(2) 1232. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 8, AUGUST 2008. Fig. 1.. Block structures of the (a) general SC-FDE system, (b) WirelessMAN-SCa PHY, and (c) sub-block processing, for B = 2, D = 3, and P = 2.. zn,i = e. j2πni (1+δ)ε/N. 1 N. N/2. . ej2πnk/N ej2πni δk/N Hk,i Xk,i + vn,i , 0 ≤ n ≤ N − 1,. Zk,i = λN (ϕkk )ejπ[2i(N +NG )+2NG +N −1]ϕkk /N Hk,i Xk,i + Ck,i + Vk,i , − N/2 + 1 ≤ k ≤ N/2,. (2). Z k,l = λNP (ϕkk )ejπ[2l(NP +NG )+4NG +NP −1]ϕkk /NP H k,i Ak + C k,l + V k,l , − NP /2 + 1 ≤ k ≤ NP /2,. (5). conjugate product of two consecutive blocks as Yk,i. (1). k=−N/2+1. =. ∗ Zk,i Zk,i−1. ≈. ∗ ej2π(N +NG )ϕkk /N |Hk,i |2 Xk,i Xk,i−1 + noise.(3). If Xk,i and Xk,i−1 are known, the phase ϕkk can be obtained ∗ by taking the argument of Yk,i /Xk,i Xk,i−1 , and ε and δ can be extracted by applying the linear regression [7] to a straight line ϕkk ≈ ε + δk. However, Xk,i and Xk,i−1 are mixtures of the time-domain data and pilot symbols and, hence, are unknown. To solve this difficulty, the sub-block processing is introduced. C. Sub-Block Processing The block structure of the WirelessMAN-SCa PHY [4] is depicted in Fig. 1(b), where the unique word (UW) possesses the constant amplitude zero auto-correlation (CAZAC) property. The CP and pilot word, respectively, consist of one and P UWs. To simplify the system design, the length of the burst data is an integer multiple of the length of a UW, i.e., N − NP = DNU . As shown in Fig. 1(c), for sub-block processing, the block boundaries are shifted right by NG samples, and each block of length N + NG is partitioned into B sub-blocks, each of length NP +NG . More specifically, the ith block is partitioned into B sub-blocks with sub-block indices l = iB + 0, iB + 1, · · · , iB + (B + 1), for i ≥ 0. In each pilot sub-block, i.e., the last sub-block of each block, there are P + 1 UWs; the first UW acts as the CP, while the last P UWs act as the pilot word. Since the pilot word {am , 0 ≤ m ≤ NP − 1} consists of P identical UWs possessing the CAZAC property,. √ the amplitudes of pilot symbols are |am | = ES for all m, and the squared amplitudes of pilot subcarrier symbols are 2 P NU ES , if k ∈ A, |Ak |2 = (4) 0, else, where Ak is the NP -point DFT of am and A = {k : k = qP and − NU /2 + 1 ≤ q ≤ NU /2}. Subsequently, we analyze the effects of the CFO and SFO for the pilot sub-blocks with indices l ∈ L, where L = {l : l = iB + (B − 1), for i ≥ 0}. First, the mth time-domain sample in the lth sub-block is x([l(NP +NG)+2NG +m]TS ) = xm,l , for −NG ≤ m ≤ NP − 1 and l ∈ L, where xm,l = am , for 0 ≤ m ≤ NP − 1, and xm,l = am+NP , for −NG ≤ m ≤ −1. Considering the effects of the CFO and SFO, defining ϕqk ≡ (1 + δ)(ε + q) − k and ε ≡ εNP /N , following the derivations given in (1) and (2), and using the NP -point DFT, one can derive the frequency-domain samples of a pilot subblock as (5), which is shown below Fig. 1, where H k,i is the channel frequency response, C k,l is the ICI, and V k,l ∼ CN (0, NP N0 ) is the noise. By neglecting the ICI for small ε and δ, one can write the conjugate product of two consecutive pilot sub-blocks as Y k,i. =. Z k,iB+(B−1) Z ∗k,(i−1)B+(B−1). ≈. ej2π(N +NG )ϕkk /NP |H k,i |2 |Ak |2 + noise, (6). provided that H k,i ≈ H k,i−1 . Then the total frequency offsets ϕkk can be estimated on the non-zero pilot subcarriers, i.e., ϕ ˆ kk,i =. NP arg(Y k,i ), for k ∈ A, 2π(N + NG ). where arg(· ) denotes the argument of a complex number.. Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 00:51 from IEEE Xplore. Restrictions apply.. (7).

(3) CHIANG et al.: JOINT ESTIMATION OF CARRIER-FREQUENCY AND SAMPLING-FREQUENCY OFFSETS FOR SC-FDE SYSTEMS. Fig. 2.. 1233. The proposed receiver architecture.. III. J OINT E STIMATION OF CFO AND SFO For small ε and δ, the product term εδ in ϕkk is neglected. Thus the joint one-shot estimation of the CFO and SFO can be achieved by employing ϕkk = ε + δk and performing LS regression on the estimates of the total frequency offsets given in (7). To avoid the effects of the direct-current (DC) offset and phase non-linearity, the subcarriers at DC and the band edges are not used in the regression and only the non-zero pilot subcarriers with indices k ∈ K are employed, where K = {k : k = ±P, ±2P, · · · , ±QP }. From our experience, a rule of thumb is to choose Q as Q = 0.4NU , where ·. denotes the floor function. The SWLS estimator of the CFO and SFO collects ϕ ˆ kk,i from the non-zero pilot subcarriers, assigns each pilot subcarrier a different weight wk,i , and performs LS regression via [8] wk,i ϕ ˆ kk,i N k∈K · , (8) εˆi = NP wk,i k∈K. δˆi =. kwk,i ϕ ˆ kk,i 2 , k wk,i. k∈K. (9). k∈K. where wk,i = |H k,i |2 and is used to compensate for different levels of fading encountered at different subcarriers. During each block, the channel frequency response is, in general, unknown before synchronization takes place. However, we can simply use the channel estimate of the last block in the 2 ˆ synchronization for the present block, i.e., wk,i = |H k,i−1 | ˆ and wk,0 = 1, where H k,i can be obtained by using a pilotaided channel estimation based on the sub-block structure [10]. Noteworthily, the SWLS estimator reduces to the LLS estimator, provided that equal weights (i.e. wk,i = 1) are utilized. As depicted in Fig. 2, to smooth the fluctuation of one-shot estimates εˆi and δˆi , we incorporate a closed-loop tracking [9] into the proposed receiver architecture, i.e., εˆi = εˆi−1 + γε εˆi ,. εˆ0 = 0,. (10). + γδ δˆi , δˆi = δˆi−1. δˆ0 = 0,. (11). where γε and γδ are step sizes taking on the values between (0, 1). With the closed-loop tracking, the magnitudes of the residual CFO and SFO decrease and converge to small values. IV. P ERFORMANCE A NALYSIS In this section, assuming small ε and δ, we prove that the LLS estimator is unbiased and derive its MSE. For the AWGN channel (i.e. H k,i = 1), substituting (4) into (6) and ignoring the product of two noise terms (for high signal-to-noise ratio (SNR) region), one can rewrite (6) as Y k,i = ej2π(N +NG )ϕkk /NP P 2 NU ES + W k,i , for k ∈ K, (12) where W k,i is the noise distributed as CN (0, 2P 3 NU2 ES N0 ). We define Y˜ k,i ≡ Y k,i /P 2 NU ES and α ≡ exp[−j2π(N + NG )ε/NP ], and write their product as ˜ k,i , αY˜ k,i = ej2π(N +NG )δk/NP + αW. (13). ˜ where W k,i ∼ CN (0, 2N0 /P ES ). By substituting (7) into (8) and using arg(Y˜ k,i ) = arg(Y k,i ), the estimate is rewritten as εˆi =. N arg(Y˜ k,i ). 4πQ(N + NG ). (14). k∈K. Then, in light of (13) and (14), the estimation error of the CFO can be derived as eε,i. ≡ = ≈ ≈. εˆi − ε. N arg(αY˜ k,i ) 4πQ(N + NG ) k∈K N (αY˜ k,i ) 4πQ(N + NG ) k∈K N ˜ ), (αW k,i 4πQ(N + NG ). (15). k∈K. where (· ) denotes the imaginary part of a complex number, the third line is due to the approximation of θ ≈ sin θ for small θ, and the last line is due to the assumption of small δ. Using ˜ ) ∼ N (0, N0 /P ES ), it can (15) and recognizing that (αW k,i be shown that εˆi is unbiased, i.e., E[eε,i ] = 0, and its MSE is. Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 00:51 from IEEE Xplore. Restrictions apply..

(4) 1234. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 8, AUGUST 2008. Fig. 3.. RMSEs of the closed-loop tracking in multipath fading channels for ε = 0.05 and δ = −20 ppm. TABLE I S YSTEM PARAMETERS. derived as MSE(ˆ εi ). ≡ E[e2ε,i ] =. 8π 2 P Q(N. N2 . + NG )2 · ES /N0. (16). Similarly, one can show that δˆi is unbiased and its MSE is MSE(δˆi ). ≡ E[e2δ,i ] =. (17) NU2 . 4π 2 P (2Q3 + 3Q2 + Q)(N + NG )2 · ES /N0. Moreover, by assuming that the tracking error is small enough to allow for loop linearization [9, Ch. 3], the MSEs of εˆi and δˆi in the steady state are derived as: MSE(ˆ εi ) =. γε2 MSE(ˆ εi ), 2 − γε. (18). MSE(δˆi ) =. γδ2 MSE(δˆi ). 2 − γδ. (19). V. N UMERICAL R ESULTS The performance of the proposed receiver depicted in Fig. 2 has been evaluated with the system parameters summarized in Tab. I and the SUI-4 channel model specified in [11]. Fig. 3 demonstrates the root-mean-squared errors (RMSEs) of the LLS and SWLS estimators with the closed-loop tracking in quasi-static and time-varying multipath Rayleigh fading channels, where the CFO and SFO are ε = 0.05 subcarrier spacings and δ = −20 ppm, respectively. The RMSEs of the LLS estimator with the closed-loop tracking in the AWGN channel are also provided as the benchmarks. Due to the. weighting based on the squared-magnitude of the subcarrier frequency response, the SWLS estimator is robust to the channel frequency-selectivity and is of a smaller RMSE than the LLS estimator in both quasi-static and time-varying channels. For the time-varying channels, due to the additional ICI caused by the channel variation within a block duration and the failure of the assumption, H k,i ≈ H k,i−1 [cf. (6)], the performances of both LLS and SWLS estimators are degraded and the RMSE floors present. VI. C ONCLUSIONS Based on the sub-block processing, the frequency-domain joint estimation of the CFO and SFO for SC-FDE systems was proposed. Both the LLS and SWLS regressions were considered, where the latter was shown to be robust to the channel frequency-selectivity. For the AWGN channel, the LLS estimator was proven to be unbiased and the MSEs. Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 00:51 from IEEE Xplore. Restrictions apply..

(5) CHIANG et al.: JOINT ESTIMATION OF CARRIER-FREQUENCY AND SAMPLING-FREQUENCY OFFSETS FOR SC-FDE SYSTEMS. were derived. Finally, the proposed receiver was tested in both quasi-static and time-varying multipath fading channels and its superiority has been demonstrated in the numerical results. R EFERENCES [1] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, no. 2, pp. 100–109, Feb. 1995. [2] D. Falconer, S. L. Ariyavisitakul, A. B. Seeyar, and B. Eidson, “Frequency domain equalization for single-carrier broadband wireless systems,” IEEE Commun. Mag., vol. 40, no. 4, pp. 58–66, Apr. 2002. [3] Y. Li and G. L. St¨uber, Orthogonal Frequency Division Multiplexing for Wireless Communications. New York: Springer, 2006. [4] Air Interface for Fixed Broadband Wireless Access Systems (IEEE Std 802.16-2004), June 2004. [5] H. Yu, M. S. Kim, and J. Y. Ahn, “Carrier frequency and timing offset tracking scheme for SC-FDE systems,” in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications, pp. 1–5, Sept. 2003.. 1235. [6] K. Shi, E. Serpedin, and P. Ciblat, “Decision-directed fine synchronization in OFDM systems,” IEEE Trans. Commun., vol. 53, no. 3, pp. 408–412, Mar. 2005. [7] I. H. Hwang, H. S. Lee, and K. W. Kang, “Frequency and timing period offset estimation technique for OFDM systems,” IEE Electron. Lett., vol. 34, no. 6, pp. 520–521, Mar. 1998. [8] P. Y. Tsai, H. Y. Kang, and T. D. Chiueh, “Joint weighted least-squares estimation of carrier-frequency offset and timing offset for OFDM systems over multipath fading channels,” IEEE Trans. Veh. Technol., vol. 54, no. 1, pp. 211–223, Jan. 2005. [9] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum, 1997. [10] P. H. Chiang, G. L. St¨uber, D. B. Lin, and H. J. Li, “Pilot-aided fine synchronization for SC-FDE systems on multipath fading channels,” in Proc. IEEE Int. Conf. on Communications, pp. 2853–2858, June 2007. [11] Channel Models for Fixed Wireless Applications (IEEE 802.16a-03/01), June 2003.. Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 00:51 from IEEE Xplore. Restrictions apply..

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