2013 8th International Conference on Communications and Networking in China (CHINACOM)

Adaptive Joint Frame Synchronization and Carrier Frequency Offset Estimation in OFDM Systems

Dandan Zhang, Xu Zhang, Bin Zhong, Zhongshan Zhang* and Keping Long Institute of Advanced Network Technology and New Services (ANTS) and

Beijing Engineering and Technology Research Center for Convergence Networks and Ubiquitous Services University of Science and Technology Beijing (USTB), Beijing, China 100083

Email: zhangmo6881526@hotmail.com;zhangxuustb@yeah.net; zhongbin-1982@ 163 .com; { zhangzs,longkeping}@ustb.edu.cn

Abstract-An adaptive joint frame synchronization and carrier frequency offset estimation scheme in orthogonal frequency division multiplexing (OFDM) systems is proposed in this paper. Compared with conventional algorithm, the proposed one can reduce the Cramer-Rao lower bound (CRLB) to the minimum without changing the total energy consumption. In the proposed scheme, a variable parameter M is introduced in generating training sequence, where M is a function of the maximum multipath delay of the multipath channel. By adaptively adjusting M, the optimum training sequence can be obtained. Accurate frame synchronization and carrier frequency offset acquisition can be performed simultaneously in the proposed scheme. An adaptive tracking algorithm in the proposed scheme is also needed to estimate the remaining carrier frequency offset after acquisition. By using the proposed optimum training sequence, considerable performance improvement in the proposed tracking algorithm over Moose algorithm can be obtained.

I. INTRODUCTION

Orthogonal Frequency Division Multiplexing (OFDM) is a fundamental technique for high-speed optical communications, next-generation communication systems, the third-generation partnership project (3GPP) in the form of its long-term evo­lution (LTE) and many other communication systems [1]­[4]. OFDM system is widely used for its high utilization of spectrum and good performance of resistance to multipath fading to replace the complex and expensive adaptive equal­izer [5], [6]. With the rapid development of Digital Signal Processing (DSP), OFDM has attracted much more attention as a high-speed transmission technology which can eliminate intersymbol interference (lSI) caused by multipath to the greatest extent.

However, OFDM systems are very sensitive to carrier frequency offset, which is caused by Doppler Shift. Car­rier frequency offset will destroy the orthogonality among sub carriers and significantly degrade the performance of the OFDM receiver [7], [8]. Signal amplitude corresponding to each carrier in the output of the filters is cut down by carrier frequency offset. Also, carrier frequency offset introduces intercarrier interference (lCI) from the nonorthogonal carriers. The reduction of signal-to-noise (SNR) has been caused by these effects. Many estimation algorithms have been proposed for carrier frequency offset estimation in OFDM systems [9]­[15].

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Conventional algorithms transmit two or more identical patterns at the transmitter and utilize time-correlation property between the same structure. Carrier frequency offset can be estimated by estimating the phase rotations between these patterns at the receiver [3], [7], [8], [16]. Most of these are maximun likelihood frequency offset estimation algorithms based on synchronization symbols [17], [18]. In fact, the sample sequence arrangement in each pattern in a training symbol sequence needn't to be identical. That is to say, the arrangement in the patterns of training symbols other than the first one can be arbitrary. The carrier frequency offset can also be estimated based on this non-identical-pattern training symbols, provided that the pattern of the training symbol/sequence is known by the receiver.

In this paper, an optimized sample arrangement scheme in a training sequence is proposed. Based on the proposed training sequence, high accuracy carrier frequency offset estimators can be derived in both the Additive White Gaussian Noise (AWGN) and the multipath fading channels. Based on the proposed training sequence, the CRLB can be reduced to the minimum, and the total energy consumption don't change at the same time.

The rest of the paper is organized as follows. An optimized training sequence in the AWGN channel is proposed in Section II. Section III presents the modification of the proposed training sequence in the multipath fading channels. Adaptive optimization of M in real systems will be discussed in Section IV. Simulation results are given in Section V with different optimum M, and conclusions are drawn in Section VI.

II. TRAINING SEQUENCE DESIGN FOR FLAT FADING

CHANNELS

OFDM input signals are complex numbers of some signal constellation. After Inverse Discrete Fourier Transform (lDFT) and Parallel-to-Serial (PIS) operations, the resultant signals will be transmitted in wireless channels. The relationship between the transmitted and the received signal is:

j27r1=:k

r(k) = s(k) x e-----W- + w(k) k ~ 0 (1)

where s(k) denotes the kth transmitted sample, r(k) denotes its received version, c denotes the carrier frequency offset

978-1-4799-1406-7 © 2013 IEEE

normalized to a subcarrier spacing of OFDM symbols, w(k) represents the additive white Gaussian noise (AWGN) and N denotes the number of subcarriers.

Assume two identical samples separated by (h-l) samples are transmitted at the transmitter, and at the receiver, an effective estimator can be derived based on these two received samples as:

A N x arg{r(k + h)r*(k)} (2) c = k:::: 0, h:::: 1

2nh

where r(k) and r(k + h) denote the former and the later received samples respectively.

In conventional estimators, two or more identical patterns are usually utilized to compose a training symbol/sequence [9], [16]. In fact, the replicated training symbols can arrange their samples in arbitrary sequence rather than identical pattern to that of the first one.

In this paper, a training sequence includes only two training symbols is assumed, and samples in each training symbol are considered Independent and Identical Distributed (lID) complex Gaussian random variables with zero mean.

Lemma 1: A training sequence includes two training sym­bols where the samples of the second one are copied from the first one and are arranged in arbitrary sequence (either identical to or different from that in the first one) is utilized for carrier frequency offset estimation. If the following two conditions are satisfied:

1. The samples of the first training symbol are Independent and Identical Distributed (lID) complex Gaussian random variables of zero mean

2. The second training symbol is the reverse repeat of the first one

then the minimum Cramer-Rao Lower Bound (CRLB) is ob­tained.

Proof: The CRLB for a training sequence including two training symbols can be derived as:

Var{8} > ~~~------~~------------N~l E { B2In!(Y(i),Y(i+hi))}

-.L-~ y(i),y(i+hi) Be2 ,/;=0

N 2

(3) where 8 denotes the estimation error, 'Pi = arg{y*(i) . y(i + h ·)} - SNR - ~ - E{ls(i)12} dE· . {.}

< ,'f - - (J'~ - E{lw(i)12} an y«),y«+h i )

denotes the expectation with respective to (y(i), y(i + hi)). For a given first symbol, variance errors can be reduced by

simply adjusting sample sequence in the replicated training symbol. The minimization of V ar{ 8} can be illustrated as:

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(4)

where d i E [N + h - 1, 2N + h - 2] and d i is unique for different i.

For a fixed h, if di = 2N + h - 2 - i is satisfied for each i, i.e., the second symbol is the reverse repeat of the first

one, then Min (Ntl i· di ) is obtained, and the minimized <=0

variance error is:

Min(V ar{ 8}) = 4n2(3h2 + 6(N _ 1~: + 4N2 - 6N + 2)"'( (5)

In order to make the estimator work correctly, the phase rotations between each pair of corresponding samples in a training sequence should be limited in (-n, n), which requires h not too large. When h = 1, the training sequence is reduced to the one proposed in [11]. In the following discussion, without loss of generality, we assume h = 1.

III. TRAINING SEQUENCE OPTIMIZATION FOR MULTIPATH

FADING CHANNELS

The Central-Symmetric structure (the second half is the reverse repeat of the first half) of a training sequence is helpful in further reducing the CRLB as compared to the training sequence with Identical-Repeat structure (proposed in [9]) in flat-fading channels. In the multipath channels, however, a performance floor appears at high SNR because of the reduction in the effective received Signal-to-Inteiference-plus­Noise-Ratio (SINR) [11].

Now we meet a fundamental tradeoff in training sequence design: training sequence with Central-Symmetric structure outperforms training sequence with Identical-Repeat structure in frame synchronization and carrier frequency offset esti­mation error in the flat-fading channels, however, with per­formance degradation in the multipath channel. Compared to it, training sequence with Identical-Repeat structure has little power loss when performs synchronization in the multipath channel. Integrating benefits of both structures is the key point in training sequence optimization.

In this section, we propose a new method to optimize the training sequence design. Like in [9], the proposed training sequence is also composed of two Identical-Length training symbols, where the data in the second training symbol are copied from the first one. Different from [9], we generate the first training symbol and logically sub-divide it into M > 1 Length-Q (Q = t't > 1 where we assume N is times of M)

sub-blocks, say, Sub-block 1, Sub-block 2, ... , Sub-block M. The second training symbol is generated by padding these Sub-blocks with reverse order, i.e., Sub-block M, Sub-block M - 1, ... , Sub-block 1, as illustrated in Fig.l(a). At the receiver, carrier frequency offset can be estimated based on the phase rotation between each pair of corresponding Sub­blocks. When in multipath fading channels, the first L samples in each Sub-block will be interfered by its preceding Sub-block (L is the maximum multipath delay normalized to samples), as illustrated in Fig.l(b). These Interference Parts can not be used in carrier frequency offset estimation just because they will introduce some biases in the estimation result. Only Useful Parts are used for carrier frequency offset estimation, and a Best Linear Unbiased Estimator (BLUE) is given by:

error. On the other hand, for a given L, increasing M will increase the proportion of Interference Part in each Sub-block, which as a result increases the power loss in estimation. In order to achieve a performance advantage over Moose algo­rithm with respect to estimation accuracy, Var{ SCM;LlIBLUE} should be strictly smaller than 41r}NI' (the CRLB of Moose algorithm), which leads to:

L M 2N-N Q _ N < 4M3 - M < 4 - 4M

and the maximum allowable Mis:

Mmax = l~J -1

where l x J means the maximum integer part of x.

(12)

(13)

eA-1£ SCM;LlIBLUE = ITA-II

For a given L, in order to obtain the M that m1ll1-(6) mizes Var{sCM;LlIBLUE}, we can take partial derivative to

Var{sCM;LlIBLUE} with respect to M and setting the result to zero, which results in:

where both 1 [11 ... IV and £ [sIILS2IL··· sMIL]T are M x 1 column vectors, A = diag {Var{slld··· Var{sMld} is a M x M diagonal matrix, and

Mxarg { Pt1 rCk+2QC M-Pl+Ql.r*Ckl} k=(p-l)Q+L

21r[2CM -p)+l] 1 :<:; p :<:; M,O :<:; L < Q

(7)

represents the carrier frequency offset sub-estimator derived by estimating the phase rotations between Sub-block p of the first training symbol and its replication of the second training symbol.

By using method proposed in [6] we know that SplL is conditionally unbiased, and its variance error is given by:

1. _ (1 + 1.) . e-1]

V {A } "-' 1] 1] (8) ar eplL = 87f2[2(M _ p) + IJ2Q2

where'T) = CQ:;/:;?2. Given high SNR, Var{spld can be approximated as:

Var{spld ----+ 41r2[2CM-pl::::J2Q2CQ-Lh (9)

and SCM;LlIBLUE can be simplified as:

M

3 L: [2(M - p) + 1]2sp1L p=1

SCM;L l 1 BLU E = ---'---M-("-4-M---::-2 ---1""")-- (10)

In order to make the proposed estimator work correctly, 1101 < 2C2t:i-g should be satisfied.

The RLB of variance error for the proposed BLUE is given by:

___ 1_ - I T A- 1 I > 3M - 41r2C4M2_1)( ~-L h

Var{SCM;LlIBLUE} (11)

There is a potential tradeoff in determining M: On the one hand, larger M is welcome in Eq.(11) to reduce the variance

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MminVar

l3 'N-N----;. /1=1=6L=2 4L + 4L V + 123 N2 +

= l \fKJ N N VI 16£2 j 4L - 4L + 123N2

(14) Based on Eq.(13) and Eq.(14), we can obtain the optimum

M for the proposed training sequence as:

Mopt = Min{Mminvar; Mmax} (15)

For flat-fading channels, i.e., L = 0, from Eq.(12) to Eq.(15) we know that Mopt = N and Eq.(5) is met.

Note that since the first L samples of each Sub-block in a received training sequence are not used, the computational complexity of the proposed estimator is L// smaller than that of Moose algorithm.

IV. ADAPTIVE OPTIMIZATION OF M IN REAL SYSTEMS

In real systems, M should be adaptively adjusted in order to accommodate the proposed training sequence to the current wireless channel. When M = N, the proposed training sequence reduces to that proposed in [11], where a joint frame synchronization and carrier frequency offset estimation scheme was proposed. Each tap in a multipath channel can also be detected correctly by using the timing metric proposed in [11], which is helpful in optimizing M in the proposed scheme.

Synchronization can be logically sub-divided into two phases [13], [19]-[21]. The first phase is called Initial Acquisi­tion, where both timing synchronization and carrier frequency offset acquisition can be performed. The second phase is called Adaptive Tracking, where the remaining carrier frequency offset after acquisition will be estimated from time to time. Initial Acquisition can be performed as follows. Firstly, the transmitter will set M = N to generate a training sequence

and transmit it. At the receiver, a timing metric is designed as:

I:~: r(2N - 1 - k + ())r*(k + ()) I Mo(c) = 2N-l

L Ir(k+()) 12

(16)

k=O

where () is a time index. Based on it, the Initial Acquisition can be represented as:

{e ; t} = arg max {Mo(c f-4 cacq)} (17) {O; €acq }

N -I

L f(2N-l-k+O)f*(k+O)

where Mo(c f-4 cacq) = k = O

2 N -I and L If(k+O)1 2

k = O - j2rrkc(l,c q

f(k) = r(k) . e N denotes the Cacq-compensated receive sequence. Note that in multipath channels, when Ic-cacql « 1 is satisfied, one local peak of Mo(c f-4 cacq) implies one detected tap. After Initial Acquisition, the receiver will feed­back parameters to the transmitter, such as the maximum delay and the carrier frequency offset acquisition result. And then Adaptive Tracking starts. The transmitter will pre-compensate the carrier frequency offset by using the acquisition result and make sure that the remaining carrier frequency offset is well within the Tracking range of the proposed scheme, and then adjust M according to the current wireless channel to generate a new training sequence.

V. SIMULATION RESULTS

In this paper, quasi-static block fading wireless channels are assumed, i.e., we assume that the channel impulse response does not change during one training sequence period. A wire­less system operating at 5GHz, with bandwidth of lOMHz and DFf length of 2048 is assumed. Two scenarios are illustrated in Table.I.

In this section, the performance of the estimator using the proposed optimum training sequence is compared to that of Moose's algorithm [9] with a cyclic-prefix length of 16. Simulation results are shown in Figs.2 and 3.

In Scenario I, there are two multipath taps and the maximum discrete-time delay is 2 samples. In this scenario, the optimum M is 8. Performance advantages over Moose algorithm can be obtained in the proposed algorithm with M = 2,4 , 8. As illustrated in Fig.2, the simulations are performed to ob­tain considerable improvement over Moose algorithm. When M = 2, e.g., the performance improvement of the proposed algorithm over Moose algorithm is about l.4dB, or about 1.7dB when M = 8 in the proposed algorithm.

In Scenario II, the maximum discrete-time multi path delay increases to 8 samples, and the corresponding optimum M in the proposed algorithm is 4. When M is set to be larger than 4, e.g., M = 8, the performance will degrade compared to when M = 4, although now performance advantage over Moose algorithm is still achievable. In Fig.3, e.g., the performance improvement of the proposed algorithm when l'I/I = 4 over Moose algorithm is about 0.9dB, however, it will reduce to

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~---------v----------~---------v----------/ Training Symbol I Training Symbol 2

(a) Training Sequence Structure

lntcrfcrala Useful Inlafamcc Usefu l Inlafamcc Useful lntcrfcn,na Useful

.. Pr.. ill Part .. ~ Part .. ~ .... Part Ii ~~~.",.>----,Pw1-"'---""

Sub-block I I ... Sub-block M ub-blockM I ... Sub-blockl I I Sub-block I I ... I Sub-block M I Sub-block M I ··· I Sub-block I I

Sub-block ! Sub-block M Sub-blockM Sub-block 1 I

(b) Multipath Illustration

Fig. 1. The illustration of the optimized training sequence.

OFT Length = 2048

c Moose Estimator + Proposed: M=2 x Proposed: M=4 • Proposed: M=8

--- CRLB: Moose

,,,, ,,,,

. .. . CRLB: Proposed (M=2) - CRLB: Proposed (M=4) - CRLS: Pr M=8

10' L---~---7--~~--~5----~6 --~~--~---7--~'0

SNR (dB)

Fig. 2. Performance comparison of Moose 's algorithm and the proposed algorithm in Scenario I multipath channel.

about 0.7dB when M is set to 8 in the proposed algorithm. The result can verify that we should make a tradeoff in determining the optimum M in real systems.

VI. CONCLUSION

In this paper, an adaptive joint frame synchronization and carrier frequency offset estimation scheme in OFDM systems is proposed. The proposed scheme extends algorithm proposed in [11], and the training sequence generation here can be adaptively optimized according to the current wireless chan­nel. Considerable performance improvement in the proposed scheme over Moose algorithm can be obtained, with regard to either estimation accuracy or computational complexity.

OFT Length = 2048

o

o Moose Estimator + Proposed: M=2 x Proposed: M=4 • Proposed: M=8

--- CRLS: Moose .... CRLS: Proposed (M=2) _.- CRLS: Proposed (M=4) - CALS: Pr M=8

1 ~L---~----~--~----~--~----~----~--~~~ 1 5 6

SNR (dB) 10

Fig. 3. Perfonnance comparison of Moose 's algorithm and the proposed algorithm in the Scenario II multipath channel.

TABLE I WIRELESS CHANNELS WITH CENTRAL FREQUENCY=5GHz,

BANDWIDTH=10MHz, DFT LENGTH=2048

Subcarrier Modulation QPSK Scenario Delay (Ms) 0 I 0.1

I Power (dB) 0 I -4.3 CFO (Hz) 488.3

Subcarrier Modulstion QPSK Scenario Delay (MS) o I 0.2 I 0.4 I 0.7

II Power (dB) o I -4.3 I -8.68 I -17.38 CFO (Hz) 488.3

ACKNOWLEDGEMENT

The authors would like to thank the anonymous reviewers for their time and effort on improving this paper.This work was supported by the National Natural Science Foundation of China (No. 61172050), Program for New Century Excellent Talents in University (NECT-12-0774), the National Basic Research Program of China (2012CB315905), the Foundation of Beijing Engineering and Technology Center for Conver­gence Networks and Ubiquitous Services, and National Key Projects (2012ZX03001029-005 and 2012ZX03001032-003). The corresponding Author is Dr. Zhongshan Zhang.

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