HWANG et al.: OFDM AND ITS WIRELESS APPLICATIONS: A SURVEY 1677
It can easily be seen that ML estimation is the same as least-
square (LS) estimation in this case. Direct calculation yields
that LS estimation of the channel frequency response vector is
ˆ
H
LS
= S
−1
p
x. (7)
To reduce the computational complexity of ML or LS channel
estimation, S
−1
p
in (7) can be calculated offline. Estimation in
this case neither needs nor exploits the information of channel
statistics. Consequently, estimation performance is usually not
good enough; thus, it is often applied as initial estimation.
By exploiting channel statistics, channel estimation can sig-
nificantly be improved. With the correlation matrix of the
channel frequency response vector R
H
= E{HH
H
}, linear
minimum mean square error (LMMSE) channel estimation
can be obtained. For LMMSE estimation, channel frequency
responses are regarded as random variables. Estimation of the
channel frequency response vector is found to minimize the
mean square error (MSE), i.e., E
ˆ
H − H
2
. Consequently
ˆ
H
LMMSE
= R
H
R
H
+ σ
2
n
S
p
S
H
p
−1
−1
S
−1
p
x
= R
H
R
H
+ σ
2
n
S
p
S
H
p
−1
−1
ˆ
H
LS
.
Compared with ML or LS estimation, LMMSE estimation has
much better performance; however, it requires channel statistics
and has higher computational complexity.
With a minor modification, the principle of the aforemen-
tioned LMMSE estimation can also be used to estimate channel
frequency responses other than pilot subchannels as in PACE,
which is also called optimal interpolation in [45] and [46].
Training or pilot symbols can be designed to facilitate chan-
nel estimation. For example, if they are with constant modulus,
that is, |s
k
|
2
= σ
2
s
for k =0,...,N − 1, then S
p
S
H
p
= σ
2
s
I,
σ
2
n
(S
p
S
H
p
)
−1
=(σ
2
n
/σ
2
s
)I, and no matrix inversion operation
is required. Furthermore, if the training or pilot symbols are
identical, that is, s
k
= s for k =0,...,N − 1, then even S
p
=
sI and S
−1
p
=(1/s)I. As indicated in [54], the coefficient
matrix for LMMSE estimation can be computed with the help
of transform domain processing, such as Fourier transform or
singular value decomposition. The correlation matrix of the
channel frequency response vector is determined by the power
delay profile (PDP) of the wireless channel. Therefore, if the
accurate PDP is known in advance, then the coefficient matrix
for LMMSE estimation can be calculated offline. When the
accurate PDP is unknown in advance, different solutions have
been suggested. For example, channel estimation may set up
coefficients according to a uniform or an exponential PDP [21],
[46], [54], [56].
Estimation performance can further be improved by a 2-D
LMMSE estimator exploiting the time-domain correlation of
channels in addition to the frequency-domain correlation. To
reduce the computational complexity of the 2-D LMMSE esti-
mator, several methods have been proposed, and some of them
bear the same spirit of the 1-D algorithms discussed before. By
exploiting the separable feature of channel correlation at time
and frequency domains [45], [56], 2-D LMMSE estimation is
decoupled into two cascade 1-D LMMSE estimations. Another
solution partitions the 2-D time–frequency region into several
small regions and performs LMMSE estimation by only consid-
ering the correlation among adjacent subchannels in each small
region [54].
1) Pilot-Aided Channel Estimation: Using pilot tones to
estimate channel coefficients was first proposed in [57]. The
two major issues of pilot-aided channel estimation are pilot
design and interpolation.
The optimal design for the pilot pattern, power allocation,
and number of pilots has extensively been studied [57]–[63],
which critically depends on a proper criterion and the channel
model. The impact of pilots on system performance for time-
varying channels has first been analyzed in [58]. The optimal
pilot design for frequency-selective channels has been inves-
tigated in [59] and [60], whereas that for doubly selective
channels has been investigated in [61]. The pilots have been
designed to minimize the MSE of channel estimation [57]
or CRB [62], maximizing the channel capacity [59]–[61] and
minimizing the symbol error rate [63]. An extensive review on
the topic has been addressed in [47].
As indicated before, LMMSE estimation can be applied for
joint channel estimation and interpolation. However, it requires
channel statistics and high computational complexity [45],
[46]. This motivates us to develop low-complexity interpolation
algorithms.
Two of the simplest ways are piecewise constant and linear
interpolation [2], [64]. However, more pilots are required for
them to achieve acceptable performance in frequency-selective
channels. If the statistics of channel variation either in the
frequency domain (PDP) or in the time domain (Doppler
spectrum) are known apriori, a high-order polynomial can be
applied to accurately fit wireless channels [64], [65].
In addition to the linear and high-order polynomial-based in-
terpolation, pilot-aided channel estimation can also be based on
FFT [46], [66]. FFT-based channel estimation essentially uses
a low-pass filter as an interpolater, and the filter is implemented
in a transform domain [55]. Instead of using a low-pass filter,
we may catch significant taps in the transform domain and turn
off those trivial taps, which can improve the performance of
channel estimation, particularly when the SNR of the system
is low. However, when turning off trivial taps or using a low-
pass filter, a useful component may also be removed, which will
cause a large estimation error for those subchannels on the edge
[55], [56].
2) DDCE: For DDCE, CSI at the preamble block(s) is first
estimated and then used to demodulate and detect the symbols
at the next data block. CSI can be tracked by using detected
symbols or data, either hard decision or soft decision, as shown
in [67]–[69]. For systems with error-correction coding, redun-
dancy in coding can be exploited by iteratively performing soft
symbol decision and channel estimation [68], [70].
A major problem of DDCE is error propagation, which is
particularly severe for a system with a large Doppler frequency.
This can be solved by periodically inserting training blocks
[71]. Comb pilots may also be applied instead of training blocks
[46], and channel tracking or prediction can be used for further
performance improvement [72].
For coded OFDM systems, expectation–maximization algo-
rithms can improve estimation performance by exploiting the
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