Power Allocation for Block-Fading Channels with Arbitrary Input Constellations

Transcription

1 254 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS VOL. 8 NO. 5 MAY 2009 Power Allocation for Block-Fading Channels with Aritrary Input Constellations Khoa D. Nguyen Alert Guillén i Fàregas and Lars K. Rasmussen Astract We consider power allocation strategies for aritrary input channels with average and -to-average power ratio PAPR constraints. We are focusing on systems with a fixed and finite input constellation as encountered in most practical systems. Generalizing previous results we derive the optimal power allocation scheme that imizes the outage proaility of lock-fading channels with aritrary input constellations suject to PAPR constraints. We further show that the signal-to-noise ratio exponent for any finite -to-average power ratio is the same as that of the -power limited prolem resulting in an error floor. We also derive the optimal power allocation strategies that maximize the ergodic capacity for aritrary input channels suject to average and PAPR constraints. We show that capacities with -to-average power ratio constraints even for small ratios are close to capacities without -power restrictions. For oth delay-limited and ergodic lock-fading channels the optimal power allocation strategies rely on the first derivative of the input-output mutual information which may e computationally prohiitive for efficient practical implementation. To overcome this limitation we develop suoptimal power allocation schemes that resemle the traditional water-filling technique. The suoptimal power allocation schemes significantly reduce computational and storage requirements while enjoying imal performance losses as compared to optimal schemes. Index Terms Power allocation lock-fading channel coded modulation imum-mean squared error diversity outage proaility channel capacity -to-average power constraints. I. INTRODUCTION AMajor design challenge in wireless communication is to effectively deal with the varying nature of the channel commonly referred to as fading []. When knowledge of the channel fading coefficients also known as channel state information CSI is availale at the transmitter power allocation schemes can e employed to improve performance [2] [4]. Manuscript received Feruary ; revised June and Septemer ; accepted Octoer The associate editor coordinating the review of this paper and approving it for pulication was I.-M. Kim. K. D. Nguyen is with the Institute for Telecommunications Research University of South Australia Mawson Lakes Boulevard Mawson Lakes 5095 South Australia Australia dangkhoa.nguyen@postgrads.unisa.edu.au. A. Guillén i Fàregas is with the Department of Engineering University of Camridge Trumpington Street Camridge CB2 PZ UK alert.guillen@eng.cam.ac.uk. L. K. Rasmussen was with the Institute for Telecommunications Research University of South Australia Mawson Lakes SA 5095 Australia. He is now with the Communication Theory Laoratory School of Electrical Engineering and the ACCESS Linnaeus Center Royal Institute of Technology Osquldas väg 0 SE Stockholm Sweden Lars.Rasmussen@ieee.org. This work was presented in part at the IEEE International Symposium on Information Theory Toronto Canada July This work has een supported y the Australian Research Council under ARC grants RN DP and DP Digital Oject Identifier 0.09/TWC /09$25.00 c 2009 IEEE Transmitter CSI can e otained y reusing the receiver CSI for transmission in time-division duplex TDD systems [2] or from a dedicated feedack channel [3] providing perfect or quantized CSI [4][5]. In a delay-limited system codewords are transmitted over channels with B degrees of freedom where B is finite. Example scenarios are transmission over slowly varying channels or OFDM transmission over frequency-selective channels. The channel is conveniently modeled as a lock-fading channel [6] [7] where each codeword is transmitted over B corresponding flat-fading locks. In this case the maximal achievale rate is a random variale depending on the channel realization. For most fading statistics the channel capacity is zero since there is a non-zero proaility that any positive rate is not supported y the channel. A relevant performance measure in this case is the information outage proaility [7] which is the proaility that communication at a target rate R is not supported y the channel. The outage proaility is also a lower ound on the word-error proaility for communicating at rate R [8]. The optimal power allocation prolem has een investigated in [8] for channels with Gaussian inputs and in [9] [0] [] for channels with aritrary input constellations. References [8] [0] consider systems with per-codeword and average power constraints and show that systems with average constraints perform significantly etter than systems with constraints. However systems with average constraints may employ large possily infinite power. As a compromise the optimal power allocation strategy for Gaussian input channels enforcing oth and average power constraints is considered in [8] while the use of quantized power levels is investigated in [2]. For transmission over a fast-varying fading channel the fading statistics are revealed within each codeword and the channel is ergodic i.e. it has infinite degrees of freedom B. In this case adaptive techniques aim at maximizing the ergodic channel capacity which is the maximum data rate that can e transmitted over the channel with vanishing error proaility [5]. Optimal power allocation schemes such as water-filling for channels with Gaussian inputs [8] [5] and mercury/water-filling for channels with an aritrary input [9] have een developed for systems with average power constraints. The work in [3] derives the optimal power allocation strategy for Gaussian input channels with oth and average power constraint which results in a variation to the classical water-filling algorithm [5]. In this paper we consider power allocation strategies for aritrary input channels with average and -to-average power ratio PAPR constraints. We are focusing on cases with

2 NGUYEN et al.: POWER ALLOCATION FOR BLOCK-FADING CHANNELS WITH ARBITRARY INPUT CONSTELLATIONS 255 a fixed and finite input constellation as encountered in most practical systems. Generalizing the results in [8] [9] yields the optimal power allocation scheme that imizes outage proaility for transmission with aritrary inputs over a lockfading channel suject to PAPR constraints. The optimal power allocation schemes that maximize the ergodic capacity for aritrary input channels suject to average and PAPR constraints are also derived. In oth cases the optimal power allocation strategies rely on the first derivative of the inputoutput mutual information which may e computationally prohiitive for practical implementation. We therefore develop suoptimal power allocation schemes for systems with average and PAPR constraints that significantly reduce the complexity while enjoying imal performance losses from optimality. The paper is organized as follows. Sections II and III descrie the system model and the information theoretic framework. Section IV discusses power allocation algorithms for imizing the outage proaility of delay-limited lockfading channels while algorithms for maximizing the ergodic capacity is given in Section V. Concluding remarks are given in Section VI. The following notation is used in the paper. Expectation with respect to a random variale ξ is denoted E ξ [ ] while expectation with respect to ξ Ris E ξ R [ ]. Componentwise inequalities are denoted. The exponential equalities fξ. = Kξ d indicates that lim ξ fξξ d = K with exponential inequalities similarly defined. Finally fξ denotes maxfξ 0}. II. SYSTEM MODEL Consider transmission over a channel consisting of B locks of L channel uses in which lock =...B undergoes an independent fading gain h corresponding to a power fading gain γ h 2. Assume that h = h...h B and γ = γ...γ B are availale at the receiver and the transmitter respectively. Suppose the transmit power is allocated following the rule pγ =p γ...p B γ.then the corresponding complex ase-and equivalent is y = p γh x z =...B where y C L x L with C eing the signal constellation set are the received and transmitted signals in lock respectively and z C L is the additive white Gaussian noise AWGN vector with independently identically distriuted circularly symmetric Gaussian entries Z N C 0. Assume that the signal constellation of size 2 M satisfies x x 2 =2 M then the instantaneous received signal-tonoise ratio SNR at lock is given y p γγ. We consider systems with the following power constraints: Peak power : pγ p γ P B Average power : E γ [ pγ ] P av. For the fully-interleaved ergodic case the channel model can e otained from y letting B and L =.Due to ergodicity power allocation for lock is only dependent on γ. For simplicity of notation denote pγ as the transmit power corresponding to the power fading gain γ. The following power constraints are considered: Peak power : pγ P Average power : E γ [pγ] P av. Here we study the performance of systems with and average power constraints [8] [3] as well as systems with a -to-average power ratio constraint P P av PAPR. The fading gain h isassumedtohaveanakagami-mdistriuted magnitude and uniformly distriuted phase perfectly compensated for. The proaility density function pdf of the fading gain h is f h ξ = 2mm ξ 2m e mξ2 =...B Γm where Γa is the Gamma function Γa = 0 t a e t dt. The pdf of the power fading gain is f γ γ = mm γ m e mγ γ 0. 2 Γm The Nakagami-m distriution represents a large class of practical fading statistics. In particular we can recover the Rayleigh fading y setting m =and approximatethe Ricean fading with parameter K y setting m = K2 2K []. III. OUTAGE PROBABILITY AND ERGODIC CAPACITY Let I ρ e the input-output mutual information of an AWGN channel with input constellation and received SNR ρ. Given a channel realization γ and a power allocation scheme pγ satisfying the power constraint P theinstantaneous input-output mutual information of the delay-limited lock-fading channel given in is I B pγ γ = B I p γ. 3 For a fixed transmission rate R communication is in outage when I B pγ γ <R. The outage proaility which is a lower ound to the word error proaility is given y pγpr PrI B pγ γ <R. 4 Also the capacity of an ergodic fading channel with input constellation and power allocation rule pγ is given y C E γ [I pγγ]. With Gaussian inputs we have that I G ρ =log 2 ρ while for coded modulation over uniformly-distriuted fixed discrete signal constellations we have that [ ] E log 2 e ρx x Z 2 Z 2 I ρ =M 2 M x x where the expectation is taken over Z N C 0. In deriving optimal power allocation schemes a useful measure is the first derivative of the mutual information I ρ with respect to the SNR [9] [0]. From [6] we have that d dρ I ρ = log 2 MMSE ρ

3 256 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS VOL. 8 NO. 5 MAY 2009 where MMSE ρ is the imum mean-square error MMSE in estimating an input symol in transmitted over an AWGN channel with SNR ρ. For Gaussian inputs MMSE G ρ = while for a general constellation we have that [9] ρ MMSE ρ = 2 M x x 2 π C x xe y ρx 2 2 x e y ρx 2 dy. The mutual information and its first derivative for itinterleaved coded modulation BICM using the classical noniterative BICM decoder proposed y Zehavi in [7] can also e calculated as shown in [8]. IV. OUTAGE PROBABILITY MINIMIZATION A. Peak Power Constraints For power constraint P the power allocation prolem is [8] p opt γ =arg pγ P pγ 0 pγp R. 5 The solution is given y [9] [] p opt γ = } MMSE MMSE 0 ηγ 6 γ for =...Bwhereη is chosen such that the power constraint is met with equality. The optimal power allocation schemes in 6 involves an inverse MMSE function which may e excessively complex for practical implementation. Moreover the MMSE function provides little insight into the effects of system parameters. We propose suoptimal power allocation schemes similar to waterfilling that tackle oth drawacks leading to or losses in outage performance as compared to the optimal solution. The complexity of the solution in 6 is due to the complex expression of I ρ. Low complexity approximations of the solution can e otained y replacing I ρ in 5 with a simpler expression. For Gaussian input channels with I G ρ =log 2 ρ solving 5 leads to the simple waterfilling scheme [5]. This motivates approximating I ρ of systems with discrete input constellations y I ρ log 2 ρ log 2 } 7 where is a design parameter. Intuitively approximates a threshold SNR eyond which I ρ does not increase significantly with ρ. Details on choosing will e given later in the section. We otain a suoptimal power allocation scheme given y γ = arg max pγ P pγ 0 I p γ. 8 Since I p γ =log 2 for p γ a solution of 8 can e otained y solving γ = arg max pγ P 0 p γ...b log 2 p γ. 9 By applying the Karush-Kuhn-Tucker KKT conditions [9] we have that } = η γ 0 γ for =...Bwhereη takes the largest value such that γ P. The resulting power allocation scheme is similar to water-filling except for the truncation of the allocated power at γ. We refer to this scheme as truncated water-filling. The outage performance otained y the truncated water-filling scheme depends on the choice of. Proposition : Consider transmission over the lock-fading channel defined in with input constellation and the truncated water-filling power allocation scheme γ given in 0. Assume that the power fading gains follow the distriution given in 2. Then for large P the outage proaility γp R is given y P md R γp R. = K where d R B R I. Proof: See the Appendix. Following [0] [20] the truncated water-filling otains the optimal outage diversity when d R = dr B R M i.e. I BR B B R R. 2 M Therefore y letting the truncated water-filling power allocation scheme given in 0 which now ecomes the classical water-filling algorithm for Gaussian inputs provides optimal outage diversity at any transmission rate. For any rate R such that B R M is not an integer we can design a truncated water-filling scheme that otains optimal diversity y choosing R. With the results aove we choose as follows. For a transmission rate R such that B R M is not an integer we perform a simulation to compute the outage proaility otained y truncated water-filling with various R andpickthe that gives the est outage performance. The dashed lines in Figure illustrate the performance of the otained schemes for lock-fading channels with B =4and QPSK input under Rayleigh fading. At all rates of interest the truncated water-filling schemes suffer only or losses in outage performance as compared to the optimal schemes solid lines especially at high SNR. We also oserve a remarkale difference with respect to pure water-filling for Gaussian inputs dotted lines. As a matter of fact pure waterfilling performs worse than uniform power allocation. For rate R such that B R M is an integer 2 requires for optimal outage diversity. Therefore especially at high operating SNR needs to e relatively large to maintain diversity. While I ρ saturates at M I ρ saturates at log 2. Hence a large results in a large discrepancy etween I ρ and I ρ which in turns causes a large outage performance loss of the truncated water-filling scheme. For =5 the su-optimality of the truncated water-filling scheme is illustrated y gap etween the dashed and the

4 NGUYEN et al.: POWER ALLOCATION FOR BLOCK-FADING CHANNELS WITH ARBITRARY INPUT CONSTELLATIONS PoutpPR R =.7 =0 PoutpPR R =0.4 =3 R =0.9 =6 R =.4 =0 0 4 R =0.5 R =.0 R = P db Fig.. Outage performance of various short-term power allocation schemes for QPSK-input lock-fading channels with B = 4 and Rayleigh fading. The solid-lines represent the optimal scheme; the solid lines with represent uniform power allocation; the dashed lines and dashed-dotted lines represent truncated water-filling and its corresponding refinement respectively; the dotted lines represent the classical water-filling scheme P db Fig. 2. Outage performance of various short-term power allocation schemes for QPSK-input lock-fading channels with B = 4 and Rayleigh fading. The solid-lines represent the optimal scheme; the solid lines with represent the uniform power allocation; the dashed lines and dashed-dotted lines correspondingly represent the truncated water-filling and its refinement with =5. solid lines in Figure 2. In the extreme case where pure water-filling we oserve a significant loss in outage performance as illustrated in Figure. To reduce this loss we propose a refined truncated water-filling scheme p ref γ which is ased on a more accurate approximation I ref log ρ = 2 ρ ρ α κ log 2 ρa κ log 2 a} otherwise where κ and a are chosen such that in a db scale κ log 2 ρ a is a tangent to I ρ at a given point ρ 0 ; α is chosen such that κ log 2 αa =log 2 α and is a design parameter. Parameters of I ref γ for some schemes are collected in Tale I. Similar to the truncated water-filling case p ref = γ κη α γ η γ } 3 where η takes the largest value such that p ref γ P. The refined scheme provides additional gain over the truncated water-filling scheme especially when the transmission rate requires a relatively large to maintain the outage diversity. The dashed-dotted lines in Figure 2 show the outage performance of the refined truncated water-filling scheme for lock-fading channels with B =4 and QPSK input under Rayleigh fading. The outage performance of the refined truncated water-filling scheme is close to the outage performance of the optimal case even at rates where the Singleton ound is discontinuous. B. Average Power Constraint Under an average power constraint the power allocation prolem is p opt av =arg E[ pγ ] P av pγ 0 PrI B pγγ R. 4 From [8] [0] the solution p opt av γ of 4 is given y p opt opt γ opt γ s av γ = 5 0 otherwise where γ is the transmission strategy that satisfies the rate constraint with imum power and s is given y opt γ =arg I B γγ R γ 0 γ 6 s =maxs : P opt γs P av } 7 with P γs E γ s [ γ ] eing the average power consumed y using given a power constraint s. The outage proaility is given y [0] p opt av γp avr = p opt γs R. For completeness opt γ is given y [0] = MMSE γ opt MMSE 0 ηγ } 8 with η chosen such that I B opt γ γ =R. The threshold s in 7 is fixed for a given P av and fading statistic f γ γ. Consequently the complexity of the scheme p opt av γ is governed y the complexity of opt γ. Therefore suoptimal alternatives can e employed in 5 to reduce the complexity of p opt av γ. Following the approach in Section IV-A we consider γ =arg B log 2 γ BR 0 γ...b Applying the KKT conditions we have = ηγ γ γ. 9 } 20

5 258 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS VOL. 8 NO. 5 MAY 2009 TABLE I PARAMETERS ρ 0 κa AND α FOR THE REFINED POWER ALLOCATION SCHEME. Modulation Scheme QPSK 8-PSK 6-QAM 64-QAM CM BICM CM BICM CM BICM CM BICM ρ κ a α where η is chosen such that B log 2 γ =BR. The power allocation rule γ does not meet the rate constraints since log 2 γ I γ. By adjusting η we otain tw = ηγ γ } 2 where η is chosen such that I B tw γ γ = R. The truncated water-filling scheme for systems with average power constraints can e otained y employing tw γ instead of opt γ in 5 av γ = tw γ tw γ s tw 0 otherwise where tw γ is given in 2 and s tw satisfies 22 s tw =maxs : P tw γs P av }. 23 The achieved outage proaility is given y γp av R=Pr tw γ >s tw. The following Proposition helps analyzing the performance the power allocation rule av γ. Proposition 2: Consider transmission at rate R over the lock-fading channel given in with power allocation scheme av γ and long-term power constraint P av = P tw γs. Then independent of the fading statistics the outage proaility satisfies Pr tw γ >s= γsr. 24 Proof: Given a γ tw γ γ are detered y an increasing function of η. Therefore if tw γ > s γ wehave γ tw γ which induces I B tw γ γ <R. Similarly I B γ γ <Rinduces tw γ >s. Therefore from Proposition under Nakagami-m fading p tw av γ P tw γsr. = Ks md R. 25 As shown in [0] the large-snr performance of power allocation with average power constraint is detered from the large-snr ehavior of the corresponding power allocation scheme with -power constraints. Using 25 and applying the result from [0] the delay-limited capacity of systems employing av γ is positive if md R > while if md R < the outage diversity with respect to P av is given md y R md R. Optimal outage diversity is then guaranteed if d R =dr. Poutp av PavR R =0.5 =4dB R =.0 =6dB 2 =7.5dB R =.5 =9dB 2 =9dB R =.7 =9dB 2 =9dB P av db Fig. 3. Outage performance of various long-term power allocation schemes for QPSK-input lock-fading channels with B = 4 and Rayleigh fading. The solid-lines represent the optimal scheme; the dashed lines and dasheddotted lines correspondingly represent the long-term truncated water-filling av γ with and its refinement p ref av γ with 2. Similar to Section IV-A we propose the refinement of the truncated water-filling scheme which is otained y employing the structure in 5 with ref γ where ref = κη α η γ γ γ } 26 and η is chosen such that I B ref γ γ =R. Detering η for the power allocation rules in 2 and 26 requires evaluating or taulating I ρ whichmaye computationally complex. This can e avoided y using an approximation Ĩ ρ of I ρ. LetΔR max ρ Ĩ ρ I ρ then a suoptimal γ satisfies the rate constraint when η is chosen such that B Ĩ γ =BR ΔR. Following [2] we propose Ĩ ρ =M e cρc 2 c3.using numerical optimization to imize the mean-squared-error etween I ρ and Ĩ ρ the parameters c c 2 c 3 for various modulation schemes are showed in Tale II. The performance of the suoptimal schemes is illustrated y the dashed lines in Figure 3. As we oserve the performance of the truncated water-filling scheme is very close to that of the optimal scheme; the refined truncated water-filling scheme provides additional gains especially for systems operate at rates close to M.

6 NGUYEN et al.: POWER ALLOCATION FOR BLOCK-FADING CHANNELS WITH ARBITRARY INPUT CONSTELLATIONS 259 TABLE II OPTIMIZEDc c 2 AND c 3 PARAMETERS FOR THE APPROIMATION OF Ĩ ρ. Modulation Scheme QPSK 8-PSK 6-QAM 64-QAM CM BICM CM BICM CM BICM CM BICM c c c ΔR C. Peak-to-Average Power Ratio Constraints For systems with average power P av and -to-average power ratio constraint PAPR the power allocation rule is suject to oth average power constraint P av and power constraint P =PAPR P av the optimal power allocation scheme solves [8] ˆpγ =arg pγ P E[ pγ ] P av pγ 0 PrI B pγ γ <R 27 Following the arguments in [8] the optimal power allocation rule ˆp opt γ is given y ˆp opt opt γ opt γ ŝ γ = 28 0 otherwise where opt γ is given in 8 and ŝ =s P } with s defined as in 7. We oserve that depending on P av and the PAPR which is fixed one of the power constraints is redundant and the outage performance is dependent on the remaining constraint. In particular we have that ˆp opt γp av R = p opt γp R s >P p opt av γp av R s 29 P. Consequently the outage proaility can also e evaluated as ˆp opt γp av R =max p opt γp R p opt av γp avr }. 30 Suoptimal schemes ased on truncated and refined truncated water-filling follow directly from the approach descried aove. For example the truncated water-filling solution ˆ γ is otained from 22 y replacing s tw with ŝ tw =P s tw }.ForlargeP av we have the following. Proposition 3: Consider transmission at rate R over the lock-fading channel given in with power allocation scheme ˆp opt γ or ˆ γ. Assume input constellation of size 2 M. Further assume that the power fading gains γ follow the Nakagami-m distriution given in 2. Then for large P av and any PAPR < the outage proaility ehaves like ˆp opt γp av R. = KP mdr av 3 ˆ γp av R. = K P md R av. 32 Proof: For sufficiently large P we have that [0] p opt γp R. = K P mdr. 33 Let P opt γs e the average power constraint as a function of the threshold s in the allocation scheme p opt av γ in 5. Asymptotically with s [0] d ds P opt γs =. K drs dr. Therefore from L Hôpital s rule we have for any PAPR PAPR P opt γs d lim = lim s s s ds PAPR P opt γs =0. It follows that for any PAPR there exists an s 0 and the corresponding average power constraint P 0 = P opt γs 0 such that s 0 =PAPR P 0 and s>p opt γs PAPR if P opt γs >P 0. Consequently ˆp opt γp av R = p opt γ PAPR P av R for P av >P 0. Thus together with 33 at large P av wehave ˆp opt γp av R. = Pout p opt γ PAPR P av R. = K PAPR mdr Pav mdr. By noting that γp R =. K P md R the proof for the suoptimal scheme ˆ γ follows using the same arguments as aove. The threshold P 0 in the proof is the average power constraint such that the threshold s in 7 satisfies s =PAPR P 0. Equivalently P 0 satisfies opt γ df γ γ =P 0 34 γ: opt γ PAPR P 0 where F γ γ is the joint pdf of γ = γ...γ B. We therefore have that ˆp opt γp av R = p opt γ PAPR P av R for P av >P 0. Thus for asymptotically large P av the outage proaility for systems with a PAPR constraint is detered y the outage proaility of systems with power constraint P =PAPR P av. As a consequence we have that the delay-limited capacity [22] is zero for any finite PAPR. For simplicity we first consider the outage performance of systems with B =under Nakagami-m fading statistic. Let F γ γ e the cumulative distriution function cdf of γ. For P av >P 0 s>papr P av the outage proaility is p opt γ PAPR P av R I = F R γ. PAPR P av For P av <P 0 s<papr P av s in 5 is otained from mi R Γ m mi R = P av Γm s

7 2520 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS VOL. 8 NO. 5 MAY PoutˆpPavR PAPR = 0dB PAPR = 0dB PoutˆpPavR PAPR = 20dB 0 4 PAPR = 0dB PAPR = 0dB 0 5 PAPR = P av db Fig. 4. Outage proaility for systems with PAPR constraints over Nakagamim lock-fading channels B =m =R = 6-QAM inputs. The solid and dashed lines correspondingly represent outage proaility of systems with coded modulation and BICM. 0 5 PAPR = PAPR = 5dB P av db Fig. 6. Outage proaility for systems with and average power constraints using 6-QAM input constellation over Nakagami-m lock-fading channels with B =4m =R =3and -to-average power ratio PAPR. The solid and dashed lines correspondingly represent outage proaility of systems with optimal and truncated water-filling schemes with =9dB. PoutˆpPavR PAPR = 0dB PAPR = 0dB PAPR = PAPR = 5dB P av db Fig. 5. Outage proaility for systems with PAPR constraints over Nakagamim lock-fading channels B =4m =R =3 6-QAM inputs. The solid and dashed lines correspondingly represent outage proaility of systems with coded modulation and BICM. aility can e otained y the outage proaility of systems with only a power constraint P av PAPR. Simulation results for a 6-QAM input Rayleigh fading channel with B =4locks at rate R =3are given in Figure 5. In oth cases the outage proaility at high P av resulting from the optimal power allocation scheme is governed y the power constraints and therefore the optimal outage diversity is given y the Singleton ound. The outage performance of systems with 6-QAM inputs Rayleigh fading channel with B =4R =3 employing the truncated water-filling scheme is illustrated in Figure 6. As pointed out in Section IV-A relatively high 9 db is required to maintain the optimal outage diversity up to the P av of interest. In this case the refined truncated water-filling power allocation scheme as discussed in the previous section can e used to reduce the gap from optimality. For rates where smaller can e used small gaps from the optimal outage proaility are oserved as for systems with and with average power constraints. and the outage proaility is given y p opt av γp avr I = F R γ. s The analytical result for B =is illustrated in Figure 4 for a 6-QAM input Rayleigh fading channel at rate R =.We oserve that as we increase the PAPR constraint the error floor occurs at lower error proaility values and eventually at values elow a target quality-of-service error rate. We also oserve that the loss incurred y BICM is imal. For systems with B> analytical results are not availale in closed form. However from 30 the outage proaility of systems with PAPR constraints can e otained from systems with power constraints and systems with average power constraints separately. Moreover at high P av the outage pro- V. ERGODIC CAPACITY MAIMIZATION We now consider the capacity of the ergodic channel where B is sufficiently large to reveal the statistics of the channel within one codeword. For a given power allocation rule pγ the ergodic capacity of the channel is C = E γ [I pγγ] = I pγγf γ γdγ. 35 γ>0 A. Average Power Constraint For a system with an average power constraint P av the power allocation prolem is p opt γ = arg max E γ[pγ] P av pγ 0 E γ [I pγγ]. 36

8 NGUYEN et al.: POWER ALLOCATION FOR BLOCK-FADING CHANNELS WITH ARBITRARY INPUT CONSTELLATIONS 252 The solution is given y [9] p opt γ = γ MMSE MMSE 0 η } 37 γ where η is chosen such that E γ [p opt γ] = P av. The resulting capacity is η C opt = I MMSE f γ γdγ. 38 γ η MMSE 0 As efore the suoptimal truncated water-filling scheme can e otained y solving γ = arg max E γ [log 2 pγγ]. 39 E γ[pγ] P av 0 pγ γ Using the KKT conditions we have that γ = γ η } 40 γ where η is chosen such that E γ [ γ] = P av. The resulting capacity is C tw η = I ηγ f γγdγ I F γ. η η 4 B. Peak-to-Average Power Constraint For systems with a PAPR constraints the optimal power allocation rule is given y p opt papr γ = arg max E γ[0 pγ] P av pγ P E γ [I pγγ] 42 where P =PAPR P av. Applying the KKT conditions the optimal power allocation is p opt papr γ = P γ MMSE MMSE 0 η }} γ where η is chosen such that E γ [ p opt paprγ ] = P av. The suoptimal power allocation rule ased on the truncated water-filling algorithm is papr γ = arg max Letting αγ = scheme is given y E γ[pγ] P av 0 pγ P γ } P γ papr γ = αγ E γ [log 2 pγγ]. } then a truncated water-filling η } 43 γ [ where η is chosen such that E γ p tw papr γ ] = P av. It can e seen that if η P or η η P 43 is equivalent to 40. Therefore the resulting ergodic capacity is given in 4. Otherwise let a = η P and = P then the resulting ergodic capacity can e written as a Cpapr tw = I ηγ f γ γdγ /η a I P γf γ γdγ F γ I. C P av db Fig. 7. Capacity of ergodic fading channel with m = 6-QAM coded modulation inputs and average power constraint. The dashed line represents capacity of the unfaded AWGN channel and the solid dashed-dotted and dotted lines correspondingly represent capacities with optimal truncated water-filling and uniform power allocation. C P av db Fig. 8. Capacity of ergodic fading channel with m = 6-QAM coded modulation inputs and PAPR = 3dB. The dashed line represents capacity of the unfaded AWGN channel and the solid dashed-dotted and dotted lines correspondingly represent 6-QAM capacities with optimal truncated waterfilling and uniform power allocation. C. Numerical Results Numerical results for the ergodic capacity of Rayleigh fading channels with 6-QAM inputs are presented in Figures Figures 7 and 8 show the performance of the truncated water-filling scheme with average power constraints and PAPR constraints respectively where has een chosen to maximize capacity at each P av. The results show that the performance of the truncated water-filling scheme is close to optimal for oth systems with average power constraints and systems with PAPR constraints. Figure 9 shows the ergodic capacity for various PAPR constraints. Loss in capacity due to PAPR constraints occurs at low and high P av. Still the loss is imal even with relatively small PAPR 4dB.

9 2522 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS VOL. 8 NO. 5 MAY 2009 C P av db Fig. 9. Capacity of ergodic fading channel with m = 6-QAM coded modulation inputs and PAPR constraints. The solid line with crosses represents capacity of the unfaded AWGN channel; the dotted line represents the capacity with uniform power allocation and the solid dashed-dotted and dashed lines correspondingly represent 6-QAM capacities with PAPR = 4 db. VI. CONCLUSIONS We have studied power allocation schemes under average and -to-average power constraints for ergodic and delay-limited lock-fading channels with aritrary input distriutions. We propose suoptimal schemes with low computational and storage capailities while still perforg close to optimal. In the delay-limited lock-fading case we have shown that the suoptimal scheme maintains the optimal outage diversity in most cases. A refined truncated water-filling scheme is proposed to reduce the performance loss when optimal outage diversity cannot e otained y truncated water-filling. For systems with PAPR constraints the optimal and suoptimal solutions can e easily computed from the corresponding solutions with independent and average power constraints. The asymptotic performance for finite PAPR is always detered y the power and the exponent is therefore given y the exponent of systems with power constraints. In the ergodic case the truncated water-filling scheme yields insignificant loss compare to the optimal scheme for the entire SNR range. Also even small PAPR values entail imal loss to the ergodic capacity. Let APPENDI I ρ I ρ> 0 otherwise. Since γ wehavethat 44 I ptw γ I B γ γ I BP γ 45 We further lower ound I B γ γ y the following. Proposition 4: Consider the truncated water-filling scheme given in 0. We have that I ptw γ I P γ 46 for any channel realization γ wherei ρ is given in 44. Proof: According to 44 I P γ is non-zero only P if γ P. Therefore we need to prove that if γ then γ for all realization of γ. Without loss of generality assume that γ... γ B.Ifγ B < P 46 is certainly true. Otherwise there exists a k k B such that γ k < P γ k... γ B. Consider the following two cases. If B γ < BP then from 0 = γ =...B. Otherwise from 0 the power allocation solution is } = η γ =...B 47 γ where η is chosen such that B ptw = BP. Since γ k P wehavefrom0thatptw γ γ k P = k...b. Therefore k = =k kp. 48 then for =...k Now suppose η< γ k γ < γ k γ k = γ k P. Thus k ptw η <kpwhich contradicts to 48. Therefore assumption η< γ k is invalid. We then conclude that η γ k γ = k...b. Therefore from 47 = γ = k...b. Thus in all cases we have = γ if γ P. This concludes the proof of the proposition. Therefore from 45 we have that B Pr I BP <BR γp R B Pr I P γ <BR With similar arguments to the analysis in [20] we have that with Nakagami-m fading B Pr I. BP γ <BR = K l P md R B Pr I P. γ <BR = K u P md R which leads to γp R =. K P md R. REFERENCES [] J. G. Proakis Digital Communications 4th ed. McGraw Hill 200. [2] R. Knopp and G. Caire Power control and eamforg for systems with multiple transmit and receive antennas" IEEE Trans. Wireless Commun. vol. pp Oct

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Hanly and D. N. C. Tse Multiaccess fading channels part II: delay-limited capacities" IEEE Trans. Inform. Theory vol. 44 no. 7 pp Nov Khoa D. Nguyen S 06 received the B.E. in Electrical Engineering at the University of Melourne Victoria Australia in Since 2006 he has een pursuing the Ph.D. degree in electrical engineering at the Institute for Telecommunications Research University of South Australia Australia. He was a summer research scholar at the Australian National University in 2004 and held a visiting appointment at University of Camridge United Kingdom in His research interests are in the areas of information theory and error control codes. He has mainly worked on adaptive techniques and theoretical limits of communications with practical constraints. Alert Guillén i Fàregas S0 M 04was orn in Barcelona Catalunya Spain in 974. He received the Telecommunication Engineering degree and the Electronics Engineering degree from Universitat Politécnica de Catalunya Barcelona Catalunya Spain and the Politecnico di Torino Torino Italy respectively in 999 and the Ph.D. degree in communication systems from Ecole Polytechnique Fédérale de Lausanne EPFL Lausanne Switzerland in From August 998 to March 999 he conducted his Final Research Project at the New Jersey Institute of Technology Newark NJ USA. He was with Telecom Italia Laoratories Italy from Novemer 999 to June 2000 and with the European Space Agency Noordwijk The Netherlands from Septemer 2000 to May 200. During his doctoral studies from 200 to 2004 he was a Research and Teaching Assistant at the Moile Communications Department Institut Eurécom Sophia-Antipolis France. From June 2003 to July 2004 he was a Visiting Scholar at the Communications Theory La at EPFL. From Septemer 2004 to Novemer 2006 he was a Research Fellow at the Institute for Telecommunications Research University of South Australia Mawson Lakes Australia. During June-July 2005 and June 2006 he held a visiting appointment at Ecole Nationale Supérieure des Télécommunications Paris France. Since 2007 he has een a Lecturer in the Department of Engineering University of Camridge Camridge U.K. where he is also a Fellow of Trinity Hall. He has held visiting appointments at Ecole Nationale Supérieure des Télécommunications Paris France Universitat Pompeu Fara Barcelona Spain and the University of South Australia Australia. His specific research interests are in the area of communication theory information theory coding theory digital modulation and signal processing techniques particularly with wireless terrestrial and satellite applications. Dr. Guillén i Fàregas is currently an Editor for IEEE Transactions on Wireless Communications. He received a predoctoral Research Fellowship from the Spanish Ministry of Education to join ESA. He received the Young Authors Award of the 2004 European Signal Processing Conference EUSIPCO 2004 Vienna Austria and the 2004 Nokia Best Doctoral Thesis Award in Moile Internet and 3rd Generation Moile Solutions from the Spanish Institution of Telecommunications Engineers. He is also a memer of the ARC Communications Research Network ACoRN and a Junior Memer of the Isaac Newton Institute for Mathematical Sciences. Lars K. Rasmussen S 92 M 93 SM 0 was orn on March in Copenhagen Denmark. He got his M. Eng. in 989 from the Technical University of Denmark Lyngy Denmark and his Ph.D. degree from Georgia Institute of Technology Atlanta Georgia USA in 993. From 993 to 995 he was a Research Fellow at the Institute for Telecommunications Research ITR University of South Australia Adelaide Australia. From 995 to 998 he was a Senior Memer of Technical Staff with the Centre for Wireless Communications at the National University of Singapore Singapore. From 999 to 2002 he was an Associate Professor at Chalmers University of Technology Göteorg Sweden where he maintained a part-time appointment until From 2002 to 2008 he held a position as Research Professor at ITR University of South Australia where he was the leader of the Communications Signal Processing research group the Convenor of the Australian Research Council ARC Communications Research Network ACoRN and a co-founder of Cohda Wireless Pty Ltd Adelaide Australia. He has held visiting positions at University of Pretoria Pretoria South Africa Southern Poro Communications Sydney Australia and Aalorg University Aalorg Denmark. He now holds a position as Professor in Communications Theory School of Electrical Engineering and the ACCESS Linnaeus Center at the Royal Institute of Technology Stockholm Sweden. He is a Senior Memer of the IEEE a memer of the IEEE Information Theory and Communications Societies and served as Chairman for the Australian Chapter of the IEEE Information Theory Society He is an associate editor for IEEE TRANSACTIONS ON COMMUNICATIONSand was a guest editor for IEEE JOURNAL ON SELECTED AREAS IN COMMU- NICATIONS He was also a memer of the organizing committees for the IEEE 2004 International Symposium on Spread Spectrum Systems and Applications Sydney Australia and the IEEE 2005 International Symposium on Information Theory Adelaide Australia. His research interests include adaptive coding and modulation multiuser communications coding for fading channels and cooperative communications for wireless networks.