On the optimality of the null subcarrier placement for blind carrier offset estimation in OFDM systems Wu, Y.; Attallah, S.; Bergmans, J.W.M.

Transcription

1 On the optimaity of the nu subcarrier pacement for bind carrier offset estimation in OFDM systems Wu, Y.; Attaah, S.; Bergmans, J.W.M. Pubished in: IEEE Transactions on Vehicuar Technoogy DOI: 0.09/TVT Pubished: 0/0/2009 Document Version Pubisher s PDF, aso known as Version of Record (incudes fina page, issue and voume numbers) Pease check the document version of this pubication: A submitted manuscript is the author's version of the artice upon submission and before peer-review. There can be important differences between the submitted version and the officia pubished version of record. Peope interested in the research are advised to contact the author for the fina version of the pubication, or visit the DOI to the pubisher's website. The fina author version and the gaey proof are versions of the pubication after peer review. The fina pubished version features the fina ayout of the paper incuding the voume, issue and page numbers. Link to pubication Genera rights Copyright and mora rights for the pubications made accessibe in the pubic porta are retained by the authors and/or other copyright owners and it is a condition of accessing pubications that users recognise and abide by the ega requirements associated with these rights. Users may downoad and print one copy of any pubication from the pubic porta for the purpose of private study or research. You may not further distribute the materia or use it for any profit-making activity or commercia gain You may freey distribute the URL identifying the pubication in the pubic porta? Take down poicy If you beieve that this document breaches copyright pease contact us providing detais, and we wi remove access to the work immediatey and investigate your caim. Downoad date: 25. Ju. 208

2 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY to the cassic two-stage scheme, provided that the SNR is sufficienty high. For exampe, given a target BER of 0 5, the three-stage receiver using SD (N cand = 32) is capabe of achieving a performance gain of 2.5 db over its two-stage counterpart in an upink (8 4) SDMA/OFDM 4-QAM system. Furthermore, an additiona 2-dB performance gain can be attained with the aid of the nove centershifting-based SD amagamated with an IrCC. REFERENCES [] L. Hanzo, M. Munster, B. J. Choi, and T. Keer, OFDM and MC-CDMA for Broadband Muti-User Communications, WLANs and Broadcasting. Piscataway, NJ: IEEE Press, [2] E. Viterbo and J. Boutros, A universa attice code decoder for fading channes, IEEE Trans. Inf. Theory, vo. 45, no. 5, pp , Ju [3] B. M. Hochwad and S. ten Brink, Achieving near-capacity on a mutipeantenna channe, IEEE Trans. Commun., vo. 5, no. 3, pp , Mar [4] K. Wong, C. Tsui, R. S. K. Cheng, and W. Mow, A VLSI architecture of a K-best attice decoding agorithm for MIMO channes, in Proc. IEEE Int. Symp. Circuits Syst., May 2002, vo. 3, pp [5] W. H. Mow, Maximum ikeihood sequence estimation from the attice viewpoint, IEEE Trans. Inf. Theory, vo. 40, no. 5, pp , Sep [6] A. M. Chan and I. Lee, A new reduced-compexity sphere decoder for mutipe antenna systems, in Proc. IEEE Int. Conf. Commun., Apr./May 2002, vo., pp [7] A. Wofgang, J. Akhtman, S. Chen, and L. Hanzo, Reducedcompexity near-maximum-ikeihood detection for decision feedback assisted space time equaization, IEEE Trans. Wireess Commun., vo. 3, no. 7, pp , Ju [8] Y. Xie, Q. Li, and C. N. Georghiades, On some near optima ow compexity detectors for mimo fading channes, IEEE Trans. Wireess Commun., vo. 6, no. 4, pp , Apr [9] J. Boutros, N. Gresset, L. Brune, and M. Fossorier, Soft-input softoutput attice sphere decoder for inear channes, in Proc. IEEE Goba Teecommun. Conf., Dec. 2003, vo. 3, pp [0] J. Wang, S. X. Ng, L. L. Yang, and L. Hanzo, Combined seriay concatenated codes and MMSE equaization: An EXIT chart aided perspective, in Proc. IEEE Veh. Techno. Conf. Fa, Sep. 2006, pp. 5. [] S. ten Brink, Convergence behavior of iterativey decoded parae concatenated codes, IEEE Trans. Commun., vo. 49, no. 0, pp , Oct [2] T. Cui and C. Teambura, An efficient generaized sphere decoder for rank-deficient MIMO systems, IEEE Commun. Lett., vo. 9, no. 5, pp , May [3] L. Hanzo and T. Keer, OFDM and MC-CDMA: A Primer. Hoboken, NJ: Wiey, [4] M. Tucher, A. C. Singer, and R. Koetter, Minimum mean squared error equaization using aprioriinformation, IEEE Trans. Signa Process., vo. 50, no. 3, pp , Mar [5] M. Tücher and J. Hagenauer, Exit charts of irreguar codes, in Proc. Conf. Inf. Sci. Syst., 2002, pp CD-ROM. [6] M. Tücher, Design of seriay concatenated systems depending on the bock ength, IEEE Trans. Commun., vo. 52, no. 2, pp , Feb [7] A. Ashikhmin, G. Kramer, and S. ten Brink, Extrinsic information transfer functions: Mode and erasure channe properties, IEEE Trans. Inf. Theory, vo. 50, no., pp , Nov [8] J. Kiewer, S. X. Ng, and L. Hanzo, Efficient computation of exit functions for nonbinary iterative decoding, IEEE Trans. Commun., vo. 54, no. 2, pp , Dec [9] J. Kiewer, A. Huebner, and D. J. Costeo, On the achievabe extrinsic information of inner decoders in seria concatenation, in Proc. IEEE Int. Symp. Inf. Theory, Seatte, WA, Ju. 2006, pp [20] S. X. Ng, J. Wang, and L. Hanzo, Unveiing near-capacity code design: The reaization of Shannon s communication theory for MIMO channes, in Proc. IEEE Int. Conf. Commun., May 2008, pp On the Optimaity of the Nu Subcarrier Pacement for Bind Carrier Offset Estimation in OFDM Systems YanWu, Student Member, IEEE, Samir Attaah, Senior Member, IEEE, and J. W. M. Bergmans, Senior Member, IEEE Abstract Liu and Turei proposed a bind carrier frequency offset (CFO) estimation method for orthogona frequency-division mutipexing (OFDM) systems, making use of nu subcarriers. The optima subcarrier pacement that minimizes the Cramer Rao bound (CRB) of the CFO estimation was reported by Ghogho et a. In this paper, we study the optimaity of the nu subcarrier pacement from another perspective. We first show that the SNR of the CFO estimation using nu subcarriers is a function of the nu subcarrier pacement. We then formuate the CFO-SNR optimization for the nu subcarrier pacement as a convex optimization probem for sma CFO vaues and derive the optima pacement when the number of subcarriers is a mutipe of the number of nu subcarriers. In addition, we show that the SNR-optima nu subcarrier pacement aso minimizes the theoretica mean square error in the high SNR region. When the number of subcarriers is not a mutipe of the number of nu subcarriers, we propose a heuristic method for the nu subcarrier pacement that sti achieves good performance in the CFO estimation. We aso discuss the optimaity of the nu subcarrier pacement in practica OFDM systems, where guard bands are required at both ends of the spectrum. Index Terms Bind carrier offset estimation, convex optimization, orthogona frequency-division mutipexing (OFDM). I. INTRODUCTION Orthogona frequency-division mutipexing (OFDM) is known to be more sensitive to carrier frequency offset (CFO). For an OFDM system with CFO, we can write the received time-domain signa in the foowing form []: y m = EW P H m s m e j2πφ 0(m )(+N g/n ) + n m. () Here, we use superscript m to indicate the OFDM symbo index. E = diag(,e j2πφ 0/N,...,e j2π(n )φ 0/N ) is a diagona matrix containing CFO φ 0, which we assume to be normaized with respect to subcarrier spacing 2π/N. In a practica OFDM system, there are some subcarriers that do not carry any data. They are caed nu subcarriers, whereas the data-carrying subcarriers are simpy caed data subcarriers. Let P out of N subcarriers be the data subcarriers. Then, W P is an N P submatrix that is obtained from the N N inverse discrete Fourier transform (DFT) matrix W N. H m is a diagona matrix containing the channe frequency response, s m is a P vector containing the transmitted data in the mth OFDM symbo, N g denotes the ength of the cycic prefix, and n m is an additive white Gaussian noise (AWGN) vector. Manuscript received September 30, 2007; revised Apri, First pubished September 2, 2008; current version pubished Apri 22, The review of this paper was coordinated by Prof. H.-C. Wu. Y. Wu is with the Signa Processing Systems Group, PT 3.27, Department of Eectrica Engineering, Technische Universiteit Eindhoven, 5600 MB Eindhoven, The Netherands (e-mai: y.w.wu@tue.n). S. Attaah is with the Schoo of Science and Technoogy, SIM University, Singapore (e-mai: samir@unisim.edu.sg). J. W. M. Bergmans is with the Signa Processing Systems Group, PT 3.06, Department of Eectrica Engineering, Technische Universiteit Eindhoven, 5600 MB Eindhoven, The Netherands (e-mai: J.W.M.Bergmans@tue.n). Coor versions of one or more of the figures in this paper are avaiabe onine at Digita Object Identifier 0.09/TVT /$ IEEE Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on February 9,200 at 07:38:7 EST from IEEE Xpore. Restrictions appy.

3 20 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009 Liu and Turei [] presented a bind CFO estimation method based on the received signa on the nu subcarriers. Let us define m = [ m,2 m,...,l m ] as the nu subcarrier indexes in OFDM symbo m and L = N P as the number of nu subcarriers. It was shown that the CFO estimate can be obtained from the minimization of the cost function, which is given as M J (z) = w H Z y m 2 (2) m= m where M is the tota number of OFDM symbos that are used for the CFO estimation, w H is the th row of the DFT matrix, and Z = diag(,z,z 2,...,z (N ) ). The optima CFO estimate is normay obtained through a search over the range of possibe CFO vaues. The nu subcarrier pacement that guarantees the identifiabiity of the CFO estimation method in [] was studied in [2] and [3]. In [3], it is reported that the nu subcarrier pacement that minimizes the Cramer Rao bound (CRB) is achieved by pacing them with even spacing across the whoe OFDM symbo. In this paper, we study the optimaity of the nu subcarrier pacement using a very different approach compared to [3]. We found that the SNR of the CFO estimation using the method in [] is a function of the nu subcarrier pacement. Therefore, we want to find the optima pacement of nu subcarriers such that the SNR of the CFO estimation is maximized. We formuate the SNR maximization probem of the nu subcarrier pacement and derive the optima soution from a convex optimization procedure for sma CFO vaues. We found that when the number of subcarriers is divisibe by the number of nu subcarriers, the exact optima nu subcarrier pacement can be found. We aso prove that the SNR-optima nu subcarrier pacement is aso optima in minimizing the theoretica MSE, which is given in [4], of the CFO estimation. Interestingy, this is the same nu subcarrier pacement that minimizes the CRB in [3]. Therefore, the main contribution of this paper does not ie in the finding of a new optima nu subcarrier pacement. This paper has rather contributed additiona theoretica insights on why the nu subcarriers shoud be paced eveny. We show that the eveny spaced nu subcarrier pacement not ony minimizes the CRB but aso maximizes the SNR and minimizes the theoretica MSE of the CFO estimation. When the number of subcarriers is not divisibe by the number of nu subcarriers, it is difficut to prove the optimaity of the nu subcarrier pacement due to the integer constraint on the optimization variabes. However, we wi show a heuristic procedure on how to pace the nu subcarriers where good performance can sti be achieved. We extend the optimization probem to a practica OFDM system where guard bands are required at both ends of the spectrum. In this case, if given a few more nu subcarriers that can be freey inserted in the OFDM symbo, we show how to pace them to guarantee the SNR optimaity in the CFO estimation. We show that for practica OFDM systems with guard bands, the introduction of a few extra nu subcarriers eads to much better performance of the bind CFO estimation. II. PLACEMENT OF NULL SUBCARRIERS BASED ON THE CFO-SNR MAXIMIZATION Given a CFO vaue of φ 0, the received signa on nu subcarrier i of OFDM symbo m can be written as r m i = N n=0,n i h m n s m n C m n i (φ 0 )+n m i = ICI m i (φ 0 )+n m i (3) where h m i and s m i are the channe response and the transmitted data on subcarrier i of OFDM symbo m, respectivey. ICI m i (φ 0 ) is the Inter- Carrier Interference (ICI) due to the CFO of φ 0,andn m i is an AWGN noise. The vaue of Ck m(φ 0) is given by [5] Ck m (φ 0 )= sin [π(k + φ ( ( 0)] N sin [ π (k + φ N 0) ] exp jπ(k + φ 0 ) )) N exp (j2πφ 0 (m )( + N g /N )). (4) Using (3), the cost function in (2), which is the summation of the received signa power over a the nu subcarriers, can be equivaenty rewritten as J (φ) = M m= L N 2 h m n s m n Cn m i (φ 0 φ)+n m i. (5) n=0,n / m Correspondingy, the estimate of the CFO is given by ˆφ =argminj (φ). (6) φ Note that the received signa on a nu subcarrier i in (3) is the sum of ICI m i and n m i.ici m i is the usefu signa term that we can use for the estimation of CFO φ 0,andn m i is the noise term, which is uncorreated with ICI m i. Therefore, using (3), we can define an objective function, so caed SNR CFO, as foows: ( M L E ) ICI m i (φ 0 ) 2 m= SNR CFO = ( M ) (7) L E n m i 2 m= where E denotes statistica expectation. Note that the objective function can be interpreted as the SNR of the CFO estimation. The power of the ICI on subcarrier i in OFDM symbo m can be written as E { N ICI m i (φ 0 ) 2 = E h m n s m n 2 sin 2 [π(n i + φ 0 )] N 2 sin [ 2 π (n N i + φ 0 ) ]. n=0,n / Notice that the ICI power for the mth OFDM symbo depends ony on the signas in OFDM symbo m and is not affected by other OFDM symbos. As the noise in OFDM symbo m is aso independent from the noise in other OFDM symbos, the SNR CFO optimization for M OFDM symbos can be independenty performed on each OFDM symbo. Therefore, the optimization ony needs to be performed for one OFDM symbo. From now on, for ease of notation, we wi drop OFDM symbo index m. In this case, nu subcarrier pacement that maximizes estimation SNR CFO in (7) can be found by ( L ) =argmax(snr CFO ) = arg max E ICI i (φ 0 ) 2 =argmax =argmax L L { N n=0,n / { N n=0,n / E h n s n 2 sin 2 [π(n i + φ 0 )] N 2 sin [ 2 π (n N i + φ 0 ) ] sin [ 2 π (n N i + φ 0 ) ] as E h n s n 2 = E{ h n 2 E{ s n 2 is independent of the nu subcarrier pacement. The numerator sin 2 [π(n i + φ 0 )] is equa to sin 2 (πφ 0 ) and is aso independent of the nu subcarrier pacement. In practice, φ 0 is normay modeed as a random variabe with a uniform (8) Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on February 9,200 at 07:38:7 EST from IEEE Xpore. Restrictions appy.

4 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY distribution between [ θ, θ). In this case, the cost function can be rewritten as =argmax =argmax 2θ +θ θ { L L N n=0,n / [ N n=0,n / ] sin [ 2 π (n N i + φ 0 ) ] dφ 0 N 2θ π f(n i) where f(k) is given by [ ( π ) ( π )] f(k) = cot N (k θ) cot N (k + θ) for k = (N ),...,,,...,N. (0) Note that k = n i 0 for n/. It coud be easiy shown that function f(k) is periodic with period N, i.e., f(k) =f(k + N). Therefore, for the subsequent optimization, we ony need to consider function f(k) over one period, i.e., k =, 2,...,N. Another property of f(k) is that it is an even function of k, i.e., f(k) =f( k) for any integer k. Discarding the constants, we can rewrite (9) in the foowing form: { L N =argmax f(n i ) f(n i ) f(0). n=0 n,n i () The third term in (), i.e., f(0), is independent of and, hence, can be dropped. Using the periodicity of f(k), it can be easiy shown that the first term in the summation N n=0 i) in () is aso independent of i. Therefore, the cost function in () can be simpified to { L L =argmin f( i j ). (2) j=,j i Notice that the new cost function in (2) depends ony on the spacing, not the absoute positions, of the nu subcarriers. Let us define the spacing between the ith and (i +)th nu subcarriers as k i = i+ i for i =, 2...,L and k L = N + L.We further define p i,m = m k j=0 [(i+j ) mod L]+ for i =, 2,...,L and m =, 2,...,L. Here, we use [i mod L] for integers i and L to denote the integer remainder of i/l. The subscript i indicates the k index of the first term in the summation because [(i +0 ) mod L]+=i. The subscript m indicates the tota number of terms in the summation. Therefore, p i,m is actuay the spacing between the ith nu subcarrier and its mth neighboring nu subcarrier to the right in the cycic sense. Therefore, the contribution to the tota cost function due to a particuar nu subcarrier i is the summation of f( i j ) from a its L neighboring nu subcarriers, i.e., L f( j=,j i i j ).Asp i,m is the spacing between the ith nu subcarrier and its mth neighboring nu subcarrier, we can write L f( j=,j i i j )= L f(p m= i,m). Summing this over a the L nu subcarriers, i.e., L possibe vaues of i, the new cost function can be written as { L L L L J (k,k 2,...,k L )= f(p i,m )= f(p i,m ) (3) with L k i = N. m= m= (9) The optimization probem in (3) has a variabes being integers. Such integer programming probems are difficut to sove anayticay. Therefore, we first reax the constraints on a k i s being integers and assume them to be rea positive numbers. This approach has been commony used in finding the optima bit aocations for mutiuser or muticarrier systems (see, for exampe, [6]). For ease of anaysis, we aso assume that θ< so that k θ>0 and k + θ<nare satisfied for a possibe vaues of k. It can be easiy shown that if the above condition is satisfied, (d 2 /dk 2 )f(k) > 0 for <k<n. Therefore, f(k) is a convex function for <k<n, andθ<. According to Jensen s inequaity [7], if f(k) is convex for k,k 2,...,k L,and given λ,λ 2,...,λ L with λ + λ λ L =,then f(λ k + + λ L k L ) λ f(k )+ + λ L f(k L ). (4) By setting λ = λ 2 = = λ L =(/L), wehave ( ) L L ( mn f(p i,m ) f p i,m = f L L L ). (5) Here, we make use of L p i,m = mn because L k i = N.The equaity in (5) hods for a given m when a the p i,m s for i =, 2,...Lare equa. When k = k 2 = = k L = N/L, the equaity in (5) hods for a vaues of m. Therefore, we obtain L J (k,k 2,...,k L ) m= { ( mn L f L ). (6) The right-hand side of (6) is independent of k i s and is the ower bound of the cost function. Therefore, this cost function is minimized when k = k 2 = = k L = N/L. This means that the nu subcarriers shoud be paced eveny spaced across the whoe OFDM symbo. If N/L is an integer, the nu subcarriers shoud be paced N/L apart to maximize SNR CFO. Therefore, in system design, when we can freey choose the number of nu subcarriers L, we shoud aways choose L such that N/L is an integer to ensure the optimaity of the nu subcarrier pacement. However, for systems where N is not divisibe by L, it turns out to be difficut to prove the optimaity of a particuar nu subcarrier pacement because of the integer constraints on the vaues of k i s. For rea number k i s, we know that, to maximize SNR CFO, the spacing between the nu subcarriers shoud be the same. In the foowing, we propose a heuristic method in pacing the nu subcarriers as eveny as possibe for integer k i s. Let k = N/L and k u = N/L, where k and k u are both integers. Here, we use x to denote the argest integer that is smaer than or equa to x, whereas we use x to denote the smaest integer that is arger than or equa to x. We know that to achieve cose to even spacing between the nu subcarriers, a the k i vaues shoud be chosen as either k or k u. Next, we determine how many k i s shoud take the vaue k and how many k i s shoud take the vaue k u, and we use n and n u to denote the two numbers, respectivey. The vaues of n and n u can be obtained by soving { n + n u = L (7) n k + n u k u = N. Now, the probem of pacing the nu subcarriers is equivaent to pacing these n k s and n u k u s as eveny as possibe. It is obvious that if we pace a the k s consecutivey and a the k u s consecutivey, the spacing between the nu subcarriers is not going to be very even. They shoud be aternativey paced in some way. Without oss of generaity, et us assume that n n u.if(n /n u )=q is an integer, we shoud Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on February 9,200 at 07:38:7 EST from IEEE Xpore. Restrictions appy.

5 22 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009 TABLE I HEURISTIC NULL SUBCARRIER PLACEMENT WHEN N IS NOT DIVISIBLE BY L (n >n u) [8], the toerance of the transmit and receive center frequency shoud be ±20 ppm. Therefore, the worst case frequency offset is 40 ppm, which is about 200 khz for a 5.2-GHz center frequency. This worst case frequency offset corresponds to the vaue of θ =0.66. Moreover, for indoor appications, due to ow mobiity and high carrier frequency (5 GHz for the IEEE 802.a system), the CFO due to the Dopper shift is negigibe. Therefore, this is a vaid assumption in practice, particuary for indoor wireess LAN-based appications due to the high-quaity osciators that are currenty used. III. PLACEMENT OF NULL SUBCARRIERS BASED ON THE THEORETICAL MSE MINIMIZATION TABLE II HEURISTIC NULL SUBCARRIER PLACEMENT FOR L =4TO FOR N =64OFDM SYSTEMS In this section, we prove that the SNR-optima nu subcarrier pacement is aso optima in minimizing the MSE of the CFO estimation in the high SNR region. Let us set Δφ = φ 0 ˆφ as the CFO estimation error. The inear approximation of Δφ can be obtained as [4] Δφ = J (φ) φ φ=φ 0 2 J (φ) φ 2 φ=φ 0. (9) group qk s foowed by one k u into one group and pace n u of such groups as iustrated in Tabe I. Otherwise, we et q = n /n u. In this case, we shoud have two kinds of pacing groups. The type group consists q k s foowed by one k u, and the type 2 group consists q +k s foowed by one k u. The number of type groups g and the number of type 2 groups g u can be obtained by soving { g + g u = n u g q + g u (q +)=n. (8) These two types of groups shoud be paced aternativey. A summary of this heuristic pacement method is given in Tabe I. The nu subcarrier pacement for L =4to L =nu subcarriers for an OFDM system with N =64 subcarriers using the proposed heuristic method is isted in Tabe II. For the case of L =5and 6, we have verified that the nu subcarrier pacement using the heuristic method is the same as the optima pacement that is obtained through an exhaustive computer search. Note that our previous derivation is based on the assumption that θ< to ensure that f(k) is convex for k =, 2,...,N. Thisis a vaid assumption for most indoor communication systems that are operating at the 2.4- and 5-GHz bands, such as wireess oca area network (LAN) systems [8]. According to the IEEE 802.a standard After some agebraic manipuations, the estimation error can be expressed as in (20), shown at the bottom of the page. Assuming the noise on different subcarriers to be independent and identicay distributed (i.i.d.), with zero mean and variance σn, 2 we can show that E n (Δφ) =0. Therefore, the inearized estimator is unbiased. This aso means that the MSE of the CFO estimation is equa to the variance of Δφ. Let us aso assume that transmitted signa s m k is aso i.i.d., with zero mean and unit variance, and the channe is appropriatey normaized such that the channe component on each subcarrier has zero mean with unit variance, i.e., E(h m n )=0and E( h m n 2 )=for n =0,,...,N and m =,...,M. We obtain that the MSE of CFO estimation in the high SNR region is given by E n,s,h [ (Δφ) 2 ] = 2π 2 M L N 2 σn 2 N m= n=0,n / m sin 2 [ π N (n m i )]. (2) Now, et us ook at the nu subcarrier optimization probem again. The nu subcarrier pacement that minimizes the MSE can be formuated as ( [ =argmin ]) En,s,h (Δφ) 2 =argmax N ( M L N m= n=0,n / m sin [ 2 π (n N m)] i ). (22) If L (/(sin 2 [(π/n)(n m n=0,n / m i )])) is maximized for every vaue of m, i.e., for each OFDM symbo, then the cost function Δφ = M L M R N m= k=0,k / m L N m= n=0,n / N [ ] ( ) exp jπ n m i h m n s m n n m N i sin[ N π (n m i )] k n N ) h m k (hm n ) s m k (sm n ) π exp( jπ n=0,n / m N sin( N π (k m i )) sin( N π (n m i )) (20) Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on February 9,200 at 07:38:7 EST from IEEE Xpore. Restrictions appy.

6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY in (22) is maximized. Therefore, the optimization probem over M OFDM symbos is equivaent to the optimization in one OFDM symbo, which is given by =argmax ( L N n=0,n / g(n i ) ) (23) where g(k) =(/(sin 2 [(π/n)(n i )])). It is straightforward to show that g(k) is aso periodic with period N. Using a simiar approach as we have done in Section II, the optimization probem in (23) can be simpified to =argmin { L L j=,j i g( i j ). (24) It can aso be shown that g(x) is a convex function of rea number x for <x<n. Therefore, the optimization probem in (24) is essentiay the same as the optimization probem in (2), as g(x) and f(x) are both convex. This means that the optima soutions to the two optimization probems are the same. Thus, we have proven the foowing. Proposition: The nu subcarrier pacement that maximizes the SNR of the CFO estimation defined in (7) aso minimizes the MSE of the CFO estimation in the high SNR region given in (2). TABLE III SNR-OPTIMAL FREE NULL SUBCARRIER PLACEMENT FOR IEEE 802.a SYSTEMS Section II because that soution requires the k i s to be equa for a i =, 2,...,L, which means that the nu subcarriers shoud be eveny paced across the whoe OFDM symbo. This is impossibe for our probem, as we do not have the freedom to freey pace a the nu subcarriers due to the fixed positions of the guard nu subcarriers. As a resut, the cosed-form optima soution for (25) is difficut to find. However, in practice, the number of free nu subcarriers L n must be kept sma to minimize the oss of the transmission data rate, as they occupy the usefu spectrum of the data subcarriers. Therefore, it is usuay possibe to resort to a computer search to find the optima pacement of these subcarriers offine. Tabe III shows the optima pacement of L n free nu subcarriers with different L n vaues for an IEEE 802.a compiant system, which is obtained by the computer search. In such a system, there are a tota of N =64subcarriers. Subcarriers [ 3: 27, 27:32] are used as guard bands, i.e., L =5 and L 2 =6. Here, the θ vaue used is 0.5. We can see that the pacement is cose to the eveny spaced pacement. IV. PLACEMENT OF NULL SUBCARRIERS BASED ON PRACTICAL CONSIDERATIONS For practica OFDM systems, it is usuay necessary to consecutivey pace some nu subcarriers at both ends of the spectrum as guard bands. These are caed the guard nu subcarriers. In this section, we show that, given the fixed positions of the guard nu subcarriers, if there are a few nu subcarriers to freey pace in the OFDM symbo for the purpose of CFO estimation (we ca these nu subcarriers as free nu subcarriers), how shoud we pace them to achieve optimaity in the SNR sense? Let us study an OFDM system having two guard bands with L and L 2 nu subcarriers, respectivey, at both ends of the spectrum. Suppose that we have L n free nu subcarriers that we can freey pace between subcarrier N/2+L and N/2 L 2 +. The whoe set of a the nu subcarriers becomes =[, 2,..., L, L +,..., L +L n, L +L n+,..., L ] where L = L + L 2 + L n is the tota number of nu subcarriers. Again, we define k i = i+ i as the spacing between the i+ th nu subcarrier and the th nu subcarrier, and p i,m = m k j=0 [(i+j ) mod L]+ as the spacing between the ith nu subcarrier and its mth neighboring nu subcarrier to the right in the cycic sense. Foowing the simiar procedures as in Section II, we coud obtain the cost function for the OFDM system with the guard band as J (k L,k L +,...,k L +L n )= L L f(p i,m ) (25) m= subject to L k i = N. Comparing (25) with (3), we can see that the summation is sti taking over a the L nu subcarriers, incuding the guard and free nu subcarriers. However, for this probem, we woud not be abe to reach the same optima soution as in V. S IMULATION RESULTS Computer simuations were performed for an OFDM system with 64 subcarriers and a ength-6 cycic prefix. According to the specifications that are given in IEEE 802.a, there are nu subcarriers that are consecutivey paced from subcarriers 27 to 37 [8]. To achieve a fair comparison, we are aso using nu subcarriers in our simuations. We ony use one OFDM symbo for the CFO estimation, i.e., M =. We use channe mode A of the HiperLan II channe modes [9] in a the simuations. It is a mutipath Rayeigh fading channe with an exponentia power deay profie and a root-mean-square deay spread that is equa to one moduation symbo interva. To assess the performance of the proposed nu subcarrier pacement, we define the estimation MSE as MSE = N s N s (φ 0 ˆφ) 2 (26) where ˆφ and φ 0 represent the estimated and true CFOs, respectivey, and N s denotes the tota number of Monte Caro trias. A comparison between the MSE that is obtained through simuations and the theoretica MSE that is obtained from (2) is depicted in Fig.. The SNR on the x-axis is the SNR of the received signa and not the CFO estimation SNR that we are trying to optimize. The CFO vaue that we use in the simuation is uniformy distributed between 0.5 and We compare the theoretica MSE and the MSE that is obtained from simuations for both the consecutive nu subcarriers that are paced from 27 to 37 according to IEEE 802.a, and the proposed nu subcarrier pacement according to Tabe II. From the comparison, we can see that the theoretica MSE approximates the actua MSE very cosey for the proposed scheme for an SNR that is arger than 0 db. Fig. 2 shows the performance of the bind carrier offset estimation using the method in [] with nu subcarriers that are paced with different spacings. The proposed scheme paces the nu subcarriers according to Tabe II. We can see that with the proposed nu subcarrier Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on February 9,200 at 07:38:7 EST from IEEE Xpore. Restrictions appy.

7 24 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009 Fig. 3. SER performance of systems using CFO estimation with different nu subcarrier pacements. Fig.. Comparison between the theoretica MSE and the MSE that is obtained from simuations. Fig. 4. MSE performance of the CFO estimation for OFDM systems with guard bands and a different number of optimay paced free nu subcarriers. Fig. 2. MSE performance of the CFO estimation using different nu subcarrier pacements. pacement, the CFO estimation accuracy is improved significanty. The performance gain, compared to the consecutive nu subcarrier pacement, is as arge as 0 db. We can aso see that the further apart the nu subcarriers are paced, the better the MSE performance wi become. Athough we coud not prove the optimaity of the nu subcarrier pacement that is obtained from the heuristic method in Tabe I, from the resuts, we can see that it sti eads to very good performance in the CFO estimation. The symbo error rate (SER) performance is shown in Fig. 3 for quaternary phase-shift keying moduations. From the SER performance, we can see a performance gain of 3.5 db compared to the consecutive nu subcarrier pacement. Fig. 4 shows the improvement in the CFO estimation that is achieved by introducing a few optimay paced free nu subcarriers besides the guard nu subcarriers. The system foows the IEEE 802.a specifications with guard nu subcarriers. The CFO vaue that we used in the simuation is, again, uniformy distributed between 0.5 and We can see that, by introducing two extra free nu subcarriers, the performance of the CFO estimation coud be improved by 5 db compared to using guard nu subcarriers aone. The performance can be further improved by introducing more free nu subcarriers. The gain, on the other hand, becomes smaer as the number of free nu subcarriers increases. VI. CONCLUSION In this paper, we have formuated the nu subcarrier pacement probem for bind CFO estimation in an OFDM system using the SNR CFO maximization criterion. We have showed that for sma CFO vaues, this eads to a convex optimization probem, and the optima pacement is achieved by pacing the nu subcarriers eveny across the OFDM symbo. We have proved that this optima nu subcarrier pacement aso minimizes the theoretica MSE, which is an accurate approximation of the MSE of the CFO estimation in the high SNR region. For systems where the number of subcarriers is divisibe by the number of nu subcarriers, this optima pacement can be achieved. Otherwise, based on a heuristic procedure, we have showed how to pace the nu subcarriers such that good performance Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on February 9,200 at 07:38:7 EST from IEEE Xpore. Restrictions appy.

8 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY in the CFO estimation can sti be achieved. We have aso studied the optima free nu subcarrier pacement probem for practica OFDM systems with guard bands. We have demonstrated that the proposed nu subcarrier pacement significanty improves the performance of the CFO estimation. ACKNOWLEDGMENT The authors woud ike to thank Prof. T. J. Lim from the University of Toronto for the insightfu discussion that they had with him and for his vauabe suggestions. REFERENCES [] H. Liu and U. Turei, A high-efficiency carrier estimator for OFDM communications, IEEE Commun. Lett., vo. 2, no. 4, pp , Apr [2] X. Ma, C. Tepedeeniogu, G. Giannakis, and S. Barbarossa, Non-dataaided carrier offset estimators for ofdm with nu subcarriers: Identifiabiity, agorithms, and performance, IEEE J. Se. Areas Commun., vo. 9, no. 2, pp , Dec [3] M. Ghogho, A. Swami, and G. Giannakis, Optimized nu-subcarrier seection for CFO estimation in OFDM over frequency-seective fading channes, in Proc. IEEE GLOBECOM, Nov. 200, vo., pp [4] U. Turei, D. Kivanc, and H. Liu, Experimenta and anaytica studies on a high-resoution OFDM carrier frequency offset estimator, IEEE Trans. Veh. Techno., vo. 50, no. 2, pp , Mar [5] K. Sathananthan and C. Teambura, Probabiity of error cacuation of OFDM systems with frequency offset, IEEE Trans. Commun., vo. 49, no., pp , Nov [6] C. Y. Wong, R. Cheng, K. Lataief, and R. Murch, Mutiuser OFDM with adaptive subcarrier, bit, and power aocation, IEEE J. Se. Areas Commun., vo. 7, no. 0, pp , Oct [7] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, [8] Part : Wireess LAN Medium Access Contro (MAC) and Physica Layer (PHY) Specifications: High-Speed Physica Layer in the 5 GHz Band, IEEE Std. 802.a-999, Sep Suppement to IEEE Std , pp [9] J. Medbo, H. Haenberg, and J.-E. Berg, Propagation characteristics at 5 GHz in typica radio-lan scenarios, in Proc. IEEE Veh. Techno. Conf. Spring, May 999, vo., pp Performance of Variabe-Power Adaptive Moduation With Space Time Coding and Imperfect CSI in MIMO Systems Xiangbin Yu, Shu-Hung Leung, Wai Ho Mow, Senior Member, IEEE, and Wai-Ki Wong Abstract The performance anaysis of muti-input muti-output (MIMO) systems with M-ary quadrature ampitude moduation (MQAM) and a space time bock code (STBC) over fat Rayeigh fading channes for imperfect channe state information (CSI) is presented. In this paper, the optimum fading gain switching threshods for attaining maximum spectrum efficiency (SE) subject to a target bit-error rate (BER) and an average power constraint are derived. It is shown that the Lagrange mutipier in the constrained SE optimization does exist and is unique for imperfect CSI and for singe-input singe-output (SISO) systems under perfect CSI. On the other hand, the Lagrange mutipier wi be unique if the existence condition for MIMO under perfect CSI is satisfied. Numerica evauation shows that the variabe-power (VP) adaptive moduation (AM) with STBC provides better SE than its constant-power (CP) counterpart. Index Terms Adaptive moduation (AM), muti-input muti-output (MIMO) system, space time coding, spectrum efficiency (SE), variabe power (VP). I. INTRODUCTION Adaptive moduation (AM) is a powerfu technique for improving the spectrum efficiency (SE), which can take advantage of the timevarying nature of wireess channes to transmit data at higher rates under favorabe channe conditions and to maintain the bit error rate (BER) by varying the transmit power and symbo rate under poor channe conditions [] [3]. The mutipe-antenna approach is another we-known SE technique with diversity and/or coding gain. In particuar, space time coding in a mutiantenna system provides effective transmit diversity for combating fading effects [4], [5]. Therefore, the effective combination of AM and mutipe-antenna technique has received much attention in the iterature [6]. Most of the above systems, however, empoy constant-power (CP) AM schemes. This CP approach restricts the systems performance because the freedom of a variabe power (VP) has been ignored. VP in AM has been considered in [7] [9], which are mosty singeinput singe-output (SISO) systems. For the optimization of the SE for most of the aforementioned schemes, the Lagrange mutipier technique is used to integrate constraints to the optimization probem. However, the existence and uniqueness of the Lagrange mutipier have yet to be studied in the iterature. Furthermore, no practica agorithm for computing the Lagrange mutipier has been deveoped. The notations we use throughout this paper are as foows. Bod uppercase and owercase etters denote matrices and coumn vectors, respectivey. The superscripts ( ) H, ( ) T, and ( ) denote the /$ IEEE Manuscript received March 2, 2007; revised Apri 5, 2008, May 26, 2008, and Juy, First pubished Juy 25, 2008; current version pubished Apri 22, This work was supported in part by the China Postdoctora Science Foundation under Grant and in part by CityU under Grant The review of this paper was coordinated by Dr. C. Yuen. X. Yu was with the City University of Hong Kong, Kowoon, Hong Kong. He is now with the Nanjing University of Aeronautics and Astronautics, Nanjing 2006, China (e-mai: yxbxwy@nuaa.edu.cn). S.-H. Leung and W.-K. Wong are with the City University of Hong Kong, Kowoon, Hong Kong (e-mai: eeeugsh@cityu.edu.hk; eewkwong@ ciytu.edu.hk). W. H. Mow is with the Hong Kong University of Science and Technoogy, Kowoon, Hong Kong (e-mai: ecewhmow@ece.ust.hk). Digita Object Identifier 0.09/TVT Authorized icensed use imited to: Eindhoven University of Technoogy. Downoaded on February 9,200 at 07:38:7 EST from IEEE Xpore. Restrictions appy.