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Signal Processing 88 (2008) 10021016
A novel simplified channel tracking method for MIMOOFDM
systems with null sub-carriers
Hyoung-Goo Jeona, Erchin Serpedinb,
aDepartment of Information Communication Engineering, Dongeui University, Busan, Republic of KoreabDepartment of Electrical and Computer Engineering, Texas A&M University College Station, TX 77843-3128, USA
Received 10 April 2007; received in revised form 7 August 2007; accepted 19 October 2007
Available online 26 November 2007
Abstract
This paper proposes an efficient scheme to track the time variant channel induced by multi-path Rayleigh fading in
mobile wireless multiple input multiple outputorthogonal frequency division multiplexing (MIMOOFDM) systems with
null sub-carriers. In the proposed method, a blind channel response predictor is designed to cope with the time variant
channel. The proposed channel tracking scheme consists of a frequency domain estimation approach that is coupled with a
minimum mean square error (MMSE) time domain estimation method, and does not require any matrix inverse
calculation during each OFDM symbol. The main attributes of the proposed scheme are its reduced computational
complexity and good tracking performance of channel variations. The simulation results show that the proposed method
exhibits superior performance than the conventional channel tracking method [Y.G. Li, N. Seshadri, S. Ariyavisitakul,
Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels, IEEE J. Sel. Areas
Commun. 17 (March 1999) 461471] in time varying channel environments. At a Doppler frequency of 100 Hz and bit
error rates (BER) of 104, signal-to-noise power ratio (Eb=N0) gains of about 2.5 dB are achieved relative to theconventional channel tracking method [Y.G. Li, N. Seshadri, S. Ariyavisitakul, Channel estimation for OFDM systems
with transmitter diversity in mobile wireless channels, IEEE J. Sel. Areas Commun. 17 (March 1999) 461471]. At a
Doppler frequency of 200 Hz, the performance difference between the proposed method and conventional one becomes
much larger.
r 2007 Elsevier B.V. All rights reserved.
Keywords: Channel; Estimation; MIMO; OFDM; Tracking; Fading; Doppler
1. Introduction
The multiple input multiple output (MIMO)
technique represents an efficient method to increase
data transmission rate without increasing band-
width since different data streams are transmittedfrom each transmit antenna [1]. Recently, orthogo-
nal frequency division multiplexing (OFDM) has
been effectively used for transmitting high speed
data in multi-path fading channel environments. In
OFDM, the high speed data stream is processed in
parallel and transmitted by N (in general, a power
of 2) orthogonal sub-carriers. The high spectral
efficiency of OFDM and its robustness to multi-
path fading channel environments are the main
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0165-1684/$- see front matterr 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.sigpro.2007.10.017
Corresponding author. Tel.: +1 979 458 2287;
fax: +1979 8624630.
E-mail addresses: [email protected] (H.-G. Jeon),
[email protected] (E. Serpedin).
http://www.elsevier.com/locate/sigprohttp://dx.doi.org/10.1016/j.sigpro.2007.10.017mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.sigpro.2007.10.017http://www.elsevier.com/locate/sigpro -
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reasons for its widespread usage in high bit-rate
transmissions such as digital audio broadcasting
(DAB), digital video broadcasting (DVB) and
wireless local area networks (WLAN) [2]. The
combined transmission method of MIMOOFDM
has attracted a lot of attention as a new datatransmission method in high speed data rate
systems. In MIMOOFDM receivers, the estimated
channel frequency response is used to separate the
mixed signals received from multiple antennas. An
important aspect is the fact that the performance of
MIMOOFDM receivers highly depend on the
accuracy of the channel estimator.
Thus far, numerous studies for channel estima-
tion in MIMOOFDM systems have been reported
(see e.g., [310]). Among the most notable results, Li
proposed a MMSE channel estimation method [3]
that exhibits good accuracy. However, this methodis computationally very complex due to the inverse
matrix calculation. In [6], by exploiting the correla-
tion of the subcarrier responses, Minn et al.
proposed a low complexity channel estimation
method which reduced the inverse matrix size by
half. However, Minn et al. method may cause
channel estimation errors in large delay spread
environments. Li also proposed a simplified channel
estimation method which required no matrix inver-
sion [4]. However, as mentioned in [6], if null sub-
carriers are used, Qiin of [4] would not be theidentity matrix, and there may be some performance
degradation according to the number of null sub-
carriers. Since real OFDM systems have null sub-
carriers in the guard band, a low complexity channel
estimation method considering null sub-carriers is
still needed.
As a possible solution to these problems, we are
proposing a novel simplified channel tracking
method that relies on a blind channel predictor.
The proposed method does not require prior
channel information or matrix inversion calculation
at all. In addition, the proposed method can
effectively track the nonlinear time varying channel
by using a piecewise linear model. To reduce its
computational complexity while maintaining a good
tracking accuracy, the proposed channel tracking
scheme is built by coupling a frequency domain
estimation approach with an MMSE time domain
channel estimation approach.
The remainder of this paper is organized as
follows. In Section 2, the MIMOOFDM system
and channel model are briefly described. Section 3
introduces the proposed channel tracking method.
In Section 4, the mean square error (MSE) and
computational complexity of the proposed channel
tracking scheme are assessed. The performance of
the proposed method is corroborated by computer
simulations in Section 5. Finally, Section 6 con-
cludes the paper.
2. Channel and MIMOOFDM system description
The channel impulse response of the mobile
wireless channel [3,6] can be modeled by
ht; t XL1k0
aktdt tk, (1)
where akt denotes the complex gain of the kth
path, tk represents the delay of the kth path, L is the
number of the multi-paths in the channel and dtstands for the impulse function. The frequency
response at time t is given by
Ht;f9
Z11
ht; t ej2pft dt XL1k0
akt ej2pftk.
2
Considering the motion of the mobile station, the
path gains akts are modeled to be independent
wide-sense stationary, narrow band complex Gaus-
sian processes and to have different average powers
s2k. With tolerable leakage, the channel frequencyresponse can be expressed as [3]
Hl; k9HlTf Tg; kDf XL01n0
hl; nWknN , (3)
where hl; n9hlTfTg; nts, WN9 expj2p= N,L0 stands for the channel length and depends on the
time dispersion of the wireless channel, N is the
number of tones and the fast Fourier transforma-
tion/inverse fast Fourier transformation (FFT/
IFFT) size, Tf
and Df denote the OFDM symbol
period and sub-carrier spacing of the OFDM
system, respectively, Tg represents the guard time
Tg9Tf=4 and ts is the sample interval given byts 1=NDf.
In a MIMOOFDM system, the output signal at
each Rx (receive) antenna is a mixed signal
consisting of the data streams coming from all Tx
(transmit) antennas. If the channel response
does not change during one OFDM symbol
and the cyclic prefix is longer than the channel
response length, the receive signal at the jth Rx
antenna can be expressed in the frequency domain
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as follows:
Rjl; k XNti1
Hijl; kXil; k Wjl; k,
j 1; . . . ; Nr; 0pkpN 1, 4
where Hijl; k is the channel frequency responsecorresponding to the kth sub-carrier and the lth
OFDM symbol transmitted between the ith Tx
antenna i 1; . . . ; Nt and the jth Rx antenna.Also, let N, Nr and Nt denote the number of sub-
carriers, the number of Rx antennas, and the
number of Tx antennas, respectively. Xil; k de-notes the data transmitted from the ith Tx antenna
on the kth sub-carrier at the lth OFDM symbol.
Wjl; k represents the additive white Gaussian noise(AWGN) at the jth receiver antenna, with zero
mean and variance s2n, and is assumed to beuncorrelated for different js, ks, or ls. Under the
assumption that the channel stays constant within
one OFDM symbol duration but the channel
changes from symbol to symbol, we will develop a
channel tracking scheme with improved perfor-
mance relative to the conventional scheme [3]. The
computer simulations, which assume realistic Ray-
leigh fading conditions (that are not limited to the
block fading assumption) [11], corroborate the
superior performance of the proposed channel
tracking scheme. The indices n and k denote timeand frequency-domain indices, respectively. The
symbols ~a, a and a denote the temporally estimated
value, the estimated value and the predicted value of
the variable a, respectively. In this system model,
time synchronization is assumed to be perfect, and
the maximum likelihood (ML) detection method is
used. Tx antennas transmit a long preamble
consisting of two training symbols before data
transmission mode, as WiBro and WLAN systemsdo [2,12]. It is assumed that in the data transmission
mode, Nd OFDM symbols are transmitted con-
secutively in each Tx antenna. For unbiased
performance comparison of channel tracking algo-
rithms, no channel coding is used.
3. Proposed channel estimation method
Since the wireless channel is time-variant, it is
necessary to track the channel response continu-
ously. In addition, since the received signal at eachRx antenna in MIMOOFDM systems is a multi-
ple-input single-output (MISO) signal, a time
domain channel estimation cannot be directly
applied on the received signal. In this paper, we
propose a low complexity adaptive channel estima-
tion method based on a blind channel prediction
scheme that is suitable for time variant channel
environments. The conceptual block diagram of
the proposed channel tracking scheme is shown in
Fig. 1. Before channel estimation, the frequency
domain MISO signal received at the jth Rx antenna(Rjl; k) is converted into the desired single-inputsingle-output (SISO) signal (See Section 3.2 for the
definition of desired SISO signal) by canceling the
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FFTrj[l,n] Rj[l,k]
Xi[l,k]
Sij[l,k] Hij[l,k] Hij[l,k]
hij[l,n]
MISO
to
SISOconversion
Freq.domain
Channel
Est.
IFFT FFT
Timedomain
Channel
Estimation
Post ML
detector
SISOMISO
Hij[l,k]Pre-ML
detector
Channel
predictor
Delay
device
Fig. 1. Block diagram of the proposed channel tracking method.
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other interfering signals coming from the other Tx
antennas. When pre-demodulating and converting
the MISO signal Rjl; k into the desired SISOsignal, the predicted channel response Hijl; k isused to cope with the time variant channel instead
of using the previously estimated channel responseHijl 1; k. Once the SISO signal is obtained,temporal channel estimation in frequency domain
is carried out to remove the inverse matrix calcula-
tion required in the following time domain channel
estimation. In the time domain channel estimation
block, the channel impulse response is obtained by
minimizing a MSE cost function, considering the
presence of null sub-carriers.
3.1. A blind channel response predictor design
In this paper, we design a blind channel predictor
by exploiting a piecewise linear model for the time
varying channel. Fig. 2 shows an example of the
time varying channel frequency response at the kth
sub-carrier. The channel frequency response of each
sub-carrier, Hijl; k, varies nonlinearly with time.However, the adjacent channel frequency responses
Hijl; k and Hijl 1; k present a certain correla-tion with each other and this relation is expressed as
Hijl; k Hijl 1; k Dijl; k, (5)
where Dijl; k denotes the difference between thechannel frequency responses corresponding to lth
and l 1th symbols at the kth sub-carrier. If a
piecewise linear model [13] is used during the short
time of some OFDM symbol periods as shown in
Fig. 2, the nonlinear time varying channel frequency
response Hijl; k can be treated as a linear model.Using the piecewise linear model, let us assume
that Hijl; k varies linearly during the time of theconsecutive MOFDM symbols. The variable Mcan
be set according to the channel response changing
rate. For example, a channel with a short coherence
time will have a small M, and vice versa. In the
piecewise linear model, the condition Dijl; k Dijl 1; k is assumed. Therefore, from (5) we inferthat Hijl; k Hijl 1; k Dijl 1; k. If we
know^
Hijl 1; k and^
Dijl 1; k at the timeinstant corresponding to the lth symbol, thenthe predicted channel response Hijl; k can beobtained as
Hijl; k9Hijl 1; k Dijl 1; k. (6)
Referring to Fig. 2 of the piecewise linear model,
Dijl 1; k can be expressed by using the previouslyestimated channel responses, since 2Dijl 1; k Hijl 1; k Hijl 3; k and Dijl 1; k Hijl 2; k Hijl 3; k. In the case where the
channel response varies linearly during the Mconsecutive OFDM symbols, therefore, (6) can be
expressed by the linear combination of the M 1
previously estimated channel responses as follows:
Hijl; k9XM1m1
omHijl m; k
XM1m1
omHijl m; k Zijl m; k, 7
where om is a weight value and Zijl; k9Hijl; k
Hijl; k and denotes the random channel esti-mation error with zero mean and variance s2e (see
Section 4.2). Let us define DHijl; k9Hijl; k Hijl; k as the channel response prediction randomerror. The weight values can be found by minimiz-
ing the following MSE cost function:
xl; k9EfjDHijl; kj2g
E Hijl; k XM1m1
omHijl m; k
28>>>>>>:
(15)
where index k 0 denotes the DC component and g
stands for the number of null sub-carriers in the
guard band. Note that estimating the channel
response in the frequency domain removes the need
of calculating an inverse matrix in the next step of
time domain channel estimation employing an
MMSE technique. The time domain channel esti-
mation is performed during the next step, byconsidering all the null sub-carriers used in the
guard band. The time domain channel estimate
hijl; n can be found by minimizing the followingMSE cost function:
Cfhijl; ng; i 1; . . . ; Nt
9XN1k0
Yijl; kXL0n0
hijl; nWnkN Zk
2
. 16
hijl; n is the estimated channel impulse response
and can be determined by solving
qCfhijl; ng
qhijl; n09
1
2
qCfhijl; ng
qRhijl; n0 j
qCfhijl; ng
qIhijl; n0
( )
0, 17
where R and I denote the real and imaginary
parts of a complex number, respectively, and
n0 0; 1; . . . ; L0. Direct solving (17) results in
XN1k0
Yijl; k XL0n0
hijl; nWknN Zk !
ZkWkn0N 0.
(18)
Define
qn9XN1k0
ZkWknN , (19)
yijl; n9XN1k0
Yijl; kZkWknN . (20)
Then, (18) can be expressed as
XL01n0
hijl; nqn0 n yijl; n0 (21)
for i 1; . . . ; Nt and n0 0; 1; . . . ; L0. Eq. (21) canbe expressed in matrix form as
Qhijl yijl, (22)
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where
Q9
q0 q1 q1 L0
q1 q0 q2 L0
..
. ...
...
qL0 1 qL0 2 q0
0BBBBB@
1CCCCCA
,
(23)
hijl9hijl; 0; hijl; 1; . . . ; hijl; L0 1T, (24)
yijl yijl; 0;yijl; 1; . . . ;yijl; L0 1T. (25)
Hence, the channel impulse response can be
estimated by
hijl Q1yijl. (26)
Since matrix Q is a time invariant constant matrixdetermined by the null sub-carrier pattern, the
inverse matrix Q1 can be pre-calculated and stored
in the memory before beginning the channel
tracking. Therefore, no explicit calculation of Q1
is required for every OFDM symbol. However, [3]
and [6] require inverse matrix calculation for every
OFDM symbol.
4. MSE calculation and complexity comparison
4.1. MSE calculation
In this section, we will derive the MSE of the
proposed channel estimation scheme. From (20),
one infers that
yijl; n XN1k0
XL01m0
hijl; mWmkN Vijl; k
!ZkWnkN
XL01m0
hijl; mqn m vijn; 0pnpL0,
27
where vijn PN1
k0 Vijl; kZkWnkN . Eq. (27) can
be expressed in matrix form as
yijl Qhijl vijl, (28)
where
hijl hijl; 0; hijl; 1; . . . ; hijl; L0 1T, (29)
vijl vjl; 0; vjl; 1; . . . ; vijl; L0 1T. (30)
From (28), the channel impulse response estimate
corresponding to Tx antenna iand Rx antenna jcan
be expressed as
hijl Q1yijl hijl Q
1vijl. (31)
The MSE of the channel impulse response estimate
can be given by
MSEl9Efkhijl hijlk2g
EfQ1vijlHQ1vijlg
TracefQ1EfvijlvijlHgQ1Hg. 32
A generic entry of EfvijlvijlHg is given by
Efvijl; n1vijl; n2g E
XN1k10
Vijl; k1Zk1Wn1k1N
!(
XN1
k10
Vijl; k2Zk2Wn2k2N
!)
XN1
k1;k20
EfVijl; k1Vijl; k2g
Zk1Zk2Wn1k1n2k2N , 33
where n1; n2 1; 2; . . . ; L0. If we substitute (14) into(33), Eq. (33) can be rewritten as
Efvijl; n1vijl; n2g
XN1k0
PNtm1;maijXml; kj
2
jXil; kj2
s2p
(
s2n
jXil; kj2
)ZkW
n1n2kN , 34
where s2p9EjDHijl; kj2 s2e
PM1m1 o
2m and DHij
l; k and Wjl; k are assumed to be independent ofeach other. Hence, if a constant modulus modula-
tion is used,
EfvijlvijlHg Nt 1s
2p
s2njXil; kj
2
Q: 35
If no null sub-carriers are used, since Q NI,
MSEl L0N
Nt 1s2
p 1
SNR
. (36)
Given a SNR, the MSEl is dependent on the
prediction error, the channel response length and
the number of Tx antennas.
4.2. Mean and variance of random variable Zijl; k
For convenient comprehension, let us assume
that BPSK is used for Xil; k, then EXil; k 0.Xml; k, DHijl; k, Wjl; k and Xil; k are indepen-
dent of each other. It is clear that EXml; k 0,
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EXil; k 0 and EWjl; k 0. Therefore, it isalso clear that EVijl; k 0, where Vijl; k is givenby (14). From (31), the channel estimation error in
time domain can be expressed as Ehijl hijl
Q1Evijl. Since Evijn PN1k0 EVijl; kZk
W
nk
N 0, Q
1
Evijl 0. It means that the meanof the channel estimation error in time domain is
zero. Since the channel frequency response estimate
Hijl; k is obtained by performing Fourier trans-formation for the channel impulse response estimate
hijl; n, the mean of the channel estimation error infrequency domain is also zero. That is, EZijl; k EHijl; k Hijl; k 0.
The MSE of the channel frequency response in
the kth sub-carrier is defined by EHijl; kHijl; k
2 s2e . The total MSE of the channel
impulse response in time domain is equal to
MSEl given by (36). When the channel impulseresponse hijl; n is Fourier transformed to obtainthe channel frequency response Hijl; k, the totalMSE in time domain is equal to the total MSE in
frequency domain. Therefore, if there are N sub-
carriers in the OFDM system, then s2e MSEl=N.
4.3. Computational complexity comparison
In this section, the computational complexities of
the schemes proposed in [3,6] and the method
proposed herein are compared briefly. The complex-ity comparison will be focused on the channel
estimation based on the decision-directed estimation
method, as [6] did. Since Q1 can be pre-calculated,
Q1 calculation is not included in the complexity
comparison. Since Lis simplified method [4] may
cause estimation error in MIMOOFDM systems
with null sub-carrier and employing non-constant
modulus modulations, Lis simplified method will
not be discussed in this comparison. In the case of
two Tx and Rx antennas and N sub-carriers, the
channel estimation complexity for each method is
given in Table 1 (refer also to [6]). In Table 1, FFTN
denotes the number of multiplications required forthe FFT operation with size N. invL0 L0 stands
for the number of multiplications required for L0
L0 matrix inversion. Nu denotes the number of the
sub-carriers used. When Nu N, no null sub-
carrier is used. The number of FFT operations for
Li method [3] can be easily obtained by referring to
Fig. 3 in [3]. Considering Eqs. (15) and (16) in [3],
the number of multiplications required for calculat-
ing hijl can be derived straightforwardly. The
calculation amount for Minn method [6] can be
obtained by considering the similarity with Li
method. Note that when Nu N and constantmodulation is used, Qiin is a identity matrix. In
that case, Lis method needs calculating only
invL0 L0 instead of calculating inv2L0 2L0.
From Table 1, we can see that the proposed method
has the lowest complexity among these methods,
regardless of the non-constant modulation and the
presence of null sub-carriers.
5. Performance evaluation
Computer simulations are carried out to evaluatethe performance of the proposed method. Two Tx
and two Rx antennas are used for the MI-
MOOFDM system. There are a total of 128 sub-
carriers so that the FFT/IFFT size is 128. The DC
component sub-carrier is not used, and 10 and 9
sub-carriers on each end of the spectral band,
respectively, are used as guard band. The rest of 108
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Table 1
Evaluation of computational complexity for each method
Condition Method No. of complex multiplications and divisions
Constant modulus with Nu N Ref. [3] 3N 2L02 5FFTN invL0 L0
Ref. [6] 4:5N 2L02 3FFTN=2 invL0 L0
Proposed 3N 2L02 5FFTN
Constant modulus with NuaN Ref. [3] 3N 2L02 5FFTN inv2L0 2L0
Ref. [6] 4:5N 2L02 3FFTN=2 invL0 L0
Proposed 3N 2L02 5FFTN
Non-constant modulus Ref. [3] 5N 2L02 5FFTN inv2L0 2L0
Ref. [6] 6N 2L02 4FFTN=2 2 invL0 L0
Proposed 3N 2L0
2 5FFTN
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sub-carriers are used to transmit data. The OFDM
symbol rate is 10 Ksps, and the symbol period is
100ms, including the guard time of 20ms. The
channel length L0 is assumed to be 18. Modulation
in sub-carriers is QPSK. The carrier frequency is
2.4 GHz. The multi-path Rayleigh fading channel
assumes two rays with equal gain, and each ray has
six multi-path delay taps. Each signal path is
assumed to undergo an independent Rayleigh
fading. The rms delay spread is 1:82ms. We used
the Rayleigh fading channel simulator (Jakessinusoid sum method) openly published in
Ref. [11]. The Doppler frequencies 40, 100 and
200 Hz are used to represent different mobile
environments. After completion of channel estima-
tion by using the training signal, the system state is in
data transmission mode. In data transmission mode,
channel tracking for 20 consecutive OFDM symbols
is carried out continuously, using a decision directed
method in which the demodulated data is used as the
reference data. The performance of the system is
measured by the estimators MSE and bit error rates
(BER), each averaged over 100,000 OFDM blocks.
For unbiased comparison, no channel coding is used
in this simulation. In order to track the time varying
channel response, Kalman filter method may be used
for MIMOOFDM systems as Komninakis did [14].
However, the Komninakis method should calculate
the inverse matrix in every OFDM symbol to obtain
Kalman gain matrix and thereby the calculation
amount increases significantly. For this reason, we
compare the proposed method with Kalman filter
estimator, using scalar Kalman filter in each sub-
carrier [15].
Fig. 3 shows an example of the proposed channel
tracking and the nonlinear time variant channel
frequency response H11l; k simulated at thegiven multi-path channel parameters, l 1; 2; . . . ;100, k 10, and maximum Doppler frequency
fd 200 Hz. In Fig. 3, solid line is the channel
response tracked by the proposed method at
Eb=N0 15 dB. Fig. 3 shows that the nonlinearchannel response is well tracked by the proposed
method. The performance simulation results are
shown in Figs. 411. Fig. 4 shows the MSE of theproposed method at the conditions ofM 4, 5 and 6,
and at the fixed SNR ofEb=N0 25 dB. The range ofthe normalized Doppler frequency (fdTs) is given
from 0 to 0:03. In the case of fdTso0:02, theperformance of the proposed scheme (M 4) is the
worst among all the investigated channel tracking
methods. The reason is that in decision directed mode,
the performance of the proposed channel response
estimator is very highly affected by the prediction
error caused by the first channel prediction just after
two training OFDM symbols. The prediction error
causes the demodulation error which results in MSE
performance degradation and propagates into the
next channel estimation. As mentioned before, when
two training OFDM symbols are received, only two
channel responses Hij1; k and Hij2; k areobtained, and Hij3; k; . . . ; Hij1 M; k are as-sumed to be equal to Hij2; k. Therefore, from (10),the MSEs for the first channel prediction are
proportional to xl; k / 1:8s2e when M 4, xl; k /s2e when M 5 and xl; k / 0:68s
2e when M 6,
respectively. There is a larger difference in MSE
between M 4 and 5 than between M 5 and 6.
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0 10 20 30 40 50 60 70 80 90 1001
0.5
0
0.5
1
1.5
2
OFDM symbol index
Real{H11(l,10)}
Eb/No= 15 dB
fd= 200 HzEb/No= 15 dB
fd= 200 Hz
0 10 20 30 40 50 60 70 80 90 1001.5
1
0.5
0
0.5
1
OFDM symbol index
Image{H11(l,1
0)}
Channel response
Estimated channel response
Channel response
Estimated channel response
Fig. 3. An example of channel frequency response Hijl; k and tracking at fd 200Hz, k 10 and Eb=N0 15dB.
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The noise effect in channel prediction can be
reduced by increasing M, as shown in Fig. 4.
However, increasing M beyond a certain limit may
result in performance degradation due to the
increased sensitivity to channel time variations.
Note that MSEM 6 is larger than MSEM
5 at fdTs40:015. We can see that when M 5, the
proposed method shows the best performance in
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0 0.005 0.01 0.015 0.02 0.025 0.03104
103
102
101
fd*T
s
MSE
Proposed(M=4)
Proposed(M=5)
Proposed(M=6)
Li original
Fig. 4. MSE performance as a function of fdTs at a given M.
0 0.005 0.01 0.015 0.02 0.025 0.03104
102
103
100
101
101
fd*T
s
MSE
Eb
/No
= 25 dB
Proposed CE
No predict
FDE
Li original
Fig. 5. MSE performance as a function of fdTs for each channel tracking method.
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Fig. 4. Hereafter, M is set to 5 in all the simulations
for performance evaluation. When M 5 and the
OFDM symbol period is 100 ms, the time duration
for which a piecewise linear model is assumed is
500ms. Fig. 5 shows the MSE for each method as a
function offdTs at the conditions: Eb=N0 25 andM 5. As we can see from Fig. 5, the MSE of the
proposed method increases more slowly than the
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0 5 10 15 20 25 30
104
101
102
103
101
100
102
Eb/No[dB]
MSE
Proposed CE
No predict
FDE
FED + Km filter
Li original
Fig. 6. MSE performance at fd 40Hz.
0 5 10 15 20 25 30
107
104
105
106
102
101
103
100
Eb/No[dB]
BER
Proposed CE
Perfect CE
No predict
FDE
FED + Km filter
Li original
Fig. 7. BER performance at fd 40Hz.
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other methods as fdTs increases. On the other hand,
MSE of Lis original method increases rapidly when
fdTs40:015.
Figs. 6, 8 and 10 show MSE performances at
Doppler frequencies of 40, 100 and 200 Hz, respec-
tively. Figs. 7, 9 and 11 show BER performances at
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0 5 10 15 20 25 30
104
101
102
103
101
102
100
103
Eb/No[dB]
MSE
Proposed CE
No predict
FDE
FED + Km filter
Li original
Fig. 8. MSE performance at fd 100Hz.
0 5 10 15 20 25 30
104
105
106
102
101
103
100
Eb/No[dB]
BER
FED + Km filter
Proposed CE
Perfect CE
No predictFDE
Li original
Fig. 9. BER performance at fd 100Hz.
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Doppler frequencies of 40, 100 and 200 Hz, respec-
tively. In these figures, BER performance curves for
perfect channel estimation are given to show the
performance in the ideal channel estimation case. In
these figures, FDE denotes the frequency domain
channel tracking method of Ref. [10]. In order to
compare the effect of channel prediction, BER and
MSE curves for the no predict tracking method are
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0 5 10 15 20 25 30
104
101
102
103
101
100
102
Eb/No[dB]
MSE
Proposed CE
No predict
FDE
FED + Km filter
Li original
Fig. 10. MSE performance at fd 200Hz.
0 5 10 15 20 25 30
106
104
10
5
102
101
103
100
Eb/No [dB]
BER
Proposed CE
Perfect CE
No predictFDEFED + Km filterLi original
Fig. 11. BER performance at fd 200Hz.
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also drawn in these figures. No predict means the
proposed method without the channel prediction
function. If no prediction is used, the previously
estimated channel value Hijl 1; k is used asthe current channel value Hijl; k. In this case, the
channel estimation error is given by Hijl; kHijl 1; k Dijl; k Zijl; k. When the channelprediction is used, the channel estimation error is
given by DHijl; k Hijl; k Hijl; k. The MSEof no prediction is given by EHijl; kHijl 1; k
2 VarDijl; k s2e . The MSE of
prediction is EjDHijl; kj2 1:5s2e as we can see
from (10). Therefore, the performance of predic-
tion is better than that of no prediction as long as
VarDijl; k40:5s2e is satisfied. VarDijl; k % 0 at
a low Doppler frequency ands2e is inversely propor-
tional to the Eb=N0 such that s2e % 0 at a high
Eb=N0. Therefore, the lower the Doppler frequencyis in the wireless channel, the higher Eb=N0 isrequired to satisfy the condition of VarDijl; k40:5s2e . That is the reason why the performance ofno prediction is better than that of prediction at
fd 40 Hz. Note that in Fig. 6, the performance
gap in Eb=N0 between no prediction and predic-tion is getting narrow with increase of Eb=N0. If wecan know the information about Eb=N0 and theDoppler frequency, either no prediction or pre-
diction can be selected to improve the performance
of channel estimator, based on such information.At a Doppler frequency of 40 Hz, there is a little
difference in the MSE and BER performance
between the proposed method and Lis original
method. At fd 100 Hz and BER of 104, the
performance improvement provided by the pro-
posed method is about 2.5 dB in Eb=N0 comparedwith that of Lis original method. However, at a
Doppler frequency of 200 Hz, the BER performance
difference becomes much larger when compared
with those of other methods. As we can see from
Figs. 9 and 11, due to the inter sub-carrier
interference (ICI) [16], BER performance at a
Doppler frequency of 200 Hz is worse than that of
100 Hz.
From these figures, we can observe that the
channel estimation error of Lis original method is
very large at a given low Eb=N0. At a given lowEb=N0, BER is high and the demodulated dataXl; k is erroneous. Since Lis original methodcalculates the inverse of a matrix made of erroneous
demodulated data Xl; k, the channel estimationerror is amplified by noise and the erroneous inverse
matrix. In the worst case of low Eb=N0 (less than
5 dB), the channel estimation error may diverge when
tracking the channel response. On the other hand,
since the proposed method uses the known inverse
matrix Q1 which is a constant time-invariant
matrix, the effect of the erroneous demodulated data
is much less significant than that of Lis originalmethod. The simulation results show that as expected
the proposed method does not diverge.
6. Conclusions
This paper proposed a novel simplified channel
tracking method to reduce the computational
complexity and improve the tracking performance
in time varying channel environments. In the
proposed method, a blind channel response pre-
dictor is designed to cope with the time variant
channel. The proposed channel tracking scheme
consists of a frequency domain estimation approach
that is coupled with an MMSE time domain
estimation method, and does not require any matrix
inverse calculation during each OFDM symbol. By
converting the MISO signal into a SISO signal and
performing temporal channel estimation in the
frequency domain before beginning time domain
channel estimation, no matrix inversion is required
anymore. The simulation results show that the
proposed method exhibits superior performance
than Lis original method in time varying channelenvironments. At a Doppler frequency of 100 Hz
and BER of 104, signal-to-noise power ratio
(Eb=N0) gains of about 2.5 dB are achieved relativeto Lis original method. At a Doppler frequency of
200 Hz, the performance difference between the
proposed method and conventional one becomes
much larger.
Acknowledgment
This work was partially supported by researchproject 07-03 funded by Electronics and Telecom-
munications Research Institute (ETRI) in Korea.
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