WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 Published online 21 August 2009 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/wcm.833 Uplink carrier frequency offset estimation for WiMAX OFDMA-based ranging Yonghua Lin1∗,† , Da Fan1,2 , Qing Wang1 and Jianwen Chen1 1 IBM China Research Lab, Beijing 100094, China 2 Department of Electronic Engineering, Tsinghua University, Beijing 100084, China Summary Ranging is one of the most important processes in the mobile worldwide interoperability for microwave access (WiMAX) standard, for resolving the uplink synchronization and near/far problems. In this paper, we focus on the multi-user carrier frequency offset (CFO) estimation in both initial ranging and periodic ranging. After the analysis of some existing ranging methods, we propose two algorithms based on the correlation properties of pseudo noise (PN) sequences in time domain and frequency domain respectively. The root mean square error (RMSE) performance is evaluated in both additive white Gaussian noise (AWGN) channel and multi-path fading channel. Simulation results show that the proposed frequency-domain cross-correlation method performs better than the proposed time-domain cross-correlation method, and is more robust to multi-user interference and residual timing offset. Copyright © 2009 John Wiley & Sons, Ltd. KEY WORDS: Ranging; OFDMA; WiMAX 1. Introduction Broadband wireless access (BWA) systems have attracted much attention in industry for providing flexible and easy deploymenta solutions to wireless high-speed communications. A technology developed to fulfil these characteristics, standardized by IEEE, is 802.16. It is commonly referred to as worldwide interoperability for microwave access (WiMAX) [1]. Two versions of WiMAX have been defined, the first is based on IEEE 802.16-2004 and is optimized for fixed and nomadic access, the second version is designed to support portability and mobility, and will be based on the IEEE 802.16e amendment to the standard [2]. ∗ † One of the most interesting PHY modes supported by WiMAX standard is orthogonal frequency division multiple access (OFDMA) PHY mode. In OFDMA, subcarriers are grouped into subchannels which are assigned to multiple users for simultaneous transmissions. But OFDMA inherits the weakness of OFDM, being sensitive to symbol timing offset (STO) and carrier frequency offset (CFO). STO is induced by the round trip delay (RTD) between the subscriber station (SS) and the base station (BS) [3], which results in intersymbol interference (ISI). CFO is caused by Doppler effects and/or poor oscillator alignments. In such case, it will destroy the orthogonality among different subcarriers, resulting in intercarrier interferences (ICI) and Correspondence to: Yonghua Lin, IBM China Research Lab, Beijing 100094, China. E-mail: linyh@cn.ibm.com Copyright © 2009 John Wiley & Sons, Ltd. UPLINK CARRIER FREQUENCY OFFSET ESTIMATION multiuser interferences (MUI) [3]. Though each SS could establish initial synchronization with the BS by using the downlink preamble, the uplink signal arriving at the BS may be plagued by residual synchronization errors due to Doppler shifts and propagation delays [3]. If the users are not synchronized with the receiver, they will interfere with each other and the BS will not be able to recover individual signals of each user. In WiMAX, the issues of uplink synchronization and near/far problems are addressed by a process called ‘ranging’, and there are following kinds of ranging are defined:
Initial ranging (IR) for any ranging subscriber station


(RSS) that wants to synchronize to the system for the first time, handover (HO) ranging to support mobility and perform handoff from one access point (AP) to another, periodic ranging (PR) to update and track variations in STO and CFO, bandwidth request (BR) ranging to request access to the shared spectrum resource. In OFDMA-based ranging, RSS randomly chooses a ranging time-slot and a ranging code which is modulated by binary phase shift keying (BPSK) modulation. Then the ranging transmission shall be performed during one OFDM symbol (in periodic ranging) or two consecutive OFDM symbols (in initial ranging). More consecutive symbols can also be used for increasing the probability of code detection [2]. After separating colliding codes and extracting information about timing, frequency and power, the BS will broadcast the identified ranging codes with the needed adjustments information (e.g., timing, frequency, and power) and a status notification (e.g., success, continue, and abort). The ranging process at BS mainly includes STO and CFO estimation, and power estimation. This paper mainly considers the CFO estimation for initial ranging and periodic ranging processes in WiMAX OFDMA model. In 802.16d/e, it requires that a precision of 2% subcarrier spacing should be maintained. Frequency and timing recovery for single-user OFDM has been widely discussed in many literatures (e.g., References [3–6]). However, they cannot be used in the uplink of a multi-user system which needs to separate different users at the BS before their timing and frequency offsets are estimated respectively. Some synchronization methods for multi-user OFDM systems cannot separate different ranging users when multiple users collide in the same ranging channel (e.g., References [3,7–11]). Furthermore, these methods are only suitable for specific subcarrier-assignment Copyright © 2009 John Wiley & Sons, Ltd. 1445 schemes, such as subband based or interleaved, and cannot be used for WiMAX OFDMA uplink. Though several ranging methods for WiMAX OFDMA model have been analyzed (e.g., References [12–15]), only code detection, timing offset estimation were discussed except CFO estimation. In Reference [16], the frequency offset can be acquired by correlating FFT output samples of two consecutive OFDMA symbols at ranging subcarriers. However, this method will be invalid in the case of multiple ranging users as it cannot distinguish between them. Other ranging methods were proposed in the literatures (i.e., References [17–19]). However, their signal models are not consistent with the current WiMAX standard. Hence, they are not considered in this work. In this paper, we propose two CFO estimation methods for WiMAX OFDMA-based initial ranging and periodic ranging. The proposed algorithms are evaluated using theoretical analysis and computer simulations over additive white Gaussian noise (AWGN) and time-dispersive channel with multi-user interference. Simulation results show that the proposed frequency-domain cross-correlation method performs better than the proposed time-domain cross-correlation method even in the case of multiple users simultaneously existing in one ranging time-slot. It is more robust to residual timing offset resulting from the STO estimation error. The remainder of the paper is organized as follows. Section 2 introduces the signal model. The proposed CFO estimation methods for initial ranging and periodic ranging are discussed in Section 3 and 4, respectively. The effect from residual timing offset is analyzed in Section 5. Performance analysis is provided in Section 6, and conclusions are finally drawn in Section 7. 2. Signal Model Our system model is mainly based on the IEEE 802.16e standard [2]. We consider an OFDMA uplink system with N subcarriers. After assigning direct current (DC) and guard subcarriers, the remaining subcarriers, Nd , are grouped into 0 subchannels. Each subchannel has P = Nd /Q subcarriers. The subcarriers assigned to each subchannel are not necessarily adjacent as they are chosen randomly. Each user in uplink is assigned to one or more subchannels. For OFDMA-based ranging, one ranging time-slot is N2 (6 or 8) subchannels by N1 (1, 2, 3, or 4) OFDMA symbols as shown in Figure 1. In WiMAX ranging, IR and HO use the Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm 1446 Y. LIN ET AL. (u) (u) (u) [XR,m (0), XR,m (1), · · · , XR,m (N − 1)]T , according to (u) XR,m (k) Fig. 1. Ranging slot allocation. =
Sm (r), 0, if k = ind(r) otherwise (1) where ind(r) is the index of the rth ranging subcarrier. The length of the cyclic prefix (CP) is equivalent to Ng samples, and assumed to be longer than the maximum channel delay spread. After N-point inverse fast Fourier transform (IFFT) and CP insertion at the transmitter, the nth element of the time-domain ranging signal of the uth ranging user is given by for initial ranging  N−1  (u) 1    XR,0 (k)ej2πk(n−Ng )/N  N   k=0    n = 0, · · · , N − 1 s (u) xR (n) = N−1    (u) 1  XR,1 (k)ej2πk(n−Ns )/N  N   k=0    n = Ns , · · · , 2Ns − 1 (2) for periodic ranging (u) Fig. 2. Initial ranging and periodic ranging transmission. (a) Initial ranging transmission over two consecutive OFDMA symbols. (b) Periodic ranging transmission over three consecutive OFDMA symbols. same signal structure, and PR and BR use another signal structure. So in this paper, only IR and PR are discussed. N2 is set to 6. N1 is set to 2 for IR and 3 for PR, respectively. A ranging channel is composed of one or more groups of six adjacent subchannels. As shown in Figure 2(a), for initial ranging, same ranging code with length-R is modulated and transmitted in the ranging channel during one ranging time-slot composed of two OFDMA symbols. For periodic ranging, the RSSs modulate three consecutive ranging codes on the ranging subchannel for a period of three OFDMA symbols (one code per symbol) as shown in Figure 2(b). Moreover, a set of pseudo noise (PN) sequences are selected as ranging codes. In the ranging time-slot randomly selected by the uth RSS, the ranging code transmitted in the mth OFDM symbol is, firstly, modulated by BPSK modulation, which is denoted by Sm = [Sm (0), Sm (1), · · · , Sm (R − 1)]T , where m = 0, 1 for IR and m = 0, 1, 2 for PR. The signal is then mapped onto the N subcarriers (u) of an OFDM symbol in frequency domain, XR,m = Copyright © 2009 John Wiley & Sons, Ltd. xR (n) = N−1 1  (u) XR,m (k)ej2πk(n−mNs −Ng )/N , N k=0 n = mNs , · · · , (m + 1)Ns − 1 (3) where m = 0, 1, 2, and Ns = N + Ng . Note that the second symbol has a cyclic postfix instead of a CP in initial ranging as shown in Figure 2(a). Similarly, the data signal of the vth data subscriber station (DSS) is denoted by (v) xD (n) = N−1 1  (v) CD,m (k)ej2πk(n−mNs −Ng )/N N k=0 n = mNs , · · · , (m + 1)Ns − 1 (4) (v) where m = 0, 1 for IR and m = 0, 1, 2 for PR, CD,m (k) is the kth subcarrier signal transmitted by the vth DSS in the mth OFDM symbol. Because different SSs will have different location, the corresponding transmission delays (dR,u for uth RSS and dD,v for vth DSS) in units of OFDM samples are different. The maximum possible relative delay dR,max for the RSS is the roundtrip transmission delay at the cell boundary, which can be found from the knowledge of cell radius in practice. The maximum delay dD,max for DSSs is determined by the timing requirement of the ranging process. Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm UPLINK CARRIER FREQUENCY OFFSET ESTIMATION We consider a multipath Rayleigh fading channel with L sample-spaced taps. The channel impulse responses (CIRs) for uth RSS and vth DSS (denoted by (u) (v) hR (l) and hD (l), l = 0, · · · , L − 1, respectively) are assumed to be constant over one ranging time-slot and nonzero only for l = 0, · · · , L − 1. Note that Ng for RSSs should be designed such that Ng ≥ dR,max + L. So the channel output samples for uth RSS and those for vth DSS should be presented by (u) yR (n) = L−1  hR (l)xR (n − l − dR,u ) L−1  hD (l)xD (n − l − dD,v ) (u) (u) (5) (v) (v) (6) l=0 (v) yD (n) = l=0 NR and ND are used to denote the number of RSSs and DSSs accessing at the same time. Then the nth received signal sample at the BS can be expressed as y(n) = N R −1 u=0 (u) yR (n) + N D −1 1447 ing the phase in time domain and frequency domain, respectively. We assume that NR is less than the number of initial ranging codes Nc and the ranging code transmitted by all RSSs are different. 3.1. Cross-correlation in Time Domain To detect the ranging codes and estimate the timing offsets, one approach would to be cross-correlate the received signal with all possible ranging codes in time domain [12]. However, the CFO estimation scheme was not addressed. In this section, a multi-user CFO estimation algorithm is proposed based on the results of code detection and STO estimation in Reference [12]. It uses the phase rotation of the cross-correlation between the received signal and the time-domain reference ranging signal. The cross-correlation is defined by Ri (d) = N s −1 (i) xR (n)y∗ (d + n) n=0 (v) yD (n) + z(n) = Si + Ii + ID + Nz (7) (8) v=0 where {z(n)} are independent and identically distributed (i.i.d.), circularly symmetric complex Gaussian noise samples with zero mean and variance σz2 . In the following analysis, we will use the similar method in References [5,15] to analyze the proposed algorithms. Firstly, the channel is assumed to be nondispersive, and the received signals from different users will only be affected by AWGN. Then, we will evaluate the performance of the proposed algorithms under AWGN and time-dispersive channel, respectively. IR has different symbol structure from PR, so we will first introduce the CFO estimation methods for IR, and then discuss their effectiveness for PR. where Si = N s −1 Ii = N R −1  (i) (u) xR (n) yR (d + n) ∗ u=0 u=i N s −1 N D −1 N s −1  (i) (v) xR (n) yD (n + d) ∗ n=0 ID = v=0 Nz = N s −1  (i) (i) xR (n) yR (d + n) n=0 n=0 (i) xR (n)z∗ (n + d) ∗ (9) n=0 3. CFO Estimation for OFDMA-based Initial Ranging In this section, two CFO estimation methods are proposed for WiMAX OFDMA-based initial ranging. The correlation properties of PN codes are adopted for extracting the desired ranging user and suppressing the interference signals of the other ranging users and data users. Frequency offset will result in phase rotation between two adjacent received samples and at the same subcarrier between two successive repeated symbols. CFO could be estimated by comparCopyright © 2009 John Wiley & Sons, Ltd. In the above equation, Ns = N + N g , (i) d = 1, 2, · · · , dmax,R , xR (n) is the ith time-domain reference ranging signal, i = 1, 2, · · · , Nc , and (·)∗ represents conjugate. From Equation (9), Si is the expected signal term, Ii is the interference term introduced by other ranging codes, ID is the interference term introduced by DSSs’ signals, and Nz is the Gaussian noise term. According to the correlation property of PN codes, i.e., the cross-correlation of different ranging codes is quite small compared with auto-correlation, Ii is very low and approaching zero. From Equations (2)–(4), since the random variables Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm 1448 Y. LIN ET AL. (i) (v) xR (n) and yD (n) are given by the sum of N i.i.d. ran(i) (v) dom variables with zero mean, both xR (n) and yD (n) asymptotically become complex Gaussian with zero mean for large N by the central limit theorem [20–22], (i) (v) and the uncorrelated samples xR (n) and yD (n) become independent Gaussian random variables, due to the fact that uncorrelated Gaussian random variables are statistically independent. Moreover, according to the law of large numbers the mean of N i.i.d. random variables with the same mean will tend to approach to the expected value for large N. As a result, we have the following approximation: ID = Ns Ns −1  1  (i) (v) xR (n) yD (n + d) Ns N D −1 v=0 = Ns v=0  φn = φ(0, n) − φ ξ (i) , d̂R,i + n  = −φ ξ (i) , d̂R,i + n ∗ n=0 N D −1   (i) E xR  (v) · E ∗ yD where f (i) is the CFO between the ith ranging user and the uplink receiver. f is the subcarrier spacing, and ξ (i) is defined as the normalized CFO (NCFO) of the ith ranging user. (i) The phase rotation difference between xR (n) and (i) yR (n) is a function of the normalized CFO and their time delay. After the correction of timing offset estimated using method in Reference [12], we assume there are no other significant phase distortion effects. For the ith ranging user, the phase rotation difference between (i) (i) xR (n) and yR (d̂R,i + n) can be written as (15) (10) ≈0 Ns −1 1  (i) xR (n)z∗ (n + d) Ns n=0   (i) = Ns · E xR · E∗ (z) N z = Ns · (11) ≈0 where E(·) represents expectation. Then Ri (d) can be rewritten as where d̂R,i is the estimated transmission delay (timing offset) of the ith RSS, n = 0, 1, · · · , Ns − 1. The phase rotation difference can be used to determine the normalized CFO ξ (i) . The phase of Ri (d̂R,i ) (Equation (8)) represents the sum of all phase rotation difference between the (i) time-domain reference ranging signal xR (n) and the received samples y(d̂R,i + n). We have ∠Ri (d̂R,i ) = ∠ N s −1 xR (n)y∗ (d̂R,i + n) N s −1  (i) (i) xR (n) yR (d̂R,i + n) (i) n=0 Ri (d) = N s −1 (i) xR (n)y∗ (d + n) ≈∠ n=0 ≈ N s −1 n=0 n=0 (i)  ∗ (i) xR (n) yR (d + n) (12) (i) (i) N s −1 (i) 2 xR (n) exp(φn ) n=0 For simplicity, we assume that the ranging users are under AWGN channel. The received signal of the ith ranging user can be rewritten as yR (n) = xR (n − dR,i )ej2πξ =∠ ∗ (i) n/N (13) =∠ N s −1   exp −φ ξ (i) , d̂R,i + n =∠ N s −1 exp n=0 n=0 From Equation (13), the carrier frequency offset of the ith ranging user, f (i) will result in a phase rotation φ(f (i) , n) = ∠ exp −j2π(d̂R,i + n)ξ (i) N −jπηξ (i) N  sin(πNs ξ (i) /N) · sin(πξ (i) /N)  (16)  φ f (i) , n = 2πf (i) n/Nf = 2πξ (i) n/N,  φ ξ (i) , n = φ(f (i) , n) (14) Copyright © 2009 John Wiley & Sons, Ltd. where η = 2d̂R,i + Ns − 1. Then the normalized CFO of the i-th ranging user, ξ (i) can be estimated by the Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm UPLINK CARRIER FREQUENCY OFFSET ESTIMATION 1449 Fig. 3. Block diagram of the proposed frequency offset estimator for initial ranging. following equation: ξ̂ (i) = − can be rewritten as ∠Ri (d̂R,i ) · N η·π (17) N R −1 y(n + Ns ) = (u) yR (n + Ng )ej2πξ (u) u=0 + 3.2. Cross-correlation in Frequency Domain (u) (u) (u) (18) where n = 0, · · · , N − 1, ξ (u) is the normalized CFO of the uth ranging user. Then the received signal at BS Copyright © 2009 John Wiley & Sons, Ltd. (v) yD (n + Ns ) + z(n + Ns ) v=0 Though several ranging methods based on crosscorrelation in frequency domain have been analyzed (e.g., [13–15]), only code detection, timing offset estimation were discussed except CFO estimation. In Reference [16], the frequency offset can be acquired by correlating FFT output samples of two consecutive OFDMA symbols at ranging subcarriers. However, this method will be invalid in the case of multiple ranging users as it cannot distinguish between them. In this section, we propose a novel CFO estimation algorithm as shown in Figure 3. The STOs of all ranging users are first estimated by using the frequency-domain cross-correlation method [16]. Then, the estimates of all STOs are fed to CFO estimator to estimate the CFOs of all ranging users. At last, the CFO of each ranging user is estimated through multiplying the complex conjugate of the sum of the first FFT window by the sum of the second FFT window. In initial ranging, the second ranging symbol is created by repeating the first ranging symbol as shown in Figure 2(a). It is assumed that the channel is constant during the transmission of the two OFDMA symbols, which corresponds to ‘slowfading’ in the radio frequency channel. Without taking into account the RSS’s transmission delay, the received ranging signal has the following characteristics: yR (n + Ns ) = yR (n + Ng )ej2πξ N D −1 (19) After the CP removal and FFT processing, the received signal y(n) in the ranging time-slot is converted into frequency-domain signal Ym (k), m = 0, 1. The first OFDMA symbol will is given by   Y0 (k) = FFTN y(n + Ng ) = N R −1 u=0 +  (u) FFTN yR (n + Ng ) N D −1 v=0  (v) FFTN yD (n + Ng )   + FFTN z(n + Ng ) = N R −1 (u) YR,0 (k) + u=0 N D −1 (v) YD,0 (k) + Z0 (k) v=0 (20) Then the second OFDMA symbol is given as follows: Y1 (k) = FFTN (y(n + Ns )) = N R −1 u=0  (u) (u) ej2πξ FFTN yR (n + Ng ) Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm 1450 Y. LIN ET AL. + N D −1 v=0 Y1′ (k) = e  (v) FFTN yD (n + Ns ) = e j2πξ (u) (u) YR,0 (k) + u=0 N D −1 (i) ×e (21) where n  = 0, · · · , N − 1, k = 0, 1, · · · , N − 1, and −j2πnk/N for any function f (n). FFTN ≡ N−1 n=0 f (n)e (u) YR,m (k) is the received frequency-domain ranging signal in the mth OFDMA symbol from the uth ranging (v) user, and YD,m (k) denotes the received frequencydomain signal in the mth OFDMA symbol from the vth data user, and m = 0, 1. Assume dR,u is the timing offset between the received signal from uth user and uplink receiver, and Ỹ is the signal without timing offset to local receiver. According to a timing offset in time domain corresponds to phase offset in frequency domain, Equations (20) and (21) could be further described with the consideration of timing offset as Y0 (k) = e N D −1 Y1 (k) = (u) ỸR,0 (k) (v) YD,0 (k) + Z0 (k) (u) (u) (22) −j2πkdR,u N (u) ỸR,0 (k) j2πkd̂R,i N (v) YD,1 (k) + e j2πkd̂R,i N Z1 (k) (25) where d̂R,i is the estimated timing offset of the ith RSS using the frequency-domain correlation method in Reference [16]. To extract the signal of the ith user in frequency domain, we use the local reference ranging (i) (i) (i) (i) code XR = [XR (0), XR (1), · · · , XR (N − 1)]T for (i) (i) frequency-domain correlation (because XR,0 = XR,1 , (i) let XR represent the ith RSS’s ranging signal in frequency domain). The results of the correlation have the following vector expression: T  (i) = ỸR,0 + ej2πξ e ej2πξ v=0 v=0 N R −1 e T (i)  E(d̂R,i )Y0 XR u=0 + j2πk(d̂R,i −dR,u ) N N D −1 + −j2πkdR,u N N R −1 (i) u=0 u=i (v) YD,1 (k) + Z1 (k) v=0 N R −1 Y1 (k) = ej2πξ ỸR,0 (k) + + FFTN (z(n + Ns )) N R −1 j2πkd̂R,i N (u) (i) XR + N R −1  u=0 u=i N D −1  (v) E(d̂R,i )YD,0 v=0 ỸR,0 (k) (u) E(d̂R,i − dR,u )ỸR,0 T N D −1 (v) YD,1 (k) + Z1 (k) (26) (23) v=0 To elimate the impact of timing offset on the i-th user, we will process Equations (22) and (23) as Y0′ (k) = e = j2πkd̂R,i N (i) ỸR,0 (k) + + N D −1 e j2πkd̂R,i N N R −1 ej2πξ u=0 u=i e j2πk(d̂R,i −dR,u ) N + (u) ỸR,0 (k) N D −1  v=0 u=0 u=i +  T (i) E(d̂R,i )Y1 XR  (i) (i) = ej2πξ ỸR,0 Y0 (k) N R −1 (i) XR T (i)  (i) XR + E(d̂R,i )Z0 XR u=0 + T (u)  T (i) XR (u) E(d̂R,i − dR,u )ỸR,0 (v) E(d̂R,i )YD,1 T T (i) XR  T (i) (i) XR + E(d̂R,i )Z1 XR (27) (v) YD,0 (k) + e j2πkd̂R,i N Z0 (k) v=0 (24) Copyright © 2009 John Wiley & Sons, Ltd. j2πd̂R,i j2π(N−1)d̂R,i N ) where E(d̂R,i ) = diag(1, e N , · · · , e is a diagonal matrix to do timing correction for the ith RSS in frequency domain, Ym = [Ym (0), Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm UPLINK CARRIER FREQUENCY OFFSET ESTIMATION (u) 1451 Ym (1), · · · , Ym (N − 1)]T , ỸR,m = [ỸR,m (0), ỸR,m (1), 4.1. (u) (v) (v) (v) · · · , ỸR,m (N − 1)]T , YD,m = [YD,m (0), YD,m (1), · · · , (v) YD,m (N − 1)]T , Zm = [Zm (0), Zm (1), · · · , Zm (N − 1)]T , and m = 0, 1. Because the proposed cross-correlation method in time domain requires only one OFDM symbol, it could be used for both initial ranging and periodic ranging without any adjustments. (u) (u) In Equations (26) and (27), by considering the crosscorrelation property of PN codes in the second term, by (i) noting that XR (k) = 0 at data subcarriers in the third term, and also by ignoring the impact of Gaussian noise with zero mean in the fourth term, we can therefore remove the last three terms on the right hand side of the two equations. Then, the estimated ξ (i) could be derived by ξ̂ (i) = arg  T (i) T (i)  E(d̂R,i )Y1 XR / E(d̂R,i )Y0 XR 4.2. Cross-correlation in Time Domain Cross-correlation in Frequency Domain In periodic ranging, not only the ranging codes transmitted in successive OFDM symbols are different (i) (i) (i) (XR,0 = XR,1 = XR,2 ), but also CP is adopted rather than cyclic postfix. After the CP removal and FFT processing of the received samples y(n), the received frequency-domain signal is as follows: 2π N+Ng −1 (28) N R −1  Ym (k) = n=Ng Two CFO estimation methods for OFDMA-based initial ranging have been analyzed in this session. To sum up, the cross-correlation process in frequency domain could provide two advantages over the crosscorrelation process in time domain. One is that, there will be no interference from synchronized DSS’s signal to the ranging channel, because they are orthogonal in frequency domain. The other one is, this method makes better use of the auto-correlation/crosscorrelation properties of PN codes which are originally sent over frequency domain [15]. 4. Algorithm Analysis for Periodic Ranging In periodic ranging, the SS can send a transmission in one of the following ways [2]: (a) Modulating one ranging code on the ranging subchannel for a period of one OFDMA symbol. (b) Modulating three consecutive ranging codes on the ranging subchannel for a period of three OFDMA symbols (one code per symbol). To perform the frequency offset estimation in frequency domain, we need more than one consecutive symbols in algorithms. So the system will select (b) for transmission, as shown in Figure 2(b). The algorithm analysis in Section 3 is based on the symbol structure of initial ranging. It is necessary to verify its effectiveness in periodic ranging with different symbol structure. Copyright © 2009 John Wiley & Sons, Ltd. + (u) yR (n + mNs ) u=0 N D −1 (v) yD (n + mNs ) + z(n + mNs ) v=0 × e−j2πnk/N = N R −1 (u) YR,m (k) + u=0 N D −1 (v) YD,m (k) + Zm (k) v=0 (29) (u) (v) where m = 0, 1, 2, YR,m (k) and YD,m (k) are the mth frequency-domain OFDM signal of the uth RSS and the vth DSS, respectively. Considering the timing offset, Equation (29) could be further described as Ym (k) = N R −1 e −j2πkdR,u N (u) ỸR,m (k) u=0 + N D −1 e −j2πkdD,v N (v) ỸD,m (k) + Zm (k) (30) v=0 where Ỹ is the signal without timing offset to local receiver. From Equation (30), the timing-offset correction of the ith ranging user is given by Ym′ (k) = e = j2πkd̂R,i N Ym (k) (i) ỸR,m (k) + N R −1 e j2πk(d̂R,i −dR,u ) N (u) ỸR,m (k) u=0 u=i Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm 1452 Y. LIN ET AL. + N D −1 e j2πk(d̂R,i −dD,v ) N (v) ỸD,m (k) + e j2πkd̂R,i N + ejθ0 e Zm (k) j2π(mNs +Ng )ξ (i) N N−1  (i) (i) (l)cl−k , (l)HR,m XR,m l=0 l=k v=0 (31) To extract the signal of the ith user in frequency domain, we use the local reference rang(i) ing code XR,m for frequency-domain correla(i) (i) (i) (i) tion. XR,m = [XR,m (0) XR,m (1) · · · XR,m (N − 1)]T (i) and XR,m (k) = 0 at data subcarriers. The vector expression of frequency-domain correlation for periodic ranging is given by (i) [E(d̂R,i )Ym ]T XR,m  T (i) (i) = ỸR,m XR,m + N R −1  E u=0 u=i + N D −1  v=0  N−1  (i) XR,m (i) 1 1 − ej2πξ = j2πξ (i) N 1−e N  (v) T (i) E(d̂R,i − dD,v )ỸD,m XR,m =e (32) (i) (n)e−j2πkn/N ỹR,m n=0 = N−1  n=0  × ejθ0 e = N−1 j2πnl 1  (i) (i) (l)e N X (l)HR,m N l=0 R,m  j2π(n+mNs +Ng )ξ (i) N e −j2πnk N N−1 1 jθ0 j2π(mNs +Ng )ξ(i)  (i) (i) N (l) XR,m (l)HR,m e e N l=0 × N−1  e j2πn(l−k+ξ (i) ) N n=0 = N−1 1 jθ0 j2π(mNs +Ng )ξ(i)  (i) (i) N (l)cl−k XR,m (l)HR,m e e N l=0 = ejθ0 e j2π(mNs +Ng )ξ (i) N N−1 1  j2πnξ(i) e N N n=0  (u) T d̂R,i − dR,u ỸR,m  ỸR,m is the signal without timing offset to local receiver. Following Reference [23], its element could be written as (i) (k) = ỸR,m where θ0 is the phase rotation between the phase of the receiver local oscillator and the carrier phase at the start of the received signal, ξ (i) is the ith rang(i) ing user’s normalized CFO, and HR,m (l) denotes the channel frequency response on the lth subcarrier of the ith ranging user during the qth OFDMA block. The complex weighting coefficients, c0 and cl−k are given by c0 =  T (i)   + E d̂R,i Zm XR,m (i) (33) (i) (i) (k)c0 (k)HR,m XR,m Copyright © 2009 John Wiley & Sons, Ltd. cl−k = jπ(N−1)ξ (i) N   sin πξ (i)  (i) N · sin πξN N−1 1  j2πn(l−k+ξ(i) ) N e N n=0 (i) 1 1 − ej2π(l−k+ξ ) j2π(l−k+ξ (i) ) N N 1−e   (i) (i) jπ(N−1)(l−k+ξ ) sin π(l − k + ξ ) N  =e (i) ) N · sin π(l−k+ξ N = (34) Note that the coefficients have the following periodical property, cl−k = cN+l−k . The first term in Equation (33) represents an attenuated and rotated version of the wanted ranging signal. The second term corresponds to the inter-carrier interference (ICI) resulting from carrier frequency offset. From Equation (34), the frequency shifting will result in a change in amplitude and phase of the wanted ranging signal given by c0 , which depends on the normalized frequency offset ξ (i) but is independent of k. In other words, all subcarriers experience the same degree of attenuation and rotation of the wanted ranging signal. Statistically, the ICI term in Equation (33) can be modeled as Gaussian noise since it is the result of an addition of N − 1 random variables (Central Limit Theorem), where N is generally large [20,21,24]. So Equation (33) Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm UPLINK CARRIER FREQUENCY OFFSET ESTIMATION 1453 can be rewritten as
In CFO estimation for periodic ranging, different N−1  ranging codes for the two consecutive symbols will be used when correlation in frequency domain. In CFO estimation for periodic ranging, to calculate the normalized CFO ξ (i) , the factor 1/2π(1 + g) should be used instead of 1/2π. (i) ỸR,m (k) = (i) ỹR,m (n)e−j2πkn/N
n=0 = ejθ0 e j2π(mNs +Ng )ξ (i) N (i) (i) XR,m (k)HR,m (k)c0 (i) + VR,m (k) (35) (i) where VR,q (k) is the additional noise modeled by the ICI disturbance on the kth subcarrier. Similar to the analysis of Equations (26) and (27), we can ignore the effect of the other ranging users, data users and noise, so the CFO of the ith ranging user could be estimated by 1 2π(1 + g)      T (i) (i) ỸR,0  E d̂R,i Y1 XR,1 · arg     T (i) ·  (i) E d̂R,i Y0 XR,0 ỸR,1 ξ̂ (i) = So, this proposed CFO estimation method with crosscorrelation in frequency domain could be used in both initial ranging and periodic ranging scenario. 5. Effect of Residual Timing offset After the imprecise timing correction due to the STO estimation error, the residual timing offset will cause an impact on CFO estimation. Here, we will provide a rough estimation of this impact. T  (i) XR,0  T (i)  XR,1 1 2π(1 + g)   N−1  (i)     HR,0 (k)   E d̂R,i Y1 T X(i)   k=0 R,1 · arg    ·    E d̂R,i  Y0 T X(i) N−1  (i) R,0 HR,1 (k) = 5.1. Cross-correlation in Time Domain From Equation (13), after the timing correction with STO estimation error, the received signal of the ith ranging user is given by (i) (i) yR (n + d̂R,i ) = xR (n + d̂R,i − dR,i )ej2πξ (i) = xR (n + dR,i )ej2πξ (i) (n+d̂ )/N R,i (i) (n+d +d )/N R,i R,i (38) k=0 (36) where g = Ng /N. We assume the channel is flat. Moreover, the neighboring OFDMA symbols have the same channel frequency response on the same subcar(i) (i) riers, i.e., HR,0 (k) = HR,1 (k), k = 0, 1, · · · , N − 1. Then Equation (36) can be rewritten as ξ̂ (i)    T (i)  E d̂R,i Y1 XR,1 1 · arg    =  T (i)  2π(1 + g) E d̂R,i Y0 X R,0 (37) Comparing the method for initial ranging in Equation (28) and that for periodic ranging in Equation (37), there will be only minor differences in processing.
For periodic ranging channel, the CP between the first and second OFDM symbol should be removed. Copyright © 2009 John Wiley & Sons, Ltd. where dR,i = d̂R,i − dR,i is the STO estimation error. Comparing Equations (38) and (13), when dR,i = 0, the STO estimation error leads to the received signal (i) out of synchronization with the reference signal xR (n), which impairs the ranging code’s orthogonality in time domain. On the other hand, the residual timing offset introduces a new phase rotation 2πξ (i) dR,i /N. Therefore, this method will cause some performance degradation. Figure 4 shows the auto-correlation of the timedomain ranging signal versus residual STO values for STO = 10 and NCFO = 0.2. It can be seen that the auto-correlation performance will degrade sharply with only one sample’s timing error. 5.2. Cross-correlation in Frequency Domain With perfect timing correction, the absolute values of the first items in Equations (26), (27), and (32) are Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm 1454 Y. LIN ET AL. Fig. 4. Auto-correlation performance of the time-domain ranging signal with residual STO values. Fig. 5. Normalized results of frequency-domain correlation with residual STO values. denoted by residual STO values for STO = 10 and NCFO = 0.2. It can be seen that the result of frequency-domain correlation will decrease quickly when the residual STO is larger than 4. P1 =  (i) ỸR,0 T (i) XR,0 (39) But if there is residual timing offset caused by STO estimation error, P1 will become P1′ = =    (i) T (i) E d̂R,i − dR,i ỸR,0 XR,0  (i) E(dR,i )ỸR,0 T (i) XR,0 6.1. Simulation Parameters (40) For simplicity, we assume that the channel is flat and has a constant gain of one. Subcarriers of the ranging channel is contiguous starting from 0, i.e., k = 0, 1, · · · , R − 1, where R is the length of ranging codes. Using Equation (35), the ratio of P1′ to P1 is P1′ P1 c0 · = R−1 k=0 e R−1 n=0 = j2πkdR,i N c0 | sin(πRdR,i /N)| |R · sin(πdR,i /N)| (41) Equation (41) could be explained as, with the residual timing offset, the signal of the ith ranging user will be decreased to P1′ /P1 of the original signal. For example, when N = 1024, R = 144 and dR,i is 5, P1′ /P1 = 0.3636. So, timing offset correction is very important in CFO estimation. Figure 5 shows normalized results of frequency-domain correlation with Copyright © 2009 John Wiley & Sons, Ltd. 6. Simulation Results In the simulation, the OFDMA system parameters are selected from Reference [2]. The uplink bandwidth is 3 MHz, the subcarrier frequency spacing f is 3.28 kHz, N = 1024. We use QPSK format for DSS. Within the 1024 subcarriers, there are 92 guard subcarriers on the left-side band and 91 on the right, and 1 DC subcarrier residing on index 512. The remaining 840 subcarriers are partitioned into Q = 35 subchannels. The ranging channel is composed of six adjacent subchannels and spanning 144 subcarriers per OFDMA symbol. The performances of proposed algorithms are evaluated in both AWGN channel and time-dispersive channel. For the time-dispersive channel, we use one of the Stanford University Interim (SUI) channel models defined by IEEE 802.16 working group for assessing technologies for broadband fixed wireless applications [25]. SUI-3 channel model with three paths is considered in our simulation. The number of sample-spaced channel taps, L, is set to 4. Channels of different users are generated independently. We consider a cell radius of 5 km which gives the maximum transmission delay (round trip) dR,max ≈ 34 ␮s = 114 samples. Ng is set to 128 samples satisfiying the condition dR,max < Ng − L. The timing requirement based on Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm UPLINK CARRIER FREQUENCY OFFSET ESTIMATION Reference [2] is that all uplink OFDM symbols should arrive at the BS within an accuracy of ±25% of the minimum guard-interval or better. In Reference [2], Ng can be 1/4, 1/8, 1/16, or 1/32 of N, and hence, the timing offset should be within ±8 samples. So the dD,max equals to eight samples in our case. Similarly, the frequency offset should be within a tolerance of maximum 2% of the subcarrier spacing. We assume that Nc = 128 ranging codes are assigned for initial ranging and the maximum number of RSSs in one ranging time-slot, NR , is set to 15. Because six subchannels are allocated to ranging subchannel, the maximum number of DSSs accessed simultaneously, ND , is 29. To quantify the performance of the CFO estimation, the normalized root mean square error (NRMSE) of the estimates is used, and it is defined as ENRMSE   NR      1  (u) (u) 2 = ξ̂ρ − ξρ NR 1455 Fig. 6. The RMSE performance of the CFO estimation versus SNR over AWGN channel. The number of RSSs is NR = 1, 2, 5, 15, respectively. (42) ρ=1 u=1 (u) where  is the total number of Monte Carlo tests, ξρ is (u) the normalized CFO of the uth ranging user, and ξ̂ρ is (u) the estimate of ξρ . The subscript (·)ρ denotes the index of the Monte Carlo test. The NRMSE is computed after averaging over all participating ranging users for 10 000 independent Monte Carlo tests. In each simulation run, the timing offset for RSSs and DSSs are generated randomly from the interval [0, dR,max ] and [0, dD,max ], respectively. The normalized CFOs of RSSs and DSSs uniformly distribute in the range of [−0.2, 0.2] and [−0.02, 0.02]. In order to evaluate the effect of data users on the performance of proposed algorithms, we consider the following conditions of 0 DSS, 15 DSSs, and 29 DSSs in one ranging time-slot. improves as SNR increases. But both methods’ performances degrade as the number of RSSs increases. The powers of the expected ranging signal and interferences increase as the SNR increases, which resulting in no obvious performance improvements of the two methods. The NRMSE performances of the two methods are still less than 2% especially in multiple RSSs condition, which satisfies the timing requirement defined in Reference [1]. Figure 7 depicts the NRMSE performance of the frequency offset estimation as a function of SNR in ‘SUI3’ time-dispersive channel. The condition is the same as that in Figure 6. From Figure 7, the time-domain 6.2. Performance of Frequency Offset Estimation Figure 6 shows the NRMSE performance of the frequency offset estimation as a function of SNR in AWGN channel. We consider the condition of precise timing and 0 DSS in one ranging time-slot. The number of RSSs is NR = 1, 2, 5, 15,respectively. In fairness, the SNRs of all RSSs are the same. From Figure 6, The proposed frequency-domain cross-correlation (CC) method performs better than the time-domain cross-correlation method. When the number of RSSs is NR = 1, the proposed two methods have the similar performances and the NRMSE of each method Copyright © 2009 John Wiley & Sons, Ltd. Fig. 7. The RMSE performance of the CFO estimation versus SNR over ‘SUI-3’ time-dispersive channel. The number of RSSs is NR = 1, 2, 5, 15, respectively. Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm 1456 Y. LIN ET AL. Fig. 8. The RMSE performance of the CFO estimation versus number of RSSs over AWGN channel. SNR = 20 dB. cross-correlation method can not meet the system requirement and the performance of frequency-domain cross-correlation method also declines slightly. For the frequency-domain cross-correlation method, in order to satisfy the system requirement that the normalized CFO must be less than 2%, the number of RSSs in one ranging time-slot can not be more than 5. The following simulations will give the NRMSE performance curves as the number of RSSs changes under different channel models, we can obtain the exact number of RSSs supported by the two methods in one ranging time-slot. Figure 8 shows the NRMSE performance of the frequency offset estimation versus the number of RSSs over AWGN channel. The performances of proposed methods are evaluated in the absence/presence of residual timing offset. We assume that all SSs have the same SNR of 20 dB and the residual timing offsets are i.i.d. for different RSSs in the range of [−8, 8]. From Figure 8, it can be seen that the frequencydomain cross-correlation method is more robust to the DSSs’ interferences than the time-domain crosscorrelation method. When there is no residual timing offsets, the NRMSEs of the two CFO estimation methods are less than 2% even in multiple RSSs and DSSs condition, such as NR = 15 and ND = 29. But the method of frequency-domain cross-correlation has smaller NRMSE than that of time-domain crosscorrelation. The key reason is that the PN code is Copyright © 2009 John Wiley & Sons, Ltd. transmitted in frequency domain originally and thus its auto-correlation/cross-correlation properties will be impaired in time domain. It can also be seen from Figure 8 that the NRMSE performance of the time-domain cross-correlation method degrades, and cannot satisfy the system requirement when there are residual timing offsets induced by the STO estimation error. However, the proposed frequency-domain cross-correlation method still has a much better performance and is robust to the residual timing offset. For example, the NRMSE is less than 2% when the number of RSSs is less than 5 in one ranging time-slot. Figure 9 depicts the NRMSE performance of the frequency offset estimation versus the number of RSSs over ‘SUI-3’ time-dispersive channel. The effect from residual STO is also evaluated at SNR = 20 dB. The error floor in Figure 9 clearly shows the performance degradation in the time-dispersive channel as compared with the corresponding curves in AWGN channel. Moreover, the time-domain cross-correlation method can not meet the system requirement. It is mainly because the derived result for the timedomain cross-correlation method is only suitable for a flat fading channel, it cannot directly apply to the time-dispersive channel environment. Without the need of this assumption, the frequency-domain crosscorrelation method gives a much better performance Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm UPLINK CARRIER FREQUENCY OFFSET ESTIMATION 1457 Fig. 9. The RMSE performance of the CFO estimation versus number of RSSs over ‘SUI-3’ time-dispersive channel. SNR = 20 dB. under the time-dispersive channel. For the frequencydomain cross-correlation method, in order to satisfy the system requirement that the normalized CFO must be less than 2%, the number of RSSs in one ranging time-slot can not be more than 8 in the absence of residual STO while 2 in the presence of residual STO, respectively. Since the probability of 3 or more RSSs choosing the same ranging slot is small [18], the proposed frequency-domain cross-correlation method is also suitable for time-dispersive channel. 7. Conclusion We have presented two multi-user CFO estimation algorithms for WiMAX OFDMA-based initial ranging and periodic ranging. The proposed algorithms are based on the correlation properties of the PN codes in time domain and frequency domain, respectively. The NRMSE performance is evaluated for both AWGN channel and time-dispersive channel. The simulation results show that the frequency domain cross-correlation method is more robust to the interferences from the other ranging users and data users and also performs better than the time-domain crosscorrelation method over multi-path fading channel and in the presence of residual timing offset. Hence, the proCopyright © 2009 John Wiley & Sons, Ltd. posed algorithms can be used in practical WiMAX BS receivers and other OFDMA-based wireless systems. References 1. WiMAX Forum. Fixed, nomadic, portable and mobile applications for 802.16-2004 and 802.16e WiMAX networks. White Paper, November 2005. 2. IEEE P802.16Rev2/D5. Part 16: Air Interface for Broadband Wireless Access Systems. June 2008. 3. Morelli M, Kuo C-CJ, Pun M-O. Synchronization techniques for orthogonal frequency division multiple access (OFDMA): a tutorial review. In Proceedings of the IEEE, vol. 95, no. 7, July 2007; 1394–1427. 4. Schmidl TM, Cox DC. Robust frequency and timing synchronization for OFDM. 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Authors’ Biographies Yonghua Lin is a Research Staff Member in System Software and Networking Department at the IBM China Research Lab. She received B.S. and M.S. degrees in Information and Communication from Xi’an Jiaotong University in 2000 and 2003, respectively. She subsequently joined IBM at China Research Lab, where she has worked on multiple Copyright © 2009 John Wiley & Sons, Ltd. projects related to multicore processors (general multicore processor and network processors) and networking (high-end router, IPTV media gateway, and mobile base station). She is leading an IBM Research group for appliance and infrastructure for wireless access network in next generation. She is an author or coauthor of 20 patents and 10 technical papers in related domain. Ms. Lin is a member of Institute of Electrical and Electronics Engineers and the Association for Computing Machinery. Da Fan received the B.S. and M.S. in Communication Engineering from PLA Information Engineering University, Zhengzhou, China, and Ph.D. in Electronic Engineering from Tsinghua University, Beijing, China, in 2000, 2003, and 2009, respectively. As an intern, currently he is working at IBM China Research Laboratory (CRL). His research interests include wireless communications and mobile networks, broadband multi-carrier/OFDM techniques, dynamic resource allocation and management, wireless MAN (WiMAX/802.16). He has published more than 20 papers on Communication and Signal Processing fields, and held two patents. Qing Wang, obtained Ph.D. from School of EEE , Nanyang Technological University, Singapore in 2003. Her research field is Statistical Signal Processing, especially for sinusoidal signal detection and estimation. She got M. E. and B. E both from Northwestern Polytechnical University, China in 1999 and 1992. The majors are Control Engineering and Electronic Engineering respectively. She joined IBM China Research Lab (CRL) in 2004. Before joined CRL, she was an Electronic Engineer in an institute for 4 years (1992–1996). In CRL, she works on wireless network cloud, WiMAX basestation design, wireless communication system on general IT platform, IMS system design, and IPTV system design. Her research interests include software defined radio, wireless broadband access, array signal processing, media accelerating on multicore system, and streaming data processing. She is an IEEE member from 2003 and a member of Communication Society. Jianwen Chen received the B.S. in information and electronics engineering from Huazhong University of Science and Technology, Wuhan, China in 2002. He received the Ph.D. in electronics engineering from Tsinghua University, Beijing, China in 2007. His research interests focused on video compression algorithms and video processing systems, wireless transmission algorithms and systems. He joined IBM China Research Center, Beijing, China in 2007. His current major focus is on wireless video algorithms and systems, programming model and parallel computation framework design and so on. Wirel. Commun. Mob. Comput. 2010; 10:1444–1458 DOI: 10.1002/wcm