IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 1 Synchronization, Channel Estimation, and Equalization in MB-OFDM Systems Y. Li, Student Member, IEEE, H. Minn, Senior Member, IEEE, and R. M. A. P. Rajatheva, Senior Member, IEEE
Abstract— This paper addresses preamble-based low com- synchronization method, while UWB systems demand a low plexity synchronization, channel estimation and equalization complexity approach. for Zero-padded (ZP) MB-OFDM based UWB systems. The proposed synchronization method consists of sync detection, The MB-OFDM systems from [2] [3] have the following coarse timing estimation, fine timing estimation, and oscillator distinctive characteristics if compared to conventional OFDM frequency offset estimation. The distinctive features of MB- systems: 1) different channel responses and channel energies OFDM systems and the interplay between the timing and carrier across different bands, 2) different carrier frequency offsets frequency hopping at the receiver are judiciously incorporated across different bands, 3) the use of zero padding (ZP) instead in the proposed synchronization method. In order to apply the low complexity one-tap frequency-domain equalizer, the required of cyclic prefix (CP), 4) the interplay between the timing circular convolution property of the received signal is obtained and the frequency hopping (a mismatched timing point at the by means of an overlap-add method after the frequency offset receiver will yield a mismatched frequency hopping and hence compensation. The proposed low complexity channel estimator a significant performance degradation). These characteristics for each band is developed by first averaging the over-lap- provide diversity components as well as additional design con- added received preamble symbols within the same band and then applying time-domain least-squares method followed by the straints, and hence, should be taken into account in the designs discrete Fourier transform. We develop an MMSE equalizer and of synchronization, channel estimation, and equalization. its approximate version with low complexity. We also derive the There exist several preamble-based synchronization and probability density functions of the UWB channel path delays, channel estimation methods for regular OFDM systems (e.g., and using them we present several optimization criteria for our see [8]–[11] and references therein), but only a few counter- proposed synchronization, channel estimation, and equalization. The effectiveness of our proposed methods and optimization parts for MB-OFDM-based UWB systems [12]–[15]. These criteria are confirmed by computer simulation results. existing works have overlooked most of the distinctive features of MB-OFDM systems. The works in [13] and [14] applied Index Terms— BLUE, channel estimation, equalization, fre- quency offset, OFDM, synchronization, timing offset, UWB. the concept of utilizing the first significant channel tap for timing improvement, which was previously proposed in [10] [11] for regular OFDM systems, in MB-OFDM systems. [13] I. INTRODUCTION considered a ZP MB-OFDM system with the same frequency NE of the promising ultra-wide band (UWB) tech- offset and channel impulse response for all bands, while O nologies is multi-band orthogonal frequency division [14] considered a CP MB-OFDM system with independent multiplexing (MB-OFDM) [1] which has been proposed for channels across different bands and no frequency offset. the IEEE 802.15.3a and ECMA standard [2] [3]. OFDM is a The work in [12] developed best linear unbiased estimation relatively mature technology and has been adopted in digital (BLUE) based oscillator frequency offset (OFO) estimators broadcasting, wireless LAN and MAN standards. OFDM has for MB-OFDM systems by exploiting some of the MB-OFDM several advantages such as low complexity equalization in characteristics, but other synchronization, channel estimation dispersive channels and the spectral scalability/adaptability. and equalization tasks were not addressed and the interplay However, OFDM is more susceptible to nonlinear distortion between the timing and the frequency hopping was not consid- at the transmit power amplifier [4] and more sensitive to ered. An MMSE-based one-tap equalizer for ZP MB-OFDM synchronization errors [5] [6]. It is shown in [7] that the MB- systems was recently proposed in [15], but no optimization OFDM based UWB receivers are quite sensitive to carrier for ZP MB-OFDM systems was attempted, and the method is, frequency estimation errors. Hence, the application of OFDM as will be shown in this paper, just an approximate MMSE. technology in UWB systems requires a high performance The overlap-add (OLA) method was proposed in [16] for the Manuscript received June 5, 2007; revised September 23, 2007; accepted low complexity equalization in ZP OFDM systems, but the November 6, 2007. The associate editor coordinating the review of this paper effects of OLA window lengths and timing offsets were not and approving it for publication was E. Serpedin. This paper was presented considered. in parts at ICC 2007. Y. Li and H. Minn are with the Department of Electrical Engineering, In this paper, we propose low-complexity synchronization, University of Texas at Dallas, P.O. Box 830688, EC 33, Richardson, TX channel estimation and equalization methods and correspond- { } 75080 USA (e-mail: yxl044000, hlaing.minn @utdallas.edu). ing parameter optimizations for ZP MB-OFDM systems based R. M. A. P. Rajatheva is with the Asian Institute of Technology, PO Box 4, Klong Luang 12120, Pathumthani, Thailand (e-mail: [email protected]). on the preamble of [2] [3]. The main contributions of this The work of H. Minn was supported by the Erik Jonsson School Research paper are as follows. Excellence Initiative, the University of Texas at Dallas, USA. Digital Object Identifier 10.1109/T-WC.2008.070601 1) We derive the probability density functions (pdf) and the 1536-1276/08$25.00 c 2008 IEEE 2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008
statistics of the channel path delays of the IEEE 802.15 UWB channel model from [17]. Note that these pdfs are also useful in evaluating or optimizing the energy capture of other UWB systems. 2) We present distinctive characteristics of the ZP MB- OFDM systems which are worthy to be taken into account into the system design but have been overlooked by the existing works. 3) We present how to optimize the parameter settings Fig. 1. The structure of the preamble pattern 1 from [2] and [3] (Also of the proposed synchronization, channel estimation, shown are the three preamble parts used in our synchronization method. and equalization methods by utilizing the distinctive The preamble symbols of carrier 2 and 3 within Part-a are not used in our characteristics of the ZP MB-OFDM systems and the synchronization method. channel path delay pdfs we derive. Note that existing works neither exploited all of these characteristics nor addressed all synchronization, channel estimation and 24 periods with an inverted polarity for the last period in equalization tasks together. each band. The channel estimation sequence is composed of 6 4) We derive an exact MMSE equalizer for ZP MB-OFDM identical periods of the frequency-domain channel estimation systems, and its low-complexity implementation. Our sequence. The structures of the preamble pattern 1 in each of MMSE equalizer reduces to the equalizer in [15] after the three different bands are shown in Fig. 1 (see [3] [12] for a high SNR approximation. other patterns). 5) Our investigation provides new insights for improving The MB-OFDM-based systems from [2] and [3] use ZP the receiver functionalities with low complexity. For guard intervals consisting of Npre prefix and Nsuf suffix zero example, it reveals different performance sensitivities samples, instead of the conventional CP guard interval. An to different parameters, and provides guidelines in set- OFDM symbol including the guard interval has a total of M = ting these parameters for robustness and complexity- N +Ng samples where Ng = Npre+Nsuf and, N is the number reduction. of the OFDM sub-carriers. Between ith and (i +1)th OFDM The rest of this paper is organized as follows. Section II symbols in [2] and [3], there are 37 guard samples consisting describes the signal and channel models. Section III introduces of Nsuf =32suffice samples and Npre =5prefix samples. the statistics of the UWB channels. The proposed synchro- The preamble sample index 0 corresponds to the first non-zero nization, channel estimation, and equalization methods are sample of the first preamble symbol. Denote the low-pass- presented in Section IV, and the parameter optimizations of equivalent time-domain samples (sampled at N/T =1/Ts, N the proposed methods are described in Section V. Simulation times the sub-carrier spacing) transmitted in the q-th frequency results and discussions are provided in Section VI, and the band by {sq(k)} and their corresponding frequency-domain paper is concluded in Section VII. symbols by {s˜q(n)}. During the m-th symbol interval, i.e., Notations: The superscripts T and H represent the transpose k ∈ [mM − Npre, (m +1)M − Npre − 1], {sq(k)} can be all and the Hermitian transpose, respectively. The kth row, nth zeros if no signal is transmitted in the qth band. X X · column element of a matrix is denoted by [ ]k,n. E[ ] The IEEE 802.15.3a UWB RF channel model described in represents a statistical expectation. The N × N unitary DFT [17] is given by F f f f F matrix is denoted by =[ 0, 1,..., N−1], while K is the sub-matrix of F consisting of the first K columns and Lh K F (:,i : k) represents [f , f +1,...,f ]. The corresponding i i k hRF(t)=X α δ(t − T − τ ) (1) frequency-domain variable of a time domain variable x is k,l l k,l l=0 k=0 denoted by x˜. {·} represents the real part of {·}.TheN ×N identity matrix is denoted by I and e represents the ith i where T , τ , and X are random variables representing column of I. l k,l the delay of the l-th cluster, the delay (relative to the l- th cluster arrival time) of the k-th multipath component of II. SIGNAL AND CHANNEL MODELS the l-th cluster, and the log-normal shadowing, respectively. We consider the MB-OFDM-based UWB systems from The channel coefficients are defined as a product of small- [2] [3] where the carrier frequency is hopped within a pre- scale and large-scale fading coefficients, i.e., αk,l = pk,lξlβk,l ± defined set of carrier frequencies {fq} (corresponding to where pk,l takes on equiprobable 1 to account for signal { } disjoint frequency bands) from symbol to symbol, according inversion due to reflections, and ξlβk,l are log-normal to a time-frequency code. The MB-OFDM system from [2] distributed path gains. Then we have E [αk,l(Tl,τk,l)] = 0 2 − Γ − | | Tl/ τk,l/γ and [3] has 4 different preamble patterns (for 4 pico-nets), and E αk,l(Tl,τk,l) =Ω0e e . Γ and γ are the each associated with a different time-frequency code. Each cluster decay factor and ray decay factor, respectively. Details preamble pattern is constructed by a synchronization sequence of the channel models are referred to [17]. portion followed by a channel estimation sequence portion. In this paper, we consider a low-pass-equivalent system that The synchronization sequence is composed by successively absorbs the carrier-frequency hopping into the channel impulse repeating a time-domain preamble sequence (symbol) over response (CIR). The sample-spaced low-pass-equivalent CIR LI et al.: SYNCHRONIZATION, CHANNEL ESTIMATION, AND EQUALIZATION IN MB-OFDM SYSTEMS 3 for the q-th band is given by Λ and λ are the cluster arrival rate and the ray arrival rate, respectively. The m-th moment of Zk,l is obtained as Lh K − 2 ( + ) m j πfq Tl τk,l − − − m d ΦZk,l (s) hq(n)=X αk,le p(nTs Tl τk,l t0) E[z ]= | =0 k,l dsm s l=0 k=0 (2) m i −1 m −i−1 m −i −m+i where the effect of the combined transmit and receive filter = (−l − n)(−Λ) (−k − u)(−λ) . =0 i =0 =0 with the impulse response p(t) whose span is [−t0,t0] has i n u (6) been included in the CIR, and the delay t0 is inserted for the causality. Applying the residue theory, we obtain the pdf of Zk,l as { q} { q − } ∞ Define tl = tl (i): i =0,...,N 1; l =1,...,Lq −sz q f (z )= Φ (s)e k,l ds where {t (i):i =0, 1, ..., N − 1} denotes the time-domain Zk,l k,l Zk,l l −∞ sample index set corresponding to the l-th non-zero preamble l k (l+k) (R1 + R2)Λ λ (−1) k =0or l =0 symbol period in the q-th band with N ≤ N ≤ N +Nsuf . L = q δ(z ) k =0&l =0 is the number of nonzero preamble symbols in the q-th band k,l (7) and depends on the preamble pattern and the band index q.Let { } where xq(k) denote the low-pass-equivalent time-domain channel ⎧ −1 i−1 ( + ) ⎪ l −Λzk,l l m=0 k m output signal samples corresponding to the q-th band. Then ⎪ (−1) e =0 ⎨⎪ i i!(l−i−1)! the corresponding low-pass-equivalent time-domain received (−k−i) (l−i−1) q (Λ − λ) z u(zk,l) k =0& l =0 samples {r¯ (t (i))} in the q-th band can be expressed as 1 R1 = k,l q l ⎪ (−1)l (l−1) −Λ ⎪ z e zk,l u(z ) k =0&l =0 ⎩⎪ (l−1)! k,l k,l q 2 q ( ) q q j πυq tl i /N 0 l =0 r¯q(tl (i)) = e xq(tl (i)) + n(tl (i)) (3) ⎧ −1 i−1 ( + ) q ⎪ k −λzk,l k m=0 l m where {n(t (i))} are zero-mean independent and identically- ⎪ (−1) e =0 l ⎨⎪ i i!(k−i−1)! distributed circularly-symmetric complex Gaussian noise sam- (−l−i) (k−i−1) (Λ − λ) z u(zk,l) k =0& l =0 2 q 2 R2 = k,l ples with variance σ = E{|n(t (i))| }, and υq is the ⎪ (−1)k (k−1) − l ⎪ z e λzk,l u(z ) k =0& l =0 normalized (by the subcarrier spacing) carrier frequency offset ⎩⎪ (l−1)! k,l k,l (CFO)oftheq-th band which is related to the normalized OFO 0 k =0, υ by with u(·) being the unit step function. Since {Tl} and {τk,l} are independent, we have υq = bq υ, for q =1, 2, 3. (4)
fTl,τk,l (x, y)=ftl (x)fτk,l (y) (8) For the frequency synthesizer in [1], we have [b1,b2,b3]= 13 15 17 where fTl (x)=fZ0,l (x) and fτk,l (y)=fZk,0 (y). 16 , 16 , 16 (c.f. [12]). Since the receiver does not know Next, we find the covariance matrix of the CIR for each in advance when the preamble will arrive, we denote the band. Let L denote the maximum length of the sample-spaced low-pass-equivalent received signal as {rq(k)} which contains CIR vector of each band and hq denote the L × 1 CIR vector noise-only samples followed by the received preamble samples of the qth band. Depending on the actual length of each {r¯q(m)}. The number of the initial noise-only samples is a channel realization, hq may contain some zero samples. The random variable depending on the arrival time of the preamble covariance matrices of hq are the same for all bands as will and the receiver’s sync detection window timing. be apparent in the following. Hence, we simply denote the covariance matrix of hq by Ch. From (2), we can calculate 2 the element of Ch as III. STATISTICS OF THE UWB CHANNEL MODELS ∞ ∞ Lh K 2 ∆ Ch(m, n)= E |αk,l(Tl,τk,l)| We first derive the pdf of the delay Zk,l = Tl + τk,l for the 0 0 =0 =0 k-th ray of the l-th cluster of the UWB channel, from which l k × p(mTs − Tl − τk,l − t0)p(nTs − Tl − τk,l − t0) the pdfs of Tl and τk,l are obtained. × f (T ,τ ) dT dτ (9) Note that T0 =0, τ0,l =0; and Zk,l can be expressed as Tl,τk,l l k,l l k,l the sum of inter-arrival times of the first l clusters and the first where we have used the statistical independence among k rays of the lth cluster. Since inter-arrival times of the UWB {αk,l(Tl,τk,l)}. 2 channel paths are independent exponential random variables, Substituting (7)-(8) and E |αk,l(Tl,τk,l)| = − −T /Γ −τ /γ Zk,l is the sum of (l +k 2) independent exponential random Ω0e l e k,l into (9) gives variables. The moment generating function of Zk,l is given by Lh K T T −Tl/Γ −τk,l/γ Ch(m, n)= Ω0e e Λ l λ k 0 0 Φ (s)= , ∀k, l except k = l =0. l=0 k=0 Zk,l − − Λ s λ s × p(mTs − Tl − τk,l − t0) p(nTs − Tl − τk,l − t0) (5) × fZ0,l (Tl)fZk,0 (τk,l) dTl dτk,l (10)
1An arbitrary carrier phase offset can be absorbed into the CIR and hence 2We neglect the shadowing parameter X in this paper and hence realiza- it is not included explicitly in the signal model. tions of {αk,l} are not normalized. 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 which can be numerically evaluated. We will use the above A. Sync Detection and Coarse Timing Estimation channel covariance matrix in optimizing our proposed methods The carrier frequency of the receiver is initially set to in Section V. Note that the pdfs of the UWB channel path the one that corresponds to the first preamble symbol, and delays we derive are also useful in calculating or optimizing denote the corresponding band index by q¯. During the sync the energy capture of other UWB systems. detection process, the receiver carrier frequency hopping is not performed since it requires timing information which is IV. PROPOSED SYNCHRONIZATION,CHANNEL unavailable yet. We use the first two preamble symbols in the ESTIMATION AND EQUALIZATION q¯th band to detect the arrival of the preamble. Then the sync In this paper, we consider sync detection, timing offset esti- detection metric is given by mation, OFO estimation, channel estimation and equalization M(k)=|R ¯(1,k,N )| . (13) for the packet-based MB-OFDM system in [2] and [3]. We a,q assume that the receiver knows what preamble the transmitter For complexity reduction4 and energy saving, the computation uses but not the OFO and the arrival time of the preamble. of M(k) is performed only at k =0, ∆, 2∆, ..., where ∆ The whole interval of the 30 preamble symbols with the (an integer) is a design parameter. { q } M sample index set tl (i) is divided into three disjoint parts Once (k) becomes larger than a pre-defined threshold – part-a, part-b, and part-c – with the corresponding sample MSD, say at k = kSD, the receiver decides that the preamble { q } { q } { q } index sets ta,l(i) , tb,l(i) , and tc,l(i) , respectively. All has been detected and starts finding the coarse timing point the distances of the preamble symbol pairs within part-x are as follows: denoted by {dx(m):m =1, 2,...} where x is a, b,or kct = arg max {M(k)} (14) c representing the corresponding part. For example, for the kSD≤k≤kSD+W1−1 preamble pattern 1, we may set the first part as the first 6 where W1 is the length of the window within which the coarse symbol interval, the second part as the next 18 symbol interval, timing point is searched and W1 can be set to M.Alower and the third part as the last 6 symbol interval (see Fig. 1). In { q } { q } { q } { q complexity version of the above coarse timing estimator can this case, ta,l(i) = tl (i):l =1, 2 , tb,l(i) = tl (i): q q be implemented by finding the point k1 corresponding to the l =3, 4, 5, 6, 7, 8}, {t (i)} = {t (i):l =9, 10}, {db(m)} c,l l maximum of the metric values at k = kSD, kSD +∆1, kSD + = {3M,6M,9M,12M,15M}. We also define dx(0) = 0 for 2∆1, ... , and then the point corresponding to the maximum convenience. of the metric values at k1 −∆1/2, k1 −∆1/2 +1, ..., Define the correlation term with a correlation distance k1 + ∆1/2. Here, ∆1 (an integer) is a design parameter. dx(m) between non-zero received preamble symbols within Based on the coarse timing point kct, the receiver determines part-x in the q-th band as at what time instant it should hop the carrier frequency to R (m, kt,N ) demodulate the rest of the preamble. x,q − ∗ = ( 1) rq (k)rq(k + dx(m)) (11) B. Fine Timing Offset Estimation {k, k + dx(m)} ∈{ q } The fine timing estimation is based on the preamble part-b kt + tx,l(i) while the coarse timing estimation is based on part-a. Hence, where the correct points for the coarse timing point kct and the fine { }∈{ q } timing point kft correspond to the beginning of part-a and part- 1, if k + dx(m) kt + tb,8(i) = (12) b, respectively. We denote the beginning of part-b by kref .The 0, else proposed fine timing estimator is described by 5 note that i =0, 1,...,N − 1. is a parameter introduced 3 by the burst cover sequence (see [2] [3] for details). During kft = −η + arg max |Rb,q(1,i,N )| ∈( + − + + ) part-a where the sync detection and coarse timing estimation i kct Da W2,kct Da W3 q=1 are performed, the receiver carrier frequency hopping is not (15) performed. During the remaining parts, the receiver carrier where Da is the length (in samples) of part-a and η is a fixed frequency is hopped according to the preamble pattern. The timing adjustment. For CP-based OFDM systems, η should be fine timing and the OFO estimations are based on the received set such that the timing point is most of the time within the preamble symbols within part-b, while channel estimation task ISI-free part of the CP [10]. We will describe how to set η is based on part-c 3. The reason for the use of only part- for the considered ZP OFDM system in Section VI. b for the fine timing estimation is to allow processing time so that timing adjustment (for frequency hopping instants) C. OFO Estimation and Compensation based on the fine timing estimation can be made at the end We adopt the BLUE-based OFO estimator from [12] with of the preamble. The receiver tasks are performed in the a few differences. In [12] the OFO estimator utilizes the first following order: sync detection, coarse timing estimation, fine 21 preamble symbols, while in this paper the OFO estimator timing estimation, OFO estimation and compensation, channel estimation, equalization and data detection. 4M2(k) can also be used in place of M(k). 5Note that although more correlation terms can be included in the fine tim- 3Note that preamble symbols from part-b can also be used to improve the ing estimation, for low complexity implementation we just use the correlation channel estimation. terms with the correlation distance db(1). LI et al.: SYNCHRONIZATION, CHANNEL ESTIMATION, AND EQUALIZATION IN MB-OFDM SYSTEMS 5 is based only on part-b allowing processing time and resource Next, CFO compensation for the qth band is performed for prior tasks of sync detection and coarse timing estimation. starting from preamble part-b as Furthermore, the correlation window length for each preamble ∆ −j2πmυb q /N symbol is fixed at N samples in [12] but is set to N in this rq(kft + m)e = r˘q(kft + m)form =0, 1,.... paper which will be optimized for better performance. (26) The OFO estimator is given by
3 1T C−11 D. Channel Estimation q=1 θ bqvq υ = q (16) 3 T −1 2 The use of zero-amplitude guard samples in MB-OFDM =1 1 C 1 b q θq q UWB systems [2] [3] instead of conventional CP guard Hq samples saves transmit power and allows a sufficient amount where υq = ωq(m)θq(m) (17) of time for the transmitter and receiver to switch between m=1 carrier frequencies. To obtain the circular convolution prop- N erty required for the frequency-domain one-tap equalization, θq(m)= angle{Rb,q(m, kft,N )} (18) 2πdb(m) we apply the OLA method [16] to the frequency-offset- compensated received signal. Specifically, for every l-th non- with ωq(m) being the m-th component of the weighting vector zero OFDM symbol in the qth band after CFO compensation, −1 −1 −1 ≤ ≤ { ω = C 1 1T C 1 . (19) the OLA method adds M0(0 M0 Nsuf ) samples r˘q(kft+ q θq θq lM +N), r˘q(kft +lM +N +1),...,r˘q(kft +lM +M0 +N −1)} C { { − } In (19), θq is the covariance matrix of θq(m):m = to r˘q(kft +lM), r˘q(kft +lM +1),...,r˘q(kft +lM +M0 1) , } 1 (l) 1,...,Hq , Hq is a design parameter (see [12]) and denotes resulting in an N × 1 vector yq with the k-th element given the all-one column vector, whose size should be apparent from by the expression. Two approaches (Methods A and B) for the ⎧ ⎨⎪r˘q(kft + lM + k) calculation of {Cθ } are given in [12]. Due to the different q (l) window length in this paper, the covariance matrices {C } y (k)= +r˘q(kft + lM + k + N), if 0 ≤ k ≤ M0 − 1 θq q ⎩⎪ become slightly different from those in [12]. For example, r˘q(kft + lM + k), else. for preamble pattern 1 or 2, denoting the number of non-zero (27) preamble symbols within the combined part-b in the qth band The OLA length M0 need not be the same for preamble sym- ¯ by Lq, and applying Method A, we obtain bol and data symbol. We use Mp and Md to denote the OLA length for preamble symbol and data symbol, respectively. C (m, n)=g(m, n) θ⎧q After applying the OLA method, the preamble symbols in ( ¯ − ) 2 ⎪ Lq m N σ ¯ ⎪ m + 2 ( ) , if m = n & m ˆ ˆ˜ ˆ˜ ˆ˜ T by√ taking the DFT of hq as [hq(0), hq(1),...,hq(N −1)] = spreading) is obtained as NF L hˆq. H H −1 T T T Y˜ m = H Cs˜ H Cs˜ V [y˜ 2 y˜ 2 ] (34) lξ nξ lξ, m nξ, m H H Hl Cs˜H + Cn Hl Cs˜H E. Equalization V ξ lξ ξ nξ where = H H Hn Cs˜H Hn Cs˜H + Cn 1) One-tap Zero-Forcing (ZF) Equalizer: In general, data ξ lξ ξ nξ (35) can be transmitted with or without spreading using a time- frequency code. Suppose the mth OFDM data symbol (after The matrix inverse operation indicates that the MMSE spreading if applied) is transmitted in the qth band, and let equalizer is much more complicated than the one-tap ZF y˜q,m(i) denote the corresponding received symbol on the equalizer. We provide a much simplified computation of the ith subcarrier after performing CFO compensation, the OLA inverse in (33) and (34) in Appendix. method, and DFT. If the data are not spread by a time- 3) One-tap MMSE Equalizer: Although the MMSE equal- frequency code, the frequency-domain one-tap ZF equalizer izer can be simplified, the complexity is still higher than the output decision variable for the ith subcarrier data of the mth one-tap ZF equalizer. If we neglect the extra noise introduced OFDM data symbol is given by by the OLA method, then the noise can be approximated as the white Gaussian noise with variance σ2. Under this ˜ (m) ˆ˜ ∗ |ˆ˜ |2 Y (i)=hq(i) y˜q,m(i)/ hq(i) . (30) circumstance, the MMSE equalizers in (33) and (34) can be simplified to the same form as proposed in [15]. For the UWB If data spreading is performed using a time-frequency code [2] system without data spreading, the output of the approximated [3] such as each data symbol is transmitted on two consecutive one-tap MMSE equalizer is given by symbol intervals using a frequency hopping pattern of [l0 n0 l1 n1 l2 n2](e.g.,[123123]),thentheZFequalizer output (m) ˜ ∗ ˜ 2 2 2 Y˜ (i)=hˆq(i) y˜q,m(i)/|(hˆq(i)| + σ /E[|s˜(i)| ]). (36) decision variable for the mth data (before spreading) on the n i-th subcarrier is obtained as For the system using data spreading, the output of the approx- ˜ ∗ ˜ ∗ imated one-tap MMSE equalizer is given by hˆ (i) y˜ 2 (i)+hˆ (i) y˜ 2 +1(i) ( ) lξ lξ, m nξ nξ, m Y˜ m (i)= (31) ˆ˜ ∗ ˆ˜ ∗ ˜ 2 ˜ 2 hlξ (i) y˜lξ,2m(i)+hnξ (i) y˜nξ,2m+1(i) |hˆ (i)| + |hˆ (i)| (m) lξ nξ Y˜ (i)= . (37) |ˆ˜ |2 |ˆ˜ |2 2 | | 2 hlξ (i) + hnξ (i) + σn/E[ x˜(i) ] where ξ = m mod 3 in the band indexes lξ and nξ. 2) MMSE Equalizer: A one-tap MMSE-based equalizer for Furthermore, if the frequency-domain data are QPSK sym- MB-OFDM systems has been proposed in [15] by assuming bols [2] [3], the approximated one-tap MMSE equalizer is that the frequency-domain noise remains white after applying equivalent to the one-tap ZF equalizer we introduced in the OLA method. However, the covariance matrix of the (30) and (31). The reason is that the detection of QPSK frequency-domain noise after applying the OLA method is symbols only depends the phase of the received signal, and C 2 I F F H given by n = σ ( N + Md Md ), which is no longer the noise variance term in the expression of the approximated diagonal. Hence the MMSE equalizer in frequency domain one-tap MMSE equalizer will not affect the received phase is no longer a one-tap equalizer, and the equalizer proposed information. in [15] is not an MMSE equalizer in the strict sense. In this section, we will introduce the MMSE equalizer for the MB- V. O PTIMIZING THE PROPOSED METHODS OFDM systems. First we assume the data are not spread. Define This section presents how to optimize the proposed methods by designing the window lengths involved in several tasks. H ˆ˜ ˆ˜ ˆ˜ − T q = diag [hq(0), hq(1),...,hq(N 1)] . (32) Recall that we consider Nsuf =32zero-amplitude suffix sam- ples and Npre =5zero-amplitude prefix samples. We assume The frequency-domain MMSE equalizer output decision vari- that the maximum channel dispersion is Nsuf =32samples, able for the mth OFDM data symbol on the qth band is given and the interval of the Npre =5prefix samples is reserved by for switching the carrier frequency. Hence, the considered −1 correlation window length range is [N,N+Nsuf ] = [128, 160] Y˜ H C H C HH C y m = q s˜ q s˜ q + n ˜q,m (33) and the OLA window length range is [0, 32]. where Cs˜ represents the covariance matrix for transmit data in the frequency-domain. If the data are independent QPSK A. Window Length for Timing Estimation C signals in frequency-domain, s˜ is a diagonal matrix with Define “1”s on the transmitted subcarrier and “0”s on the guard subcarriers [2] [3]. β(i, T1,N )=Ra,1(1,T1 + i +1,N ) − Ra,1(1,T1 + i, N ). If data are spread using a time-frequency code and the (38) frequency hopping pattern of [l0 n0 l1 n1 l2 n2], the MMSE Then for a given channel output signal {x1(k)}, equalizer output decision variable for the mth data (before {β(i, T1,N )} are independent complex Gaussian random LI et al.: SYNCHRONIZATION, CHANNEL ESTIMATION, AND EQUALIZATION IN MB-OFDM SYSTEMS 7 ⎧ ⎪ F (:, (M : L − k − 1))H (M + k )(F (:, (N − L + M + k : N − 1)))H s˜ ⎪ d ft a d ft d ft q ⎪ + F (:, (N − k : N − 1))H (k )(F (:, (0 : k − 1)))H s˜ if k > 0&M + k variables with we develop our optimization for CIR estimation under perfect synchronization first. Next, we incorporate the effect of timing E[β(i, T1,N )] offset into our design. ∗ ∗ 1 − 1 = x1(i + N )x (i + N + da(1)) x1(i)x (i + da(1)) After applying the OLA method with window length Mp (39) on the CFO-compensated averaged received preamble sym- 2 2 2 Var[β(i, T1,N )] = σ (|x1(i)| + |x1(i + da(1))| bol, the MSE of the length-L CIR estimation under perfect 2 2 4 synchronization can be derived as + |x1(i + N )| + |x1(i + N + d (1))| )+2σ . (40) a 2 H Then we use the following criterion to design the correlation MSEh(L ,Mp) = trace (Ch)+σ trace Υ Υ(I + T ) window length N for timing estimation t ΥH ΥSC¯ S¯ − ΥSC¯ ⎧ ⎫ +trace h 2 trace h . ⎨N/ 2 ⎬ |E[β(i, T1,N )]| (44) Nt = arg max 1 − Q ≤ ≤ + ⎩ ⎭ N N N Ng i=0 Var[β(i, T1,N )] H −1 H where T =[e1,...,eM , 0 ×( − )], Υ =(S S) S (41) ¯ ¯ ¯ p N N Mp and S = SL −S∆. S∆ is an N ×L matrix, and for L>Mp, where Q(·) is the Gaussian tail probability, and T1 for each N ¯ S∆(Mp +1 : L, Mp +1 : L)=S∆, and the other elements of is chosen such that [T1,T1 + N + N/2 − 1] is approximately ¯ ¯ S∆ are all zeros. If L ≤ Mp, S∆ is the all-zero matrix. S∆ kref ,kref N Nsuf symmetric with respect to [ + + ]. This criterion is an (L − M ) × (L − M ) matrix with the (n, k)th element can be approximately considered as the probability of the p p given by s(N − 1 − k + n) if k ≥ n, and zero if k C discuss this issue in Section VI. where θq |N is given in (20). Then the best window length Nf for our OFO estimator is determined by D. Optimizing the Data OLA Length Nf = arg min {Var{υq|N }} . (43) ≤ ≤ + N N N Nsuf The OLA method increases the noise power if compared to the CP-based systems since the noise terms are also C. CIR Estimation Window Length and Preamble OLA Length added up. In this sense, we prefer a smaller OLA length The channel taps with largest delays may contain negligible Md (for data) which will decrease the extra noise power. channel energy and neglecting them in the CIR estimation However, the shortening of the OLA length Md may destroy will suppress noise from those neglected taps and hence the circular convolution property and in this circumstance the may improve the CIR estimation performance. Hence, in this OLA method with one-tap frequency-domain equalizer will section, we derive channel estimation MSE obtained with the introduce the inter-carrier interferences (ICI). For example, CIR estimation window length L , and find the optimum L with perfect timing, if Md is less than the order of the CIR, which yields the minimum channel estimation MSE. Since fine there will be ICI. Note that the timing offset also affects timing and frequency synchronization have been performed, the optimal value of Md. In this section, we will find the 8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 0 10 optimum Md which minimizes the combined interference and SNR = 15dB extra noise power introduced by the OLA method. SNR = 10dB After applying the OLA method, the received i-th subcarrier SNR = 5dB −2 data can be expressed as 10 − ICI e y˜q(i)=˜xq(i) x˜q (i)+˜nq(i)+˜n(i) (46) −4 10 where x˜q(i)=h˜q(i)˜sq(i), n˜(i) is the frequency-domain noise e sample (the same statistics as in CP-based systems), n˜q(i) is the extra frequency-domain noise sample due to the OLA −6 ICI 10 method, and x˜q (i) is the ICI term on the i-th subcarrier due to the OLA method. Denote the corresponding vectors ICI e ICI e s˜ x˜ n˜ n˜ −8 of s˜q(i), x˜q (i), n˜(i), and n˜q(i) by q, q , , and q, 10 respectively. Note that we have window length metrics for timing e n˜ = F (:, (0 : Md − 1)) q −10 10 × n(kft + lM + N : kft + lM + N + Md − 1). (47) For simplicity in presentation, we assume in this section that −12 the perfect timing point index (for kft)is0 (i.e., kref =0). 10 xICI 130 135 140 145 150 155 160 Then, the ICI vector ˜q can be expressed as (48), where timing window length N’ Ha(x) is an (L−x)×(L−x) matrix with the (n, k)th element Fig. 2. The timing correlation window length design metric. being equal to hq(L − 1 − k + n) if k ≥ n, and zero if k (a) (b) 0.9 100 M = 30 d N’=128, t = 5Ts 20 0 M = 25 90 0.8 d N’=132, t0 = 5Ts M = 20 N’=144, t = 5T d 0 s 80 15 N’=160, t = 5Ts 0.7 M = 15 0 ) d 2 N’=132, t = 3T 70 M = 10 0 s d N’=132, t = 2T 0.6 10 0 s 60 N’=128, t0 = 5Ts 50 0.5 N’=132, t = 5Ts 5 0 40 N’=144, t0 = 5Ts 0.4 N’=160, t0 = 5Ts Mean timing offset (sample) 30 0 Variance of timing offset (sample 20 0.3 interference plus extra noise power −5 10 0.2 0 0 5 10 15 20 0 5 10 15 20 SNR (dB) SNR (dB) 0.1 −30 −20 −10 0 10 20 30 η =0 Fig. 3. The simulation results of the fine timing offset estimation with . timing offset (sample) −7 10 Fig. 5. The effects of timing offset and data OLA length Md. reference (N’ = 132) proposed (N’ = 128) ence which is obtained by correlating adjacent OFDM symbols proposed (N’ = 132) proposed (N’ = 144) and averaging on each band. For the reference method, the proposed (N’ = 160) estimated CFOs on the three bands are converted to OFO −8 estimates (see (4)) and averaged to get an OFO estimate. It 10 can be seen that the proposed OFO estimators outperform the reference estimator by nearly 2dB SNR advantage. Fig. 5 presents the average interference plus extra noise power Ξ for several OLA lengths (Md) and timing offsets. Ξ increases significantly when the timing offset lies outside a − −9 certain range [a, b], e.g., a = 8,b =5for Md =20, and MSE of normalized OFO estimator 10 this range increases with Md but b is observed to be fixed around t0/Ts. For a particular Md, the timing offset should be within the corresponding range [a, b] in order to minimize the interference plus extra noise power. This range is similar 0 5 10 15 20 SNR (dB) to the ISI-free timing offset range in CP-based OFDM systems but there are two differences. First, in CP-based systems the Fig. 4. The simulation results of the OFO estimation. above timing offset range is [−Ng + L − 1, 0] and positive timing offsets definitely introduce interference, while in ZP OFDM systems the above timing offset range includes some (see the curves with solid line in the figure) gives the mean positive timing offsets. Second, for a timing offset within timing offset (in samples) between (t0/Ts) and 2+(t0/Ts) the corresponding best timing offset range, there is neither for SNRs of practical interest. interference nor extra noise for CP-based systems but there is 8 The OFO estimation MSE results are given in Fig. 4. always extra noise and possibly interference for ZP systems. Both BLUE variance and MSE results are quite insensitive to The timing adjustment parameter η should be chosen such that different correlation window lengths. It can be seen from Fig. 4 the timing offset is most of the time within the above range that different window lengths give virtually the same MSE [a, b]. Based on our observations in Fig. 3 and Fig. 5, our performance. Since using a smaller length requires smaller suggestion is to set η approximately equal to σt +(t0/Ts) complexity, a good design choice is to use the best timing where σt is the standard deviation of the fine timing estimator. estimator correlation window length for the OFO estimator We set η =10in our simulation with t0/Ts =5. window length as well, i.e., (Nf = Nt = 132). We also In Fig. 6, we present the channel estimation MSE given present the performance of a conventional estimator as a refer- in (44) versus the channel estimation window length L for different values of the OLA length Mp at SNR of 10 dB. 8The plots of the BLUE variances versus correlation window lengths and For our considered channel, the best parameter setting for the the corresponding discussion are skipped due to the space limitation (see [18] ∗ for details). CIR estimation under perfect synchronization is ([L ]perfect = 10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 0 10 (a) (b) [M ] = 35 p perfect timing offset = −10 [Mp] = 5 timing offset = −5 perfect −1 10 [M ] = 25 timing offset = 0 p perfect [M ] = 15 p perfect −1 10 [M ] = 10 p perfect −1 10 uncoded BER −2 10 channel estimation MSE −3 10 average interference plus extra noise power on one subcarrier −2 10 5 10152025303540 0 102030 0102030 [L’] perfect M M d d Fig. 6. The effects of CIR estimation length [L ]perfect and preamble OLA Fig. 7. The effect of data OLA length Md (SNR = 10 dB). (a) Proposed length [Mp]perfect on the channel estimation MSE (SNR = 10dB) under design metric for Md (b) The uncoded BER performance versus Md. perfect synchronization. −1 10 ∗ 23, [Mp]perfect = 10). The MSE performance degradation due one−tap equalizer to the use of a smaller L than [L ]∗ is more significant perfect MMSE equalizer than that due to the use of a larger L . When taking into account the random timing offset ∆k = kft − kref ,theL ∗ should be chosen larger than [L ]perfect so that most of the ∗ time the CIR estimation window contains the first [L ]perfect actual CIR taps. Mp should also be increased accordingly. A ∗ − −2 good design is to set L =[L ]perfect + σt E[∆k] +2 10 ∗ and Mp =[Mp]perfect + σt − E[∆k] +2. With t0/Ts =5, Nt = 132, and η =10,wehaveσt − E[∆k] 5 and hence a good choice is L =30and Mp =17. At SNR of 15 dB uncoded BER 9 ∗ ∗ , we obtain [L ]perfect =25and [Mp]perfect =12, and hence L =32and Mp =19. Since [L ]perfect and [Mp]perfect are quite insensitive to the SNR, we can use the same L and Mp for different SNRs in our design. Fig. 7(a) presents the plot of Ξ/N versus M at SNR −3 d 10 =10dB with different fixed timing offset values10 of 0, −5 − and 10 samples, respectively. We observe that with perfect 0 5 10 15 20 timing the best Md is 13, while with a timing offset ∆k (a SNR (dB) negative number due to the timing advancement) the best Md is 13−∆k. In Fig. 7(b) we present the corresponding uncoded Fig. 8. The uncoded BER performance of the UWB systems using one- tap equalizer and MMSE equalizer with Md =20, assuming perfect timing, BER simulation results for the QPSK modulated signals with OFO and channel estimations. one-tap frequency-domain equalization. The BER curves are consistent with the interference plus extra noise power curves in Fig. 7(a). Note that choosing a larger value of Md is In Fig. 8, we present the uncoded BER performances for the more robust to random timing offsets. Our suggestion is to UWB systems using one-tap equalizer and MMSE equalizer set Md a few samples (say 2) larger than the best Md value assuming perfect timing, OFO and channel estimations. The − corresponding to a timing offset around E[∆k] σt . With BER advantage of the MMSE equalizer is marginal, indicating t0/Ts =5, Nt = 132, and η =10, this timing offset would the one-tap equalizer as a preferred choice due to its low − be around 5 and hence a good choice for Md is 20, which complexity advantage. is close to M =17, and hence we may set M = M =20 p p d Finally, Fig. 9 and Fig. 10 present the uncoded BER for simplicity. performance. The “reference” in the figures represents the system performance obtained by using conventional methods 9The corresponding plots are omitted due to the space limitation. 10We only consider non-positive timing offsets since the timing adjustment and parameters, i.e. correlation-based timing estimator with yields negative timing offset most of the time. Nt = 160, the conventional CFO estimator mentioned before LI et al.: SYNCHRONIZATION, CHANNEL ESTIMATION, AND EQUALIZATION IN MB-OFDM SYSTEMS 11 −1 −1 10 10 reference proposed (one−tap) proposed (MMSE) reference, SNR = 0dB uncoded BER one−tap, SNR = 0dB −2 10 −2 uncoded BER 10 MMSE, SNR = 0dB reference, SNR = 5dB one−tap, SNR = 5dB MMSE, SNR = 5dB one−tap (spread), SNR = 0dB one−tap (spread), SNR = 5dB 5 101520253035 M d 0 5 10 15 SNR (dB) Fig. 9. The uncoded BER performance of the proposed and reference methods with Nt = Nf = 132, L =28, η =10versus the OLA length Fig. 10. The uncoded BER performance of the proposed and reference N = N = 132 L =28 η =10 M = M =20 (Md = Mp) when SNRs = 0, 5 dB. methods with t f , , ,and d p . for Fig. 4, frequency domain least-square channel estimator, estimation performance is more sensitive to the correlation and one-tap ZF frequency-domain equalizer. The proposed window length than the frequency offset estimation. Hence, method uses Nt = Nf = 132, and L =28. although the best correlation window lengths for timing and In Fig. 9, we present the uncoded BER performance versus frequency offset estimation are not the same, we may use Md (= Mp) at SNR = 0, 5 dB (the performance plots for the same correlation window length designed for the timing SNR = 10, 15 dB are skipped due to space limitation; see estimation when performing both tasks. The best overlap-add [18] for details). Compared with the reference method, our window lengths for preamble and data are observed to be proposed methods obtain nearly 2 dB SNR advantage. We approximately the same, and the use of the overlap-add length observe that the optimum values of Md slightly vary with larger than the best value gives a more robust performance different SNR values and our design of Mp = Md =20gives than the use of a smaller length. The MMSE equalizer for the approximately the best result across the SNRs of interest. And ZF-OFDM systems is no longer a one-tap frequency-domain the performance difference between the one-tap ZF equalizer equalizer, but the latter gives almost the same performance as and the MMSE equalizer is small. The BER curve for the time- the former, and hence is a better choice for practical systems. frequency spread data is shown as “(spread)” in Fig. 9. We The simulation results corroborate the effectiveness of our observe that at the cost of data rate reduction, the spreading proposed synchronization, channel estimation and equalization improves the error performance significantly. methods and corresponding optimization criteria. In Fig. 10, we present the uncoded BER performance versus SNR. We use M (= M )=20for both proposed d p APPENDIX and reference methods. The proposed method gives about Ψ H C HH C 2 dB SNR advantage at uncoded BER of 10−2. The BER In (33), we find that the matrix q s˜ q + n , performance advantage of the proposed method becomes which needs to be inverted, is a patterned matrix with the form Ψ D 2 Md f f H D H C HH 2I more obvious at higher SNR, which may be ascribed to its = +σ k=1 k k , where = q s˜ q +σ N is {f } × better synchronization performance and the fact that the BER a nonsingular diagonal matrix, k are N 1 DFT vectors. performance is dominated by synchronization errors at higher Applying a similar approach from Theorem 8.3.3 in [19], the SNR and by the noise at lower SNR. inverse of Ψ can be simplified by −1 Md Ψ−1 H C HH 2I D−1f D−1f H VII. CONCLUSIONS = q s˜ q + σ N + γ˜k k( k) k=1 We have presented low complexity synchronization, channel (55) estimation and equalization methods for MB-OFDM based − 2 2f T D−1f ∗ UWB systems by utilizing the distinctive features of MB- where γ˜k = σ /(1 + σ k k). By applying (55) OFDM systems. We have derived the probability density we avoid the non-diagonal matrix inverse and reduce the functions (pdfs) of the UWB channel path delays and using complexity of the MMSE Equalizer. these pdfs we present how to optimize the synchronization, The MMSE equalizer for the system with the data spreading channel estimation and equalization methods. The timing needs the inverse of the matrix V defined in (35), which is 12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 given by [19] [16] B. Muquet, Z. Wang, G. Giannakis, M. de Courville, and P. Duhamel, −1 −1 −1 “Cyclic prefixing or zero padding for wireless multicarrier transmis- {V }11 =[V 11 − V 12V 22 V 21] (56) sions?” IEEE Trans. Commun., vol. 50, no. 12, pp. 2136–2148, Dec. {V −1} −V −1V V − V V −1V −1 2002. 12 = 11 12[ 22 21 11 12] (57) [17] A. Molisch, J. Foerster, and M. Pendergrass, “Channel models for −1 −1 −1 −1 ultrawideband personal area networks,” IEEE Wireless Commun., vol. 10, {V }21 = −V 22 V 21[V 11 − V 12V 22 V 21] (58) no. 6, pp. 14–21, Dec. 2003. −1 −1 −1 {V }22 =[V 22 − V 21V 11 V 12] (59) [18] T. Jacobs, Y. Li, H. Minn, and R. Rajatheva, “Synchronization in MB- OFDM-based UWB systems,” in Proc. IEEE ICC, June 2007, pp. 1071– H H V 11 H C ˜H C V 12 H C ˜H V 21 1076. where lξ s lξ + n, lξ s nξ , H H [19] F. Graybill, Matrices with Applications in Statistics. Wadsworth H C ˜H and V 22 H C ˜H + C . V 12 and V 21 nξ s lξ nξ s nξ n Publishing, 1983. are diagonal matrices, and V 11 and V 22 have the same pattern as Ψ. Hence, the computation of the inverse can be simplified Ψ−1 Yinghui Li (S’05) received the B.E. and M.S. using the method we proposed for . degrees in Electrical Engineering from the Nanjing University of Aeronautics and Astronautics, Nan- REFERENCES jing, China, in 2000 and 2003, and the Ph.D. degree in Electrical Engineering from the University of [1] A. Batra, J. Balakrishnan, G. Aiello, J. Foerster, and A. Dabak, “Design Texas at Dallas in 2007. From 2004 to 2007, she of a multiband OFDM system for realistic UWB channel environments,” was a Research Assistant with the Information and IEEE Trans. Microwave Theory Tech., vol. 52, no. 9, pp. 2123–2138, Communications Systems Laboratory at UTD. Since Sept. 2004. 2008, she has been with LitePoint Inc, CA, USA. [2] Multi-band OFDM physical layer proposal for IEEE 802.15 Task Group Her research interests include signal processing al- 3a, IEEE Std., Mar. 2004. gorithms and training signal designs for broadband [3] High rate ultra wideband PHY and MAC standard, ECMA Std., 2005. wireless communications. [4] D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun., vol. 48, no. 10, pp. 1755–1764, Oct. 2000. Hlaing Minn (S’99-M’01-SM’07) received his B.E. [5] T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitivity of OFDM degree in Electronics from Yangon Institute of systems to carrier frequency offset and Wiener phase noise,” IEEE Trans. Technology, Yangon, Myanmar, in 1995, M.Eng. Commun., pp. 191–193, Feb.-Mar.-Apr. 1995. degree in Telecommunications from Asian Institute [6] H. Steendam and M. Moeneclaey, “Synchronization sensitivity of multi- of Technology (AIT), Thailand, in 1997 and Ph.D. carrier systems,” European Commun., ETT special issue on multi-carrier degree in Electrical Engineering from the University spread spectrum, vol. 52, pp. 834–844, May 2004. of Victoria, Victoria, BC, Canada, in 2001. [7] Y. Zhou, A. Karsilayan, and E. Serpedin, “Sensitivity of multiband ZP- He was with the Telecommunications Program OFDM ultra-wide-band and receivers to synchronization errors,” IEEE in AIT as a laboratory supervisor during 1998. He Trans. Signal Processing, vol. 55, no. 2, pp. 729–734, Jan. 2007. was a research assistant from 1999 to 2001 and [8] P. Moose, “A technique for orthogonal frequency division multiplexing a post-doctoral research fellow during 2002 in the frequency offset correction,” IEEE Trans. Commun., vol. 42, no. 10, pp. Department of Electrical and Computer Engineering at the University of 2908–2914, Oct. 1994. Victoria. He has been with the University of Texas at Dallas, USA, since [9] H. Minn and P. Tarasak, “Improved maximum likelihood frequency 2002, and currently he is an Associate Professor. His research interests include offset estimation based on likelihood metric design,” IEEE Trans. Signal wireless communications, statistical signal processing, error control, detection, Processing, vol. 54, no. 6, pp. 2076–2086, June 2006. estimation, synchronization, signal design, cross-layer design, and cognitive [10] H. Minn, V. Bhargava, and K. Letaief, “A robust timing and frequency radios. He is an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, synchronization for OFDM systems,” IEEE Trans. Wireless Commun., and the JOURNAL OF COMMUNICATIONS AND NETWORKS. vol. 2, no. 4, pp. 822–839, July 2003. [11] H. Minn, V. K. Bhargava, and K. B. Letaief, “A combined timing and Nandana Rajatheva received the B.Sc. degree in frequency synchronization and channel estimation for OFDM,” IEEE Electronic and Telecommunication Engineering with Trans. Commun., vol. 54, no. 3, pp. 416–422, Mar. 2006. first class honors from the university of Moratuwa, [12] Y. Li, T. Jacobs, and H. Minn, “Frequency offset estimation for MB- Moratuwa, Sri Lanka, in 1987, and the M.Sc., OFDM-based UWB systems,” in Proc. IEEE ICC, vol. 10, June 2006, Ph.D. degrees from the University of Manitoba, pp. 4729–4734. Winnipeg, MB, Canada in 1991 and 1995 respec- [13] C. W. Yak, Z. ling Lei, S. Chattong, and T. T. Tjhung, “Timing synchro- tively. Currently he is an Associate Professor in nization and frequency offset estimation for ultra-wideband (UWB) multi- the Telecommunications Field of Study, School of band OFDM systems,” in Proc. IEEE Intl. Symp. on Personal, Indoor and Engineeing and Technology, Asian Institute of Tech- Mobile Radio Commun., vol. 1, 11-14 Sept. 2005, pp. 471–475. nology (AIT), Thailand. He was with the University [14] C. Berger, S. Zhou, Z. Tian, and P. Willett, “Precise Timing for of Moratuwa, Sri Lanka before joining AIT where he Multiband OFDM in a UWB System,” in Proc. IEEE International was promoted to Professor in Electronic & Telecommunication Engineering Conference on Ultra-Wideband, Sept. 2006, pp. 269–274. in June 2003. From May 1996 to Dec.2001, he was with TC-SAT as an [15] S. Traverso, I. Fijalkow, and C. Lereau, “Improved equalization for Associate Professor. His research interests include wireless communications UWB multiband OFDM,” in Proc. Information and Communication with applications in OFDM and MIMO systems, signal processing and coding Technologies 2006 (ICTTA ’06), vol. 2, Apr. 2006, pp. 2634–2638. techniques in communications.