SpringerWPC2013

Wireless Pers Commun (2013) 71:1159–1174
DOI 10.1007/s11277-012-0867-0
RLS-Based Estimation and Tracking of Frequency Offset
and Channel Coefﬁcients in MIMO-OFDM Systems
M. Jamalabdollahi · S. Salari
Published online: 7 October 2012
© Springer Science+Business Media New York 2012
Abstract In this paper, we investigate the problem of carrier frequency-offset (CFO) syn-
chronization and channel estimation in multiple-input multiple-output orthogonal frequency-
division multiplexing systems operating over unknown frequency-selective fading channels.
We ﬁrst propose a novel joint CFO and channel estimator, assuming time-domain training
blocks are available. The proposed joint estimator consists of two recursive least-square
(RLS) algorithms which iterate their estimated CFO and CIR values. We then derive a more
precise pilot-aided RLS algorithm to estimate the residual frequency synchronization errors
or track small CFO changes. With this, the accuracy of channel estimation is also enhanced.
The analysis and simulation results show that, the proposed estimation and tracking scheme
which is fully compatible with the existing standards is able to attain fast convergence, high
stability, and ideal performances as compared with relevant Cramer–Rao bounds in all ranges
of signal-to-noise ratio. Moreover, it can work well for wide tracking range up to ±0.5 of
the subcarrier spacing.
Keywords Channel estimation · Frequency synchronization · MIMO ·
OFDM · RLS algorithm
1 Introduction
Next-generation wireless communication systems are required to support high-data-rate
applications. A novel technique that has attracted a lot of attention is multiple-input multiple-
output (MIMO) combined with orthogonal frequency division multiplexing (OFDM) [1,2].
MIMO offers extraordinary throughput without additional power consumption or bandwidth
M. Jamalabdollahi · S. Salari (B)
Faculty of Electrical and Computer Engineering, K.N. Toosi University of Technology,
P. O. Box: 16315-1355,Tehran 1431714191, Iran
e-mail: salari_soheil@yahoo.com
M. Jamalabdollahi
e-mail: jamalabdollahi@ee.kntu.ac.ir
123

1160 M. Jamalabdollahi, S. Salari
expansion [3], and OFDM introduces overlapping but orthogonal narrowband sub-channels
to convert a frequency selective fading channel into a non-frequency selective one. Moreover,
OFDM avoids inter-symbol interference (ISI) by means of cyclic prefix (CP) [4].
Coherent detection which constitutes the best demodulation principle implies an accu-
rate channel impulse response (CIR) estimation algorithm at the receiver. However, a basic
challenge in MIMO-OFDM system design is an increase in the computational complexity
of CIR estimation algorithm when the number of transmit/receive antennas increases [5]. In
addition, MIMO-OFDM exhibits pronounced sensitivity to carrier frequency offset (CFO),
which introduces inter-carrier interference (ICI) [6]. Hence, the combination of CFO and
CIR estimation leads to particularly complex problems in MIMO-OFDM systems due to the
number of unknowns [7].
In many MIMO-OFDM systems, CIR and CFO estimation are treated separately with the
aid of different training blocks. Generally, the CIR estimation is performed after frequency
synchronization and assumes frequency synchronization is perfectly achieved [2,5,8]. Unfor-
tunately, in practice, such an assumption is rarely valid in the presence of noise [9]. The
residual CFO due to frequency synchronization error will degrade the performance of CIR
estimation significantly [10]. A reasonable solution to this problem is to estimate CFO and
CIR simultaneously based on the common training blocks, which provides a better perfor-
mance at the cost of a higher computational complexity [10,11]. This issue has received a
lot of attentions in OFDM context and several solutions are now available (see, e.g., [11–14]
and references therein). However, all of them are designed for single-input single-output
(SISO)-OFDM systems rather than MIMO-OFDM systems.
Recently, to obtain CFO and CIR simultaneously, expectation-maximization (EM)-based
methods have been received a lot of attention (see, e.g., [15–19] and references therein).
Nguyen-Le et al. [20] which is an extension of their proposed technique in [11], have also
developed a vector recursive-least-square (VRLS)-based scheme for joint synchronization
and channel estimation in MIMO-OFDM systems. Based on the extended H∞ filter (EHF),
[21] have proposed a new method for the problem of joint CFO and channel estimation in
MIMO-OFDM systems. Compared to the conventional extended Kalman filter (EKF)-based
method, the EHF-based method does not require any knowledge of the noise distributions,
so it is more flexible and robust to the unknown disturbances. However, high computational
complexity is the main common drawback of most cited works. Therefore, some approx-
imations and/or simplifications are employed to reduce the computational complexity. For
instance, Salari et al. [19] which is an extension of [18], proposed a simpler method based on
the Taylor series expansion. Carefully observing the simulation results of most cited papers,
it is perceived that, when the initial frequency offset is beyond ±0.2 of subcarrier spacing,
the mean square error (MSE) of both CFO and CIR estimators are started to move away
from the Cramer–Rao bounds (CRBs) as the signal-to-noise ratio (SNR) is increased. Thus,
their tracking ranges are well below half or ±0.5 of the subcarrier spacing. This is due to the
less-accurate results arising from the approximations and/or simplifications made at large
initial frequency offset values.
This paper is our effort to establish a new effective scheme for CFO synchronization and
CIR estimation in MIMO-OFDM systems. We first propose a novel iterative joint CFO and
CIR estimator, assuming time-domain training blocks are available. The proposed joint esti-
mator consists of two RLS algorithms which iterate their estimated CFO and CIR values.
Then, to further improve the CFO estimation accuracy, we derive a more precise pilot-aided
RLS algorithm to estimate the residual errors or track small changes. With this, the accuracy
of channel estimation is also enhanced. Furthermore, to benchmark the performance of the
proposed scheme, the CRBs are derived for both CFO and CIR estimators. It is worthy to
123

RLS-Based Estimation and Tracking of Frequency Offset 1161
mention that the proposed CFO synchronization and CIR estimation scheme is fully compat-
ible with the existing OFDM standards. It is also capable to accommodate any space–time
coded (STC)-OFDM transmission. Based on simulation results, the performance of the joint
ML estimator is quite satisfying. Simulation results show that both CFO and CIR MSEs
tightly close to the CRBs, even at high SNR values rather than departing from CRBs as the
SNR is increased. Moreover, the proposed scheme can work well for initial frequency offset
up to 0.5 of the subcarrier spacing. Therefore, it can be used to cover a wide tracking range.
The rest of the paper is organized as follows. In the next Section, we describe the MIMO-
OFDM signal model. Section 3 deals with the time-domain joint estimation of frequency-
offset and MIMO channel coefficients. The pilot-aided CFO tracking algorithm is derived in
Sect. 4. CRBs for both channel and frequency offset estimators, algorithm complexity and
simulation results are discussed in Sect. 5 and finally, the main results are summarized in
Sect. 6.
2 System Model
We consider an MIMO-OFDM communication system with K subcarriers, N transmit (TX)
and M receive (RX) antennas, signaling through an L-tap frequency-selective fading channel.
A block diagram of the transmitter is shown in Fig. 1. The MIMO-OFDM transmitter
consists of N OFDM transmitters [8], among which each input stream is being grouped by
the serial-to-parallel (S/P) converter to yield N sequences of KD data symbols, and then, each
branch in parallel performs training symbol insertion, and K-point inverse discrete Fourier
transformation (IDFT) and adds a CP of length Kg before the final TX signal is up-converted
to radio frequency (RF) centered at fc and transmitted.
Let us define the mth OFDM block to be transmitted from uth transmit antenna as Xu,m =
[Xu,m(1), . . . , Xu,m(K)]T
K×1, where Xu,m(k) denotes the mth transmitted symbol from the
uth transmit antenna at the kth subcarrier. The corresponding time-domain symbol is given
by
xu,m(n) =
1
√
K
K
k=1
Xu,m(k)ej 2πk
K n
, n = 1, . . . , K (1)
Like practical applications, data transmission is carried out in a burst manner. To facilitate
quick synchronization, each burst is preceded with PShort + PLong known training blocks
Fig. 1 Block diagram of MIMO-OFDM Transmitter
123

Short
Training
Blocks
First long
Training
Block
Second long
Training
Block
1-th
OFDM
Block
2-th
OFDM
Block
…
225-th
OFDM
Block
Preamble Segment Data Segment
Fig. 2 Typical burst structure based on the IEEE 802.11a standard
(the preamble segment). The preamble is carefully designed to provide enough information
for good burst detection, frequency offset estimation, symbol timing and channel estima-
tion.1 The data segment spans PD OFDM blocks. Typically, in addition to KD data symbols,
each OFDM block also includes KP inserted pilots to help update/track channel estimation
and synchronization, where KD + KP ≤ K. Figure 2 illustrates a typical burst structure
(e.g., IEEE 802.11a standard [22]) with K = 64, KD = 48, KP = 4, PD = 225 and
PLong = 2.
It is assumed that the fading process remains static during each transmitted burst but it
varies from one burst to another, and the fading processes associated with different trans-
mit–receive antenna pairs are uncorrelated. The Lth order frequency-selective fading channel
from the uth transmit antenna to the vth receive antenna is denoted by
hv,u = hv,u(0), . . . , hv,u (L − 1)
T
L×1
, 1 ≤ u ≤ N, 1 ≤ v ≤ M (2)
where hv,u(l) is the fading coefﬁcient of the lth tap associated with the uth transmit and the
vth receive antennas.
At each receive antenna, a superposition of faded signals from all transmit antennas plus
noise is received. Received signals are down converted to baseband with the local oscillators
centered at ˆfc. After S/P conversion and CP removal, the nth received sample of the mth
OFDM block at the vth receive antenna is determined by [19]:
rv,m(n) = ej 2πε
K (n+nm)
N
u=1
L−1
l=0
hv,u (l) xu,m(n − l) + wv,m(n) (3)
where ε is the CFO normalized to the subcarrier spacing, n = 1, . . . , K, nm = Kpreamble +
Kg + (m − 1)(K + Kg), (Kpreamble denotes the preamble length) and wv,m(n) is a zero-
mean, additive white, complex Gaussian distributed noise sample with variance σ2
w. The
presence of CFO entails loss of orthogonality among subcarriers, which results in ICI and
hence bit-error rate (BER) degradation. The goal is thus estimating ε and multiplying the
received time domain signal with e− j 2πε
K (n+nm)
. The estimation of CFO will be described in
next sections.
Denoting Hv,u(k) = L−1
l=0 hv,u(l)e− j 2πkl
K , the corresponding frequency-domain sample
can be expressed as [20]:
Yv,m(k) =
1
K
K
n=1
ej 2πε
K (n+nm)
N
u=1
K
i=1
Hv,u(i)Xu,m(i)ej 2π(i−k)
K n
+ Wv,m(k) (4)
where Wv,m(k) represents the frequency-domain zero-mean, additive white, complex
Gaussian distributed noise sample with variance σ2
W . In (4), the term i = k corresponds
to the subcarrier of interest, whereas the other terms with i = k represent the ICI. Hence:
1 Assuming ideal synchronization in time, the frame detection and symbol timing will not be addressed in
this paper.
123

Fig. 3 Block diagram of MIMO-OFDM receiver using the proposed frequency synchronization and channel
estimation scheme
Yv,m(k) =
1
K
N
u=1
Hv,u(k)Xu,m(k)
K
n=1
ej 2πε
K (n+nm)
+
1
K
N
u=1
K
i = 1,
i = k
Hv,u(i)Xu,m(i)
K
n=1
ej 2πε
K (n+nm)
ej 2π(i−k)
K n
IC I
+Wv,m(k) (5)
3 Joint CFO and CIR Estimation
Usually, channel estimation is carried out after frequency synchronization and assumes fre-
quency synchronization is perfectly achieved. The residual CFO due to frequency synchro-
nization errors will degrade the performance of channel estimation significantly [10]. If the
same training blocks are used to simultaneously perform the CFO and CIR estimation, sig-
nificant beneﬁts may be realized. Motivated by this fact, we propose a novel scheme for joint
CFO and CIR estimation.
As illustrated in Fig. 3, the proposed scheme consists of two RLS algorithms. The ﬁrst one
(CIR estimator) takes as input the time-domain received training symbols from M receive
antennas and ˆε(i−1) [Eq. (19)], (which is fed back by the second RLS algorithm at the previous
iteration2) and computes as output the h
(i)
v [Eq. (9)], (which is sent to the second RLS algo-
rithm). The second RLS algorithm (CFO estimator) takes as input the time-domain received
training symbols from M receive antennas and h
(i)
v . It computes the ˆε(i) and then feeds it
back to the CIR estimator and thus completes one iteration. At the end of the last iteration,
2 We set the initial estimates ˆε(0) = 0 and h
(0)
v = 0.
123

the final CFO and CIR estimate values, ˆε and hv, are output by the joint estimator. In the
following, we describe the performed calculations in each iteration of the joint estimator.
Based on the signal model in (3), the least-square (LS) estimate for hv =
[hT
v,1, . . . , hT
v,N ]T
N L×1(v = 1, . . . , M) is obtained by minimizing the following cost function
[23]:
ξ h
(i)
v =
i
p=1
λ
i−p
1 e(p)
v
2
(6)
where h
(i)
v and λ1 denote the estimated value of hv at the ith iteration and the forgetting
factor, respectively. In the sequel to this section we concentrate on the preamble and omit the
temporal index m for notation simplicity. From (3), we obtain
e(i)
v = e− j 2π ˆε(i−1)
K (i+L)
rv (i + L) − xT
(i + L) h
(i−1)
v , v = 1, . . . , M (7)
x (i + L) = [x1 (i + L) , . . . , x1 (i + 1) , . . . , xN (i + L) , . . . , xN (i + 1)]T
N L×1 (8)
where ˆε(i−1) denotes the CFO estimate at the (i − 1)th iteration of. The final expression for
h
(i)
v is then obtained as [23]:
h
(i)
v = h
(i−1)
v + k
(i)
h e(i)
v , v = 1, . . . , M (9)
where k
(i)
h is defined by
k
(i)
h =
P
(i−1)
h x∗ (i + L)
λ1 + xT (i + L) P
(i−1)
h x∗ (i + L)
(10)
and
P
(i)
h =
1
λ1
P
(i−1)
h − k
(i)
h xT
(i + L)P
(i−1)
h (11)
We will shortly explain how h
(i)
v is then used by the second RLS algorithm to find ˆε(i).
To exploit the RLS approach in finding ˆε(i), the following weighted sum squared error
should be minimized:
ξ(ˆε(i)
) =
i
p=1
λ
i−p
2 e(p)
ε
2
(12)
in (12), λ2 represents the forgetting factor of the second RLS algorithm. From (3), we obtain
e(i)
ε = r(i + L) − f(ˆε(i)
, x(i + L)) (13)
where
r (i + L) = [r1 (i + L) , . . . ,rM (i + L)]T
M×1 (14)
f ˆε(i)
, x(i + L) = f1 ˆε(i)
, x(i + L) , . . . , fM ˆε(i)
, x(i + L)
T
M×1
(15)
fv ˆε(i)
, x(i + L) = ej 2π ˆε(i)
K (i+L)
xT
(i + L)h
(i)
v , 1 ≤ v ≤ M (16)
In order to make use a linear RLS, a first-order Taylor series approximation is applied to the
non-linear estimation error. Therefore
123

e(i)
ε ≈ r (i + L) − f(ˆε(i−1)
, x (i + L)) +
∂f(ˆε(i−1), x (i + L))
∂ ˆε(i−1)
(ˆε(i)
− ˆε(i−1)
) (17)
where
∂f ˆε(i−1), x (i + L)
∂ ˆε(i−1)
= j
2π(i + L)
K
f(ˆε(i−1)
, x (i + L)) (18)
Subsequently, the final expression for ˆε(i) is obtained as:
ˆε(i)
= ˆε(i−1)
+ Re e(i)
ε
T
k(i)
ε (19)
where e
(i)
ε and k(i)
ε , respectively, are defined as:
e(i)
ε ≈
r (i + L) − f ˆε(i−1), x (i + L)
1 +
∂fT
ˆε(i−1), x (i + L)
∂ ˆε(i−1)
k(i)
ε
(20)
k(i)
ε =
P
(i−1)
ε
∂f∗
ˆε(i−1), x (i + L)
∂ ˆε(i−1)
λ2 +
∂fT
ˆε(i−1), x (i + L)
∂ ˆε(i−1)
P(i−1)
ε
∂f∗
ˆε(i−1), x (i + L)
∂ ˆε(i−1)
(21)
and
P(i)
ε =
1
λ2
P(i−1)
ε − k(i)
ε
∂fT
ˆε(i−1), x (i + L)
∂ ˆε(i−1)
P(i−1)
ε (22)
Based on (9) and (19), we form a novel method for joint CFO and CIR estimation.
Proposed estimator iteratively provides the channel and frequency offset estimates. To regain
the orthogonality of sub-carriers, ˆε is removed from time-domain received samples in (3)
before applying the discrete Fourier transform (DFT) operation, as shown in Fig. 3. However,
the hv will be used by tracking algorithm [See Eqs. (28) and (29)].
4 Tracking Using Pilots
The frequency offset requires constant tracking. This is aided by inserting known pilot sym-
bols at fixed positions. In this section, to further improve the CFO estimation accuracy, we
derive a more precise pilot-aided RLS algorithm to estimate the residual errors or track small
CFO values. With this, the accuracy of channel estimation is also enhanced.
In detail, the pth pilot of the mth received OFDM block at the vth receive antenna can be
expressed as:
Yv,m(kp) =
1
K
N
u=1
Hv,u(kp)Xu,m(kp)
K
n=1
ej 2π ˇε
K (n+nm)
+ ICI + Wv,m(kp) (23)
where ˇε denotes the frequency offset estimation error, ε − ˆε. Using the same strategy as that
given in the previous subsection, the final expression for ith estimate of ˇε is obtained as3:
ˇε(i)
= ˇε(i−1)
+ Re e(i)
T
k(i)
(24)
3 We set the initial estimate ˇε(0) = 0.
123

where e(i) is defined by
e(i)
≈
Ym kp − f ˇε(i−1), Xu,m(kp)
1 +
∂fT
ˇε(i−1), Xu,m(kp)
∂ ˇε(i−1)
k(i)
(25)
Ym(kp) = [Y1,m(kp), . . . , YM,m(kp)]T
M×1 (26)
f(ˇε(i−1)
, Xu,m(kp)) = [ f1(ˇε(i−1)
, Xu,m(kp)), . . . , fM (ˇε(i−1)
, Xu,m(kp))]T
M×1 (27)
fv(ˇε(i)
, Xu,m(kp)) =
1
K
N
u=1
Hv,u(kp)Xu,m(kp)
K
n=1
e
j
2π ˇε(i)
K
(n + nm)
(28)
Hv,u(kp) =
L−1
l=0
hv,u(l)e
− j
2πl
K
kp
(29)
It is noted that the use of 1
K
K
n=1 ej 2π ˇε(i)
K n
≈ sinc(ˇε(i))ejπ ˇε(i)
results in an approximate
expression for (28) that was used in [11]. Moreover,
k(i)
=
P(i−1) ∂f∗
(ˇε(i−1)
, Xu,m(kp))
∂ ˇε(i−1)
λ3 +
∂fT
(ˇε(i−1)
, Xu,m(kp))
∂ ˇε(i−1) P(i−1) ∂f∗
(ˇε(i−1)
, Xu,m(kp))
∂ ˇε(i−1)
(30)
where λ3 represents the forgetting factor of the RLS algorithm, and
P(i)
=
1
λ3
P(i−1)
− k(i) ∂fT
(ˇε(i−1), Xu,m(kp))
∂ ˇε(i−1)
P(i−1)
(31)
As before mentioned, imperfect frequency synchronization degrades the accuracy of chan-
nel estimation, which in turn, affects the overall receiver performance. Consequently, after
CFO tracking is performed, the channel estimation which is performed without considering
ˇε can be refined [as stated in Eqs. (7)–(11)].
5 Complexity and Simulation Results
5.1 Cramer–Rao Bounds
In this subsection, to benchmark the performance of the proposed scheme, we evaluate the
CRBs for both CFO and CIR estimators.
Let us define
Am =
⎡
⎢
⎢
⎢
⎣
x1,1,m x1,2,m · · · x1,N,m
x2,1,m x2,2,m · · · x2,N,m
...
...
...
...
xK,1,m xK,2,m · · · xK,N,m
⎤
⎥
⎥
⎥
⎦
K×N L
(32)
where
xn,u,m = xu,m (n) , xu,m (n − 1) , . . . , xu,m (n − L + 1) (33)
123

We can rewrite (3) into the following equivalent form
rv,m (n) = ej 2πε
K (n+nm)
Am (:, n) hv + wv,m (n) , n = 1, . . . , K, v = 1, . . . , M (34)
where Am(:, n) denotes the nth row of the matrix Am and hv = [hv,1hv,2 . . . hv,N ]T
N L×1
(hv,u is deﬁned in Eq. (2)). Based on (34), the mth OFDM block at the vth receive antenna,
rv,m = [rv,m(1) . . .rv,m(K)]T
K×1,can be expressed as
rv,m = Fm (ε) Amhv + wv,m, v = 1, . . . , M (35)
where Fm(ε) = diag ej 2πε
K (1+nm) . . . ej 2πε
K (K+nm)
K×K
and wv,m = [wv,m (1) . . . wv,m
(K)]T
K×1.
Stacking vectors rv,m of (35) in a column to form the vector rm and vectors hv to obtain
vector h, (35) can be reformulated as
⎡
⎢
⎢
⎢
⎣
r1,m
r2,m
...
rM,m
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Fm (ε) Am . . . 0
...
...
...
0 . . . Fm (ε) Am
⎤
⎥
⎦ ×
⎡
⎢
⎢
⎢
⎣
h1
h2
...
hM
⎤
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎣
w1,m
w2,m
...
wM,m
⎤
⎥
⎥
⎥
⎦
(36)
or in the matrix form
rm = (IM ⊗ Fm (ε) Am) h + wm (37)
where ⊗ denotes the Kronecker product and wm is a noise vector of the mth OFDM block
which is assumed to be zero-mean circularly symmetric complex-value Gaussian with covari-
ance matrix Σv = σ2
v .IMK .
Let η = [εhRhQ]T stands for the parameter vector of interest where hR and hQ, respec-
tively, denote the real and imaginary parts of h. Under all the assumptions are made, the
received block rm, is a complex-valued circularly symmetric Gaussian vector with mean
μ = (IM ⊗ Fm (ε) Am) h and covariance matrix Σv = σ2
v .IMK . For this kind of problems, it
is known that the Fisher information matrix (FIM) for estimation of ηTσ2
v is block diagonal,
that is, the estimation of σ2
v is decoupled from that of η [19,24]. Therefore we only consider
the FIM for η, which we denote by FI. The latter is given by
FI =
2
σ2
v
Re
∂μH
∂η
∂μ
∂ηT
(38)
The CRB is obtained as the inverse of the FI. Using the same principle as those given in
[19,24], we obtain the ﬁnal expression for CRB of ε as
C RB (ε) =
K2σ2
v
8π2
hH
IM ⊗ AH
m Dm.ΠADmAm h
−1
(39)
where
ΠA = IK − Am AH
m .Am
−1
AH
m (40)
Dm = diag {1 + nm, 2 + nm, . . . , K + nm} (41)
Note that nm = Kpreamble + Kg + (m − 1) K + Kg .
123

Table 1 Computational complexity comparison
Algorithm Real products Real additions
Joint estimator
14(N L)2 + 12M N L + 8N L
+14M2 + 32M + 1
12(N L)2 + 12M N L + 2N L
+12M2 + 12M − 2
Tracking algorithm
14M2 + 14(N L)2 + 8M N L
+4MK + 8N L
+2M N + 38M
12M2 + 12(N L)2 + 8M N L
+4MK + 2N L + 2M N
+14M − 2
VRLS
4(2LM N + 1)2(M + 2)
+4(2LM N + 1)(2LM N + 2M + 1)
+M(2N + 15LN + 4L + 13)
+5K + 2N + 39
4(2LM N + 1)2(M + 1)
+4(2LM N + 1)(3LM N + 0.5M + 0.5)
+M(4LM N + 4LN + 4L + 2N + 3)
+2K + 2N + 17
MLE#1 8M N LK + 4MK NPoint + 8MK 8M N LK + 4MK NPoint − 2NPoint
MLE#2 4M N LK + 7MK 4M N LK + 2MK − 2
It is worth mentioning that the channel coefﬁcients are estimated with the aid of preamble
sequences. Therefore, in this case, Dm = diag {1, 2, . . . , KPreamble} and
Am =
⎡
⎢
⎢
⎢
⎣
x1,1,m x1,2,m · · · x1,N,m
x2,1,m x2,2,m · · · x2,N,m
...
...
...
...
xK,1,m xK,2,m · · · xK,N,m
⎤
⎥
⎥
⎥
⎦
K×N L
(42)
where K = KPreamble. Using the same approach of [19,24], we obtain for any unbiased
estimate h of h
E[(h − h)(h − h)H
] ≥
σ2
v
2
(2λ + γ −1
ββH
) = C RB(h) (43)
where
λ = IM ⊗ AH
m .Am
−1
(44)
β = IM ⊗ AH
m .Am
−1
AH
m .Dm.Am h (45)
γ = hH
IM ⊗ AH
m Dm.ΠADmAm h (46)
5.2 Algorithm Complexity
To evaluate the computational complexity of the proposed scheme, we calculate the num-
ber of real multiplications and real additions, which are needed to implement it. We also
compare the computational complexity of the proposed scheme with that of the EM-based
estimators [18,19] (denoted by “MLE#1” and “MLE#2”) and that of the VRLS-based esti-
mator [20] (denoted by “VRLS”). The overall operations are summarized in Table 1, where
NPoint denotes the number of points searched over the uncertainly interval.
123

5.3 Simulation Results
In this subsection, simulation results are presented to demonstrate the effectiveness of the
proposed scheme. In all simulations, a 2 × 2 MIMO-OFDM communication system with
K = 64 subcarriers is considered. In addition, we set the OFDM-related simulation param-
eters based on the IEEE 802.11a un-coded systems [22]. The signal constellation of the
quadrature phase shift keying (QPSK) is employed for the OFDM blocks of KD = 48 data
symbols and KP = 4 equally spaced pilot tones of the same power. For each transmit antenna,
a burst format of two long identical training blocks (PLong = 2) and PD = 225 data OFDM
blocks (as shown in Fig. 2) is used in the simulation. For each transmit–receive antenna pair,
we considered a Rayleigh fading channel with L = 7 taps and RMS delay spread of 50 (ns)
that are randomly generated for each OFDM frame.
Inallinvestigations,theCFOisnormalizedtothesubcarrierspacingandrandomlyselected
in the range [−0.5, 0.5]. The performance of the CFO estimator is evaluated with the MSE
between the estimated and the true frequency values. The MSE is averaged for each SNR
over 2000 packet Monte-Carlo runs. Similar to [20], the performance of CIR estimation is
evaluated with the normalized MSE (NMSE). In the case of channel estimation, similar to
[25], the normalized MSE (NMSE) between the estimated and the true channel responses is
deﬁned as
NMSEC I R =
M
v=1 hv − hv
2
M
v=1 hv
2
(47)
In the following, SNR means the ratio of the average received signal power to the average
noise power as
SNR =
M
v=1 E N
u=1
L−1
l=0 hv,u (l) xu,m (n − l)
2
M
v=1 E wv,m(n)
2
(48)
5.3.1 Example 1 (Performance of CFO Estimator)
First, we study the performance of time-domain CFO estimator derived in (19). In Fig. 4,
we depict the MSE of the time-domain CFO estimator versus the number of time-domain
training symbols for different forgetting factors. From Fig. 4, it is clearly seen that increas-
ing the number of time-domain training symbols improves the performance of the CFO
estimator. Furthermore, we deduce that the proposed CFO estimator in (19) achieves the
best performance with the forgetting factor λ = 0.95 and the regularization parameter
γ = 10.
Next, we consider the performance of the frequency-domain CFO tracking algorithm
derived in (24). The forgetting factor and regularization parameter are respectively chosen
as λ = 0.995 and γ = 10. In Fig. 5, we plot the MSE of the tracking algorithm versus
SNR. From Fig. 5, it is observed that as the number of OFDM blocks increases, the MSE
performance of the CFO tracking algorithm improves. Another observation is that, using
more OFDM blocks, the tracking algorithm is able to attain fast convergence to the optimum
performance.
Finally, we compare our two-step CFO estimation scheme with the EM-based CFO esti-
mator proposed in [18] (denoted by “MLE#1”) and the RLS-based CFO estimator proposed
123

0 10 20 30 40 50 60 70 80 90 100 110 120
10
-3
10
-2
10
-1
Number of time-domain training symbols, i
MSE(Frequency)
Forgetting Factor = 0.98
Fig. 4 MSE of the CFO estimate by the proposed joint estimator (SNR = 10 dB)
0 2 4 6 8 10 12 14 16 18 20
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
Signal-to-Noise Ratio, dB
MSE(Frequency)
Number of OFDM Blocks = 1
Fig. 5 MSE of the CFO estimate by the proposed pilot-aided estimator. Various numbers of OFDM blocks
are considered
in [20] (denoted by “VRLS”) in both accuracy and computational complexity aspects. For
MLE#1, we search over 5000 points equally spaced over the CFO interval [−0.5, 0.5].
In Fig. 6, the MSE performances of all mentioned algorithms are plotted versus SNR. The
averaged CRB derived in (39) is also shown as a bench mark. From Fig. 6, we deduce
that our scheme outperforms the proposed methods in [18,20] in all ranges of SNR. Fur-
thermore, MSE performance of our scheme is quite comparable to CRB in all ranges of
123

0 2 4 6 8 10 12 14 16 18 20
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
MSE(Frequency)
MLE#1
VRLS
Proposed Method
CRB
Fig. 6 CFO accuracy comparison for a 2×2 MIMO-OFDM systems (Number of OFDM Blocks = 50)
200 400 600 800 1,000
10
5
10
6
10
7
10
8
10
9
Number of subcarriers, K
TotalnumberofRealAdditionsandProducts
MLE#1, Iter 10
VRLS, 50 OFDM Blocks
VRLS, 25 OFDM Blocks
Proposed Method, 50 OFDM Blocks
Proposed Method, 25 OFDM Blocks
MLE#2, Iter 10
Fig. 7 Computational complexity comparisons for a 2×2 MIMO-OFDM system
SNR. Figure 7 compares the total required operations (real additions plus real products)
involved in the mentioned estimators as a function of the number of subcarriers (K). In Fig. 7,
the total required operations for the EM-based CFO estimator proposed in [19] (denoted
by “MLE#2”) is also plotted. The curves are computed from the results in Table 1 with
Npoint = 5, 000, L = 7, and N = M = 2. It is observed that the computational complexity
123

0 2 4 6 8 10 12 14 16 18 20
10
-4
10
-3
10
-2
10
-1
10
0
NormalizedMSE(Channel)
VRLS
MLE#1
Proposed Method
CRB
Fig. 8 MSE performance of channel estimator versus SNR for a 2×2 MIMO-OFDM system
of MLE#1 and VRLS is much higher than our algorithm. Compared with MLE#2, our scheme
has slightly higher computational complexity. However, in exchange for the price paid for
the computational complexity, we have gained much superior MSE performance.
Simulation results show that the performance of our proposed CFO estimator is quite
satisfying and it can be used to cover a wide tracking range up to ±0.5 of the subcarrier
spacing.
5.3.2 Example 2 (Performance of Channel Estimator)
In this example, we test the performance of our channel estimator derived in (9). In Fig. 8,
the simulated NMSE of our channel estimator is presented and compared with the CRB in
(43) and channel estimators proposed in [18] (denoted by “MLE#1”) and [20] (denoted by
“VRLS”). From Fig. 8, it is observed that the NMSE of our channel estimator outperforms
the MLE#1 and VRLS in all ranges of SNR. In addition, the NMSE of our channel estimator
is close to CRB even at high SNR values. This is because that our channel estimator uses the
estimated CFO values in (24) to form the error function derived in (7).
6 Conclusions
In this paper, with the aid of training symbols in both time and frequency domains, we
proposed a novel two-step scheme for frequency synchronization and channel estimation in
multi-antenna OFDM systems. The performance of the proposed estimation and tracking
scheme is benchmarked with CRBs, and investigated by computer simulations. Simulation
results show that the proposed CFO and CIR estimators achieve almost ideal performances
compared with the CRBs in all ranges of SNR with a wide tracking range.
123

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Author Biographies
M. Jamalabdollahi was born in Tehran, Iran on September 11, 1980.
He received the B.Sc. degree in Electrical Engineering from University
of Mazandaran, Babol, Iran, in 2004 and the M.Sc. degree from K.N.
Toosi University of Technology, Tehran, Iran, in 2011. From October
2006 he is working on Mobile Communications Technologies Co. in
Software Radio Unit, Tehran, Iran. His main research interests include
MIMO-OFDM(A), multiuser MIMO-OFDM, sensor network and PHY
layer implementation of wireless communications.
S. Salari was born in Mashhad, Iran on March 27, 1976. He received
the B.Sc. degree in electrical engineering from Shahid Bahonar Univer-
sity, Kerman, Iran, in 1998 and the M.Sc. degree from K.N. Toosi Uni-
versity of technology, Tehran, Iran, in 2002. He has also been awarded
the degree of Ph.D. by the K.N. Toosi University of technology, Teh-
ran, Iran, in 2007. Since 2007, he has been assistant professor at K.N.
Toosi University of Technology with major interest in wireless sensor
networks, MIMO-OFDM communication and cognitive radio systems.
123

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