Optimal blockwise subcarrier allocation policies in single-carrier FDMA uplink systems

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1 Optimal blockwise subcarrier allocation policies in single-carrier FDMA uplink systems Antonia Masucci, Elena Veronica Belmega, Inbar Fijalkow To cite tis version: Antonia Masucci, Elena Veronica Belmega, Inbar Fijalkow Optimal blockwise subcarrier allocation policies in single-carrier FDMA uplink systems EURASIP Journal on Advances in Signal Processing, SpringerOpen, 4, 4 (), pp76 <86/ > <al-3879> HA Id: al-3879 ttps://alarcives-ouvertesfr/al-3879 Submitted on 8 Apr 6 HA is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers arcive ouverte pluridisciplinaire HA, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés

2 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 ttp://aspeurasipjournalscom/content/4//76 RESEARCH Open Access Optimal blockwise subcarrier allocation policies in single-carrier FDMA uplink systems Antonia Maria Masucci,*, Elena Veronica Belmega and Inbar Fijalkow Abstract In tis paper, we analyze te optimal (blockwise) subcarrier allocation scemes in single-carrier frequency division multiple access (SC-FDMA) uplink systems witout cannel state information at te transmitter side Te presence of te discrete Fourier transform (DFT) in SC-FDMA/ortogonal frequency division multiple access OFDMA systems induces correlation between subcarriers wic degrades te transmission performance, and tus, only some of te possible subcarrier allocation scemes acieve better performance We propose as a performance metric a novel sum-correlation metric wic is sown to exibit interesting properties and a close link wit te outage probability We provide te set of optimal block-sizes acieving te maximum diversity and minimizing te inter-carrier sum-correlation function We derive te analytical closed-form expression of te largest optimal block-size as a function of te system s parameters: number of subcarriers, number of users, and te cyclic prefix lengt Te minimum value of sum-correlation depends only on te number of subcarriers, number of users and on te variance of te cannel impulse response Moreover, we observe numerically a close strong connection between te proposed metric and diversity: te optimal block-size is also optimal in terms of outage probability Also, wen te considered system undergoes carrier frequency offset (CFO), we observe te robustness of te proposed blockwise allocation policy to te CFO effects Numerical Monte Carlo simulations wic validate our analysis are illustrated Keywords: SC-FDMA/OFDMA; Subcarriers allocation; Cannel frequency diversity; Cyclic prefix induced; Correlation; Carrier frequency offsets Introduction Due to its simplicity and flexibility to subcarrier allocation policies, single-carrier frequency division multiple access (SC-FDMA) as been proposed as te uplink transmission sceme for wireless standard of 4G tecnology suc as 3GPP long-term evolution (TE) -3 SC-FDMA is a tecnique wit similar performance and essentially te same general structure as an ortogonal frequency division multiple access (OFDMA) system A remarkable advantage of SC-FDMA over OFDMA is tat te signal as lower peak-to-average power ratio (PAPR) tat guarantees te transmit power efficiency at te mobile terminal level 4 However, similarly to OFDMA, SC-FDMA sows sensitivity to small values of carrier frequency offsets (CFOs) generated by te frequency misalignment *Correspondence: antoniamasucci@inriafr ETIS/ENSEA - University of Cergy Pontoise - CNRS, 6 Avenue de Ponceau, 954 Cergy, France INRIA Paris-Rocquencourt, e Cesnay Cedex, France between te mobile users oscillators and te base station 5-7 CFO is responsible for te loss of ortogonality among subcarriers by producing a sift of te received signals causing inter-carrier interferences (ICI) In tis work, we sow tat te SC-FDMA uplink systems witout CFO and wit imposed independent subcarriers attain te same cannel diversity gain for any subcarrier allocation sceme However, due to te discrete Fourier transform (DFT) of te cannel, correlation between te subcarriers is induced, and tus, a degradation of te transmission performance occurs Terefore, tere exist some allocation scemes tat are able to acieve an increased diversity gain wen coosing te appropriate subcarrier allocation block-size In te uplink SC-FDMA transmissions, users spread teir information across te set of available subcarriers Subcarrier allocation tecniques are used to split te available bandwidt between te users In te case in wic no cannel state information (CSI) is available at te transmitter side, te most popular allocation sceme is te 4 Masucci et al; licensee Springer Tis is an Open Access article distributed under te terms of te Creative Commons Attribution icense (ttp://creativecommonsorg/licenses/by/4), wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly credited

3 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page of 7 ttp://aspeurasipjournalscom/content/4//76 blockwise allocation in 8 In tis blockwise allocation sceme, subsets of adjacent subcarriers, called blocks, are allocated to eac user, (see Figures,, and 3) In particular, we call mono-block allocation te sceme wit te maximum block-size given by te ratio between te number of subcarriers and te number of users, illustrated in Figure Te interleaved allocation sceme is a special case in wic subcarriers are uniformly spaced at a distanceequaltotenumberofusers(block-sizeb is equal to one), as sown in Figure 3 Te interleaved allocation is usually considered to benefit from frequency diversity (IEEE 86) 9 However, robustness to CFO can be improved by coosing large block-sizes since tey better combat te ICI In te case of full CSI, an optimal blocksize as been proposed for OFDMA systems in as a good balance between te frequency diversity gain and robustness against CFO In tis paper, we study te optimal block-size allocation scemes, in te case of an uplink SC-FDMA witout CSI To te best of our knowledge, te closest works to ours are references, A subcarrier allocation sceme wit respect to te user s outage probability as been proposed in for OFDMA/SC-FDMA systems wit and witout CFO In particular, te autors of propose a semi-interleaved subcarrier allocation sceme capable of acieving te diversity gain wit minimum CFO interference However, te autors analyze only te case in wic every user in te system transmits one symbol spread to all subcarriers, and as a consequence, teir diversity results are restricted to te considered model and wit a low data rate We point out tat our main contributions wit respect to consist in te following: we consider a more general model; we analyze all possible subcarrier allocation block-sizes; and we find te analytical expressions of te optimal blockwise allocation scemes tat acieve maximum diversity Moreover, we provide an analytical expression of te correlation between subcarriers and we analyze its effects on te system transmission s performance More precisely, in tis work, we propose a new allocation policy based on te minimization of te correlation between subcarriers In particular, in order to optimize te block-size subcarrier allocation, we propose a new performance metric, ie, te sum-correlation function tat we define as te sum of correlations of eac subcarrier wit respect to te oters in te same allocation sceme Te introduction of te sum-correlation function as a performance metric is motivated by te fact tat te correlation generated by te DFT implies tat some allocation scemes acieve a iger diversity gain tan oters Te interestofteproposedapproacisduetotefacttat it allows us to find te exact expression of te block-sizes tat acieve a iger diversity gain It turns out tat te minimum sum-correlation is acieved by block-size allocation policies tat lie in a set composed of all block-sizes tat are inferior or equal to a given tresold depending explicitly on te system s parameters: te number of subcarriers,tenumberofusers,andtecyclicprefixlengt Furtermore, we find te minimum value of te sumcorrelation function Tis value guarantees to acieve te maximum diversity gain, and wat is more remarkable, it depends only on te number of subcarriers, number of users, and te variance of te cannel impulse response We also provide interesting properties of te individual sum-correlation terms: te auto-correlation term (ie, te correlation between te subcarrier of reference and itself) depends on te lengt of cyclic prefix; te correlations between te subcarrier of reference and te ones tat are spaced from it of a distance equal to a multiple of te ratiobetweentenumberofsubcarriers,andtecyclic prefixes are equal to zero Te most interesting property of te proposed sum-correlation function is te close link to te outage probability and tus to te diversity gain Numerically, we observe tat te maximum diversity or te minimum outage allocation coincides wit te one minimizing our sum-correlation function Moreover, we observe tat wen te SC-FDMA system undergoes CFO, we ave te robustness to CFO for practical values of CFO Tis means tat wen te CFO goes to zero, te CFO sum-correlation can be approximated wit te sum-correlation defined in te case witout CFO Tis analysis as been done similarly to in wic coded Figure Mono-block allocation sceme Block-size b 8, 6 subcarriers, and users (subcarriers allocated to user u denoted by arrow markers and to user u by diamond markers)

4 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 3 of 7 ttp://aspeurasipjournalscom/content/4//76 Figure Blockwise allocation sceme Block-size b 4, 6 subcarriers, users (subcarriers allocated to user u denoted by arrow markers and to user u by diamond markers) OFDMA systems are analyzed Uncoded OFDMA cannot exploit te frequency diversity of te cannel; terefore, te use of cannel coding wit OFDMA in reduces te errors resulting from te multipat fading environment recovering te diversity gain Coding is not needed in SC-FDMA since it can be interpreted as a linearly precoded OFDMA system 4 We underline tat, wit respect to in wic te results ave been briefly announced, in tis paper: ) we provide a deeper and a more detailed teoretical analysis; ) we consider a new performance metric, not identical to te one analyzed in, wic takes into account te lengt of te cannel impulse response and a general power delay profile wic allow us to generalize our previous results in bot cases, wit and witout CFO; 3) novel simulation results are presented in order to validate tese new results Te difficulty of our analytical study is related to te discrete feasible set of allocation block-sizes and also to te objective function (ie, te sum-correlation function we propose) wic is closely linked wit te outage probability wose minimization is still an open issue in most non-trivial cases 3 However, we provide extensive numerical Monte Carlo simulations tat validate our analysis and all of our claims Te sequel of our paper is organized as follows In Section,wepresentteanalyticalmodelofteSC- FDMA system witout CFO In Section 3, we define a novel sum-correlation function and its properties; moreover, we find te optimal block-sizes for a subcarrier allocation sceme minimizing te subcarrier correlation function and we sow te numerical results tat validate our analysis We present te SC-FDMA system wit CFO in Section 4 We define te corresponding sum-correlation function and we observe its robustness against CFO Numerical results tat validate tis analysis are also presented At last, in Section 5 we conclude te paper System model witout CFO We consider a SC-FDMA uplink system were mobile users communicate wit a base station (BS) or access point In te case in wic te system is not affected by CFOs, te users are syncronized to te BS in time and frequency domains No CSI is available at te transmitter side Te total bandwidt B is divided into subcarriers and we denote by M (were x is te integer part of x) te number of subcarriers per user Notice tat we coose as an integer power of two in order to optimize te DFT processing To provide a fair allocation of te spectrum among te users (fair in te sense tat te number of allocated subcarriers is te same for all users), notice tat te number of not-allocated carriers is M < << wic is a negligible fraction of te total available spectrum Witout loss of Figure 3 Interleaved allocation sceme Block-size b, 6 subcarriers, users (subcarriers allocated to user u denoted by arrow markers and to user u by diamond markers)

5 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 4 of 7 ttp://aspeurasipjournalscom/content/4//76 generality and also to avoid complex notations a,wewill assume in te following tat is also a power of two and tat M Te signal at te input of te receiver DFT was expressed in as follows: y y y (u) (u) N u (u) (u) + n n n were is te lengt of te cyclic prefix Te vector (u) (u),, (u) is te cannel impulse response wose dimension is lower tan or equal to Te elements k are te symbols at te output of te inverse discrete Fourier transform (IDTF) given by F (u) () b x(u) () wit F te -size inverse DFT matrix, x (u) F N P x (u) Nu were F N P Nu is te -size DFT matrix, and x (u) is te vector of te M-ary symbols transmitted by user u Te vector x (u) does not ave a particular structure, contrary to were it is assumed to be equal to x wic Nu means tat one symbol is spread to all subcarriers Te symbol (u) b is te subcarrier allocation matrix wit only one element equal to in eac column wic occurs at rows tat represent te carriers allocated to user u according to te considered block-size b β {,, } Tesetβ is composed of all divisors of, tis guarantees a fully utilized spectrum Te SC-FDMA can be viewed as a pre-coded version of OFDMA since te -size DFT matrix does not affect te cannel diversity Discarding in te signal at te input of te receiver DFT te components corresponding to te cyclic prefix and rearranging te terms, we get y y } {{ } y (u) (u) (u) (u) (u) (u) (u) }{{} + (u) circ n n were (u) circ is a circulant matrix Denoting r Fy, we ave found tat te received signal at te BS after te -size DFT is given by: r F (u) x(u) + Fn circ F (u) b (3) H (u) (u) b x(u) + ñ (4) were H (u) F (u) circ F is te diagonal cannel matrix of user u wit te diagonal (k, k)-entry given by H (u) k m (u) m e jπmk/, (5) and ñ Fn is te additive Gaussian noise wit variance σ n I Terefore, over eac subcarrier k,,, we ave r k m (u) m e jπmk/ (u) k,k x(u) k + n k Note ten tat H (u) is diagonal tanks to te assumption on te cannel impulse response lengt being sorter tan te cyclic prefix However, te diagonal entries (5) are correlated wit eac oter 3 Minimization of te subcarriers sum-correlation Frequency diversity occurs in OFDMA systems by sending multiple replicas of te transmitted signal at different carrier frequencies Te idea beind diversity is to provide independent replicas of te same transmitted signal at te receiver and appropriately process tem to make te detection more reliable Different copies of te signal sould be transmitted in different frequency bands, a condition wic guarantees teir independence Te subcannels given in (5) are correlated and no more tan a -order frequency diversity gain can be possible since tere are only independent cannel coefficients, ie,

6 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 5 of 7 ttp://aspeurasipjournalscom/content/4//76 (u),, (u), 4 Intuitively, we can imagine tat tere are groups of frequencies wic are identical Eac user retrieves te maximal diversity if it as at least blocks of size b wic implies b Terefore, users can acieve full diversity wen te block-size is witin te coerent bandwidt In tis work, we propose an original approac to find te block-size tat guarantees te maximum diversity Tis approac is based on te minimization of te subcannels/subcarrier sum-correlation function In te sequel, we define a measure of correlation between subcarriers, tat we call sum-correlation function, and we derive its properties Moreover, we find te set of optimal block-sizes b β wic minimizes tis correlation in order to minimize te effects tat it produces 3 Properties of te sum-correlation function Assuming tat te cannel impulse responses are independent but distributed accordingly to te complex Gaussian distribution CN(, σ (u) ) 4, we define for eac user te sum of correlations of eac subcarrier wit respect to te oters in te same allocation sceme as follows: Ɣ u,m (b) H m (u) H(u) c u E c u C u E σ (u) H (u) m + + σ (u) c u C u c u m e πj ( ) (m c u ) E H m (u) H(u) c u c u C u were m C u is te reference subcarrier and C u { k,,3,, } bnu sin π (m c u ) sin π (m c u) (6) {(k )b +(u )b + i, i {,, b}} Te set C u is composed of all indices of subcarriers allocated to user u given a block-size b allocation sceme (7) Te ratio b represents te number of blocks tat can be allocated to eac user given a block-size b Weconsider, terefore, tat te total subcarriers are divided into b large-blocks tat contain b subcarriers corresponding to te blocks, one for eac user, of size bte set C u is te union of indices of subcarriers allocated to user u in all tese large-blocks Inside of te large-block of index k, te indices of te b subcarriers allocated to user u are (k )b + (u )b + i, i {,, b}, were (k )b corresponds to te previous k large-blocks and (u )b corresponds to te previous allocated users (,,, u ), seefigure4tefunctionɣ u,m (b) is te sum of te correlations between subcarriers tat are in te same allocation sceme Considering te subcarriers m and c u in C u, we denote tedistancebetweentemby d : m c u (k k )b + i i (8) { } wit k, k N,, 3,, p b and i, i {,, b} We define te function if d f (d) sin π d e πj ( ) d (9) oterwise sin π d Te next result guarantees tat te function f (d) is independent of te user index Proposition Given te parameters,,and, te value f (d) of te function in (9) for any d m c u in (8) is independent of te user index u Proof: Tedependenceoff onteuserindexu is expressed by te term (m c u ), representing te distance between two subcarriers in te same allocation sceme, were m, c u C u Ifmand c u are in two different blocks tere exist two indices k and k in tat {,, 3,, } b suc m (k )b + (u )b + i () c u (k )b + (u )b + i () Figure 4 Set of subcarriers Te total subcarriers are divided into b large blocks Te kt large block contains b subcarriers, wic are divided into block of size b, one for eac user

7 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 6 of 7 ttp://aspeurasipjournalscom/content/4//76 wit i and i in {,, b} Terefore, te distance m c u ( k k ) b + i i () does not depend on te user index u Ifm and c u are in tesameblock,k k and te same reasoning olds Tis guarantees te independence of te function f (d) on te particular user u Te independence between f (d) and user index u comes from te fact tat given a block-size b,tesetofdistances between subcarriers are te same for all users We observe tat te function f (d) as te following circularity property Proposition (Circularity of f(d)) Given te number of subcarriers and for any distance d between subcarriers given in (8), we ave f (d + ) f (d) (3) Proof: From (9), f sin π (d + ) d + e πj ( ) sin π (d + ) ejπ d e jπ e jπ d e jπ e jπ d e jπ e jπ d e jπ e jπ d e jπ e jπ d e jπ ejπ d e jπ d e jπ d e jπ d e jπ sin e πj ( ) sin π d f (d) π d d ( ) d (d+ ) jπ e d e jπ e jπ d e jπ We consider te following sum-correlation metric: Ɣ(b) m C u Ɣ u,m (b) (4) Tanks to Proposition and Proposition, it can be expressed as follows: were Ɣ(b) C u σ (u) f (d) N p d D b (5) σ (u) + f (d) (6) d D b d { } d kbnu + i, i {,, b } D b { } k,, b (7) Given a subcarrier of reference, witout loss of generality, te set D b represents te set of te distances b between te subcarrier of reference and all te oter subcarriers in te same allocation sceme (te k factor represents ere te distance between te large-blocks) Tis definition is consistent since te function f (d) as te circularity property wit respect to Tis means tat it does not matter wic subcarrier of reference we consider Terefore, te sum-correlation function defined in (5) is independent on te reference subcarrier Tis guarantees tat te next results old for eac user in te system We observe tat tere is no correlation between te subcarrier of reference and oter carriers wic are spaced from it at a distance equal to a multiple of Itisobvious from te definition of te function f (d) tat it is equal to zero wen te distance d is a multiple of te ratio : ( f r N ) p, r N (8) Tis is wat we observe in Figure 5, in wic we plot te absolute value of te function f (d), witm, 3, and 4 We observe tat tis function is equal to zero for all te multiples of 8 We ave seen tat te function f (d) equals zero for all multiples of and tat te distance d can be expressed in function of te block-size b (see (8) and (7)) In te following, we provide te expression of te block-size b suc tat d Proposition 3 Given a fixed distance d between subcarriers, te corresponding block-size is equal to b Proof: Notice tat, given our allocation policy in Figure 4, not all te distances can be acieved for any possible block-size We consider an arbitrary distance d in D b

8 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 7 of 7 ttp://aspeurasipjournalscom/content/4//76 f(d) d Figure 5 Te function f (d),witm, 3,,and 4 In tis case, 8 and we are interested to find te block-size tat ensures d : d kb + i wit i {,, b }, { } and k,, b r wit r First, we analyze te case wen b Intiscase,we observe tat te condition d is satisfied if k and i Tismeanstatforallb tere are two subcarriers in te same block suc tat teir distance is More interesting is te case wen b < Fromte fact tat i {,, b } and b <,inordertoave te distance d,weavejusttoconsidertecase i Ten, it is obvious tat we ave d equal to wen b Tis means tat te block-size is b 3 Novel blockwise allocation sceme In te next Teorem, we give te set of optimal block-sizes tat minimize te sum-correlation function Ɣ(b)Tisset is given by β {,,, } β Teorem We consider our uplink system wit subcarriers, users { and a cannel impulse response lengt Given β N,, p, }, we ave Te elements in te set β minimize te sum-correlation function Ɣ(b): β arg min Ɣ(b) (9) b β Te optimal value of te sum-correlation function depends only on te system parameters Ɣ(b ) σ (u) Nu, b β () Proof: TeproofisgiveninteAppendix5 Proposition 4 In te case witout CSI, assuming tat is not known{ at te transmitter } side, we propose to use β restricted to,, Proof: We observe tat {,, {,, } is included in,, } Intuitively, tis means tat, in a more realistic scenario in wic only te knowledge of and not of te cannel lengt is available, we can still provide te subset of optimal block-sizes We observe tat, in te case witout CSI, te minimum value of te sum-correlation function depends only on te number of subcarriers, number of users, and te variance of te cannel impulse response and tat te largest optimal block-size is given by b max, () wic is a function of system s parameters: number of subcarriers, number of users, and cyclic prefix lengt In a more general scenario in wic te cannel lengt is different for eac user, ie, (u), te optimal block-size maximizing te sum-correlation function is a difficult problem and an open issue However, in a realistic scenario in wic tese parameters (u) are not known, te system planner would assume te worse case scenario and approximate tem wit te lengt of te cyclic prefix Since te lengt of te cyclic prefix is bigger, te cosen block-lengt is suboptimal and given by () 33 Numerical results: diversity and sum-correlation In tis section, te aim is to igligt te close relationsip between diversity gain, outage probability and te sum-correlation function We define te outage

9 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 8 of 7 ttp://aspeurasipjournalscom/content/4//76 probability of te system under consideration as te maximum of te outage probabilities of te users: P out max P (u) out () u { } were P (u) out Pr C (u) b < R wit R is a fixed target transmission rate and C (u) b is te instantaneous mutual information of te user u defined in te following We consider te transmitted symbols in () distributed accordingly to te Gaussian distribution suc tat E x (u) x (u)h ITe user instantaneous acievable spectral efficiency assuming single-user decoding at te BS 5 in te case witout CFO is as follows: C (u) b B log det I + H(u) (u) b Nu Iσ n + H (v) (v) B v v u m C u log b H(v) H(u) ( + ) σn H m (u) (3) An explicit analytical relation between te sumcorrelation function and te outage probability is still an open problem Te major issue is te fact tat te distribution of te mutual information is very complex and closed-form expressions for te outage probability are not available in general For example, Emre Telatar s conjecture on te optimal covariance matrix minimizing te outage probability in te single-user MIMO cannels 3 is yet to be proven We propose a new metric, te sumcorrelation function, and sow by simulations tat tere is an underlying relation between te sum-correlation function and te outage probability Indeed, it is intuitive tat, in SC-FDMA systems, correlation among te subcarriers decreases te diversity gain and, tus, te transmission reliability decreases 6,7 Tis explains tat te outage probability increases wen te correlation among subcarriers is increasing Tis connection as been validated via extensive numerical simulations Te interest beind tis connection is tat te sum-correlation function as a closed-form expression allowing us to perform a rigorous analysis and to find te blockwise subcarrier allocation minimizing te sum-correlation wic is consistent wit te optimal blockwise subcarrier allocation minimizing te outage probability Te following results illustrate numerically tis connection 33 Uncorrelated subcarriers We consider te case of a SC-FDMA system wit independent subcarriers Since te subcarriers are independent te correlation between tem is zero, wic means tat te sum-correlation Ɣ(b) is equal to zero for any blocksize b In te next simulation, we observe tat we obtain te same performance in terms of te outage probability regardless of te particular allocation sceme and te block-size, see Figure 6 Altoug tis scenario is unrealistic from a practical standpoint, it is important to notice tat, in tis case, tere are no privileged block-sizes to acieve better diversity gain In Figure 6, we plot te outage probability in te SC-FDMA system wit independent subcarriers (subcannels) generated by complex Gaussian distribution wit respect to SNR for te scenario 64,, and fixed rate R bits/s/hz In particular, in tis case wit independent subcarriers, we consider te matrix H (u) in (4) to be diagonal wit entries H (u) k iid CN(, σ ) It is clear tat for any block-size (ence, for any subcarrier allocation sceme) we obtain te same performance in terms of outage probability Terefore, tere are not any privileged block-size allocations to acieve better diversity gain Tis motivates and strengtens our observation tat te subcarrier correlation as a direct impact on te outage probability 33 Correlated subcarriers In tis section, we consider a more interesting and realistic SC-FDMA system given in (4) For simplicity and P out 3 b b b4 b8 b6 b3 4 8 Figure 6 Outage probability for a SC-FDMA wit independent cannels wit 64 and

10 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 9 of 7 ttp://aspeurasipjournalscom/content/4//76 lack of space-related reasons, te simulations presented ere ave been done for te particular case Numerous oter simulations were performed in te general case, wic confirm te teoretical result of Teorem In Figure 7, we ave plotted wit respect to te blocksize b, te sum-correlation function Ɣ(b) in te SC-FDMA system witout CFO for te scenario 4, 4and σ () 5, σ () 5, σ (3) 5, σ (4) 3 Te illustrated markers represent te values of te function Ɣ(b) for te given coice of te parameters of te system We observe tat te minimal values of Ɣ(b) are obtained for te block-sizes b β {,, 4}Inparticular, b β {,, 4} we ave Ɣ(b ) σ (u) Nu 47 In Figure 8, we use Binary Pase Sift Keying (BPSK) modulation in te following scenario: 64,, 8, and σ () σ () We observe tat te optimal block-sizes are in β {,, 4} (but ere, we just plot te smallest and te biggest values) for te BER wic confirms tat tese block-sizes optimize also te sum-correlation function we ave proposed In Figure 9, we use BPSK modulation, 64,, 4, and σ () σ () Te optimal block-sizes are givenintesetβ {,, 4, 8} In Figure, we use BPSK modulation in te following scenario: 8,, and 8 In tis case, we consider an exponential power delay profile wic means tat σ (u) e τ/ wit τ {,,, } e τ/ τ Bit Error Rate b * b max 4 b3 Figure 8 Bit error rate for a SC-FDMA system witout CFO for te scenario 64,,and 8,andσ () σ () Te optimal block-sizes are b β {,, 4} Te teoretical results are confirmed since te optimal block-sizes are in β {,, 4, 8} In Figure, we evaluate te outage probability P out in te SC-FDMA system witout CFO for te scenario 4, 4, and R bits/s/hz We observe tat te optimal block-sizes are te ones tat correspond to te outage probabilities wic ave a iger 9 b Γ(b) Bit Error Rate * b 8 max b Figure 7 Function Ɣ(b) wit 64, 4, 4,and σ () 5, σ () 5, σ (3) 5, σ (4) 3 For any b β {,, 4},weaveƔ(b ) ( ) b Figure 9 Bit error rate for a SC-FDMA system witout CFO for te scenario 64,, 4,andσ () σ () Te optimal block-sizes are b β {,, 4, 8}

11 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page of 7 ttp://aspeurasipjournalscom/content/4//76 b * b max 8 Bit Error Rate 3 b64 P out b b b4 * b max Figure Bit error rate for a SC-FDMA system witout CFO for te scenario 8,, 8, and exponential power delay profile Te optimal block-sizes are b β {,, 4, 8} 3 b6 b3 4 8 Figure Outage probability for a SC-FDMA system witout CFO wit 64,,and 4 Te optimal block-sizes are b β {,, 4, 8} decreasing rate as a function of te SNR We see tat te curves wit b β {,, 4} (in tis case b max ) are overlapped and tey represent te lower outage probability Tese block-sizes are te same tat minimize te sum-correlation function (see Figure 7) In Figure, we plot te outage probability for te SC- FDMA system witout CFO for te scenario 64,, 4, and R bits/s/hz so tat b max In tis case in wic te subcarriers are correlated, we observe tat te curves wit b β {,, 4, 8} ave a iger diversity Many oters simulations, canging te values of te parameters (in particular, and ), ave been performed, and similar observations were made Moreover, we ave done simulations coosing te following as a performance metric: P out 3 4 b b * b max 4 b8 b6 4 8 Figure Outage probability for a SC-FDMA system witout CFO wit 64, 4,and 4 Te optimal block-sizes are b β {,, 4} P out,b P (u) out,b (4) Te same observation can be made wit tis outage metric 4 Robustness to CFO In tis section, we analyze te case of SC-FDMA systems wit CFO and te effect of CFO on te optimal block-size We define te sum-correlation function and we sow its robustness to CFO We start by describing in details te system model 4 System model If te system undergoes CFOs, te signal at te input of te receiver DFT is given in (5), and it was introduced in,

12 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page of 7 ttp://aspeurasipjournalscom/content/4//76 y y y (u) (u) (u) (u) δ (u) ( + ) δ (u) + n n n (5) Te diagonal elements δ (u) k are te frequency sift coefficients given by δ (u) jπkδf c (u) T k e, k {,, + (u) δf c },were is te normalized CFO of user u By discarding te cyclic prefix symbols and rearranging te terms in (5), we ave y y (u) circ δ (u) ( + ) δ (u) }{{} + δ (u) n n Te received signal at te BS after te DFT is (6) r CFO H (u) (u) (u) b x(u) + ˆn, (7) were te matrix (u) Fδ (u) F represents te effect of CFO on te interference among subcarriers In particular, we ave te (l, k) element of (u) : (u) l,k i e jπiδf / e jπi(l k)/ sin ( π(δf + k l) ) ( e πj sin π (δf + k l) ) (δf +k l) (8) In te sequel, we denote H (u) H (u) (u), wic is no longer a diagonal matrix 4 Diversity versus CFO in subcarrier allocation We consider te following inter-carrier correlation function: Ɣu,m CFO (b, δf ) E H m (u) m,m,, H (u) c u E H m (u) H(u) c u m,m c u,k k C u H (u) E m m,m m,k k C u + E H m (u) H(u) c u m,m c u,k k C u c u m H (u) E m m,m H + (u) E m m,m m,k + k C u k m + c u m + + k C u k m E H m (u) c u m,m c u,c u N p E H m (u) c u m,m c u,k k C u c u m k c u σ (u) sin (πδf ) sin ( N π p δf ) ( e πj sin(π(δf + k m)) sin( N π + p (δf + k m)) + σ (u) N p c u m ) (k m) sin(πδf ) sin( π δf ) e πj ( ) (m c sin π u) (m c u ) sin (πδf ) sin ( π δf ) + k C u k c u N p sin(πδf ) sin(π(δf + k c u )) sin( N π p δf ) sin( N π p (δf + k c u )) sin π (m c u) e πj (k c u) (9) were m C u is te reference( subcarrier, δf represents te ) CFO of user u, and H (u) c u cu,,, H c (u) u cu, H (u) c u represents te c u -t row of te matrix H (u) We define te sum-correlation metric as follows: Ɣ CFO (b, δf ) Ɣu,m CFO m C u (b, δf ) (3) In te following, we provide an approximation of te correlation function Ɣm,u CFO (b, δf ) in wic te dependance on te CFO values δf is taken into account In particular,

13 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page of 7 ttp://aspeurasipjournalscom/content/4//76 we consider te second order Taylor approximation of (b, δf ) wen δf Ɣ CFO m,u Ɣ CFO m,u (b, δf ) ƔCFO m,u + dɣcfo m,u (b,) + (b,)δf dδf d Ɣ CFO m,u d(δf ) (b,)(δf ) Tis first term Ɣm,u CFO (b,) is given by Ɣ CFO u,m (b,) σ (u) + σ (u) c u m N p + N p σ (u) + σ (u) N p + N p k C u k m (3) e πj (k m) + e πj ( ) (m c sin π u) (m c u ) k C u k c u c u m sin π (m c u ) sin π (m c u) sin π (m c u) e πj (k c u) e πj ( ) (m c u) (3) Indeed, tis expression corresponds exactly to Ɣ m,u (b), ie, te sum-correlation function in te case witout CFO in (6) Te first derivative of Ɣm,u CFO (b, δf ) wit respect to δf computed in (b,) is dɣ CFO m,u dδf (b,) σ (u) + σ (u) N k C p u k m c u m N k C p u k c u e πj (k m) πcos(π(k m)) sin( π (k m)) + e πj ( ) (m c sin π u) (m c u ) e πj sin π (m c u) (k c u) π cos(π(k c u )) sin( π (k c u )) (33) Te second derivative of Ɣm,u CFO (b, δf ) computed in (b,) is d Ɣm,u CFO d(δf ) (b,) σ (u) π ( N p ) e πj (k m) k C u k m π cos π (k m) sin π + (k m) + Terefore, we ave Ɣ CFO m,u (u) σ (b, δf ) σ (u) c u m e πj ( ) (m c u) sin π (m c u ) sin π (m c u) π ( ) k C u k c u π cos π (k c u) sin π (k c u ) e πj (k c u) (34) + σ (u) sin π (m c u ) e πj ( ) (m c u) + sin π (m c u) + σ (u) N p e πj (k m) cos(π(k m)) sin( π k C u (k m)) + k m + σ (u) c u m e πj ( ) (m c sin π u) (m c u ) sin π (m c u) e πj (k c u) cos(π(k c u )) sin( N π p (k c u )) k C u k c u + σ (u) ( N p ) N p cos π (k m) + sin π (k m) + σ (u) c u m N p ( e πj k C u k m πδf ) (k m) e πj ( ) (m c sin π u) (m c u ) sin π (m c u) ( N p ) e πj (k c u) k C u k c u cos π (k c u) ( ) sin π π (δf ) (k c u ) (35)

14 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 3 of 7 ttp://aspeurasipjournalscom/content/4//76 We observe in te above approximation te presence of te predominant term represented by te sum-correlation Ɣ m,u (b) witout CFO and te first and te second derivatives of Ɣm,u CFO (b, δf ) wic are multiplied by te CFO value δf and δf, respectively Tis last terms may result in a different solution for te optimal block-size or block-sizes tat optimize te sum-correlation function tan te solution for te case wit no CFO We observe tat te first and te second derivatives represent a complex function tat implicitly depends on b Finding te optimal blocksize or block-sizes in an analytical manner, as done in te case wit no CFO, seems very difficult if at all possible and is left for future investigation Wen te system undergoes CFO, te carrier correlation and CFO affect te system performance simultaneously We ave proposed in te largest optimal block-size b max as te unique optimal block-size: Since CFO yields a diversity loss, in te presence of moderate values of CFO, te optimal block-size allocation is b max We ave found tat te optimal block-sizes tat acieve maximum diversity are te ones tat minimize te correlation between subcarriers Moreover, larger blocksizes are preferable to combat te effect of ICI Also, since b max is te largest block-size between te ones minimizing te correlation, it is also te one tat minimizes te negative effects caused by te presence of CFO Terefore, b max represents a good tradeoff between diversity and CFO Moreover, te observation is validated also by numerical simulations illustrated in te next subsection Γ CFO (b,δ f) δ f δ f δ f3 δ f b Figure 3 Te correlation function Ɣ CFO (b, δf ) as function of b for te scenario: 64,, 8,andσ () 5, σ () 5, δf {,, 3, 4} Te optimal block-size minimizing te correlation function is b max 4 In Figure 6, we use BPSK modulation in te following scenario: 64,, and 4 We consider te same model proposed in were one symbol is spread over all subcarriers Te CFO is independently and uniformly generated for eac user in, Weobservetat 43 Numerical results: CFO impact In Figure 3, we plot te correlation Ɣ CFO (b, δf ) for te scenario: 64,, 8, and σ () 5, σ () 5 Te considered CFO values are δf {,, 3, 4} Weobservetatb max 4isteblocksize tat acieves te minimum value of te correlation function Ɣ CFO (b, δf ), validating our conjectured optimal block-size In te next two simulations, we use BPSK modulation and 8, Nu, and 8 Figure 4 illustrates te bit error rate (BER) curves for a SC-FDMA system wit CFO independently and uniformly generated for eac user in, 3 We observe tat for tese low CFOs we ave te optimal block-sizes given by β {,, 4, 8} Figure 5 illustrates te BER curves for a SC-FDMA system wit CFO independently and uniformly generated for eac user in, Weobservetat,intiscase,weave a unique optimal block-size given by βmax 8 Tis validates our observations, ie, wen te CFO s values are increasing, te best tradeoff between diversity and CFO is represented by te largest block-size of our proposed set b max Bit Error Rate 3 b * b max 8 b SBNR(dB) Figure 4 Bit error rate for a SC-FDMA system wit CFO for te scenario 8,,and 8 Te CFO of eac user is independently uniformly generated in, 3 Te optimal block-sizes are b β {,, 4, 8}

15 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 4 of 7 ttp://aspeurasipjournalscom/content/4//76 b b * b max 8 * b 4 max Bit Error Rate b64 Bit Error Rate b Figure 5 Bit error rate for a SC-FDMA system wit CFO for te scenario 8,,and 8 Te CFO of eac user is independently uniformly generated in, Te optimal block-sizes is b max Figure 7 Bit error rate for a SC-FDMA system wit CFO for te scenario 64,,and 8 Te CFO is independently and uniformly generated for eac user in, 5 Te optimal block-sizes are b β {,, 4} te optimal block-sizes are in β {,, 4, 8} for te BER wic confirms tat our analysis is valid for te model proposed in In Figure 7, we use BPSK, 64,, and 8 and an exponential power delay profile Te CFO is independently and uniformly generated for eac user Bit Error Rate 3 b * b 8 max b Figure 6 Bit error rate for a SC-FDMA system wit CFO for te scenario 64,,and 4 Te CFO is independently and uniformly generated for eac user in, Te optimal block-sizes are b β {,, 4, 8} in, 5 Te set of optimal block-sizes given by β {,, 4} as sown in te figure For different and larger CFO values, as considered in 8, we notice tat an error floor is obtained due to te effect of CFO interference Tus, in suc cases, optimizing te block-size is not very relevant as all possibilities obtain suc poor results in terms of BER In te next simulation, we consider te following scenario: 64, 4, and 8 In te Figure 8, we plot te outage probability of an SC-FDMA system wit CFO (marker lines) against te outage probability of te SC-FDMA system witout CFO (dased lines) Te CFO for eac user is independently uniformly generated in δf,, andterater is taken equal to bits/s/hz We can see tat te curves in te CFO case fit very well te outage probability curves witout CFO In particular, tey appear in a decreasing order of block-size Tis validates our analytical analysis on te approximation of te CFO sum-correlation function to te case witout CFO wen te CFO goes to zero Moreover, we observe tat in te two cases we ave te same optimal block-sizes set, given by β {, } 5 Conclusions In tis work, we ave provided te analytical expression of te set of optimal sizes of subcarrier blocks for SC- FDMA uplink systems witout CFO and witout cannel state information Tese optimal block-sizes allow us to minimize te sum-correlation between subcarriers and to acieve maximum diversity gain We ave also provided

16 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 5 of 7 ttp://aspeurasipjournalscom/content/4//76 P out 3 4 b CFO b* max CFO b4 CFO b8 CFO b6 CFO b b* max b4 b8 b6 4 8 Figure 8 Outage probability for a SC-FDMA system wit and witout CFO Outage probability for a SC-FDMA system witout CFO (appearing in decreasing order of block-size from up to down) and wit CFO (marker lines) for te scenario 64, 4,and 8 Te CFO of eac user is independently uniformly generated in, Te optimal block-sizes are b β {, } te analytical expression of te sum-correlation between subcarriers induced by SC-FDMA/OFDMA Moreover, we ave found an explicit expression of te largest optimal block-size wic minimizes te sum-correlation function depending on te system s parameters: number of subcarriers, number of users, and te cyclic prefix lengt Interesting properties of tis novel sum-correlation function are also presented It turns out tat te minimal sum-correlation value depends only on te number of subcarriers, number of users, and te variance of te cannel impulse response We validate via numerical simulations tat te set of optimal block-sizes acieving maximum diversity minimizes te outage probability in te case witout CFO Also, in te case were te system undergoes CFO, we consider a sum-correlation function wic is robust to CFO Robustness is induced by te fact tat wen te CFO goes to zero, te CFO sum-correlation can be well approximated by te sum-correlation function defined in te case witout CFO Terefore, we propose b max a good tradeoff between diversity and CFO since it represents te unique optimal block-size tat acieves maximum diversity All tese results and observations ave been validated via extensive Monte Carlo simulations Endnotes a If we do not take tis assumption into account, we would ave to use Ñ p M instead of to denote te actual allocated number of carriers and instead of as te number of carriers per user b Here we use te word distance as synonym of difference and not for Euclidean distance Appendix Proof of Teorem Proof: From te definition of te set D b, we can write te function Ɣ(b) as follows: Ɣ(b) σ (u) N f (d) u d D b σ (u) σ (u) σ (u) b i k bnu k + f (kb + i) b f (kb ) + k i k b f (kb ) + First of all, we analyze te term: b i and, in particular, its it term k f (kb + i) k k k i f (kb + i) k f (kb + i) (36) k f (kb +i) ( ) sin π (kb + i) e jπ ( ) sin π (kb + i) e jπ kbnu e jπ i e jπ kbnu e jπ i ( α i z k) (kb+i) α i z k (37) wit α i : e jπ i and z : e jπ bnu Usingte decomposition of a geometric series of radius α i z k,we furter obtain k f (kb + i) k l (α i z k) l

17 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 6 of 7 ttp://aspeurasipjournalscom/content/4//76 and, inverting te two sums, we ave k f (kb + i) l α l i k (z l) k N p bnu + αi l bn (zl ) k u l k (38) Now, we look at te term k f (kb ) in Equation (36) and observe tat, by using te same reasoning, we can write k f (kb ) N p bnu + b l k (z l ) k (39) In wat follows, we consider two different cases: (a) Te case in wic z l In tis case, we observe tat l {,,, } and k α l i (zl ) k α l i k (z l ) k z bnu z l αi l e jπl e jπ bl ifz l (4) k (z l ) k (4) Terefore, using Equations (38), (39), (4), and (4), Equation (36) becomes Ɣ(b) σ (u) σ (u) σ (u) + ( ) + (b ) N p b b b N p b Nu (4) (b) Te case in wic z l We observe tat if tere exists an integer l in {,,, } suc tat z l, ten we ave and k α l (z l ) k k α l () k α l b (43) (z l ) k b k (44) Terefore, from te definition of z e jπ bnu,weave tat z l wen N nu p Z + Witout loss of generality, we look at te smallest integer in Z +,andweseetat b l l (45) b Hence, since l {,,, } we ave tat < (46) b wic is equivalent to b > Terefore, wen b >,wecanaveatleastonesumofteform k α l (z l ) k > (and k (z l ) k > ) From Equations (38), (39), and (4), we can conclude tat Ɣ(b) > σ (u) Nu, b > (47) To conclude our proof, from te analysis of cases (a) and (b), we can state te following result: Ɣ m () Ɣ m and Ɣ m (b) >Ɣ m for all b > Competing interests Te autors declare tat tey ave no competing interests σ (u) Nu Acknowledgements Te work of te first autor as been done wile se was wit ETIS/ENSEA - University of Cergy Pontoise - CNRS aboratory, Cergy-Pontoise, France (48) (49)

18 Masucci et al EURASIP Journal on Advances in Signal Processing 4, 4:76 Page 7 of 7 ttp://aspeurasipjournalscom/content/4//76 Received: 7 February 4 Accepted: 6 November 4 Publised: 6 December 4 References HG Myung, J im, DJ Goodman, Single carrier FDMA for uplink wireless transmission IEEE Veicular Tecnol (3),3 38 (6) AF Molisc, A Mammela, Taylor D P, Wideband Wireless Digital Communication (Prentice Hall PTR, Upper Saddle River, NJ, USA, ) 3 H Ekstrom, Tecnical solutions for te 3G ong-term Evolution IEEE Commun Mag 44(3),38 45 (6) 4 HG Myung, in Proceedings of te 5t European Signal Processing Conference Introduction to Single Carrier FDMA (Poznan, Poland, 7), pp PH Moose, A tecnique for ortogonal frequency division multiplexing frequency offset correction IEEE Trans Commun 4(),98 94 (994) 6 H Sari, G Karam, I Jeanclaude, in Proceedings of te 6t Tirrenia International Worksop on Digital Communications Cannel equalization and carrier syncronization in OFDM systems (Tirrenia, Italy, 993), pp 9 7 Y Zu, B etaief, in Proceedings of Global Telecommunication Conference CFO estimation and compensation in single carrier interleaved FDMA systems (Honolulu, Hawaii, USA, 9), pp 5 8 A Sol, A Klein, in Proceedings of te 5t European Signal Processing Conference Comparison of localized, interleaved, and block-interleaved FDMA in terms of pilot multiplexing and cannel estimation (Poznan, Poland, 7) 9 Koffman, V Roman, Broadband wireless access solutions based on OFDM access in IEEE 86 IEEE Commun Mag 4(4), 96 3 () B Aziz, I Fijalkow, M Ariaudo, in Proceedings of Global Telecommunications Conference (GOBECOM ) Tradeoff between frequency diversity and robustness to carrier frequency offset in uplink OFDMA system (Houston, Texas, ), pp 5 SH Song, G Cen, KB etaief, ocalized or interleaved? A tradeoff between diversity and CFO interference in multipat cannels IEEE Trans Wireless Commun (9), () AM Masucci, I Fijalkow, EV Belmega, in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC) Subcarrier allocation in coded OFDMA uplink systems: Diversity versus CFO (ondon, United Kingdom, 3) 3 E Telatar, Capacity of multi-antenna gaussian cannels Eur Trans Telecommun, (999) 4 D Tse, P Viswanat, Fundamentals of Wireless Communications (Cambridge University Press, New York, NY, USA, 4) 5 W Yu, W Ree, S Boyd, JM Cioffi, Iterative water-filling for gaussian vector multiple-access cannels Inform Teory, IEEE Trans 5(), 45 5 (4) 6 Zeng, DNC Tse, Diversity and multiplexing: a fundamental tradeoff in multiple-antenna cannels IEEE Trans Inform Teory 49(5), (3) 7 M Godavarti, A Hero, in Proceedings of IEEE International Conference on Acoustic, Speec and Signal Processing Diversity and degrees of freedom in wireless communications (Orlando, F, USA, ), pp M-O Pun, M Morelli, CCJ Kuo, Maximum-likeliood syncronization and cannel estimation for OFDMA uplink transmissions IEEE Trans Commun 54(4), (6) doi:86/ Cite tis article as: Masucci et al: Optimal blockwise subcarrier allocation policies in single-carrier FDMA uplink systems EURASIP Journal on Advances in Signal Processing 4 4:76 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 Hig visibility witin te field 7 Retaining te copyrigt to your article Submit your next manuscript at 7 springeropencom