An Average Cramer-Rao Bound for Frequency Offset Estimation in Frequency-Selective Fading Channels

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH An Average Cramer-Rao Bound for Frequency Offset Estimation in Frequency-Seective Fading Channes Yinghui Li, Haing Minn, Senior Member, IEEE, and Jianqiang Zeng, Student Member, IEEE Abstract Severa variations of Cramer-Rao bounds for carrier frequency offset CFO estimation in frequency-seective fading channes have been used to benchmark practica estimators performance or to design training signas for CFO estimation. Among them, the extended Mier-Chang bound EMCB provides a tighter bound than the for ocay unbiased estimators. However, there is no cosed-form expression of the EMCB for the CFO estimation in frequency-seective fading channes with an arbitrary training signa. In this etter, we derive a cosed-form exact average the EMCB vaid for any training signa and any signa structure for the CFO estimation over frequency-seective Rayeigh fading channes with uncorreated or arbitrariy correated taps. The accuracy and generaity of the proposed average expression, and its advantages over the existing expressions are corroborated by numerica and simuation resuts. Index Terms Cramer-Rao bound, frequency offset estimation, frequency-seective fading, Rayeigh fading. I. INTRODUCTION THE carrier frequency offset CFO between the transmitter and receiver can cause a severe performance degradation [1], [], and hence severa CFO estimators, e.g., [3] [6], have been proposed in the iterature. Due to the presence of other unknown parameters e.g., channe impuse response, phase offset, the existing CFO estimators can be categorized into two types a direct CFO estimation, e.g., [3], [5], and a joint estimation of CFO and channe impuse response or generaized maximum ikeihood approach, e.g., [4], [7] [10]. To benchmark performance of these estimators, severa variations of Cramer-Rao bound were derived in [4], [11] [16]. Among them, the extended Mier-Chang bound EMCB provides a tighter bound than the, the hybrid, the modified, Mier-Chang bound, and asymptotic for ocay unbiased estimators 1 [13]. When considering a joint estimation of CFO and channe impuse response, the corresponding s of the CFO and the channe response ony concern with the particuar reaization of channe and CFO. In particuar, the of CFO estimation in this case depends on the particuar channe reaization and hence we Manuscript received May 4, 008; revised August, 008 and November 15, 008; accepted November 7, 008. The associate editor coordinating the review of this etter and approving it for pubication was Y. J. Zhang. Y. Li is with the LitePoint Corp. e-mai: yinghui.i@itepoint.com. H. Minn and J. Zeng are with the Department of Eectrica Engineering, University of Texas at Daas UTD e-mai: jianqiang.zeng, haing.minn@utdaas.edu. This work was supported in part by the Erik Jonsson Schoo Research Exceence Initiative, the University of Texas at Daas. Digita Object Identifier /TWC those estimators that are unbiased for a vaues of the nuisance parameters. In some particuar cases, a specificay designed training sequence may make the decouped from the channe reaization [17] but the wi sti depend on the energy of that channe reaization, and hence an averaging over the random snap-shot channe energy woud sti be required /10$5.00 c 010 IEEE wi ca it a snap-shot. If an average performance bound of CFO based on the joint estimation is desired, the above snap-shot needs to be averaged over the channe fading. This average corresponds to the extended Mier-Chang bound, and the Monte Caro simuation is used to compute this average e.g., see [13] and [6] as no cosed-form expression is avaiabe. The or simiar bounds are aso usefu in designing training signas [7], [8], [16], [18], but not a of them are anayticay tractabe. The use of average in [8] provides better training signas than the use of asymptotic in [7]. The average is derived by means of an approximation in [8] for periodic training signas with cycic prefix CP structure. The existing cosed-form average expression in [8] hods for neither an arbitrary training signa nor a zeropadded ZP signa structure. To the best of our knowedge, the cosed-form exact average expression for the CFO estimation in frequency-seective Rayeigh fading channes for an arbitrary training signa with any structure CP or ZP has not been derived in the iterature, and is what we propose in this etter. The rest of this paper is organized as foows. Section II describes the signa mode, the, the, and the snap-shot. Section III derives the cosed-form exact average for an arbitrary training signa. Section IV provides performance comparison and discussions, and Section V concudes the paper. We use the foowing notations. The superscripts T and H denote the transpose and the Hermitian transpose, whie tr stands for the trace operation. diagx 0, x 1,..., x K 1 representsak K diagona matrix whose diagona eements are given by x 0, x 1,..., x K 1. E[ ] stands for the expectation whie E x [ ] denotes the expectation over the random vector x. I K is the K K identity matrix. II. SIGNAL MODEL AND THE SNAP-SHOT Let us consider a wide-sense-stationary quasi-static frequency-seective Rayeigh fading channe characterized by its ow-pass-equivaent channe impuse response CIR vector h =[h0,h1,...,hl 1] T,whereL is the number of the sampe-spaced T s -spaced CIR sampes. h is a zero-mean compex Gaussian random vector with a covariance matrix given by C h = E[hh H ] = U h diagλ h 0,λ h 1,...,λ h L 1, 0,...,0U H h = U H h Σh U h 1 where Σ h is the L L diagona matrix with non-increasing eigen-vaues of C h, L is the number of nonzero eigen-vaues

2 87 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 010 λ h k of C h,and U h is constituted by the first L coumns of the unitary matrix U h.define υ as the carrier frequency offset normaized by 1/NT s where N is the number of training signa sampes excuding the guard sampes. Consider a transmitted training sequence sn :n = L +1, L +,...,0, 1,...,N 1. For CP based training structures, N = N, andsn :n = L +1, L +,..., 1 are the L 1 CP sampes. For ZP based training structures[19], N = N + L 1, andsn :n = L +1, L +,..., 1 and sn :n = N,N +,...,N+L are the nu prefix 3 sampes of the training signa and those of the foowing data signa, respectivey. Then the received signa vector can be expressed in matrix form as r = ΩυSh + n where r = [r0, r1,..., rn 1] T, h = [h0, h1,..., hl 1] T, n = [n0, n1,..., nn 1] T, Ωυ =diag1,e jπυ/n,..., e jπn 1υ/N, [S] k, = sk for 0 k N 1, 0 L 1, andn is the zero-mean white compex Gaussian noise vector with the covariance matrix σ ni N. For Rayeigh fading channes, pr υ, the probabiity density function pdf of r given υ is zero-mean compex Gaussian with the covariance matrix C r = ΩυSC h S H Ω H υ+σ ni N 3 and the of the direct CFO estimation can be obtained as [14] N = 8π trr 1 ΛRΛ trλ 4 where R = SC h S H + σn I N and Λ = diag0, 1,...,N 1. The corresponding maximum ikeihood estimator 4 MLE of υ which maximizes pr υ is aso known as Bayesian MLE and it requires the knowedge of C h and σn, and hence it is unreaizabe for practica systems with unknown C h and σn. On the other hand, both υ and h can be jointy estimated based on ML principe as in [4], [7] [10]. The snap-shot for the υ derived from the joint estimation of υ and h is given by [4] σn h = 8π h H S H ΛI N BΛSh 5 where B = SS H S 1 S H. 6 The notation h is used to refect that the is just for a given channe reaization h. The is given by [11][13] = σ n N 8π trs H ΛI N BΛSC h. 7 III. AVERAGE In this section, we present the of the CFO estimation based on the ML joint CFO and channe estimation in mutipath Rayeigh fading channes. We derive the cosed-form 3 They can aso be in the form of nu suffix. 4 which is given as ˆυ ML = arg υ minr H ΩυR 1 Ω H υ r. expression of the averaged over a mutipath Rayeigh fading channe as With the definition of = E h [ h ]. 8 Z = h H S H ΛI N BΛSh, 9 the snap-shot can be expressed as h = α Z = Z 10 where α = N σ n 8π, and the average can be obtained as E Z [ Z ].Define h = Q = Σ 1/ h Σ 1/ h U H h h, 11 U H h S H ΛI N BΛS U h Σ1/ h = UΣ Q U H 1 where U is a unitary matrix and Σ Q = diagλ 0,λ 1,,λ K 1, 0,...,0. Then 9 can be written as Z = h H Q h = g H Σ Q g = K 1 k=0 λ k gk 13 where g = U H h. In a mutipath Rayeigh fading channe, both h and g are zero-mean compex Gaussian random vectors with the covariance matrix I L,andZ is a sum of weighted Chi-square random variabes. Depending on the eigen-vaues of Q, we divide the whoe space of possibe training signas into two cases: 1 Case I: Q has at east two distinctive eigen-vaues. The moment generating function of Z in 13 is given by[0] m Φ Z s = p Z ze sz dz = 1 s κ 14 0 where m is the number of nonzero distinct eigen-vaues of Q and κ is the mutipicity of. By the inverse Lapace transform, the pdf of Z can be expressed as [1] p Z z = κ k=1 A k 1 f k z/ 15 where f n is a Chi-square pdf with n degrees of freedom DOFs, and f n x = xn/ 1 e x, 0 <x< 16 Γn/ m A k = λ i κi k b k 17 b k = i=1 κ k 1 j 1=0 j 1 1 j j =0 j 1 j 3=0 j1 1 j 1 j 3 κ k 1 j 1 A κ k 1 j 1 A j1 1 j 18 A j 1 j3 κ k!

3 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH = m k=1,k = A n = 1 n+1 n! 1 1 κk 19 λ k k=1,k = 1 κ k 1 n+1. λ k 0 After some manipuation, the average is given by α = im β 0 β z p Zzdz 1 A 1 β = α im E 1 β 0 k= κ A k + 1 F k β/ k 1 where F n x is the cumuative distribution function of a Chi-square random variabe with n DOFs, and E 1 x is the exponentia integra function defined by E 1 x x e u u du = C nx k=1 x k k k! 3 with C being the Euer s constant. Based on, an approximate average for periodic training signas with CP structure can be obtained as in [9]. In the foowing we wi derive the cosed-form exact average expression for an arbitrary training signa with any structure CP or ZP. According to the property of the Lapace transform, we have im p Zz = im sφ Zs. 4 z 0 + s From 15, the eft side of 4 can be written as im z 0 + p Zz = im z 0 + A 1 e z/ = A 1. 5 Substituting 14 into the right side of 4, we have im sφ Zs = im s m 1 s κ =0, m > 1 6 s s and from 4-6 we obtain A 1 =0. 7 Substituting 7 and 3 back to, we obtain the average as k A 1 = α im C n β βλ β 0 k k! k=1 κ A β k + 1 f k udu k 1 k= 0 = N σn A 1 κ A k 8π n +. 8 k 1 k= Case II: Q has K identica eigen-vaues λ. or average simuation average proposed Fig. 1. The comparison of the proposed average, simuated average and the for a CP-based training signa which yieds a distinct eigen-vaues for Q. In this case, we have K 1 Z = λ gk. 9 k=0 Then, Z is a Chi-square random variabe with DOF K, and the pdf of Z is given by p Z z = zk 1 ΓKλ K e z/λ 30 where Γ is the gamma function. Substituting 30 into 1, we obtain the average as = N σn 1 8π K 1λ. 31 From 8 and 31, we observe that the average is determined ony by the eigen-vaues of Q, which is cosey reated with the training signa matrix S and the channe covariance matrix C h. For a given C h, the average expressions 8 and 31 provide an easy computation of the exact average for the CFO estimation in a frequencyseective Rayeigh fading channe for any training signa with any structure CP or ZP. They aso enabe an easy numerica comparison of performance of different training signas. IV. PERFORMANCE COMPARISON We assume N = 64 and L = 16 in our simuation. We evauate the s in both uncorreated and correated mutipath Rayeigh fading channes. The uncorreated channe has 8 sampe-spaced taps and an exponentia power deay profie with 3 db per tap decaying factor. The correated sampe-spaced channe is the Channe B of the indoor office ITU-R channe mode[]. We verify the accuracy of our cosed-form exact average expressions by comparing them with the Monte Caro

4 874 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 010 a non ZAC b CAZAC or average simuation average proposed average Simuation average Simuation Fig.. The comparison of the proposed average, the simuated average and the simuated for a CP-based training signa where three of the eigen-vaues of Q are identica. Fig. 4. The comparison of the average s obtained by simuation, by the proposed expression and by the existing approximate expression in [8], as we as the and for different training sequences with ZP structure in uncorreated mutipath Rayeigh fading channes. a non ZAC b CAZAC or average simuation average proposed average average Simuation Simuation Fig. 5. The comparison of the average s obtained by simuation, by the proposed expression and by the existing approximate expression in [8] as we as the and the for different training sequences with ZP structure in correated mutipath ITU fading channes. Fig. 3. The comparison of the proposed average, simuated average and the for a CP-based training signa which yieds a identica eigen-vaues for Q. simuation resuts of the average for different nonperiodic training signas with CP structure in Figs Aso incuded for the comparison are the mean-square error simuation resuts of the generaized maximum ikeihood CFO estimation proposed in [4]. Fig. 1 presents the resuts for the case where a eigen-vaues of Q are distinct. Fig. corresponds to the case where three of the eigen-vaues of Q are identica, whie Fig. 3 shows the resuts for the case with a identica eigen-vaues of Q. Our cosed-form expressions exacty match with the Monte Caro resuts for a the cases of the considered training signas. Note that the estimator s cosey foows the exact average except at very ow SNR. Fig. 4 and Fig. 5 pot the proposed exact average, the existing approximate average [8], the cacuated by 7, the cacuated by 4, and the Monte Caro resuts for severa ZP periodic training signas [19] in mutipath fading channes with uncorreated taps and correated taps, respectivey. The existing approximate average expression yieds sighty arger vaues than the Monte Caro resuts for the constant ampitude zero auto-correation CAZAC training sequence, and the performance gap becomes significanty arger for the non-zac training sequence. The gaps of the and the to the Monte Caro resuts are aso arge for both the CAZAC and non-zac training sequence, which indicates that and the are rather oose

5 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH athough their computation may be simper than our average. It is interesting to note in Fig. 4 and Fig. 5 that the existing approximate average, which shoud represent a ower bound, may be even arger than the estimator s, which underines importance of the accurate average. Our cosed-form average expression provides an exact match to the Monte Caro resuts for both correated and uncorreated mutipath fading channes. These resuts highight the accuracy and generaity of our cosed-form average expressions and their advantages over the existing expressions. Note that the CFO estimation performance based on the ML joint CFO and channe estimation does not approach to the 4 even for arge SNR and training signa ength. This is not contradictory to the we-known fact that MLE approaches to asymptoticay. The in this case corresponds to the Bayesian MLE which requires the knowedge of the channe covariance matrix and the noise variance. Without those knowedge, the Bayesian MLE which maximizes pr υ cannot be reaized. On the other hand, the ML joint CFO and channe estimation does not require those knowedge, and as iustrated in Figs. 1-3, its performance is matched by our average expressions. V. CONCLUSIONS We have derived a cosed-form exact average averaged over the channes of the CFO estimation over mutipath Rayeigh fading channes with uncorreated or arbitrariy correated taps for any training signa with any structure CP or ZP. In contrast to the imited appicabiity and/or inaccuracy of the existing average expression, and the extensive Monte Caro simuation run time required by other existing bounds, our cosed-form expressions provide an easy computation of the exact average without any imitation to the training signas and structures. Simuation resuts corroborate that our resuts cosey match with the mean-square error performance of a practica generaized maximum ikeihood estimator and are much more accurate than other existing bounds. Our average expressions provide a usefu metric for evauating exact estimation performance bound in fading channes and for designing corresponding training signas. REFERENCES [1] T. Poet, M. Van Bade, and M. Moenecaey, BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise, IEEE Trans. Commun., vo. 43, no. 34, pp , [] H. Steendam and M. Moenecaey, Synchronization sensitivity of muticarrier systems, European Commun., vo. 5, pp , May 004. [3] M. Morei and U. Mengai, An improved frequency offset estimator for OFDM appications, IEEE Commun. Lett., vo. 3, no. 3, pp , [4], Carrier-frequency estimation for transmissions over seective channes, IEEE Trans. Commun., vo. 48, no. 9, pp , 000. [5] H. Minn, P. Tarasak, and V. 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