Improved Maximum Likelihood Frequency Offset Estimation Based on Likelihood Metric Design

Transcription

1 1 Iproed Maxiu ielihood Frequency Offset Estiation Based on ielihood Metric Design Hlaing Minn*, Meber, IEEE and Poraate Tarasa, Meber, IEEE Abstract For eerging high data-rate counication systes in highly dispersie channels such as ultra-wideband systes, possible frequency offsets could be larger than the estiation range of the existing ethods using training signals with identical parts or repetitie training signals (i.e., the training signals are coposed of seeral identical sub-blocs or are obtained by repeating a training sub-bloc for seeral ties). This paper presents a noel iproed axiu lielihood frequency offset estiator which can handle at least twice the estiation range of the existing ethods using training signals with identical parts and achiees a better estiation perforance. Based on the lielihood etric, a new design etric is introduced which is a pair-wise error probability (PEP) between the correct frequency offset point and a trial frequency offset point. The proposed PEP etric gies ore theoretical insights on the perforance of practical axiu lielihood estiators. How to design the PEP to achiee both a larger estiation range and a better estiation perforance in fading channel enironents is also presented and the corresponding estiator ipleentation is described. Keywords synchronization, estiation, frequency offset, lielihood etric design. I. INTRODUCTION Frequency synchronization is an essential tas at a counication receier. For pacet-based systes such as G, 3G and beyond-3g cellular systes, wireless ANs, wireless MANs, etc., training signal based frequency offset estiation and copensation are typically perfored. The scope of this paper encopasses all pacet-based single-carrier as well as ulti-carrier systes except CDMA-based systes. There are seeral existing wors on frequency offset estiation, e.g., [1]- [11]. They are ainly based on a correlation ter of the training signal in tie-doain or frequency-doain [1]- [], axiu-lielihood principle [5] [], a Bayesian approach [], a cobination of correlation ters in a sub-optial way [8], or a cobination of correlation ters using best linear unbiased estiation principle []- [11]. Most of the eploy repetitie training signals consisting of seeral identical parts (or in the for of cyclic prefixes) which also yield low coplexity estiators. In [], a axiu lielihood frequency offset estiation ethod (ME#1) was presented based on a joint estiation of frequency offset and channel ipulse response. With a proper training signal, ME#1 can handle absolute frequency offsets This wor was supported in part by the Eri Jonsson School Research Excellence Initiatie, the Uniersity of Texas at Dallas. H. Minn is with the Departent of Electrical Engineering, School of Engineering and Coputer Science, Uniersity of Texas at Dallas, Richardson, TX , U.S.A. (hlaing.inn@utdallas.edu). P. Tarasa is with the Departent of Electrical Engineering, Korea Adanced Institute of Science and Technology, Korea, (ptarasa@stein.aist.ac.r). *Contact Author less than half of the sybol rate which is the axiu possible estiation range for any estiator operating on sybolrate receied signal saples. To reduce the ME#1 s ery high coplexity, [] also presented ME# which utilized a periodic training signal with a period of saples (the nuber of channel taps). The coplexity of ME# is approxiately ties that of ME#1 but the corresponding estiation range of ME# is reduced to ties the sybol rate. Since the estiation range of ME# is inersely propotional to the nuber of channel taps, ME# cannot be applied to systes possible frequency offsets are larger than the aboe range. For exaple, in ultra-wideband (UWB) systes, the nuber of channel taps can be quite large and possible frequency offsets (due to ery high carrier frequency and/or low cost deices) can be larger than the estiation range of ME#. Siilarly, the aboe-entioned existing ethods with repetitie training signals experience the sae proble. For highly-dispersie channel enironents, deeloping frequency offset estiators which can handle all possible frequency offsets with reasonable coplexity is a challenging proble which has not been addressed in the literature. As UWB systes becoe ore proinent, the aboe proble becoes an iportant issue. Hence, in this paper, we address this issue and deelop a noel iproed axiu lielihood frequency offset estiation ethod which can handle a larger frequency offset with a coparable coplexity. Based on the lielihood etric, a new design etric naed pair-wise error probability (PEP) is introduced. The training signal consisting of seeral identical sub-blocs are designed based on the PEP etric in order to achiee the estiation range extension. The proposed design also brings in estiation perforance iproeent. The proposed schee can be applied to singlecarrier as well as ulti-carrier systes except CDMA systes 1. The proposed approach can be related to the optial periodic training signal design for frequency offset estiation presented in [1]. The approach fro [1] is based on the Craer-Rao lower bound (CRB) and it does not proide inforation on whether a practical frequency offset estiator will achiee the iproeent projected in the CRB. On the other hand, the proposed approach in this paper is based on the lielihood etric of a practical axiu lielihood estiator. It proides ore insights for practical axiu lielihood estiators and it ensures the iproeent in practice. The rest of the paper is organized as follows. Section II describes signal odel and Section III presents the proposed 1 For CDMA systes, the proposed concept could still be applied in downlin but ulti-user interference and ulti-user training signal designs would also need to be taen into consideration in uplin.

2 r e p r Nw o ~ X X p X X p approach for iproed axiu lielihood estiator with extended estiation range. Siulation results and discussions are gien in Section I and finally the paper is concluded in Section. II. SIGNA MODE For coplexity reduction, consider an arbitrary training signal consisting of seeral identical sub-blocs (say, subblocs). Each training sub-bloc is coposed of training saples,..., is the nuber of saple-spaced channel taps. The locations of the identical sub-blocs are defined by the tie-indices of the first saples of the sub-blocs, naely,,..., for,..., are not consecutiely located, i.e.,. If two adjacent sub-blocs, then there can be null saples or non-zero saples between the two sub-blocs. The non-zero saples between the two training sub-blocs could coney soe control inforation or could be used as training saples for other synchronization tass. The first sub-bloc seres as a cyclic prefix (CP). Siilarly, the -th sub-bloc will sere as a CP if. At the receier, the obseration ector for frequency offset estiation is fored fro the receied training saples by reoing the CPs (the first sub-bloc and all other training sub-blocs for which! ) and the null or data saples between any two training sub-blocs. Suppose in the obseration ector there are " sub-blocs with the corresponding tie-indices $#, #,..., #&% #('). Fig. 1 depicts seeral training signal structures and corresponding construction of obseration ectors (coposed of $#( ) shaded or unshaded blocs represent transitted training signal sub-blocs and blan spaces between training sub-blocs represent null or non-zero saples for other purposes. Consider a coplex baseband receied obseration ector * (coposed of the receied training sub-blocs with tieindices $#( ) gien by *, !/$/$/. - "1 3( (1) the indices of - are with reference to the obseration ector (not the receied saple tie-indices). Then we can express * as * 5 3 8:;< =5 3?>;< @ 1 3( (3) < + A. A.!/$/$/. A "B 3( () 5 3 DCE&F GH I?J$K?M N$OQP 3. I?J$KR?M N$O3P 3. /$/$/. I J$K?STHR?M N$O P 3 (5) P 3 + U. I?J$W$N.!/$/$/. I?J$WUM X O(N () Y 3 UZ #( () + 8 U[ \ M ] O&^.!_ _ "1 U. ` B U/ (8) We will use saple and sybol interchangeably. In the aboe equations, is the acb carrier frequency offset noralized by the saple rate, A are independent and identically distributed (iid) zero-ean circularlysyetric coplex Gaussian noise saples each haing a ariance of are the channel tap gains?e assued to reain constant during the training bloc, and X denotes a odulo- operation. The superscripts f a,, and g represent the conjugate, the transpose and the conjugate transpose operations, respectiely. 8 Note that consists of ",h identical sub-atrices which are designed to be of full ran. In practice, the exact nuber of channel taps ay not be nown and it can ary as well. Hence, should represent an upper bound of the nuber of channel taps. In this case, the channel ector contains actual channel taps appended with zero-alue taps and the signal odel reains the sae. III. THE PROPOSED METHOD We consider a joint estiation of and as in []. The lielihood function is gien by i *j. 3Z d % XlHn d p * 5 8 Uq () and are trial alues of and in finding/searching the best alues (of and ) that axiize the lielihood function and represents a Euclidean nor-square. For a fixed, wep find that axiizes the lielihood function i *j!. by * 5 differentiating 8 with respect to and equating the result to zero. p The corresponding estiate is gien by r Q8ts!8 8ts 5 s * (1) 8 has been designed to hae a full ran. After substituting (1) into (), the axiu lielihood estiate of is gien by =F u?gf x (11) x * s 5?y 5 s * / (1) y In the aboe equation, is a "zh " projection atrix gien by {}~ /$/$/ ~ y 8 Q8 s 8 8 s ". /$/$/ ~ ~ X h is an (()-(13)) is the sae as [] except that (13) identity atrix. The aboe deriation and the foration of the obseration ector * are different fro []. After soe siplification and dropping unnecessary factors, we obtain x = ƒ % % ˆ ] 1. # I?J M(KŠM w N$O K? &M w NO&OŒ= 1?Ž (1)

3 ; ; ; ; d : denotes the real part of and. # X - #1; - 1; (15) Ž,+....$/$/$/Q.. ". U..$/$/$/?. U. "...$/$/$/?. " U. " 3( (1),+ I J M(K?M NO w K?M N$O&O w. I J M(K??M NO w KR?M N$O&O w.$/$/$/. I?J M(K?M NO w K?STHRM NO&O w. I?J M(KR?M NO w KR?M N$O&O w.$/$/$/?. I?J M(KR?M NO w K?STHRM NO&O w. I?J M(K M NO w K M N$O&O w.$/$/$/. I?J M(K?STHRQM N$O w K?STHR?M N$O&O w?/ (1) Now, we introduce a new ariable to be used in deeloping our proposed ethod as x 3 x = 1U+ 3 33Žt/ (18) We will design the lielihood etric to achiee a larger estiation range and a better estiation perforance. This will be accoplished by eans of the design on the training subbloc locations. We will use + _ as our perforance easure in the design. This easure indicates how liely a trial point will be chosen as the frequency offset estiate between the actual frequency offset and the trial point. In other words, this probability can be considered as a pairwise error probability (PEP) of frequency offset estiation the exact frequency offset and any other frequency offset trial point constitute the pair. Note that when, this probability equals to one and it does not represent PEP but for conenience, + _ will be referred to as the PEP etric throughout the paper. as a Gaussian rando ariable (see Appendix-A for justification). The ean and the ariance of are, respectiely, gien by We can approxiate M N$O w M M N$O M N$O&O w (M M N$O M N$O&O w (1) M NO w M M N$O M N$O&O w M O3M O (M M N$O M NO&O w M M N$O M NO&O w M O3M O(M M N$O M NO&O w M M N$O M NO&O w M OM O (M M N$O M NO&O w () Then our design perforance easure, PEP, is gien by + _ +! #"!$ 3 %'& C+ 3)( (1)?e Std[ ] denotes the standard deiation of and is the Gaussian tail probability. One can note fro the aboe equation that + 3 $ /Std+ 3 can be used as a design perforance easure as well. Next, we calculate the ean and the correlation of. # required in (1) and (). After a direct (but lengthy) calculation, we obtain + $ + $ $ +. # 3 $ ; d '* + # (). #. 3 $ $ d * + H; * +# $ d * + ; * +# ;,+ &<B. #... (3). #. 3 $ $ d * + ; * +# $ d * +# ; * + ;,+ &<B.. #.. () X /.. and + &<. #... 1 ; B 5 d. if # = 3 d. if # =,, #8. if, #, #. $!- otherwise. 3 (5) Note that in deeloping our design easure we hae considered a fixed channel output sub-bloc energy $ or a fixed instantaneous (snap-shot) signal-to-noise ratio SNR' = $ d. In practical fading channels, $ or SNR' will fluctuate. For the estiation to be robust in fading enironents, we will use SNR' = db in our design. Different alues of actual noralized frequency offset siply result in shifted ersions (in the -axis) of the etrics, + 3 $ %'& C + 3, and + _ but they do not change the shapes of the etrics (see Appendix-A for the proof). The shifting of the etrics will not affect the frequency offset estiation range which is deterined by the distance in -axis between the etric pea corresponding to the actual frequency offset and the adjacent etric pea with coparable etric alue (etric nulls in place of etric peas for the etric + 3 $ %'& C + 3 ). Hence, in our lielihood etric design, without loss of generality, we set =. By plotting + _ ersus for different training sub-bloc locations, one can design the training sub-bloc locations that achiee a larger frequency offset estiation range (without abiguity) and a better frequency offset estiation ean-square error () perforance. The expected alue of x or the ratio + $ 3 /Std+ 3 can also be used as the design perforance easures. Using + _ is ore inforatie in that it indicates how liely a trial frequency offset point will be chosen as opposed to the correct frequency offset. To illustrate our proposed approach, let us consider a syste identical training sub-blocs are transitted and ;: (see Fig. 1). The obseration ector will hae a saller nuber of sub-blocs " (less training energy used in the estiation) if there are ore non-consecutie groups of sub-blocs, i.e., if there are ore for which. Hence, we consider a schee with two non-consecutie groups of training sub-blocs (there is only one for which ). The two groups are separated by < saples. The first group contains = ; sub-blocs and the second has " = ; sub-blocs. The -th and = ; -th subblocs sere as CPs and are reoed at the receier. Hence, if <, the obseration ector has " >: training sub-blocs with the corresponding tie indices 3 gien by #('@? for?, 1,..., =, and #A R ; < = ; :; for =,..., " =. If < =, the obseration ector contains " CB consecutie sub-blocs with the corresponding tie indices #(':;??: for, 1,..., ", which is the conentional training structure used in []- [11]. Our objectie in this illustratie exaple is to find < which gies a larger frequency offset estiation range and a better estiation than the conentional approaches (corresponding to < = ). 3 The tie index of the first transitted sub-bloc is assued to be DFE.

4 In Figs. -, we present effects of <, =, ", and SNR' on seeral etrics. In Fig., the noralized ean alues of the lielihood etric x for < fro = to are presented " B for < and " :, = for <. The corresponding plots of + $ 3 /Std+ 3 and PEP are shown in Figs. 3 and, respectiely. The etric lobe centered around = (correct frequency offset) will be called ainlobe and the other lobes will be referred to as sidelobes. For the etric + 3 $ /Std+ 3, the etric null at (correct frequency offset) will be called ain-null and the = other nulls will be referred to as side-nulls. At is an integer, the ean alues of the lielihood etric are the sae for < hence liiting the estiation range of the conentional approaches to... Siilarly, the alues of + 3 $ /Std+ 3 are all zeros at z (is / at ) and the PEPs are all.5 at (is 1 at ). In other words, for <, the etric sidelobe peas (or side-nulls) hae the sae (or alost the sae) alues as the ainlobe pea (or ain-null) and this fact liits the unabiguious frequency offset estiation range. For < in Figs. -, the alues of all etrics (the ean of lielihood etric, + $ 3 /Std+ 3, and the PEP) change = = at soe or all points of,, hence opening up the possibility of estiation range extension. In other words, the sidelobe peas (or side-nulls) adjacent to the ainlobe pea (or ain-null) tae on alues which are sufficiently distanced fro the ainlobe-pea alue (or side-null alue), hence increasing the unabiguious frequency offset estiation or, the estiation range becoes ties that of < but due to relatiely large sidelobe peas within the range, its estiation perforance could be affected at low SNR. Of particular interest is < case whose estiation range is twice of the range with < and its sidelobe peas within the range are relatiely sall, range. For exaple, for < hence ensuring high accuracy of estiation. z The effects of < for soe integer are presented in Figs. 5-. Note that < = corresponds to the conentional structure [] [11] [13] and < for a nonzero positie integer corresponds to the structures considered in [1]. For these structures (corresponding to <, 1, 3 in the figures), the estiation range is.. and a larger < gies a sharper etric ainlobe (or ain-null) resulting in a better estiation perforance. Howeer, a larger < introduces new sidelobes of the lielihood or PEP etric whose alues increase as < the + $ 3 /Std+ 3 etric whose alues decrease as < increases (new side-nulls for increases), which ay liit the use of a ery large < for low SNR. We obsere that the training structure design by iniizing the CRB as in [1] is in fact aing the lielihood etric sharper around the correct frequency offset. On the other hand, the training design by iniizing the CRB ay not reeal the feasibility of the axiu lielihood estiator. For exaple, with a ery large < at low SNR, the axiu lielihood estiator would not gie a reliable result due to large sidelobe The larger ean alue of the lielihood etric for is due to the fact that one ore training sub-bloc is used in the estiation for. peas of the lielihood etric within the estiation range. This fact cannot be deduced fro the iniu CRB design fro [1]. For < with an odd integer (i.e., <,, and 3 in Figs. 5-), the estiation range is doubled and siilar discussion applies a larger < gies a sharper lielihood etric ainlobe but a ery large < ay not be gie a reliable estiate due to the increased sidelobe peas. Next, the ipacts of different = alues on the PEP are shown in Fig. 8. It is obsered that = " gies the sharpest PEP etric but its sidelobe peas are larger. It is worth-noting that iniizing the CRB [1] gies the sae result of = ". The PEPs for = " ; and = " : are obsered to be the sae. In Fig., the effects of different " and SNR' on the PEP are depicted = " is used. A larger " or SNR' results in a sharper PEP with saller sidelobes. Hence, using a larger ", we can lower the PEP sidelobes in order to ensure accurate estiation at low SNR' (due to fading) for a larger < is an odd positie integer. Siilarly, if a larger estiation range is required, the sidelobe peas associated with soe appropriate < can be lowered by using a larger " (for exaple, the sidelobe peas for < in Fig. can be lowered by a larger " to achiee an estiation range of.. ). In practical pacet-based wireless counications, if the signal arriing at the receier is in deep fade (below the receier sensitiity) or the instantaneous signal to noise ratio is ery low, the receier will not be able to detect the signal. Hence, the frequency offset estiator s perforance under such conditions is irrelaent to practical systes and typical ean-square error () perforance easure ay not reflect the exact perforance for practical systes at low SNR. Therefore, we introduce a practical estiator perforance easure naed practical which represents the eansquare estiation error gien that the receied signal power is aboe a threshold (related to the receier sensitiity). We used SNR' = db as our threshold for practical. Next, we address what sidelobe pea leel of PEP (in other words, what alue of " ) gies a reliable result with an extended estiation range. Suppose the sidelobe pea of the PEP is at SNR' = db. Then the contribution of the sidelobe to the practical is approxiately which should be less than the practical for < = (the reference ) to suppress the sidelobe proble. In design, the CRB at SNR' = db for < z (denoted by CRB ) can be used in place of the reference. Recall that " for < = is larger than " for < by one. The paraeter " for < cases can be chosen such that the corresponding PEP sidelobe pea is less than CRB. For exaple, if CRB is h and C:, the allowable for the suppression of the sidelobe proble could be or saller fro which " can easily be deterined. Following a siilar approach fro [1], we obtain the snapshot CRB of frequency offset estiation at a gien SNR' for an obseration ector consisting of " sub-blocs with the

5 r. 5 corresponding tie-indices $#, #,..., #&% as % : Z % - # M STHR % ˆ O () which is used to calculate CRB. Note that the proposed design yields a saller CRB than the conentional training structures. Proof of this fact and the deriation of CRB are gien in Appendix-B. A. Ipleentation In the following, we discuss an ipleentation of the axiu lielihood frequency offset estiator for < ;.. are non-negatie integers with if, )B U.$/$/$/. if, and is an integer. For a new ector of length as follows BA "!$#% &"'( *) +-,/. ( Š ) +-,*1-3+ 5" 8/:<;"=*>-;@? ; C3;ED D D ;$FG$HIC () J is the suation of all correlation ters. # Y which hae the sae phase factor ˆ 3 Y 3 UZ, (see ()). If, then " (if 1 training sub-blocs are transitted, " CB in this case). If and 1, then " ; (if 1 training sub-blocs are transitted, " ;: in this case). If, then " ; ; H; (if 1 training sub-blocs are transitted, " : in this case). Then the frequency offset estiate is gien by F u?gf `+ U MONNP/ Q (8) K MONNP/Q SR T MONNP"Q U MONNP/ gies the K point FFT of. In the aboe ipleentation, the trial alues of are X MONNP MONNP. MONNP ; U.$/$/$/?. MONNP. Note that KB and K is a power of for low-coplexity FFT ipleentation. If necessary, a quadratic interpolation can be applied to + 1 r X MONNP r 1; r,, X MONNP to fine-tune the frequency estiate. A larger K is associated with a larger coplexity while giing a better estiation accuracy especially if the quadratic interpolation is not perfored. With the quadratic interpolation, a suitable choice of K for low coplexity, while giing no noticeable degradation in the estiation accuracy for SNR of practical interest, would be W XŸ Z M M\[ O ] W XŸ Z or M M\[ O ]. Note that for, this ipleentation is exactly the sae as []. B. Coplexity The estiation coputational coplexities associated with the conentional and the proposed approaches are presented in ; Table I for a training signal consisting of = identical subblocs. The paraeter ^ accounts for the coplexity reduction due to the zero inputs to the FFT []. The nuber of non-zero FFT inputs "_ for the proposed approach depends on < and " _ is at ost `= and can be saller than that. For C:, " : (=1 for < ) and K a for < and K a z for <, the nuber of real ultiplications for < U/ is about `` ties, for < is about / B ties that of < and the nubers of real additions are for < = U/ a, ties and for < =, / Bb ties that of <. Recall that ME#1 [] has coplexity about ties that of < z and hence our proposed approach would be a better choice for the estiation range. I. SIMUATION RESUTS AND DISCUSSIONS A ultipath Rayleigh fading channel with >: taps and an exponential power delay profile with a 3 db per tap decay factor is considered. Channel gains are assued to reain constant oer the whole training signal. In order to hae the sae frequency resolution in the estiation, K a is used for < and K a is used for < ; / a is a non-negatie integer. To decouple the effect of FFT bin resolution in perforance coparison, we set frequency offsets on the FFT grids which would gie optiistic results but does not affect the perforance coparison. In Fig. 1, the s and practical s are presented for different alues of the training signal separation distance which is within the < with a frequency offset ` frequency offset estiation ranges for all < (all approaches). At high SNR, a larger < gies a better perforance due to a sharper lielihood (or PEP) etric ainlobe. At low SNR, < has the sallest due to the sallest sidelobe peas of its lielihood (or PEP) etric. Fro Figs. 5-, it can be obsered that < has a saller sidelobe peas for an een integer than an odd (note that the estiation range for an een is half of that for an odd ). This fact translates into a better for an een at low SNR in Fig. 1. At low SNR, practical is saller than which indicates that the conentional perforance at low SNR is pessiistic. Soe <B can gie a better practical than < = depending on the sidelobe peas of the associated lielihood (or PEP) etric. In Fig. 11, the s and practical s obtained with different < are presented for ` `. Since is larger than the estiation range of the existing approaches (corresponding to < with being a non-negatie integer), their corresponding s or practical s are ery high. Due to the extended estiation range, the proposed approach using still gies a reliable estiate. < ; / a Next, we discuss how to circuent the proble for < haing a larger practical than the conentional structure (< ) at low SNR. In fact, if we certainly now that the frequency offset is liited to.., the aboe sidelobe proble can easily be aoided for ost of the cases by liiting the search range to.. since the larger sidelobe peas of the lielihood (or PEP) etric are around.. That eans the sidelobe proble associated with soe < in Fig. 1 can be relieed by liiting the search range. If the frequency offset can be larger than but less than, then the search range cannot be liited as aboe. In this case, " can be increased to aoid this sidelobe proble as discussed in the preious section. Fro CRB and. Fig., we can easily chec that " ;: and for < will encounter the sidelobe proble while " : will aoid the proble. In Figs. 1 and 13, the and practical perforance obtained with " : are presented for `.

6 . > H D H D D D and ` `, respectiely. As expected, the sidelobe proble is aoided for practical. In Figs. 1-1, we present a coparison of the proposed ethod with different paraeters and the existing ethods fro [5] and [] in ters of the estiation and the estiation range. The training signal for [5] is generated by -point IFFT of a length- Golay copleentary sequence and then repeating it once and adding a cyclic prefix of 8 saples. Hence for [5], 13 training saples are transitted and 18 training saples are used in the frequency offset estiation. The training signal for [] is coposed of 1 identical sub-blocs (including the CP) of 8 saples each (total 13 saples) and 1 sub-blocs (18 saples) are used in the estiation. For the proposed ethod, two sets of paraeters are considered: " 3a (1 identical sub-blocs of 8 saples each (total 13 saples) are transitted), = :, and " (18 identical sub-blocs are transitted), = " #:. For each set, < and are used. The estiation ranges of [5], [], and the proposed ethod are :.,.., and. :., respectiely. Fro the siulation results in Figs. 1-1, we can also obsere these estiation ranges. The estiation range of [5] is ery sall een copared with that of [] and the proposed ethod s estiation range is twice that of []. In ters of practical perforance, [] has a better perforance than [5] while the proposed ethod outperfors both [5] and [].. CONCUSIONS For eerging high data-rate counication systes in highly dispersie channels such as ultra-wideband systes, possible frequency offsets could be larger than the estiation range of the existing ethods using training signals with seeral identical sub-blocs. This paper addressed this issue and presented a axiu lielihood estiator with an extended estiation range as well as an iproed estiation perforance. The range extension and the estiation perforance iproeent are accoplished by designing the lielihood etric or a new design etric which is a pair-wise error probability (PEP) between the correct frequency offset point and a trial frequency offset point. The proposed new PEP etric gies ore theoretical insights on the perforance of practical axiu lielihood estiators. At coparable coplexity with the sae training oerhead aount, the proposed ethod at least doubles the estiation range and also iproes the estiation perforance. APPENDIX-A In this appendix, we show that and hence, + $ 3 /Std+ 3 and PEP etrics just depend on!. By this fact, we can conclude that a change in alue will siply result in a shift in the etrics (in the -axis) which does not affect the estiation range. Hence, in our design for estiation range extension, we can siply set =. Equation (15) can be expressed as ) ( *) +-,/. ( Š ) +-,,. 1 8/:<;/=*> /? > /? > /: /? > *=? > *=? > /:? >? > () A 1; A 1; I J M(KM N$O W$N O hae the sae statistics as A. Siilarly, the eleents of 3 can be gien by ) ( Š ) +-,/. ( ) +-,, ) ( Š ) +-,/. ( ) +-,, ) ( Š ) +-,/. ( ) +-,, ) ( Š ) +S. +-,/. ( ) +. +-,, (3) C$H By substituting () and (3) into (18), expressed as can be "!#. #. ) ( Š ) +-,/. ( ) +-,, ) ( Š ) +-,/. ( ) +-,,&! %!%$ 8/:<;"=*>('!#. #. ) ( Š ) +S. +-,/. ( ) +. +-,,. 1! %! C\H /? > /? > /:? > /? ) > *=? > *=? > /:? > just depend ` Equation (31) proes that the statistics of on. Next, we present a justification for the Gaussian approxiation of. For SNR alues of practical interest, the last ter A # ; A ; fro (31) is negligible if copared to the reaining ters. Hence, we can approxiate as,!#. #. ) ( Š ) +S. +-,/. ( ) +. +-,,. 1 >+*! %! C$H ) /? > 8 /? > (3) /:? > /? > *=? > For gien, and, are deterinistic ariables and A are Gaussian rando ariables. in (3) is just a linear cobination of Gaussian rando ariables and hence, it is a Gaussian rando ariable. The insignificant ter neglected in (3) can be expressed as. #. #. / ) ( Š ) +S. +-,/. ( ) +. +-,,.-! %! C$H 1 ). (33) *=? > /:? > 3 Note that there are no coon Gaussian noise ters in and for. Since A # are iid Gaussian rando ariables (with zero-ean, ariance d ), are also iid rando ariables. This paper considers a highly dispersie channel and hence, the nuber of channel taps is large. Then by the Central liit theore, the ter in (33) (and hence ) can be approxiated as a Gaussian rando ariable. In brief, for SNR of practical interest, the ter fro (33) can be neglected and in this case is exactly a Gaussian rando ariable. If the (insignificant) ter fro (33) is included, can be approxiated as a Gaussian rando ariable by eans of the Central liit theore. APPENDIX-B In this appendix, we first derie the snap-shot CRB of the frequency offset estiation at a gien SNR' for an obseration ector consisting of " sub-blocs with the corresponding tie-indices $#, #,..., #&%. Then we proe that the proposed design gies a saller CRB than the conentional (31)

7 # = = = > > = ) " ) D = > ; training structures consisting of seeral consecutie identical sub-blocs. Following the sae approach fro [], we obtain the snapshot CRB for a gien as d s ~ % X y (3) (35) ;E= C3;D D D ;$= HBC3; ;"= C3; D DSD ;"= H C3;D D D ; = #. ;$= #. C3;D D D ;$= #. (3) After soe anipulation in (35) and substituting H (13) C into (3), we obtain $ ; D D D ; = #. & (3) 1 1 #. #. 1 (38) = =!" 1 1 #. $# #. 1 %& ' (3) 8 = h is a atrix with eleents + 8 U[ = M ] O&^ and (` = diag z, 1,...,. Substituting s!8ts the aboe equations into (3) together with the definitions 8 $ and SNR' = $ d gies the snap-shot CRB as in (). The snap-shot CRB for a gien depends on the snap-shot (instantaneous) SNR and hence, the notation CRB has been used throughout the paper. Define ) Then we hae )+* % #3 - % # / () % ) : Z / (1) ), In the following, we calculate for the conentional structure denoted by and for the proposed design denoted by. In the proposed design, the training signal is separated into two groups of the sae length and the two transitted training signal groups are distanced by < saples. Suppose ; that = training sub-blocs are transitted. For the conentional structure, we hae " ; =, # for =, 1,..., = )-*. After soe siplification in (), we obtain = = ; ; / `= () ` For the proposed design, we hae " =, #& for =, 1,..., = and # A ; ; ; = < = ; ; = < _ for =, 1,..., = < _ ; <. After )., soe calculation, we obtain = ; ; ; = = < _ `< _ ; ` H/ (3) = E[g( )] E[g( )] d= 1 d=1 1 d= 1 d= d= 1 d=5 1 d= 1 d= Fig.. The effects of the training signal group separation distance on the noralized ean alue of the lielihood etric for,..., E D/. The actual frequency offset corresponds to 1. Any other1 with noralized 35 $8 1 close to one causes an abiguity in estiation. (E : ) );, )-* )., Substituting < _ );, )-* into (3) yields. Since < _ and increases with < _, we hae proed that which eans (see (1)) the proposed design gies a saller snapshot CRB (and hence a saller aerage CRB, aeraged oer the channel fading statistics) than the conentional training structure does. REFERENCES [1] F. Classen and H. Meyr, Frequency synchronization algoriths for OFDM systes suitable for counication oer frequency selectie fading channels, IEEE TC, June 1, pp [] F. Daffara and O. Adai, A new frequency detector for orthogonal ulticarrier transission techniques, IEEE TC, July 15, pp [3] T. M. Schidl and D. C. Cox, Robust frequency and tiing synchronization for OFDM, IEEE Trans. Coun., pp , Dec. 1. [] T. Keller,. Piazzo, P. Mandarini, and. Hanzo, Orthogonal frequency diision ultiplex synchronization techniques for frequency-selectie fading channels, IEEE J-SAC, pp. -18, June 1. [5] P.H. Moose, A technique for orthogonal frequency diision ultiplexing frequency offset correction, IEEE Trans. Coun., pp. 8-1, Oct. 1. [] J-J. an de Bee, M. Sandell and P.O. Börjesson, M estiation of tie and frequency offset in OFDM systes IEEE Trans. Signal Proc., pp , July 1. [] M.G. Hebley and D. P. Taylor, The effect of diersity on a burst-ode carrier-frequency estiator in the frequency-selectie ultipath channel, IEEE Trans. Coun., pp , Apr. 18. [8] M. uise and R. Reggiannini, Carrier frequency acquisition and tracing for OFDM systes, IEEE Trans. Coun., pp , No. 1. [] M. Morelli and U. Mengali, Carrier-frequency estiation for transissions oer selectie channels, IEEE Trans. Coun., pp , Sept.. [1] H. Minn, P. Tarasa and. K. Bhargaa, OFDM frequency offset estiation based on BUE principle, IEEE TC (Fall), Sept., pp [11], Soe issues of coplexity and training sybol design for OFDM frequency offset estiation ethods based on BUE principle, IEEE TC (Spring), Apr. 3, pp [1] H. Minn, S. Xing, and. K. Bhargaa, Optial periodic training signal for frequency offset estiation in frequency selectie fading channels, IEEE ICC, June, pp [13] IEEE AN/MAN Standards Coittee, Wireless AN ediu access control (MAC) and physical layer (PHY) specifications: High-speed physical layer in the 5 GHz band, IEEE Standard 8.11a, 1.

8 # 8 TABE I ESTIMATION COMPUTATIONA COMPEXITY FOR A TRAINING SIGNA CONSISTING OF IDENTICA SUB-BOCKS # real ultiplication )E / 1 # real addition )E / D 1 / D ) NNP,1 ) # NNP. *, ) NNP, Conentional! " / # Proposed D/ (at ost) # for $E for $E% (>) U = (a) 1 l l 1 3 l 5 l 3 l 8 l 5 (b) l l 1 l l 3 l l 5 l l l 8 d =, U = (c) l l 1 l l 3 l l 5 l l d d >, sub-bloc excluded in the obseration ector sub-bloc included in the obseration ector Fig. 1. Seeral training structures for & / total transitted identical training signal sub-blocs (defined by ' ( *) ) and the corresponding obseration ector s sub-blocs (defined by '+ ) ). (a) An arbitrary training signal structure, (b) The existing training signal structure (Our considered training signal structure with ), (c) Our considered training signal structure with -,. E[Z( )]/Std[Z( )] E[Z( )]/Std[Z( )].... d= d=1 d= d= ratio. 8 1 /Std. 8 1 for.... d= d=5 d= d=.... Fig. 3. The effects of the training signal group separation distance on the,..., E/D/. The actual frequency offset corresponds to 1. Any other 1 3 with. 8 1 /Std. 8 1 close to zero causes an abiguity in estiation. (E : ) P[Z( ) ] P[Z( ) ] 1 d= d=1 d= d= d= 1 d=5 1 d= 1 d= Fig.. The effects of the training signal group separation distance on the pair-wise error probability (PEP) etric for,..., E D /. The actual frequency offset corresponds to 1. Any other 1 with PEP close to.5 causes an abiguity in estiation. (E : )

9 E[g( )] E[g( )] d= d=1 d=3 d= d= d= Fig. 5. The effects of the training signal group separation distance on the ean alue of lielihood etric for $E% $ is an integer. The actual frequency offset corresponds to 1. (E : ) P[Z( ) ] = 1 = = = d = P[Z( ) ] = 1 = = = d = Fig. 8. The effects of (the nuber of training sub-blocs in the first group) on the pair-wise error probability etric ( total nuber of training sub-blocs of the two groups is fixed. The actual frequency offset corresponds to 1.) (E : ) E[Z( )]/Std[Z( )] E[Z( )]/Std[Z( )] d= d=1 d= d= d= d= ratio. 8 1 /Std. 8 1 for $E% $ is an integer. The actual Fig.. The effects of the training signal group separation distance on the frequency offset corresponds to 1. (E : ) P[Z( ) ] U = 1 U = 18 = U/ d = SNR i = db P[Z( ) ] SNR i = db SNR i = 5 db SNR i = 1 db = d = Fig.. The effects of total nuber of training sub-blocs and SNR on the pair-wise error probability etric. The actual frequency offset corresponds to 1. (E : ) P[Z( ) ] 1 d= d=1 d=3 1 3 d= d= d=8 d=1 d=1 d= 1 3 d= d= d=8 d=1 d=1 d= P[Z( ) ] d= d= d= Fig.. The effects of the training signal group separation distance on the pair-wise error probability etric for $E% $ is an integer. The actual frequency offset corresponds to 1. (E : ) Fig = = 3/ = = 3/ Estiation perforance obtained with different training separation distance for a noralized frequency offset less than / % )E. (E : )

10 1 1 d = d = d = 1 d = 1 d = 1 d = d = d = 1 d = 1 d = SNR = 5 db SNR = 5 db 1 1 [5] [] U=15, d= U=15, d=1 U=1, d= U=1, d=1 Fig. 11. = = 3/ = = 3/ Estiation perforance obtained with different training separation distance for a noralized frequency offset between / % )E and / %)E. (E : ) Fig. 1. Coparison of estiation and estiation range for seeral ethods at SNR = 5 db 1 d = d = d = 1 d = 1 d = 1 d = d = d = 1 d = 1 d = SNR = 1 db SNR = 1 db 1 1 U = 18 = = 3/ 1 1 U = 18 = = 3/ [5] [] U=15, d= U=15, d=1 U=1, d= U=1, d= Fig. 1. Estiation perforance obtained with a larger (for suppressing sidelobe peas) for a noralized frequency offset less than / % )E. (E : ) Fig. 15. Coparison of estiation and estiation range for seeral ethods at SNR = 1 db 1 1 d = d = d = 1 d = 1 d = U = 18 = = 3/3 1 1 d = d = d = 1 d = 1 d = U = 18 = = 3/3 1 1 SNR = 15 db 1 1 SNR = 15 db [5] [] U=15, d= U=15, d=1 U=1, d= U=1, d= Fig. 13. Estiation perforance obtained with a larger (for suppressing sidelobe peas) for a noralized frequency offset between / % )E and / %)E. (E : ) Fig. 1. Coparison of estiation and estiation range for seeral ethods at SNR = 15 db