IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, … · design for MIMO-OFDM CFO estimation in...
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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.
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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 .
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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)
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4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, PP. 1244-1254, APR. 2008
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)
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JIANG et al.: FREQUENCY OFFSET ESTIMATION AND TRAINING SEQUENCE DESIGN FOR MIMO OFDM 5
[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,
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6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, PP. 1244-1254, APR. 2008
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.
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JIANG et al.: FREQUENCY OFFSET ESTIMATION AND TRAINING SEQUENCE DESIGN FOR MIMO OFDM 7
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)
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8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, PP. 1244-1254, APR. 2008
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) =
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JIANG et al.: FREQUENCY OFFSET ESTIMATION AND TRAINING SEQUENCE DESIGN FOR MIMO OFDM 9
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)
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10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, PP. 1244-1254, APR. 2008
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.
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[M (µ,µ′,µ)(∆i)]l,l < γ, if r(∆i) ∈Q−⌊Q/2⌋
⋃
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{
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[M (µ,µ′,µ)(∆i|r(εi − εi) = r(iµ′ − iµ))]l,l
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= sin2{π[εf + r(εi − εi)− r(iµ′ − iµ)]/Q}/sin2(πεf/Q)∣
∣
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Q−⌊Q/2⌋⋃
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{
(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)
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JIANG et al.: FREQUENCY OFFSET ESTIMATION AND TRAINING SEQUENCE DESIGN FOR MIMO OFDM 11
-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.
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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.
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12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, PP. 1244-1254, APR. 2008
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