In this paper, we analyze the effect of the coherence bandwidth of wireless channels on leakage suppression methods for discrete Fourier transform (DFT)based channel estimation in orthogonal frequency division multiplexing (OFDM) systems. Virtual carriers in an OFDM symbol cause orthogonality loss in DFTbased channel estimation, which is referred to as the leakage problem. In order to solve the leakage problem, optimal and suboptimal methods have already been proposed. However, according to our analysis, the performance of these methods highly depends on the coherence bandwidth of wireless channels. If some of the estimated channel frequency responses are placed outside the coherence bandwidth, a channel estimation error occurs and the entire performance worsens in spite of a high signaltonoise ratio.
I. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is a predominant transmission technology for ubiquitous wireless broadband networks because it can mitigate severe effects of frequency selective fading and provide good spectrum efficiency
[1

4]
. There have been many attempts to apply the multipleinput multipleoutput (MIMO) technology to OFDM transmission systems, such as
[5
,
6]
.
For coherent detection of the received data symbols in OFDM transmission, channel frequency responses (CFRs) must be estimated and equalized. One of the OFDM channel estimation methods is pilotaided channel estimation (PACE) where pilot symbols are assigned to pilot subcarriers, which are multiplexed with data subcarriers, and channel estimation for data symbols is performed by interpolation techniques. There are three basic factors affecting the performance of the PACE method. These are pilot patterns, estimation methods, and signal detection. The choice of these factors depends on OFDM system specifications and wireless channel conditions.
In PACE, CFRs at pilot subcarriers are estimated using least squares (LS) or minimum mean square error (MMSE) estimators, and then, interpolation techniques are performed in order to estimate CFRs at data subcarriers by using the estimated CFRs at pilot subcarriers, as shown in
Fig. 1
. As an interpolation technique of PACE methods, discrete Fourier transform (DFT)based interpolation is often used. However, it may have the leakage problem caused by virtual carriers in an OFDM symbol. In order to deal with the leakage problem, some suppression methods were proposed in
[7

11]
. In particular, optimal and suboptimal linear estimators were proposed to estimate the equally spaced virtual carriers in
[11]
, but whose performances are very sensitive to the coherence bandwidth of wireless channels. Therefore, in this study, optimal and suboptimal linear estimators are reviewed and theoretically analyzed in terms of coherence bandwidth.
Pilotaided channel estimation for orthogonal frequency division multiplexing (OFDM) systems.
The rest of this paper is organized as follows: in Section II, DFTbased channel estimation and its leakage problem are described. Section III reviews optimal and suboptimal linear estimators and analyzes the effect of coherence bandwidth on them. In Section IV, numerical results are presented to show the performance degradation due to the effect of coherence bandwidth. Conclusions are presented in Section V.
II. OFDM CHANNEL ESTIMATION
 A. DFTBased Channel Estimation
DFTbased interpolation is an efficient interpolation technique because of its good performance and low complexity
[8
,
9]
. The PACE method using DFTbased interpolation is called DFTbased channel estimation, as shown in
Fig. 2
, where an inverse DFT (IDFT) operation is executed first to obtain the estimated channel impulse responses (CIRs) by using the LSestimated CFRs at pilot subcarriers, and then, the estimated CIRs are transformed back into the frequency domain by the DFT operation to obtain the final CFRs at data subcarriers.
Discrete Fourier transform (DFT)based channel estimation. LSCE: leastsquares channel estimation.
 B. Virtual Carriers and Leakage Problem
In most commercialized OFDM systems, virtual carriers are exploited to ease the implementation of spectral masking filters and ensure guard bands to avoid interferences between adjacent systems
[10]
.
However, these virtual carriers have a bad influence on the performance of DFTbased channel estimation. In other words, they may cause leakage effects. Virtual carriers correspond to rectangular windowing in the frequency domain, which results in the convolution of CIRs with the sinc function in the time domain. Hence, the channel taps of CIRs are leaked to one another. Further, timedomain windowing is performed to reduce the noise and interference, which causes spectral leakage or Gibbs phenomenon, as shown in
Fig. 3
.
Description of the reason for leakage effects.
In order to review the performance of DFTbased channel estimation and the leakage problem from virtual carriers, the normalized mean square error (NMSE) is represented as follows: Let the number of total subcarriers be
N
=
N_{U}
+
N_{V}
+ 1, where
N_{U}
+ 1 and
N_{V}
are the number of useful subcarriers and the number of virtual carriers, respectively. The number of cyclic prefix samples is defined as
N_{G}
. The received symbol at the
k
th subcarrier is represented by
where
X
[
k
] is a transmitted symbol,
W
[
k
] is a circularly symmetric complex Gaussian noise with zero mean and variance
σ
^{2}
, and
H
[
k
] is a CFR represented by
where
h
[
l
] is the
l
th channel gain of an
L
× 1 circularly symmetric complex Gaussian CIR vector
h
with zero mean and covariance matrix
C
_{h}
=
E
[
hh
^{H}
]. The pilot symbols are assigned to estimate the CFR at pilot subcarriers
where
i_{m}
= −
N_{U}
/2 +
mD_{f}
denotes a subcarrier location,
D_{f}
denotes the minimum pilot spacing,
P
[
m
] denotes a pilot symbol with a binary phase shift keying constellation, and
N_{P}
denotes the number of pilot subcarriers.
The received pilot vector with entries
Y
[
i_{m}
] is given by
where
X
is a diagonal matrix with
P
[
m
] on its diagonal, and
H
_{P}
is a CFR vector,
W
is a noise vector with zero mean and covariance matrix
σ
^{2}
I
_{P}
(
I
_{P}
is an
N_{P}
×
N_{P}
identity matrix), and
F
is a DFT matrix with entries
where 0 ≤
m
≤
N_{P}
− 1 and 0 ≤
n
≤
L
− 1. After the LS channel estimation at pilot subcarriers, the observation vector is represented by
where the noise vector
is statistically equivalent to
W
since
E
[
] =
0
_{P×1}
and E[
] =
σ
^{2}
I
_{P}
. Here,
0
_{P×1}
denotes an
N_{P}
× 1 zero vector. The observation vector
Z
can be rewritten as
to address the leakage effects
where
is an
M
× 1 observation vector, and
M
=
N_{b}
+
N_{P}
+
N_{f}
is the number of pilot subcarriers including leftside virtual carriers (
N_{b}
) and rightside virtual carriers (
N_{f}
) with minimum pilot spacing. Here,
0
_{b×1}
and
0
_{f×1}
denote an
N_{b}
× 1 zero vector and an
N_{f}
× 1 zero vector, respectively.
H
_{b×1}
and
H
_{f×1}
denote an
N_{b}
× 1 ideal CFR vector and an
N_{f}
× 1 ideal CFR vector, respectively. In addition,
is a DFT matrix with entries
where
= −
N_{U}
/2 + (
m
−
N_{b}
)
D_{f}
, 0 ≤
m
≤
M
− 1, and 0 ≤
n
≤
L
− 1. The IDFT operation transforms
into the estimated CIR vector
in (9)
Then, the DFT operation transforms
into the estimated CFR vector
in (10)
where
=
M
I
_{L}
, and
H
=
Gh
. Here,
G
is a DFT matrix with entries
where −
N_{U}
/2 ≤
m
≤
N_{U}
/2 and 0 ≤
n
≤
L
− 1 . Note that the third term in (10) denotes the leakage. Now, the error covariance matrix of
H
can be given by
where
U
=
F
^{H}
F
,
P
=
G
^{H}
, and
V
=
E
[
]. To reveal the channel covariance matrix
C
_{h}
,
V
is rewritten as
where
_{v}
denotes a DFT matrix with entries related to
H
_{v}
. The NMSE performance can be presented as
where the first term denotes the noise effects and the second term denotes the leakage effects.
III. EFFECT OF COHERENCE BANDWIDTH ON LEAKAGE SUPPRESSION METHODS
 A. Review of Leakage Suppression Methods
In order to minimize the NMSE in (14), optimal and suboptimal linear estimators were proposed in
[11]
. They can be expressed as
Then, optimal or suboptimal linear estimators are used to estimate
=
K
_{opt}
Z
or
=
K
_{sub}
Z
, respectively. Their error covariance matrices can be given by
where
V
_{opt}
and
V
_{sub}
are represented as
Finally, their NMSEs are given by
From (21) and (22), we find that the NMSE performance improvement depends on the second terms
V
_{opt}
and
V
_{sub}
, respectively.
 B. Effect of Coherence Bandwidth
Although optimal and suboptimal linear estimators provide good performance with moderate complexity, their performance may decrease according to wireless channel conditions such as coherence bandwidth.
Coherence bandwidth is a statistical measurement of the range of frequencies over which the channel can be considered flat
[12]
. The coherence bandwidth can be approximately defined as
where
τ_{max}
is the maximum delay spread of a wireless channel and
T_{s}
is the sampling time of an OFDM system. If the coherence bandwidth is divided by the subcarrier spacing Δ
f
= 1/
NT_{s}
, the number of subcarriers in the range of the coherence bandwidth can be obtained. For instance,
where we assume
N
= 512 and
L
= 20. If the minimum pilot spacing is
D_{f}
= 8, three
D_{f}
spaced virtual carriers will be in the range of the coherence bandwidth, as shown in
Fig. 4
.
Coherence bandwidth and minimum pilot spacing.
Therefore, if optimal and suboptimal linear estimators try to estimate CFRs outside the range of the coherence bandwidth, they may experience a decrease in their performance.
IV. NUMERICAL RESULTS
Computer simulations have been run to analyze leakage suppression methods such as optimal and suboptimal linear estimators in terms of the coherence bandwidth of wireless channels. We consider the OFDM system using QPSK modulation in the 1.25MHz bandwidth at 2.3 GHz with
N_{G}
= 32 ,
N
= 512 ,
N_{U}
+ 1 = 481 ,
N_{V}
= 31 ,
D_{f}
= 8 ,
N_{P}
= 61, and
M
= 64. The multipath fading channel is assumed to have an exponential power delay profile with
L
= 20.
Fig. 5
shows the NMSE performance of DFTbased channel estimation in the case of no virtual carriers. It has constant values determined by the noise variance or the signaltonoise power ratio (SNR). In contrast,
Fig. 6
shows the NMSE performance in the case of virtual carriers (namely, conventional DFTbased channel estimators), which is theoretically plotted according to the total subcarriers index by using (14). The first term in (14) strongly depends on the noise variance, whereas the second term leads to the leakage. In addition, the second term is the dominant factor in performance degradation, particularly at edge subcarriers.
Normalized mean square error (NMSE) performance in case of no virtual carriers. SNR: signaltonoise ratio.
Normalized mean square error (NMSE) performance of the conventional method (signaltonoise ratio [SNR] = 20 dB).
Fig. 7
shows the performance improvement resulting from the suboptimal linear estimator by using
σ
^{2}
= 40 in (16). The first term is the same as that in
Fig. 6
, but the second term is highly improved when compared with that in
Fig. 6
.
Normalized mean square error (NMSE) performance of the suboptimal method (signaltonoise ratio [SNR] = 20 dB).
Fig. 8
illustrates the NMSE performance of the suboptimal linear estimator with
σ
^{2}
= 40 in (16) as the number of pilot subcarriers (
N_{P}
) decreases, which means that the number of
D_{f}
spaced virtual carriers to be estimated increases. Further,
Fig. 9
illustrates the NMSE performance of the suboptimal linear estimator with
σ
^{2}
= 10. By comparing
Fig. 8
with
Fig. 9
when
N_{P}
= 61, 59, 57, we find that the suboptimal linear estimator with
σ
^{2}
= 40 is superior to that with
σ
^{2}
= 10. Due to the high SNR assumption in
[13]
, these results are reasonable. However, when
N_{P}
= 55, 53, the NMSE performances in
Fig. 8
are more severely degraded.
Normalized mean square error (NMSE) performance of the suboptimal method (σ^{2} = 40).
Normalized mean square error (NMSE) performance of the suboptimal method (σ^{2} = 10).
These results denote the cases in which the
D_{f}
spaced virtual carriers to be estimated are out of the range of the coherence bandwidth. In other words, the
D_{f}
spaced virtual carriers outside the coherence bandwidth are sensitive to the SNR mismatch. In addition, the SNR mismatch of the suboptimal linear estimator with
σ
^{2}
= 40 is more critical than that of the suboptimal linear estimator with
σ
^{2}
= 10, when the actual SNR is 20 dB. Hence, the channel estimation errors outside the coherence bandwidth resulting from the SNR mismatch cause the entire performance degradation.
V. CONCLUSIONS
In this study, the effect of the coherence bandwidth of wireless channels on leakage suppression methods such as the optimal and suboptimal linear estimators for OFDM channel estimation was analyzed. The NMSE performances of these methods were very sensitive to the coherence bandwidth of wireless channels. If some of the estimated CFRs were placed out of the range of the coherence bandwidth, a severe channel estimation error occurred at edge subcarriers and the entire NMSE performance decreased. Further, the SNR mismatch of the suboptimal linear estimators was more critical in these cases.
BIO
Junhui Zhao
received his M.E. and Ph.D. in Electrical Engineering from Southeast University, China, in 1998 and 2004, respectively. He was a visiting researcher in Yonsei University in 2004. From 2004 to 2008, he was an associate professor at Macau University of Science and Technology, Macau, China. He is currently a professor at the School of Electronics & Information, Beijing Jiaotong University, China. His research interests include cognitive radio and wireless localization.
Ran Rong
received her M.E. and Ph.D. degrees from Chonbuk National University and Yonsei University, South Korea, in 2004 and 2009, respectively. From 2009 to 2010, she was a senior researcher at the Electronics & Telecommunication Research Institute (ETRI), South Korea, working on IEEE 802.16m standardization. She was a research associate in HKUST from 2010 to 2011, and is currently an assistant professor at the School of Electrical and Computer Engineering, Ajou University. Her research interests include massive MIMO, compressive sensing, and cognitive radio.
ChangHeon Oh
received his B.S. and M.S.E. in Telecommunication and Information Engineering from Korea Aerospace University in 1988 and 1990, respectively. He received his Ph.D. in Avionics Engineering from Korea Aerospace University, in 1996. From February 1990 to August 1993, he was with Hanjin Electronics Co. From October 1993 to February 1999, he was with the CDMA R&D Center of Samsung Electronics Co. Since March 1999, he has been with the School of Electrical, Electronics and Communication Engineering, Korea University of Technology and Education, where he is currently a professor. His research interests are in the areas of wireless communications, mobile communication, and wireless sensor networks with particular emphasis on wireless localization.
Jeongwook Seo
received his B.S. and M.S. from the Department of Telecommunication and Information Engineering, Korea Aerospace University, Gyeonggi, Korea, in 1999 and 2001, respectively, and his Ph.D. from the School of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 2010. From 2001 to 2013, he was with Network Convergence Research Center in Korea Electronics Technology Institute, Seoul, Korea. He is currently an assistant professor at the Department of Information and Communication Engineering, Namseoul University, Cheonan, Korea. His research interests include statistical signal processing for communications and networking, protocol design for machinetomachine/Internet of Things, and nextgeneration broadcasting and communication systems.
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