IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, … · design for MIMO-OFDM CFO estimation in frequency selec-tive fading channels is still an open problem. 2 IEEE TRANSACTIONS

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, PP. 1244-1254, APR. 2008 1

Frequency Offset Estimation and

Training Sequence Design for MIMO OFDMYanxiang Jiang, Student Member, IEEE, Hlaing Minn, Member, IEEE, Xiqi Gao, Senior Member, IEEE,

Xiaohu You, and Yinghui Li Student Member, IEEE

Abstract— This paper addresses carrier frequency offset (CFO)estimation and training sequence design for multiple-inputmultiple-output (MIMO) orthogonal frequency division multi-plexing (OFDM) systems over frequency selective fading chan-nels. By exploiting the orthogonality of the training sequences inthe frequency domain, integer CFO (ICFO) is estimated. With theuniformly spaced non-zero pilots in the training sequences andthe corresponding geometric mapping, fractional CFO (FCFO) isestimated through the roots of a real polynomial. Furthermore,the condition for the training sequences to guarantee estimationidentifiability is developed. Through the analysis of the corre-lation property of the training sequences, two types of sub-optimal training sequences generated from the Chu sequence areconstructed. Simulation results verify the good performance ofthe CFO estimator assisted by the proposed training sequences.

Index Terms— MIMO-OFDM, frequency selective fading chan-nels, training sequences, frequency offset estimation.

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) trans-

mission is receiving increasing attention in recent years due to

its robustness to frequency-selective fading and its subcarrier-

wise adaptability. On the other hand, multiple-input multiple-

output (MIMO) systems attract considerable interest due to

the higher capacity and spectral efficiency that they can

provide in comparison with single-input single-output (SISO)

systems. Accordingly, MIMO-OFDM has emerged as a strong

candidate for beyond third generation (B3G) mobile wide-

band communications [1].

It is well known that SISO-OFDM is highly sensitive to

carrier frequency offset (CFO), and accurate estimation and

Manuscript received October 18, 2006; revised February 5, 2007;accepted February 10, 2007. The associate editor coordinating thereview of this manuscript and approving it for publication was Dr.Defeng (David) Huang. The work of Yanxiang Jiang, Xiqi Gaoand Xiaohu You was supported in part by National Natural Sci-ence Foundation of China under Grants 60496311 and 60572072,the China High-Tech 863 Project under Grant 2003AA123310 and2006AA01Z264, and the International Cooperation Project on Be-yond 3G Mobile of China under Grant 2005DFA10360. The workof Hlaing Minn and Yinghui Li was supported in part by the ErikJonsson School Research Excellence Initiative, the University ofTexas at Dallas, USA. This paper was presented in part at the IEEEInternational Conference on Communications (ICC), Istanbul, Turkey,June 2006.

Yanxiang Jiang, Xiqi Gao and Xiaohu You are with the Na-tional Mobile Communications Research Laboratory, Southeast Uni-versity, Nanjing 210096, China (e-mail: {yxjiang, xqgao, xhyu}@seu.edu.cn).

Hlaing Minn and Yinghui Li are with the Department of ElectricalEngineering, University of Texas at Dallas, TX 75083-0688, USA(e-mail: {hlaing.minn, yinghui.li}@utdallas.edu).

compensation of CFO is very important [2]. A number of

approaches have dealt with CFO estimation in a SISO-OFDM

setup [2]–[7]. According to whether the CFO estimators use

training sequences or not, they can be classified as blind

ones [3] [4] and training-based ones [2], [5]–[7]. Similar to

SISO-OFDM, MIMO-OFDM is also very sensitive to CFO.

Moreover, for MIMO-OFDM, there exists multi-antenna in-

terference (MAI) between the received signals from different

transmit antennas. The MAI makes CFO estimation more

difficult, and a careful training sequence design is required for

training-based CFO estimation. However, unlike SISO-OFDM,

only a few works on CFO estimation for MIMO-OFDM

have appeared in the literature. In [8], a blind kurtosis-based

CFO estimator for MIMO-OFDM was developed. For training-

based CFO estimators, the overviews concerning the necessary

changes to the training sequences and the corresponding CFO

estimators when extending SISO-OFDM to MIMO-OFDM

were provided in [9], [10]. However, with the provided training

sequences in [9], satisfactory CFO estimation performance

cannot be achieved. With the training sequences in [10], the

training period grows linearly with the number of transmit

antennas, which results in an increased overhead. In [11],

a white sequence based maximum likelihood (ML) CFO

estimator was addressed for MIMO, while a hopping pilot

based CFO estimator was proposed for MIMO-OFDM in

[12]. Numerical calculations of the CFO estimators in [11]

[12] require a large point discrete Fourier transform (DFT)

operation and a time consuming line search over a large set of

frequency grids, which make the estimation computationally

prohibitive. To reduce complexity, computationally efficient

CFO estimation was introduced in [13] by exploiting proper

approximations. However, the CFO estimator in [13] is only

applied to flat-fading MIMO channels.

When training sequence design for CFO estimation is

concerned, it has received relatively little attention. It was

investigated for single antenna systems in [14], where a white

sequence was found to minimize the worst-case asymptotic

Cramer-Rao bound (CRB). Recently, an improved training

sequence and structure design was developed in [15] by

exploiting the CRB and received training signal statistics. In

[16], training sequences were designed for CFO estimation

in MIMO systems using a channel-independent CRB. In [17],

the effect of CFO was incorporated into the mean-square error

(MSE) optimal training sequence designs for MIMO-OFDM

channel estimation in [18]. Note that optimal training sequence

design for MIMO-OFDM CFO estimation in frequency selec-

tive fading channels is still an open problem.

http://arxiv.org/abs/1703.07482v1

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, PP. 1244-1254, APR. 2008

In this paper, we propose a new training-based CFO es-

timator for MIMO-OFDM over frequency selective fading

channels. To avoid the large point DFT operation and the time

consuming line search and also to guarantee the estimate preci-

sion, we propose to first estimate integer CFO (ICFO) and then

fractional CFO (FCFO). Next, we relate FCFO estimation to

direction of arrival (DOA) estimation in uniform linear arrays

(ULA). Then, we propose to exploit a geometric mapping to

transform the complex polynomial related to FCFO estimation

into a real polynomial. Correspondingly, FCFO is estimated

through the roots of the real polynomial. In the derivation

of the CFO estimator, we also develop the identifiability

condition concerning the training sequences. Furthermore,

exploiting the well-studied results for DOA estimation, we

transform the problem of training sequence design for CFO

estimation into the study of the correlation property of the

training sequences, and propose to construct them from the

Chu sequence [19].

The rest of this paper is organized as follows. In Section

II, we briefly describe the MIMO-OFDM system model. The

proposed CFO estimator including ICFO and FCFO estimation

is presented in Section III. The aspect concerning training

sequence design is addressed in Section IV. Simulation results

are shown in Section V. Final conclusions are drawn in Section

VI.

Notations: Upper (lower) bold-face letters are used for

matrices (column vectors). Superscripts ∗, T and H denote

conjugate, transpose and Hermitian transpose, respectively.

(·)P denotes the residue of the number within the brackets

modulo P . ℜ(·) and ℑ(·) denote the real and imaginary parts

of the enclosed parameters, respectively. ⌊·⌋, || · ||2, E[·] and

⊗ denote the floor, Euclidean norm-square, expectation and

Kronecker product operators, respectively. sign(·) denotes the

signum function and sign(0) = 1 is assumed. [x]m denotes

the m-th entry of a column vector x. x(m) denotes the m-

cyclic-down-shift version of x for m > 0 and |m|-cyclic-

up-shift version of x for m < 0. diag{x} denotes a diagonal

matrix with the elements of x on its diagonal. [X]m,n denotes

the (m,n)-th entry of a matrix X . FN and IN denote the

N × N unitary DFT matrix and the N × N identity matrix,

respectively. ekN denotes the k-th column vector of IN . 1Q(0Q) and 0P×Q denote the Q×1 all-one (all-zero) vector and

P × Q all-zero matrix, respectively. JQ denotes the Q × Qexchange matrix with ones on its anti-diagonal and zeros

elsewhere. Unless otherwise stated, 0 ≤ µ ≤ Nt − 1 and

0 ≤ ν ≤ Nr − 1 are assumed.

II. SYSTEM MODEL

Let us consider a MIMO-OFDM system with Nt transmit

antennas, Nr receive antennas and N subcarriers. Suppose the

training sequence transmitted from the µ-th antenna is denoted

by the N × 1 vector tµ. Before transmission, this vector is

processed by an inverse discrete Fourier transform (IDFT),

and a cyclic prefix (CP) of length Ng is inserted. We assume

that Ng ≥ L − 1, where L is the maximum length of all the

frequency selective fading channels. We further assume that

all the transmit-receive antenna pairs are affected by the same

CFO. Define

DN (ε) = diag{[1, ej2πε/N , · · · , ej2πε(N−1)/N ]T },

where ε is the frequency offset normalized by the subcarrier

spacing. Suppose the length-L channel impulse response from

the µ-th transmit antenna to the ν-th receive antenna is denoted

by the L × 1 vector h(ν,µ). Then, after removing the CP at

the ν-th receive antenna, the N × 1 received vector yν can be

written as [12]

yν =√Nej2πεNg/NDN (ε)

Nt−1∑

µ=0

{

FHN diag{h(ν,µ)}tµ

}

+wν , (1)

where

h(ν,µ) = FN [e0N , e1N , · · · , eL−1

N ]h(ν,µ),

and wν is an N×1 vector of additive white complex Gaussian

noise (AWGN) samples with zero-mean and equal variance of

σ2w.

The goal of this paper is to design the training sequences

{tµ}Nt−1µ=0 and estimate ε from the observation of {yν}Nr−1

ν=0 .

III. FREQUENCY OFFSET ESTIMATION FOR MIMO OFDM

Let Q = N/P with (N)P = 0. Design

(C0) 0 ≤ i0 < i1 < · · · < iµ < · · · < iNt−1 < Q.

Define

Θq = [eqN , eq+QN , · · · , eq+(P−1)Q

N ], 0 ≤ q < Q.

Let sµ denote a length-P sequence whose elements are all

non-zero. Then, we propose to construct the training sequence

transmitted from the µ-th antenna as tµ = Θiµ sµ (C1).

Without loss of generality, the total energy allocated to training

is supposed to be split equally between the transmit antennas,

i.e., ||sµ||2 = N/Nt (C2). Note that the entire training symbol

is utilized by our proposed CFO estimator.

A. ICFO Estimation

Let y = [yT0 ,yT1 , · · · ,yTν , · · · ,yTNr−1]

T denote the NrN×1 cascaded vector from the Nr receive antennas. Then, y can

be written as

y =√Nej2πεNg/N{INr

⊗ [DN (ε)S]}h+w, (2)

where

h = [hT0 ,hT1 , · · · ,hTν , · · · ,hTNr−1]

T ,

hν = [(h(ν,0))T , (h(ν,1))T , · · · , (h(ν,µ))T , · · · , (h(ν,Nt−1))T ]T ,

S = FHdiag{[sT0 , sT1 , · · · , sTµ , · · · , sTNt−1]T }F ,

F = [Θi0 ,Θi1 , · · · ,Θiµ , · · · ,ΘiNt−1]TFN ,

F = [e0Nt⊗ΘT

i0 , e1Nt

⊗ΘTi1 , · · · , e

µNt

⊗ΘTiµ ,

· · · , eNt−1Nt

⊗ΘTiNt−1

]{INt⊗ [FN [IL,0L×(N−L)]

T ]},w = [wT

0 ,wT1 , · · · ,wT

ν , · · · ,wTNr−1]

T .

JIANG et al.: FREQUENCY OFFSET ESTIMATION AND TRAINING SEQUENCE DESIGN FOR MIMO OFDM 3

From (2), the ML estimation of ε can be readily obtained

[20] as follows

ε = argmaxε

{

yH{INr⊗[DN (ε)S(SH S)−1SHDN (−ε)]}y

}

,

(3)

where

S = FHdiag{[sT0 , sT1 , · · · , sTµ , · · · , sTNt−1]T }.

With condition (C0), which implies the orthogonality of the

training sequences in the frequency domain, matrix inversion

involved in (3) can be avoided, and (3) can thus be simplified

as follows

ε = argmaxε

{

||{INr⊗ [FDN (−ε)]}y||2

}

. (4)

Actually, with condition (C0), our proposed training sequences

can be treated as frequency-division multiplexing (FDM) pilot

allocation type sequences [18].

Let l denote the pilot location vector which is given by

l =Nt−1∑

µ=0eiµQ . Then, we have as follows.

Theorem 1: With ε, ε ∈ (−⌊Q/2⌋, Q − ⌊Q/2⌋], ε = εuniquely maximizes the cost function in (4) for any h(ν,µ)(6=0L) with the following condition

(C3) (N −NtP ) ≥ NtP & P ≥ L &

(1Q − l)T l(q) > 0, ∀q ∈ {1, 2, · · · , Q− 1}.

Proof: See Appendix I.

It follows immediately that the corresponding estimation is

identifiable for ε ∈ (−⌊Q/2⌋, Q−⌊Q/2⌋] if condition (C3) is

satisfied.

From (4), the CFO can be estimated by exploiting fast

Fourier transform (FFT) interpolation and a time consuming

line search over a large set of frequency grids. But this

approach is computationally complicated and the estimate

precision also depends on the FFT size used. To reduce

complexity, ε is divided into the ICFO εi and the FCFO εf .

The ICFO εi can be estimated by invoking only the N point

FFT as follows

εi = argmax−⌊Q/2⌋<εi≤Q−⌊Q/2⌋

{

||{INr⊗ [FDN (−εi)]}y||2

}

. (5)

The discussion on the uniqueness of εi = εi in maximizing

the estimation metric for any h(ν,µ)(6= 0L) is provided in

Appendix II. Other ICFO estimators may also be used. Once

the ICFO is estimated, ICFO correction can be readily carried

out as follows

y = e−j2πεiNg/N [INr⊗DN (−εi)]y. (6)

B. FCFO Estimation

The FCFO εf is estimated based on y. Assume εi = εiand substitute (2) into (6). Then, by exploiting condition (C0)

and condition (C1), which implies the periodic property of the

training sequences, y can be expressed in an equivalent form

as shown in (7) at the bottom of the page. In (7),

βµ = εf + iµ,

v = e−j2πεiNg/N [INr⊗DN (−εi)]w.

It follows from (7) that the estimation of εf is equivalent to

the estimation of the Nt different equivalent CFOs {βµ}Nt−1µ=0 .

Furthermore, exploiting the periodic property of the training

sequences again, we can stack y into the Q × NrP matrix

Y = [Y0,Y1, · · · ,Yν , · · ·YNr−1], where

[Yν ]q,p = [((eνNr)T ⊗ IN )y]qP+p, 0 ≤ q < Q, 0 ≤ p < P.

Then, (7) can be expressed in the following equivalent form

Y = BX + V , (8)

where

B = [b0, b1, · · · , bµ, · · · , bNt−1],

bµ = [1, ej2πβµ/Q, · · · , ej2πβµq/Q, · · · , ej2πβµ(Q−1)/Q]T ,

X = [X0,X1, · · · ,Xν , · · · ,XNr−1],

Xν = [x(ν,0),x(ν,1), · · · ,x(ν,µ), · · · ,x(ν,Nt−1)]T ,

x(ν,µ) =√Pej2πεfNg/NDP (βµ)

×FHP diag{sµ}ΘT

iµFN [IL,0L×(N−L)]Th(ν,µ),

and V is the Q×NrP matrix generated from v in the same

way as Y . From (8), we can see that there exists an inherent

relationship between FCFO estimation and DOA estimation in

ULA [21] with P , Q and Nr satisfying NrP > Q (C4).

The covariance matrix of Y can be estimated by RY Y =Y Y H/(NrP ). Let L denote the Q × Q unitary column

conjugate symmetric matrix which is given by

L =[1− (Q)2]√

2

[

IQ/2 jIQ/2JQ/2 −jJQ/2

]

+(Q)2√

2

I(Q−1)/2 0(Q−1)/2 jI(Q−1)/2

0T(Q−1)/2

√2 0T(Q−1)/2

J(Q−1)/2 0(Q−1)/2 −jJ(Q−1)/2

.

Then, with the aid of L, the complex matrix RY Y can be

transformed into a real matrix [22] as follows

RrY Y = 1/2 ·LH(RY Y + JQR

∗Y Y JQ)L

= ℜ(LHRY Y L). (9)

The eigen-decomposition of the real matrix RrY Y

can be

y =√Nej2πεfNg/N

{

INr⊗ {[DN (β0),DN (β1), · · · ,DN (βµ), · · · ,DN(βNt−1)]

× (INt⊗ 1Q ⊗ FH

P )diag{[sT0 , sT1 , · · · , sTµ , · · · , sTNt−1]T }F }

}

h+ v, (7)


obtained as

RrY Y

= EXΛXEHX

+EV ΛV EHV, (10)

where

ΛX = diag{[λ0, λ1, · · · , λNt−1]T },

ΛV = σ2wIQ−Nt

,

λ0 ≥ λ1 ≥ · · · ≥ λNt−1 > σ2w,

and EX and EV contain the unitary eigen-vectors that span

the signal space and noise space, respectively. Let

z = ej2πβ/Q,

a(z) = [1, z, · · · , zQ−1]T .

Then, by exploiting a(z), a polynomial of degree 2(Q − 1)with complex coefficients can be obtained [23] as follows

f(z) = aT (z)JQAa(z), (11)

where

A = L[IQ −EXEHX]LH .

Correspondingly, {βµ}Nt−1µ=0 can be indirectly estimated by

calculating the pairwise roots of f(z) = 0 which are closest

to the unit circle.

Note that by its definition, z is always located on the

unit circle. By analysis, we find that the following geometric

mapping exists,

g(z) = cot(πβ/Q) = j(z + 1)/(z − 1). (12)

It follows immediately that g(z) is a monotonic reversible

mapping with real values for β ∈ [−0.5, Q − 0.5). Hence,

we have z(g) = (g+ j)/(g− j). Then, a(z) can be expressed

as a function of g as

a(z) = (g − j)1−Qd(g), (13)

where

d(g) = [(g − j)Q−1, (g + j)(g − j)Q−2, · · · , (g + j)Q−1]T .

Since the elements of d(g) are polynomials of degree (Q−1)with respect to g, d(g) in (13) can be further expressed as

d(g) = Φa(g), (14)

where Φ is the Q×Q coefficient matrix. By straight-forward

calculation, the elements of Φ are obtained as shown in

(15). In (15), Cq′′

q = q!/[(q − q′′)!q′′!]. Then, from (15), we

immediately obtain

[Φ]q,q′ = [Φ]∗Q−1−q,q′ & (Q)2ℑ{

[Φ](Q−1)/2,q

}

= 0,

∀q, q′ ∈ {0, 1, · · · , Q− 1}, (16)

which shows that Φ is a column conjugate symmetric matrix

no matter Q is even or odd.

By exploiting (13) and (14) in (11), the following equivalent

polynomial can be obtained (ignoring the constant items)

f r(g) = aT (g)JQAra(g), (17)

where

Ar = JQΦHL[IQ −EXEH

X]LHΦ.

Due to the column conjugate symmetric property of L and

Φ, we have (18) as shown at the bottom of the page. Hence,

ΦHL becomes a real matrix no matter Q is even or odd.

Moreover, EX is also a real matrix. Accordingly, f r(g) is

transformed into a polynomial with real coefficients. The roots

of f r(g) = 0 can be obtained by the fast root-calculating

algorithms for real polynomials in [24], whose computational

complexities are much less than those for complex ones.

After the roots of f r(g) = 0 are obtained, the FCFO can

be readily estimated according to the following steps.

1) Find the Nt pairwise roots of f r(g) = 0 whose imagi-

nary parts have the smallest absolute values, {ℜ(gµ)±jℑ(gµ)}Nt−1

µ=0 .

2) Calculate the equivalent CFOs corresponding to the Ntroots,

βµ = (Q/π × acot[ℜ(gµ)])Q . (19)

3) Calculate the FCFO εfµ corresponding to each βµ,

εfµ =

βµ − i0, βµ ∈ ((i0 − ǫth)Q, i0 + ǫth)

βµ − i1, βµ ∈ (i1 − ǫth, i1 + ǫth)· · · · · ·βµ − iNt−1, βµ ∈ (iNt−1 − ǫth,

(iNt−1 + ǫth)Q)

,

(20)

where ǫth denotes a threshold which is predefined to

avoid the ambiguous estimation and 0.5 < ǫth < 1,

[Φ]q,q′ = jQ−1−q′min{q,q′}∑

q′′=max{0,q+q′−Q+1}

{

Cq′′

q Cq′−q′′

Q−1−q(−1)Q−1−q−q′+q′′}

, 0 ≤ q, q′ ≤ Q− 1, (15)

[ΦHL]q,q′ =

[1−(Q)2]Q/2+(Q)2(Q−1)/2−1∑

q′′=0

{

[Φ]∗q′′,q[L]q′′,q′ + [Φ]q′′,q[L]∗q′′,q′}

+ (Q)2[Φ](Q−1)/2,q[L](Q−1)/2,q′

= 2ℜ

[1−(Q)2]Q/2+(Q)2(Q−1)/2−1∑

q′′=0

{

[Φ]∗q′′,q[L]q′′,q′}

+ (Q)2[Φ](Q−1)/2,q[L](Q−1)/2,q′ ,

∀q, q′ ∈ {0, 1, · · · , Q− 1}. (18)


[a, b) has the usual meaning for a < b, and [a, b) =[0, b) ∪ [a,Q) for a > b.

4) Obtain the final FCFO by averaging {εfµ}Nt−1µ=0 ,

εf =1

Nt

Nt−1∑

µ=0

εfµ. (21)

C. Computational Complexity

The computational load of our proposed CFO estimator

mainly involves the N point FFT, the eigen-decomposition

of RrY Y

and the root-calculation for f r(g) = 0, which

require 4N log2N , 9Q3 and 64/3 · (Q − 1)3 real additions

or multiplications [24] [25], respectively. Compared with the

direct CFO estimate from (4), which requires a large point

DFT operation and an exhaustive line search, our proposed

estimator has significantly lower complexity especially with

a relatively small Q. Furthermore, by applying the approach

in [26], the complexity of our proposed estimator can be

further decreased by calculating the roots from the first-order

derivative of f r(g) = 0, but at the cost of a slight performance

degradation at low signal-to-noise ratio (SNR).

IV. TRAINING SEQUENCE DESIGN

With our design conditions in the previous section, the

training sequences are determined completely by the base

sequences {sµ}Nt−1µ=0 with fixed P , Q and {iµ}Nt−1

µ=0 . Let

RXX denote the covariance matrix of X . It is pointed out in

[21] [22] that the optimal performance can be achieved with

uncorrelated signals, i.e., diagonal matrix RXX . Since RXX

can be estimated by RXX = XXH/(NrP ), it is expected

that good performance can be achieved with diagonal matrix

RXX . Making RXX to be a diagonal matrix is equivalent to

making [RXX ]µ,µ′

∣

∣

µ6=µ′= 0. Define

µ,µ′ = (iµ − iµ′)/Q,

sµ =1√QFHP sµ.

Assume that the channel taps remain constant during the

training period. Then, [RXX ]µ,µ′ can be expressed as

[RXX ]µ,µ′ =

Q

Nr

Nr−1∑

ν=0

L−1∑

l=0

L−1∑

l′=0

{

[h(ν,µ)]l[T(µ,µ′)]l,l′ [h

(ν,µ′)]∗l′}

, (22)

where

[T (µ,µ′)]l,l′ = (s(l)µ )TDP (µ,µ′Q)(s(l′)µ′ )∗.

Since the elements of h(ν,µ) are random variables,

[RXX ]µ,µ′

∣

∣

µ6=µ′= 0 can be achieved by making the training

sequences satisfy

[T (µ,µ′)]l,l′ = 0, if µ 6= µ′ & 0 ≤ l, l′ ≤ L− 1. (23)

It follows immediately that the optimal training sequences

should satisfy the condition in (23). However, with condition

(C0), µ,µ′ is definitely a decimal fraction, which greatly

complicates the satisfaction of the condition in (23) with

proper training sequences. To ease the above problem, we

construct two types of sub-optimal training sequences which

can make the off-diagonal elements of RXX as small as

possible. It has been proven in [15] that training sequence

with the zero auto-correlation (ZAC) property is optimal for

CFO estimation in SISO frequency selective fading channels.

Besides, constant amplitude ZAC (CAZAC) sequence (e.g.,

[19], [27] and references therein) is often a preferred choice

for training. Therefore, we propose to construct the training

sequences from a length-P Chu sequence s with its element

given by

[s]p = ejπvp2/P , 0 ≤ p ≤ P − 1,

where v is coprime to P , and P is supposed to be even.

Let sµ =√

Q/NtFPs(µM) with M = ⌊P/NI⌋ and NI ≥

Nt. Then, we refer to the so-constructed training sequences

as TS 0. Let sµ =√

Q/NtFPs. Then, we refer to the so-

constructed training sequences as TS 1. Note that the TS 1

training sequences are equivalent to the so-called repeated

phase-rotated Chu (RPC) sequences [28]. Define

pµ,l = (1 −m)µM + l,

where m = 0 for TS 0, and m = 1 for TS 1. Then, from the

above constructions we have

[T (µ,µ′)]l,l′ =1

Nt(−1)v(pµ,l−pµ′,l′ )+1ejπv(p

2

µ,l−p2

µ′,l′)/P

×e−jπ(P−1)[v(pµ,l−pµ′,l′ )−µ,µ′ ]/P sin(πµ,µ′)

/ sin{π[v(pµ,l − pµ′,l′)−µ,µ′ ]/P}. (24)

It follows immediately from (24) that

∣

∣[T (µ,µ′)]l,l′∣

∣

pµ,l−pµ′,l′ 6=0

≪∣

∣[T (µ,µ′)]l,l′∣

∣

pµ,l−pµ′,l′=0< P/Nt. (25)

We see that∣

∣[T (µ,µ′)]l,l′∣

∣ achieves its maximum for TS 0 when

(µ − µ′)M = l′ − l and for TS 1 when l = l′. Assume that

the channel energy is mainly concentrated in the preceding

M channel taps and the first channel tap is the dominant one.

Then, it can be inferred from (22) and (25) that the value

of∣

∣[RXX ]µ,µ′

∣

∣

µ6=µ′for TS 0 is very small, and it is much

smaller than that for TS 1 in the same channel environment. In

this sense, we can say that TS 0 is superior to TS 1, which will

be verified through simulation results in the following section.

Note that our proposed training sequence structure is similar

to the one introduced recently in [16], and the identifiability

of our CFO estimator, however, cannot be guaranteed with the

training sequences in [16].

V. SIMULATION RESULTS

To evaluate our CFO estimator’s performance with the pro-

posed training sequences for MIMO-OFDM, a number of sim-

ulations are carried out. Throughout the simulations, a MIMO

OFDM system of bandwidth 20MHz operating at 5GHz with

N = 1024 and Ng = 64 is used. Each channel has 4 inde-

pendent Rayleigh fading taps, whose relative average-powers

and propagation delays are {0,−9.7,−19.2,−22.8}dB and

{0, 0.1, 0.2, 0.4}µs, respectively. For the training sequences,


we set P and Q to 64 and 16, respectively. The normalized

CFO ε is generated within the range (−⌊Q/2⌋, Q− ⌊Q/2⌋].For description convenience, we henceforth refer to the pro-

posed real polynomial based CFO estimator as RPBE.

In the following, we use the average CRB (avCRB), which

corresponds to the extended Miller and Chang bound (EMCB)

[15] [29], to benchmark the performance of RPBE. The

average CRB or EMCB is obtained by simply averaging the

snapshot CRB over independent channel realizations, which

can be calculated straight-forwardly [20] as follows

CRBε =

Nσ2w

8π2hHXHB[INrN −X (XH

X )−1XH ]BXh

, (26)

where

X = INr⊗ S,

B = INr⊗ diag{[Ng, Ng + 1, · · · , Ng +N − 1]T }.

In order to verify our analysis concerning training sequence

design with the Chu sequence, we construct the random

sequences (RS) by generating the NtP pilots randomly, and

compare the MSE performances of RPBE with TS 0, TS 1

and RS. Also included for comparison are the performances

of the CFO estimators with their training sequences in [9]

and [10]. In Fig. 1, we present the corresponding simulation

results with Nt = 3 and Nr = 2. It can be observed that

the performance of RPBE with TS 0 is slightly better than

that with TS 1, which coincides with the analytical results

concerning training sequence design in the previous section.

It can also be observed that the performances of RPBE with

TS 0 and TS 1 are better than that with RS, which should

be attributed to the good correlation property of TS 0 and TS

1. Actually, the performance improvements are more evident

when the frequency selective fading channels with large delay

spreads are considered. Furthermore, we observe that the

0 4 8 12 16 2010-6

10-5

10-4

10-3

10-2

MS

E

SNR (dB)

sequences in [9] sequences in [10] TS 0 TS 1 RS avCRB (TS 0) avCRB (TS 1) avCRB (RS)

Fig. 1. CFO estimation performance for different training

sequences with Nt = 3 and Nr = 2.

performances of RPBE with TS 0 and TS 1 are far better than

that of the CFO estimator in [9], and almost the same as that

of the CFO estimator in [10]. Noting that the overhead of the

training sequences presented in [10] grows linearly with the

number of transmit antennas, we can find certain advantages

of the proposed training sequences. We also observe that the

average CRB for TS 0 is slightly smaller than that for TS 1.

Since the average CRBs for TS 0 and TS 1 are very close,

only one curve is plotted subsequently.

Depicted in Fig. 2 is the MSE performance of RPBE as a

function of SNR for different Q with Nt = 3 and Nr = 2.

Studying the curves in Fig. 2, we can see that the performance

of RPBE with the same training sequences degrades with

smaller Q provided that Q > Nt and P ≥ L, which agrees

well with the analysis in [21]. We can also see that a smaller

Q yields a larger average CRB, and the MSE performance

follows the same trend as the average CRB.

In Fig. 3, we illustrate how the number of transmit antennas

Nt affects the performance of RPBE. It can be observed that

the MSE performance for TS 1 deteriorates with increased Ntdue to the influence of MAI, whereas the MSE performance

for TS 0 degrades slightly, which should be ascribed to the

better correlation property of TS 0. Another observation is

that the number of transmit antennas has little impact on the

average CRB.

Fig. 4 shows how the number of receive antennas Nr affects

the performance of RPBE. We can observe that the MSE

performance of RPBE is substantially improved for both TS 0

and TS 1 by increasing Nr. We have the similar observation

for the average CRB. These observations imply that it is more

efficient for performance improvement of RPBE to increase

the number of receive antennas than the number of transmit

antennas.

VI. CONCLUSIONS

In this paper, we have presented a training sequence assisted

CFO estimator for MIMO OFDM systems by exploiting the

0 4 8 12 16 2010-6

10-5

10-4

10-3

MS

E

SNR (dB)

TS 0 (P=64, Q=16) TS 1 (P=64, Q=16) TS 0 (P=128, Q=8) TS 1 (P=128, Q=8) avCRB (P=64, Q=16) avCRB (P=128, Q=8)

Fig. 2. CFO estimation performance for different Q with

Nt = 3 and Nr = 2.


0 4 8 12 16 2010-6

10-5

10-4

10-3

M

SE

SNR (dB)

TS 0 (Nt=2) TS 1 (Nt=2) TS 0 (Nt=3) TS 1 (Nt=3) TS 0 (Nt=4) TS 1 (Nt=4) avCRB(2x2) avCRB(3x2) avCRB(4x2)

Fig. 3. CFO estimation performance for different numbers of

transmit antennas with Nr = 2.

0 4 8 12 16 20

10-6

10-5

10-4

10-3

MS

E

SNR (dB)

TS 0 (Nr=2) TS 1 (Nr=2) TS 0 (Nr=3) TS 1 (Nr=3) TS 0 (Nr=4) TS 1 (Nr=4) avCRB(3x2) avCRB(3x3) avCRB(3x4)

Fig. 4. CFO estimation performance for different numbers of

receive antennas with Nt = 3.

properties of the training sequences and an efficient geometric

mapping. We have developed the required conditions for

the training sequences to yield estimation identifiability, and

proposed sub-optimal training sequences constructed from the

Chu sequence. Our proposed estimator and training sequences

yield better or similar estimation performance with much

smaller training overhead than existing methods for MIMO-

OFDM systems.

APPENDIX I

This appendix presents the proof of Theorem 1. Substitute

(2) into the cost function in (4) and ignore the noise item.

Define ∆ = ε − ε. Then, the cost function in (4) can be

expressed as

P(∆) =

NhH{INr⊗ [SHDN (−∆)FH FDN (∆)S]}h. (27)

Let

S = diag{[sT0 , sT1 , · · · , sTµ , · · · , sTNt−1]T }.

Define

D(∆) = NhH [INr⊗ (FH SHSF )]h−P(∆).

Then, it follows immediately that the maximum of P(∆)corresponds to the minimum of D(∆) with respect to ∆.

Suppose

0 ≤ ω < Q−Nt & 0 ≤ zω < Q & zω 6= iµ &

zω = zω′ , iff. ω = ω′.

Then, D(∆) can be expressed as

D(∆) = dH(∆)d(∆), (28)

where

d(∆) = [dT0 (∆),dT1 (∆), · · · ,dTν (∆), · · · ,dTNr−1(∆)]T ,

dν(∆) = CSF hν ,

C =

C(0,0) · · · C

(0,Nt−1)

... C(ω,µ)

...

C(Q−Nt−1,0)· · · C

(Q−Nt−1,Nt−1)

,

C(ω,µ) =

√NΘT

zωFNDN (∆)FHN Θiµ .

Correspondingly, we have

D(∆) ≥ 0 & D(0) = 0 &

D(∆) = 0, iff. d(∆) = 0Nr(N−NtP ). (29)

Therefore, for any h(ν,µ)(6= 0L), proving that ε = ε is the

unique value to maximize P(∆) with ε, ε ∈ (−⌊Q/2⌋, Q −⌊Q/2⌋] is equivalent to proving that d(∆) 6= 0Nr(N−NtP ) for

any ∆ ∈ (−Q, 0) ∪ (0, Q).

Consider the following two cases:

1) When ∆ is not an integer: We establish immediately

from the definition of C(ω,µ) that it is a column-wise circulant

matrix. Denote the square matrix formed by the first NtP rows

of C by CP (recall that (N − NtP ) ≥ NtP in our design

condition). Then, by exploiting the relation between circulant

Nt−1∏

µ=1

µ−1∏

µ′=0

{

(bµ − bµ′)(aµ′ − aµ)}

=

Nt−1∏

µ=2

µ−1∏

µ′=1

(bµ − bµ′)

×Nt−1∑

µ′′=0

{

(−1)µ′′ ·

Nt−1∏

µ=1,µ6=µ′′

µ−1∏

µ′=0,µ′ 6=µ′′

(aµ′ − aµ) ·Nt−1∏

µ=1

(aµ′′ − bµ) ·Nt−1∏

µ=0,µ6=µ′′

(aµ − b0)

}

. (34)


matrix and DFT matrix [25], CP can be decomposed as

CP =√P (INt

⊗ FHP )Λ(INt

⊗ FP ), (30)

where

Λ =

Λ(0,0) · · · Λ(0,Nt−1)

... Λ(µ,µ′)...

Λ(Nt−1,0)· · · Λ(Nt−1,Nt−1)

,

Λ(µ,µ′) = diag{c(µ,µ′)},c(µ,µ

′) = FPC(µ,µ′)e0P .

The element of c(µ,µ′) can be obtained from its definition as

follows

[c(µ,µ′)]p =

1√Qc(µ,µ

′)(1− ej2π∆)ej2π(iµ′−zµ+∆)(−p)P /N

6= 0, 0 ≤ p ≤ P − 1, (31)

where

c(µ,µ′) = aµ/(aµ − bµ′),

aµ = ej2πzµ/Q,

bµ′ = ej2π(iµ′+∆)/Q.

It follows from (32) that rank{Λ(µ,µ′)} = P . Let

Λp = [INt⊗ (epP )

T ]Λ[INt⊗ (epP )

T ]T ,

which denotes the Nt × Nt sub-matrix of Λ. Then, we can

decompose Λp as follows

Λp =1√Q(1− ej2π∆)Λa

pCNtΛbp, (32)

where

Λap = diag{a−(−p)P /P

0 , · · · , a−(−p)P /PNt−1 },

Λbp = diag{b(−p)P/P0 , · · · , b(−p)P /PNt−1 },

and CNtis the Nt×Nt square matrix with its element given

by [CNt]µ,µ′ = c(µ,µ

′). From the definitions of aµ and bµ′ ,

we have

aµ 6= 0 & aµ 6= bµ &

aµ 6= aµ′ , bµ 6= bµ′ , aµ 6= bµ′ , ∀µ 6= µ′. (33)

Moreover, we also have (34) as shown at the bottom of the

page. Then, with the assumption that Nt and Q are not very

large (for example Nt ≤ 8, Q = 16), the determinant of CNt

can be obtained with its definition as follows

det{CNt} =

Nt−1∏

µ=1

µ−1∏

µ′=0

Nt−1∏

µ′′=0

(bµ − bµ′)(aµ′ − aµ)aµ′′

(aµ − bµ′)(aµ′ − bµ)(aµ′′ − bµ′′)

6= 0, (35)

which shows that rank{CNt} = Nt. From (32), we immedi-

ately obtain that rank{Λp} = Nt. With the special diagonal

structure of Λ, we establish that rank{Λ} = NtP and then

rank{C} = NtP . Exploiting the following relationship

rank{A0}+ rank{A1} − n ≤ rank{A0A1}≤ min{rank{A0}, rank{A1}}, (36)

where A0 has n columns and A1 has n rows, we further

establish that rank{CSF } = NtL for any non-integer ∆(recall that P ≥ L in our design condition). Hence, for any

h(ν,µ)(6= 0L), we immediately obtain from its definition that

dν(∆) 6= 0N−NtP and then d(∆) 6= 0Nr(N−NtP ).

2) When ∆ is an integer: By exploiting the structure of C,

dν(∆) can be transformed into the following equivalent form

dν(∆) = [(d(ν,0)(∆))T , (d(ν,1)(∆))T ,

· · · , (d(ν,ω)(∆))T , · · · , (d(ν,Q−Nt−1)(∆))T ]T , (37)

where

d(ν,ω)(∆) =Nt−1∑

µ=0

{

C(ω,µ)diag{sµ}ΘT

iµFN [IL,0L×(N−L)]Th(ν,µ)

}

.

For any integer ∆ ∈ (−Q, 0) ∪ (0, Q), with our design

condition we have

(1Q − l)T l(∆) > 0. (38)

Without loss of generality, suppose (iµ + ∆)Q = zω. Then,

together with condition (C0), which implies iµ = iµ′ iff. µ =µ′, we have

C(ω,µ) =

√NIP & C

(ω,µ′) = 0P×P , if µ′ 6= µ. (39)

Hence,

d(ν,ω)(∆) =√Ndiag{sµ}ΘT

iµFN [IL,0L×(N−L)]Th(ν,µ). (40)

With the condition P ≥ L, we immediately obtain

d(ν,ω)(∆) 6= 0P and then d(∆) 6= 0Nr(N−NtP ) for any

h(ν,µ)(6= 0L).Combining the above two cases, we draw the conclusion

that d(∆) 6= 0Nr(N−NtP ) for any h(ν,µ)(6= 0L) and any ∆ ∈(−Q, 0) ∪ (0, Q). This completes the proof.

APPENDIX II

In this appendix, we will discuss the uniqueness of εi = εiin maximizing the estimation metric for any h(ν,µ)(6= 0L). For

the case that ε = εi, it follows immediately from Theorem 1

that the uniqueness of εi = εi in maximizing the estimation

metric for any h(ν,µ)(6= 0L) is guaranteed. Therefore, in the

following, we only consider the case that ε 6= εi, i.e., εf 6= 0.

Define ∆i = ε − εi. Then, the estimation metric P(∆i)can be written into an equivalent form as shown in (41) at the

bottom of the next page. In (41),

M (µ,µ′,µ′′)(∆i) = SHµ Gµ′(∆i)Sµ′′ ,

Gµ(∆i) = DN (−∆i)FHN ΘiµΘ

TiµFNDN (∆i),

Sµ =√NFH

N Θiµdiag{sµ}ΘTiµFN [IL,0L×(N−L)]

T .

From the definition of Gµ(∆i), we can obtain

Gµ(∆i) =


G(0,0)µ (∆i) · · · G

(0,Q−1)µ (∆i)

... G(q,q′)µ (∆i)

...

G(Q−1,0)µ (∆i) · · · G

(Q−1,Q−1)µ (∆i)

,

(42)

where

G(q,q′)µ (∆i) = Q−1ej2π(q−q

′)(iµ−∆i)/Q · IP .

Define

sµ =1√QFHP sµ.

Then, we can also obtain that Sµ is an N × L column-wise

circulant matrix with [DQ(iµP )1Q] ⊗ [DP (iµ)sµ] being its

first column vector. Hence, we have

[M (µ,µ′,µ′′)(∆i)]l,l′ = α(µ, µ′′; l, l′)/Q

×{Q−1∑

q=0

Q−q−1∑

q′=0

{

ejq′ψejqθ + ej(q+q

′)ψe−jqθ}

−Q−1∑

q=0

ejqψ}

,

(43)

where

α(µ, µ′′; l, l′) = ej2π(liµ−l′iµ′′ )/N · (s(l)µ )HDP (iµ′′ − iµ)s

(l′)µ′′ ,

θ = 2π(iµ′ − iµ −∆i)/Q,

ψ = 2π(iµ′′ − iµ)/Q.

When µ = µ′′, imposing the condition that the elements

of sµ have constant amplitude, which translates to the time-

domain zero auto-correlation (ZAC) property of sµ [15], from

(43) we immediately obtain (44) as shown at the bottom of

the page. When µ 6= µ′′, from (43) we obtain (45) as shown

at the bottom of the page. Define

ζ(µ,µ′,µ′′)(∆i) =

∣

∣

∣[M (µ,µ′,µ)(∆i)]l,l/[M

(µ,µ′,µ′′)(∆i)]l,l′∣

∣

∣.

Then, we immediately establish

ζ(µ,µ′,µ′′)(∆i) = P/[Nt · |α(µ, µ′′; l, l′)| ]

× | sin(ψ/2)| · | cot(ψ/2)− cot(θ/2)|. (46)

The following remarks are now in order.

Remark 1: Both [M (µ,µ′,µ)(∆i)]l,l and [M (µ,µ′,µ′′)(∆i)]l,l′are the period-Q functions with respect to ∆i. Fix the true

CFO ε and vary the candidate ICFO εi between (−⌊Q/2⌋, Q−⌊Q/2⌋]. Then, we have

∆i ∈ [ε−Q+ ⌊Q/2⌋, ε+ ⌊Q/2⌋) ⊂ (−Q,Q). (47)

For description convenience, we employ r(∆i) to normalize

∆i from [ε−Q+⌊Q/2⌋, ε+⌊Q/2⌋) to [−⌊Q/2⌋, Q−⌊Q/2⌋),where

r(∆i) = ∆i − [sign(∆i −Q+ ⌊Q/2⌋)+ sign(∆i + ⌊Q/2⌋)]Q/2. (48)

Therefore, [M (µ,µ′,µ)(∆i)]l,l achieves its minimum 0 when

r(∆i) ∈ {−⌊Q/2⌋,−⌊Q/2⌋+1, · · · , Q−⌊Q/2⌋−1}\{r(iµ′−iµ)} and its maximum N/Nt when r(∆i) = r(iµ′−iµ). While∣

∣[M (µ,µ′,µ′′)(∆i)]l,l′∣

∣ achieves its minimum 0 when r(∆i) ∈{−⌊Q/2⌋,−⌊Q/2⌋+ 1, · · · , Q− ⌊Q/2⌋ − 1}.

Remark 2: Both [M (µ,µ′,µ)(∆i)]l,l and [M (µ,µ′,µ′′)(∆i)]l,l′

are the continuous functions with respect to ∆i. For any γ > 0(for example γ = 10−2), there exists δ > 0 (for example

δ = 0.1) which makes the relationships as shown in (49) and

(50) at the bottom of the next page hold.

Remark 3: Assume 0 < |εf | < 0.5. Then, for r(εi − εi) ∈{−⌊Q/2⌋,−⌊Q/2⌋+ 1, · · · , Q − ⌊Q/2⌋ − 1}, we have (51)

as shown at the bottom of the next page.

Remark 4: ζ(µ,µ′,µ′′)(∆i) is the period-Q continuous func-

tion with respect to ∆i, which achieves its minimum 0 when

r(∆i) = r(iµ′ − iµ′′) and its maximum +∞ when r(∆i) =r(iµ′ − iµ). Impose the condition that |α(µ, µ′′; l, l′)| < 1/Ntfor µ 6= µ′′. Then, there exists χ > 0 (for example χ = 5)

which makes the relationship as shown in (52) at the bottom

of the next page hold.

We use an example as shown in Fig. 5 to illustrate the above

remarks.

P(∆i) =

Nr−1∑

ν=0

Nt−1∑

µ=0

Nt−1∑

µ′=0

{

(h(ν,µ))HM (µ,µ′,µ)(∆i)h(ν,µ)

}

+

Nr−1∑

ν=0

Nt−1∑

µ=0

Nt−1∑

µ′=0

Nt−1∑

µ′′=0,µ′′ 6=µ

{

(h(ν,µ))HM (µ,µ′,µ′′)(∆i)h(ν,µ′′)

}

, (41)

M (µ,µ′,µ)(∆i) =

{

P

Nt+

2P

NtQℜ{

[(Q− 1)ejθ −Qej2θ + ej(Q+1)θ]/(1− ejθ)2}

}

·IL

=P

NtQ· 1− cos(Qθ)

1− cos θ· IL. (44)

[M (µ,µ′,µ′′)(∆i)]l,l′ = α(µ, µ′′; l, l′)/Q · (1− ejQθ)2/[(1− ejθ)(ejψ − ejθ)ej(Q−1)θ]

= α(µ, µ′′; l, l′)/Q · 1− cos(Qθ)

cos(ψ/2)− cos(θ − ψ/2). (45)


Assume

E{[h(ν,µ)]∗l [h(ν′,µ′)]l′} = 0, ∀(ν, µ) 6= (ν′, µ′).

Then, it follows from Remark 2 and 4 that (53) and (54) as

shown at the bottom of the page can be obtained, respectively.

According to its definition, [M (µ,µ′,µ)(∆i)]l,l must be one of

the Q possible values whatever iµ or iµ′ is when we vary

εi from (−⌊Q/2⌋+ 1) to (Q− ⌊Q/2⌋). From Remark 3, we

immediately obtain that

[M (µ,µ′,µ)(εf )]l,l

/

[M (µ,µ′,µ)(εf + (iµ′ − iµ)Q)]l,l ≫ 1,

if (iµ′ − iµ)Q > 1, (55)

and the items involved in the right hand side of (54) achieve

their maximum when r(εi − εi) = r(iµ′ − iµ) for any

h(ν,µ)(6= 0L). Then, we can establish that there are NrNtitems involved in the right hand side of (54) which achieve the

maximum when εi = εi (i.e., µ = µ′), and at most Nr(

Nt −min

1≤q≤Q−1

{

(1Q − l)T l(q)} )

items that achieve the maximum

when εi 6= εi (i.e., µ 6= µ′). Due to the random snapshot

channel energies, a closed form of training design conditions

to yield the uniqueness of εi = εi is intractable. However, it

follows from condition (C3) and the above analysis that the

reliability of the uniqueness can be maximized by designing

the training sequences to maximize minµ′ 6=µ

{

(iµ′ − iµ)Q}

and

min1≤q≤Q−1

{

(1Q − l)T l(q)}

.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers

for their valuable comments which helped to improve the

quality of the paper greatly.

REFERENCES

[1] G. L. Stuber, J. R. Barry, S. Mclaughlin, Y. Li, M. A. Ingram,and T. G. Pratt, “Broadband MIMO-OFDM wireless communications,”Proceedings of the IEEE, vol. 92, no. 2, pp. 271–294, Feb. 2004.

[2] P. Moose, “A technique for orthogonal frequency division multiplexingfrequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908–2914, Oct. 1994.

[3] J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of timeand frequency offset in OFDM systems,” IEEE Trans. Signal Processing,vol. 45, pp. 1800–1805, July 1997.

[4] U. Tureli, H. Liu, and M. D. Zoltowski, “OFDM blind carrier offsetestimation: ESPRIT,” IEEE Trans. Commun., vol. 48, pp. 1459–1461,Sept. 2000.

[M (µ,µ′,µ)(∆i)]l,l < γ, if r(∆i) ∈Q−⌊Q/2⌋

⋃

q=−⌊Q/2⌋

{

(q − δ, q) ∪ (q, q + δ)}

\{

(r(iµ′ − iµ)− δ, r(iµ′ − iµ) + δ)}

\{

(−⌊Q/2⌋ − δ,−⌊Q/2⌋)∪ (Q− ⌊Q/2⌋, Q− ⌊Q/2⌋+ δ)}

, (49)

∣

∣[M (µ,µ′,µ′′)(∆i)]l,l′∣

∣< γ, if r(∆i) ∈Q−⌊Q/2⌋

⋃

q=−⌊Q/2⌋

{

(q − δ, q) ∪ (q, q + δ)}

\{

(−⌊Q/2⌋ − δ,−⌊Q/2⌋)∪ (Q− ⌊Q/2⌋, Q− ⌊Q/2⌋+ δ)}

. (50)

[M (µ,µ′,µ)(∆i|r(εi − εi) = r(iµ′ − iµ))]l,l

/

[M (µ,µ′,µ)(∆i|r(εi − εi) 6= r(iµ′ − iµ))]l,l

= sin2{π[εf + r(εi − εi)− r(iµ′ − iµ)]/Q}/sin2(πεf/Q)∣

∣

r(εi−εi) 6=r(iµ′−iµ)> 1. (51)

ζ(µ,µ′,µ′′)(∆i) > P · | sin(ψ/2)| · | cot(ψ/2)− cot(ψ/2 + πδ/Q)| > χ,

if r(∆i) ∈{

(−⌊Q/2⌋, r(iµ′ − iµ′′)− δ) ∪ (r(iµ′ − iµ′′) + δ,Q− ⌊Q/2⌋)}

. (52)

P(∆i).= 0, if r(∆i) ∈

Q−⌊Q/2⌋⋃

q=−⌊Q/2⌋

{

(q − δ, q) ∪ (q, q + δ)}

\{

Nt−1⋃

µ,µ′=0

{

(r(iµ′ − iµ)− δ, r(iµ′ − iµ) + δ)}}

\{

(−⌊Q/2⌋ − δ,−⌊Q/2⌋)∪ (Q− ⌊Q/2⌋, Q− ⌊Q/2⌋+ δ)}

, (53)

P(∆i).=

Nr−1∑

ν=0

Nt−1∑

µ=0

Nt−1∑

µ′=0

{

(h(ν,µ))HM (µ,µ′,µ)(∆i)h(ν,µ)

}

,

if r(∆i) ∈Q−⌊Q/2⌋−1

⋃

q=−⌊Q/2⌋

{

[q + δ, q + 1− δ]}

⋃

{

Nt−1⋃

µ,µ′=0

{

(r(iµ′ − iµ)− δ, r(iµ′ − iµ) + δ)}}

. (54)


-8 -4 0 4 810-6

10-3

100

103

106

r( i)

[M( ' )( i)]l,l

|[M( ' '')( i)]l,l'|

( ' '')( i)

Fig. 5. [M (µ,µ′,µ)(∆i)]l,l,∣

∣[M (µ,µ′,µ′′)(∆i)]l,l′∣

∣ and

ζ(µ,µ′,µ′′)(∆i) versus r(∆i) with P = 64, Q = 16, Nt = 3,

iµ = 3, iµ′ = 7, iµ′′ = 4.

[5] M. Morelli and U. Mengali, “An improved frequency offset estimatorfor OFDM applications,” IEEE Commun. Lett., vol. 3, pp. 75–77, Mar.1999.

[6] D. Huang and K. B. Letaief, “Carrier frequency offset estimation forOFDM systems using null subcarriers,” IEEE Trans. Commun., vol. 54,no. 5, pp. 813–823, May 2006.

[7] ——, “Enhanced carrier frequency offset estimation for OFDM usingchannel side information,” IEEE Trans. Wireless Commun., vol. 5,no. 10, pp. 2784–2793, Oct. 2006.

[8] Y. Yao and G. B. Giannakis, “Blind carrier frequency offset estimationin SISO, MIMO and multiuser OFDM systems,” IEEE Trans. Commun.,vol. 53, no. 1, pp. 173–183, Jan. 2005.

[9] A. N. Mody and G. L. Stuber, “Synchronization for MIMO OFDMsystems,” in Proc. IEEE Globecom’01, vol. 1, Nov. 2001, pp. 509–513.

[10] A. van Zelst and T. C. W. Schenk, “Implementation of a MIMO OFDMbased wireless LAN system,” IEEE Trans. Signal Processing, vol. 52,pp. 483–494, Feb. 2004.

[11] O. Besson and P. Stoica, “On parameter estimation of MIMO flat-fading channels with frequency offsets,” IEEE Trans. Signal Processing,vol. 51, no. 3, pp. 602–613, Mar. 2003.

[12] X. Ma, M. K. Oh, G. B. Giannakis, and D. J. Park, “Hopping pilotsfor estimation of frequency-offset and multi-antenna channels in MIMOOFDM,” IEEE Trans. Commun., vol. 53, no. 1, pp. 162–172, Jan. 2005.

[13] F. Simoens and M. Moeneclaey, “Reduced complexity data-aided andcode-aided frequency offset estimation for flat-fading MIMO channels,”IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1558–1567, June 2006.

[14] P. Stoica and O. Besson, “Training sequence design for frequency offsetand frequency-selective channel estimation,” IEEE Trans. Commun.,vol. 51, no. 11, pp. 1910–1917, Nov. 2003.

[15] H. Minn, X. Fu, and V. K. Bhargava, “Optimal periodic training signalfor frequency offset estimation in frequency-selective fading channels,”IEEE Trans. Commun., vol. 54, no. 6, pp. 1081–1096, June 2006.

[16] M. Ghogho and A. Swami, “Training design for multipath channel andfrequency offset estimation in MIMO systems,” IEEE Trans. Signal

Processing, vol. 54, no. 10, pp. 3957–3965, Oct. 2006.

[17] H. Minn, N. Al-Dhahir, and Y. Li, “Optimal training signals for MIMOOFDM channel estimation in the presence of frequency offset and phasenoise,” IEEE Trans. Commun., vol. 54, no. 10, pp. 1754–1759, Oct.2006.

[18] H. Minn and N. Al-Dhahir, “Optimal training signals for MIMO OFDMchannel estimation,” IEEE Trans. Wireless Commun., vol. 5, no. 5, pp.1158–1168, May 2006.

[19] D. Chu, “Polyphase codes with good periodic correlation properties,”IEEE Trans. Inform. Theory, vol. 18, pp. 531–532, July 1972.

[20] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation

Theory. NJ: Prentical-Hall, 1993.

[21] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoustics, Speech, and Signal Processing,vol. 5, pp. 720–741, May 1989.

[22] M. Pesavento, A. B. Gershman, and M. Haardt, “Unitary root-MUSICwith a real-valued eigendecomposition: a theroretical and expreimentalperformance study,” IEEE Trans. Signal Processing, vol. 48, no. 5, pp.1306–1314, May 2000.

[23] Y. X. Jiang, X. Q. Gao, X. H. You, and W. Heng, “Training sequenceassisted frequency offset estimation for MIMO OFDM,” in Proc. IEEE

ICC’06, vol. 12, June 2006, pp. 5371–5376.[24] W. H. Press, Numerical Recipes in C++: the Art of Scientific Computing.

Cambridge: Cambridge University Press, 2002.[25] G. H. Golub and C. F. Van Loan, Matrix Computations. The John

Hopkins University Press, 1996.[26] F. Gao and A. Nallanathan, “Blind maximum likelihood CFO estimation

for OFDM systems via polynomial rooting,” IEEE Signal Processing

Lett., vol. 13, no. 2, pp. 73–76, Feb. 2006.[27] W. H. Mow, “A new unified construction of perfect root-of-unity se-

quences,” in Proc. IEEE Spread-Spectrum Techniques and Applications,vol. 3, Sept. 1996, pp. 955–959.

[28] J. Coon, M. Beach, and J. McGeehan, “Optimal training sequencesfor channel estimation in cyclic-prefix-based single-carrier systems withtransmit diversity,” IEEE Signal Processing Lett., vol. 11, no. 9, pp.729–732, Sept. 2004.

[29] F. Gini and R. Reggiannini, “On the use of Cramer-Rao-like bounds inthe presence of random nuisance parameters,” IEEE Trans. Commun.,vol. 48, no. 12, pp. 2120–2126, Dec. 2000.

Yanxiang Jiang (S’03) received the B.S. degreein electrical engineering from Nanjing University,Nanjing, China, in 1999, and the M.E. degree inradio engineering from Southeast University, Nan-jing, China, in 2003. He is currently working towardthe Ph.D. degree at the National Mobile Communi-cations Research Laboratory, Southeast University,Nanjing, China.

He received the NJU excellent graduate honorfrom Nanjing University in 1999. His research inter-ests include mobile communications, wireless signal

processing, parameter estimation, synchronization, signal design, and digitalimplementation of communication systems.

Hlaing Minn (S’99-M’01) received his B.E. degreein Electronics from Yangon Institute of Technol-ogy, Yangon, Myanmar, in 1995, M.Eng. degree inTelecommunications from Asian Institute of Tech-nology (AIT), Pathumthani, Thailand, in 1997 andPh.D. degree in Electrical Engineering from theUniversity of Victoria, Victoria, BC, Canada, in2001.

He was with the Telecommunications Program inAIT as a laboratory supervisor during 1998. He wasa research assistant from 1999 to 2001 and a post-

doctoral research fellow during 2002 in the Department of Electrical andComputer Engineering at the University of Victoria. Since September 2002, hehas been with the Erik Jonsson School of Engineering and Computer Science,the University of Texas at Dallas, USA, as an Assistant Professor. His researchinterests include wireless communications, statistical signal processing, errorcontrol, detection, estimation, synchronization, signal design, and cross-layerdesign. He is an Editor for the IEEE Transactions on Communications.

Xiqi Gao (SM’07) received the Ph.D. degreein electrical engineering from Southeast University,Nanjing, China, in 1997. He joined the Departmentof Radio Engineering, Southeast University, in April1992. Now he is a professor of information sys-tems and communications. From September 1999to August 2000, he was a visiting scholar at Mas-sachusetts Institute of Technology, Cambridge, andBoston University, Boston, MA. His current researchinterests include broadband multi-carrier transmis-sion for beyond 3G mobile communications, space-

time wireless communications, iterative detection/decoding, signal processingfor wireless communications.

Dr. Gao received the Science and Technology Progress Awards of the StateEducation Ministry of China in 1998 and 2006. He is currently serving as aneditor for the IEEE Transactions on Wireless Communications.


China 863-FuTURE Expert Committee. He has published two books and over20 IEEE journal papers in related areas. His research interests include mobilecommunications, advanced signal processing, and applications.

Yinghui Li (S’05) received the B.E. and M.S.degree in Electrical Engineering from the NanjingUniversity of Aeronautics and Astronautics, Nan-jing, China, in 2000 and 2003, respectively. Sheis currently working toward the Ph.D. degree inElectrical Engineering at University of Texas atDallas. Her research interests are in the applicationsof statistical signal processing in synchronization,channel estimation and detection problems in broad-band wireless communications.

Xiaohu You received the M.S. and Ph.D. degreesin electrical engineering from Southeast University,Nanjing, China, in 1985 and 1988, respectively.

Since 1990 he has been working with NationalMobile Communications Research Laboratory atSoutheast University, where he holds the ranks ofprofessor and director. From 1993 to 1997 he wasengaged, as a team leader, in the development ofChina’s first GSM and CDMA trial systems. He wasthe Premier Foundation Investigator of the ChinaNational Science Foundation in 1998. From 1999 to

2001 he was on leave from Southeast University, working as the chief directorof China’s 3G (C3G) Mobile Communications R&D Project. He is currentlyresponsible for organizing China’s B3G R&D activities under the umbrellaof the National 863 High-Tech Program, and he is also the chairman of the

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