Subcarrier allocation in coded OFDMA uplink systems: Diversity versus CFO

Transcription

1 Sucarrier allocation in coded OFDMA uplin systems: Diversity versus CFO Antonia Maria Masucci, Inar Fijalow, Elena Veronica Belmega To cite this version: Antonia Maria Masucci, Inar Fijalow, Elena Veronica Belmega Sucarrier allocation in coded OFDMA uplin systems: Diversity versus CFO IEEE 24th Annual International Symposium on Personal, Indoor, and Moile Radio Communications PIMRC), Sep 213, United Kingdom pp , 213 <hal > HAL Id: hal Sumitted on 17 Sep 213 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are pulished or not The documents may come from teaching and research institutions in France or aroad, or from pulic or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, puliés ou non, émanant des étalissements d enseignement et de recherche français ou étrangers, des laoratoires pulics ou privés

2 Sucarrier allocation in coded OFDMA uplin systems: Diversity versus CFO Antonia Maria Masucci, Inar Fijalow, Elena Veronica Belmega ETIS/ENSEA - Université de Cergy Pontoise - CNRS 6 Avenue de Ponceau, 9514 Cergy, France masucci@enseafr, inarfijalow@enseafr, elmega@enseafr Astract In this paper, we analyze the optimal locwise sucarrier allocation scheme in coded Orthogonal Frequency Division Multiple Access OFDMA) uplin systems without channel state information at the transmitter side Due to the Discrete Fourier Transform, the sucarrier gains in OFDMA systems are correlated and this correlation plays a determinant role ce it causes a degradation of the transmission performance As a consequence, there exist some sucarrier allocation schemes which are preferale to achieve higher diversity gains We propose the minimization of an inter-carrier sum-correlation function defined as the sum of correlations of each sucarrier with respect to the others in the same allocation scheme We show the roustness of the proposed locwise allocation policy to the effect of carrier frequency offsets CFO) Moreover, we propose the loc-size which provides the est trade-off etween diversity and CFO for practical values of CFO Finally, numerical simulations validating our results are presented I INTRODUCTION The increag demand for high transmission data rates has lead to the development of highly adaptive communication systems Due to its flexiility, OFDMA has een proposed as the downlin transmission scheme for wireless standards of 4G technology WiMAX, LTE), 1 Recent wors analyze OFDMA uplin transmission with respect to the analysis of frequency mismatches due to the sensitivity to carrier frequency offsets CFO), 2 CFO, generated y the frequency misalignments etween the moile users oscillator and the ase station, causes the loss of orthogonality among sucarriers y producing a shift of the received signal caug intercarrierinterference ICI) OFDMA uplin systems without CFO and with independent sucarriers achieve channel diversity gain y ug any allocation scheme However, due to the Discrete Fourier Transform, the OFDMA systems undergo correlated sucarriers that imply a degradation of the transmission performance We are therefore looing at the sucarrier allocation schemes that are liely to otain an increased diversity gain, when choog the loc-size CFO increases the loss of diversity due to ICI In OFDMA uplin transmission, users spread their information across the set of availale sucarriers Sucarrier allocation techniques are used to split the availale andwidth etween the users enaling each user to have a unique access In the case with no channel state information CSI) availale, the most used allocation schemes are the locwise and interleaved allocations, 3 In the locwise allocation scheme, susets of adjacent sucarriers are allocated to each user The interleaved allocation scheme is a special case where sucarriers are uniformly spaced from each other at a distance equal to the numer of users loc-size is equal to one) The interleaved allocation is usually considered to enefit from frequency diversity when no CSI IEEE 8216) 4 However, roustness to CFO should e improved y larger loc-sizes In the case with full CSI, the authors in 5 propose an optimal loc-size as a good alance etween the frequency diversity gain and roustness against CFO A semi-interleaved sucarrier allocation scheme in OFDMA/SC-FDMA systems capale of achieving the diversity gain with minimum CFO interference has een proposed in 6 restricted to the specific case where each user in the system transmits one symol spread on all sucarriers In 7, we study the optimal locsize allocation schemes for SC-FDMA Single Carrier - FDMA) uplin systems without CFO and without CSI at the transmitter We loo at the effects of correlation on the transmission performance, we provide an analytical expression of the correlation etween sucarriers and, analyzing all possile locwise allocation schemes in function of the loc-size, we find the analytical expressions of the optimal locwise allocation schemes that achieve maximum diversity In this wor, we analyze the case of uplin coded OFDMA systems with CFO and without CSI In general, uncoded OFDMA can not exploit the frequency diversity in the channel, therefore the use of channel coding with OFDMA will reduce the errors resulting from the multipath fading environment recovering the diversity gain 8 In order to enhance the frequency diversity, we propose to minimize the correlation etween sucarrier gains similarly to 7 We define the sum-correlation function as the sum of correlations of each sucarrier with respect to the others in the same allocation scheme and we show the roustness of the proposed locwise allocation policy to CFO When the values of CFO increase, ecause of its negative effects, the smallest loc-sizes are no longer optimal as opposed to 7, the case without CFO We propose a unique optimal loc-size that minimizes the sumcorrelation Numerical Monte Carlo simulations validate this value as the est trade-off etween diversity and CFO II SYSTEM MODEL We consider an OFDMA uplin system with N u moile users communicating with a ase station BS) or access point No CSI is availale at the transmitter side The total

3 andwidth B is divided into sucarriers and we denote y M = /N u the numer of sucarriers per user We tae equal to an integer power of two in order to optimize the Discrete Fourier Transform DFT) procesg Moreover, we assume oth the numer of users N u and the loc-size to e integer powers of two to have fair in the sense that the numer of allocated sucarriers is the same for all users) allocation schemes among the users and a fully utilized spectrum The signal efore the DFT at the receiver is given in 1), in the next page, y 5, where L is the length of the cyclic prefix The vector =,, hu) represents the discrete channel impulse response of user u The elements are the symols at the output of the Inverse DFT IDTF) given y = 1 = F 1 Π u) x u) 2) with F 1 the IDFT matrix, x u) is the signal transmitted y user u and Π u) is the sucarrier allocation matrix whose elements are equal to 1 for the carriers allocated to user u and equal to for the carriers allocated to user v with { v u according } to the considered loc-size β = 1, 2, 4,, N u = {2, 2 1, 2 2 } log,, 2 Nu The diagonal elements are the frequency shift coefficients given y = e j2δf u) c T u) δf c T, {,, + L 1}, where is the normalized CFO of user u Discarding in the signal at the input of the receiver DFT, given in 1), the last L components corresponding to the cyclic prefix and rearranging the terms we have: y 1 and y 1 circ u=1 δ 1)u) }{{} N u = circ = 1 + Denoting y r = Fy, the received signal after the DFT r = u=1 n 1 n N u H u) u) Π u) x u) + ˆn, 3) where ñ = Fn is the 1 additive Gaussian noise with variance σni, 2 H u) = F circ F 1 is the diagonal channel matrix of user u with the, )-entry given y H u) = m= hu) m e j2m/, the matrix u) = F F 1 represents the loss of orthogonality induced y CFO In particular, we have the l, ) element of u) : u) l, = 1 N p 1 e j2iδf/ e j2il )/ i= = 1 δf + l)) δf + l) )e j 1 1 ) δf+ l) In the sequel, we denote H u) H u) u) Note that in 3) several terms due to cyclic prefix were not taen into account 5, that have almost no effect unless very large cyclic prefix is considered III SUM-CORRELATION: CASE WITHOUT CFO In order to optimize the frequency diversity when the system is not affected y CFO, we define a measure of correlation etween sucarriers, that we call sum-correlation function We note that when the system is not affected y CFO the matrix u) in 3) is equal to I A Sum-correlation minimization Assuming that there is no CFO in the system and the channel impulse responses m are independent ut distriuted accordingly to the same complex Gaussian distriution CN, σh 2 ), we consider the following sum-correlation function, 7: Γ m ) = H m u) H c u) u = E E H u) m 2 + = σ2 h L + σ2 h c u m E H m u) H c u) u L m c u ) e m c u ) ) j m cu) for all u N u, where m C u is the reference sucarrier and C u = { 1)N u + u 1) + i, i {1,, }} { } 1,2,3,, Nu 5) The set C u is composed of all indices of sucarriers allocated to user u given a loc-size allocation scheme Sucarrier User th 4) N u 1 N u 1 N u 1 N u Fig 1: The total sucarriers are divided into N u large-locs The -th large-loc contains N u sucarriers, which are divided into N u loc of size, one for each user The ratio N u represents the numer of locs that can e allocated to each user given a loc-size We consider,

4 y 1 Nu y = u=1 y L +) 1 n 1 + n n L L 1) therefore, that the total sucarriers are divided into N u large-locs that contain N u sucarriers corresponding to the N u locs, one for each user, of size The set C u is the union of indices of sucarriers allocated to user u in all these largelocs Inside of the large-loc of index, the indices of the sucarriers allocated to user u are 1)N u +u 1)+i, i {1,, }, where 1)N u corresponds to the previous 1 large-locs and u 1) corresponds to the previous allocated users 1, 2,, u 1), see Figure 1 The function Γ u,m ) is the sum of the correlations etween sucarriers that are in the same allocation scheme In the next Theorem, we give the set of optimal loc-sizes that minimize the sum-correlation function given in 4) Theorem 1 7): We consider our uplin system with no CFO and sucarriers, { N u users} and a cyclic prefix of size L Given β 1, 2, 4,,, we have the optimal allocation set β = arg min β N ul Γ m ) and Γ m ) = σ2 h N u for any loc-size β and for any reference sucarrier m C u We oserve that the minimum value of the sum-correlation function depends only on the numer of users and the variance of the channel impulse response, and that the largest optimal loc-size max = N ul, is a function of system s parameters: numer of sucarriers, numer of users and cyclic prefix length B Numerical Validations In this section, we present some numerical validations In particular, we show that β in Theorem 1 minimizes the outage proaility This explains that the outage proaility, which is a measure of the reliaility of the system 9, increases when the correlation among sucarriers is increag The outage proaility of the system is defined as the maximum of the { outage} proailities of the users { in the system } P out = max u P u) out where P u) out = P r C u) < R u) with R u) a fixed spectral efficiency at user u and C u) is the user instantaneous achievale spectral efficiency assuming gleuser decoding at the BS 1, given y C u) = N u B log 2 I+H u) Π u) H u) N u Iσ n+ 2 v=1 v u H v) Π v) H v) 1 considering the transmitted symols in 2) distriuted accordingly to the gaussian distriution such that E x u) x u)h = I In Figure 2, we evaluate the outage proaility P out for the scenario N u = 4, L = 8 and R u) = 5 its/s/hz We oserve that the optimal loc-sizes are the ones that correspond to the outage proailities which have a higher decreag speed as a function of the SNR We have that the curves with β = {1, 2} in this case max = = 2) overlap and they represent the lower outage proaility, hence the optimal loc-sizes P out =1 max =2 =4 =8 = SNRdB) Fig 2: Outage proaility for an OFDMA system without CFO with = 64, N u = 4 and L = 8 The optimal loc-sizes are β = {1, 2} IV ROBUSTNESS TO CFO Similarly to the approach in Section III, we extend the definition of the sum-correlation function to the case in which the system undergoes CFO: Γ CF O u,m, δf) E = Hu) m E C u = σ2 h 3 L C u + σ2 h 1 e c u m u) H c u H u) m H u) c u m, c u, 2δf + ) 2m) ) + 2 δf + ) 2 m L m cu) m c u ) m c u ) j ) C u e j ) 1 1 m c u) cos m c u)) cos 2δf + ) m + c u )) ) ) 2 cos m c u ) cos δf + ) m + c u ) 6)

5 where m C u is the reference sucarrier, δf represents the ) u) CFO of user u and H m = H m u) m,1,, H m u) m, represents the m-th row of the matrix H u) In the next Proposition, we study the ehavior of the CFO sum-correlation function when the CFOs go to zero We show that this function can e approximated with the sumcorrelation function defined in the case without CFO in 4) This implies that we have roustness to CFO Proposition 1: When the CFO coefficient δf ecomes very small δf ) the sum-correlation function defined in 6) tends to the sum-correlation function defined in the case without CFO for any loc-size, ie lim δf Γ CF u,m O, δf) = Γ m ) Proof: We denote y S m,cu, δf) 1 C u e j ) 1 1 m c u) cos m c u )) cos 2δf + ) m + c u )) ) ) cos m c u ) cos 2 δf + ) m + c u ) In what follows, we consider the case m = c u ce it represent the predominant term in the sum-correlation function Then, we have S m,m, δf) = 1 2 C u 7) 2δf + ) 2m) ) 2 C u δf + ) 2 m = 1 2δf) 2 δf + 2 m)) = 1 2δf) )+ 2 2 δf + 2δf) ) 8) 2 m δf + 2 m) We oserve that when δf ecomes very small, we have: and also: m 2δf) ) 2 δf δf 2δf) ) 2 δf + 2 m) δf, This implies that the S m,m, δf) in 8) tends to one when δf goes to zero: S m,m, δf) 1 9) δf This means that when the CFO δf goes to zero we can approximate the CFO sum-correlation function with the sumcorrelation function defined in 4) In practical cases in which the CFOs tae moderate values, we propose a unique optimal loc-size that achieves diversity: Since CFO yields a diversity loss, in the presence of moderate values of CFO the optimal loc-size allocation is max = /N u L This means that, in presence of CFO, max represents the good trade-off etween CFO and diversity We have found that the optimal lo-sizes that achieve maximum diversity are the ones that minimize the correlation etween sucarriers and we now that, in general, in the presence of CFO in the system larger loc-sizes are preferale to oppose the effect of CFO Aove, in presence of CFO, we propose the largest optimal loc-size max as the loc-size that represents the est tradeoff etween diversity and CFO: ce it is the largest etween the loc-sizes that achieve diversity it etter contrasts the effect of CFO This oservation is validated y extensive numerical simulations one of which is illustrated in the next section V NUMERICAL VALIDATIONS In Figure 3, we plot the outage proaility of an OFDMA system with CFO marer lines) against the outage proaility of an OFDMA system without CFO dashed lines) The CFO for each user is independently uniformly generated in, 1 and R u) = 1its/s/Hz We can see that the curves in the CFO case fit very well the outage proaility curves without CFO In particular, they appear in decreag order of loc-size This validates our analytical analysis on the approximation of the CFO sum-correlation function to the case without CFO when the CFO goes to zero Moreover, we oserve that in the two cases we have the same optimal loc-sizes set, given y β = {1, 2} In the next two simulation results, we use QPSK modulation, = 128 and L = 32 We use a concatened coding, which consists in cascading two or more simple codes in a serial manner, where the resulting composite code is equivalent to a much longer code in performance and lower in decoding complexity In particular, we use the following pair: a Reed Solomon code, RS64, 48, 8), as the outer code with a rate- 1/2 convolutional coding with constraint length 7 and code generators 133, 171) in octal notation as the inner code Figure 4 illustrates the Bit Error Rate BER) curves for an OFDMA system with CFO independently and uniformly generated for each user in, 1 We oserve that for these low values of CFO we have the optimal loc-sizes given y β = {1, 2} Thus, a similar oservation as in section III can e made Figure 5 illustrates the BER curves for an OFDMA system with CFO independently and uniformly generated for each user in, 3 We oserve that in this case we have a unique optimal loc-size given y max = 2 This means that when the CFO values are increag the optimal compromise etween diversity and CFO is represented y the largest locsize in the set proposed without CFO, max Many others simulations, for different values of the parameters, have een performed and similar oservations were made Moreover, more powerful coding schemes guarantee a gain more important

6 1 1 1 =32,16,,1 =1 CFO max =2 CFO =4 CFO =8 CFO =16 CFO =32 CFO =1 max = =1 max =2 =32 P out 1 2 =4 =8 =16 =32 BER SNR SNRdB) Fig 3: Outage proaility for an OFDMA system without CFO dashed lines, appearing in decreag order of loc-size from up to down) and with CFO marer lines) for the scenario = 64, N u = 2 and L = 16 The CFO of each user is independently uniformly generated in, 1 The optimal loc-sizes are β = {1, 2} Fig 5: Bit Error Rate for an OFDMA system with CFO for the scenario = 128, N u = 2 and L = 32 The CFO of each user is independently uniformly generated in, 3 The optimal locsize is max = 2 BER =1 =2 max = SNRdB) Fig 4: Bit Error Rate for an OFDMA system with CFO for the scenario = 128, N u = 2 and L = 32 The CFO of each user is independently uniformly generated in, 1 The optimal locsizes are β = {1, 2} VI CONCLUSIONS This wor has provided an analysis of the sucarriers loc allocation policies for coded OFDMA uplin systems without channel state information and with CFO We have extended the inter-carrier sum-correlation function given in 7 for the case without CFO) and we have provided a est loc-size allocation policy minimizing the sum-correlation as trade-off etween the diversity gain and CFO effects We have also shown the roustness to CFO effects through the analysis of our proposed sum-correlation function All these results have een validated via extensive Monte Carlo simulations REFERENCES 1 A Molisch, Wideand Wireless Digital Communication, Upper Saddle River: Prentice Hall, 21 2 H Sari, G Karam, and I Jeanclaude, Channel equalization and carrier synchronization in OFDM systems, in Proceedings of the 6th Tirrenia International Worshop on Digital Communications, Tirrenia, Italy, Sep 1993, pp A Sohl and A Klein, Comparison of localized, interleaved, and loc-interleaved FDMA in terms of pilot multiplexing and channel estimation, in Proceedings of the 15th European Signal Procesg Conference, Poznan, Poland, Sep 27 4 L Koffman and V Roman, Broadand wireless access solutions ased on OFDM access in IEEE 8216, IEEE Communication Magazine, pp 96 13, Apr 22 5 B Aziz, I Fijalow, and M Ariaudo, Trade off etween frequency diversity and roustness to carrier frequency offset in uplin OFDMA system, in Proceedings of Gloal Telecommunications Conference GLOBECOM 211), Dec 211, Houston, Texas, pp S Song, G Chen, and K Letaief, Localized or interleaved? a tradeoff etween diversity and CFO interference in multipath channels, IEEE Transactions on Wireless Communications, vol 1, no 9, pp , Sep A Masucci, E Belmega,, and I Fijalow, Optimal locwise sucarrier allocation policies in gle carrier FDMA uplin systems, Sumitted to IEEE Transactions on Signal Procesg, Mar M Tonello and R Bernardini, Analysis of the achievale timefrequency diversity gains in coded OFDM, in Proceedings of the International Conference on Telecommunications, Beijing, China, Jun 22, pp L Zheng and D Tse, Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels, IEEE Transactions on Information Theory, vol 49, no 5, pp , May 23 1 W Yu, W Rhee, S Boyd, and J Cioffi, Iterative water-filling for gaussian vector multiple-access channels, Information Theory, IEEE Transactions on, vol 5, no 1, pp , Jan 24