel realizati ons, respect i vely . 7 0 5 10 15 20 25 30 35 10 1 10 2 10 3 Average throughput (bits/OFDM symbol) Average SNR (dB) 0 5 10 15 20 25 30 35 10 −2 10 −1 10 0 Average transmit power (mW) d a t a 1 d a t a 2 d a t a 3 Analytical, n o p owe r constraint Simulati on, no p ower constraint Simulati on, P th = 0 . 1 mW Average transmit power Average throughput Fig. 1: A verage throughput and av erage transmit power as a function of averag e SNR, wit h and without a po wer constraint. 5 10 15 20 25 30 −20 −15 −10 −5 0 Average SNR (dB) Objective function in (3) Proposed solution Exhaustive search N = 4 N = 6 N = 8 Fig. 2: Objective function for the pro- posed algo rithm and the e xhaustiv e search when N = 4, 6, and 8. 0 5 10 15 20 25 30 35 10 0 10 1 10 2 10 3 Average SNR (dB) Average throughput (bits/OFDM symbol) d a t a 1 P r op o s e d no power constraint P th = 0.1 mW Proposed algorithm Wyglinski [2] Fig. 3: A verage throughput as a f unction of a verage S N R for the proposed algo- rithm and W yglinski’ s algorithm i n [2]. 200 400 600 800 1000 1200 1400 10 −2 10 −1 10 0 10 1 Average throughput (bits/OFDM symbol) Average transmit power (mW) E-BER, Liu [3] U-BER, Liu [3] Mahmoo d [4] Proposed, no power constraint Fig. 4: A verage transmit power as a function of av erage throughput for the proposed algorithm and the algorithms in [3] and [4]. dB, one finds that bo th the av erage throug hput and transmit power incre a se as th e SNR increases, wh ereas for an av erage SNR ≥ 24 dB, th e transmit power satura tes while the throu ghput co ntinues to in c rease. This observation can be explained as fo llows. The r elation be twe en b i and P i in (10) implies that in creasing the numbe r o f bits at the low range of b i (that exists at low average SNR values) oc c urs at the expense of additiona l transmit power , while increasing the n umber of bits at th e h igh rang e of b i (that exists at high average SNR values) occurs at negligib le increase in th e transmit power . Ac cording ly , fo r lower values of the average SNR, increasing the average thr o ughpu t is accom panied b y a correspo nding increase in the transmit power . On the other h and, for higher values of the av erage SNR, the av erage transmit power saturate s an d th e av erage throug hput is increased. By con sid e ring a tota l power constraint, P th = 0 .1 mW , at lower SNRs, when the to tal tran smit p ower is below the threshold , th e average transmit p ower and throug hput a re similar to their respective values for th e no power constraint case. As the SNR increases, th e tr a n smit power reaches the power threshold an d the average throug hput is r educed acco rdingly . Fig. 2 compares the objectiv e fun ction achieved with the propo sed algor ithm and the exhau sti ve search that finds the discretized g lo bal optimal allocation for the prob lem in (3). Results ar e presented for P th = 5 µ W and N = 4, 8 6, an d 8; a small num ber of subcarrier s is c hosen, such th a t the exhaustive search is feasib le. As can be seen, the propo sed algor ithm app roaches the optimal results of th e exhaustive search, an d , hence, provid es a close- to -optimal solution. C. P erformance Comparison with Alg orithms in the Literatur e In Fig. 3, the throug hput ach iev ed b y the pr o posed algorithm is comp ared to that o btained by W yglinski’ s algorithm [2] f or the same o p erating condition s, with a nd without co nsidering the total p ower con straint. For a fair compar ison, the unifor m power allocation used by th e allocation schem e in [2] is computed by dividing the av erage transmit p ower alloca te d by our algorith m by the to tal numbe r of subcarriers. As sh own in Fig. 3, the propo sed algorithm p rovides a significan tly hig her thr oughp ut th an the sche me in [2] for low average SNRs. This result demon strates that optim a l allocation of transmit power is cr ucial fo r low power budgets. Fig. 4 compare s th e average transmit power obta in ed by the prop osed algorithm, in th e case of no power co nstraint, with the op timum power allocation o f Liu and T ang [ 3], a variation called E -BER [3] that assum es an equa l BER per subcarr ier , and th e algorith m of Ma h mood and Belfiore [4]. After matching the o perating cond itions, on e c a n see th at the propo sed allo cation scheme assign s less average power than th e schemes in [3] and [4] to achieve the same av erage BER and thro u ghput. Th e different results between [3] and [4] (wh ile both guarantee the same fixed throug hput) are mainly because the algo rithms in [3] allocate the same number o f b its p er subcarrier, while the algorithm in [4] allocates a different nu m ber of bits per su bcarrier, which is intuitively m ore efficient. The imp roved p erforma n ce of the propo sed joint bit and power allocation alg orithm does no t come at the co st of additional co mplexity . Its com putational complexity is of O ( N ) when the initial value of α results in an in acti ve power constraint, which is similar to that of Liu’ s alg orithm. Other wise, it is of O ( N log ( N )) , which is lo wer than that of W yglinski’ s O ( N 2 ) algorith m and sig nificantly lower than O ( N !) of th e exhaustive search . I V . C O N C L U S I O N In this letter , we prop osed a novel alg orithm that jointly maxim izes th e thr oughp ut and minim izes the tra n smit power g iv en constrain ts on the BER per subcarrier and the total tran sm it power . Closed-for m expressions were derived for th e close-to- optimal b it and p ower allocation s per sub carrier, average throug hput, and average transmit power . Simulation r esults dem onstrated that the prop osed algorithm outperf orms d ifferent allocation schemes th at separately max imizes the throug hput o r minimizes the tran smit power , u n der the same op erating conditio ns, while requirin g similar or r educed co mputation al effort. Additionally , it was shown that its per formanc e app roaches that of th e exha u sti ve search with significan tly lower comp lexity . R E F E R E N C E S [1] K. Fazel and S. Kaiser , Multi-c arrier and Spre ad Spect rum Systems: fr om OFDM and MC-CDMA to LTE and W iMAX . John W iley & Sons Inc, 2008. [2] A. 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