IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002
67
Simplified Channel Estimation for OFDM Systems
With Multiple Transmit Antennas
Ye (Geoffrey) Li, Senior Member, IEEE
Abstract—Multiple transmit-and-receive antennas can be used
in orthogonal frequency division multiplexing (OFDM) systems
to improve communication quality and capacity. In this paper,
we present two techniques to improve the performance and
reduce the complexity of channel parameter estimation: optimum
training-sequence design and simplified channel estimation. The
optimal training sequences not only simplify the initial channel
estimation, but also attain the best estimation performance. The
simplified channel estimation significantly reduces the complexity
of the channel estimation at the expense of a negligible performance degradation. The effectiveness of the new techniques is
demonstrated through the simulation of an OFDM system with
two-transmit and two-receive antennas. The space-time coding
with 240 information bits per codeword is used for transmit
diversity. From the simulation, the required signal-to-noise ratio is
only about 9 dB for a 10% word error rate for a channel with the
typical urban- or hilly-terrain delay profile and a 40-Hz Doppler
frequency.
Index Terms—MIMO systems, OFDM, parameter estimation,
transmit diversity.
I. INTRODUCTION
M
ULTIPLE transmit-and-receive antennas can be used
with orthogonal frequency division multiplexing
(OFDM) to improve the communication capacity and quality
of mobile wireless systems. Channel parameters are required
for diversity combining, coherent detection, and decoding.
For OFDM systems with multiple transmit antennas, such
as permutation [1]–[3] and space-time coding [4], [5] based
transmit diversity, different signals are transmitted from different transmit antennas simultaneously. Consequently, the
received signal is the superposition of these signals, which
gives rise to challenges for channel estimation. In this paper, we
investigate training-sequence design and parameter estimation
simplification techniques for OFDM with multiple transmit
antennas.
Training sequences are used in wireless communication systems to obtain initial estimation of channel parameters, timing,
and frequency offset. For multiple transmit antenna systems,
training sequences should be designed to decouple the interantenna interference for channel estimation. A common conjecture is to use orthogonal sequences at different transmit antennas. However, orthogonality is not sufficient for good paManuscript received November 18, 1999; revised July 24, 2000 and October
30, 2000; accepted August 10, 2000. The editor coordinating the review of this
paper and accepting it for publication is A. Czylwik.
The author is with the School of Electrical and Computer Engineering,
Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (e-mail:
[email protected]).
Publisher Item Identifier S 1536-1276(02)00180-0.
rameter estimation of dispersive fading channels. Instead, the
optimal training sequences for different training antennas are
shown to be local orthogonal, i.e., for any starting position they
are orthogonal over the minimum set of elements. Furthermore,
this also results in simplified channel estimation.
When OFDM systems are used for data transmission, pilotsymbol-aided or decision-directed estimation must be used to
track channel variations. For systems with only one transmit antenna, channel parameter estimation [6]–[11] has been successfully used to improve the performance. For systems with multiple transmit antennas, a channel estimator has been developed
in [5] by using the correlation of the channel parameters at different frequencies. However, it requires the inversion of a large
matrix to decouple the inter-antenna interference. Hence, a simplified channel estimator is desired to reduce the complexity.
In this paper, we present two reduced-complexity channel estimation techniques for OFDM systems with multiple transmit
antennas: optimum training sequences and simplified estimation. The first technique carefully selects the training sequences
to eliminate interantenna interference. The second greatly simplifies the channel estimation at the expense of a negligible performance degradation.
The rest of this paper is organized as follows. In Section II,
we briefly introduce OFDM systems with multiple transmit
antennas, discuss the channel model, and describe the basic
principle of the channel estimator. We then present optimum
training-sequence design in Section III. Next, in Section IV,
we introduce the simplified channel estimation and analyze its
performance. Finally, we demonstrate the performance of the
new techniques by computer simulation in Section V.
II. MULTIPLE TRANSMIT ANTENNAS FOR OFDM IN MOBILE
CHANNELS
Before introducing the new techniques, we briefly describe
an OFDM system with multiple transmit antennas, the statistics
of mobile channels, and the principle of the original estimator
developed in [5].
A. Transmit Diversity for OFDM Systems
An OFDM system with two transmit and two receive antennas is shown in Fig. 1. Though the figure emphasizes transmit
diversity, the techniques developed in this paper can be directly
applied to any OFDM system with multiple transmit antennas.
At time , a data block
is transformed into two different signals
at the transmit diversity processor, where , ,
and are the number of subchannels of the OFDM systems, subchannel (or tone) index, and antenna index, respectively. Each of
1536–1276/02$17.00 © 2002 IEEE
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Fig. 1.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002
Transmit diversity for OFDM.
these signals forms an OFDM block. The transmit antennas sifor
multaneously transmit OFDM signals modulated by
. For space-time-coding-based transmit diversity [5],
the transmit diversity processor is a space-time encoder, while
it may include an error-correction encoder and a permutation
operation if permutation diversity [3] is used.
At the receiver, the discrete Fourier transform (DFT) of the
received signal at each receive antenna is the superposition of
two distorted transmitted signals. The received signal at the th
receive antenna can be expressed as
where
(1)
where
denotes the channel frequency response
where
at the th tone of the th OFDM block, corresponding to
the th transmit and the th receive antenna. The statistical
characteristics of wireless channels are briefly described in
denotes the additive complex Gaussian
Section II-B.
noise on the th receive antenna and is assumed to be zero
mean with variance . The noise is uncorrelated for different
’s, ’s, or ’s.
To achieve transmit diversity gain and detect the transmitted
signal, channel state information is needed. In [5], we developed
a channel estimator by exploiting the time- and frequency-domain correlations of the channel parameters. After describing
the channel statistics in Section II-B, we summarize the principle of the estimator in Section II-C.
B. Channel Statistics
The complex baseband representation of the mobile wireless
channel impulse response can be described by [12]
(2)
where is the delay of the th path
is the corresponding
complex amplitude
is the shaping pulse whose frequency
response is usually a square-root raised-cosine Nyquist filter.
’s are wide-sense
Due to the motion of the vehicle,
stationary (WSS) narrow-band complex Gaussian processes,
which are independent for different paths.
From (2), the frequency response at time is
For OFDM systems with proper cyclic extension and timing, it
can be shown in [5], [9] that with tolerable leakage, the channel
frequency response can be expressed as
(4)
and
.
,
, and
in the above
expression are the block length, symbol duration, and tone
spacing, respectively. They are related by
and
, where
is the duration of the cyclic
’s, for
, are WSS
extension. In (4),
narrow-band complex Gaussian processes.
is the number of
nonzero taps of the channel impulse response sampled at a rate
. The average power of
and
(
) depend
of
on the delay profiles of the wireless channels. If an OFDM
is
system is reasonably designed, then
less than, but very close to .
C. Basic Principle of Channel Estimation
For an OFDM system with multiple transmit antennas, every
tone at each receive antenna is associated with multiple channel
parameters, which makes the channel estimation difficult. Fortunately, channel parameters for different tones of each channel
are correlated and a channel estimator has been developed in [5]
based on this correlation.
Following(4), the frequency response at the th tone of the
th block corresponding to the th transmit antenna can be expressed as1
(5)
, we only need to estimate
.
Hence, to obtain
From the previous section, the signal from each receive antenna can be expressed as
(6)
(3)
1The
r [n; k]
index j for different receive antennas is omitted from
and w [n; k].
H
[n; k]
,
LI: SIMPLIFIED CHANNEL ESTIMATION FOR OFDM SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS
for
’s, for
mation of
and all . If the transmitted signals
are known,2
, the temporal esti, can be found by [5]
(7)
or
(8)
69
with
and
, it is suffiTo find
for
since
only
cient to find
’s for
, where
if
.
consists of
Note that
is the DFT of
. Therefore,
with
such that
for all
there is no
’s since it will result in
and so
,
for all .
Let
(10)
where
is the temporal estimation of channel parameter
vector, defined as
for some with
checked that
. Then it can be directly
(11)
and
,
,
and
are definded as
for
.
Equation (11) implies that
and
.
Therefore,
In general, for systems with the number of transmit antennas,
, less than or equal to
, let
and
(12)
(9)
for
, where
and
the largest integer no larger than . Then for any
denotes
respectively.
From (8), we can see that a
matrix inversion is
. In order to rerequired to get the temporal estimation of
duce the computational complexity, the significant-tap-catching
(STC) estimator has been proposed in [5]. However, it is still
very complicated.
III. OPTIMUM TRAINING SEQUENCES
In this section, we investigate the optimum training sequences
for channel estimation in OFDM with multiple transmit antennas. For simplicity, we assume that the modulation results in
a constant-modulus signal. However, with only minor modification, the results discussed here are applicable to any modulation
format.
and from (9)
For constant-modulus modulation,
(13)
Note that
; therefore
(14)
and
(15)
for any , where
denotes the unit impulse function. Consequently,
, where is a
identify ma’s, are chosen such that
trix. If the training sequences,
for
, then, from (8),
and no matrix inversion is required for channel estimation. The
questions are if such a sequence exists and how the training sequence affects the performance of the channel estimation. To
answer these questions, we first construct the training sequences
and then demonstrate that the constructed training sequences actually make the channel estimation attain the best performance.
’s for
be any training sequence
Let
at training time that is good for timing and frequency synchronization and possibly other properties in OFDM systems.
2During the training period, the transmitted signals for each antenna are
known. When the system is in data transmission mode, decoded data are used
to generate the reference signals.
Consequently,
for
or
(equivalent to
), which results in
for all
. If
,
.
Hence,
for all
.
It should be indicated that the above optimum training-sequence design approach is not applicable to those systems with
.
Fig. 2 shows how an optimum training sequence works for an
OFDM system with four transmit antennas. The OFDM system
has 128 tones and the length of the channel impulse responses
corresponding to all transmit antennas is less than 32, as shown
in (12) and
’s for
in Fig. 2(a). Then,
are shown as in Fig. 2(b). As indicated before
(16)
70
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002
(a)
(b)
Fig. 2. OFDM with optimum training sequence. (a) jh
for
and
[n; l]j
’s. (b) p
[n; l]
’s.
. Furthermore
(17)
for
and
. Therefore, by carefully
selecting the relative phases between the training sequences for
different transmit antennas, the effect of the channels correis shifted to
sponding to different transmit antennas on
different regions, so the parameters for different channels can
be easily estimated without using any matrix inversion.
for
, the estimator
From [5], only when
attains its lower MSE bound. Therefore, the optimum training
sequences can also make the estimator achieve the best performance.
We have presented the optimum sequence design for the
training period of the decision-directed channel estimation.
The above discussion can be easily extended to pilot sequence
design for pilot-symbol-aided channel estimation or channel
estimation in single carrier systems. For pilot-symbol-aided
channel estimation in OFDM systems [6], [11] with the pilot
tones scattered into different times and frequencies, the pilot
sequences are two-dimensional (2-D). The optimum sequence
LI: SIMPLIFIED CHANNEL ESTIMATION FOR OFDM SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS
Fig. 3.
71
Simplified channel estimator for transmit diversity.
design strategy described above can be directly used there.
Furthermore, it is more flexible in this case since the relative
phases of the pilot sequences for different transmit antennas
can be shifted in a 2-D plane.
is known, then
can be
From the above equations, if
estimated without any matrix inversion. However, neither
or
is known.
Denote
’s for
the robust estimation of channel
parameter vectors at time , i.e.,
IV. SIMPLIFIED CHANNEL ESTIMATION
In the previous section, we introduced optimum training-sequence design for decision-directed channel estimation for
OFDM systems with multiple transmit antennas. Using the
optimum training sequences, the temporal channel estimation
during the training period can be simplified and can achieve
the best performance. However, when the systems are in the
’s at
data transmission mode, the transmitted signals
different transmit antennas are determined by the random data
to be transmitted; therefore, their relative phases cannot be
chosen using the above technique. Hence, some simplification
technique is desired to reduce the complexity of channel parameter estimation for OFDM with multiple transmit antennas.
A. Algorithm
Here, we develop simplified channel-estimation algorithm.
Though we discuss OFDM systems with only two transmit antennas, the proposed algorithm can be easily extended to those
OFDM systems with over two transmit antennas.
From (7), we can see that
From the discussion in the previous section, for systems with
and,
constant modulus modulation,
therefore
(18)
where ’s (
) are the coefficients of the robust channel
.
estimator [5], [9] and their Fourier transform denoted by
,
If robust estimation of channel parameter vectors at time
, and
are used to substitute
and
in the right sides of (18), then
(19)
The substitution reduces the computational complexity of the
channel estimation. However, it may also cause some performance degradation but, from the following analysis and computer simulation, the performance degradation is negligible.
The simplified channel estimation with optimum training sequence is shown in Fig. 3 and summarized in Table I.
The STC and the estimation enhancing techniques developed
in [5] can be also used together with the simplified estimation
described here to further reduce the complexity and improve the
performance.
B. Performance Analysis
Here, we analyze the performance of the simplified channel
estimation. In particular, we investigate the mean-square error
(MSE) of the temporal estimation for simplified channel estimator.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002
TABLE I
SIMPLIFIED CHANNEL-ESTIMATION ALGORITHM
Note that if
satisfies the optimum training-sequence
, then from [5] both the
condition in Section III, i.e.,
original and the simplified channel estimators are the same and
both attain the same best performance bound
(a)
(20)
MSE
’s
When the system is in the data transmission mode,
are random variables, which are assumed to be independent for
different ’s, ’s, and ’s. Based on this assumption, we have
for
. On the other hand, from [5]
(b)
Fig. 4. Performance comparison of the simplified and the original estimator
for the TU channels with (a) f = 40 Hz and (b) f = 200 Hz, respectively.
where the
vector
represents the effect of the additive Gaussian noise, whose elements are independent identically
distributed complex Gaussian random variables with zero mean
. Therefore, (19) becomes
and variance
It is shown in the Appendix that the MSE of the temporal
estimation is
MSE
MSE
(21)
Since both
and
are zero mean
(22)
Therefore, the simplified temporal estimation is an unbiased estimator.
(23)
where MSE is the MSE of robust channel estimation that exploits both the time- and frequency-domain correlations
and is the Doppler frequency of the channel. MSE
in the above is caused by using the noisy estimated parameters in the previous symbol duration in simplified channel
estimation. Usually, MSE is much less than the noise level
) and, therefore, compared with the original es(MSE
timation; the performance degradation of the simplified estimation is negligible when the square of normalized Doppler fre). Furtherquency is much less than the noise power (
more, the simplified estimator does not require a large matrix
LI: SIMPLIFIED CHANNEL ESTIMATION FOR OFDM SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS
73
inversion and, therefore, it has lower complexity and also is numerically stable.
V. PERFORMANCE EVALUATION BY COMPUTER SIMULATION
To demonstrate the performance of the simplified channel
estimation, computer simulations have been conducted for the
space-time coding based tracnsmit diversity. Before presenting
the simulation results, we first describe the parameters of the
simulated OFDM systems.
A. Parameters
Channels with the typical urban (TU) and hilly terrain (HT)
delay profiles [12] and a Doppler frequency of 40 or 200 Hz are
used to represent different mobile environments. Two transmit
and two receive antennas are used for diversity. The links between different transmit or receive antennas are independent.
However, they have the same global statistics.
The system parameters are the same as those in [3] and [5].
The entire channel bandwidth, 800 kHz, is divided into 128 subchannels. The four subchannels on each end are used as guard
tones and the rest (120 tones) are used to transmit data. To
make the tones orthogonal to each other, the symbol duration
is chosen to be 160 s. An additional 40 s guard interval is
used to provide protection from intersymbol interference caused
by channel multipath delay spread. This results in a total block
s and a subchannel symbol rate
K
length
symbols/s.
A 16-state space-time code [14] with 4 PSK is used in the
system. Each data block, containing 236 bits, is coded into two
different blocks, each of which has exactly 120 symbols to form
an OFDM block. The channel parameter estimation approaches
developed in this paper are used to provide estimated parameters for decoding. Overall, the described system can transmit
data at a rate of 1.18 Mbits/s over an 800–kHz channel, i.e., the
transmission efficiency is 1.475 bits/s/Hz.
B. Simulation Results
The system performance is measured by the word error rate
(WER), which is averaged over 10 000 OFDM blocks. Figs. 4
and 5 show WER versus signal-to-noise ratio (SNR) for the
simplified and the original seven-tap STC estimators for the
TU and the HT channels, respectively. From Figs. 4 and 5, the
simplified estimators almost have the same performance as the
corresponding original estimators for channels with different
delay profiles and Doppler frequencies, even though the simplified estimators have much a lower complexity. In particular, for
both the TU and HT channels with a lower Doppler frequency
(
Hz), the required SNR for a 10% WER is about
8.5 dB for the simplified and original enhanced and nonenhanced estimators. As the Doppler frequency increases, the performance gap between the enhanced and nonenhanced estimaHz, the enhanced estimators
tors increases. When
have about a 1.5-dB SNR improvement for the TU channel and
a 1.2-dB SNR improvement for the HT channel over the nonenhanced estimators. However, there is still a 3.5- to 4-dB SNR
(a)
(b)
Fig. 5. Performance comparison of the simplified and the original estimator
for the HT channels with (a) f = 40 Hz and (b) f = 200 Hz, respectively.
gap for the systems using the ideal and the estimated channel
parameters, which implies there is still room for improving the
performance of the OFDM systems by improving the accuracy
of the channel estimation.
VI. CONCLUSION
Channel parameter estimation is an important task in OFDM
systems. In this paper, we have presented criteria for optimum
training-sequence design for OFDM systems with multiple
transmit antennas and have also simplified the channel parameter estimators developed previously. Using the design criteria,
we can construct training sequences that not only optimize, but
also simplify the channel estimation during the training period.
The simplified estimator has similar performance to that in [5],
but with much lower complexity. The techniques proposed here
can be used in OFDM with multiple transmit-and-receive antennas for diversity or multiple input/multiple output (MIMO)
systems for high-rate wireless data access.
74
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002
APPENDIX
MSE BOUND OF THE SIMPLIFIED ESTIMATION
The MSE of the temporal estimation is
MSE
(A1)
(A4)
In this appendix, we will evaluate the three terms in the above
inequality.
To evaluate the first term, a useful statistical property of
has to be introduced. From the statistical assumptions
on
where
. The last inequality in the above derivation
is demonstrated in [15].
Similar to the derivation of the first term, we can show that
the second term is
MSE
(A5)
where MSE is the MSE of the channel estimators that exploits
the time- and frequency-domain correlations. It can be directly
checked that
(A6)
Hence
MSE
MSE
(A7)
(A2)
ACKNOWLEDGMENT
’s for different ’s or ’s
The above identity implies that
’s are determined by the transare uncorrelated. Since
mitted signals, they are independent of
’s and
The author would like to thank N. R. Sollenberger and L. J.
Cimini for their insightful comments.
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if
(A3)
otherwise.
Therefore
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75
Ye (Geoffrey) Li (S’93–M’95–SM’97) was born
in Jiangsu, China. He received the B.S.E. and
M.S.E. degrees from the Department of Wireless
Engineering, Nanjing Institute of Technology,
Nanjing, China, in 1983 and 1986, respectively, and
the Ph.D. degree from the Department of Electrical
Engineering, Auburn University, Auburn, AL, in
1994 .
From 1986 to 1991, he was a Teaching Assistant
and then a Lecturer with Southeast University, Nanjing, China. From 1991 to 1994, he was a Research
and Teaching Assistant with Auburn University. From 1994 to 1996, he was a
Post-Doctoral Research Associate with the University of Maryland at College
Park. From 1996 to 2000, he was with AT&T Laboratories-Research, Red Bank,
NJ. Since August 2000, he has been an Associate Professor with the School of
Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta.
His current research interests include statistical signal processing and wireless
mobile systems with an emphasis on signal processing in communications.
Dr. Li was a Guest Editor for special issues of SIGNAL PROCESSING FOR
WIRELESS COMMUNICATIONS for the IEEE JOURNAL ON SELECTED AREAS
IN COMMUNICATIONS and is currently serving as an Editor for WIRELESS
COMMUNICATION THEORY for the IEEE TRANSACTIONS ON COMMUNICATIONS.