Optmal Perodc Tranng Sgnal for Frequency Offset Estmaton n Frequency Selectve Fadng Channels Hlang Mnn, Member, IEEE and Shaohu Xng Department of Electrcal Engneerng School of Engneerng and Computer Scence Unversty of Texas at Dallas, Rchardson, TX 7583, USA Emal: hlang.mnn@utdallas.edu Vjay. Bhargava, Fellow, IEEE Department of Electrcal and Computer Engneerng Unversty of Brtsh Columba Vancouver, B.C., V6T Z4, Canada E-mal: vjayb@ece.ubc.ca Abstract Ths paper addresses optmal perodc tranng sgnal desgn for frequency offset estmaton n frequency selectve multpath Raylegh fadng channels. For a fxed transmtted tranng sgnal energy and wthn a fxed length block, the optmal perodc tranng sgnal structure (the optmal locaton of dentcal tranng sub-blocks) and the optmal tranng sub-block sgnal are presented. The optmalty s based on the mnmum Cramer-Rao bound (CRB) crteron. Based on the snap-shot CRB, the optmal perodc tranng structure s derved. The optmal tranng sub-block sgnal s obtaned by utlzng the average CRB and the receved tranng sgnal statstcs. The optmal tranng structure wth optmal tranng sgnals acheves substantal performance mprovement over non-optmal tranng structure wth non-optmal tranng sgnals. I. INTRODUCTION Tranng sgnals are commonly used n communcatons systems for tmng synchronzaton, frequency synchronzaton and channel estmaton. Tranng sgnal desgn s an mportant ssue snce a proper desgn can sgnfcantly mprove the estmaton performance and can drastcally reduce the estmaton complexty. In [], perodc tranng sequences for channel equalzaton were addressed and zero autocorrelaton (ZAC) sequences were shown to be optmal for the cyclc convoluton type channel equalzaton. Perodc and aperodc tranng sequences for least-square channel estmaton were consdered n [] where best sequences obtaned by computer search were presented for several channel lengths and tranng sequence lengths. Several desgn crtera and correspondng methods for effcent computer search of optmal sequences for channel estmaton were presented n [3]-[4]. Constructons of optmal complex sequences were dscussed n [5]-[6] and references theren. Optmal tranng sequences and plot tones for OFDM channel estmaton were addressed n [7]. Recent treatments on the optmal tranng structure and tranng sgnal desgn for channel estmaton can be found n [8], [9] and references theren. Tranng sequence desgn for tmng synchronzaton was dscussed n [] n the context of MS sgnal. The best pattern (+ or - sgns) of repettve tranng sub-blocks for tmng synchronzaton were presented n []. Regardng tranng desgn for frequency synchronzaton, [] addressed optmal tranng sgnals n AWGN channel by usng the CRB. Optmal tranng sgnal desgn for frequency offset estmaton n frequency selectve channels s not an easy task. Recently, [3] ncely addressed ths problem by applyng a mnmax approach based on asymptotc CRB. The channel gans reman constant wthn the tranng block and the asymptotc CRB s obtaned by settng the tranng block length (n samples) N. For a fxed channel energy, the mnmax approach mnmzes the asymptotc CRB for the worst-case channel response. The optmalty of the tranng sgnal from [3] s lmted by the mnmax approach and the asymptotc CRB but [3] does provde a neat soluton to a long standng problem. It s also noted that due to the fxed channel energy constrant, the channel fadng effect s not ncluded n the tranng sgnal desgn of [3]. The optmal tranng sgnal desgn for frequency offset estmaton n frequency selectve fadng channel s stll an open problem whch wll be addressed n ths paper. For the ease of estmaton complexty, we consder perodc tranng sgnal consstng of several dentcal sub-blocks. Perodc tranng sgnals have been extensvely used n practce (e.g., GSM, IEEE 8.a, 8.5, 8.6a). In ths paper, we wll answer the followng queston: For a fxed transmtted tranng sgnal energy, what s the optmal perodc tranng structure (the locaton of the tranng sub-blocks) wthn a fxed-length block and what s the optmal tranng sgnal (of a sub-block) that gves the mnmum CRB of the frequency offset estmaton n frequency selectve multpath Raylegh fadng channels?. It wll be addressed by two steps. In the frst step, we wll fnd the optmal tranng structure for a fxed block length and a fxed receved tranng energy. In the second step, we wll nvestgate the optmal tranng sgnal (of a subblock) for a fxed transmtted tranng energy n a frequency selectve multpath Raylegh fadng channel. The combnaton of the results from the two steps wll gve the soluton to the above queston. IEEE Communcatons Socety 488
The rest of the paper s organzed as follows. Secton II descrbes the sgnal model and the CRB. Secton III presents the optmal perodc tranng structure. In Secton IV, the optmal tranng sgnal (of a sub-block) s addressed. Smulaton results and dscussons are provded n Secton V and conclusons are gven n Secton VI. II. SIGNAL MODEL AND CRB We consder a wde-sense statonary uncorrelated scatterng frequency selectve Raylegh fadng channel characterzed by L taps wth uncorrelated complex baseband tap gans h(), h(),..., h(l ) and tap-spacng of symbol duraton T s. The channel s assumed to be quas-statc where the tap gans reman essentally constant over the block length NT s.the complex baseband receved sgnal sampled at the symbol rate can be expressed by r(n) =x(n)e jπnv + w(n), n =,,..., N () where v s the carrer frequency offset normalzed by the symbol rate /T s, {w(n)} are the uncorrelated samples of a zero-mean complex Gaussan nose process each havng a varance of σ n, and {x(n)} are the channel output sgnal samples correspondng to the transmtted sgnal samples {s n : n = L +, L +,..., N } and can be gven by x(n) = h(k) s n k, n =,,..., N. () In matrx form, the receved sgnal vector s expressed as r =Λ(v)Sh + w (3) where r = [r(), r(),..., r(n )] T, h= [h(), h(),..., h(l )] T, w= [w(), w(),..., w(n )] T, Λ(v) = dag{, e jπv,..., e jπ(n )v } s a dagonal matrx, S s an N L matrx wth entres S j = s j, N, j L, and the superscrpt T denotes the transpose operaton. The covarance matrces for w and h are gven by C w = E[ww H ] = σ ni N and C h = E[hh H ] = dag{σ h,σ h,..., σ h } where trace(c h) s assumed to be unty and I N s a N N dentty matrx. The superscrpt H represents the Hermtan transpose. If {s k } are known tranng sgnal samples, then for a gven channel realzaton h, the condtonal CRB (or the snap-shot CRB) for the estmaton of v based on the receved vector r s gven by [4] CRB h = σ n 8π h H S H M (I N B)MSh where M = dag{,,..., N } and B = S(S H S) S H s a projecton matrx. If the whole tranng block of length N+L samples s constructed by repeatng (P +) tmes the same tranng sgnal sub-block of length L samples, the snap-shot CRB for the obtaned perodc tranng sgnal can be smplfed as [4] where CRB h = (4) 3 SNR π (PL 3 )(P ) =CRB SNR. (5) SNR = E/L σ n and E denotes the energy of a receved tranng sub-block. The snap-shot CRB depends only on SNR for fxed values of (6) sub-block ndex L N Receved Tranng Samples 3 U- P- P- Fg.. An arbtrary perodc tranng structure llustratng the sub-blocks wth ndexes [,,..., U ] used n the estmaton. (The shaded subblocks serve as CPs and hence are not ncluded n the observaton vector of the estmaton) P and L. Hence, CRB h of (5) can be consdered as CRB SNR, the CRB condtoned on the snap-shot SNR of the receved tranng sgnal. III. OPTIMAL PERIODIC TRAINING STRUCTURE Perodc tranng sgnals are commonly used n practce due to ther advantage of complexty reducton n estmaton. For example, usng a perodc tranng block, whch conssts of (P +) sub-blocks of length L samples each, rather than a nonperodc tranng block of the same length N + L =(P +)L would reduce the estmaton complexty approxmately by a factor of L [4]. For the same reason, we consder perodc tranng sgnals n ths paper. Our nterest s to fnd optmal perodc tranng structure that gves the mnmum CRB. The CRB s derved for a jont estmaton of the frequency offset and the channel mpulse response as n [4]. The problem s formulated as follows. For a block wth a fxed length of (N + L) samples, whch may contan tranng sgnals only or both tranng and data sgnals, and for a fxed receved tranng sgnal energy E, what s the best perodc tranng structure? Ths problem s dvded nto two subproblems. In the frst sub-problem, we wll nvestgate the best structure consstng of V dentcal tranng sub-blocks,.e., we have to fnd the best locatons of V dentcal tranng sub-blocks wthn the block of length N + L samples. Note that E = V E and V (P +) f N + L =(P +)L. Inthe second sub-problem, we wll fnd the best value of V. Note that f E = E/(P +), then V = P +. The frst receved sub-block whch serves as a cyclc prefx part to absorb the channel dsperson, s dscarded n the estmaton. Let U be the number of tranng sub-blocks used n the estmaton. Then U = P n ths case. In general, f all V sub-blocks are consecutvely located as one group, U = V. If V sub-blocks form two groups of consecutve sub-blocks, where data sgnals are located between the two groups, then U = V. If there are G groups of consecutve tranng subblocks, we have U = V G. Hence, formng more groups of consecutve tranng sub-blocks (a group may contan one or more consecutve sub-blocks) wll result n more loss of tranng energy used n the estmaton. We wll denote ths tranng energy loss as the energy loss n the cyclc prefxes. A general tranng structure can be defned by the locaton ndex vector J =[,,..., U ] of the U sub-blocks used n the estmaton (see Fg. ). The ndexes wthn the block are from to P and hence, we have, U P and k < m for k<m. The correspondng snap-shot CRB can IEEE Communcatons Socety 489
be gven by (detals are skpped due to space lmtaton) CRB SNR (J) = SNR 8π L 3 ( U k ( U n= n) U ). (7) For a general tranng structure consstng of V =( +) dentcal tranng sub-blocks, each havng a receved sub-block energy E = E/( +), the best tranng structure can be obtaned by J + = arg mn J + CRB SNR (J +) (8) where the vector J + s of varable length U, U ( +), correspondng to the locaton ndexes of U subblocks used n the estmaton out of + tranng sub-blocks. For each fxed value of U, there are several dfferent J + s correspondng to dfferent locaton ndex vectors. Based on the snap-shot CRB n (7) and the result of the optmal tranng sgnal desgn n the AWGN channel from [], the optmal tranng structure can be gven by ts locaton ndex vector as follows: J + =[,,...,,P, P +,...,P ]. (9) The correspondng snap-shot CRB can be expressed as CRB SNR (J + )= 3 SNR 4π L 3 (4 +3P 6P ). () If the block contans an odd number of tranng sub-blocks, say +3 sub-blocks wth the receved sub-block energy of E = E/( +3), the optmal structure wll take ether of the followng two forms: J +3 = [,,...,,P, P +,...,P ] () J +3 = [,,...,,,P,P,...,P ]. () Both structures wll gve the same performance. It can be easly checked from (7) that J + gves a smaller snap-shot CRB than J +3. Hence, t s suffcent to consder an even number of tranng sub-blocks,.e., ( +). Next, we nvestgate what value of s the best for a block wth a fxed length of (P +)L samples, a fxed total receved tranng energy E and a fxed nose varance σn. Eq.() can be expressed as CRB SNR (J + ) = 3σ n/e + L π 4 3 6P +(3P ) = 3σ n/e f() (3) L π where f() = + 4 3 6P +(3P ). (4) Hence, the best value of s determned by = arg mn (P )/ f() (5) whch can be easly evaluated numercally. A close form soluton can be obtaned but due to space lmtaton, t s not ncluded. Note that snce f() s ndependent of SNR, the best value holds for any SNR and hence, for fadng channels as well. IV. OPTIMAL TRAINING SUB-BLOC SIGNAL In the prevous secton, we desgn the optmal perodc tranng structure for a fxed block length and a fxed receved tranng energy. Although the channel dsperson effect s ncluded n the desgn, the channel fadng effect s excluded from the desgn due to the condton of the fxed receved tranng energy. In ths secton, we wll nvestgate the effect of the channel fadng on the average CRB and we wll fnd optmal tranng sgnal wthn a tranng sub-block so that the average CRB s mnmzed. We consder a frequency selectve multpath Raylegh fadng channel. The problem can be formulated as follows: For a fxed transmtted energy of a perodc tranng sgnal composed of several dentcal tranng sub-blocks, what s the best tranng sub-block sgnal that mnmzes the average CRB n a frequency-selectve multpath Raylegh fadng channel? The multpath channel fadng causes fluctuaton of the receved tranng energy (although the long-term average or the expected value of the receved tranng energy s fxed) whch n turn affects the average CRB. To mnmze the average CRB, the tranng sgnal should be desgned such that the receved tranng energy fluctuaton s mnmzed. Ths fact wll be proved later. In the context of a perodc tranng sgnal, only one tranng sub-block s needed to be consdered n the tranng sgnal desgn. Defne the followng: Z = x(k) = E. (6) Then for a gven perodc tranng structure, we know from (7) that CRB SNR = α Z =CRB Z (7) where α = σ n 8π L ( U k ( U n= n) U ). (8) In a multpath Raylegh fadng channel, Z can be well approxmated by a Gamma random varable wth the parameter n. Note that x(k) s a ch-square random varable. The sum of ch-square random varables s often represented by another ch-square random varable (see [5] and references theren). The Gamma dstrbuton s a generalzaton of the ch-square dstrbuton n that n s an nteger n the latter but can be any postve real number n the former. In our case, n. The mean and varance of Z are gven by E[Z] =nσ and σz =nσ 4 where the values of n and σ depend on the tranng sgnal but E[Z] =nσ = s k s a constant equal to the transmtted energy of one tranng sub-block (assumng the total channel power transfer gan s unty). Let p Z(z) denote the probablty densty functon of Z. The average CRB n the multpath Raylegh fadng channel can be gven by CRB = = CRB Z p Z(z) dz (9) α z σ n/ Γ(n/) zn/ e z σ dz () α E[Z] σz. () /E[Z] IEEE Communcatons Socety 49
Eq.() ndcates that the larger receved tranng energy fluctuaton (larger σ Z) causes the larger CRB n the multpath Raylegh fadng channel. In other words, the tranng sgnal whch gves the mnmum fluctuaton of the receved tranng energy s the optmal tranng sgnal. After some calculaton, the varance of Z s gven by σ Z = + + (C S(k, k)) E[ h(k) 4 ] m= n=,n m = l=,l C S(m, n)c S(n, m)σ h m σ h n C S(, )C S(l, l)σ h σ h l (E[Z]) () where C S(m, n) s the (m, n)-th element of C S = S H S whch represents the perodc autocorrelaton of the tranng sgnal {s k : k =,,...,L }. Note that C S(n, n) = s k for all n and C S(m, n) =CS(n, m). From (), t s clear that σz s mnmzed when C S(l, k) = for l k. In other words, the tranng sgnal whch possesses zero perodc autocorrelaton for any non-zero correlaton lag (usually referred to as ZAC sgnal) mnmzes the receved tranng energy fluctuaton n the multpath Raylegh fadng channel. Combnng ths result wth that of () ndcates that the ZAC tranng sgnals are optmal for frequency offset estmaton n multpath Raylegh fadng channels. V. SIMULATION RESULTS AND DISCUSSIONS We assume that the channel gans reman unchanged over a block of length 4L samples (.e., P = 39) where the channel length L s 6 samples. The channel s assumed to have a power delay profle wth a 3 db per tap decayng factor. We frst evaluate varous tranng structures whch have the same total transmtted tranng energy E Tx and are composed of dentcal tranng sub-blocks of length L samples each. Structure# denotes a structure where the frst sub-blocks are tranng sgnals. Structure# represents a structure where all 4 sub-blocks are tranng sgnals (sub-block energy n ths case wll be smaller than that n Structure#). The Optmal structure s the one where the frst and the last + subblocks are tranng sgnals. The tranng sub-block sgnal used s the one from IEEE 8.a (OFDM). Note that Structure# corresponds to the tranng sgnal proposed n [6]. The frequency offset estmaton method of [7] (Method-B) s used n the evaluaton of the tranng structures. For very low SNR, the ntal frequency offset compensaton of [7] s done by the method of [6] wth 64-pont FFT. The results are presented for dfferent values of γ = E Tx /(6 σ n). For the optmal structure wth the optmal value ( =4),γ equals to SNR. Fg. shows the frequency offset estmaton performance n terms of the snap-shot CRBs (7), the average CRBs ((9) evaluated by smulaton) and MSE (smulaton results) for dfferent tranng structures n the multpath Raylegh fadng channel. Both CRBs and MSEs ndcate the same nformaton on the tranng structures. Structure# performs better than Structure# but at the cost of more system resource (tme). The optmal structures wth = 4 and = 9 perform sgnfcantly better than Structure# and Structure#. The optmal structure wth =9utlzes the same system resource (tme) as Structure#. The optmal structure wth =4uses less system resource (tme). Although smaller needs less system resource (tme), t may not gve a better estmaton performance. The best value of, denoted by,nterms of estmaton performance s gven by (5). For the consdered parameters, =4. The smulaton results and the snap-shot CRBs are presented n Fg. 3 for the optmal structure wth dfferent values of. They agree wth the theoretcal result of (5) ndcatng =4. Next, usng the optmal tranng structure wth the best value of, we nvestgate several tranng sub-block sgnals. The consdered sgnals are the OFDM tranng sub-block sgnal from IEEE 8.a, several constant ampltude sequences ncludng a m-sequence (-bt augmented to have an even length) and a constant ampltude ZAC sequence (CAZAC) and two arbtrary non-zac sequences. The snap-shot CRB (whch s ndependent of the tranng sub-block sgnal), the average CRBs and the MSEs for dfferent tranng sub-block sgnals are presented n Fg. 4. Among them, the ZAC sequence s the best. Snce the IEEE 8.a tranng sub-block sgnal and the m-sequence have correlaton propertes very close to that of ZAC, ther performance are almost the same as ZAC sequence (not dstngushable n the fgure). The performances of two arbtrary non-zac sequences are worse than the ZAC sequence, as expected. The optmal tranng sub-block (ZAC sequence) mproves about.5 to db n CRB or MSE performance over the two arbtrary non-zac sequences. For the consdered parameters, the proposed optmal tranng structure has approxmately 9 db mprovement n CRB or MSE performance over the conventonal consecutve perodc tranng structure such as Structure#. VI. CONCLUSIONS We have presented an optmal perodc tranng sgnal desgn for frequency offset estmaton n frequency selectve multpath Raylegh fadng channels. The optmalty s based on the mnmum average CRB n the frequency selectve fadng channel wthn the framework of a fxed total transmtted tranng sgnal energy and a fxed block length over whch the channel gans reman unchanged. The tranng desgn s addressed n terms of the optmal perodc tranng structure (the optmal locaton of dentcal tranng sub-blocks wthn the block) and the optmal tranng sub-block sgnal. Sgnals havng zero autocorrelaton (ZAC) property experence the mnmum receved energy fluctuaton n frequency selectve Raylegh fadng channels whch translates nto the mnmum average CRB. Hence, the optmal tranng sub-block s a ZAC sgnal. Ths result also provdes the mssng proof of the optmalty or near-optmalty of several tranng sgnals adopted n many standards. The optmal perodc tranng structure conssts of ( +) dentcal tranng sub-blocks, IEEE Communcatons Socety 49
each havng length L samples (the channel length) wthn the block of length (P +)L samples. The frst + tranng subblocks are located at the begnnng and the rest resdes at the end of the block. The value of s a functon of P and can easly be calculated. The optmal perodc tranng structure acheves a sgnfcant estmaton performance mprovement over the conventonal consecutve perodc tranng structure. ACNOWLEDGMENT Ths work was supported n part by the School of Engneerng and Computer Scence at the Unversty of Texas at Dallas and n part by the Natural Scences and Engneerng Research Councl (NSERC) of Canada. Snap shot CRB 8 9 Structure# Structure# Optmal (=9) Optmal (=4) MSE or Average CRB 7 8 9 MSE (Structure#) MSE (Structure#) MSE (Optmal, =9) MSE (Optmal, =4) CRB (Structure#) CRB (Structure#) CRB (Optmal, =9) CRB (Optmal, =4) REFERENCES [] A. Mlewsk, Perodc sequences wth optmal propertes for channel estmaton and fast start-up equalzaton, IBM Journal of Research and Development, Vol. 7, No. 5, Sept. 983, pp. 46-43. [] S.N. Crozer, D.D. Falconer and S.A. Mahmoud, Least sum of squared errors (LSSE) channel estmaton, IEE Proceedngs-F, Vol. 38, No. 4, Aug. 99, pp. 37-378. [3] C. Tellambura, M.G. Parker, Y.J. Guo, S.J. Shepherd and S.. Barton, Optmal sequences for channel estmaton usng dscrete Fourer transform technques, IEEE Trans. Commun., Vol. 47, No., Feb. 999, pp. 3-38. [4] W. Chen and U. Mtra, Tranng sequence optmzaton: Comparson and an alternatve crteron, IEEE Trans. Commun., Vol. 48, No., Dec., pp. 987-99. [5] Wa H. Mow, A New Unfed Constructon of Perfect Root-of-Unty Sequences, Proc. Spread Spectrum Technques and ts Applcatons (ISSSTA 96), Manz, Germany, pp. 955-959, 996. [6] J.C.L. Ng,.Ben Letaef, and R.D. Murch, Complex Optmal Sequences wth Constant Magntude for Fast Channel Estmaton Intalzaton, IEEE Trans. on Commun., Vol. 46, N. 3, Mar. 998, pp. 35-38. [7] J.H. Manton, Optmal tranng sequences and plot tones for OFDM systems, IEEE Communcatons Letters, Vol. 5, No. 4, Aprl, pp. 5-53. [8] M. Dong and L. Tong, Optmal desgn and placement of plot symbols for channel estmaton, IEEE Trans. Sgnal Processng, Vol. 5, No., Dec., pp. 355-369. [9] X. Ma, G.B. Gannaks and S. Ohno, Optmal tranng for block transmssons over doubly selectve wreless fadng channels, IEEE Trans. Sgnal Processng, Vol. 5, No. 5, May 3, pp. 35-365. [] R. Dabora, J. Goldberg and H. Messer, Inherent lmtatons n dataaded tmng synchronzaton of contnuous phase modulaton sgnals over tme-selectve fadng channels, IEEE Trans. Sgnal Processng, Vol. 5, No. 6, June, pp. 47-48. [] H. Mnn, V.. Bhargava and. Ben Letaef, A robust tmng and frequency synchronzaton for OFDM systems, IEEE Trans. Wreless Commun., Vol., No. 4, July 3, pp. 8-839. [] H. Mnn and S. Xng, An optmal tranng sgnal structure for frequency offset estmaton, submtted to IEEE Trans. Commun. [3] O. Besson and P. Stoca, Data-aded frequency offset estmaton n frequency selectve channels: Tranng sequence selecton, Proc. of IEEE Intl. Conf. on Acoustcs, Speech, and Sgnal Processng (ICASSP ),Vol. 3, May, pp. 445-448. [4] M. Morell and U. Mengal, Carrer-frequency estmaton for transmssons over selectve channels, IEEE Trans. Commun., Vol. 48, No. 9, Sept., pp. 58-589. [5] Q.T. Zhang and D.P. Lu, A smple capacty formula for correlated dversty Rcan fadng channels, IEEE Communcatons Letters, Vol. 6, No., Nov., pp. 48-483. [6] J. L, G. Lu and G. B. Gannaks, Carrer frequency offset estmaton for OFDM-based WLANs, IEEE Sgnal Processng Letters, Vol. 8, No. 3, Mar., pp. 8-8. [7] H. Mnn, P. Tarasak and V.. Bhargava, OFDM frequency offset estmaton based on BLUE prncple, Proc. IEEE Vehcular Tech. Conf. VTC (Fall), Vancouver, Canada, Sept., pp. 3-34. 5 5 5 5 Fg.. Performance of dfferent perodc tranng structures n the multpath Raylegh fadng channel MSE or Snap shot CRB 3.5 3.5.5 4 x 9 3 4 5 6 7 8 9 γ = db MSE CRB.6 x.4..8.6.4. γ = db 3 4 5 6 7 8 9.4 x..8.6.4 γ = db. 3 4 5 6 7 8 9 Fg. 3. Performance at dfferent values of for the optmal tranng structure contanng (+) sub-blocks MSE 8 9 Non ZAC # Non ZAC # CAZAC m sequence OFDM sgnal 5 5 Average CRB 8 9 5 5 Non ZAC # Non ZAC # CAZAC m sequence OFDM sgnal Snap shot CRB Fg. 4. Performance of dfferent tranng sgnals (of a sub-block) wth the optmal tranng structure (=4) n the multpath Raylegh fadng channel IEEE Communcatons Socety 49