MULTIPLE transmit-and-receive antennas can be used

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 MULTIPLE 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 pa- Manuscript 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: liye@ece.gatech.edu). 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

68 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002 Fig. 1. Transmit diversity for OFDM. these signals forms an OFDM block. The transmit antennas simultaneously transmit OFDM signals modulated by for. 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 denotes the channel frequency response 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 Section II-B. denotes the additive complex Gaussian 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] 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. Due to the motion of the vehicle, s are wide-sense stationary (WSS) narrow-band complex Gaussian processes, which are independent for different paths. From (2), the frequency response at time is (1) (2) where 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 where 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 extension. In (4), s, for, are WSS narrow-band complex Gaussian processes. is the number of nonzero taps of the channel impulse response sampled at a rate of. The average power of and ( ) depend on the delay profiles of the wireless channels. If an OFDM system is reasonably designed, then is 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 as 1 Hence, to obtain, we only need to estimate. From the previous section, the signal from each receive antenna can be expressed as (4) (5) (6) (3) 1 The index j for different receive antennas is omitted from H [n; k], r [n; k] and w [n; k].

LI: SIMPLIFIED CHANNEL ESTIMATION FOR OFDM SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS 69 for and all. If the transmitted signals s, for are known, 2, the temporal estimation of, can be found by [5] or where is the temporal estimation of channel parameter vector, defined as and,, and are definded as and (7) (8) To find with and, it is sufficient to find for since only consists of s for, where if. Note that is the DFT of. Therefore, there is no with such that for all s since it will result in and so, for all. Let (10) for some with. Then it can be directly checked that (11) Equation (11) implies that for. Therefore, and. In general, for systems with the number of transmit antennas,, less than or equal to, let (12) respectively. From (8), we can see that a matrix inversion is required to get the temporal estimation of. In order to reduce the computational complexity, the significant-tap-catching (STC) estimator has been proposed in [5]. However, it is still very complicated. (9) for, where and denotes the largest integer no larger than. Then for any 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. For constant-modulus modulation, and from (9) Note that and ; therefore (13) (14) for any, where denotes the unit impulse function. Consequently,, where is a identify matrix. If the training sequences, s, are chosen such that 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. Let s for be any training sequence at training time that is good for timing and frequency synchronization and possibly other properties in OFDM systems. 2 During 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. (15) 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 Fig. 2(a). Then, in (12) and s for 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 [n; l]j s. (b) p [n; l] s. for and. Furthermore (17) for and. Therefore, by carefully selecting the relative phases between the training sequences for different transmit antennas, the effect of the channels corresponding to different transmit antennas on is shifted to different regions, so the parameters for different channels can be easily estimated without using any matrix inversion. From [5], only when for, the estimator 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 71 Fig. 3. 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. 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 data transmission mode, the transmitted signals s at 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 constant modulus modulation, and, therefore (18) From the above equations, if is known, then can be estimated without any matrix inversion. However, neither or is known. Denote s for the robust estimation of channel parameter vectors at time, i.e., 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.

72 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 condition in Section III, i.e.,, then from [5] both the original and the simplified channel estimators are the same and both attain the same best performance bound (a) MSE (20) When the system is in the data transmission mode, s 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] Fig. 4. (b) Performance comparison of the simplified and the original estimator for the TU channels with (a) f =40Hz 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 and variance. Therefore, (19) becomes It is shown in the Appendix that the MSE of the temporal estimation is MSE MSE (23) Since both and are zero mean (21) (22) Therefore, the simplified temporal estimation is an unbiased estimator. 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 (MSE ) and, therefore, compared with the original estimation; the performance degradation of the simplified estimation is negligible when the square of normalized Doppler frequency is much less than the noise power ( ). Furthermore, 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 length s and a subchannel symbol rate K 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 estimators increases. When Hz, the enhanced estimators 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 Fig. 5. (a) (b) Performance comparison of the simplified and the original estimator for the HT channels with (a) f =40Hz 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) 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 (A4) 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) The above identity implies that s for different s or s are uncorrelated. Since s are determined by the transmitted signals, they are independent of s and Therefore if otherwise. (A3) ACKNOWLEDGMENT The author would like to thank N. R. Sollenberger and L. J. Cimini for their insightful comments. REFERENCES [1] S. M. Alamouti, A simple transmit diversity scheme for wireless communications, IEEE J. Select. Areas Commun., vol. 16, pp. 1451 1458, Oct. 1998. [2] A. F. Naguib, N. Seshadri, V. Tarokh, and S. Alamouti, Combined interference cancellation and ML decoding of block space-time coding, in Proc. 1998 IEEE Global Telecommunication Conf., Sydney, Australia, Nov. 1998, pp. 7 15. [3] Y. (G.) Li, J. Chuang, and N. R. Sollenberger, Transmit diversity for OFDM systems and its impact on high-rate data wireless networks, IEEE J. Select. Areas Commun., vol. 17, pp. 1233 1243, July 1999. [4] D. Agarwal, V. Tarokh, A. Naguib, and N. Seshadri, Space-time coded OFDM for high data rate wireless communication over wideband channels, in Proc. 48th IEEE Vehicular Technology Conf., Ottawa, Canada, May 1998, pp. 2232 2236. [5] Y. (G.) Li, N. Seshadri, and S. Ariyavisitakul, Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels, IEEE J. Select. Areas Commun., vol. 17, pp. 461 471, Mar. 1999.

LI: SIMPLIFIED CHANNEL ESTIMATION FOR OFDM SYSTEMS WITH MULTIPLE TRANSMIT ANTENNAS 75 [6] P. Hoeher, S. Kaiser, and P. Robertson, Two-dimensional pilot-symbolaided channel estimation by Wiener filtering, in Proc. 1997 IEEE Int. Conf. Acoustics, Speech, and Signal Processing, Munich, Germany, Apr. 1997, pp. 1845 1848. [7] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Börjesson, OFDM channel estimation by singular value decomposition, IEEE Trans. Commun., vol. 46, pp. 931 939, July 1998. [8] V. Mignone and A. Morello, Cd3-OFDM: A novel demodulation scheme for fixed and mobile receivers, IEEE Trans. Commun., vol. 44, pp. 1144 1151, Sept. 1996. [9] Y. (G.) Li, L. J. Cimini, and N. R. Sollenberger, Robust channel estimation for ofdm systems with rapid dispersive fading channels, IEEE Trans. Commun., vol. 46, pp. 902 915, July 1998. [10] Y. (G.) Li and N. R. Sollenberger, Adaptive antenna arrays for OFDM systems with co-channel interference, IEEE Trans. Commun., vol. 47, pp. 217 229, Feb. 1999. [11] Y. (G.) Li, Pilot-symbol-aided channel estimation for OFDM in wireless systems, IEEE Trans. Veh. Technol., vol. 49, pp. 1207 1215, July 2000. [12] R. Steele, Mobile Radio Communications. Piscataway, NJ: IEEE Press, 1992. [13] L. J. Cimini, B. Daneshrad, and N. R. Sollenberger, Clustered OFDM with transmit diversity and coding, in Proc. 1996 IEEE Global Telecommunication Conf., London, U.K., Nov. 1996, pp. 703 707. [14] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless communication: Performance analysis and code construction, IEEE Trans. Inform. Theory, vol. 44, pp. 744 765, Mar. 1998. [15] A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. 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.