Research Article A Hybrid Single-Carrier/Multicarrier Transmission Scheme with Power Allocation

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1 Hidawi Publishig Corporatio EURASIP Joural o Wireless Commuicatios ad etworkig Volume 08, Article ID 68032, pages doi:055/08/68032 Research Article A Hybrid Sigle-Carrier/Multicarrier Trasmissio Scheme with Power Allocatio Dailo Zaatta Filho, Luc Féty, 2 ad Michel Terré 2 Sigal Processig Laboratory for Commuicatios DSPCom), State Uiversity of Campias UICAMP), Campias, SP, Brazil 2 Laboratory of Electroics ad Commuicatios, Coservatoire atioal des Arts et Métiers CAM), 754 Paris, Frace Correspodece should be addressed to Dailo Zaatta Filho, dailoz@decomfeeuicampbr Received May 07; Accepted 6 August 07 Recommeded by Luc Vadedorpe We propose a flexible trasmissio scheme which easily allows to switch betwee cyclic-prefixed sigle-carrier CP-SC) ad cyclicprefixed multicarrier ) trasmissios This scheme takes advatage of the best characteristic of each scheme, amely, the low peak-to-average power ratio PAPR) of the CP-SC scheme ad the robustess to chael selectivity of the scheme Moreover, we derive the optimum power allocatio for the CP-SC trasmissio cosiderig a zero-forcig ZF) ad a miimum mea-square error MMSE) receiver By takig the PAPR ito accout, we are able to make a better aalysis of the overall system ad the results show the advatage of the scheme for flat ad mild selective chaels due to their low PAPR ad that the scheme is more advatageous for a arrow rage of chaels with severe selectivity Copyright 08 Dailo Zaatta Filho et al This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited ITRODUCTIO Orthogoal frequecy divisio multiplexig OFDM) is already used i digital radio DAB), digital televisio DVB), wireless local area etworks eg, IEEE 802a/g ad HIPERLA/2), broadbad wireless access eg, IEEE 8026), digital subscriber lies DSL) ad certai ultra wide bad UWB) systems eg, MBOA) Recetly, it has also bee proposed for future cellular mobile systems [] By implemetig a iverse fast Fourier trasform IFFT) at the trasmitter ad a fast Fourier trasform FFT) at the receiver, OFDM coverts a selective chael, which presets itersymbol iterferece ISI), ito parallel flat subchaels, which are ISI-free, with gais equal to the chael s frequecy respose values To elimiate iterblock iterferece IBI) betwee successive IFFT-processed blocks, a cyclicprefix CP) of legth o less tha the chael order is iserted at each block by the trasmitter This CP coverts the liear chael covolutio ito circular covolutio At the receiver, the CP is discarded, which elimiates IBI The resultig chael covolutio matrix is circulat ad is diagoalized by the IFFT- ad FFT-matrices see, eg, [2]) Although OFDM results i simple trasmitters ad receivers, eablig simple equalizatio schemes, it has some drawbacks, amog which we ca cite high peak-to-average power ratio PAPR), sesitivity to carrier frequecy offset, ad the fact that it does ot exploit the chael diversity [3] as the more importat oes To circumvet these problems, the use of a cyclic-prefixed sigle-carrier CP-SC) modulatio was proposed to take advatage of, o the oe had, the simplicity of the OFDM modulatio ad, o the other had, the low PAPR, the frequecy offset robustess, ad the iheret exploitatio of the chael diversity of the SC modulatio Several works compare the performace of OFDM ad CP-SC eg, [4 0]) It is worth metio that the log term evolutio LTE) of the uiversal mobile telephoe system UMTS) is cosiderig the use of the OFDM for dowlik, but CP-SC for the uplik, maily due to PAPR issues [], sice power amplifiers have a little dyamic rage ad ted to saturate sigals with high PAPR These saturatios are harmful to the OFDM sigal ad, usually, a power back-off is ecessary to cotrol the resultig oliear distortio itroduced by the power amplifier [2]

2 2 EURASIP Joural o Wireless Commuicatios ad etworkig Here, we cosider that the trasmitter has partial chael state iformatio CSI), i terms of the sigal-to-oise ratio SR) of each subchael I this sceario, it is well kow that for OFDM it is possible to allocate power ad bits across the subchaels i order to maximize the rate [3] This solutio is the practical implemetatio of the water-fillig approach, which maximizes the capacity of a frequecy selective chael [4] Ideed, i this case, OFDM exploits the differeces i the SR across subchaels Hece, eve without codig, the OFDM scheme is able to exploit the chael diversity, i cotrast to the case of o CSI, where COFDM coded OFDM) must be employed [5] Buildig o our previous work [6], we propose a power allocatio approach to CP-SC trasmissio, whe a liear zero-forcig ZF) or a miimum mea-square error MMSE) receiver is used The aim is to beefit from the chael kowledge at the trasmitter while maitaiig a low PAPR i order to reduce ecessary power back-off, resultig i more power available for the trasmissio The differece of the proposed power allocatio i this work, i cotrast to [7], is that here we propose power allocatio schemes for the CP-SC schemes, while usig a classical power allocatio for OFDM, referred hereafter as cyclic-prefixed multicarrier ) scheme Moreover, we compare the performace of both trasmissio schemes takig ito accout the PAPR, i terms of the peak trasmissio power eeded for a give trasmit rate or o a fixed saturatio rate with power backoff, i cotrast to[5], where the compariso is performed takig ito accout the mea trasmit power Although for a wide variety of chaels the CP-SC schemes outperform the scheme, for highly selective chaels the approach leads to a better performace [6] The mai cotributio of this work is that we propose a trasmitter/receiver scheme for trasmittig either a CP-SC or a sigal, with o chages i the trasceiver structure but oly chagig oe matrix at each side We also show, for some simulated chaels, the optimum poit for switchig from oe scheme to the other i order to remai optimal i terms of the peak trasmissio power eeded for a give trasmit rate Furthermore, for both strategies, we derive the optimal power allocatio to maximize capacity, give a mea trasmit power level The rest of this paper is orgaized as follows Sectio 2 presets the proposed trasmit ad receiver schemes, icludig the geeratio, equalizatio, ad the derivatio of the obtaied SR for the sigle-carrier schemes ad the multicarrier scheme The optimum power allocatio for these schemes is derived i Sectio 3, where we also derive the achievable bit rate obtaied with this allocatio We assess the performace of each scheme i Sectio 4 by meas of umerical simulatios Coclusios are draw i Sectio 5 otatios Bold upper lower, resp) letters deote matrices vectors, resp); ) H deotes the Hermitia traspose cojugate traspose); Ai, j) deotes the i, j) etry of the matrix A; ad tr{a} deotes the trace of the matrix A We always idex matrix ad vectors etries startig from 2 TRASMISSIO SCHEMES The trasmit scheme used i this work is show i Figure We ote that this is a flexible trasmit scheme i the sese that it ca be used to geerate a CP-SC sigal as well as a sigal by chagig the trasmit switch matrix Q Moreover, this scheme also icludes a power allocatio matrix P, which is resposible for allocatig power to the trasmit symbols carried by differet subcarriers i the case ad for coformig the pulses that will carry the trasmit symbols i the CP-SC case Figure 2 shows the receiver scheme, composed of a FFT, a liear frequecy-domai equalizer W, ad the receive switch matrix Oce agai, this scheme allows the receptio of both waveforms by correctly choosig the receive switch matrix Z I the sequel, we show how to choose Q ad Z i order to geerate either a CP-SC or a sigal ad we also obtai the equalizer W for each scheme 2 Sigle-carrier trasmissio I order to geerate a CP-SC sigal, we use the trasmit scheme show i Figure, with the trasmit switch matrix Q equal to the FFT matrix F The trasmitted sigal vector x, before the cyclic extesio, ca be writte as x F H PQs F H PFs, ) where F is the orthoormal DFT discrete Fourier trasform)matrixofsize, s is the legth )vectoroftrasmitted symbols, ad the power allocatio matrix P is a diagoal matrix ote that the trasmit matrix give by T F H PF is, bycostructio,acirculatmatrix,whichimpliesthateach trasmit symbol s) is carried by a circulat-delayed versio of the same trasmit pulse give by ay colum of T), i exactly the same maer as i a classical SC modulatio Moreover, this is a adaptive scheme, sice the trasmit pulse ca be chaged by chagig the power allocatio matrix P The received sigal, after removig the cyclic prefix ad FFT, ca be writte as r FHF H PFs + F, 2) where H represets the effect of the composite chael resultig from the cascade of the aalog trasmissio chai ad the physical chael, ad is the additive oise vector at the receiver, assumed to be zero-mea, Gaussia, ad white with power 2 Thaks to the isertio of the cyclic prefix at the trasmitter ad its removal at the receiver, the chael H is give by a circulat matrix with its first colum give by the composite chael impulse respose appeded by zeros The orthoormal DFT matrix of size is defied by its elemets as F, k) / )e j2π[k ) )/],for,, ad k,,, ad it has the followig properties F H F I ad FF H I

3 Dailo Zaatta Filho et al 3 s) Serial to parallel Trasmit Power IFFT switch allocatio matrix Q P F H + L Parallel to serial x) x Figure : Trasmit scheme y) Serial to parallel + L Frequecy Receive FFT domai switch equalizer matrix F W H Z Parallel to serial ŝ) r Figure 2: Receiver scheme if eeded) Recallig that the DFT matrix diagoalizes ay circulat matrix [2], we ca write C FHF H, 3) where C is a diagoal matrix composed of the DFT of the composite chael impulse respose, which is equivalet to the frequecy respose of the composite chael at the frequecies of the differet subcarriers Hece 2) simplifies to r CPFs + F 4) I the receiver, the receive switch matrix is set to be the IFFT matrix, so that Z F H ad the estimated sigal vector at the receiver is give by ŝ F H W H r F H W H CPFs + F H W H F, 5) where W is a diagoal matrix At this poit, we are able to compute the equalizer W I the sequel, we aalyze two differet criteria to obtai this equalizer, amely the zeroforcig ZF) criterio ad the miimum mea-square error MMSE) criterio 22 ZF receiver The ZF criterio aims to completely cacel the ISI itroduced by the chael The ZF receiver is performed by multiplyig the received sigal by the iverse of the overall chael i the frequecy domai to compesate for the frequecy selectiveess, leadig to a ISI-free sigal at the receiver By ispectio of 5), we have that the zero-forcig equalizer is give by W H ZF CP) P C C P, 6) where the secod equality comes from the fact that both C ad P are diagoal matrices Usig this equalizer, the estimated sigal vector at the receiver reads ŝ ZF s + F H W H ZFF s + F H C P F 7) The trasmitted sigal is the perfectly recovered without ISI), but the oise that corrupts the decisio has a covariace matrix give by R ZF 2 F H P P H C C H F 8) It appears that this is a circulat matrix ad thus the variaces of the oise give by the diagoal elemets of R ZF ) that corrupts each symbol i the block are the same ad are give by 2,ZF 2 tr{ F H P P H C C H F } 2 tr{ P P H C C H} 2, p i c i where the secod equality comes from the matrix property tr{ab} tr{ba}, p i Pi, i) 2 is the power allocated to the ith subcarrier ad c i Ci, i) 2 is the squared chael gai at subcarrier i Hece, we ca write the decisio SR for the ZF receiver as SR ZF 2 s 2 9) ), 0) p i c i where 2 s is the power of the trasmitted symbols s)

4 4 EURASIP Joural o Wireless Commuicatios ad etworkig 23 MMSE receiver The use of the MMSE criterio is justified by the fact that miimizig the mea-square error MSE) leads to the maximizatio of the decisio SR, which is iversely proportioal to the bit error rate BER) Hece, by miimizig the MSE, oe should expect to decrease the BER The optimum MMSE solutio is the give by the Wieer solutio [7] W MMSE Rrr P rs, ) where R rr is the correlatio matrix of the received sigal r ad P rs is the cross-correlatio matrix betwee the received sigal r ad the desired sigal vector s,whereeachcolumof P rs correspods to the cross-correlatio vector betwee the received sigal ad the respective elemet of the desired sigal By usig 4), we ca write the correlatio matrix R rr as R rr E { rr H} E { CPF ss H F H P H C H} +E { F H F H} 2) 2 s CC H PP H + 2 I, where we have used the fact that the trasmitted symbols are iid with power 2 s, that is, E{ss H } 2 s I The cross-correlatio vector is give by P rs E { rs H} CPF E { ss H} + F E { s H} 2 s CPF, 3) sice the oise ad the sigal s are idepedet Isertig 2) ad3) ito ), we ca compute the MMSE equalizer, give by W MMSE Rrr P rs 2 s 2 s CC H PP H + 2 I ) 4) CPF We ote that this equalizer depeds o the chael through its frequecy respose C) ad o the trasmit pulse, which depeds o P By replacig W MMSE i 5)with4), the estimated symbols are give by ŝ MMSE As + B, 5) where A 2 s F H CC H PP H 2 s CC H PP H + 2 I) F ad B 2 s F H P H C H 2 s CC H PP H + 2 I) F From 5) we ca compute the desired sigal power ad the equivalet oise ad ISI power First, let us cosider oly the ifluece of the desired sigal It appears that the gai betwee s ad its estimatio ŝ is give by the diagoal elemets of the matrix A, which is a circulat matrix Hece, all diagoal elemets of A are equal ad ca be writte as Ai, i) { tr 2 s F H CC H PP H 2 s CC H PP H + 2 I ) } F tr { 2 s CC H PP H 2 s CC H PP H + 2 I ) } 2 s p i c i 2 s p i c i + 2 6) The, the power of the desired sigal at ay istat is give by P d 2 s 2 s p i c i 2 s p i c i + 2 )2 7) We ca also compute the power of the estimated sigal P e ), defied as the power of the desired sigal P d ) plus the power of the ISI P ISI ) The power of the estimated sigal is give by the diagoal elemets of the covariace matrix of the estimated sigal, which, due to the fact that it is also a circulat matrix, is give by P e P d + P ISI tr{ E [ Ass H A H]} { [ 2 s tr 2 s CC H PP H 2 s CC H PP H + 2 I ) ] 2 } 8) ) s p i c i s 2 s p i c i + 2 Fially, we ca compute the power of the oise that corrupts the desired sigal, give by the diagoal elemets of the covariace matrix of the equivalet oise From 5), we ca write the equivalet oise covariace matrix as R MMSE 2 2 s ) 2F H CC H PP H 2 s CC H PP H + 2 I ) 2 F 9) which is also a circulat matrix Therefore, it allows to express the variace of the equivalet oise by P tr{ R MMSE } 2 ) 2 2 s pi c i 2 s p i c i ) ) Oce the quatities P e P d + P ISI ad P have bee defied, we ca express the sigal to sigal-plus-iterfereceplus-oise ratio SSIR) of the estimated sigal as SSIR MMSE P d P d 2) P d + P ISI + P P e + P As show i the appedix, this SSIR is give by SSIR MMSE 2 s p i c i 2 s p i c i ) The equivalet decisio SR for the MMSE receiver is give by SR MMSE SSIR MMSE SSIR MMSE [ 2 ) s p i c ] i 2 s p i c i )

5 Dailo Zaatta Filho et al 5 24 Multicarrier trasmissio To geerate a sigal, we simply set the trasmit switch matrix equal the idetity matrix, Q I, so that the trasmit symbols s) are directly carried by the differet subcarriers, after power allocatio The trasmitted sigal vector x, before the cyclic extesio, is the give by x F H PQs F H Ps 24) The received sigal, after removig the cyclic prefix ad FFT, reads r CPs + F 25) At the receiver, we set Z I ad the estimated sigal vector is give by ŝ W H r W H CPs + W H F 26) Each subcarrier is the idepedetly equalized by applyig the gai Wi, i) at the receiver This gai ca be obtaied either by usig a ZF or a MMSE criterio, resultig i the same performace So, cosiderig the ZF criterio, we have that ad the estimated sigal reads W H OFDM C P, 27) ŝ OFDM s + C P F 28) The resultig SR at subcarrier i is the give by SR OFDM i 3 POWER ALLOCATIO 2 s p i c i 2 29) The goal of this sectio is to fid the optimal power allocatio matrix P to maximize the achievable rate subject to a costat trasmit power ad a give scheme The costrait o the trasmit power is related to the values of the coefficiets p i ad ca be expressed as p i 30) This costrait implies that the power of the trasmitted sigal x) is the same of the symbols s), give by 2 s The coefficiets p i are oly resposible for distributig this trasmit power across the subcarriers I the ext two sectios, we derive the optimum power allocatio for the CP-SC ZF ad MMSE receivers obtaied i Sectio 2, ad i the followig sectio, we cosider the case 3 scheme For the CP-SC schemes, maximizig the achievable bit rate implies the maximizatio of the decisio SR From 0)we see that, i order to maximize the SR for the ZF receiver, we oly eed to miimize the term betwee paretheses, which is the oise ehacemet iheret to the ZF receiver Hece, we ca write the power allocatio problem as mi pi st p i c i p i 3) This problem ca be solved by the use of Lagrage multipliers The Lagrage cost fuctio is the give by J ZF + λ p i ), 32) p i c i where λ is the Lagrage multiplier The optimum solutio is obtaied by settig the derivative of J ZF with respect to p i ) to zero These derivatives are give by J ZF ) p 2ci λ 33) i pi Ad thus, by makig J ZF / p i 0, we fid p i ci 34) λ The value of λ ca be computed so that the costrait of costat trasmit power is respected leadig to p i λ λ, 35) ci ) 36) ci The optimum power allocatio for the ZF receiver is the give by ) p opt,zf i 37) ci c i By replacig the optimal powers p i i 0), we obtai the optimum decisio SR for the ZF receiver as SR opt ZF 2 s 2 ) 2 38) ci I possessio of this SR, we ca readily obtai the achievable bit rate per trasmitted symbol for the ZF receiver scheme, give by C opt ZF log s ) ) ci

6 6 EURASIP Joural o Wireless Commuicatios ad etworkig 32 scheme It is straightforward to see that maximizig the SR of the estimated symbols after the MMSE receiver is equivalet to maximize the SSIR of these symbols, give by 22) The costraied maximizatio of the SSIR ca thus be writte as max p i SSIR st p i, 2 s p i c i 2 s p i c i ) which ca also be solved by usig Lagrage multipliers The Lagrage cost fuctio is give by J 2 s p i c i MMSE 2 s p i c i λ p i ), 4) where λ is the Lagrage multiplier The derivative of J MMSE with respect to the powers p i is J MMSE 2 s c i 2 p 2 λ, 42) i 2 s p i c i + ) 2 ad the optimum powers p i ca be foud by makig J MMSE / p i 0 as follows: J MMSE 2 s c i 2 p 2 λ 0 43) i 2 s p i ci 2 + ) 2 After some maipulatios, we ca rewrite 43)as p i [ 2 λ 2 2 ]+ s c i 2, 44) s c i where [a] + is equal to a if a 0 ad is equal to 0 otherwise Equatio 44) shows that the optimal powers p i follow a water-fillig priciple [3, 4] These optimal values ca thus be obtaied by adjustig the water-level / λ to respect the power costrait ad the computig the optimal powers usig 44) It is importat to highlight that, sice we are dealig with powers, the values p i must all be oegative, which explais the use of the operator [ ] + If we assume that all terms betwee brackets i 44) are oegative, that is, all subcarriers are used i the trasmissio, we ca obtai the value of λ aalytically as il 2 / 2 s c i λ + il 2 / 2 ), 45) s c i adtheoptimalpowersread opt, MMSE pi + il 2 / 2 s c i ) il 2 / 2 s c i ) 2 2 s c i 2 2 s c i 46) evertheless, if some p i are egative, the optimum solutio is obtaied by droppig the subcarriers where p i < 0ad computig 46) agai for this ew subset of subcarriers This process is repeated util all powers are oegative ad the fial subset of used subcarriers is called Ω It is worth highlightig that both summatios of 46) areow carried o the subset Ω By usig these optimal powers i 23), we obtai the optimum decisio SSIR for the MMSE receiver as ) 2 /) SSIR opt MMSE i Ω 2 / 2 s c i Ω + i Ω 2 / 2 ), 47) s c i where Ω is the cardiality of Ω Hece, after some maipulatio, the achievable bit rate per trasmitted symbol for the MMSEreceiverschemeisgiveby C opt MMSE + 2 i Ω log 2 2 s c i ) Ω + ) 2 i Ω 2 + ) 2 2 s c i i Ω 2 s c i 48) 33 scheme I the case of trasmissio, the optimum power allocatio to maximize the achievable rate is give by the well kow water-fillig solutio [3, 4] Followig the same algorithm for fidig the subset of used subcarriers Ψ, the optimalpowerallocatioisgiveby opt, pi Ψ + Ψ ) 2 k Ψ 2 2 s c i 2 i Ψ, 49) s c i where Ψ is the cardiality of Ψ The achievable bit rate per trasmitted symbol 2 for the scheme is give by C opt log ) ) /Ψ + /Ψ i Ψ 2 / 2 )) s c i 2 s c i 2 i Ψ 2 50) 4 SIMULATIO RESULTS We cosider the proposed trasmit ad receive schemes, show i Figures ad 2, with 256 subcarriers I order to assess the performaces of the proposed techique, we cosider a first-order FIR chael, described by oe zero placed at This chael is ormalized so that its eergy is uitary, resultig i hz) z + 2 5) 2 Here symbol deotes each oe of the samples i the block ad ot the global symbol the block itself)

7 Dailo Zaatta Filho et al PTX/ 2 db) ccdf) Figure 3: Mea trasmit power eeded to trasmit 4 bits/symbol as a fuctio of the selectiveess of the chael PAPRdB) Figure 4: Complemetary cumulative distributio fuctio ccdf) of the trasmit power for 4 bits/symbol I the followig, we first compare the performace of the sigle-carrier schemes ad ) with that of the more classical water-filled as a fuctio of the selectiveess of the chael, expressed by the parameter For each chael, we compute the optimum power allocatio to achieve a give ormalized rate i bits/symbol) for atargetberof0 3 We have limited the modulatio cardiality to 30 bits/symbol We observe that the scheme is able to achieve ay rate from 0 to 30 bits/symbol, whereas the CP-SC schemes ca oly achieve iteger rates I order to uderstad the behavior of the schemes as a fuctio of the selectiveess of the chael, we have plotted the mea trasmit power eeded to trasmit 4 bits/symbol as a fuctio of the parameter, show i Figure 3 Itisworth otig that the value of 4 bits/symbol was chose to preset the graphics, but the aalysis ad coclusios are the same for ay other chose value We ca see i 3 that both CP-SC schemes perform very close to the scheme for low values of low selectivity) ad preset a power loss that icreases with the parameter Moreover, we see that the CP- SC-ZF scheme degrades quicker tha the for > 09 due to the oise ehacemet iheret to the ZF receiver However, as discussed earlier, the mea trasmit power is ot the oly performace idicator ad the PAPR must be take ito accout for a better aalysis of the overall system performace I order to characterize the behavior of the PAPR of each scheme, we cosider the complemetary cumulative distributio fuctio ccdf) of the trasmit power for the proposed schemes, as show i Figure 4 for some represetative values of ote that the value of the ccdf for a give PAPR is equivalet to the probability that the trasmit sigal is above this PAPR, which ca be see as the probability of saturatio give a back-off equal to this PAPR We observe that both CP-SC schemes have similar PAPR distributio for values of up to 09, but the scheme presets higher PAPR with high probability with respect to the scheme, sice the optimum power allocatio geerated higher allocated powers i the subcarriers with low gai O the other had, whe compared to the multicarrier scheme, the scheme shows sigificat gais i terms of PAPR for the whole rage of values of This gai icreases whe the selectiveess of the chael decreases ad also whe the saturatio probability decreases Figure 5 shows the value of the PAPR as a fuctio of the selectiveess of the chael for o saturatio ad a probability of saturatio of % We ca see that, for the scheme, the PAPR is roughly costat ad does ot chage with the selectiveess of the chael Whe we cosider the maximum trasmit power the o saturatio case), this PAPR is of db), which is the size of the FFT However, practical systems work with a give saturatio rate, which is admissible without icurrig i sigificat performace loss If we cosider a probability of saturatio of %, the PAPR decreases to 65 db, remaiig idepedet of The CP-SC schemes start from a PAPR of 0 db for the flat chael 0) ad preset a icrease of this PAPR as a fuctio, which is higher for the o saturatio case, as expected Oce agai, we see that the scheme exhibits a higher PAPR tha, beig comparable or higher tha that of for high values of Fially, we ote that the PAPR of is always lower tha that of for both the cosidered cases By takig the PAPR ito accout, Figure 6 shows the performace i terms of the peak trasmit power for o saturatio ad a probability of saturatio of % We observe that the scheme demads a roughly costat peak power, regardless of the chael selectivity ad that, by allowig a probability of saturatio of %, oe

8 8 EURASIP Joural o Wireless Commuicatios ad etworkig PAPR db) 5 0 PAPR db) a) b) Figure 5: PAPR for a) o saturatio ad b) % of saturatio as a fuctio of the selectiveess of the chael for 4 bits/symbol ote that the PAPR-axis values are differet Ppeak/ 2 db) Ppeak/ 2 db) a) b) Figure 6: Peak trasmit power eeded to trasmit 4 bits/symbol as a fuctio of the selectiveess of the chael for a) o saturatio ad b) % of saturatio ca gai more tha 5 db O the other had, the CP- SC schemes demad a expoetial icrease of the peak power to maitai the same trasmit rate for more selective chaels This behavior comes from both the icrease of the mea trasmit power eeded to achieve the same rate ad from the icrease i the PAPR for higher values of evertheless, the CP-SC schemes outperform the CP- MC scheme for a wide rage of less selective chaels, that is, for the o saturatio case, is always better tha ad is better for values of lower tha about 098, ad for the more practical case of a probability of saturatio of %, the CP-SC schemes are approximately equivalet, beig better tha for < 077

9 Dailo Zaatta Filho et al 9 th % saturatio 0% saturatio % saturatio 0% saturatio umber of bits/symbol ΔC SC-MC)/MC %) % saturatio Δ 530 db) 0% saturatio Δ 48 db) % saturatio Δ 282 db) 0% saturatio Δ 02 db) SR db) Figure 7: Threshold th for switchig from a sigle-carrier scheme to a multicarrier oe as a fuctio of the umber of trasmit bits/symbol ad differet saturatio rates Below this threshold, the CP-SC schemes outperforms the scheme Figure 8: Relative capacity of the CP-SC schemes with respect to the scheme) as a fuctio of the SR for differet degrees of saturatio for 07 The value betwee parethesis is the differetial back-off betwee the scheme ad the CP-SC schemes Hece, we see that the CP-SC schemes are advatageous over the scheme for a wide rage of chaels, with the exact threshold depedig o the acceptable saturatio rate The proposed hybrid trasmissio scheme is based o the choice of the trasmissio scheme betwee a siglecarrier ad a multicarrier scheme i order to make better use of the available trasmissio peak power Figure 7 shows this threshold as a fuctio of the ormalized trasmit rate for differet saturatio rates As expected, the outperforms the scheme for small data rates ad both schemes are equivalet for large data rates, sice the required SR for large data rates is high, decreasig the ifluece of the oise Also, as expected, the threshold icreases with the decrease of the saturatio rate, favorig the CP- SC schemes over the oe We ote the asymptotic behavior of the threshold, which ca be used as a rule of thumb i the desig of practical systems usig a hybrid trasmissio scheme 4 Capacity results By usig the capacity results from Sectio 3 ad the PAPR levels obtaied by simulatio i the first part of this sectio, we ca ow compare the achievable bit rate per trasmitted symbol of the proposed schemes subject to the same saturatio rates To do so, we compute the capacity of each scheme usig a suitable power back-off to respect the desired saturatio rate Figure 8 shows the capacity of both CP-SC schemes with respect to the scheme, i percetage, for a mild chael 07) We observe that, as expected from the aalysis of Figure 7, the scheme outperforms the oe for all saturatio rates except for 0% Also, as ΔC SC-MC)/MC %) % saturatio 0% saturatio 00% saturatio 0% saturatio Figure 9: Relative capacity of the CP-SC schemes with respect to the scheme) as a fuctio of the selectiveess of the chael for a SR of db expected, the scheme performs poorly i the low SR regio due to the oise ehacemet ad is equivalet to the for high SR For this chael, the CP- SC-MMSE scheme achieves from % to more tha 80% capacity gai i the low SR regio for saturatio rates equal or lower tha % The gai for a typical applicatio varies from 0% to 25% for db ad saturatio rates equal or lower tha % The higher gais obtaied i the low SR regio are due to the fact that, i this regio, the power

10 0 EURASIP Joural o Wireless Commuicatios ad etworkig gai due to a lower back-off becomes more advatageous tha the better immuity to selective chaels of the CP- MC We ow cosider a typical coditio, amely SR of db, ad assess the capacity gai as a fuctio of the selectiveess of the chael, as show i Figure 9 We observe gais from % up to 90% for flat chaels by usig a sigle-carrier scheme istead of multicarrier i this case, whe takig the PAPR ito accout From this figure, we ca also obtai the thresholds th for switchig from oe scheme to aother as 064, 079, 084, ad 087 for saturatio rates of 0%, %, 0%, ad 00%, respectively We observe a agreemet betwee these values ad the asymptotic oes from Figure 7 5 COCLUSIO We have proposed a flexible trasmissio scheme which easily allows to switch betwee cyclic-prefixed sigle-carrier CP-SC) ad cyclic-prefixed multicarrier ) trasmissios by chagig a matrix at the trasmitter ad oe at the receiver This scheme takes advatage of the best characteristic of each scheme, amely the low PAPR of the CP-SC scheme ad the robustess to chael selectivity of the scheme Moreover, we have derived the optimum power allocatio for the CP-SC trasmissio cosiderig a zero-forcig ZF) ad a miimum measquare error MMSE) receiver By doig so, we were able to make a fair compariso betwee ad CP-SC whe the trasmitter has partial chael state iformatio CSI) By takig the PAPR ito accout for a better aalysis of the overall system, the simulatios results show the advatage of the CP-SC schemes, i particular of the CP-SC- MMSE scheme for flat ad mild selective chaels due to their low PAPR O the other had, the scheme is more advatageous for a arrow rage of chaels with severe selectivity We have also derived the capacity of the proposed schemes with optimal power allocatio The simulatio results show typical gais of about % to 50% whe switchig to the scheme for chaels that do ot preset a high selectivity APPEDIX From 2), the MMSE SSIR ca be expressed as SSIR MMSE 2 2 ) 2 s p i c i s 2 s p i c i + 2 ) 2 ) 2 2 s p i c i s 2 s p i c i s pi c i 2 s p i c i ) A) By simplifyig the term 2 s,wecarewritea) asfollows SSIR MMSE 2 ) 2 s p i c i 2 s p i c i + 2 ) 2 2 s p i c i 2 s p i c i s p i c i 2 s p i c i ) A2) By covertig the two terms i the deomiator of A2) to the commo deomiator, we have SSIR MMSE /) 2 s p i c i / 2 s p i c i + 2 ) ) 2 /) [ ] ) 2 2 ) s p i c i s p i c i / 2 s p i c i + 2 ) 2 /) 2 s p i c i / 2 s p i c i + 2 ) ) 2 /) ] [ 2 s p i c i 2 s p i c i + 2 ) / 2 s p i c i + 2 ) 2 /) 2 s p i c i / 2 s p i c i + 2 ) ) 2 /) 2 s p i c i / 2 s p i c i + 2 ) 2 s p i c i 2 s p i c i + 2 ACKOWLEDGMETS A3) This work was partially supported by RRT Frech atioal Research etwork i Telecommuicatios), through project BILBAO, ad by CPq Brazilia Research Coucil) ad FAPESP The State of São Paulo Research Foudatio) REFERECES [] Z Wag, X Ma, ad G B Giaakis, OFDM or sigle-carrier block trasmissios? IEEE Trasactios o Commuicatios, vol 52, o 3, pp , 04 [2] P J Davis, Circulat Matrices, Joh Wiley & Sos, ew York, Y, USA, 979 [3] H Sari, G Karam, ad I Jeaclaude, A aalysis of orthogoal frequecy-divisio multiplexig for mobile radio applicatios, i Proceedigs of the 44th IEEE Vehicular Techology Coferece VTC 94), vol 3, pp , Stockholm, Swede, Jue 994 [4] D Falcoer, S L Ariyavisitakul, A Beyami-Seeyar, ad B Eidso, Frequecy domai equalizatio for sigle-carrier broadbad wireless systems, IEEE Commuicatios Magazie, vol 40, o 4, pp 58 66, 02 [5] J Louveaux, L Vadedorpe, ad T Sarteaer, Cyclic prefixed sigle carrier ad multicarrier trasmissio: bit rate compariso, IEEE Commuicatios Letters, vol 7, o 4, pp 80 82, 03 [6] H Sari, G Karam, ad I Jeaclaude, Trasmissio techiques for digital terrestrial TV broadcastig, IEEE Commuicatios Magazie, vol 33, o 2, pp 00 09, 995

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