MULTICARRIER MODULATIONS (MCM) have attracted. FDM-Structured Preamble Optimization for Channel Estimation in MIMO-OQAM/FBMC Systems

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1 FDM-Strutured Preamble Optimization for Channel Estimation in MIMO-OQAM/FBMC Systems Wenfeng Liu, Da Chen, Kai Luo, Tao Jiang, and Daiming Qu arxiv:8376v [sit Feb 8 Abstrat In this paper, we onsider the problem of preamble design in multiple-input multiple-output (MIMO) systems employing offset quadrature amplitude modulation based filter bank multiarrier (OQAM/FBMC) and propose a preamble optimization method for the frequeny division multiplexing (FDM)- strutured preamble Speifially, we formulate an optimization problem to determine the frequeny division multiplexed preambles, where the objetive is to minimize the mean square error (MSE) of the hannel estimation, subjet to the onstraint on the transmit energy For two transmit antennas, we find the relationship between preambles and the intrinsi interferene from neighboring symbols to ahieve the minimum hannel estimation MSE, and derive the optimal losed-form solution For more than two transmit antennas, the onstrained preamble optimization problem is nononvex quadrati Therefore, we onvert the original optimization problem into a quadratially onstrained quadrati program (QCQP) and obtain the suboptimal solution by relaxing the nononvex onstraint Simulation results demonstrate that, in terms of MSE and bit error rate (BER) performanes, the proposed method outperforms the onventional FDM preamble design method at all signal-to-noise ratio (SNR) regimes and outperforms the interferene approximation method-omplex (IAM-C) preamble design method at low to medium SNR regimes with lower preamble overhead Index Terms OQAM/FBMC, MIMO, hannel estimation, frequeny division multiplexing (FDM), preamble optimization I INTRODUCTION MULTICARRIER MODULATIONS (MCM) have attrated a lot of attention due to the apability to effiiently ope with frequeny seletive hannels Muh of the attention in the present literature emphasizes on the use of the onventional orthogonal frequeny division multiplexing (OFDM) However, the OFDM system uses retangular window on eah subhannel, whih leads to high out-ofband radiation Moreover, the OFDM system sarifies data transmission rate beause of the insertion of yli prefix (CP) To remedy the problems of the OFDM system, the offset quadrature amplitude modulation based filter bank multiarrier (OQAM/FBMC) has attrated inreasing attention [ [7 Compared with the onventional OFDM system, the OQAM/FBMC system provides lower sidelobes through the use of well-shaped prototype filter and higher useful data rate due to the fat that OQAM/FBMC does not require the CP W Liu, D Chen, K Luo, T Jiang, and D Qu are with Shool of Eletroni Information and Communiations, Wuhan National Laboratory for Optoeletronis, Huazhong University of Siene and Tehnology, Wuhan 4374, China ( liuwenfeng@hustedun; henda@hustedun; kluo@hustedun; taojiang@ieeeorg; qudaiming@hustedun) Furthermore, OQAM/FBMC brings advantages suh as robustness to narrow-band interferene and arrier frequeny offset Due to the above superiorities over OFDM, OQAM/FBMC is being onsidered as a promising tehnique for ognitive radio [8, professional mobile radio (PMR) evolution [9, and 5G ellular networks [, [ As another promising tehnique in future ommuniation systems, multiple-input multipleoutput (MIMO) is able to inrease the system throughput and the link reliability [, [3 Therefore, it is believed that the suessful ombination of MIMO and OQAM/FBMC is important and is envisioned to ahieve higher spetral effiieny [4 [6 In MIMO-OQAM/FBMC ommuniation systems, the hannel estimation plays a signifiant role in the reovery of data symbols Typially, training symbols are employed at the transmitter to perform hannel estimation In the literature, several training shemes and assoiated estimation methods for MIMO-OQAM/FBMC systems have been proposed, whih an be mainly divided into two ategories, ie, the sattered pilots-based methods [7 [9 and the preamble-based methods [ [4 Generally speaking, the sattered pilotsbased methods are used to trak the hannel variations in fast fading environments, while the preamble-based methods are more suitable for time invariant hannels with better hannel estimation performane In this paper, we onsider the time invariant hannels with low frequeny seletivity and fous on the preamble-based methods in MIMO-OQAM/FBMC systems The appropriate preamble design is ruial for hannel estimation in MIMO-OQAM/FBMC systems Different from the OFDM system, the orthogonality ondition of the OQAM/FBMC system only holds in the real field, whih auses intrinsi imaginary interferene between data symbols and preamble symbols [5, [6 Note that, the OFDM system provides orthogonality in the omplex field and there is no imaginary interferene between symbols Therefore, the onventional preamble design methods in the OFDM system annot be diretly used in the OQAM/FBMC system [7, [8 Moreover, when OQAM/FBMC is ombined with MIMO tehniques, the multi-antenna interferene should also be onsidered, whih makes the preamble design more ompliated The typial preamble design method, interferene approximation method (IAM), has been proposed and widely investigated in MIMO-OQAM/FBMC systems due to its simpliity and effiieny [, [ The key idea is to approximate the intrinsi interferene from neighboring symbols and onstrut This work has been submitted to the IEEE for possible publiation Copyright may be transferred without notie, after whih this version may no longer be aessible

2 omplex-valued pseudo-pilots at all frequenies to estimate the hannel However, sine the preamble overhead per antenna inreases linearly with the number of transmit antennas, the IAM method enounters a onsiderable loss in spetral effiieny, whih is unbearable espeially for a relatively large number of transmit antennas To redue the preamble overhead, some other effetive methods with shorter preambles were proposed in MIMO-OQAM/FBMC systems, whih inlude the time domain hannel model-based preamble design method [ and the sparse preamble design methods with only three OQAM/FBMC symbols being needed per antenna for hannel estimation [3, [4 The frequeny-division multiplexing (FDM) based preamble design was first introdued in [9 for MIMO-OFDM systems and this idea reappeared reently in [3, [4 in the ontext of MIMO-OQAM/FBMC systems, whih shows great hannel estimation performane with low preamble overhead In the FDM method, the preamble symbols from different transmit antennas are frequeny division multiplexed As a onsequene, the hannels from eah of the transmit antennas an be estimated separately without inter-antenna interferene Compared with the IAM method, the FDM method requires only three olumns of preamble overhead and hene ahieves a signifiant improvement in spetral effiieny However, the preamble in the existing FDM method is obtained without theoretial analysis and its optimality annot be guaranteed, resulting in a signifiant performane loss in hannel estimation Therefore, the preamble optimization for the FDM-strutured preamble in MIMO- OQAM/FBMC systems needs further investigations This paper aims to design the optimal FDM-strutured preamble sequene to enhane the estimation auray The key idea is to formulate a preamble optimization problem by minimizing the mean square error (MSE) of the hannel estimation, with the onstraint on the training energy at the MIMO-OQAM/FBMC transmitter In the ase of two transmit antennas, we derive the losed-form solution for the preamble optimization problem by exploring the relationship between preambles and the intrinsi interferene from neighboring symbols to ahieve the minimum MSE In the ase of three or more transmit antennas, the preamble optimization problem is nononvex quadrati, whih auses the optimal solution hard to obtain Therefore, we onvert the original optimization problem into a quadratially onstrained quadrati program (QCQP) and relax the nononvex onstraint of the optimization problem to obtain the suboptimal solution In order to verify the validity of the proposed preamble optimization method, and ompare the MSE and bit error rate (BER) performanes with the onventional FDM and IAM methods, simulations are onduted under and 4 4 MIMO onfigurations, respetively Simulation results show that the proposed method ahieves better performane than the onventional FDM method at all signal-to-noise ratio (SNR) regimes and outperforms the IAM-real (IAM-R) and IAM-omplex (IAM-C) methods (two variants of IAM) at low to medium SNR regimes with lower preamble overhead The remainder of the paper is organized as follows In Setion II, the MIMO-OQAM/FBMC system model is presented The onventional IAM and FDM preamble design methods are reviewed in Setion III Then, the preamble optimization problem for the FDM-strutured preamble and the different solving proesses that orrespond to different number of transmit antennas are proposed in Setion IV Setion V ompares the MSE and BER performanes of different preamble design methods through simulations, followed by the onlusions in Setion VI The following notations are employed in this paper Bold lower-ase letters denote olumn vetors and bold upper-ase letters are used for matries For the matrix A, notations A T, A H and Tr(A) indiate its transpose, onjugate transpose and trae operations, respetively R{ } is the real part of a omplex-valued number and E( ) is the expetation operator We use I n and n to denote an n n identity matrix and a zero matrix, respetively The notation C m n represents a set of m n matries with omplex entries Finally, j = II SYSTEM MODEL A OQAM/FBMC System Model The baseband disrete-time signal at the output of an OQAM/FBMC synthesis filter bank (SFB) an be written as [3, [3 s[k = M m= n Z [ a m,n g k n M e jπmk/m e jϕm,n, () }{{} g m,n[k where M is the number of subarriers, a m,n denotes the realvalued symbol onveyed by the subarrier of index m during the symbol time of index n, and g[k is a symmetrial realvalued prototype filter satisfying the perfet reonstrution ondition [3 ϕ m,n is an additional phase term given by ϕ m,n = ϕ +(π/)(m+n) mod π, where ϕ an be arbitrarily hosen Here we set ϕ = We assume that the OQAM/FBMC signal is transmitted through a hannel that varies slowly with time, and its delay spread is signifiantly shorter than the symbol interval [4 This implies that the hannel transfer funtion over eah subarrier band may be approximated by a flat gain Let H m,n denotes this gain for the mth subarrier at the nth symbol interval With the slow varying assumption, we omit the subsript n from H m,n for the sake of brevity In addition, a omplex additive white Gaussian noise (AWGN) with zero mean and variane σ is assumed to be introdued at the hannel output Aordingly, we an express the analysis filter bank (AFB) output at the mth subarrier and nth OQAM/FBMC symbol as [3 y m,n = H m m,n + η m,n, () where η m,n originates from the hannel noise and m,n = a m,n + ja (i) m,n is the virtually transmitted symbol at (m, n), with ja (i) m,n being the intrinsi imaginary interferene from the neighboring frequeny-time (FT) points With a well loalized pulse g[k in time and frequeny, it an be assumed that the intrinsi imaginary interferene mostly originates from the first-order neighboring FT points [3 Let us denote ζm,n p,q = g m,n [kgp,q[k (3) k=

3 3 a mn, Transmit antenna OQAM/FBMC Modulation, N r H m, H m Reeive antenna OQAM/FBMC Demodulation y mn, a ˆm, n N a ț mn OQAM/FBMC Modulation N, Nt H m t Nr, Nt H m MIMO Channel Nr OQAM/FBMC Demodulation y N, r mn MIMO Equalization N aˆ, t mn Fig The MIMO-OQAM/FBMC system model Then, the imaginary interferene term ja (i) m,n an be approximated as ja (i) m,n a m+p,n+q ζ m,n m+p,n+q, (4) (p,q ) Ω where the neighborhood Ω = {(p, q ) p, q {,, } and (p, q ) (, )} and ζ m,n m+p,n+q represents the ontribution of a m+p,n+q to the imaginary interferene ja m,n (i) It is noteworthy that for a well-designed prototype filter g[k, ζ m,n m+p,n+q is pure imaginary for any (p, q ) (, ) and ζ m,n m+p,n+q = for (p, q ) = (, ) Thus, for (p, q ) (, ), in the remainder of this paper, we refer to the terms ζ m,n m+p,n+q as the imaginary interferene oeffiients Note that, the intrinsi imaginary interferene ould be removed by taking the real part after hannel equalization In this paper, sine we mainly fous on the preamble design issues, the equalization and operation of taking real part are not disussed B MIMO-OQAM/FBMC System Model The MIMO-OQAM/FBMC system model is depited in Fig, where the transmitter and the reeiver are equipped with N t and N r antennas, respetively Sine the hannel estimation performane of MIMO-OQAM/FBMC systems would not differ in the ase of hannel oding, we only onsider the unoded senario below At the transmitter side, the symbols spatially multiplexed on the mth subarrier at the nth time index are denoted by a m,n = [ a m,n, a T m,n,, am,n Nt, eah element of whih is transmitted at different antennas after the orresponding OQAM/FBMC modulation At the reeiver side, the link of eah transmit and reeive antenna pair is degraded by multipath fading and ontaminated with AWGN For eah given FT position (m, n), let Hm r,t be the frequeny response of the hannel between the tth transmit antenna and the rth reeive antenna and ηm,n r be the noise omponent at the rth reeive antenna By assuming perfet time and frequeny synhronization, the demodulated symbol of the rth reeive antenna an be obtained by extending () to the MIMO ase as [6, [ N t ym,n r = Hm r,t t m,n + ηm,n, r r N r, (5) t= where t m,n represents the orresponding virtually transmitted symbol at the tth transmit antenna Aording to (4), t m,n an be written as t m,n = a t m,n+ a t m+p,n+q ζ m,n m+p,n+q, t N t (p,q ) Ω (6) We denote the demodulated symbol vetor by y m,n = [ y m,n, ym,n, T, ym,n Nr, the virtually transmitted vetor by m,n = [ m,n, T m,n,, m,n Nt and the additive noise vetor by η m,n = [ ηm,n, ηm,n, T, ηm,n Nr Thus, the equation (5) an be expressed as where H m = y m,n = H m m,n + η m,n, (7) Hm, Hm, Hm,Nt Hm, Hm, Hm,Nt H Nr, m Hm Nr, Hm Nr,Nt is the MIMO hannel frequeny response (CFR) at that FT point III THE CONVENTIONAL PREAMBLE DESIGN METHODS IN MIMO-OQAM/FBMC SYSTEMS Sine the orthogonality ondition of the OQAM/FBMC system only holds in the real field, whih auses intrinsi imaginary interferene to preamble symbols at the reeiver, preamble design in the OQAM/FBMC system is more diffiult than that in OFDM Worse is that there also exists multiantenna interferene when OQAM/FBMC is extended to the MIMO ase In this setion, we takle the preamble design problem in MIMO-OQAM/FBMC systems and give a brief review of the onventional preamble design methods A IAM Method IAM preamble design method has drawn muh attention due to its simpliity and effiieny [ The IAM family that inludes IAM-R, IAM-imaginary (IAM-I), IAM-C, and extended IAM-C (E-IAM-C) were firstly proposed in the SISO system based on approximating the unknown intrinsi interferene, where three olumns of preamble symbols are plaed in front of the data frame [3 [34 Correspondingly, (8)

4 4 j j j j j j j j j j j j j j j j a b Fig Transmitted pilots at the SFB input for the IAM-C method, with (a) and (b) orresponding to the two transmit antennas in the MIMO ase, the IAM preambles were onstruted by repeating the SISO preamble N t times for eah transmit antenna, with some hanged signs being inserted simultaneously to guarantee the orthogonality of different antennas [ Taking N t = as an example, Fig depits the IAM-C preamble at the SFB input with M = 8 subarriers Aording to (7), the demodulated symbols at times n =, 3 are obtained as [, [ [ ym, y m,3 = Hm [ m, m,3 m, m,3 + [ η m, η m,3 Under the assumption that the intrinsi interferene mostly originates from the first-order FT neighbors, the intrinsi interferene to the pilots are omputed approximately and hene the so-alled pseudo-pilots are onstruted for hannel estimation We an easily see that the pseudo-pilots satisfy m, = m,3 = m, = m,3 = m, where m an be alulated as in the SISO ase Substituting it into (9) we get (9) [ ym, y m,3 = Hm m A + [ η m, η m,3, () where A is the orthogonal matrix [ A = Hene, the least squares estimate of the CFR matrix at the mth subarrier an be obtained as Ĥ m = [ y m, y m,3 A m = H m + [ η m, η m,3 A () m In the IAM method, all subarriers are used to diretly estimate the orresponding CFR values It is worth noting that the length of the IAM preamble overhead is N t + (3N t for E-IAM-C) olumns, where N t olumns are used for hannel estimation Therefore, the IAM method enounters a onsiderable loss in spetral effiieny, espeially for a relatively large number of transmit antennas B FDM Method To improve the spetral effiieny, the FDM preamble design method with the sparse type has been proposed by sharing the subarriers among the antennas This subsetion provides a brief review of the FDM method presented in [4 In the FDM method, the pilots are designed speially to guarantee that the pseudo-pilots at different subarriers are of the frequenydivision multiplexed type As a onsequene, the hannels at the ative subarriers an be estimated separately without inter-antenna interferene The frequeny response values at the missing subarriers are then found via interpolation of neighboring subhannels Fig 3 depits an example for the ase N t = and M = 8, in whih t m,n is the pseudo-pilot (ie, the ideal AFB output) orresponding to the tth transmit antenna,, 4, 6,, 3, 5, 7, ( a) ( b) Fig 3 Reeived pilots at the AFB output for the FDM method, with (a) and (b) orresponding to the two transmit antennas and being the guard symbols As an be seen in Fig 3, nonzero pseudo-pilots for hannel estimation are distributed on odd/even subarriers of the middle olumn, with the rest of them being nulled Aording to (5), the orresponding demodulated symbols at the rth reeive antenna an be expressed as { ym, r Hm r, m, + η r = m,, m even, Hm r, m, + ηm,, r () m odd Therefore, the least squares estimates of Hm r, and Hm r, given by Ĥm r, = yr m, = H r, m, Ĥm r, = yr m, = H r, m, m + ηr m, m, m + ηr m, m, are, m even, (3), m odd, (4) respetively It is lear from (3) and (4) that the CFR values at the ative subarriers are perfetly estimated irrespetive of the additive noise Aordingly, the CFR values at the missing subarriers an be easily alulated by linear interpolation of neighboring subhannels It is worth pointing out that the preamble overhead of the FDM method is fixed three olumns of symbols, whih shows overwhelming superiority in spetral effiieny ompared to the IAM method However, the preamble in the existing FDM method is obtained without theoretial analysis and there is

5 5 no literature onerning the preamble optimization in terms of MSE performane urrently To enhane the estimation auray, it is neessary to design the orresponding optimal preamble for the FDM method IV PROPOSED PREAMBLE OPTIMIZATION METHOD In this setion, we propose a preamble optimization method for the FDM-strutured preamble to improve the hannel estimation performane in MIMO-OQAM/FBMC systems Aording to (3) and (4), it is lear that the FDMstrutured preamble optimization results from: (i) pseudopilots at the ative subarriers of maximum magnitudes to ontrol the estimation noise power, and (ii) pseudo-pilots at the missing subarriers of zero values so that there is no interferene on the orresponding points at the reeiver side To alulate the pseudo-pilot at eah FT point, one needs to for the neighbors (p, q ) Ω as in (4) It an be easily shown that the imaginary interferene oeffiients follow a speifi pattern for any employed prototype filter [ For odd n, know the imaginary interferene oeffiients ζ m,n m+p,n+q for the first-order FT neighbors are detailed in Table I The quantities δ, β, γ are onstants for a ertain system and an be omputed previously Note that, there might be some sign hanges for other definitions of ϕ [ the values of ζ m,n m+p,n+q TABLE I VALUES OF THE IMAGINARY INTERFERENCE COEFFICIENTS ζ m,n m+p,n+q ζ m,n m+p,n+q q = q = q = p = jδ jβ jδ p = jγ jγ p = jδ jβ jδ A Preamble Optimization Problem Given the symmetri and mutual isolation nature of pseudopilots distribution shown in Fig 3, we assume that preamble symbols are generated repeatedly on every N t adjaent subarriers For one speifi antenna, the preamble matrix at the SFB input is given as A = a a a 3 a 4 a 5 a 6 a 3Nt a 3Nt a 3Nt, (5) where a i, i =,,, 3N t represent the transmitted pilots, with a being transmitted at the ative subarrier and a 3i, i =, 3,, N t being transmitted at the missing subarriers The pseudo-pilots at orresponding points are denoted by 3i, i =,,, N t, whih an be alulated aording to (6) and the imaginary interferene oeffiients in Table I More speifially, we an express the pseudo-pilot as 3i = a 3i +ja (i) 3i, where ja(i) 3i is the intrinsi interferene from the neighboring FT points to a 3i Analogously, the preambles transmitted at other antennas an be taken as the frequeny shifted versions of those at this antenna In this paper, we formulate an optimization problem to determine the optimal preamble, where the objetive is to minimize the hannel estimation MSE, subjet to a total training energy onstraint per antenna Aording to (3) and (4), the hannel estimation MSE at the ative subarrier is given by MSE = E ( Ĥm H m ) = σ (6) It is lear from (6) that the larger the magnitude of the pseudo-pilot is, the better the hannel estimation will be Then, we onsider the training energy onstraint It should be noted that there is a differene between the energy input to the SFB and the energy of the modulated preamble at the SFB output due to the fat that the existene of omplexvalued pilot symbols would destroy the real orthogonality of the OQAM/FBMC system In our approah, more realistially, the transmit energy onstraint of the preamble is translated at the SFB output as in [35 Thus, the preamble optimization problem is formulated as min a,a,,a 3Nt MSE = σ /, (7a) subjet to 3i =, i =, 3,, N t (7b) P out ε, (7) where the onstraint (7b) means the pseudo-pilots at the missing subarriers are fored to zero and (7) refers to the SFB output energy onstraint (see Appendix A for the expression of P out ) In the following subsetions, we will disuss the different solving proesses for N t = and N t >, respetively B Closed-form Solution for N t = For two transmit antennas, the preamble matrix at the SFB input is given as a a a 3 A = a 4 a 5 a 6 a a a 3, (8) a 4 a 5 a 6 where a is transmitted at the ative subarrier and a 5 is transmitted at the missing subarrier Let a = [a, a, a 3, a 4, a 5, a 6 T C 6 and w = [jγ,, jγ, jδ,, jδ T denote the preamble vetor and the interferene weighting vetor, respetively Under the assumption of the interferene being mostly ontributed first-order neighboring FT points, the energy of the modulated preamble at the SFB output beomes P out = M ah Ba, (9)

6 6 where B = jγ jδ jγ jγ jδ jδ jγ jδ jδ jγ jδ jδ jγ jγ jδ jγ is a full-rank Hermitian matrix Besides, aording to (6), the pseudo-pilots at orresponding points are obtained as { = a + j [(a 4 + a 6 )δ + (a 3 a )γ = w H a, 5 = a 5 + j [(a + a 3 )δ + (a 6 a 4 )γ = w H Ta, () where [ 3 I T = 3 I 3 3 is a 6 6 transformation matrix Resorting to matrix notation the optimization problem (7) is modified as min = w H a, (a) a C 6 subjet to w H Ta =, (b) M ah Ba ε () Without loss of generality, we assume ε = M/ Then, the above optimization problem () beomes min w H a, (a) a C 6 subjet to w H Ta =, (b) a H Ba () As stated and proved in Appendix B, an optimal losed-form solution to () is given by a = B w (3) This solution indiates that in the ase of two transmit antenna, the optimal preamble depends only on the imaginary interferene oeffiients and an be pre-alulated based on the prototype filter g[k employed C Suboptimal Solution for N t > Given the symmetri and fast deay nature of the imaginary interferene oeffiients as in Table I, the preamble optimization problem for any N t > would be analogous Therefore, we take N t = 4 as an example and give a general solving proess In this ase, the preamble matrix of one antenna at the SFB input is given as a a a 3 A = a 4 a 5 a 6 a 7 a 8 a 9, (4) a a a where a is transmitted at the ative subarrier and a 5, a 8, a are transmitted at the missing subarriers Denoting by a = [a, a,, a T C the preamble vetor, by w = [ jδ, jβ, jδ, jγ,, jγ, jδ, jβ, jδ T the orresponding interferene weighting vetor, the energy of the modulated preamble at the SFB output beomes P out = M 4 ah Ba, (5) where B C an be pre-alulated analogously (the expression of B is omitted here for simpliity) Besides, the pseudo-pilots at orresponding points an be expressed as = w H T a, 5 = w H T a, 8 = w H (6) T a, = w H T 3 a, where T = T = T = T 3 = I 3 I I I I I I I I I I 3 I are both 9 transformation matries We assume the total training energy onstraint ε = M/4 Aording to the above derivations, the original optimization problem (7) for N t = 4 is modified as,,, min w H T a, (7a) a C subjet to w H T a =, (7b) w H T a =, w H T 3 a =, a H Ba (7) (7d) (7e) Reall that w H T i a = a H ( T H i ww H T i ) a and that x = x =, for all x C Hene, the above optimization problem (7) an be equivalently stated as min a H ( T H a C ww H ) T a, (8a) subjet to a H ( T H ww H ) T a =, (8b) a H ( T H ww H T ) a =, a H ( T H 3 ww H T 3 ) a =, a H Ba (8) (8d) (8e) It should be noted that the objetive funtion (8a) and the onstraints (8b), (8), (8d), and (8e) are both quadrati Therefore, the optimization problem (8) is a quadratially onstrained quadrati program (QCQP) Solving the general

7 7 ase of QCQP is an NP-hard problem and therefore laks omputationally effiient solution Observe that the optimal solution is hard to obtain for the above nononvex QCQP with omplex variables Therefore, in the following, we propose to onvert the original nononvex optimization into a onvex optimization by relaxing the nononvex onstraint and obtain the orresponding suboptimal solution Denoting by C i = T H i ww H T i, i =,,, 3 the orresponding oeffiient matries, we firstly perform the trae transformation on the objetive funtion and onstraints as a H C i a = Tr ( a H C i a ) = Tr ( C i aa H), i =,,, 3 (9) a H Ba = Tr ( a H Ba ) = Tr ( Baa H) (3) Correspondingly, the optimization problem (8) beomes min Tr ( C aa H), (3a) a C subjet to Tr ( C i aa H) =, i =,, 3 (3b) Tr ( Baa H) (3) In partiular, both the objetive funtion and onstraints in (3) are linear in the matrix aa H Therefore, we define a new variable X = aa H, with X being a rank one symmetri positive semidefinite matrix Then, the optimization problem (3) is rewritten as min Tr(C X), (3a) X C subjet to Tr(C i X) =, i =,, 3 (3b) Tr ( BX ), X, rank(x) =, (3) (3d) where in (3d) indiates that X is symmetri positive semidefinite Indeed, the only diffiult onstraint in (3) is the nononvex rank onstraint rank(x) = By dropping the nononvex rank-one onstraint, (3) is relaxed as min Tr(C X), (33a) X C subjet to Tr(C i X) =, i =,, 3 (33b) Tr ( BX ), X (33) (33d) The problem (33) is onvex and an be solved effetively using the onvex optimization toolbox CVX [36 Denoting by X a globally optimal solution to (33), the remaining problem is how to obtain the feasible solution ã to (3) from X Resorting to the matrix eigendeomposition, the matrix X is represented in terms of its eigenvalues and eigenvetors, ie, r X = λ i u i u H i, (34) i= where r is the rank of X and λ i is a salar, termed the eigenvalue orresponding to the eigenvetor u i Note that, the best rank-one approximation X m to X is given by X m = λ m u m u H m, (35) where λ m is the maximum eigenvalue of X Thus, the andidate solution to (3) is aordingly obtained as ã = λ m u m It should be noted that some additive map operations may also be needed to obtain a nearby feasible solution if λ m u m is not feasible, where the literature [37 an be referred for a more detailed disussion V SIMULATION RESULTS In this setion, simulations are onduted to evaluate the proposed preamble design method in MIMO-OQAM/FBMC systems We ompare the proposed method with the onventional IAM and FDM methods under and 4 4 MIMO onfigurations, respetively The simulation parameters for the MIMO-OQAM/FBMC systems are as follows There are M = 56 subarriers modulated by 4-QAM onstellations and the subarrier spaing is 94 khz Eah data frame onsists of omplex OQAM/FBMC symbols The wellknown PHYDYAS prototype filter with overlapping fator of 4 is employed beause of its strong frequeny loalization [38 The multipath hannel SUI-3 proposed by the IEEE 86 broadband wireless aess working group is adopted for our simulations [39 Meanwhile, the MIMO-OQAM/FBMC reeiver employs the LMMSE equalizer for data detetion With PHYDYAS filter, the imaginary interferene oeffiients are determined as in Table II for odd time index n Thus, the quantities in Setion IV are assigned as δ = 58, β = 393, γ = 5644 and the optimal preamble an be onstituted aordingly It is worth mentioning that the optimality of the proposed preamble is guaranteed on ondition that the intrinsi imaginary interferene only ome from the first-order neighborhood However, in the simulation experiments, the interferene to the (m, n) FT point from (m ±, n ± ) and (m ±, n ± 3) is non-negligible as detailed in Table II In onsequene, the residual interferene from data symbols would degrade the hannel estimation performane Therefore, we also propose the extended versions of the optimal preamble by following the pilot symbols with one and two olumns of zero symbols, whih are defined as G = and G =, respetively For fair omparison, the preambles of all methods are normalized to equal power at the SFB output The preamble overhead omparison among different preamble design methods is summarized in Tables III, whih shows great superiority of our proposed method ompared to the IAM variants in terms of spetral effiieny Then, the omparison of peak-to-average power ratio (PAPR) performane is onduted among different methods It is not surprising that the periodi preamble struture of the proposed method and the three IAM variants, namely IAM-R, IAM-C, and E- IAM-C, would result in high PAPR signal Nonetheless, the proposed preamble still yields the best PAPR harateristis at the SFB output (PAPR 8 db) ompared to that of the IAM-R ( 44 db), IAM-C ( 56 db), and E-IAM-C ( 65 db) preambles In the following we will ompare the MSE and BER performanes of different preamble design methods under and 4 4 MIMO onfigurations, respetively The ase of two transmit antennas is onsidered firstly Fig 4 and Fig 5 show the MSE and BER performanes

8 8 TABLE II VALUES OF THE IMAGINARY INTERFERENCE COEFFICIENTS WHEN PHYDYAS FILTER IS USED ζ m,n m+p,n+q q = 3 q = q = q = q = q = q = 3 p = 49j 5j 58j 393j 58j 5j 49j p = 668j 5644j 5644j 668j p = 49j 5j 58j 393j 58j 5j 49j TABLE III PREAMBLE OVERHEAD COMPARISON FOR CHANNEL ESTIMATION IN MIMO-OQAM/FBMC SYSTEMS Preamble overhead Transmit antennas Conventional Proposed method IAM-R IAM-C E-IAM-C FDM [4 without null G = G = N t = N t = N t N t + N t + 3N t of different preamble design methods for a MIMO- OQAM/FBMC system, respetively For the sake of ompleteness, the performane urves of the CP-OFDM system are also presented with the CP length being hosen as M 8 aording to the LTE speifiation [4 Sine the residual interferene is non-negligible as in Table II, the pseudo-pilots in (9) differ from eah other and the orret pseudo-pilots are used in the simulations Besides, the intrinsi real-valued interferenes from the imaginary-valued pilot symbols to data symbols are also onsidered for aurate data detetion BER CP OFDM IAM R IAM C E IAM C Conventional FDM [4 Proposed Proposed, G= Proposed, G= 5 5 CP OFDM IAM R IAM C E IAM C Conventional FDM [4 Proposed Proposed, G= Proposed, G= E b /N (db) MSE (db) 5 Fig 5 Comparison of BER performanes among different preamble design methods for a MIMO-OQAM/FBMC system SNR (db) Fig 4 Comparison of MSE performanes among different preamble design methods for a MIMO-OQAM/FBMC system As observed in Fig 4, the proposed method exhibits a signifiant performane gain over the onventional FDM method for all SNR regions and all methods perform better than CP-OFDM at low SNRs The well-known error floor of the IAM variants and the proposed method at high SNRs is also observed due to the unavoidable intrinsi interferene Besides, the proposed method outperforms the IAM-R and IAM-C methods at low to medium SNRs The reason is that the proposed method ontributes to the maximum magnitudes of pseudo-pilots at the ative subarriers, whih weakens the effet of noise However, in the ase of relatively higher SNRs, the noise beomes nonsignifiant and the interferene pre- anellation errors (inluding the residual intrinsi interferene whih omes from the invalidation of model () and the error brought by linear interpolation of neighboring subhannels) in the proposed method prevail, whih aounts for the redued derease speed of the proposed urve Nevertheless, we an see that the proposed preamble with G = ahieves hannel estimation MSE smaller than the IAM variants and the error floor of the proposed method is almost ompletely removed if two olumns of zero symbols are plaed immediately behind the pilots The orresponding unoded BER performane omparison and a zoomed-in version of part of BER urves are presented in Fig 5 As observed, our proposed method ahieves BER performane omparable to the IAM variants To verify the theoretial result in subsetion IV-C, the simulation is also onduted for the MIMO 4 4 onfiguration, with the simulation parameters being set up similarly For the proposed method, the approximate optimal solutions of a, a,, a are alulated as, 97,, 95,, 95,,,, 95,, 95 aording to the solving proess in subsetion IV-C Hene, the preambles transmitted at eah of the transmit antennas are

9 9 obtained by the orresponding repetition and frequeny shift operations Besides, the IAM variants are also tested for the sake of the omparison, where the hanged signs in () are seleted as a Hadamard matrix of order four [ MSE (db) CP OFDM IAM R IAM C E IAM C Conventional FDM [4 Proposed Proposed, G= Proposed, G= SNR (db) Fig 6 Comparison of MSE performanes among different preamble design methods for a 4 4 MIMO-OQAM/FBMC system BER CP OFDM IAM R IAM C E IAM C Conventional FDM [4 Proposed Proposed, G= Proposed, G= E b /N (db) Fig 7 Comparison of BER performanes among different preamble design methods for a 4 4 MIMO-OQAM/FBMC system In the 4 4 MIMO-OQAM/FBMC system, the MSE and unoded BER performanes of different preamble design methods are depited in Fig 6 and 7, respetively From the simulation results, the proposed method with the optimized preamble outperforms the onventional FDM method at all SNR regimes in terms of MSE and BER performanes Besides, the proposed method performs slightly better than the IAM-R and IAM-C methods at low to medium SNRs, whih validates the effiieny of the proposed method However, sine three fourths of subhannels are obtained by the linear interpolation of neighboring subhannels, the hannel estimation performane of the proposed method is worse than the E-IAM-C method due to the interferene pre-anellation errors Nevertheless, as is shown in Table III, the preamble overhead of the E-IAM-C method at eah transmit antenna is twelve olumns of symbols, while only three olumns of symbols are needed with the proposed method Besides, the proposed preamble with zero symbols being plaed behind ould signifiantly improve the hannel estimation MSE and BER performanes as in the two transmit antennas ase Thus, the proposed method provides a satisfatory ompromise between hannel estimation performane and preamble overhead to systems with more than two transmit antennas VI CONCLUSIONS In this paper, we formulated an optimization problem to determine the frequeny division multiplexed preambles in MIMO-OQAM/FBMC systems and gave the orresponding solutions for two transmit antennas and more than two transmit antennas, respetively In the ase of two transmit antennas, we found the relationship between preambles and the intrinsi interferene from neighboring symbols to minimize the MSE of the hannel estimation, and derived the optimal losed-form solution In the ase of more than two transmit antennas, we presented a general solving proess to obtain the suboptimal solution by relaxing the nononvex onstraint The MSE and BER performanes of different preamble design methods were ompared through simulations under and 4 4 MIMO onfigurations, respetively Simulation results have demonstrated that the proposed method with the optimized preamble shown great superiority ompared with the onventional FDM method Besides, a better hannel estimation performane an be ahieved than the IAM-R and IAM-C method at low to medium SNR regimes Although a performane worse than the E-IAM-C method was experiened, the proposed method only requires fixed three olumns of preamble symbols, resulting in onsiderable improvement in spetral effiieny ompared with the IAM variants APPENDIX A THE ENERGY OF THE MODULATED PREAMBLE AT THE SFB OUTPUT Aording to (), the energy of the modulated preamble at the SFB output an be derived as P out = = = = + k= + k= M s[ks[k {[ M m = n = M [ M m = n = m = m = n = n = + M M k= m = m = n = n = a m,n g m,n [k } a m,n gm,n [k { a m,n a m,n } g m,n [kgm,n [k a m,n a m,n ζ m,n m,n (36)

10 Let us denote by d n = [a,n, a,n, a,n,, a M,n T, n =,, the preamble vetors Define V p,q as the transmultiplexer response matrix of size M M whose elements are given by [V p,q i,j = ζ i,p j,q for i {,,, M } and j {,,, M } Then, we an easily rewrite (36) as P out = [ d H d H d H V, V, V, V, V, V, V, V, V, d d } {{ }} {{ } V d = d H Vd, (37) where d C 3M is the ombined preamble vetor and V C 3M 3M is the transmultiplexer response matrix It is noteworthy that V an be pre-alulated based on the prototype filter g[k employed APPENDIX B CLOSED-FORM SOLUTION FOR THE PROBLEM () It is lear that the matrix B in (9) is unitarily equivalent to a diagonal matrix due to its Hermitian property Let us denote by B = UΣU H, with U unitary matrix and Σ real diagonal matrix, the eigenvalue deomposition of the Hermitian matrix B Define ã = Σ / U H a as the new variable Then, the optimization problem () an be equivalently restated as d min w H UΣ / ã, (38a) ã C 6 subjet to w H TUΣ / ã =, (38b) ã H ã (38) We may as well drop the onstraint (38b) to obtain the following relaxed version of (38) min κ H ã, (39a) ã C 6 subjet to ã H ã, (39b) where κ = Σ / U H w C 6 Based on the onepts and properties of Hermitian inner produt in linear algebra, it is lear that the objetive funtion in (39) an be minimized on ondition that ã is ollinear with κ, whih means all omponents of ã are in the same ratio to the orresponding omponents of κ Thus, the optimal solution to (39) is ã = κ = Σ / U H w (4) It an be readily heked that the solution (4) also satisfies (38b), whih means κ is a feasible solution to (38) Note that, the optimal objetive value of (39) is definitely not greater than that of (38) Therefore, we ould onlude that κ yields the best objetive among all vetors that satisfy the onstraints in the problem (38) Hene, κ is also the optimal solution to (38) and the optimal solution to the original problem () is given by a = UΣ / ã = UΣ U H w = B w (4) This solution produes losed-form expression for the optimal preamble in the ase of two transmit antennas, whih depends only on the employed prototype filter REFERENCES [ B Farhang-Boroujeny, OFDM versus filter bank multiarrier, IEEE Signal Proess Mag, vol 8, no 3, pp 9-, May [ B Farhang-Boroujeny and H Moradi, OFDM inspired waveforms for 5G, IEEE Commun Surveys Tuts, vol 8, no 4, pp , 4th Quart 6 [3 X Gao, W Wang, X Xia, E K S Au, and X You, Cyli prefixed OQAM-OFDM and its appliation to single-arrier FDMA, IEEE Trans Commun, vol 59, no 5, pp , May [4 L Zhang, P Xiao, A Zafar, A Quddus, and R Tafazolli, FBMC system: An insight into doubly dispersive hannel impat, IEEE Trans Veh Tehnol, vol 66, no 5, pp , May 7 [5 L Zhang, A Ijaz, P Xiao, A Quddus, and R Tafazolli, Subband filtered multi-arrier systems for multi-servie wireless ommuniations, IEEE Trans Wireless Commun, vol 6, no 3, pp , Mar 7 [6 R Zakaria and D L Ruyet, A novel filter-bank multiarrier sheme to mitigate the intrinsi interferene: Appliation to MIMO systems, IEEE Trans Wireless Commun, vol, no 3, pp -3, Mar [7 D Kong, D Qu, and T Jiang, Time domain hannel estimation for OQAM-OFDM systems: Algorithms and performane bounds, IEEE Trans Signal Proess, vol 6, no, pp 3-33, Jan 4 [8 PHYsial layer for DYnami spetrum AesS and ognitive radio (PHYDYAS), [Online Available: [9 Enhaned multiarrier tehniques for professional ad-ho and ellbased ommuniations (EMPhAtiC), FP7 ICT projet, [Online Available: [ P Banelli, S Buzzi, G Colavolpe, A Modenini, F Rusek, and A Ugolini, Modulation formats and waveforms for 5G networks: Who will be the heir of OFDM? IEEE Signal Proess Mag, vol 3, no 6, pp 8-93, Nov 4 [ Q Li, H Niu, A Papathanassiou, and G Wu, 5G network apaity: Key elements and tehnologies, IEEE Veh Tehnol Mag, vol 9, no, pp 7-78, Mar 4 [ S Jin, W Tan, M Matthaiou, J Wang, and K K Wong, Statistial eigenmode transmission for the MU-MIMO downlink in Riian fading, IEEE Trans Wireless Commun, vol 4, no, pp , De 5 [3 R Zhang and C K Ho, MIMO broadasting for simultaneous wireless information and power transfer, IEEE Trans Wireless Commun, vol, no 5, pp 989-, May 3 [4 F Rottenberg, X Mestre, F Horlin, and J Louveaux, Single-tap preoders and deoders for multiuser MIMO FBMC-OQAM under strong hannel frequeny seletivity, IEEE Trans Signal Proess, vol 65, no 3, pp 587-6, Feb 7 [5 X Mestre and D Gregoratti, Parallelized strutures for MIMO FBMC under strong hannel frequeny seletivity, IEEE Trans Signal Proess, vol 64, no 5, pp -5, Mar 6 [6 A I Pérez-Neira, M Caus, R Zakaria, et al, MIMO signal proessing in offset-qam based filter bank multiarrier systems, IEEE Trans Signal Proess, vol 64, no, pp , Nov 6 [7 R Nissel and M Rupp, On pilot-symbol aided hannel estimation in FBMC-OQAM, in Pro IEEE Int Conf Aoust Speeh Signal Proess, Mar 6, pp [8 J Javaudin and Y Jiang, Channel estimation in MIMO OFDM/OQAM, in Pro Int Workshop Signal Proess Adv Wireless Commun (SPAWC), Jul 8, pp 66-7 [9 J Javaudin and Y Jiang, Channel estimation for iterative MIMO OFDM/OQAM transeivers, in Pro Eur Wireless Conf (EW), Jun 8, pp -6 [ E Kofidis, D Katselis, A Rontogiannis, and S Theodoridis, Preamble based hannel estimation in OFDM/OQAM systems: A review, Signal Proess, vol 93, no 7, pp 38-54, Jul 3 [ E Kofidis and D Katselis, Preamble-based hannel estimation in MIMO-OFDM/OQAM systems, in Pro IEEE Int Conf Signal Image Proess Appl (ICSIPA), Nov, pp [ E Kofidis, Preamble-based estimation of highly frequeny seletive hannels in MIMO-FBMC/OQAM systems, in Pro th Eur Wireless Conf (EW), May 5, pp -6 [3 M Nájar, C Bader, F Rubio, et al, MIMO hannel matrix estimation and traking, PHYDYAS D4 deliverable, Available: [4 S Taheri, M Ghoraishi, P Xiao, A Cao, and Y Gao, Evaluation of preamble based hannel estimation for MIMO-FBMC systems, ZTE Commun, vol 4, no 4, pp 3-, Ot 6

11 [5 D Chen, D Qu, T Jiang, and Y He, Prototype filter optimization to minimize stopband energy with NPR onstraint for filter bank multiarrier modulation systems, IEEE Trans Signal Proess, vol 6, no, pp 59-69, Jan 3 [6 W Wang, X Gao, F C Zheng, and W Zhong, CP-OQAM-OFDM based SC-FDMA: Adjustable user bandwidth and spae-time oding, IEEE Trans Wireless Commun, vol, pp , Aug 3 [7 W Liu, D Chen, D Kong, and T Jiang, Preamble overhead redution with IAM-C for hannel estimation in OQAM-OFDM systems, in Pro IEEE China Summit Int Conf Signal Inf Proess (ChinaSIP), Jul 5, pp [8 D Kong, X Xia, T Jiang, and X Gao, Channel estimation in CP- OQAM-OFDM systems, IEEE Trans Signal Proess, vol 6, no, pp , Nov 4 [9 H Minn and N Al-Dhahir, Optimal training signals for MIMO OFDM hannel estimation, IEEE Trans Wireless Commun, vol 5, no 5, pp 58-68, May 6 [3 P Siohan, C Silet, and N Laaille, Analysis and design of OQAM- OFDM systems based on filterbank theory, IEEE Trans Signal Proess, vol 5, no 5, pp 7-83, May [3 C Lélé, J-P Javaudin, R Legouable, A Skrzypzak, and P Siohan, Channel estimation methods for preamble-based OFDM/OQAM modulations, European Trans Teleommun, vol 9, no 7, pp 74-75, Mar 7 [3 C Lélé, P Siohan, and R Legouable, db better than CP-OFDM with OFDM/OQAM for preamble-based hannel estimation, in Pro IEEE Int Conf Commun (ICC), May 8, pp 3-36 [33 J Du and S Signell, Novel preamble-based hannel estimation for OFDM/OQAM systems, in Pro IEEE Int Conf Commun (ICC), Jun 9, pp -6 [34 E Kofidis and D Katselis, Improved interferene approximation method for preamble-based hannel estimation in FBMC/OQAM, in Pro Eur Signal Proess Conf (EUSIPCO), Aug, pp [35 E Kofidis, Preamble-based estimation of highly frequeny seletive hannels in FBMC/OQAM systems, IEEE Trans Signal Proess, vol 65, no 7, pp , Apr 7 [36 M Grant, S Boyd, and Y Ye, CVX: MATLAB software for disiplined onvex programming [Online Available: [37 Z Luo, W Ma, A M C So, Y Ye, and S Zhang, Semidefinite relaxation of quadrati optimization problems: From its pratial deployments and sope of appliability to key theoretial results, IEEE Signal Proess Mag, vol 7, no 3, pp -34, May [38 A Viholainen, M Bellanger, and M Huhard, Prototype filter and struture optimization, [Online Available: [39 V Ereg, K V S Hari, M S Smith, et al, Channel models for fixed wireless appliations, Teh Rep, IEEE 86 Broadband Wireless Aess Working Group, Jul [4 I Barhumi, G Leus, and M Moonen, Optimal training designs for MIMO OFDM systems in mobile wireless hannels, IEEE Trans Signal Proess, vol 5, no 6, pp 65-64, Jun 3