Timing Synchonization in High Mobility OFDM Systems Yasamin Mostofi Depatment of Electical Engineeing Stanfod Univesity Stanfod, CA 94305, USA Email: yasi@wieless.stanfod.edu Donald C. Cox Depatment of Electical Engineeing Stanfod Univesity Stanfod, CA 94305, USA Email: dcox@spak.stanfod.edu Abstact OFDM systems ae sensitive to timing synchonization eos. Utilizing pilot-aided channel estimatos in OFDM systems can futhe incease this sensitivity. This is shown in [2] whee authos have analyzed the effect of such eos on a pilotaided channel estimato in a fixed wieless envionment. They poposed an algoithm that exploits this sensitivity to impove timing synchonization without additional taining ovehead. The effect of these eos and design of suitable synchonization algoithms fo high mobility applications, howeve, have not been studied befoe. In this pape we extend the analysis in [2] to high mobility envionments. Timing synchonization becomes moe challenging fo mobile applications since powedelay pofile of the channel may change apidly due to the spoadic bith and death of the channel paths. We find analytical expessions fo channel estimation eo in the pesence of timing synchonization eos and mobility. We show that the sensitivity of the channel estimato can still be exploited to impove timing synchonization in high mobility envionments. Then we extend the algoithm poposed in [2] to high mobility applications. Finally simulation esults show the pefomance of the algoithm in high delay and Dopple spead envionments. I. ITRODUCTIO Othogonal Fequency Division Multiplexing (OFDM) systems divide the given bandwidth into naow sub-channels. By tansmitting low data ates in paallel on these sub-channels, OFDM systems can handle high delay spead envionments [1]. The pefomance of OFDM systems, howeve, is sensitive to the pefomance of the timing synchonize and channel estimato (in this pape, timing synchonization efes to the coect detection of the stat of an OFDM symbol). Futhemoe, high mobility can intoduce time-vaiations in one OFDM symbol, which uins the pefomance. Each of these issues has been mainly exploed sepaately. Reseach in timing synchonization has mainly focused on fixed wieless applications and can be pimaily categoized into two goups: Taining-based and coelation-based. The fist goup is based on tansmitting two identical symbols [7]. Mulle [5] has povided a good suvey and compaison of such algoithms. The pefomance of these methods is good but thee is a waste of bandwidth in tansmitting the taining infomation. The second categoy is based on using the edundancy of the cyclic pefix [8]. Then the stat of the symbol is whee the coelation of the stat and end data points is maximized. In the absence of delay spead, this would wok fine. Howeve, in the pesence of delay spead, the cyclic pefix would be affected by the pevious OFDM symbol esulting in pefomance degadation. Thee ae othe methods that use cyclic pefix fo coase synchonization followed by a fine tuning [9]. Howeve, Yang in [9] makes the assumption that the fist channel tap is the stongest, which may not be the case in envionments with no line of sight path. To estimate the channel in high delay spead OFDM applications, it is necessay to tansmit pilot tones. If maximum channel delay spead spans ν sampling peiods, egi et al. [6] has shown that only L = ν+1 equally-spaced pilots ae needed fo channel estimation. To get an estimate of the channel at all the sub-caies, an IFFT of length L, zeo padding and an FFT of length should be pefomed, whee epesents the numbe of sub-caies. Thee ae othe less optimum ways of intepolating the channel in between pilot sub-caies. As delay spead inceases, the pefomance of these suboptimum methods degades dastically. Theefoe they ae not the focus of this pape as we ae inteested in high delay spead envionments. Most of the wok in timing synchonization and channel estimation has looked at these issues sepaately. To undestand the effect of timing synchonization eos on channel estimation, authos in [2] took at oveall look at both issues. They showed the supe-sensitivity of the pilot-aided channel estimato to timing synchonization eos. Based on thei analysis, they poposed a obust timing synchonization algoithm that utilizes this sensitivity to coect fo timing synchonization eos without additional taining ovehead. Timing synchonization becomes moe challenging in high mobility envionments. Fo such applications, applying taining-based algoithms fo timing synchonization can incease the taining ovehead consideably. Futhemoe, utilizing coelation-based methods would esult in pefomance degadation due to high delay spead. Theefoe the method poposed by Mostofi et al. [2] can be a good candidate fo such applications. In [2], the analysis was pefomed fo fixed wieless applications and the effect of mobility was not studied. It is the goal of this pape to take mobility into account. We show that the sensitivity of the channel estimato can still be exploited to impove timing synchonization in high mobility envionments. Based on the analysis, we extend the algoithm poposed in [2] to high mobility applications. IEEE Communications Society 2402
We show both analytically and though simulation esults that the algoithm woks obustly in high delay and Dopple spead envionments. II. SYSTEM MODEL Conside an OFDM system in which the given bandwidth is divided into sub-channels and the guad inteval spans G sampling peiods. We assume that the length of the channel is always less than o equal to G in this pape. X i epesents the tansmitted data point in the i th sub-channel and is elated to time-domain sequence, x, as X i = 1 k=0 x ke j2πki. Sequences y and its FFT, Y, ae the eceived data points in time and fequency domain espectively and w i is AWG. Let T be the time duation of one OFDM symbol afte adding the guad inteval. Then, h (k) q epesents the q th channel tap at time t = k T s whee T s = T +G. A constant channel is assumed ove the time inteval k T s t<(k +1) T s with t =0indicating the stat of the data pat of the symbol. A. Ideal case: no mobility and pefect synchonization In this case, we dop the supescipt k as h (k) q will not be a function of k. IfH denotes the FFT of the channel, h, then at the i th sub-channel, we will have Y i = H i X i + W i,whee W is the FFT of w. LetH eq (i) = k h eq(k)e j2πik epesent the elationship between X i and Y i. Then in this case we will have H eq (i) =H i.letν be the maximum pedicted nomalized length of the channel delay spead. Theefoe, only L = ν +1 equally-spaced pilot tones, X pl (l i ) fo 0 i L 1, ae needed to estimate channel fequency-vaiations whee l i = i L Ĥ eq (l i )=. Then, Y l i X pl (l i ) = H eq(l i )+ W l i X pl (l i ) 0 i L 1 (1) Though an IFFT of length L, the estimate of the channel in time-domain would be ĥeq(k) = 1 L 1 L i=0 Ĥeq(l i )e j2πik L fo 0 k L 1. Then though an FFT of length, the estimate of the channel at all the sub-caies is Ĥeq(i) = j2πik ĥeq(k)e fo 0 i 1. L 1 k=0 B. Case of mobility with pefect synchonization In this case, the time-domain eceived signal, y mob,k is: y mob,k = q h (k) q x ((k q)) ϑ mob,k +w k 0 k 1 (2) The subscipt mob distinguishes the mobile case fom the pevious case and (( )) efes to a cyclic shift in the base of. By taking an FFT of y mob,k, the fequency-domain eceived signal will be: 1 Y mob,i = H i,0 X i + H i,z X ((i z)) z=1 +W i 0 i 1 ICI mob (i) (3) The second tem on the ight hand side of Eq. 3 epesents the Inte-Caie-Intefeence (ICI) intoduced by mobility. It can be easily shown that H i,z is as follows, H i,z = 1 C 1 ) g=0 g =0 h(g g e j2π (z (g g)+g i) whee C ν epesents nomalized length of the channel delay spead. In this case, H mob,eq (i) =H i,0. Following the same pocedue of the pevious sub-section, an estimate of H mob,eq (i) can be acquied using the pilots. In this case, L pilots may not be enough to estimate the channel. Diffeent methods can be used to mitigate the effect of mobility. This pape will not focus on mobility mitigation since the main objective of the pape is to achieve obust timing synchonization in the pesence of mobility. Fo moe on mobility mitigation, efe to [3]. C. Case of timing synchonization eos and mobility Conside a case that a timing eo of m sampling peiods has occued. m>0 and m<0 denote timing eos of m to the ight and left side of the stat of the OFDM symbol espectively. 1) Case of m > 0 in the pesence of mobility: In this case, an eo of m sampling peiods to the ight side has occued. Then, the tems y mob,0,y mob,1,...,y mob,m 1 ae missed and instead m data points of the next OFDM symbol ae eoneously selected. The eceived signal can thus be witten as follows: y mob,k = ϑ mob,((k+m)) γ k +s k +w k 0 k 1 (4) whee the supescipt { denotes the case of m > 0. wk 1 0 k m 1 is AWG, γk = 0 m k 1 and s k = { 0 0 k m 1 ymob,pf next. ynext (k + m) else mob,pf (k) epesents the k th sample of the output cyclic pefix of the next OFDM symbol in the pesence of mobility. It can be easily shown that the FFT of ymob,k will be as follows: signal noise Ymob,i = e j2πmi F } {{ i X i + W } mob(i)+ H mob,eq (i) intefeence Γ 0 e j2πmi ICImob (i)+icimob(i) +ISImob(i) coelated tems whee Fi = 1 2πmz z=0 e j Γ z H ((i z)), z, Γ is the FFT of γ and Wmob is AWG. ICI mob is as defined in Eq. 3. ICImob and ISI mob ae the ICI and ISI (Inte-OFDM Symbol-Intefeence) tems intoduced by timing synchonization eos. Due to the effect of mobility, thei expessions ae diffeent fom the case of fixed wieless. Using the expessions of the intefeence tems and afte a long deivation, an expession can be deived fo SIRmob, the aveage Signal to Intefeence Ratio fo the case of m>0 in the pesence of mobility [4]. (5) IEEE Communications Society 2403
2) Case of m<0 in the pesence of mobility: In this case, due to the pesence of the cyclic pefix, the numbe of data points that ae missed can be less than m [2]. If the length of the channel delay spead spans C ν sampling peiods, only d = max(c (G + m), 0) data points ae coupted due to the intefeence fom the pevious symbol. Theefoe the time-domain eceived signal in this case will be, y l mob,k = ϑ mob,((k+m)) γ l k +p k +w l k 0 k 1 (6) whee the supescipt { l denotes the case of m < 0. wk l is 0 0 k d 1 AWG, γk l = 1 d k 1 and { 0 d k 1 p k = y mob,pf (G + m + k) else. y mob,pf (k) epesents the k th sample of the output cyclic pefix of the cuent OFDM symbol in the pesence of mobility. It can be easily shown that the FFT of ymob l will be as follows: Ymob,i l = e j2πmi F l i X i + Γl 0 e j2πmi ICImob (i) + ICImob(i)+ISI l mob(i)+w l mob(i) l (7) whee Fi l is defined simila to the case of m>0, Γl is the FFT of γ l, Wmob l is AWG and Hl j2πmi mob,eq (i) =e Fi l. Similaly, an expession can be deived fo SIRmob l [4]. Timing eos fo the case of m<0 can esult in lowe intefeence than the case of m>0 (o no intefeence) due to the pesence of cyclic pefix. III. EFFECT OF TIMIG ERRORS O CHAEL ESTIMATIO I THE PRESECE OF MOBILITY In this section we exploe the effect of timing eos on the pefomance of a pilot-aided channel estimato. Conside the case of m 0. Though an IFFT of Hmob,eq, the timedomain equivalent channel will be h mob,eq (k) =f ((k+m)), with f epesenting the IFFT of F. Timing synchonization eo intoduces a otation of m sampling peiods in the base of in the equivalent channel. It can be easily poved that f has the same length as the channel delay spead. Theefoe, this otation will esult in the expansion of the channel beyond its maximum pedicted length. Fig. 1b shows the equivalent channel fo an f of length L 1 shown in Fig. 1a. As can be seen, a otation has occued and esulted in the expansion of the equivalent channel beyond the maximum pedicted length of ν. Even one eo to the ight side will esult in an equivalent channel of length 1. This will degade the pefomance of the channel estimato, as it assumes an equivalent channel that spans ν sampling peiods at maximum. To see the effect of timing eos on channel estimation analytically, conside the case that L equally-spaced pilot tones ae inseted among the sub-caies. It can be easily shown that the time-domain channel estimate can be expessed as follows: ĥ mob,eq(k) =f ((k+m)) L + u k }{{} + vk }{{} Intefeence AW G (8) f 0 f 1 0 L-m-1 -m -1 Fig. 1b Equivalent channel (m 1) f m f0 f L 1 f 1 f L 1 01 L-1 0 1 L-1 Fig. 1a Oiginal channel (length L 1) Fig. 1d Oiginal channel (length L 1) f m f L 1 f 0 f m 1 f 1 f m 1 0 L-m-1 L-1 Fig. 1c Estimated channel (m 1) f l 0 f l 0 f l L+m f0 l f l L+m f l L+m 1 0 -m L-1 L-m-1 Fig. 1e Equivalent channel (m 1) 0 -m L-1 Fig. 1f Estimated channel (m 1) As can be seen fom Eq. 8, thee ae thee factos contibuting to channel estimation eo: effect of otation, intefeence and noise. The fist facto is caused since the equivalent channel has a otation in the base of while the estimated equivalent channel has a otation in the base of L. Since L is chosen based on the maximum pedicted length of the oiginal channel, ν, it is typically consideably smalle than. Theefoe, channel estimation eo can be consideable, solely due to the fist facto. Fig. 1c shows the estimated equivalent channel fo the equivalent channel of Fig. 1b (effect of intefeence and noise is not shown on the figue). Compaing Fig. 1b and 1c, a mismatch can be obseved in the location of the fist m taps of the oiginal channel of Fig. 1a (oiginal channel efes to f in the absence of otation). Since these taps ae typically stong, this can esult in a consideable pefomance degadation of the channel estimato. To analytically assess the contibution of each of the afoementioned factos, next we deive an expession fo channel estimation eo: H mob,eq(i) = m 1 k=0 β i,k f k + U i + V i 0 i 1 (9) whee Hmob,eq epesents the fequencydomain channel estimation eo with Ui = Γ L 1 z=0 α 0 e j2πmlz i,z and Vi = L 1 z=0 α i,z W mob (lz) X pl (l z) and v. l z = L z, α i,z = 1 L ICI mob (l z)+ici mob (lz)+isi mob (lz) X pl (l z) epesenting the FFTs of u L 1 o=0 ej2πo( z L i ) and β i,k = e j2πi(k m) (1 e j2πil ). Afte a long deivation, nomalized channel estimation eo at i th sub-caie, E (m, i), can be tightly appoximated as: E (m, i) = H mob,eq (i) 2 = 4P Hmob,eq (i) 2 % sin2 ( πil ) + whee SRmob = σ2 X σ2 H 2 σ 2 W and SIRmob facto#1:otation effect 1 SIRmob + 1 SRmob (10) 1 z=m 1 z =m R n((z z )T s ) is as defined in Section II. R n(zt s ) is the IEEE Communications Society 2404
nomalized auto-coelation function of channel taps. In deiving Eq. 10, the same nomalized auto-coelation function is assumed fo all the taps. This assumption is not equied fo the analysis and design of a obust timing synchonize in the est of the pape. σx 2 and σ2 W ae aveage powes of X and W espectively. P% epesents the atio of the powe of the mismatched taps to the total powe of the channel and σh 2 epesents total channel powe. Compaed with the channel estimation eo deived in [2], otation has the same contibution as it had in the fixed wieless case. The second and thid tems on the ight hand side of Eq. 10, howeve, have slightly highe values compaed to thei coesponding tems in the fixed wieless case. Still, the effect of otation is the majo contibuto to channel estimation eo as the fist tem gets consideably high values. facto#1 does not affect pilot sub-channels. Howeve, it esults in a consideable incease of eo fo othe sub-caies paticulaly those at i = o odd ceil( 2L ), whee o odd epesents odd integes. To see the effect of facto#1, Fig.2showsE (m, i) fo the sub-caies in the middle of evey two consecutive pilots and fo diffeent levels of mobility. f d,nom efes to the pecentage of the maximum Dopple spead divided by sub-caie spacing and each channel tap has Jakes powe-spectum. As can be seen, mobility does not have a distinguishable impact on channel estimation eo fo f d,nom as high as 20% since the effect of otation is vey high. To see the contibution of the otation facto, the solid line shows the effect of the fist tem solely. As can be seen, facto#1 is almost 100% contibuto to channel estimation eo at low m.asm appoaches the length of the guad inteval, ICI and ISI intoduced by timing synchonization eo incease. Still, facto#1 contibutes to moe than 80% of channel estimation eo at m = G. To examine a case whee Dopple spead has a distinguishable impact on channel estimation eo, we incease f d,nom to 50%. It should be noted that even in case of a pefect timing synchonization, such a high Dopple level would uin the pefomance of an OFDM system dastically and theefoe is not a ealistic scenaio. Even fo such a high Dopple level, facto#1 contibutes to moe that 70% of channel estimation eo. The timing synchonization method poposed in [2] was effective as long as facto#1 is the majo contibuto to channel estimation eo. As was shown, mobility does not have a consideable impact on channel estimation in the pesence of timing eos. Theefoe, extending the method poposed in [2] should povide obust timing synchonization fo high mobility applications. Simila expessions can be deived fo the case of m<0. We will have h l mob,eq (k) = f ((k+m)) l and ĥl mob,eq (k) = f((k+m)) l L + u l k + vl k. Fig. 1e and 1f show the equivalent and estimated equivalent channel fo f l of Fig. 1d espectively. On the contay to the case of m>0 whee even one eo to the ight esulted in an equivalent channel of length 1 (see Fig. 1b), the equivalent channel length fo m<0 vaies depending on the length of the channel. Fo instance, fo a channel of length C ν, the equivalent channel length will be C m fo m 1. Theefoe fo C ν m 1, the equivalent length would still be less than o equal to ν, which poses no poblem fo the channel estimato. Futhemoe, the mismatch is in the location of the last m taps of the oiginal channel which ae not typically that stong. Depending on the length of the channel, these taps can be solely occupied by noise/intefeence. Theefoe, we see again that eos to the left side may not cause any pefomance degadation depending on the length of the channel delay spead, guad inteval and numbe of pilots. Fig. 2 Effect of mobility on channel estimation eo E 6 5 4 3 2 1 no Dopple f d,nom =10% f d,nom =20% f d,nom =50% facto#1: Effect of otation 0 0 0.2 0.4 0.6 0.8 1 m/g IV. TIMIG SYCHROIZATIO ERROR CORRECTIO We showed that pilot-aided channel estimato is supesensitive to timing synchonization eos due to the effect of otation. We also showed that in a mobile envionment, effect of otation is still the dominant cause of channel estimation pefomance degadation. Theefoe, this sensitivity can be exploited to design a synchonization algoithm that woks obustly in high mobility envionments. Afte a coase timing synchonize has detected a stat point fo the symbol (a coelation-based synchonize can be used fo this), Ĥ mob,eq can be obtained using pilots. In the pesence of timing eos, this channel estimate may be fa fom H mob,eq.call ˆXi = Y mob,i Ĥ mob,eq (i), the estimated input at ith sub-channel. Let Xi = Dec( ˆX i ) epesent the estimated input afte passing though the decision device. Define a decision-diected measue function as M = 1 i=0 ˆX i X i 2. In the pesence of channel otation, M can become vey lage. Theefoe, synchonization eo coection can be obtained iteatively by minimizing M. ote that we detect timing eos solely due to the lage impact of facto#1 on the pefomance. Theefoe, as long as facto#1 is the majo cause of pefomance loss, which is the case with high pobability, we can detect timing eos. Due to the lage contibution of otation, it is possible to pefom all the updates necessay to find the best timing coection solely in the fequency domain. Conside coecting eos to the ight. As can be seen fom Fig. 1c, the position of the last m taps of the estimated channel is diffeent fom that of the equivalent channel, whee m is unknown. Theefoe though an iteative pocess, we update the estimated channel, coecting fo one mismatched tap at a time. Then the update necessay fo coecting eos to the ight at the k th iteation IEEE Communications Society 2405
and i th sub-channel will be as follows: Ĥ (k+1), mob,eq (i) =Ĥ(k), mob,eq (i)+c 1 ĥ mob,eq(l k) e j2πik/ (11) Similaly, we will have Ĥ(k+1),l mob,eq (i) = Ĥ(k),l mob,eq (i) c 1 (k 1) e j2π(k 1)i/ fo detecting eos to the ĥ l mob,eq left, whee c 1 =1 e j2πl and Ĥ(1), mob,eq (i) =Ĥ(1),l mob,eq (i) = Ĥ mob,eq (i). In each iteation, the measue function, M (k), will be evaluated. Finally the iteation with smallest M is chosen and the coection necessay would be applied to the stat of the symbol in time-domain. V. SIMULATIO RESULTS We simulate an OFDM system in a time-vaiant envionment with high delay spead as is the case fo an SF (Single Fequency etwok) channel. We choose the following system paametes based on Siius Radio (a DAB sevice povide) second geneation system specification poposal. Input modulation is 8PSK. Bit ate is 7.3Mbps, =892 and L=223. Powe-delay pofile of the simulated channel is shown in Fig. 3. It has two main clustes each with 9 taps to epesent an SF channel. Channel delay spead is 36.5µs spanning 64% of the guad inteval. Each channel tap is geneated as a andom pocess with Rayleigh distibuted amplitude and unifomly distibuted phase using Jakes model. The auto-coelation of each tap is zeo-ode Bessel function. It was shown in [2] that the poposed algoithm can educe the eo pofile vey close to that of the pefect synchonization in a fixed wieless envionment. Hee we show that mobility has a negligible effect on the pefomance of the algoithm. We simulate Relative powe of the taps 0.1 0.08 0.06 0.04 0.02 Fig. 5 Channel powe delay pofile 36.5 micos 0 0 50 100 150 Delay as a multiple of sampling peiod two methods. Method I utilizes the taditional coelationbased timing synchonize and picks the maximum coelation point. The second method, the poposed one, utilizes method I fo initial coase synchonization followed by the poposed decision-diected timing adjustment of the pevious section. To evaluate the pefomance of these methods, we measue P eo, the pobability of making a timing eo of m sampling peiods. Fig. 4 shows the pefomance in high mobility σ 2 X σ2 H σ 2 W envionments at = 20dB. The top sub-figue shows the eo pofile of the coelation-based method in a fixed wieless envionment. As the delay spead spans 64% of the guad inteval, timing offsets of up to 36% of the guad inteval (which becomes 82 sampling points) to the left side can occu without loss of pefomance. We call this egion safe zone, as is maked on Fig. 4. It can be seen that the coelation-based method makes a consideable amount of eo out of the safe zone. The middle sub-figue shows the pefomance of the poposed method in a fixed wieless envionment. As can be seen, its eo pofile is mainly confined to the safe zone. To see the impact of mobility, Fig. 4 also shows the pefomance fo diffeent levels of mobility: f d,nom =10%, 20% and 50%. Compaed with the no Dopple case, it can be obseved that the eo pofile is not affected by high mobility. This is due to the consideable impact of facto#1 on channel estimation eo as was shown in the pevious section. P eo P eo P eo Fig. 4 Pefomance in high mobility envionments coelation based 100 50 0 50 100 150 no Dopple, poposed Safe Zone f d,nom =10%, poposed 100 50 0 50 100 150 f d,nom =20%, poposed Safe Zone f =50%, poposed d,nom 100 50 0 50 100 150 m REFERECES [1] Cimini, Analysis and Simulation of a digital mobile channel using othogonal fequency division multiplexing, IEEE Tans. Comm., vol. COMM-33, pp. 665-675, July 1985 [2] Y. Mostofi, D. Cox and A. Bahai, Effect of fame synchonization eos on pilot-aided channel estimation in OFDM: analysis and solution, 5 th intenational symposium on wieless pesonal multimedia communications, Oct. 2002, pp. 1309-1313 [3] Y. Mostofi, D. Cox and A. Bahai, ICI Mitigation fo Mobile OFDM Receives, Poceedings of IEEE 38th Intenational Confeence on Communications (ICC), May 2003, Anchoage, Alaska [4] Y. Mostofi, Timing Synchonization and ICI Mitigation fo Pilot-aided OFDM Mobile Systems, PhD Thesis, Stanfod Univesity, ov. 2003 [5] S. Mulle, On the optimality of metics fo coase fame synchonization in OFDM: a compaison, inth IEEE Intenational Symposium on PIMRC, 1998 [6] R. egi and J. Cioffi, Pilot tone selection fo channel estimation in a mobile OFDM system, IEEE Tans. Consume Electonics, vol. 44, no. 3, Aug. 98 [7] T. M. Schmidl, Synchonization algoithms fo wieless data tansmission using Othogonal Fequency Division Multiplexing (OFDM), PhD Dissetation, Stanfod univesity, June 1997 [8] J. Van de Beek, M. Sandell and M. Isaksson, Low-complex fame synchonization in OFDM systems, Fouth IEEE Intenational Confeence on Univesal Pesonal Communications, 1995 [9] B. Yang, K. Letaief and R. Cheng, Timing ecovey fo OFDM tansmission, JSAC, vol. 18, no. 11, ov. 2000 IEEE Communications Society 2406