Simultaneous Multiple Carrier Frequency Offsets Estimation for Coordinated Multi-Point Transmission in OFDM Systems

4558 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 9, SEPTEMBER 2013

Simultaneous Multiple Carrier Frequency OffsetsEstimation for Coordinated Multi-Point

Transmission in OFDM SystemsYuh-Ren Tsai, Senior Member, IEEE, Hao-Yun Huang, Yen-Chen Chen, and Kai-Jie Yang

Abstract—Orthogonal frequency division multiplexing(OFDM) combined with the coordinated multi-point (CoMP)transmission technique has been proposed to improveperformance of the receivers located at the cell border.However, the inevitable carrier frequency offset (CFO) willdestroy the orthogonality between subcarriers and inducestrong inter-carrier interference (ICI) in OFDM systems.In a CoMP-OFDM system, the impact of CFO is moresevere because of the mismatch in carrier frequencies amongmultiple transmitters. To reduce performance degradation, CFOestimation and compensation is essential. For simultaneousestimation of multiple CFOs, the performance of conventionalCFO estimation schemes is significantly degraded by the mutualinterference among the signals from different transmitters. Inthis work, our goal is to propose an effective approach thatcan simultaneously estimate multiple CFOs in the downlinkby using the composite signal coming from multiple basestations corresponding to CoMP transmission. Based on theZadoff-Chu sequences, we design an optimal set of trainingsequences, which minimizes the mutual interference and isrobust to the variations in multiple CFOs. Then, we propose amaximum likelihood (ML)-based estimator, the robust multi-CFO estimation (RMCE) scheme, for simultaneous estimationof multiple CFOs. In addition, by incorporating iterativeinterference cancellation into the RMCE scheme, we propose aniterative scheme to further improve the estimation performance.According to the simulations, our scheme can eliminate themutual interference effectively, approaching the Cramer-Raobound performance.

Index Terms—Orthogonal frequency division multiplexing(OFDM), coordinated multi-point (CoMP), carrier frequencyoffset (CFO), Zadoff-Chu (ZC) sequence.

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing(OFDM) offers a promising solution for emerging

high data rate services because of its high bandwidthefficiency and resistance to multipath fading. To preventinter-carrier interference (ICI), it is essential to maintain

Manuscript received November 23, 2012; revised April 2, 2013; acceptedJune 3, 2013. The associate editor coordinating the review of this paper andapproving it for publication was L. Deneire.

This work was supported in part by the National Science Council, Taiwan,R.O.C., under Grant NSC 100-2219-E-007-010, and under Grant NSC 101-2628-E-007-014-MY2.

Y.-R. Tsai, H.-Y. Huang, and K.-J. Yang are with the Institute of Communi-cations Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan(e-mail: [email protected]).

Y.-C. Chen is with MStar Semiconductor, Inc., Hsinchu, Taiwan.Part of this work was presented at IEEE Globecom’11, Houston, Texas,

Dec. 2011.Digital Object Identifier 10.1109/TWC.2013.071913.121851

orthogonality among all subcarriers for OFDM systems.The carrier frequency offset (CFO), which generally iscaused by oscillator mismatch or Doppler effect between thetransmitter and the receiver, will destroy the orthogonalityamong subcarriers and induce strong ICI in OFDM systems.To maintain an acceptable receiving performance, CFOestimation and compensation at a receiver is essential.Conventional CFO estimators for OFDM can be classifiedinto three categories, including pilot tone-aided (PTA)-based,cyclic prefix-based (CPB), and training symbols-based (TSB)[1]-[4]. The normalized CFO, defined as the carrier frequencyoffset divided by the subcarrier spacing, can be divided intothe integral part and the fractional part. The integral CFOcauses subcarrier cyclic shifting; while the fractional CFOresults in not only ICI, but also amplitude reduction andphase rotation in the desired signal, thus severely degradingthe receiving performance.

In the literature, some works considered the propagationchannel that different paths endure different values of CFO.Under the assumption of using training sequences with perfectcorrelation properties, Ahmed et al. proposed an approxi-mative maximum-likelihood (AML) estimator for the CFOestimation of a single source [5]. However, the estimator isfeasible only for small values of CFO. In addition, in orthog-onal frequency division multiple access (OFDMA) systems,the uplink channel is shared on a non-overlapping basis bymultiple distributed users, and thus multiple CFOs estimationis essential at the receiver of a base station (BS). In [6]-[7], the multiple CFOs estimation techniques were proposedfor the uplink of OFDMA systems. To reduce the compu-tational complexity of multi-dimensional searching in MLestimation, Haring et al. exploited the correlation properties oftraining sequences and proposed a complexity-reduced AMLestimator by converting the nonlinear cost function into apolynomial series [6]. In [7], under the assumption that thecircular shift between any two training sequences is largerthan the channel length, Wu et al. proposed a suboptimalML algorithm based on the time-domain correlation propertiesof the training sequences to reduce the complexity by usingconstant amplitude zero autocorrelation (CAZAC) sequences.Some works investigated the problem of multi-CFO estimationin cooperative OFDM systems, where a receiver estimates theCFOs of multiple signals coming from different relay stations(RSs) [8]-[10]. In [8], Jiang et al. exploited time correlationproperties of training sequences for multi-CFO estimation.

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TSAI et al.: SIMULTANEOUS MULTIPLE CARRIER FREQUENCY OFFSETS ESTIMATION FOR COORDINATED MULTI-POINT TRANSMISSION IN OFDM . . . 4559

In this scheme, multiple training symbols are required andthe training sequences used for different relays are non-overlapping in the frequency-domain to prevent severe mutualinterference due to CFO. Similarly, Zhang et al. exploitedthe time-correlation properties of CAZAC sequences andproposed a two-step multi-CFO estimation process, includingcoarse estimation and fine estimation, to enhance estimationperformance [9]. In [10], Mehrpouyan and Blostein proposedtwo computationally efficient iterative estimators based on thecorrelation-based MUltiple SIgnal Characterization (MUSIC)algorithm for multi-CFO estimation. However, the estimatorrequires multiple training symbols to obtain the correlationstatistics.

The coordinated multi-point (CoMP) transmission tech-nique has been proposed for OFDM systems to enhance thereceiving performance of multiple user equipments (UEs)located at the cell border. The principle of CoMP is touse multiple BSs/RSs to simultaneously transmit the sameinformation signal. As a result, for a specific user under theCoMP transmission, multiple signals coming from differentBSs/RSs in the same band are received in the downlink toimprove the receiving performance. In CoMP-OFDM systems,the multiple transmitters must be well synchronized to preventmutual interference. But it is generally true that different trans-mitters have different CFOs; hence, the orthogonality betweensubcarriers collapses at the receiver and the performance isdegraded. Under the scenario applying CoMP transmission, aUE receiver must estimate multiple CFOs corresponding tothe multiple receiving signals for compensation. However, thedifferent signals interfere with each other at the receiver, andthis mutual interference significantly degrades CFO estima-tion performance; therefore, it is difficult to obtain a goodestimate of the CFOs. This also implies that the conventionalCFO estimators, which aim at single CFO estimation, arenot suitable for CoMP-OFDM systems. In [11], Manolakiset al. studied the impact of synchronization impairments,including CFO and sampling frequency offset (SFO), on thereceive performance in CoMP systems. Kotzsch and Fettweisderived the metrics for evaluating the impact of time andfrequency asynchronisms in CoMP systems with large basestation distances [12]. Zarikoff and Cavers proposed in [13] amaximum likelihood (ML)-based scheme for simultaneouslyestimating multiple frequency offsets in the downlink of coor-dinated multiple input/multiple output (MIMO) systems. Thesearching algorithm based on the Newton method is feasibleonly for the scenario with a small CFO range. Accordingly, thesegmented estimators were proposed by breaking the trainingsequence into multiple short segments for the scenario witha large CFO range. Zarikoff and Cavers also worked onthe CFO compensation problem at the BSs for coordinatedmulti-cell systems [14]. In [15], by exploiting the time-domain correlation properties, Liang et al. used the optimalfrequency division multiplexed pilot tones for CFO estimationin the downlink of coordinated MIMO OFDM systems, and aparallel interference cancelation (PIC) strategy was proposedto iteratively compensate the CFO impact.

The goal of this work is to propose an effective approachthat can simultaneously estimate multiple CFOs in the down-link under the interference of multiple receiving signals. We

T1, w1, μ1

1st BS

T2, w2, μ2

TK, wK, μK

2nd BS

Kth BSUE

Tk: the training sequence of the k-th BS

wk: the CFO of the k-th BS

μk: the timing error of the k-th BS

Fig. 1. The CoMP-OFDM system model.

propose an ML-based CFO estimator for simultaneous multi-CFO estimation in CoMP-OFDM systems. We adopt the TSBapproach by using well-designed training sequences to reducethe mutual interference and achieve good estimation perfor-mance. Based on the Zadoff-Chu (ZC) sequences, we designan approach for finding an optimal set of training sequencesthat minimizes the mutual interference and is robust to thevariations in CFOs. The set of designed training sequencesare referred to as the Robust Orthogonal SEquence (ROSE)set. Using the proposed ROSE set at multiple simultane-ously transmitting BSs/RSs, we propose the robust multi-CFOestimation (RMCE) scheme for multi-CFO estimation at areceiver. In addition, after the premier multi-CFO estimatesare obtained, the mutual interference can be roughly estimatedand then used for interference cancellation. By incorporatingthe iterative interference cancellation concept, we also proposean algorithm, referred to as the RMCE with iterative interfer-ence cancellation (RMCE-IIC) scheme, to further improve theestimation performance.

The remainder of this paper is organized as follows. Sec-tion II illustrates the system model and describes the MLestimation approach. Section III proposes the ROSE set formulti-CFO estimation and presents the RMCE and RMCE-IIC schemes. Section IV shows the simulation results forperformance evaluation. Finally, the conclusion is drawn inSection V.

II. PRELIMINARIES

A. System Model

Fig. 1 shows the CoMP-OFDM system model. The consid-ered OFDM system comprises N subcarriers. By using CoMPtransmission, the signal received at a desired UE is the com-posite of the signals coming from the K BSs correspondingto CoMP transmission. The propagation channel from the k-thBS, for 1 ≤ k ≤ K , to the UE is assumed to be an independentLk-tap multipath fading channel. The channel coefficientshk,�, for 1 ≤ k ≤ K and 0 ≤ � ≤ Lk − 1, are assumedto be complex-Gaussian distributed and time-invariant duringan OFDM symbol interval. Let the time-domain discrete signaltransmitted by the k-th BS be denoted as xk[i], where i is thetime-sample index for −Ncp ≤ i ≤ N − 1 and Ncp is thecyclic prefix (CP) length. Then, after removing the CP, the


time-domain discrete signal received at the UE for multi-CFOestimation can be expressed as

y[i] =K∑

k=1

exp

(j2πiwk

N

)Lk−1∑�=0

hk,�xk[i− �− μk] + v[i], (1)

for 0 ≤ i ≤ N − 1, where −0.5 < wk < 0.5 is theresidual CFO, normalized to the carrier spacing, and μk is theinteger-valued timing error, both corresponding to the signalfrom the k-th BS; and v[i] is the zero-mean additive whiteGaussian noise (AWGN) with variance σ2

v . To reduce thesystem complexity, in existing OFDM networks the frequencyerror is normally decomposed into an integer part plus afractional part [16]. In our work, we focus on the fractionalpart of CFO, and define the CFO vector corresponding to theK BSs as w

Δ= [w1, w2, · · · , wK ]T.

Let the channel length plus the timing error be restrictedwithin a range (0, Δ] (i.e., 0 < Lk + μk ≤ Δ for 1 ≤ k ≤K), where Δ is smaller than Ncp. The received signal can bewritten in the vector form as

y = [y[0], · · · , y[N − 1]]T =∑K

k=1ΦkAkhk + v, (2)

where v = [v[0], v[1], · · · , v[N − 1]]T is the AWGN vector

with T denoting transposition; the phase rotation matrix is

Φk = diag [exp (jφk[0]) , · · · , exp (jφk[N − 1])] (3)

with φk[i] = 2πwki/N + ϕk and ϕk the phase of the firsttime sample;

hk =[0Tμk×1, hk,0, hk,1, · · · , hk,Lk−1, 0

T(Δ−μk−Lk)×1

]T(4)

is the channel vector with 0i×1 denoting an all-zero vector oflength i; and

Ak=

⎡⎢⎢⎢⎢⎢⎣

xk[0] xk[N − 1] · · · xk[N −Δ+ 1]

xk[1] xk[0] · · · xk[N −Δ+ 2]...

.... . .

...

xk[N − 1] xk[N − 2] · · · xk[N −Δ]

⎤⎥⎥⎥⎥⎥⎦ (5)

is an N × Δ circular signal matrix. For simplicity, we canrewrite (2) as

y = Ph+ v (6)

where

P = [Φ1A1, Φ2A2, · · · , ΦKAK ]N×KΔ (7)

and

h = [hT1 , h

T2 , · · · , hT

K ]T. (8)

The parameter μk is introduced to incorporate the effect ofdifferent propagation delays from different BSs in realisticenvironments. Note that, in the following derivations, thetiming error μk can be omitted under the assumption of0 < Lk + μk ≤ Δ for 1 ≤ k ≤ K .

B. Maximum-Likelihood Estimation Approach

Based on the TSB approach, the signals transmitted from allBSs for CFO estimation are known at the UE. The logarithmof the conditional probability density function (pdf) of y, givenw and h, can be written as

ln p(y|w,h) = −N2 ln(2πσ2

v)− 12σ2

v[y −Ph]H [y −Ph]

= C0 − C1[y −Ph]H [y −Ph] ,(9)

where C0 = −N ln(2πσ2v)

/2, C1 = 1

/2σ2

v , and the super-script H denotes the conjugate transposition. By taking differ-entiation at the log-likelihood function w.r.t. h and setting itto zero, the channel estimator is obtained as follows [5]:

h = (PHP)−1PHy, (10)

which is a least squares (LS) estimator of h. Substituting (10)into (9), we obtain

ln p(y|w) = C0 − C1

[y −P(PHP)

−1PHy

]H×

[y −P(PHP)

−1PHy

],

(11)

which is a function of the CFO vector w. Subsequently, theML estimator of w is obtained by

w = argmaxw

{ln p(y|w)}

= argmaxw

⎧⎨⎩C0 − C1

[y −P(PHP)

−1PHy

]H×

[y −P(PHP)

−1PHy

]⎫⎬⎭ .

(12)By expanding (12) and ignoring the terms irrelevant to w,including C0, C1, and |y|2, we can rewrite the ML estimatorof w as

w = arg maxw

{yHP

(PHP

)−1PHy

}. (13)

III. ROBUST MULTI-CFO ESTIMATION (RMCE) SCHEME

A. Complexity-Reduced ML Estimator

The multi-CFO estimator shown in (13) involves multiplerounds of matrix inversion, and thus the computational com-plexity is too high for use in real-time applications. However,making the matrix PHP ≈ cI, where c is an arbitraryconstant and I is the identity matrix, can greatly reduce thecomputational complexity. Based on (7), the matrix PHP canbe represented by K2 sub-matrices as

PHP =

⎡⎢⎢⎢⎣

B(1,1) · · · B(1,K)

.... . .

...

B(K,1) · · · B(K,K)

⎤⎥⎥⎥⎦KΔ×KΔ

. (14)

The sub-matrix B(k,k′), for 1 ≤ k, k′ ≤ K , is given by

B(k,k′) = (ΦkAk)HΦk′Ak′ , (15)

in which the entries are[B(k,k′)

]l,g

=N−1∑i=0

x∗k [〈N + 1− l + i〉N ]

×xk′ [〈N + 1− g + i〉N ] e−j

(2πiθ

k,k′N +ϕk,k′

),

(16)


for 1 ≤ l, g ≤ Δ, where θk,k′ = (wk−wk′), ϕk,k′ = ϕk−ϕk′

and 〈·〉N is the modulo-N operator.Note that the off-diagonal entries in PHP can be regarded

as the interferences coming from the same BS on differentpropagation paths or from other BSs. If the signals used bythe K BSs for multi-CFO estimation are mutual-interferencefree, all of the off-diagonal entries will be zero. In other words,for a mutual-interference free scenario using unity-amplitudetraining sequences, the entries in PHP can be expressed as

[B(k,k′)

]l,g

=

⎧⎨⎩ N, ∀k = k′ and l = g

0, otherwise. (17)

Hence, if the cross-correlation property

N−1∑i=0

x∗k [〈N + 1− l + i〉N ]xk′ [〈N + 1− g + i〉N ]

× exp [−j (2πiθk,k′/N + ϕk,k′ )]

= Nδ(k − k′)δ(l − g),

(18)

for 1 ≤ l, g ≤ Δ and 1 ≤ k, k′ ≤ K , holds for all the time-domain training sequences, we have PHP = NI. As a result,the ML estimator of w can be simplified as

w = arg maxw

{yHPPHy

/N

}. (19)

By neglecting the factor N and expanding (19) based on (7),we have

w = arg maxw

⎧⎨⎩ yHΦ1A1A

H1 Φ

H1 y + · · ·

+yHΦKAKAHKΦH

Ky

⎫⎬⎭ . (20)

By inspecting (20), the joint estimation problem can beseparated into K independent estimation sub-problems, eachof which solves the CFO estimation corresponding to thesignal from a BS. The estimator of wk for the signal fromthe k-th BS is given by

wk = arg maxwk

{yHΦkAkA

Hk Φ

Hk y

}, (21)

and the corresponding channel gain estimator is

hk = (AHk Φ

Hk y)

/N, (22)

where Φk is the matrix by substituting wk into Φk.

B. Robust Orthogonal SEquence (ROSE)

The objective of designing a training sequence set thatretains PHP = NI for all scenarios is very difficult to achieve.Theoretically, applying orthogonal sequences as the trainingsequences achieves the mutual-interference free conditionwhen no CFO exists between each BS and the UE. However,when CFOs do exist, mutual interference is inevitable and theestimation performance is severely degraded. Hence, our goalis to design a training sequence set that minimizes the mutualinterference and is robust to variations in CFOs. By applyingthe proposed training sequences, PHP ≈ NI can be retainedfor all scenarios and the complexity-reduced ML estimatorproposed in (21) can be used.

In this work, we use the ZC sequences [17] as thebases for training sequences because they have good prop-erties, including constant amplitude and zero autocorrela-tion for any non-zero circular shifts. Assuming that thesequence length N is even, a ZC sequence, denoted asZ = [Z[0], · · · , Z[N − 1]]

T, is defined by

Z[i] = exp(jMπi2/N), for i = 0, · · · , N − 1, (23)

where M is an integer parameter relatively prime to N . Notethat the odd-length ZC sequences can still be used as thebases of training sequences in our scheme for the case withan odd-valued N , which achieve the same performance andproperties as the even-length ZC sequences. Then, we cangenerate an orthogonal training sequence set for multiple BSsby circular shifting in time. Denoting the training sequencesas Tk = [Tk[0], · · · , Tk[N − 1]]

T for 1 ≤ k ≤ K , we definethe training sequence used at the k-th BS as

Tk = Z 〈Dk〉 = [Z[N −Dk], · · · , Z[N −Dk − 1]]T,(24)

where Z 〈Dk〉 is a circular-shifting version of Z by right-shifting Dk, 0 ≤ Dk ≤ N − 1, samples. Without loss ofgenerality, we assume that the first training sequence is equalto the prime ZC sequence; that is, T1 = Z and D1 = 0.

Using the training sequence set, we have the following twopropositions regarding the mutual interferences from the sameBS on different propagation paths or from other BSs.

Proposition 1: By applying the ZC training sequences, thesignals coming from the same BS on different propagation tapsare still mutually orthogonal for any non-zero CFO under amultipath fading channel.

Proof: See Appendix A.Proposition 2: By applying the ZC training sequences, the

signals coming from two different BSs cannot maintain mutualorthogonality if the two corresponding CFOs are different.

Proof: See Appendix B.Based on the derivation in Proposition 1, it can be easily

shown that (ΦkAk)HΦkAk = NI. Hence, in (14), we have

B(k,k) = NI for 1 ≤ k ≤ K . Furthermore, based on thederivation in Proposition 2, the mutual interferences betweenthe k-th and k′-th BSs correspond to the entries in the sub-matrix B(k,k′), except for the effect of the channel gainvectors hk and hk′ . Because the existence of multiple CFOsdestroys the orthogonality among the received training signals,the prerequisite for complexity reduction (i.e., PHP ≈ cI)cannot be achieved. Therefore, in the design of the ROSEset for multi-CFO estimation, we need to focus only onminimizing the mutual interferences coming from differentBSs. If B(k,k′) ≈ 0 can be achieved, the complexity-reducedML estimator in (21) can then be used. To attain the goalof minimizing the mutual interferences coming from differentBSs, we modified the criterion for searching the proposedROSE set as follows:

S∗ = argminS

∥∥∥(N × (

PHP)−1 − I

)∥∥∥F

(25)

where S = {T1,T2, · · · ,TK}, S∗ = {T∗1,T

∗2, · · · ,T∗

K}is the set of optimal training sequences, and ‖U‖F =√tr(UHU) denotes the Frobenius norm of a complex matrix

U. The set of optimal training sequences S∗ potentially can


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 150

100

150

200

Evaluated CFO w1

Log

-lik

elih

ood

M = 1, w∗ = - 0.5M = 1, w∗ = - 0.2M = 3, w∗ = - 0.5

Fig. 2. The log-likelihood evaluation under some sampled channel realiza-tions based on the multipath channel model considered in Section IV.

be found by performing an exhaustive search. To efficientlyfind S∗, we discuss the results of the scenarios with differentnumbers of involved BSs in the following text.

Based on the considered ML estimator, there is an additionalconstraint on the parameter M for selection of the ROSEset; that is, the parameter selection must satisfy M > 1.To address this problem simply, we consider the scenariowith a single BS (i.e., K = 1) under a single-path fadingchannel. We assume that the exact channel gain and CFOare, respectively, h∗ and w∗, and the applied ZC sequenceis T = [T [0], · · · , T [N − 1]]

T with M = 1, whereT [i] = exp(jπi2

/N) for i = 0, · · · , N − 1. Based on

the ML estimator in (13), the log-likelihood function w.r.t.w1 is yHP

(PHP

)−1PHy for the received signal vector

y = h∗ × ΦT, where Φ is the matrix by substituting w∗

into (3). Because there is only one BS involved, we havePHP = NI and the log-likelihood function can be simplifiedas yHPPHy

/N . For the log-likelihood function evaluation of

w1 = w∗, the processing matrix is P|w1=w∗ = ΦA and wehave the maximal log-likelihood

L(y |w1 = w∗ ) = yHPPHy/N

∣∣w1=w∗

= |h∗|2THΦHΦAAHΦHΦT/N

= |h∗|2THAAHT/N = |h∗|2N,

(26)

where A is the circular signal matrix corresponding to T,ΦHΦ = I and THAAHT = N2. Similarly, for the log-likelihood function evaluation of w1 = w∗ + 1, we have thelog-likelihood

L′(y |w1 = w∗ + 1) = yHPPHy/N

∣∣w1=w∗+1

= |h∗|2THΦHΦ′AAHΦ′HΦT/N

Δ= |h∗|2T′HAAHT′

/N,

(27)where Φ′ is the matrix by substituting w∗ + 1 into (3) and

T′ = Φ′HΦT = diag[exp (−j2πi/N)]i=0,··· ,N−1T

=[exp

(jπ(i2 − 2i)

/N

)]i=0,··· ,N−1

=[exp

(jπ(i − 1)

2/N − jπ/N

)]i=0,··· ,N−1

.

(28)

T1 = Z T1[N – 1]T1[0] T1[1] T1[N – D2]

T1[N – D2 – 1]T1[N – D2] T1[0]

… …

… …2 D=T T1[N – 1]

…

…

Z 2

Fig. 3. The relation between two training sequences T1 and T2.

Note that T′ is a time-shift version of T with an additionalphase shift −π/N , which can be merged into the channeleffect. As a result, we have T′HAAHT′ = N2 and the log-likelihood L′(y |w1 = w∗ + 1) is still equal to |h∗|2N . Thisimplies that w1 = w∗ and w1 = w∗ + 1 achieve the samelog-likelihood for M = 1. Hence, ambiguity may occur inthe log-likelihood evaluation for the region −1 ≤ w1 ≤ 1,severely degrading estimation performance. On the other hand,if a parameter M > 1 is chosen, the ambiguity does not occurin the region of interest.

Note that the above discussion still holds for a channel withmultiple taps. Fig. 2 shows the log-likelihood evaluation undersome sampled channel realizations based on the multipathchannel model considered in Section IV. For the case withM = 1, we find that there are two peaks in the region−1 ≤ w1 ≤ 1, leading to ambiguity in the CFO estimation;whereas, if M = 3, there is only one peak in the regionwithout any ambiguity.

C. ROSE for the Scenario with Two BSs

Considering the scenario with two BSs (i.e., K = 2), theapplied training sequences are T1 = Z and T2 = Z 〈D2〉 (i.e.,T2 [i] = T1 [〈i−D2〉N ]) for 1 ≤ D2 ≤ N − 1, as depictedin Fig. 3. Based on (14) and (15), PHP for K = 2 can bewritten as

PHP∣∣K=2

=

⎡⎣ N × I AH

1 ΦH1 Φ2A2

AH2 Φ

H2 Φ1A1 N × I

⎤⎦ . (29)

If all the entries in AH1 Φ

H1 Φ2A2 and AH

2 ΦH2 Φ1A1 are small

enough, PHP ≈ cI is achieved. Note that AH2 Φ

H2 Φ1A1 is

complex symmetric to AH1 Φ

H1 Φ2A2. Thus, we need to focus

only on AH1 Φ

H1 Φ2A2. Substituting (29) into (25), we rewrite

the criterion for K = 2 as

S∗|K=2 = {T∗1,T

∗2} = arg min

{T1,T2}

∥∥AH1 Φ

H1 Φ2A2

∥∥F.

(30)The entries of the mutual-interference matrix AH

1 ΦH1 Φ2A2

can be expressed as

[AH

1 ΦH1 Φ2A2

]l,g

=N−1∑i=0

(T1 [〈1− l+ i〉N ] ej(2πw1i/N+ϕ1)

)∗×T2 [〈1− g + i〉N ] ej(2πw2i/N+ϕ2),

(31)where 1≤ l, g≤Δ. Applying (23) and after some manipula-tion, we have

[AH

1 ΦH1 Φ2A2

]l,g= exp(j)

N−1∑i=0

e(−j2πi[θ1,2+M(D2+g−l)]/N),

(32)


where = (ϕ2 − ϕ1) − πM[(1− l)2 − (1−D2 − g)2

]/N

and θ1,2 = w1 − w2. Then, substituting (32) into (30), theoptimization problem can be rewritten as

S∗|K=2 = {T∗1,T

∗2}

= arg minM,D2

√Δ∑l=1

Δ∑g=1

∣∣∣∣N−1∑i=0

exp(−j2πi[θ1,2+M(D2+g−l)]

N

)∣∣∣∣2

,

(33)where |exp(j)| = 1 is ignored. Note that theoptimization in (33) relies only on the parametersM and D2. For simplicity, we define λl,g

Δ=∣∣∣∑N−1

i=0 exp (−j2πi [θ1,2 +M(D2 + g − l)]/N)∣∣∣, which

can be completely represented by using a two-variablefunction

Q(M, q) ≡∣∣∣∣∣N−1∑i=0

exp (−j2πi [θ1,2 +Mq]/N)

∣∣∣∣∣ , (34)

where q is an integer. Substituting (34) into the objectivefunction of (33) and combining the terms with the same valueof q into a single term, we define the interference metric asfollows:

(M,D2) =

√Δ∑l=1

Δ∑g=1

λ2l,g

=

√Δ−1∑

s=−Δ+1

(Δ− |s|)Q2(M,D2 + s).

(35)

Consequently, given a specific value of M > 1, we have theoptimal value of D2 as

D(opt)2 = arg min

D2

(M,D2). (36)

Because θ1,2 is a random variable unknown to the receiver,the selection of D

(opt)2 must be robust to variation of θ1,2.

Under the assumption −0.5 < wk < 0.5, the value of θ1,2 isin the range −1 < θ1,2 < 1. In (34), assuming that Mq =m×N+β for a non-negative integer m and an integer β withthe smallest value of |β|, we can simplify the representationof Q(M, q) as

Q(M, q) =

∣∣∣∣∣N−1∑i=0

exp (−j2πi [θ1,2 + β]/N)

∣∣∣∣∣ . (37)

From the viewpoint of minimizing the mutual interference,we regard the random variable θ1,2 as a disturbing term for aspecific value of β. The optimal D2 must minimize Q(M, q)for all possible values of q in (35), under the disturbance ofθ1,2. To find the optimal value D

(opt)2 , we give the following

theorem.Theorem 1: In the scenario with two involved BSs, the

optimal D(opt)2 for the training sequences minimizing the

interference metric is to follow the constraint⟨MD

(opt)2

⟩N

≈ N/2, (38)

when D1 is set to 0.Proof: See Appendix C.

0

200

400

600

800

1000

1200

0 43 85 128 171 213 255Time Shift Index D2

Mut

ual I

nter

fere

nce

Mea

sure

ℑ

w1 – w2 = 0.1w1 – w2 = 0.3w1 – w2 = 0.5

Fig. 4. The mutual interference measure �(M,D2) versus D2 under thecondition with M = 3, N = 256, Δ = 20, and K = 2.

TABLE ITHE AVAILABLE VALUES OF D

(opt)2 AND THE VALUES OF D2 YIELDING

THE PEAKS OF �(M,D2) FOR M = 3 AND 5 IN THE TWO-BS SCENARIO.

Available D(opt)2 D2 yielding peaks of �(M,D2)

M = 3 43, 128, 213 0, 85, 171

M = 5 26, 77, 128, 179, 230 0, 51, 102, 154, 205

We define the effective distance between a pair of trainingsequences as

ε =

⎧⎨⎩ 〈MD〉N , if〈MD〉N ≤ N/2

N − 〈MD〉N , if〈MD〉N > N/2. (39)

where D is the relative circular-shift value and 0 ≤ ε ≤ N/2.Based on Theorem 1, the effective distance between T1 = Z

and T2 = Z⟨D

(opt)2

⟩is ε1,2 ≈ N/2, which approaches the

maximum possible value. If the effective distance is close to0, a peak value of (M,D2) occurs. Note that there are Mpossible values of D2 that achieve 〈MD2〉N ≈ N/2 for 1 ≤D2 ≤ N−1. In Fig. 4, we show the values of (M,D2) versusD2 under the condition with M = 3, N = 256, Δ = 20, andK = 2. As we expected, (M,D2) achieves large valueswhen 〈MD2〉N is close to 0 or N (i.e., when D2 is close to0, 85, 171, and 256), and the one-sided width of each peakis about Δ. Moreover, a small value of (M,D2) can beobtained when 〈MD2〉N ≈ N/2. Correspondingly, the Mpossible values of D2 that achieve 〈MD2〉N ≈ N/2 are 43,128, and 213. In Table 1, we show the available values ofD

(opt)2 and the values of D2 yielding the peaks of (M,D2)

for M = 3 and 5.

D. ROSE for the Scenario with More Than Two BSs

Note that the results for K = 2 are not feasible forthe scenario with more than two BSs. For example, whenM = 3 and N = 256, if T1 = Z and T2 = Z 〈43〉 areselected, neither T3 = Z 〈128〉 nor T′

3 = Z 〈213〉 is afeasible training sequence for the 3-rd BS, because the cross-correlation between T2 = Z 〈43〉 and T3 = Z 〈128〉 (orT′

3 = Z 〈213〉) is very high. Hence, in the scenario withmore than two BSs, the optimal set of training sequences


0

100

200

0

100

200

0

1000

2000

500

1000

1500

2000

2500

Time Shift Index D2

Mut

ual I

nter

fere

nce

Mea

sure

ℑ

Time Shift Index D3

Fig. 5. The mutual interference measure �(M,D) versus (D2,D3) forM = 3, N = 256, Δ = 20, and K = 3 under the scenario with w1 = 0.1,w2 = −0.3, and w3 = 0.5.

must be considered jointly. By setting T1 = Z, the optimalcircular-shift set D = {D2, · · · , DK} for {T2, · · · ,TK}, isdetermined by

D(opt) ={D

(opt)2 , · · · , D(opt)

K

}= arg min

D(M,D),

(40)where the interference metric of the total mutual interferenceis defined as

(M,D) =

K−1∑m=1

K∑n=m+1

∥∥∥(ΦmAm)HΦnAn

∥∥∥F. (41)

To find the optimal set D(opt), we give the following theorem.Theorem 2: In the scenario with K involved BSs, the

optimal circular-shift set D(opt) for the training sequencesminimizing the interference metric is to follow the constraint⟨

MD(opt)k

⟩N

≈ (k − 1)N/K, for 2 ≤ k ≤ K, (42)

when D1 is set to 0.Proof: See Appendix D.

Fig. 5 shows the values of (M,D) versus (D2, D3) forM = 3, N = 256, and Δ = 20 under the scenario withK = 3, w1 = 0.1, w2 = −0.3 , and w3 = 0.5. Because weset D1 = 0, peaks occur when the effective distance between apair of sequences approaches 0; that is, 〈MD2〉N , 〈MD3〉N ,or 〈M |D2 −D3|〉N approaches 0 or N . On the other hand,the valleys occur when all the values 〈MD2〉N , 〈MD3〉N ,and 〈M |D2 −D3|〉N , approach N/3 or 2N/3. The resultsare consistent with the results drawn by Theorem 2. Table 2shows the available sets of D(opt) for M = 3 and K = 3 and4. For example, for M = 3 and K = 3, the feasible valuesof D

(opt)2 are 28, 114, and 199, and those of D

(opt)3 are 57,

142, and 228. Thus, there are a total of 9 possible pairs of(D

(opt)2 , D

(opt)3 ). Similarly, the ROSE sets for the scenario

with K = 4 can be easily derived. There are a total of 27possible sets of (D(opt)

2 , D(opt)3 , D

(opt)4 ).

E. RMCE with Iterative Interference Cancellation (RMCE-IIC) Scheme

Although the proposed ROSE set can reduce the mu-tual interference between the signals coming from different

TABLE IITHE AVAILABLE SETS OF D(opt) FOR M = 3 AND K = 3 AND 4.

K = 3 (D(opt)2 , D

(opt)3 ) : (28, 57), (28, 142), (28, 228), (114, 57),

(114, 142), (114, 228), (199, 57), (199, 142), (199, 228)

K = 4 (D(opt)2 , D

(opt)3 , D

(opt)4 ) : (21, 43, 64), (21, 43, 149),

(21, 43, 235), (21, 128, 64), (21, 128, 149), (21, 128, 235),

(21, 213, 64), (21, 213, 149), (21, 213, 235), (107, 43, 64),

(107, 43, 149), (107, 43, 235), (107, 128, 64), (107, 128, 149),

(107, 128, 235), (107, 213, 64), (107, 213, 149),

(107, 213, 235), (192, 43, 64), (192, 43, 149), (192, 43, 235),

(192, 128, 64), (192, 128, 149), (192, 128, 235),

(192, 213, 64), (192, 213, 149),(192, 213, 235)

BSs, the interference still degrades estimation performance.To further improve estimation performance, we propose anestimation algorithm that incorporates iterative interferencecancellation into the RMCE scheme, and call it the RMCE-IICscheme. Based on the current estimation of w(n)

k and h(n)k , we

reconstruct the signal received from the k-th BS as

r(n)k = Φ

(n)k Akh

(n)k , for 1 ≤ k ≤ K, (43)

where superscript (n) denotes the iteration index and Φ(n)k

is the matrix by substituting w(n)k into Φk. To eliminate the

mutual interference, the signals from other BSs are canceledfrom the original signal y to estimate the signal received fromthe k-th BS; that is,

y(n)k = y −

K∑m=1,m �=k

Φ(n)m Amh(n)

m , for 1 ≤ k ≤ K, (44)

Then, y(n)k is used as the desired signal for the estimation of

wk . According to (21) and (22), the CFO and channel gainestimators are given by

w(n+1)k = argmax

wk

{(y

(n)k )

HΦkAkA

Hk Φ

Hk y

(n)k

}(45)

and

h(n+1)k =

[AH

k (Φ(n+1)k )

Hy(n)k

]/N. (46)

Setting the maximum number of iterations as η, the RMCE-IIC algorithm is shown as follows:

RMCE-IIC Algorithm

Step 1: Based on the received signal vector y, the esti-mated CFO wk and channel gain vector hk, fork = 1, · · · , K , are obtained by using (21) and(22). The estimation results are regarded as the initialestimation (i.e., w(0)

k and h(0)k ).

Step 2: Based on w(n)k and h

(n)k , the estimated signal from

the k-th BS (i.e., y(n)k ) is obtained by using (44), for

1 ≤ k ≤ K .Step 3: Based on y

(n)k , the new estimate of CFO w

(n+1)k

and channel gain vector h(n+1)k are obtained by

using (45) and (46). Then, the iteration number nis increased by 1. If n < η, go to Step 2; otherwise,terminate the procedure.


The final values of wk and hk are set to the most recentversion of estimation.

F. Complexity Analysis

In the complexity analysis, we focus on comparing the com-plexity of the proposed RMCE scheme with the conventionalML estimator. It is noted that the computational complexityof an ML estimator depends primarily on the evaluation of itslikelihood function. Recalling the conventional ML estimatorof w in (13), the complexity of computing yHP, PHP,and

(PHP

)−1is O (NKΔ), O

(NK2Δ2

), and O

(K3Δ3

),

respectively [18]. Then, the complexity of the matrix multi-plications among the three parts, yHP,

(PHP

)−1, and PHy,

is O(K2Δ2

). In general, N > KΔ. Therefore, the overall

complexity of evaluating yHP(PHP

)−1PHy is dominated

by O(NK2Δ2

). On the other hand, in the proposed RMCE

scheme, the ML estimator of wk is given in (21). Notethat yHΦkAkA

Hk Φ

Hk y can be obtained by first determining

Ψk = AHk Φ

Hk y and then computing ΨH

k Ψk . Since Φk is a di-agonal matrix, the complexity of computing Ψk = AH

k ΦHk y is

O (NΔ). Then, the complexity of computing ΨHk Ψk is O (Δ).

Accordingly, the complexity of evaluating yHΦkAkAHk Φ

Hk y

is dominated by O (NΔ). As a result, the overall complexityof obtaining all the K estimates is dominated by O (NKΔ).Obviously, the complexity of the conventional ML estimatoris proportional to K2Δ2, while that of the RMCE scheme isproportional to KΔ.

Besides direct complexity evaluation of the ML-based CFOestimators, the iteration number required for convergence isanother important issue for complexity analysis in the iterativeinterference cancellation approaches. For the conventional MLscheme, multiple CFOs are jointly estimated, which resultsin a very slow convergence rate, e.g., tens iterations. On thecontrary, in the proposed RMCE-IIC scheme, each CFO isindividually estimated, which helps to improve the correspond-ing convergence rate; in general, RMCE-IIC converges within2 iterations to achieve the best performance. In summary,the RMCE scheme significantly reduces the computationalcomplexity when compared with the conventional ML estima-tor, especially for the case incorporating iterative interferencecancellation.

IV. SIMULATION RESULTS

In the simulations, the applied channel model is the ex-tended vehicular A (EVA) model [19] with 9 taps generated byusing the Jack’s method. The channel gains remain unchangedduring an OFDM symbol interval. We assume that the numberof subcarriers is N = 256 and the length of the CP isN/4 = 64. The maximum channel length plus the timingerror is restricted to Δ = 20, and the signals from differentBSs are assumed to have the same average received power atthe desired UE. We present the estimation performances, mea-sured in the mean square error (MSE), of the proposed RMCEand RMCE-IIC schemes, where the number of iterations isset to η = 1 for RMCE-IIC. The Cramer-Rao bound (CRB)proposed in [20] for multiple parameters estimation is alsoshown as a performance baseline, which is the best achievableperformance. All the results are obtained by averaging 30,000

SNR (dB)

Mea

n Sq

uare

Err

or (

MSE

)

0 5 10 15 20 25 30

10-2

10-3

10-6

10-4

10-5

ST-N: η = 0ST-N: η = 17ST-N/ZC: η = 0, D2 = 85ST-N/ZC: η = 17, D2 = 85ST-N/ROSE: η = 0ST-N/ROSE: η = 1RMCE: η = 0RMCE-IIC: η = 1CRB

Fig. 6. Average CFO estimation performance comparison under the two-BSscenario for M = 3, N = 256, and Δ = 20 with uniformly randomlygenerated CFOs −0.2 < wk < 0.2.

runs of simulation, where the exact CFO values are generatedrandomly in the range [-0.5, +0.5] (except for Fig. 6) and thetiming errors are randomly distributed in the range [0, 5] ofsample intervals.

Fig. 6 shows the CFO estimation performance versus thesignal-to-noise ratio (SNR) under the scenario with two BSs(i.e., K = 2). We compare our scheme with the iterativeST-N scheme proposed in [13], in which orthogonal ran-dom sequences are adopted and the estimator is based onthe conventional ML approach. The exact CFO values aregenerated randomly in the range [-0.2, +0.2], since the ST-N scheme is feasible only for small CFOs. For the proposedRMCE and RMCE-IIC schemes, the applied ROSE parametersare M = 3 and (D1, D2) = (0, 43). We also evaluate theST-N scheme by using the ZC sequences with M = 3 and(D1, D2) = (0, 85) (which is not a ROSE set), denoted asST-N/ZC. In addition, the ROSE set can also be applied to theST-N scheme to reduce the mutual interference, denoted asST-N/ROSE. We observe that the RMCE scheme significantlyoutperforms the ST-N scheme and approaches the CRB. IfRMCE-IIC is used, the performance is almost the same as theCRB, even when the number of iterations is η = 1. For theST-N scheme, a worse performance is obtained even when thealgorithm is converged (i.e., η = 17). However, if the ROSEset is adopted, the ST-N scheme has a fast convergence rateand its performance can be greatly improved. This implies thatthe mutual interference can be eliminated effectively by usingthe proposed ROSE set.

Fig. 7 shows the MSE performance of CFO estimation ver-sus the SNR for the proposed RMCE and RMCE-IIC schemes.The applied ROSE parameters are M = 3 and (D1, D2) =(0, 43) for K = 2 and (D1, D2, D3) = (0, 28, 57) for K =3. The exact CFO values are now generated randomly in therange [-0.5, +0.5]. In the scenario of K = 3, the estimationperformance is slightly degraded by more mutual interference,when compared to the scenario of K = 2. However, theproposed RMCE and RMCE-IIC schemes still perform verywell and approach the CRB.

Fig. 8 shows the average normalized MSE performance ofchannel estimation corresponding to Fig. 7, where the average


0 5 10 15 20 25 30

10-2

10-3

10-6

10-4

10-5

SNR (dB)

Mea

n S

quar

e E

rror

(M

SE

)

RMCE: η = 0, K = 3RMCE-IIC: η = 1, K = 3RMCE: η = 0, K = 2RMCE-IIC: η = 1, K = 2CRB

Fig. 7. Average CFO estimation performance of the RMCE and RMCE-IICschemes for M = 3, N = 256, and Δ = 20 with uniformly randomlygenerated CFOs −0.5 < wk < 0.5.

0 5 10 15 20 25 30

10-2

10-3

10-4

10-5

SNR (dB)

Ave

rage

Nor

mal

ized

Mea

n S

quar

e E

rror

(N

MS

E)

RMCE: η = 0, K = 3RMCE-IIC: η = 1, K = 3RMCE: η = 0, K = 2RMCE-IIC: η = 1, K = 2CRB

Fig. 8. Average channel estimation performance of the RMCE and RMCE-IIC schemes for M = 3, N = 256, and Δ = 20 with uniformly randomlygenerated CFOs −0.5 < wk < 0.5.

normalized MSE is defined as∥∥∥hk − hk

∥∥∥2/(

Lk‖hk‖2)

.

The proposed schemes still perform very well in channelestimation, achieving the performance quite close to the CRB.

Fig. 9 compares the MSE performance of using differentROSE sets for K = 2. According to Table 1, there are threepossible selections for T2 (i.e., D(opt)

2 = 43, 128, 213) whenT1 = Z. It is observed that using different ROSE sets achievesthe same CFO estimation performance for both the RMCE andRMCE-IIC schemes, which implies that different ROSE setsare equivalent in the CFO estimation.

Fig. 10 shows the MSE performance versus the differencebetween the two corresponding CFOs for K = 2 and SNR= 30 dB, by using the proposed ROSE sets or the two ZCsequences with (D1, D2) = (0, 5). Because the possible CFOrange is [-0.5, 0.5], the CFO difference θ1,2 ranges from -1to +1. For a specific value of θ1,2 = w1 − w2, the CFOsare generated randomly within [-0.5, 0.5]. For the proposedRMCE and RMCE-IIC schemes, the variation in the MSEperformance is small w.r.t. the change in θ1,2, and RMCE-IICachieves the best performance for different θ1,2. On the otherhand, if the ZC sequences with (D1, D2) = (0, 5) are used,

0 5 10 15 20 25 30

10-2

10-3

10-6

10-4

10-5

SNR (dB)

Mea

n S

quar

e E

rror

(M

SE

)

RMCE: M = 3, D2(opt) = 43

RMCE: M = 3, D2(opt) = 128

RMCE: M = 3, D2(opt) = 213

RMCE-IIC: M = 3, D2(opt) = 43, η = 1

RMCE-IIC: M = 3, D2(opt) = 128, η = 1

RMCE-IIC: M = 3, D2(opt) = 213, η = 1

Fig. 9. Performance comparison of using different ROSE sets for M = 3,N = 256, K = 2, and Δ = 20 with uniformly distributed CFOs.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

100

10-2

10-7

10-3

10-5

10-1

10-4

10-6

CFO Difference θ12

Mea

n S

quar

e E

rror

(M

SE

)

w1: D2(opt) = 43

w2: D2(opt) = 43

w1: D2 = 5w2: D2 = 5RMCERMCE-IIC: η = 1

Fig. 10. Average CFO estimation performance versus the difference betweenthe two corresponding CFOs for k = 2 and SNR = 30 dB.

strong mutual interference occurs and estimation performanceis severely degraded. However, if the CFO difference θ1,2approaches 0, the two signals cannot be distinguished andno mutual interference exists, which leads to performanceimprovement in CFO estimation using the RMCE process.Furthermore, we find that applying the iterative interferencecancellation process does not improve estimation performancewhen using the ZC sequences with (D1, D2) = (0, 5); thisis primarily because the channel estimation is destroyed bystrong mutual interference, even when the CFO differenceθ1,2 approaches 0. Fig. 11 shows the average normalizedMSE performance of channel estimation corresponding to Fig.10. We observe that the channel estimation performance isquite poor when the ZC sequences with (D1, D2) = (0, 5)are used; whereas, if the proposed ROSE sets are applied,good channel estimation performance is guaranteed. Hence,if the ZC sequences with (D1, D2) = (0, 5) are used, theiterative interference cancellation process cannot eliminate themutual interference, even when the CFO estimation is goodat θ1,2 = 0.

V. CONCLUSION

In this work, we have proposed two effective, low-complexity schemes, RMCE and RMCE-IIC, for simultane-


-1 -0.5 0 0.5 1

100

10-2

10-5

10-1

10-3

10-4

CFO Difference θ12

Ave

rage

Nor

mal

ized

Mea

n S

quar

e E

rror

(N

MS

E)

h1: D2(opt) = 43

h2: D2(opt) = 43

h1: D2 = 5h2: D2 = 5

Fig. 11. Average channel estimation performance versus the differencebetween the two corresponding CFOs for k = 2 and SNR = 30 dB.

ous estimation of multiple CFOs in CoMP-OFDM systemsover frequency-selective fading channels. The schemes arebased on the use of the proposed training sequences that cansignificantly reduce the mutual interference between signalstransmitted by different BSs and are robust to the variationsin multiple CFOs. The proposed ROSE set has the propertyof maintaining very low mutual interference for the variationsin the CFOs. The derivation of the ROSE set is flexible andsuitable for the scenarios with multiple BSs joining the CoMPtransmission. The RMCE scheme can achieve the performanceof the CRB in the low SNR to medium SNR regions, but is stilldegraded by the mutual interference in the high SNR region.However, if RMCE-IIC is applied, the performance is furtherimproved, approaching the best achievable performance inthe high SNR region, even when the number of iterationsis only 1. The simultaneous multi-CFO estimation schemescan reduce the requirement in bandwidth resource and achievevery good performance. In addition, the proposed ROSE setcan also be applied to other multi-CFO estimation schemes toreduce the mutual interference. As a result, great improvementin estimation performance and/or convergence rate can beachieved.

APPENDIX

A. Proof of Proposition 1

According to (3)-(5), the received signal vector correspond-ing to the k-th BS on the path with a delay of m samplesrelative to the first time-sample y[0] is

yk,m = hk[m]× [ejφk[0]xk [〈−m〉N ] , ejφk[1]xk [〈1−m〉N ]

, · · · , ejφk[N−1]xk [〈N − 1−m〉N ]]T

,(47)

where hk[m] is the m-th entry of hk. The cross-correlationbetween yk,m and yk,n is

yHk,myk,n = h∗

k[m]hk[n]×N−1∑i=0

x∗k [〈i−m〉N ]xk [〈i− n〉N ].

(48)By the ZC sequence properties, the equality∑N−1

i=0 x∗k [〈i−m〉N ]xk [〈i− n〉N ] = 0 holds for m = n. As

a result, we have

yHk,myk,n = h∗

k[m]hk[n]× 0 = 0, for m = n. (49)

Note that (49) holds for any pairs of (m,n) with m = n,regardless of the value of wk.

B. Proof of Proposition 2

Based on (47), the cross-correlation between yk,m andyk′,n, which, respectively, are the signals coming from thek-th and k′-th BSs, is obtained as

yHk,myk′,n = h∗

k[m]hk′ [n]×N−1∑i=0

exp(−jφk[i])x∗k [〈i−m〉N ]

× exp(jφk′ [i])xk′ [〈i− n〉N ]

= h∗k[m]hk′ [n]×

N−1∑i=0

exp [j (φk′ [i]− φk[i])]

×x∗k [〈i−m〉N ]xk′ [〈i− n〉N ] .

(50)If the two corresponding CFOs are different, we have φk[i] =φk′ [i]. As a result, in general, yH

k,myk′,n = 0 and mutualorthogonality is destroyed.

C. Proof of Theorem 1

According to (37), the value of Q(M, q) is the magnitudeof the sum of N unity-length complex vectors. When β = 0and |θ1,2| is small enough (e.g., |θ1,2| ≤ 0.1), we have|2πiθ1,2/N | � π for 0 ≤ i ≤ N − 1 and all the Ncomplex vectors congregate within a narrow angle spread.Consequently, the magnitude of the resultant vector is verylarge, implying a large value of Q(M, q). Similarly, if |β| isquite small (e.g., β = 1 or -1), some values of θ1,2 make|θ1,2 + β| � 1, which also leads to a large value of Q(M, q).By investigating the last summation term in (35), there will beat least one term of Q2(M,D2 + s) corresponding to |β| ≤ 1when 0 ≤ D2 ≤ Δ − 1, N − Δ + 1 ≤ D2 ≤ N − 1, ormNM − Δ + 1 ≤ D2 ≤ mN

M + Δ − 1 for 1 ≤ m ≤ M − 1.Hence, the interference metric (M,D2) results in a largevalue for D2 in these regions. In other words, when 〈MD2〉Nis close to 0 or N , the mutual interference becomes large.Conceptually, if |β| is a large integer, a small value of Q(M, q)can be obtained regardless of the value of θ1,2. To minimizethe impact of the disturbing term θ1,2, we shall select D2

to make all the corresponding values of |β| large enough.Hence, to minimize the interference metric, D(opt)

2 must befar away from the above-mentioned regions. In other words,D

(opt)2 follows the constraint shown in (38).

D. Proof of Theorem 2

Based on the same arguments for the scenario with twoBSs, the optimal set D(opt) must maximize the effectivedistance between each pair of training sequences to minimizethe mutual interference. To maximize the minimal effectivedistance, the selected circular-shifts must be equally spaced,leading to the set D(opt) satisfying (42). This selection ensuresthat the minimum effective distance is close to N/K.


ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their valuable comments and suggestions for improvingthe quality of our work.

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Yuh-Ren Tsai (M’01, SM’12) received the B.S.degree in electrical engineering from National TsingHua University, Hsinchu, Taiwan, in 1989, and thePh.D. degree in electrical engineering from NationalTaiwan University, Taipei, Taiwan, in 1994. From1994 to 2001, he was a Researcher in Telecommuni-cation Laboratories of Chunghwa Telecom Co., Ltd.,Taiwan. Since 2001, he has been with the Depart-ment of Electrical Engineering and the Institute ofCommunications Engineering at National Tsing HuaUniversity, Hsinchu, Taiwan, where he is currently a

Professor. His research interests include wireless transmission, wireless sensornetworks and mobile cellular systems.

Hao-Yun Huang received the B.S. degree in Com-munications Engineering from the Yuan Ze Uni-versity, Taoyuan, Taiwan, in 2007, and the M.S.degree in the Institute of Communications Engi-neering, National Tsing Hua University, Hsinchu,Taiwan, in 2010. He is currently working towardthe Ph.D. degree in the Institute of Communica-tions Engineering, National Tsing Hua University,Hsinchu, Taiwan. His research focuses on frequencysynchronization and channel estimation of OFDMand CoMP systems.

Yen-Chen Chen received the B.S. degree from theDepartment of Engineering and System Science,National Tsing Hua University, Hsinchu, Taiwan,in 2003, and the Ph.D. degree in the Institute ofCommunications Engineering, National Tsing HuaUniversity, Hsinchu, Taiwan, in 2011. He is cur-rently a principle engineer in MStar Semiconductor,Inc., Taiwan, since 2011. His research interests in-clude multimedia transmission, multicast/broadcastservices, resource optimization, and OFDM systems.

Kai-Jie Yang received the B.S. degree in ElectricalEngineering from National Chung Cheng University,Chiayi, Taiwan, in 2000, and the M.S. and Ph.D.degrees in Communications Engineering from Na-tional Tsing Hua University, Hsinchu, Taiwan, in2002 and 2009, respectively. His research interestsinclude link stability prediction and mobility estima-tion in mobile wireless networks.

Documents

Simultaneous Multiple Carrier Frequency Offsets Estimation for Coordinated Multi-Point Transmission in OFDM Systems