Preamble and pilot symbol design for channel estimation in OFDM systems with null subcarriers

Transcription

1 RESEARCH Open Access Preamble and pilot symbol design for cannel estimation in OFDM systems wit null subcarriers Suici Ono *, Emmanuel Manasse and Masayosi Nakamoto Abstract In tis article, design of preamble for cannel estimation and pilot symbols for pilot-assisted cannel estimation in ortogonal frequency division multiplexing system wit null subcarriers is studied. Bot te preambles and pilot symbols are designed to minimize te l 2 or te l norm of te cannel estimate mean-squared errors (MSE) in frequency-selective environments. We use convex optimization tecnique to find optimal power distribution to te preamble by casting te MSE minimization problem into a semidefinite programming problem. Ten, using te designed optimal preamble as an initial value, we iteratively select te placement and optimally distribute power to te selected pilot symbols. Design examples consistent wit IEEE a as well as IEEE e are provided to illustrate te superior performance of our proposed metod over te equi-spaced equi-powered pilot symbols and te partially equi-spaced pilot symbols. Keywords: Ortogonal frequency division multiplexing (OFDM), Cannel estimation, Semidefinite programming (SDP), Convex optimization, Pilot symbols, Pilot design I. Introduction Ortogonal frequency division multiplexing (OFDM) is an effective ig-rate transmission tecnique tat mitigates inter-symbol interference (ISI) troug te insertion of cyclic prefix (CP) at te transmitter and its removal at te receiver. If te cannel delay spread is sorter tan te duration of te CP, ISI is completely removed. Moreover, if te cannel remains constant witin one OFDM symbol duration, OFDM renders a convolution cannel into parallel flat cannels, wic enables simple one-tap frequency-domain equalization. To obtain te cannel state information (CSI), training OFDM symbols or pilot symbols embedded in eac OFDM symbol are utilized. Training OFDM symbols or equivalently OFDM preambles are transmitted at te beginning of te transmission record, wile pilot symbols (complex exponentials in time) are embedded in eac OFDM symbol, and tey are separated from information symbols in te frequency-domain [1-3]. If te cannel remains constant over several OFDM symbols, cannel estimation by training OFDM symbols may be sufficient for symbol detection. But in te event of cannel variation, training OFDM symbols sould be * Correspondence: ono@irosima-u.ac.jp Hirosima University, Kagamiyama, Higasi-Hirosima , Japan retransmitted frequently to obtain reliable cannel estimates for detection. On te oter and, to track te fast varying cannel, pilot symbols are inserted into every OFDM symbol to facilitate cannel estimation. Tis is known as pilot-assisted (or -aided) cannel estimation [2,4,5]. Te main drawback of te pilot-assisted cannel estimation lies in te reduction of te transmission rate, especially wen larger number pilot symbols are inserted in eac OFDM symbol. Tus, it is desirable to minimize te number of embedded pilot symbols to avoid excessive transmission rate loss. Wen all subcarriers are available for transmission, training OFDM preamble and pilot symbols ave been well designed to enance te cannel estimation accuracy, see e.g., [6] and references terein. If all te subcarriers can be utilized, ten pilot symbol sequence can be optimally designed in terms of (i) minimizing te cannel estimate mean-squared error [1,3]; (ii) minimizing te bit-error rate (BER) wen symbols are detected by te estimated cannel from pilot symbols [7]; (iii) maximizing te lower bound on cannel capacity wit cannel estimates [8,9]. It as been found tat equally spaced (equi-distant) and equally powered (equi-powered) pilot symbols are optimal wit respect to several performance measures Ono et al; licensee Springer. Tis is an Open Access article distributed under te terms of te Creative Commons Attribution License (ttp://creativecommons.org/licenses/by/2.0), wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited.

2 Page 2 of 17 In practice, not all te subcarriers are available for transmission. It is often te case tat null subcarriers are set on bot te edges of te allocated bandwidt to mitigate interferences from/to adjacent bands [10,11]. For example, IEEE a as 64 subcarriers among wic 12 subcarriers, one at te center of te band (DC component) and at te edges of te band are set to be null, i.e., no information is sent [12]. Te presence of null subcarriers complicates te design of bot te training preamble for cannel estimation and pilot symbols for pilot-aided cannel estimation over te frequency-selective cannels. Null subcarriers may render equi-distant and equi-powered pilot symbols impossible to use in practice. In te literature, several pilot symbols design tecniques for OFDM systems wit null subcarriers ave been studied [11,13-16]. In [11], a metod tat assigns equal power to all pilot subcarriers and utilizes te exaustive searc metod to obtain te optimal pilot set is proposed. However, te approac in [11] optimizes only pilot placements of te equally powered pilot symbols. Te design of pilot symbols sould take into account te placements as well as power loading. Moreover, te exaustive searc becomes intractable for large number of pilot symbols and/or active subcarriers even if te searc process is carried out during te system design pase. To address te exaustive searc problem, te partially equi-spaced pilot (PEP) sceme, wic will be referred as PEP in tis article, is discussed in [13]. Te algoritm in [13] is novel as it can be employed to design pilot symbols for bot te MIMO-OFDM as well as SISO-OFDM systems. Furtermore, te design considers bot te placements and power distribution to te pilot symbols. However, te metod does not guarantee better performance for some cannel/subcarriers configuration. In [14], equi-powered pilot symbols are studied for cannel estimation in multiple antennas OFDM system wit null subcarriers. However, tey are not always optimal even for point-to-point OFDM system. Also in [15], a proposal was made tat employs cubic parameterizations of te pilot subcarriers in conjunction wit convex optimization algoritm to design pilot symbols. However, te accuracy of cubic function-based optimizations in [15] depends on many parameters to be selected for every cannel/subcarriers configuration wic complicates te design. Pilot sequences designed to reduce te MSE of te cannel estimation in multiple antenna OFDM system are also reported in [13,16] but tey are not necessarily optimal. In tis article, we optimally allocate power to te preamble as well as design pilot symbols to estimate te cannel in OFDM systems wit null subcarriers. Even toug tere is no closed form expression relating pilot placement wit te MSE, we propose an algoritm tat takes into account bot te pilot placements and power distribution. Our design criteria are te l 2 norm as well as te l norm of te MSE of cannel estimation in frequency-domain. Contrary to [15], were it is stated tat l is superior over l 2,weverifytattereisnosignificant difference in performance between te two norms. To find te optimal power allocation, we first sow tat te minimization problem can be casted into a semidefinite programming (SDP) problem [17]. Wit SDP, te optimal power allocation to minimize our criterion can be numerically found. We also propose an iterative algoritm tat uses te designed optimal preamble as an initial value to determine te significant placement of te pilot symbols and power distribution. Finally, we present design examples under te same setting as IEEE a and IEEE e to sow te improved performance of our proposed design over te PEPs and te equi-spaced equi-powered pilot symbols. We also made comparisons between our proposed design, PEP, and te design proposed by Baxley et al. in [15] for IEEE e. We demonstrated tat our proposed design can be used as a framework to design pilot symbols for different cannel/subcarriers configurations, and it is crucial to optimally allocate power to te pilot symbols to improve te MSE and BER performances. It is also verified tat, te conventional preamble of IEEE a is comparable to te optimally designed preamble. II. Preamble and pilot symbols for cannel estimation We consider point-to-point wireless OFDM transmissions over frequency-selective fading cannels. We assume tat te discrete-time baseband equivalent cannel as FIR of maximum lengt L, and remains constant in at least one block, i.e., is quasi-static. Te cannel impulse response is denoted as { 0, 1,..., L-1 }. Since we basically deal wit one OFDM symbol, we omit te index of te OFDM symbol for notational simplicity. Let us consider te transmission of one OFDM symbol wit N number of subcarriers. At te transmitter, a serial symbol sequence {s 0, s 1,..., s N-1 } undergoes serialto-parallel conversion to be stacked into one OFDM symbol. Ten, an N-points inverse discrete Fourier transform (IDFT) follows to produce te N dimensional data, wic is parallel-to-serial converted. A CP of lengt N cp is appended to mitigate te multipat effects. Te discrete-time baseband equivalent transmitted signals u n can be expressed in te time-domain as

3 Page 3 of 17 u n = 1 N 1 s k e j2πkn N, n [0, N 1]. (1) N k=0 Assume tat N cp is greater tan te cannel lengt L so tat tere is no ISI between te OFDM symbols. At te receiver, we assume perfect timing syncronization. After removing CP, we apply DFT to te received timedomain signal y n for n Î [0, N - 1] to obtain for k Î [0, N -1] Y k = 1 N 1 y n e j2πkn N = H k s k + W k, (2) N n=0 were H k is te cannel frequency response at frequency 2πk/N given by L 1 H k = l e j2πkl N, (3) l=0 and te noise W k is assumed to be i.i.d. circular Gaussian wit zero mean and variance σ 2 w. For simplicity of presentation, we utilize a circular index wit respect to N were te index n of a sequence corresponds to n modulo N. LetK be a set of active subcarriers (i.e., non-null subcarriers), ten te cardinality of a set K can be represented as K. Take WLAN standard (IEEE a), for example, were 64 subcarriers (or slots) are available in te OFDM symbol during data transmission mode. Out of wic 48 are utilized as information symbols, 4 as pilot symbols, wile te rest except for te DC subcarrier serves as spectral nulls to mitigate te interferences from/to OFDM symbols in adjacent bands. Tus, K = {1,2,..., 26, 38, 39,..., 63} and K =52. Te detailed structure of te OFDM packet in a timefrequency grid is sown in Figure 1. At te beginning of te transmission, two long OFDM preambles are transmitted to obtain CSI (see [[18], p. 600]). In IEEE a standard, te first part of te preamble consists of 10 sort pilot symbols in 12 subcarriers equally spaced at 4 subcarriers interval, wic is not sown in Figure 1. Te second part of te preamble initiation, wic corresponds to te first two OFDM symbols of Figure 1, requires te transmission of two columns of pilot symbols in all active subcarriers in order to make precise frequency offset estimation and cannel estimation possible [18]. For cannel estimation, we place N p ( K ) pilot symbols {p 1,..., p Np } at subcarriers k 1, k 2,..., k Np K(k 1 < k 2 < < k Np ), wic are known at te receiver. We assume tat N p L so tat 8.125MHz frequency te cannel can be perfectly estimated if tere is no noise, and we denote te index of pilot symbols as K p = {k 1,..., k NP }. Let diag(a) be a diagonal matrix wit te vector a on its main diagonal. Collecting te received signals aving pilot symbols as Ỹ =[Y k1,..., Y knp ] T, (4) we obtain MHz time Figure 1 Te time-frequency structure of an IEEE a packet. Saded subcarriers contain pilot symbols. Ỹ = D Hp p + W, (5) were D Hp is a diagonal matrix wit its nt diagonal entry being H kn suc tat D Hp = diag(h k1,..., H knp ), (6) and p is a pilot vector defined as p =[p 1,..., p Np ] T. (7) From Ỹ, we would like to estimate cannel frequency responses for equalization and decoding. (In tis article, we consider only cannel estimation by one OFDM symbol but te extension to multiple OFDM symbols could be possible). Let us define K s as an index set specifying te cannel frequency responses to be estimated. In oter words, H k for k K s ave to be estimated from Ỹ. In a long training OFDM preamble, all subcarriers in K can be utilized for pilot symbols so tat K p = K. On te oter and, in pilot-assisted modulation (PSAM) [4], a few known pilot symbols are embedded in an OFDM symbol to facilitate te estimation of unknown cannel. Tus,forPSAM,weaveK s = K\K p were\represents set difference.

4 Page 4 of 17 If we can adopt equally spaced (equi-distant) pilot symbols wit equal power for cannel estimation and symbol detection, ten it can be analytically sown tat te cannel mean-squared estimation error [1,3] as well as te BER [7] are minimized, wile te lower bound on cannel capacity [8,9] is maximized. But te optimality of equi-distant and equi-powered pilot symbols does not necessarily old true wen tere are null subcarriers. In tis article, for a given K, we use convex optimization tecnique to optimally distribute power to tese subcarriers. Ten, we propose an algoritm to determine pilot set K p wit significant power to be used for PSAM. III. Mean-squared cannel estimation error Let us define F as an N N DFT matrix, te (m +1,n +1)tentryofwicise -j2πmn/n.wedenoteann L matrix F L =[f 0,..., f N 1 ] H (8) consisting of N rows and first L columns of DFT matrix F, wereh is te complex conjugate transpose operator. We also define an N p L matrix F p aving f H k n for k n K p as its nt row. Ten, we can express (5) as Ỹ = D p F p + W, (9) were te diagonal matrix D p and cannel vector are respectively defined as D p =diag(p 1,..., p Np ), (10) and = [ 0,..., L 1 ] T. (11) Let a vector aving cannel responses to be estimated, i.e., H k for k K s,be H s =[H k1,..., H k Ks ] T. (12) Similar to F p, we define a K s L matrix F s aving f H k n for k n K s as its nt row, were k n <k n if n <n. Ten, we obtain H s = F s. (13) We assume tat te mean of te cannel coefficients is zero, i.e., E{} = 0 and te cannel correlation matrix is R = E{ H }, (14) were E{ } stands for te expectation operator. Ten, since (9) is linear, te minimum mean-squared error (MMSE) estimate Ĥ s of H s is given by [19] ( 1Ỹ. Ĥ s = E{H s Ỹ H } E{ỸỸ }) H (15) It follows from (9) and (13) tat E{H s Ỹ H } = F s R F H p DH p, (16) and E{ỸỸ H } = D p F p R F H p DH p + σ w 2 I. (17) We utilize te notation A 0 (or A 0) for a symmetric matrix A to indicate tat A is positive semidefinite (or positive definite). Let us assume R 0forte simplicity of presentation. If we define te estimation error vector E s as E s = Ĥ s H s, (18) ten, te correlation matrix R e of E s can be expressed as [19] [ R e = E{E s E H s } = F s R ] 1 2 F H p pf p F H s, (19) were Λ p is a diagonal matrix given by p = D H p D p = diag(λ 1,..., λ Np ), (20) wit λ n = p kn 2 for k n K p. On te oter and, te least squares (LS) estimate of H s is found to be F s (D p F p ) Ỹ,were( ) stands for te pseudo-inverse of a matrix. Te LS estimate does not require any prior knowledge on cannel statistics and is tus widely applicable. In contrast, te second order cannel statistics R = E{ H } and te noise variance 2 are essential to compute te MMSE estimate. Wen te signal-to-noise ratio (SNR) gets larger, i.e., 2 gets smaller for a given signal power, te MMSE estimate converges to te LS estimate. In general, te LS estimate can be easily obtained from te MMSE design by setting R =0and 2 = 1. Tus,toavoid possible duplications in te derivations, we only consider te MMSE estimate. In place of te cannel frequency responses, one may want to estimate te cannel coefficient directly. Similar to (15) and (19), te MMSE estimate ĥ of and te error correlation matrix are found to be ( ĥ = E{Ỹ H 1Ỹ, } E{ỸỸ }) H (21) and { [ĥ ] } [ H E ][ĥ = R ] 1 2 F H p pf p. (22)

5 Page 5 of 17 If F H s F s = ci for a non-zero constant c, ten from (19), E{ E s 2 } = ce{ ĥ 2 }, were denotes te Euclidean norm. Te equation F H s F s = ci is attained if all pilot symbols ave te same power and are uniformly distributed in an OFDM symbol. But, tis is not always possible if tere are null subcarriers in te OFDM symbol. As sown later, even wit null subcarriers, te minimization of E{ ĥ 2 } becomes possible. Now, our objective is to find te optimal pilot symbols tat minimize a criterion function. Two important criteria are considered. One is te l 2 norm of te meansquared cannel estimation errors {r k } k Ks at data subcarriers, wic is defined as η 2 = k Ks Were r k = (trace R e ) 2, (23) r k = E{ Ĥ k H k 2 }. (24) Te oter is te maximum of {r k } defined as η =maxr k, k K s (25) wic is te l norm of {r k } k Ks. It sould be remarked tat η2 2 = E{ Ĥ s H s 2 } ce{ ĥ 2 } if F H s F s ci. To differentiate tem, we call te former te frequency-domain cannel MSE and te latter te timedomain cannel MSE. In te long preamble of IEEE a standard, equipowered pilot symbols are utilized but may not be optimal due to te existence of null subcarriers. Equi-powered pilot symbols are also investigated for cannel frequency response estimation in multiple antenna OFDM system wit null subcarriers [14]. To reduce te sum of cannel MSE for multiple antenna OFDM system, pilot symbol vector p as been designed to satisfy F H p pf p = I p in [16]. However, suc pilot sequence does not always exist. In addition, te necessary and sufficient condition for its existence witin te active subcarrier band as not yet been fully establised. IV. Pilot power distribution wit SDP For any prescribed energy to be utilized for cannel estimation, we normalize te sum of pilot power suc tat N p p k 2 = λ k =1. (26) k K p k=1 Ten, our problem is to determine te optimal λ =[λ 1,..., λ Np ] T, (27) tat minimizes 2 in (23) or in (25) under te constraint (26). We first consider te minimization 2. Te optimal power distribution can be obtained by minimizing te squared 2 in (23) wit respect to l under te constraints tat [1,...,1] λ =1, λ 0, (28) were a 0 (or a 0) for a vector signifies tat all entries of a are equal to or greater tan 0 (or strictly greater tan 0). As stated in te previous section, analytical solutions could not be found in general. As in [20], we will resort to a numerical design by casting our minimization problem into a SDP problem. Te SDP covers many optimization problems [17,21]. Te objective function of SDP is a linear function of a variable x Î R M subject to a linear matrix inequality (LMI) defined as F(x) =A 0 + M x m A m 0, (29) m=1 were A m Î R M M. Te complex-valued LMIs are also possible, since any complex-valued LMI can be written by te corresponding real-valued LMI. Since te constraints defined by te LMI are convex set, te global solution can be efficiently and numerically found by te existing routines. By re-expressing te nt row of F p as fh,ourmse n minimization problem can be stated as min λ trace ( R σ 2 w N P n=1 λ n fn fh n ) 1 R subject to [1,...,1] λ 1, λ 0, (30) were R = F H s F s. Tis problem possesses a similar form as te transceiver optimization problem studied in [22], wic is transformed into an SDP problem. Similar to te problem in [22], our problem can be transformed into an SDP form. Now let us introduce an auxiliary Hermite matrix variable W and consider te following problem: min trace (W R) W,λ subject to [1,...,1]λ 1, λ 0 n=1 (31) ( ) W R N 1 P 2 λ n fn fh n. (32)

6 Page 6 of 17 It is reasonable to assume tat te number of data carriers is greater tan te cannel lengt, i.e., K s > L, so tat R 0. For R 0, if W ( R σ 2 w ( R2 WR2 R2 R σ 2 w NP λ n f n=1 n fh n NP λ n f n=1 n fh n ) 1, ten ) 1 R1 2 [[23], p. 470]. From [[23], p. 471] it can be sown tat ( ) 1 trace (R2 WR2 ) trace R2 R NP 2 λ n fn fh n R1 2, Wic is equivalent to ( ) trace(w R) trace R N 1 P 2 λ n fn fh n R. (33) It follows tat te minimization of trace(w R)is ( acieved if and only if W = R ) 1 NP 2 λ n f n=1 n fh n, wic proves tat te minimization of trace (W R)in (31) is equivalent to te original minimization problem in (30). Similarly, it as been proved in [[23], p.472] tat R I NP n=1 λ n f n fh n I n=1 n=1 (34) W NP is positive definite if and only if R λ n f n=1 n fh n 0 ( and W R ) 1 NP 2 λ n f n=1 n fh n Tus, te constraint (32) can be rewritten as sown in (36) below. Finally, we reac te following minimization problem wic is equivalent to te original problem: min trace(w R) W,λ subject to [1,..., 1] λ 1, λ 0 R I Np n=1 λ n f n fh n I (35) 0. (36) W Tis is exactly an SDP problem were te cost function is linear in W and l, andteconstraintsareconvex, since tey are in te form of LMI. Tus, te global optimal solution can be numerically found in polynomial time [17,21]. We ave discussed te design of pilot symbols minimizing te frequency-domain cannel estimate MSE and are in general more preferable tan pilot symbols minimizing te time-domain cannel estimate MSE. Pilot symbols minimizing te time-domain cannel estimate MSE can be obtained by just replacing R wit I in (30) (cf. (23) and (22)), and apply te same design procedure used for te pilot symbols minimizing te frequency-domain cannel estimate MSE. Next, we consider te minimization of in (25), tat is, min max r k, λ k K s subject to [1,..., 1] λ 1, λ 0. Te minimization is equivalent to (37) min λ,ν ν (38) subject to (28) and r k ν for all k K s. (39) It follows from (19) tat [ r k = f H k R ] 1 2 F H p pf p f k. (40) By using Scur s complement, (39) can be written as ( R ) NP 2 n=1 λ n f n fh n f k 0, for all k K s. (41) ν f H k Since (41) is convex, te minimization problem in (38) is also a convex optimization, and can be solved numerically. Compared to te minimization of te l 2 norm, te minimization of te l norm ave K s 1 constraints, wic lowers te speed of numerical optimization. V. Pilot design As we ave seen, for a given set of subcarriers, te optimal pilot symbols are obtained by resorting to numerical optimization. In te OFDM preamble, all active subcarriers can be utilized for cannel estimation so tat we ave N p = K. On te oter and, in a pilot-assisted OFDM symbol, we ave to select pilot subcarriers and allocate power to pilot and data subcarriers. To determine te optimal set K p aving N p entries, i.e., te optimal location of N p pilot symbols, we ave to enumerate all possible sets, ten optimize te pilot symbols for eac set and compare tem. Tis design approac becomes infeasible as K gets larger. In [15], te pilot location is caracterized wit a cubic function, and an iterative pilot symbol design for LS cannel estimation as been developed by using te cubic function. Te cubic parameterization can also be applicable to our optimization. However, te parameterization depends on several parameters to be selected for every cannel/subcarriers configuration, and for eac set of parameters, te objective function as to be iteratively optimized wic complicates te design. In [20], anoter pilot selection sceme as been proposed

7 Page 7 of 17 and is reported in [15] tat for some special cases, it does not work well. In tis article, we improve te metod of [20] by introducing an iterative algoritm as follows. Let N r (i) be a positive even integer. First, we use a designed optimal preamble wit SDP and denote its l k as λ 0 1,..., λ(0) K, ten, we remove N r (0) subcarriers wit minimum power symmetrically about te center (DC) subcarrier, i.e., N r (0) /2 on every side of te central DC subcarrier. Ten, we optimize te remaining pilot symbols. Similarly, for te it iteration, after removing subcarriers corresponding to N r (i) minimum power, we optimize pilot power for te remaining set again wit SDP. Wen te iterative algoritm is completed, we will remain wit only K p subcarrier indexes and its corresponding optimal power. Our design procedure is as outlined by te pseudocode algoritm below: 1) Set i =0. 2) Obtain te optimal preamble using convex optimization and initialize temporary set K (i) p = K. 3) If N p < K (i) p, remove from K (i) p, N (i) r subcarriers wit minimum power symmetrically wit respect to te DC subcarrier, else go to step 5. 4) Optimize te power of te remaining subcarriers using SDP and go to step 3 after updating i i +1. 5) Exit. Te value of N r (i) ( 2) is not fixed. Te number of iterations can be reduced by increasing te value of N r (i). However, wen te number of removed subcarriers N r (i) is large, te proposed sceme may not work well for some cannel/subcarriers configuration as in [20], were te significant N p subcarriers of te optimized preamble are selected at once. To obtain a better pilot set for any cannel/subcarriers configuration, te number of removed subcarriers sould be kept smaller. Tere is a tradeoff between te computational complexity and te estimation performance of te resultant set. Since we can design pilot symbols off-line, we can set te minimum for N r (i) suc as N r (i) =2. N (i) r VI. Design examples In tis section, we demonstrate te effectiveness of our proposed preamble and pilot symbols designs troug computer simulations. Te parameters of te transmitted OFDM signal studied in our design examples are as in te IEEE a and IEEE e (WiMaX) standards. For IEEE a, an OFDM transmission frame wit N = 64 subcarriers is considered. Out of 64 subcarriers, 52 subcarriers are used for pilot and data transmission wile te remaining 12 subcarriers are null subcarriers [[18], p. 600]. For IEEE e standard, an OFDM transmission frame in [[24], p. 429] is considered. In a data-carrying symbol 200 subcarriers of te N = 256 subcarrier window are used for data and pilot symbols. Of te oter 56 subcarriers, 28 subcarriers are null in te lower-frequency guard band, 27 subcarriers are nulled in te upper frequency guard band, and one is te central null (DC) subcarrier. Of te 200 used subcarriers, 8 subcarriers are allocated as pilot symbols, wile te remaining 192 subcarriers are used for data transmission. In te simulations, te total power of eac OFDM frame is normalized to one, but power distribution among pilot symbols is not constrained to be uniform. Te diagonal element of cannel correlation matrix is set to be E{ m n } = cδ(m n)e 0.1n for m, n Î [0, L - 1], were δ( ) stands for Kronecker s delta,andc is selected suc tat trace R =1. A. Preamble design First, we start wit te design of preamble were all active subcarriers are considered as pilot symbols. For a given cannel lengt L, to design an OFDM preamble, we optimize all active subcarriers by minimizing te l 2 norm or te l norm of MSEs {r k } k Ks using convex optimization package in [25]. Figure 2 depicts te optimal power distribution to te IEEE a preamble designed by l 2 norm wen te SNR is 10 db and te cannel lengt L = 4. We omit te power distribution by l norm, since it is nearly identical to tat of te l 2 norm-based design. Unlike te standard preamble were equal power is allocated to all te subcarriers, our optimized preamble distribute power to te subcarriers suc tat te cannel estimate MSE is minimized. We also consider a case wen te cannel lengt L = 8. Te results in Figure 3 sow te optimized preamble at 10 db. Again, tere is no significant difference between te design wit l 2 norm and te design wit l norm. However, te computational complexity of te design wit l 2 norm is quite lower tan te computational complexity of te design wit l norm, tereby making te former more preferable to te latter. Even toug te design process is usually done in off-line, suc minor advantage may be an important factor wen designing preambles and pilot symbols for an OFDM frame wit a large number of subcarriers. Figures 2 and 3 sow tat te designed preambles are symmetric around 0. Tis is due to te symmetric nature of our objective function and its constraints. Tere are differences in power distribution to te

8 Page 8 of Pilot Preamble 0.2 λ subcarrier Figure 2 Power of te preamble and pilot symbols designed by te l 2 norm for L = N p = 4 at 10 db (IEEE a) Pilot Preamble λ Subcarrier Figure 3 Power of te preamble and pilot symbols designed by te l 2 norm for L = N p = 8 at 10 db (IEEE a).

9 Page 9 of 17 designed preambles wen L =4andL = 8, tis suggests tat in te preamble design, equi-powered subcarriers may not necessarily be optimal wen tere are null subcarriers. Tis may not be well encapsulated in te overall cannel estimate MSE. However, wen considering te cannel estimate in eac subcarrier, tere is a sligt difference between te proposed designs and te standard preamble especially at te edges of te active band. To verify tis, we compare frequency-domain cannel MSE 2 obtained by te l 2 and l norm-based design wit te standard IEEE a preamble. By varying te cannel lengt L, from 1 to 16, we numerically obtain te cannel estimate MSE for eac L. Figure 4 presents te frequency-domain cannel MSE 2, against cannel lengt L at 10 db. From te plot, it is obvious tat tere is no significant difference between te tree designs, wic suggests tat te standard preamble is almost optimal in te l 2 sense even if tere are null subcarriers. Tis is not so surprising since in te absence of null subcarriers, equi-powered preamble is optimal. Troug our design approac, we numerically corroborate tat for IEEE a, te standard preamble is nearly optimal. To demonstrate te versatility of our metod, we minimize te LS cannel estimate MSE to design preambles for te IEEE e standard. Figure 5 sows te designed preamble of IEEE e for L = 16. Similar to a, te distribution of power to te active subcarriers is not uniform. Tis furter suggests tat equi-powered preambles are not necessarily optimal for te OFDM systems wit null subcarriers. B. Pilot design We employ te algoritm developed in Section V to design pilot symbols for PSAM. Similar to te preamble design, total power of te pilot symbols are normalized to one. First, we consider an OFDM symbol wit 64 subcarriers and 4 pilot symbols, i.e., N p =4.Tiscomplies wit te IEEE a standard pilot symbols, were four equi-spaced and equi-powered pilot symbols are adopted. In general, witin an OFDM symbol, te number of pilot symbols in frequency domain sould be greater tan te cannel lengt (maximum excess delay), wic is related to te cannel delay spread (i.e., N p L) [2]. Wen N p >L, te MSE performance will be improved as long as te power of pilot symbols is optimally distributed, but te capacity (or data rate) will be degraded. Tus, in our simulations, we use N p = L. However,it sould be remarked tat except for some special cases, it still remains unclear wat value of N p is optimal. 6.5 l 2 6 l IEEE a 5.5 MSE L Figure 4 Frequency-domain cannel MSE 2 of preamble at 10 db (IEEE a).

10 Page 10 of Pilot Preamble λ subcarrier Figure 5 Power of te preamble and pilot symbols designed by te l 2 norm for L = N p = 16 (IEEE e). Figure 2 sows te pilot symbols designed by l 2 norm at 10 db wen N p = L = 4. Te designed pilot symbols are almost equi-spaced (K p = {±8, ±24)}, and te existing standard allocates te equi-powered pilot symbols at (K p = {±7, ±21}). For OFDM systems wit null subcarriers, equi-spaced pilot symbols aving te same power are not necessarily optimal. In our proposed design, te optimized power allocated to te pilot symbols is not uniformly distributed, wic suggests tat in te presence of null subcarriers, equi-powered pilot symbols may not necessarily be optimal. Tis may not be well encapsulated in te total cannel estimate MSE, but is more clearer wen considering te cannel estimate in eac subcarrier. We also illustrate te performance of our proposed algoritm by designing pilot symbols for N p = L =8. Figure 3 presents te power distribution to te designed pilot symbols at 10 db. Te pilot power distribution is found to be symmetric around 0. Tis is due to te symmetric nature of our objective functions and te fact tat pilot positions are obtained by removing te minimum power subcarriers symmetrically. Te eigt pilot symbols are located at te subcarriers K p ={±4,±12, ±19, ±26}. Pilot symbols are well distributed witin te in-band region, wic ensures nearly constant estimation in all subcarriers. We make a comparison of our proposed design, te PEPs sceme and te equi-spaced equi-power design wic we will refer to it as a reference design. Figure 6 sows te designed pilot set for eac of te tree metods. Bot of te proposed and PEP design allocate some pilot subcarriers close to te edges. For te reference design, te equi-spaced and equi-powered pilot symbols are allocated at ±3, ±9, ±15, and ±21. Tere are no pilot subcarriers close to te edges of te active band. Te lack of te pilot subcarriers at te edges of te OFDM symbol may lead to iger cannel estimation errors for te active subcarriers close to te null subcarriers. To demonstrate te effectiveness of te pilot symbols in Figure 6, we plot te cannel estimate MSE for eac active subcarrier. Te total power allocated to te pilot symbols is te same for all tree designs. Figure 7 sows te cannel estimate MSE of te tree designs. From te results, it is clear tat, bot of our proposed and te PEP design outperform te reference design, and tere is no significant difference between te proposed design and te PEP design. Te reference (equispaced equi-powered) design does a poor job of estimating cannel close to te null subcarriers, tis is due to te lack of pilot subcarriers at te edges of te OFDM symbol. Cannel estimation via extrapolation results into iger errors at te edges of te OFDM symbols if

11 Page 11 of Proposed Reference PEP λ Figure 6 Comparison of te pilot power and placement (IEEE a). Subcarrier Proposed Reference PEP NMSE(dB) L=1 L=8 L= Subcarrier Number Figure 7 Comparison of te normalized cannel estimate MSE between te proposed and PEP design (IEEE a).

12 Page 12 of 17 tere is no pilot subcarriers wit significant power at te edges of te OFDM symbols. Tis suggests tat bot te pilot power and te placements need to be carefully considered in te design. We also compare te average cannel estimate MSE of our proposed design wit te PEP design for different cannel lengt L. We evaluate te cannel estimate MSE of our proposed designs as well as te PEP sceme, by varying te cannel lengt L, from1to16. Figure 8 presents te average cannel MSE against cannel lengt L at 10 db. Te proposed designs, (l 2 and l ) exibit lesser cannel MSE tan te PEP symbols except for te trivial case wen L = 1. Tis verifies te potential of our proposed designs over te PEP sceme. Figure 8 sows tat te two design criteria (l 2 and l ) exibit nearly te same 2 for different cannel lengts L. Tis is contrary to [15], were it is stated tat te l norm-based design outperforms te l 2 norm-based design. Our results suggest tat any of te two design criteria can be used to design pilot symbols. However, it sould be noted tat l 2 norm-based design as less computational complexity as compared wit te l. Since te results obtained by te two proposed metods are almost similar for bot te preamble and te pilot symbol design, ten it is reasonable to adopt te l 2 norm-based design for te preamble as well as for te pilot symbol design. Next, we consider pilot symbol design for IEEE e by minimizing te LS cannel estimate MSE. Figure 5 sows te designed pilot symbols for L = N p = 16. Similarly, te pilot symbols are well distributed witin te active subcarrier band, and te power distribution to te pilot symbols is not uniform. Te result empasizes on adopting non-uniform power distribution for an OFDM frame wit null subcarriers. Furtermore, te result underlines te potential of our proposed sceme in designing pilot symbols for OFDM systems wit different number of subcarriers, cannel lengt, and dedicated number pilot subcarriers. We ave provided te results based on te LS estimator to verify tat, in designing pilot power and placement, minimization of bot te LS and te MMSE cannel estimate MSE, perform almost equally well. Figure 9 illustrates te power distribution to te pilot symbols for te PEP, te metod in [15], wic we will refer to it as Baxley and te proposed designs. Similar to te results in Figure 6 for IEEE a, te placement of pilot symbols for te PEP and our proposed design are almost same. However, tere is a noticeable difference in power distribution between te PEP and te proposed design for L = N p = 16. Tis suggests tat 6.5 l 2 6 l PEP 5.5 NMSE(dB) L Figure 8 Comparison of te cannel estimate MSE 2 between te proposed designs and te PEP at 10 db (IEEE a).

13 Page 13 of PEP Baxley Proposed λ subcarrier Figure 9 Comparison of te pilot power and placement (IEEE e). te MSE performance of te two designs will be different. For Baxley s metod, bot te pilot placement and te power distribution are comparable to our proposed design. Te design in [15], uses exaustive grid searc to obtain pilot set wit minimum cannel estimate MSE. Te placement of te pilot symbols depends on te searcing granularity over te predetermined domain of some optimizing parameters. Te main callenge in [15] lies in te adjustment of several parameters to obtain reasonable pilot positions. Wit te proper selection of te optimizing parameters of te cubic function, te performance of Baxley s metodiscomparable to our proposed design. Te prominence of optimal power distribution to te pilot symbols is furter depicted in Figure 10, were for te designed pilot symbols in Figure 9, we plot te normalized cannel estimate MSE for eac active subcarrier. Te total power allocated to te pilot symbols is te same for all te tree designs. Figure 10 depicts te cannel estimate MSE of te tree designs. From te results, it is clear tat, bot of our proposed and te Baxley designs outperform te PEP design, and tere is no significant difference between te proposed design and te Baxley design. Unlike te results depicted in Figure 7, were te performance of te PEP design are comparable to our proposed metod, ere te PEP design does a poor job of estimating te cannel close to te null subcarrier zone. Tis is not due to te lack of pilot subcarriers at te edges of te OFDM symbol, but due to insignificant power allocated to te pilot symbols close to te null subcarriers. Te results suggest tat, considerable MSE performance improvements can be realized by te proper distribution of power to te pilot symbols. Similarly, we make a comparison of te frequencydomain cannel MSE between te PEP, te Baxley metod, and our pilot symbol designs for IEEE e. Figure 11 presents te cannel estimate MSE of te IEEE e for N p = 16. From te results, it is clear tat tere is a significant improvement in MSE performance between our proposed and te PEP designs. Tis furter substantiates te superior performance of our proposed metod over PEP for different cannel/subcarriers configurations. Te Baxley metod performs equally well as our proposed design, wic suggests tat te metod in [15] can be used for pilot symbol designs as well. However, te cubic parameterization tecnique adopted in te Baxley metod obtains pilot placement for a given set of successive active subcarriers. Tus, for te case were some subcarriers witin te active band

14 Page 14 of Proposed Baxley PEP NMSE(dB) Subcarrier Number Figure 10 Comparison of te normalized cannel estimate MSE for N p = 16 (IEEE e). are reserved for oter applications, te metod cannot be easily adopted. For example, to avoid te cubic parameterization from selecting te central DC subcarrier, additional constraint is introduced. Tis implies tat, cubic parameterization tecniques call for some modifications wenever active subcarriers are not consecutive. Unlike [15], our proposed design can be easily employed to obtain significant pilot set for any given set of active subcarriers. A good example is in [26,27], were tone reservation (TR) tecnique is adopted for peak-to-average power ratio (PAPR) reduction. For a system tat utilizes TR metod to mitigate PAPR, te transmitter sends dummy symbols in some selected subcarriers witin te active band [26,27]. Tese reserved subcarriers cange te orientation of te data subcarriers in te active band and tereby limit direct application of te cubic parameterization tecniques in designing pilot symbols. Unlike cubic parameterization metod, PEP as well as our proposed metod can be easily adopted even wen some subcarriers witin te active band are restricted. Anoter example is in [13], were PEP sceme is used to design pilot symbols for bot te MIMO-OFDM as well as te SISO-OFDM systems. Like PEP, our proposed metod can be easily adopted in te design of te disjoint pilot symbols for MIMO-OFDM systems. For MIMO-OFDM systems tat utilizes pilot symbols, to reduce interference between te pilot symbols transmitted from different antennas, it is necessary for te pilot symbols to be ortogonal. Te ortogonality of te pilot sequences for MIMO-OFDM can be establised by ensuring tat te pilot symbols of one transmit antenna are disjoint from te pilot symbols of any oter transmit antenna or by using pase-sift (PS) codes. Baxley metod cannot be directly applied to design disjoint pilot sets for MIMO-OFDM systems wile our metod be can easily applied. In te following, we explore te BER performance gains tat could be realized if te pilot symbols in IEEE e are designed to conform wit te proposed metod. We demonstrate te efficacy of our proposed metod by comparing te BER performance of te proposed, PEP, te Baxley and te reference design. Te frequency-selective cannel wit L = N p taps is considered. Eac cannel tap is i.i.d. complex Gaussian wit zero mean and te exponential power delay profile is given by te vector r =[r 0... r L -1 ]wereρ l = Ce 1/2, and C is a constant selected suc tat L 1 l=1 ρ l = 1.

15 Page 15 of Proposed PEP Baxley 22 MSE (db) L Figure 11 Comparison of te cannel estimate MSE 2 between te proposed l 2 design, Baxley and te PEP for N p =16(IEEE e). Figures 12 and 13 depict te BER performance of te OFDM system wen using PSK modulation. For N p =8, we made a comparison between te proposed design, PEP, te Baxley, and te reference design. It is noted tat conventional standard utilizes equi-spaced equipowered pilot symbols. Figure 12 depicts te BER performance of te four designs for QPSK modulation. Te PEP, te Baxley, and te proposed design outperform te reference design. Te results sow tat, at ig SNR, tere is a sligt improvement in BER performance of te proposed design over te PEP and te Baxley design. To demonstrate te flexibility of our proposed design, we provide two comparable BER curves, Proposed 1 and Proposed 2, were te former is te BER performance of our designed pilot symbols and te latter is te BER performance of our pilot symbols optimized wen we reserve subcarriers K TR = {±100, ±76, ±47, ±14} for TR. Tis verifies tat, even for discontinuous data subcarriers, our proposed design can still be used to design significant pilot symbols, wile te Baxley design cannot be directly applicable to tis configuration. For N p = 16, we only consider PEP, te Baxley, and teproposeddesignasitisimpossibletoave16 equally spaced pilot symbols witin 200 active subcarriers. Figure 13 depicts te BER performance for QPSK, 16-PSK, and 64-PSK. Te results verify tat te proposed design provides improved BER performance over te PEP design. Tis performance gap is a result of te PEP design aving insignificant power distribution to te pilot symbols at te edges of te active subcarrier band tat leads to poor estimate of te cannels. Te performance of te Baxley design is similar to te proposed design for QPSK-modulated data. However, for 16-PSK and 64-PSK modulation, our proposed design outperforms te Baxley design. Also, for 16-PSK and 64-PSK modulation, tere is a sligt improvement in BER performance of te Baxley metod over te PEP design. Te gain attained by our proposed metod over te PEP and te Baxley design togeter wit te flexibility of te proposed tecnique in designing pilot symbols for different cannel/subcarriers configurations promotes our proposed design to be a candidate for pilot symbols design in OFDM systems wit null subcarriers. VII. Conclusion We ave addressed te design of optimal preamble as well as suboptimal pilot symbols for cannel estimation

16 Page 16 of Reference PEP Baxley Proposed 1 Proposed 2 BER SNR(dB) Figure 12 Comparison of te BER performance for different designs for N p = 8 (IEEE e). Proposed PEP Baxley 10 1 BER QPSK 16 PSK 64 PSK SNR(dB) Figure 13 Comparison of te BER performance for N p = 16 (IEEE e).

17 Page 17 of 17 in OFDM systems wit null subcarriers. First, we ave formulated te cannel estimate MSE minimization problem for a determined subcarrier set as an SDP, ten we ave employed convex optimization tecniques to obtain near optimal power distribution to a given subcarrier set. Design examples consistent wit IEEE a standard sow tat in terms of cannel estimate MSE, te long OFDM preamble wit equi-powered active subcarriers is nearly optimal. In designing te pilot symbols, we ave considered pilot placement as well as power allocation. We ave proposed an iterative algoritm to determine pilot placements and ten distribute power to te selected pilot symbols by using convex optimization tecniques. Several examples consistent wit IEEE a and IEEE e ave been provided wic sow tat te proposed metods can be used to design preamble as well as pilot symbols for cannel estimation in OFDM systems wit different frame size. We ave also verified tat our proposed metod is superior over te equispaced equi-powered pilot symbols as well as te PEP and sligtly outperforms te Baxley design. VIII. Abbreviations CP:cyclicprefix;CSI:cannelstateinformation;IDFT: inverse discrete Fourier transform; ISI: inter-symbol interference; LMI: linear matrix inequality; MMSE: minimum mean-squared error; MSE: mean-squared errors; OFDM: ortogonal frequency division multiplexing; PAPR: peak-to-average power ratio; PEP: partially equispaced pilot; PSAM: pilot-assisted modulation; SDP: semidefinite programming; SNR: signal-to-noise ratio; TR: tone reservation. IX. Competing interests Te autors declare tat tey ave no competing interests. Note Te material in tis article was presented in part at Asia-Pacific Conference on Communications Received: 15 July 2010 Accepted: 3 June 2011 Publised: 3 June 2011 References 1. R Negi, J Cioffi, Pilot tone selection for cannel estimation in a mobile OFDM system. 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EURASIP Journal on Wireless Communications and Networking :2.