Power Allocation for Fading Channels with Peak-to-Average Power Constraints

1 Po wer Allocation for Fa d ing Channels with Peak-to-A v erage Po wer Constraints Khoa D. Nguyen, Albert Guil l ´ en i F ` abreg as and Lars K. Rasmussen Abstract Power allocation with peak-to- av erage po wer ratio constraints is in vestigated fo r transmission over Nakagami- m fading channels with arbitrary input distributions. I n the case of delay-lim ited blo ck-fading channels, we find the solution to the minimum outag e power allocation scheme with peak-to-average power constraints and arb itrary input d istributions, and show that the signal-to -noise ratio expon ent for any finite peak- to-average power ratio is the same as that o f the p eak-power lim ited prob lem, resulting in an err or floor . In the case of the ergodic fully-in terleav ed ch annel, we find th e power allocation rule that yields the maximal information r ate for an arbitrary input distrib ution and show that ca pacities with peak- to-average power ratio constraints, ev en fo r small ratios, ar e very close to cap acities without peak-p ower restrictions. K. D. Nguyen and L. K. Rasmussen are with the Institute for T elecommunications Research, Univ ersity of South Australia, Mawson Lakes Boulev ard, Mawson L akes 5095 , South Australia, Australia, e-mail: dangkhoa.nguy en@postgrads.u nisa.edu.au, lars.ras mussen@unisa.e du.au. A. Guill ´ en i F ` abreg as is with the Dep artment of Enginee ring, Univ ersit y of Cambridg e, Trump ington S treet, Cambridge CB2 1PZ, UK, e-mail: guill en@ieee.org. This work was submitted in part at 2008 IEEE International Symposium on Information Theo r y , T o ronto, Canada, July 2008. This work has been supported by the Australian Research Council under ARC grants RN0459498, DP0558861 and DP088116 0. Nov ember 20, 2021 DRAFT 2 I . I N T R O D U C T I O N The ultimate goal of a wireless commu nication sy stem is to reliably transpo rt high data rates over channels with tim e-v arying transfer characteristics; commo nly t ermed fading channels [1], [2]. When channel state information (CSI), nam ely knowledge o f the chann el realizations, is not readily av ailable to t he transmit ter , error control coding and automatic-repeat-request (ARQ) techniques have bee n used e xt ensiv ely to compensate for the f adi ng characteristics of the channels [3]. In some systems, CSI can be made av ailable at the transmitter , either via a dedicated feedback link [4], or using channel reciprocity i n systems em ploying ti me-division du plex (TDD) [5]. In this case, power and rate can be adapted according to t he channel realization to further improve the rate/reliabil ity p erformance of the system. Different adaptation techniques can be employed depending on the syst em requirements and the nature of the wireless channel [2]. In th is paper , we consider power all ocation t echniques th at min imize th e word error rate over slowly-v arying fading channels, and maxi mize the ergodic capacity over fast fading channels [2]. Firstly , we consid er sy stems where codew ords are transm itted over channels with B degrees of freedom, where B is finite. Examples for such scenarios are transmi ssion ov er slowly-v arying channel, o r transmission using orthogonal d ivision mu ltiplexing (OFDM) techniques ove r fre- quency selective channels. The channel i s con veniently modeled as a blo ck-fading channel [6], [7], wh ere each codeword is transm itted over B corresponding flat fading blocks. In this case, the maxim al achiev able rate is a random variable, dependent on the channel realization. For most fading statist ics, the channel capacity is zero since there is a non-zero probability that any positive rate is not suppo rted by the channel. A relev ant performance measure in this case is the information outage probability [7], which is the probability that commu nication at a target rate R is not supp orted by the channel. The ou tage prob ability is also a lower bound of t he word error probability for comm unicating with rate R over the channel [8]. In this case, p owe r all ocation techniques aim at minimizin g the outage p robability gi ven a rate R . The optimal power allocation problem h as been in vestigated in [8] for channels with Gaussian input s, and in [9], [10], [11] for Nov ember 20, 2021 DRAFT 3 channels wit h arbitrary input constellation s. The works in [8], [10] consider systems with peak (per -codeword) power constraints and average p owe r constraints, and sho w that systems with a verage power const raints perform s ignificantly b etter than systems with peak po wer constraints. Howe ver , s ystems with aver age power constraints employ very large (possib ly infinit e) peak power , whi ch i s not feasibl e in practice. T o t his end, th e opt imal power allocation strategy for Gaussian input channels with both peak and a verage power cons traints is also deriv ed in [8]. For transm ission over a fast-v arying fading channel, the fading statistics are revealed within each codeword, and the channel is er godic, i.e. it has infinite degrees of freedom ( B → ∞ ). In this case, adaptive techniqu es aim at maxi mizing the ergodic channel capacity , which is the maximum data rate that can be transm itted ov er the channel with vanishing error probability [12]. Optimal power allocation schemes, such as water-fi lling for channels with Gauss ian input s [12], [8] and mercury/water- filling for channels w ith an arbitrary input [9], have been developed for systems wi th av erage power const raints. The work in [13] derives the optimal power allocation strategy for Gaussi an input channels with both peak and average power constraint, which results in a va riation to the classical water -filling algorith m [12]. In this paper , we consider power allocation strategies for arbit rary i nput channels with peak- to-av erage power ratio (P APR) constraint s. W e derive the opt imal power all ocation scheme that minimizes outage probability for t ransmission with arbitrary inputs ov er a block-fading chann el. The op timal power allo cation strategy that maximizes the ergodic capacity for arbitrary inpu t channels is also deriv ed. In both cases, t he optim al power allo cation strategies rely on the first deriv ative of the inpu t-output mutual inform ation, which may be computati onally prohibitive for implementati on in specific l ow-c ost s ystems. W e therefore study a subopti mal power allocation scheme, which signi ficantly reduces the com putational and storage requirements, while incurring minimal performance loss compare to the optim al scheme. The remainder of t he paper is organized as foll ows. Sections II and III describe the system model and the information theoretic frame work of the work. Section IV discusses power allocation Nov ember 20, 2021 DRAFT 4 algorithms for minim izing t he o utage probabili ty of delay-li mited block-fading channels , while algorithms for m aximizing the ergodic capacity is given in Section V. Concluding remarks are giv en in Section VI. I I . S Y S T E M M O D E L Consider transmission over a channel cons isting of B b locks of L channel u ses, i n wh ich, block b, b = 1 , . . . , B , undergoes an independent fading gain h b , corresponding to a power fading gain γ b , | h b | 2 . As sume that h = ( h 1 , . . . , h B ) and γ = ( γ 1 , . . . , γ B ) are av ailable at the recei ver and the transm itter , respectively . Supp ose the transmit power is allocated following the rule p ( γ ) = ( p 1 ( γ ) , . . . , p B ( γ )) . Then the correspond ing comp lex base-band equ iv alent is given by y b = p p b ( γ ) h b x b + z b , b = 1 , . . . , B , (1) where y b ∈ C L , x b ∈ X L , with X ⊂ C being the s ignal const ellation set, are t he receiv ed and transmitted si gnals in bl ock b , respectively , and z b ∈ C L is the addit iv e whit e Gauss ian noise (A WGN) vector with independent ly identically distributed circularly symmetric Gaussian entries ∼ N C (0 , 1) . Assume that the signal constellation X of s ize 2 M satisfies P x ∈X | x | 2 = 2 M , then the ins tantaneous received signal-to-noise ratio (SNR) at block b is g iv en by p b ( γ ) γ b . W e consi der systems with the following power constraints : P eak p o w er : h p ( γ ) i , 1 B B X b =1 p b ( γ ) ≤ P peak , Av erage p o w er : E [ h p ( γ ) i ] ≤ P a v . For the fully-interleaved er godic case, t he channel mod el can be obt ained from (1) b y letting B → ∞ and L = 1 . Due to ergodicity , power allocation for block b is only dependent o n γ b . For simplicity of no tation, denote p ( γ ) as the transmit power corresponding to the power fading Nov ember 20, 2021 DRAFT 5 gain γ . The foll owing power constraints are considered: P eak p o w er : p ( γ ) ≤ P peak , Av erage p o w er : E γ [ p ( γ )] ≤ P a v . Po wer allocation schemes for system s with peak po wer constraints and average power con - straints h a ve been studied in [8], [10] for the delay-limited channel and in [14], [13] for the er g odic channel 1 . Po wer all ocation wi th avera ge po wer constraints offers significant performance adva ntage but requires large peak powers [10 ], [8], which may proh ibit application in practical systems. In t his work, we study the performance of syst ems with peak power constraints in addi- tion to a verage power constraints [8], [13]. In particular , we consider systems wit h a con strained peak-to-averag e power ratio P APR , P p eak P a v ≥ 1 . W e consi der block-fading chann els where the fading gain h b has Nakagam i- m distributed magnitude and uni formly di stributed phase 2 . The probabil ity dens ity functi on (pdf) of | h b | of the fading gain is f | h b | ( ξ ) = 2 m m ξ 2 m − 1 Γ( m ) e − mξ 2 , b = 1 , . . . , B , where Γ( a ) i s the Gamm a function , Γ( a ) = R ∞ 0 t a − 1 e − t dt . The pdf of t he power fading gain is then giv en by f γ ( γ ) =    m m γ m − 1 Γ( m ) e − mγ , γ ≥ 0 0 , otherwise . (2) The Nakagam i- m distribution represents a large class of practical fading statisti cs. In particular , we can recover the Rayleigh fading b y setting m = 1 and approximate th e Ricean fading wit h parameter K by setting m = ( K +1) 2 2 K +1 [1]. 1 In t he literature, peak power constraints have al so been referred to as short-term power constraints, and average po wer constraints as lon g-term power constraints. 2 W e assu me that the phase wil l be perfectly compen sated du e t o the pe rfect CSI at the receiv er . Nov ember 20, 2021 DRAFT 6 I I I . O U T A G E P RO BA B I L I T Y A N D E R G O D I C C A P A C I T Y Let I X ( ρ ) be the input-output mutual information of an A WGN channel with input constellation X and recei ved SNR ρ . Given a channel realization γ and a power allocation scheme p ( γ ) satisfying th e power constraint P , the in stantaneous i nput-output mut ual informati on of the delay- limited block-fading channel given in (1) is I B ( p ( γ ) , γ ) = 1 B B X b =1 I X ( p b γ b ) . (3) For a fixed transm ission rate R , communicati on is in outage when I B ( p ( γ ) , γ ) < R . The out age probability , which is a lower bound to the word error probability , is given by P out ( p ( γ ) , P , R ) , Pr ( I B ( p ( γ ) , γ ) < R ) . (4) Besides, the capac ity of an er g odic fading channel with input constellation X and power allocation rule p ( γ ) i s giv en by C , E γ [ I X ( p ( γ ) γ )] . (5) The mut ual information I X ( ρ ) in (3) and (5) is defined as follows. W ith Gaussian input s, we hav e that I X G ( ρ ) = log 2 (1 + ρ ) , while for coded mod ulation over uniformly-distributed fixed discrete signal constellations 3 , we hav e t hat I X ( ρ ) = M − 1 2 M X x ∈X E Z " log 2 X x ′ ∈X e −| √ ρ ( x − x ′ )+ Z | 2 + | Z | 2 !# , where Z ∼ N C (0 , 1) . W e also consider syst ems with bit -interleav ed coded modulati on (BICM) using the classical non -iterativ e BICM decoder propos ed by Z eha v i in [15]. The mu tual infor- mation for a giv en labelling rule can be expressed as [16] I BICM X ( ρ ) = M − 1 2 M 1 X c =0 M X j =1 X x ∈X j c E Z " log 2 P x ′ ∈X e −| √ ρ ( x − x ′ )+ Z 2 | P x ′ ∈X j c e −| √ ρ ( x − x ′ )+ Z 2 | # , (6) 3 Although the restriction to uniform constellations is very relev ant in practice and mathematically con venient, the main results of this paper only depend on t he underlying probability distribution on X through the mutual information and its first deri vati ve. Therefore, the generalization is straightforward. Nov ember 20, 2021 DRAFT 7 where th e sets X j c contain all signal points where th e j th position i n the corresponding binary signal-point l abelling is c . In deriving optim al power allocation schemes, a useful m easure is t he first deriv ative of th e mutual information I X ( ρ ) with respect to the SNR [9], [10]. From [17] we have that, d dρ I X ( ρ ) = 1 log 2 MMSE X ( ρ ) , where MMSE X ( ρ ) is the m inimum mean-square error (MMSE) in esti mating an input symbol i n X transmitted over an A WGN channel with SNR ρ . For Gaus sian inputs, MMSE X G ( ρ ) = 1 1+ ρ , while for a general constellation X , we hav e t hat [9] MMSE X ( ρ ) = 1 2 M X x ∈X | x | 2 − 1 π Z C    P x ∈X xe −| y − √ ρx | 2    2 P x ∈X e −| y − √ ρx | 2 dy . For sys tems with BICM, the first deriv ativ e of the mutual in formation with respect to SNR is giv en by [18] 4 MMSE BICM X ( ρ ) , d dρ I BICM X ( ρ ) = M X j =1 1 2 1 X c =0  MMSE X ( ρ ) − MMSE X j c ( ρ )  . In th e remainder of t he paper , we perform analysis for the coded modul ation case. Results for the BICM case can be obtained by simply repl acing I X ( ρ ) , MMSE X ( ρ ) by I BICM X ( ρ ) and MMSE BICM X ( ρ ) , respectiv el y . I V . O U T A G E P RO BA B I L I T Y M I N I M I Z A T I O N A. P eak and A verag e P ower Constraints In this section, we re view kno wn results on peak-po wer and a verage-power constrained systems, respectiv ely , over delay-limited channels relev ant to our main results. A detailed t reatment of 4 W i th some ab use of notation, we use MMSE BICM X ( ρ ) to denote the first deri vativ e with respect to ρ of the mutual information. Ho we ver , MMSE BICM X ( ρ ) i s not the minimum mean-square error in estimating the channel input from its output, since the noise is not Gaussian due to the demod ulation pro cess. Nov ember 20, 2021 DRAFT 8 optimal and subopt imal power allo cation schemes for systems with peak-powe r and ave rage- power constraints, respecti vely , over delay-lim ited bl ock-fading channels i s g iv en in [10]. 1) P eak P ower Constraint: For systems with peak powe r cons traint P peak , t he op timal p ower allocation s cheme is the solutio n of the following problem [8]            Minimize P out ( p ( γ ) , P peak , R ) Sub ject to h p ( γ ) i ≤ P peak p b ≥ 0 , b = 1 , . . . , B (7) The s olution is given by [9], [10] p peak b ( γ ) = 1 γ b MMSE − 1 X  min  MMSE X (0) , η γ b  , (8) for b = 1 , . . . , B where η i s chosen such that the peak power constraint is met with equali ty . As shown in [10], an alternativ e optimal p owe r allocation rule for p eak power cons traint is giv en by p peak ( γ ) =      ℘ ( γ ) , if h ℘ ( γ ) i ≤ P peak 0 , otherwise , (9) where ℘ ( γ ) is the solution of the problem            Minimize h ℘ ( γ ) i Sub ject to I B ( ℘ ( γ ) , γ ) ≥ R ℘ b ≥ 0 , b = 1 , . . . , B . (10) From [10 ], ℘ ( γ ) is given by ℘ b ( γ ) = 1 γ b MMSE − 1 X  min  MMSE X (0) , 1 η γ b  , b = 1 , . . . , B (11) where η is now chos en s uch that the rate constraint is met, 1 B B X b =1 I X  MMSE − 1 X  min  MMSE X (0) , 1 η γ b  = R. Nov ember 20, 2021 DRAFT 9 The power allocation scheme g iv en in (8) i s less compl ex than the one given in (9) for systems with peak power constraint s. H owe ver , the two schemes are equiva lent in terms of outage probability , and the latter is useful for the analys is of syst ems with av erage power or P APR constraints. In t he foll owing, we onl y con sider p peak ( γ ) giv en in (9) for system s with peak power constraints. The ev aluatio n of p peak ( γ ) in (9) in volves comput ing the in verse of the function MMSE X ( ρ ) , which may be computationally prohibitive for specific practical systems. F o llowing the analysis in [10], w e obt ain a subopt imal truncated water -filling power allocati on rule p peak t w ( γ ) by replacing ℘ in (9) with ℘ t w giv en by ℘ t w b = min  β γ b ,  η − 1 γ b  +  , b = 1 , . . . , B , (12) where β is a predefined design parameter 5 and η is chosen such t hat the rate requirement is satisfied I B  p peak t w ( γ ) , γ  = R. 2) A vera ge P ower Constraint : Un der an a verage po wer constraint, the optimal po w er allocation scheme solves      Minimize Pr( I B ( p ( γ ) , γ ) < R ) Sub ject to E [ h p ( γ ) i ] ≤ P a v . (13) From [10 ], the soluti on p a v ( γ ) of (13) is given by p a v ( γ ) =      ℘ ( γ ) , h ℘ ( γ ) i ≤ s 0 , otherwise , (14) 5 The optimal v alue of β is dependent o n the transmission rate and the tar get ou tage prob abili ty . Large v alues of β guarantee the optimal outage di versity over a larg er range of transmission rate, while too large (an d too small) value s of β decrease the achie vable coding gain. See [10] for guidelines on how to find β . Nov ember 20, 2021 DRAFT 10 where ℘ ( γ ) is given in (11) and s is such that (noting that E [ h p a v ( γ ) i ] is a fun ction of s )      s = ∞ , if lim s →∞ E [ h p a v ( γ ) i ] ≤ P a v P a v = E [ h p a v ( γ ) i ] , otherwise . (15) The threshold s is a function of ℘ ( γ ) , P a v and the f adin g statistics f γ ( γ ) ; thus s is fixed and ca n be predetermined. Consequently , the complexity of th e scheme p a v ( γ ) is gove rned by the com plexity of ℘ ( γ ) . Therefore, sub optimal alternativ es of ℘ ( γ ) can be u sed t o reduce the com plexity of p a v ( γ ) . The truncated water -filli ng scheme for system s with average power constraint s p a v t w ( γ ) [10] can be obtained by employing ℘ t w ( γ ) in st ead of ℘ ( γ ) , i.e. p a v t w ( γ ) =      ℘ t w ( γ ) , h ℘ t w ( γ ) i ≤ s t w 0 , otherwise , (16) where ℘ t w ( γ ) is given in (12) and s t w satisfies      s t w = ∞ , if lim s tw →∞ E [ h p a v t w ( γ ) i ] ≤ P a v P a v = E [ h p a v t w ( γ ) i ] , otherwise . (17) B. P eak-to-A verag e P ower Ratio Constraints For systems with av erage p owe r P a v and peak-to-av erage p ower ratio P APR , the opti mal power allocation s cheme solves the foll owing problem [8],            Minimize Pr ( I B ( p ( γ ) , γ ) < R ) Sub ject to h p ( γ ) i ≤ P peak = P APR · P a v E [ h p ( γ ) i ] ≤ P a v . (18) Follo wi ng the arguments in [8], th e optimal powe r all ocation rule p ⋆ ( γ ) is as follows. Nov ember 20, 2021 DRAFT 11 Pr oposit ion 1: A solut ion to problem (18) i s g iv en by p ⋆ ( γ ) =      ℘ ( γ ) , h ℘ ( γ ) i ≤ ˆ s 0 , otherwise , (19) where ℘ ( γ ) is given in (11) and ˆ s = min { s, P peak } with s defined as in (15). Pr oof: If P a v and P peak are such that s ≤ P peak , we have ˆ s = s . Therefore, p ⋆ ( γ ) coincides with p a v ( γ ) . Furthermore, p ⋆ ( γ ) satisfies the peak power constraint since h p ⋆ ( γ ) i ≤ s ≤ P peak . Consequently , p ⋆ ( γ ) is a soluti on of (18) si nce the peak powe r constraint i s redundant. If P a v and P peak are such that s > P peak , we ha ve ˆ s = P peak < s . Therefore, p ⋆ ( γ ) coincides with p peak ( γ ) . Now , noting that E [ h p ⋆ ( γ ) i ] is an increasing function of ˆ s , we ha ve t hat E [ h p ⋆ ( γ ) i ] < E [ h p a v ( γ ) i ] ≤ P a v . Consequently , p ⋆ ( γ ) is a so lution of (18) since the av erage power constraint is redundant. Thus, in all cases, p ⋆ ( γ ) is a so lution of (18). Remark 1: From the proof, we obs erve that, depending on P a v and the P APR (which is fix ed), one of the power con straints is redundant and t he outage performance i s dependent on t he remaining cons traint. In particular we have that P out ( p ⋆ ( γ ) , P a v , R ) =      P out  p peak ( γ ) , P peak , R  , s > P peak P out ( p a v ( γ ) , P a v , R ) , s ≤ P peak . (20) Consequently , the outage probabilit y can also be ev aluated as P out ( p ⋆ ( γ ) , P a v , R ) = ma x  P out  p peak ( γ ) , P peak , R  , P out ( p a v ( γ ) , P a v , R )  = max  P out  p peak ( γ ) , P APR · P a v , R  , P out ( p a v ( γ ) , P a v , R )  . (21) The abov e expression clearly highlights that in order to c ompute the outage probability with P APR constraints, it is sufficient to transl ate the curve corresponding to the peak power constraint by P APR dB and then find the maximum between the translated curve and the curve corresponding to the av erage power constraint . Nov ember 20, 2021 DRAFT 12 W ith similar arguments to the previous section, the su boptimal truncated water -filling scheme p ⋆ t w ( γ ) for system s with P APR cons traints is giv en by p ⋆ t w ( γ ) =      ℘ t w ( γ ) , h ℘ t w ( γ ) i ≤ ˆ s t w 0 , otherwise , (22) with ˆ s t w = min { s t w , P peak } where s t w is given in (17). Th e ou tage probability of systems with P APR constraints is also given by P out ( p ⋆ t w ( γ ) , P a v , R ) = ma x  P out  p peak t w ( γ ) , P APR · P a v , R  , P out ( p a v t w ( γ ) , P a v , R )  . 1) Asymptotic Anal ysis: In this section we s tudy the asy mptotic behavior of the outage prob- ability under P APR constraint s. In particular , we study the SNR exponents, i.e., the asymp totic slope of the outage probability for large SNR. For lar g e P a v , we hav e the foll owing result. Pr oposit ion 2: Consider transmissi on at rate R o ver the b lock-fading channel given in (1) with power allocation scheme p ⋆ ( γ ) (or p ⋆ t w ( γ ) ). Ass ume input constellatio n X of size 2 M . Further assume that the power fading gains γ follow the N akagami- m di stribution given i n (2). Th en, for large P a v and any P APR < ∞ , the outage probabili ty beha ves like P out ( p ⋆ ( γ ) , P a v , R ) . = K P − md ( R ) a v (23) P out ( p ⋆ t w ( γ ) , P a v , R ) . = K β P − md β ( R ) a v , (24) where d ( R ) is the Singleton bound [19], [20], [21], [22] d ( R ) = 1 +  B  1 − R M  , and d β ( R ) is given by d β ( R ) = 1 +  B  1 − R I X ( β )  . (25) Pr oof: For sufficiently lar ge P peak we hav e that [10] P out  p peak ( γ ) , P peak , R  . = K peak P − md ( R ) peak . (26) Nov ember 20, 2021 DRAFT 13 Let P ( s ) be the av erage power constraint as a function of the thresh old s in the allocatio n scheme p a v ( γ ) in (14). Asymp totically with s , we have [10] d ds P ( s ) . = K peak d ( R ) s − d ( R ) . From L ’H ˆ opital’ s rule, we h a ve for any P APR lim s →∞ P APR · P ( s ) s = lim s →∞ d ds P APR · P ( s ) = lim s →∞ P APR · K d ( R ) s − d ( R ) = 0 . It follows that for any P APR , th ere exists an s 0 and the corresponding av erage power constraint P 0 = P ( s 0 ) such that s 0 = P APR · P 0 and s > P ( s ) · P APR if P ( s ) > P 0 . Consequently , P out ( p ⋆ ( γ ) , P a v , R ) = P out  p peak ( γ ) , P APR · P a v , R  for P a v > P 0 . Thus, together with (26), at lar ge P a v , we hav e P out ( p ⋆ ( γ ) , P a v , R ) . = P out  p peak ( γ ) , P APR · P a v , R  . = K peak P APR − md ( R ) P − md ( R ) a v (27) as s tated in (23). By noting that [10] P out  p peak t w ( γ ) , P peak , R  . = K peak β P − md β ( R ) peak , the proof for the suboptimal scheme p ⋆ t w ( γ ) follows using the sam e arguments as above. The threshold P 0 in the proof is the avera ge power constraint such that the threshold s in (15) satisfies s = P APR · P 0 . Equiv alentl y , P 0 satisfies Z γ : h ℘ ( γ ) i≤ P APR · P 0 h ℘ ( γ ) i dF γ ( γ ) = P 0 , (28) where F γ ( γ ) is the join t pdf of γ = ( γ 1 , . . . , γ B ) . W e therefore hav e that P out ( p ⋆ ( γ ) , P a v , R ) = P out  p peak ( γ ) , P APR · P a v , R  for P a v > P 0 . Therefore, for asympt otically l ar g e P a v , the outage probabi lity for s ystems with a P APR constraint is determined by the outage probabi lity o f systems with peak power cons traint P peak = P APR · P a v . As a consequence of th e above analysis, we have that the delay-lim ited capacity [23 ] is zero for any finite P APR . This is ill ustrated by examples in the next section. Nov ember 20, 2021 DRAFT 14 2) Numerical Results: F or simplicity , we first consider the outage performance of sy stems with B = 1 under Nakagami- m fading st atistic. Then, the outage probability can be numerically e va luated as foll ows. Let γ be th e power fading gain, then ℘ ( γ ) = I − 1 X ( R ) γ and (28) reduces to Z ∞ I − 1 X ( R ) P 0 · P APR I − 1 X ( R ) γ m m γ m − 1 Γ( m ) e − mγ dγ = P 0 m m Γ( m ) a Z ∞ a P APR γ m − 2 e − mγ dγ = 1 m Γ( m ) a Γ  m − 1 , m a P APR  = 1 , (29) where a , I − 1 X ( R ) P 0 and Γ( n, ξ ) is the u pper incompl ete Gamma function [24] defined as Γ( n, ξ ) , R ∞ ξ t n − 1 e − t dt . The threshol d P 0 can be obtained by solving (29) for a . For P a v > P 0 ( s > P APR · P a v ) the outage probability is giv en by P out  p peak ( γ ) , P APR · P a v , R  = Pr  γ < I − 1 X ( R ) P APR · P a v  = F γ  I − 1 X ( R ) P APR · P a v  . For P a v < P 0 ( s < P APR · P a v ) , s in (14) is obtained by sol ving mI − 1 X ( R ) Γ( m ) Γ  m − 1 , mI − 1 X ( R ) s  = P a v , and t he outage probability is given by P out ( p a v ( γ ) , P a v , R ) = Pr  γ < I − 1 X ( R ) s  = F γ  I − 1 X ( R ) s  . The analytical result for B = 1 is illustrated in Figure 1 for a 16-QAM in put, Rayleigh fading channel at rate R = 1 . W e observe that as we increase the P APR constraint, the error floor occurs at lo wer error probabili ty v alues, and eve ntually , at va lues belo w a tar g et quality-of-service error rate. W e also observe that the loss incurred by BICM is minim al. For systems with B > 1 , analytical results are not a vailable in closed form. Ho w e ver , from (21), the outage probabilit y of systems with P APR constraints can be obtained b y considerin g system s with peak p owe r constraints and systems with a verage power constraints separately . Moreove r , at Nov ember 20, 2021 DRAFT 15 high P a v , t he ou tage p robability can be obtained by t he ou tage prob ability of systems with on ly a peak power const raint P a v · P APR . Simul ation result s for a 16-QAM input , Rayleigh fading channel with B = 4 b locks at rate R = 3 are g iv en in Figure 2. In both cases ( B = 1 and B = 4 ), the outage probability at high P a v resulting from the optimal power allocation scheme is governed by the peak power constraints, and t herefore, the opt imal outage div ersit y is given by the Singleton bound. The outage performance of sy stems wi th 16-QAM inputs, Rayleigh fading channel with B = 4 , R = 3 , employing th e truncated water -filli ng scheme is illus trated in Figure 3. It foll ows from (24) that β = ∞ is required to main tain th e optim al diversity . Therefore, we need to choose β relativ ely high ( β = 19 dB) to keep t he outage performance close to optimal at outage probability 10 − 5 . The suboptim al outage diversity d β ( R ) = d ( R ) − 1 appears at lo wer outage probability . F or rates R such that B  1 − R M  is not an integer , optimal div ersi ty can be maintained with finite β , thus smaller values of β can be chosen, wh ich results in smaller performance gap b etween the truncated water -filling and the opt imal schem e. V . E R G O D I C C A P AC I T Y M A X I M I Z A T I O N W e now consider the capacity of the ergodic channel, where the number of block B i s suffi ciently lar g e to rev eal the statistics of the channel withi n one codeword. The channel model follows from (1) by letti ng B → ∞ and L = 1 . For a giv en po wer all ocation rule p ( γ ) , the er g odic capacity of the channel is C = E γ [ I X ( p ( γ ) γ )] = Z γ > 0 I X ( p ( γ ) γ ) f γ ( γ ) dγ . (30) Similarly to the pre viou s secti on, we first pre view the channel capacity under a verage power constraints before p resenting the results on the channel capacity under P APR const raints. Nov ember 20, 2021 DRAFT 16 A. A verage P o wer Constraint For a system with an ave rage power const raint P a v , the optimal power all ocation rule is given by p opt ( γ ) = arg max E γ [ p ( γ )] ≤ P a v E γ [ I X ( p ( γ ) γ )] . (31) The s olution is given by [9] p opt ( γ ) = 1 γ MMSE − 1 X  min  MMSE X (0) , η γ  , (32) where η is chosen such t hat E γ [ p opt ( γ )] = P a v . The resulting capacity is C opt = Z ∞ η MMSE X (0) I X  MMSE − 1 X  η γ  f γ ( γ ) dγ . (33) A low-complexity suboptimal solution to probl em (31) of p opt ( γ ) can be derived by approxi- mating I X ( ρ ) with the following bound I u X ( ρ ) = min { log 2 (1 + ρ ) , log 2 (1 + β ) } , (34) where β is a predefined p arameter to be optimized d epending on P a v . Th e subo ptimal power allocation s cheme is giv en by p t w ( γ ) = arg max E γ [ p ( γ )] ≤ P a v E γ [ I u X ( p ( γ ) γ )] . (35) Since I u X ( p ( γ ) γ ) = log 2 (1 + β ) if p ( γ ) ≥ β γ , the solutio n of (35) satisfies p ( γ ) ≤ β γ . Therefore, (35) i s equiv alent to p t w ( γ ) = arg max E γ [ p ( γ )] ≤ P a v p ( γ ) ≤ β γ E γ [log 2 (1 + p ( γ ) γ )] . (36) Using t he Karush-Kurhn-T ucker (KKT) conditi ons, we have that p t w ( γ ) = min  β γ ,  η − 1 γ  +  , (37) where η is chosen such t hat E γ [ p t w ( γ )] = P a v . The resulting capacity is C t w = Z β +1 η 1 η I X ( η γ − 1 ) f γ ( γ ) dγ + I X ( β )  1 − F γ  β + 1 η  . (38) Nov ember 20, 2021 DRAFT 17 B. P eak-to-A verag e P ower Constraint For sys tems wit h a P APR constraints, the optimal power allocation rule is given by p opt papr ( γ ) = arg max E γ [ p ( γ ) ] ≤ P a v p ( γ ) ≤ P p eak E γ [ I X ( p ( γ ) γ )] , (39) where P peak = P APR · P a v . A pplying the KKT conditi ons, the opt imal power allocation schem e is giv en by p opt papr ( γ ) = min  P peak , 1 γ MMSE − 1 X  min  MMSE X (0) , η γ  , (40) where η is chosen such t hat E γ  p opt papr ( γ )  = P a v . Similarly to the previous section, we derive a subopt imal power allocation rule based on t he truncated water -filling algo rithm by sol ving p t w papr ( γ ) = arg max E γ [ p ( γ )] ≤ P a v p ( γ ) ≤ min { P p eak , β γ } E γ [log 2 (1 + p ( γ ) γ )] . (41) Let α ( γ ) = min n P peak , β γ o , t hen a truncated water filling suboptim al of p opt papr is given by p t w papr ( γ ) = min  α ( γ ) ,  η − 1 γ  +  , (42) where η is chosen such that E γ  p t w papr ( γ )  = P a v . It can be seen that if η ≤ P peak or β +1 η ≤ 1 η − P p eak , (42) is equivalent to (37). Therefore, the resultin g er godic capacity is give n in (33 ). Otherwise, let a = 1 η − P p eak and b = β P p eak , then the resulting er godic capacity can be written as C t w papr = Z a 1 /η I X ( η γ − 1 ) f γ ( γ ) dγ + Z b a I X ( P peak γ ) f γ ( γ ) dγ + (1 − F γ ( b )) I X ( β ) . (43) C. Numerical Results The capacities presented in the pre vious s ections can easily be calculated using Gaussian quadrature integrations. Nu merical results for the ergodic capacity of Rayleigh fading chann els with 16-QAM inputs are presented in Figures 4, 5, 6. Fig ures 4 and 5 sho w the performance o f the Nov ember 20, 2021 DRAFT 18 truncated w ater-filling scheme with a verage po wer con straints a nd P APR c onstraints, respecti vely , where β has been chosen to maxim ize capacity at each P a v . T he result s show that the t runcated water -filling s cheme are very close t o optimal for bot h syst ems wit h av erage p owe r con straints and syst ems with P APR constraints. Th e trun cated water -filli ng scheme is therefore a p otential candidate for practical sys tem im plementation due to th e very lo w computational and storage requirements compared to the optimal s cheme. Figure 6 shows the ergodic capacity for various P APR const raints. W e observe that minim al loss in capacity i s incurred, ev en w ith relatively small P APR. V I . C O N C L U S I O N S W e hav e stu died po wer allocation schemes under P APR constraints for ergodic and delay- limited block-fading channels with arbitrary input distributions. In each case, we have comput ed the optimal sol ution and proposed a suboptim al scheme that requires lower com putational and storage capabiliti es wh ile performing close to optimal. In the delay-limit ed bl ock-fading case, we hav e shown that the optim al and subopti mal sol utions can be easi ly comput ed from the corresponding solution s w ith independent peak and av erage power constraint s. W e h a ve studied the SNR exponents, and shown that the asymptoti c performance for finite P APR is al ways determined by the peak power , and t he exponent i s therefore given by the exponent of systems with peak powe r constraints. In the ergodic case, we h a ve seen that even s mall P APR v alues entail minimal capacity loss. R E F E R E N C E S [1] J. G. P roakis, Digital Commun i cations , 4th ed. McGraw Hill, 2001. [2] D. Tse and P . V iswanath, Fundamentals of W ireless Communications . Cambridge U ni versity Press, 2005. [3] E. Biglieri, C oding for W ireless Channels . Springer , 20 06. [4] V . K. N. Lau, Y . Liu, and T .-A. Chen, “Capacity of memoryless channels and block-fad ing channels with designable cardinality-constrained channel st ate feedback, ” IE EE T rans. Inf. T heory , vol. 50, no. 9, pp. 2038–2049, Sept. 2004. Nov ember 20, 2021 DRAFT 19 [5] R. Knopp and G. Caire, “Po wer control and beamforming for systems with multiple t ransmit and receiv e antennas, ” IEEE T rans. W ir eless Comm. , v ol. 1, pp. 638–648, Oct. 2002. [6] L. H. Ozaro w , S. Shamai, and A. D. W yner , “Information theoretic considerations for cellular mob i le rad io, ” IEEE T ran s. V eh. T ech. , vol. 43, no. 2, pp. 3 59–378 , May 1994. [7] E. Biglieri, J. Proakis, and S. Shamai, “Fading channe ls: Informatic-theoretic and commu nications aspects, ” IEEE T rans. Inf. Theory , vol. 44, no. 6, pp. 26 19–2692, Oct. 1998. [8] G. Caire, G. T aricco, and E. Biglieri, “Optimal power control over fading channels, ” IE EE Tr ans. Inf. T heory , vol. 45, no. 5, pp. 1468–148 9, Jul. 1999. [9] A. Lozano, A. M. T ulino, and S. V erd ´ u, “Opitmum po wer allocation for parallel Gaussian chann els w ith arbitrary input distributions, ” IE EE T rans. Inf. Theory , v ol. 52, no. 7, p p. 3033–3051, Jul. 2006. [10] K. D. Nguyen, A. Guill ´ en i F ` abregas, and L. K. Rasmussen, “Po wer all ocation for discrete-input delay-limited fading channels, ” Sub mitted to IEEE T rans. Inf. Theory . A vailable at arXiv:0706.2033[cs.IT] . [11] G. Caire and K. R. Kumar , “Information-theoretic foundations of adapti ve coded modulation, ” P r oceedings of the IEEE , vol. 95, no. 12 , pp. 227 4–2298 , Dec. 2007. [12] T . M. Cov er and J. A. Thomas, Elements of Informa tion T heory , 2nd ed . John Wiley and Sons, 2006. [13] M. A. Kho j astepour and B. Aazhang, “The capacity of av erage and peak power constrained fad i ng ch annels with channe l side information, ” i n 2004 IEEE W ir eless Communications and Networking Conferen ce , At lanta, Georgia, USA , 24–2 9, Oct. 2004. [14] A. J. Goldsmith and P . P . V araiya, “Capa city of fading ch annels with cha nnel side information, ” IEEE Tr ans. Inf. Theory , vol. 43, no. 6, pp. 1986–19 92, Nov . 19 97. [15] E. Zehavi, “8-PSK trellis codes for a R ayleigh channel, ” IEEE T rans. Commun. , vol. 40, n o. 5, pp. 873–88 4, May 1992 . [16] G. Caire, G. T aricco, and E. Biglieri, “Bit-interleav ed coded modulation, ” IEE E T rans. Inf. Theory , vol. 44, no. 3, pp. 927–94 6, May 1998. [17] D. Guo, S. Shamai, and S. V erd ´ u, “Mutual information and minimum mean-square error in Gaussian channels, ” IEEE T rans. Inf. Theory , vol. 51, no. 4, pp. 12 61–1282, Apr . 2005. [18] A. Guill ´ en i F ` abregas and A. Martinez, “Deriv at iv e of BICM mutual information, ” IET Electronics Letters , vol. 43, no. 22, pp. 1219–122 0, Oct. 2007. [19] R. Knopp and P . A. Humblet, “On coding for block fading channels, ” I EEE T rans. Inf. Theory , vo l. 46, no. 1, pp. 189–205, Jan. 2000. [20] E. Malkam ¨ aki and H. Leib, “Coded diversity on block-fadin g channels, ” IEEE T rans. Inf. Theory , vol. 45, no. 2, pp. 771–78 1, Mar . 1999 . [21] A. Guill ´ en i F ` abregas and G. Caire, “Coded mo dulation in the block - fading chann el: Coding theorems an d code construction, ” IEEE T rans. Inf. Theory , vol. 52, no. 1, pp. 91–114, Jan. 2006. [22] K. D. Nguyen, A. Guill ´ en i F ` abregas, and L. K. Rasmussen, “ A tight l o wer bound to the outage probability of block-fading Nov ember 20, 2021 DRAFT 20 channels, ” IEEE T rans. Inf. Theory , vol. 53, no. 11, pp. 431 4–4322, Nov . 2007 . [23] S. V . Hanly and D. N. C. Tse, “Multiaccess fad ing channels-Part II : Delay-limited capacities, ” IEE E T rans. Inf. Theory , vol. 44, no. 7, pp. 2816–28 31, Nov . 19 98. [24] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with F ormulas, Graph s, and Mathematical T ables . Ne w Y ork: Dover , 1 964. Nov ember 20, 2021 DRAFT 21 0 5 10 15 20 25 30 10 −4 10 −3 10 −2 10 −1 10 0 P av (d B) P out ( p ⋆ , P av , R ) P APR = 0 d B P APR = 10 d B P APR = 20 d B P APR = ∞ Fig. 1. Outage pro bability for systems with P AP R constraints ove r Nakagami- m block-fading channels B = 1 , m = 1 , R = 1 , 16-QAM inputs. T he solid and dashed lines corresponding l y represent outage probability of systems with coded modulation and BICM. Nov ember 20, 2021 DRAFT 22 0 5 10 15 20 25 30 35 10 −4 10 −3 10 −2 10 −1 10 0 P av (d B) P out ( p ⋆ , P av , R ) P APR= ∞ P APR=15 d B P APR =10 dB P APR=0 dB Fig. 2. Outage pro bability for systems with P AP R constraints ove r Nakagami- m block-fading channels B = 4 , m = 1 , R = 3 , 16-QAM inputs. T he solid and dashed lines corresponding l y represent outage probability of systems with coded modulation and BICM. Nov ember 20, 2021 DRAFT 23 0 5 10 15 20 25 30 35 10 −4 10 −3 10 −2 10 −1 10 0 P av (d B) P out ( p ⋆ , P av , R ) P APR=0 dB P APR=10 d B P APR =15 dB P APR= ∞ Fig. 3. Outage probability for systems wi th peak and average power constraints using 16-QAM i nput constellation ov er Nakgami- m block-f ading chan nels wit h B = 4 , m = 1 , R = 3 and peak-to-a verage power ratio P APR . The solid and dashed lines correspondingly repres ent outage probability of systems with optimal and truncated water-filling schemes with β = 19 dB. Nov ember 20, 2021 DRAFT 24 −10 −5 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 P av C Fig. 4. Capacity of ergodic fading channel with m = 1 , 16 -QAM coded mod ulati on inp uts and average power constraint. The dashed line represents capacity of the unfaded A W GN channel, and the solid, dashed-dotted and dotted lines corresponding ly represent capacities wit h optimal, tr uncated water-filling and uniform po wer allocation. Nov ember 20, 2021 DRAFT 25 −10 −5 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 P av C Fig. 5. Capacity of ergodic fading channel with m = 1 , 16-QAM coded modulation inputs and P APR = 3 dB. The dashed line r epresents capacity of the unfaded A W GN ch annel an d the solid, dash ed-dotted and dotted lines co rrespondingly represent capacities with optimal, truncated water-filling and uniform p o wer all ocation. Nov ember 20, 2021 DRAFT 26 −10 −5 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 P av C Fig. 6. Capacity of ergo dic fading channel with m = 1 , 16-QAM coded modu lation inputs and P APR constraints. The solid line with crosses represents capacity of the unfaded A WGN channel; t he dotted line represents the capacity wi th uniform po w er allocation and the solid, dashed-dotted and dash ed lines correspon dingly represent capacities wi th P A PR = ∞ , 4 , 1 dB. Nov ember 20, 2021 DRAFT