Differential Space Time Block Codes Using Nonconstant Modulus Constellations

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER Differential Space Time Block Codes Using Nonconstant Modulus Constellations Chan-Soo Hwang, Member, IEEE, Seung Hoon Nam, Jaehak Chung, Senior Member, IEEE, and Vahid Tarokh, Member, IEEE Abstract We propose differential space time block codes (STBC) using nonconstant modulus constellations, e.g., quadrature amplitude modulation (QAM), which cannot be utilized in the conventional differential STBC. Since QAM constellations have a larger minimum distance compared with the phase shift keying (PSK), the proposed method has the advantage of signal-to-noise ratio (SNR) gain compared with conventional differential STBC. The QAM signals are encoded in a manner similar to that of the conventional differential STBC. To decode nonconstant modulus signals, the received signals are normalized by the channel power estimated forgoing training symbols and then decoded with a conventional QAM decoder. Assuming the knowledge of the channel power at the receiver, the symbol error rate (SER) bound of the proposed method under independent Rayleigh fading assumption is derived, which shows better SER performance than the conventional differential STBC. When the transmission rate is more than 3 bits/channel use in time-varying channels, the simulation results demonstrate that the proposed method with the channel power estimation outperforms the conventional differential STBC. Specifically, the posed method using the channel power estimation obtains a 7.3 db SNR gain at a transmission rate of 6 bits/channel use in slow fading channels. Although the performance gap between the proposed method and the conventional one decreases as the Doppler frequency increases, the proposed method still exhibits lower SER than the conventional one, provided the estimation interval is chosen carefully. Index Terms Differential encoding/decoding, differential space time block codes, space time codes. I. INTRODUCTION IT is well-known that the capacity of wireless channels increases linearly with the minimum number of transmit and receive antennas [1], [2]. To exploit this capacity, the Bell Lab. layered space time (BLAST) architecture was proposed in [3]. To retain increasing capacity with the limited cost of RF chains, an antenna subset selection was investigated in [4]. On the other hand, the space time code was used as an alternative method to achieve the capaicty limit by utilizing coding and diversity gains [5], [6]. A popular space time architecture is the coherent twoantenna space time block code (STBC), known as the Alamouti Manuscript received February 19, 2002; revised April 14, The associate editor coordinating the review of this paper and approving it for publication was Prof. Brian Hughes. C.-S. Hwang, S. H. Nam, and J. Chung are with the i-networking Laboratory, Samsung Advanced Institute of Technology, Nongseori, Kiheungeup, Yonginsi, Kyungkido, Korea ( cshwang@ieee.org; seunghoon.nam@samsung.com; jchung@samsung.com). V. Tarokh is with the Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA USA ( vahid@deas.harvard.edu). Digital Object Identifier /TSP scheme, which provides full diversity gain with a very simple transmitter and receiver structure [7]. All the aforementioned methods require channel state information (CSI) at the receiver. In general, the CSI can be estimated by utilizing training symbols, provided the channel changes slowly compared with the symbol rate. A scheme without CSI at the receiver, however, would be desirable in some situations. For example, the removal of channel estimation may reduce cost and complexity at the handset. When the fading channel changes quickly, the channel estimation overhead may deem coherent detection of space time block codes unattractive. The channel conditions may be significantly different from burst to burst, and a training sequence is required at every burst. In light of the above, noncoherent and differential space time coding methods have been developed in order to eliminate the use of training symbols [8] [11]. An attempt to achieve the non coherent channel capacity in [12] was unitary space time modulation (USTM) [8], in which the signals transmitted by antennas are orthogonal to one another. In [10], the differential USTM was developed for the application to a continuously varying fading channel. Similar differential space time code was derived using unitary group codes in [11]. A double differential STBC was proposed to improve the performance of the differential USTM in the presence of frequency offset [13]. Tarokh and Jafarkhani developed a differential STBC [9], [14] in which the differential modulation was combined with the orthogonal STBC [6]. A differential space code modulation improved the performance of STBC in the presence of interference [15]. The complexity of the differential STBC was comparable with that of the conventional coherent STBC. Differential detection inherently has a 3 db loss compared with coherent detection, unless multi-symbol detection is employed [16]. In addition, it has been shown in [14] that the differential STBC has not only low complexity but also smaller bit error rates compared with those of the USTM in case of block fading channel. The differential STBC can only be used in conjunction with constant modulus constellations. For example, when the number of transmit antenna is two, phase shift keying (PSK) modulation is employed [9]. As more bits are transmitted per channel use, e.g., for higher order signal constellations, the signal-to-noise ratio (SNR) disadvantage of -ary PSK over -ary quadrature amplitude modulation (QAM) increases [17]. For example, PSK requires additional SNR of 9.95 db compared with QAM to attain the same symbol error rate (SER) when 6 bits are transmitted per channel use. To reduce the penalty, the amplitude/phase shift keying (APSK) constellation, consisting of two X/03$ IEEE

2 2956 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003 PSK constellations, was employed in [18]. However, the performance gain over the differential STBC is marginal because its minimum distance does not increase significantly in high-order modulations. In this paper, we propose a differential STBC using nonconstant modulus constellations, which increases the minimum distance compared with that of the conventional differential STBC using PSK. The transmitted signal is modulated using QAM, encoded in a manner similar to that of the conventional differential STBC, and then normalized by the power of the previously sent symbols. The received signal is normalized by the estimated channel power, and the transmitted signal is decoded using a conventional QAM decoder. Assuming the knowledge of the channel power at the receiver, the SER bound of the proposed method under independent Rayleigh fading assumption is derived, which shows that the proposed method improves SER performance significantly compared with that of the conventional differential STBC. In addition, we develop a channel power estimation method that does not require training symbols and compare the mean square error () of the proposed power estimation with a Cramér Rao bound (CRB). The simulation results also show that the proposed method using the channel power estimation outperforms the conventional differential STBC in time-varying channels when the transmission rate is more than 3 bits/channel use. Specifically, the proposed method using the channel power estimation improves the SER performance by 7.3 db at a transmission rate of 6 bits/channel use in slow fading channels. As the Doppler frequency increases, the performance gap between the proposed method and the conventional one decreases. The proposed method still demonstrates lower SER than the conventional one by choosing the estimation interval carefully. The outline of this paper is given next. In Section II, we discuss the channel model and introduce the differential STBC. In Section III, we propose a differential STBC using nonconstant modulus modulation, compute its SER performance, and derive a channel power estimator forgoing training symbols. Section IV provides SER performance comparisons of the proposed method with the differential STBC for the slow and fast fading channels. Finally, conclusions are made in Section V. II. PRELIMINARIES In this section, we describe the channel model and summarize the results for a differential STBC. For simplicity, we assume that there are two transmit antennas and one receive antenna. on the receive antenna. Note that the subscripts denote a time index throughout the paper. The channel is modeled by where is the complex fading coefficient between the th transmit antenna and the receiver at time. We model as a zero-mean complex Gaussian random variable with variance 0.5 per real dimension. To model the time-varying channel, the channel coefficient is correlated in a time domain according to Jakes model [19]. In a time-varying channel, the speed of the channel variation is measured by, where is the Doppler frequency, and is the symbol duration. The noise variable is modeled as a zero-mean complex Gaussian random variable with variance per dimension. The transmitted symbols are normalized to have unit power, i.e.,, where denotes expectation. With the normalization, the SNR at the receiver is. B. Differential Space Time Block Code In order to put our approach in focus, we first review a differential STBC. Our presentation differs from [9] but gives the same performance and complexity. The differential STBC is a natural generalization of differential coding idea to multiple antenna systems. We consider a two transmit antenna system. At time 1, arbitrary symbols,, and at time 2, symbols, are sent, respectively, from transmit antennas 1 and 2. These symbols are unknown to the receiver and carry no information. Inductively, for, assume that the symbols and are, respectively, transmitted from antennas 1 and 2, at time, and and are transmitted from antennas 1 and 2, respectively, at time. At time, a block of bits of binary data arrives at the encoder. Each bit of binary data chooses a -ary PSK symbol, and the information-bearing symbols are denoted as, where denotes the transmit antenna index. As a result, the next two symbols for the transmission at time and are as follows: The symbols and are transmitted from antennas 1 and 2, respectively, at time, and and are transmitted, respectively, from antennas 1 and 2 at time. Assuming that the channel coefficients are constant during times, we drop the time index for channel coefficients. The received signals are then written as (1) (2) A. Channel Model Assume a communication channel consisting of two transmit antennas and one receive antenna that operates in a Rayleigh flat-fading environment. The signal arriving at the receive antenna is a superposition of the two transmitted signals and white Gaussian noise. Moreover, the two channel coefficients are independent of each other assuming a sufficient separation of transmit antennas. At time, we transmit the complex symbols and on antennas 1 and 2, respectively, and obtain, the differential de- where. At the receiver, to calculate coder computes (3) (4)

3 HWANG et al.: DIFFERENTIAL SPACE TIME BLOCK CODES 2957 where the noise TABLE I SNR DISADVANTAGE OF M-ARY PSK OVER M-ARY QAM is From (4), the receiver estimates by employing the regular -ary PSK demodulator that minimizes the difference in phase between and the points in the -ary PSK constellation. Similarly, can be obtained using. Finally, the binary data bits are recovered by using the inverse mappings and. As seen above, the scheme provides diversity gain using a very simple encoding and decoding algorithm. It is well known that -ary PSK suffers from an SNR disadvantage compared with -ary QAM [17]. In high SNR regimes, -ary PSK has a loss of the nonconstant modulus constellations, e.g., QAM, in (4), the receiver only needs to know the value of and not the values of and individually. If the receiver can estimate without training symbols, the differential STBC can even be employed in conjunction with QAM constellations. Thus, the differential STBC employing QAM constellations and the channel power estimation forgoing training symbols are developed in this subsection. The basic encoding/decoding method of the proposed STBC is similar to that of the differential STBC with some modifications. Like the differential STBC, the transmission is initialized by sending arbitrary symbols, at time 1 and, at time 2, respectively. At time, the transmission symbols and are encoded as in (6), shown at the bottom of the page, where is the QAM symbol encoding a block of bits of binary data. The difference between (2) and (6) is the presence of normalization in (6). Since is not constant, we observe that normalization by a factor of in (6) is required in the proposed scheme. Although the average power of the transmit signals remain the same without the normalization, the peak power of the transmit signals in (6) may be very large without this normalization. Similar to (4) and using the orthogonality of, the decoder computes the following equation: compared with -ary QAM. The for are tabulated in Table I. Comparing coherent -ary QAM with differential -ary PSK, the performance gap of two methods is more than 3 db. For example, 64PSK differential STBC requires an additional power of db to achieve the same SER compared with the coherent STBC using 64QAM since the penalty employing differential detection is 3 db, and the loss of using 64PSK as compared with 64QAM is 9.95 db. Thus, the use of high-order modulation for the differential STBC is not desirable. III. DIFFERENTIAL SPACE TIME BLOCK CODE USING NON CONSTANT MODULUS CONSTELLATION A. Proposed Method As shown in Section II-B, the differential STBC encodes information in and. Since the differential STBC assumes that the receiver has no knowledge of the channel coefficients and, the constellation set must be carefully designed such that all the elements of are on a unit circle. The main motivating observation made in this paper is that in order to decode (5) where the noise is defined as Assuming that the channel power is perfectly estimated, the QAM signal can be recovered from dividing (7) by the estimated values of and. Finally, the binary data bits are recovered by using the inverse mapping. The symbol can be estimated in an analogous manner. Unlike the conventional differential STBC, the receiver must estimate both and to recover the QAM modulated information signal from the received signal in (7). However, the channel power can be estimated without training symbols, which is described in detail in Section III-C. Therefore, the proposed method can still achieve the advantage of forgoing training symbols as the conventional differential STBC does. The signal power of previously transmitted symbols may be esti- (7) (6)

4 2958 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003 mated from the previous output of the decoder or by taking the auto-correlation of the received signal as, where denotes the corresponding noise term, which is similar in (7). B. SER Upper Bound In this subsection, the SER of the proposed method with 16QAM and 64QAM is analyzed. We assume that the channel coefficients are the same for four symbol times and that the channel power is perfectly estimated at the receiver. For simplicity, we drop the time index. The mean and variance of the noise component in (7) are given as TABLE II PROBABILITY MASS FUNCTION OF Since the average symbol energy of 16QAM is normalized to be 0.5, we replace in (12) with SNR in (10). The instant probability of symbol error for the proposed scheme conditioned on and is then expressed as Since the average power of each transmit symbol is 0.5, the SNR of the received signal in (7) is expressed as (9), shown at the bottom of the page. Assuming, it can be approximated as (10), shown at the bottom of the page, where (8) (11) From (6), it can be seen that the distribution of is the same as that of. If is chosen from the 16QAM constellations, then the probability mass function can be found as in Table II, assuming the uniform distribution over the 16QAM constellation. In addition, the distribution of noise in (7) is approximated as Gaussian assuming high SNR. The probability of the symbol error for -ary QAM in AWGN channel is given by [17] (13) Assuming that and are zero-mean complex Gaussian random variables with variance 0.5 per dimension, is Chi-square random variable with degree of freedom four as [17] (14) Then, the average probability of the symbol error is computed by first integrating the error probability in (13) over the fading distribution and summing it over the distribution of (15) A closed-form expression of the integration in (15) exists as (12) (16) SNR (9) SNR (10)

5 HWANG et al.: DIFFERENTIAL SPACE TIME BLOCK CODES 2959 Fig. 1. Comparison of the SER upper bounds of the proposed method and the differential STBC when 16QAM and 16PSK, respectively, are employed. where. Substituting (16) to (15), the upper bound on the average probability of the symbol error can be obtained by It can be further approximated for small value of (17) as (18) Since we normalize the average power of each transmit symbol by 0.5, the average SNR per symbol becomes, and then we can rewrite (18) as (19) At high SNR, the SER of the proposed method in a Rayleigh fading channel is asymptotically proportional to, where denotes the diversity order. From (19), we observe that the proposed method attains two level diversity. The probability of symbol error of the differential STBC scheme with PSK modulation in a Rayleigh fading channel can be calculated in a similar manner by integrating -function over Chi-square distribution. Fig. 1 shows the upper bound of the SER performance of the proposed method with 16QAM modulation. The solid line represents the upper bound given in (17), and the dash-dot line represents the upper bound for high SNR given in (18). The SER upper bounds of the differential STBC scheme with 16PSK modulation are also plotted for comparison. For example, from the bounds, the proposed method and the differential STBC require 30.5 and 33 db, respectively, to achieve SER of, and the SER performance advantage of the proposed method over the differential STBC is 2.5 db. In Section IV, using computer simulations, we show that the SER bound of the proposed method is very accurate. We also plot the SER upper bound of the proposed method using 64QAM in Fig. 2. From the bound in Fig. 2, the proposed method outperforms the differential STBC Fig. 2. Comparison of the SER upper bounds of the proposed method and the differential STBC when 64QAM and 64PSK, respectively, are employed. by 7.5 db at an SER of. We observe that the performance gain of employing the QAM becomes significant as higher order modulations are employed. C. Channel Power Estimator The conventional differential STBC removes transmission overhead due to the training symbols because a channel estimation is not required at the receiver. Similarly, in order to maintain this advantage for the proposed method, a channel power estimation method forgoing training symbols is developed. We also compare the of the proposed estimation with that of CRB using training symbols [20]. We assumed that the channel coefficients are the same during and dropped their time index. Using vector notations, the received signal in (1) for is rewritten as follows: (20) where and denote the transmitted symbol vectors from antenna 1 and antenna 2, respectively, denotes the noise vector, and denotes the number of symbols used for estimation. Let the noise be additive white Gaussian, and let and be independent Rayleigh fading channels. To derive the channel power estimation, we multiply the received signal by its Hermitian transpose as and take the expectation of (21). Since,, and,wehave The estimator for channel power is then (21) (22) (23)

6 2960 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003 Unlike the maximum likelihood (ML) channel power estimator in [21], the transmitted sequences from different antennas are not restricted to be orthogonal to each other in (23). Thus, we can estimate the sum of the channel power directly from the information bearing symbols. In addition, the estimator can be used regardless of the modulation scheme used at the transmitter. The estimator, however, is biased since the expectation of the estimation error is. The performance of the estimator is analyzed by deriving in the following. The of the estimator is (24) The detailed derivation of (24) is provided in the Appendix. Due to the second term in (24), the does not converge to zero, even if the noise variance approaches zero. The of the ML channel power estimator can be obtained from the CRB. During the derivation of the CRB, we assume that the transmitted symbol is known at the receiver. To calculate the CRB, the log likelihood function is defined as (25) where, and. The Fisher information matrix in [20] for estimating is obtained as Using the invariance property [20], the CRB of the Fisher information matrix of as (26) is derived from CRB (27) where denotes the partial derivatives of defined as. Thus, the CRB of is CRB (28) Since the distribution of both and is a complex Gaussian, by taking the expectation of (27), we can compute the of the ML estimation of channel power to be CRB (29) In Fig. 3, we illustrate the of the proposed estimator without using the training symbols and the CRB when the number of symbols used for the estimation varies from 5 to 100. Since the of the proposed estimator does not approach the CRB, the proposed estimator is not efficient. For example, at low SNR, e.g.,, or high SNR, e.g., Fig. 3. Comparison of the of the proposed estimator and CRB., the CRB is about ten times smaller than the of the proposed estimator. The of the proposed estimator, however, is comparable with the CRB at mid SNR range, e.g.,. This is because the coefficient of, which is a dominant error term at mid SNR range, is smaller than other terms. In addition, both the CRB and the of the proposed estimator are inversely proportional to because the noise variance is reduced by averaging over an symbol duration assuming slow fading channels. IV. SIMULATION RESULTS In this section, we present simulation results for the proposed method in slow and fast fading channels. For slow fading channels, we test the SER of the proposed differential STBC using the QAM constellation compared with the SER upper bound in Section III-B, assuming, and then evaluate the influence of the channel estimator on the SER by changing the number of symbols used for the estimation, i.e.,. The SER of the proposed scheme is compared with that of the differential STBC for different modulation orders. For the fast fading channels, we compare the SER of the proposed method with that of the differential STBC as varies from 10 to 10. Considering a mobile transmitting at a symbol rate of 100 khz and a carrier frequency of 1.9 GHz, the speed of the vehicle is 5.7 and 570 km/h at and, respectively. In addition, we observe the tradeoff between the tracking capability and the of the channel power estimator by changing. For the computer simulation, we utilize the modulation scheme illustrated in Section III-A, which has a rectangular constellation for QAM in [17], and modify the channel power estimator in (23) as follows: (30) In addition, we vary the number of symbols used for the estimation from five to 100. The proposed method assuming knowledge of at the receiver is also simulated as a performance

7 HWANG et al.: DIFFERENTIAL SPACE TIME BLOCK CODES 2961 TABLE III SNR GAIN OF THE PROPOSED METHOD OVER THE DIFFERENTIAL STBC IN SLOW FADING CHANNELS Fig. 4. SER of the proposed method with 8/16/64QAM in a slow fading channel (f T =10 ). The SER of differential STBC with 8/16/64PSK is plotted as a reference. bound. For the differential STBC using various, we utilize the transceiver in Section III-A with PSK constellation. A. Slow Fading Channels In Fig. 4, we compare the SER of the proposed method using channel power estimator with the differential STBC for 8, 16, and 64, assuming and. As a performance bound, we also plot the SER of the proposed method, assuming a knowledge of the channel power at the receiver. In Fig. 4, we observe that the proposed method using has a 2.6 db power gain at an SER of 10 compared with the differential STBC when 4 bits are transmitted per channel use. This result is consistent with the 2.5 db power gain obtained by the analytic upper bound in Section III-B. The SER of the proposed method with 8QAM, however, is larger than that of the differential STBC with 8PSK. In Table I, the SNR performance gap between 8PSK and 8QAM is 1.65 db. Since the proposed method suffers from SNR degradation due to the normalization of the transmitted symbol by the power of the previous symbol, 1.65 db SNR advantage over 8PSK is traded for compensating for the loss. In addition, the channel power estimation induces a further 0.2 db SNR loss; therefore, the proposed method with the channel power estimator requires 0.4 db more energy to achieve the same SER. As higher order modulations are employed, the SNR gain of the proposed method increases. For example, the proposed method attains 5.5 db and 7.9 db power gains for 5 bits/channel use and 6 bits/channel use, respectively, in Fig. 4, assuming that the channel power is known at the receiver. The SNR advantages of the proposed method over the differential STBC are summarized in Table III, assuming slow fading channels. We also consider the effect of the number of symbols used for the estimation on the SER. In Fig. 5, the SERs of the proposed method are displayed as is varied from 5 to 100. The SER increases as the number of symbols utilized for the channel power estimation decreases. For example, the performance degradation due to the channel power estimation becomes larger than Fig. 5. Simulated SER of the proposed method while the estimation interval L varies from 5 to 100 (4 bits/channel use). 5 db when. However, the SER of the proposed method using the proposed estimator is almost the same as that using the perfect channel power estimation when in Fig. 5. The SNR disadvantage due to the use of the channel power estimation (30) increases with high-order modulations. Note that the SNR loss due to the use of the channel power estimator becomes 0.6 db when 64QAM is employed in Fig. 4. B. Fast Fading Channels In Fig. 6, the SER of the proposed method employing 16QAM is compared with that of the differential STBC employing 16PSK when is and 10. For the proposed method, the SER of the proposed method using the channel power estimator and assuming a knowledge of channel power at the receiver are considered. The proposed method outperforms the differential STBC by 2.8 db in SNR at the SER of 2 by setting in Fig. 6(b). However, the SER of the proposed method using approaches that of the conventional differential STBC because the channel coefficients change during the estimation. In higher Doppler frequencies such as, the SER of the proposed method starts to exceed that of the differential STBC, even with in Fig. 6(a). In the low, the proposed method, however, achieves the SNR gain of 3.1 db over the differential STBC at the SER of 10. To illustrate the SER performances under different Doppler frequencies, Fig. 7 illustrates the SER of two methods for fixed when 4 and 6 bits are transmitted per channel use. In

8 2962 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003 performance loss due to the estimation error can reach about 4 db when the increases to 10. However, the performance degradation due to the estimation error can be reduced by choosing carefully. For both the 16QAM and 64QAM cases, the use of estimator with keeps the SER lower than that of the conventional differential STBC when 10 in Fig. 7. As the Doppler frequency increases, should be smaller because reducing enhances the tracking capability at the cost of increasing the of the estimation. Fig. 6. Comparison of the SER of the proposed method using 16QAM with a differential STBC using 16PSK when (a) f T = 10 and (b) 10. Fig. 7. Comparison of the SER of the proposed method and differential STBC in time-varying channels. (a) 24 db Eb/No and 4 bits/channel use. (b) 22.2 db Eb/N0 and 6 bits/channel use. Fig. 7(a), i.e., the 4 bits/channel use case, the proposed method with demonstrates a lower SER than the differential STBC when. For larger, the SER of the proposed method is comparable with that of the differential STBC by setting. As the modulation order increases, the performance gap between the proposed method and the differential STBC increases. In Fig. 7(b), by selecting to be 50 and 25 for and, respectively, the SER of the proposed method with 64QAM is smaller than that of the differential STBC with 64PSK because the use of 64PSK instead of 64QAM reduces the effective signal energy by about 10 db. Unlike the slow fading channel in Fig. 5, the effect of the estimation error increases the SER of the proposed method significantly. In Fig. 6, the proposed method using estimator suffers about a 1 db SNR loss even in a slow fading channel. The V. CONCLUSIONS A differential STBC using nonconstant modulus constellations has been developed, e.g., QAM, which cannot be utilized in the conventional differential STBC. Since QAM constellations have a larger minimum distance compared with PSK, the proposed method has an advantage of SNR gain compared with the conventional differential STBC. The transmitted signal is modulated using QAM, encoded in a manner similar to that of the conventional differential STBC and then normalized by the power of the previously sent symbols. The received signal is normalized by the estimated channel power, and the transmitted signal is decoded using a conventional QAM decoder. Assuming the knowledge of the channel power at the receiver, the SER bound of the proposed method under independent Rayleigh fading assumption is derived, which shows that the proposed method improves SER performance significantly compared with that of the conventional differential STBC. In addition, a channel power estimation method is developed that does not require training symbols, and the performance of the proposed power estimation is compared with the CRB. The simulation results also demonstrate that the proposed method using channel power estimation outperforms the conventional differential STBC in time-varying channels when the transmission rate is more than 3 bits/channel use. Specifically, the proposed method using channel power estimation enhances SER performance by 7.3 db at a transmission rate of 6 bits/channel use in slow fading channels. As the Doppler frequency increases, the performance gap between the proposed method and the conventional one decreases. The proposed method still exhibits lower SER than the conventional one by choosing the estimation interval carefully. APPENDIX In this Appendix, we will show that the of the proposed estimator in Section III-C is. From the definition of the (31)

9 HWANG et al.: DIFFERENTIAL SPACE TIME BLOCK CODES 2963 Using (22), the second term in (31) becomes In addition, the use of (21) expands the third term in (31) as (32) Since, the and are zero. In addition, and are zero because and are zero. Substituting by, the first term in (34) becomes (35) By substituting by, the in (34) is (36) From (32) and (33), we can reduce (31) as (33) Similarly, is the same as. can be rearranged by substituting by as Equation (34) can be expanded as Using (35) (37), the for. is (37) ACKNOWLEDGMENT The authors wish to thank the anonymous reviewers for their helpful comments and their thoughtful reviews. They greatly improved the qualities of the paper. (34) REFERENCES [1] I. E. Telatar, Capacity of multi-antenna gaussian channels, Eur. Trans. Telecommun., vol. 10, pp , Nov [2] G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Pers. Commun., vol. 6, pp , Mar [3] G. J. Foschini, Layered space-time architechture for wireless communication in fading environment when using multi-element antennas, Bell Labs. Tech. J., vol. 1, pp , Nov [4] D. A. Gore and A. J. Paulraj, MIMO anteanna subset selection with space time coding, IEEE Trans. Signal Processing, vol. 50, pp , Oct [5] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless communications: Performance criterion and code construction, IEEE Trans. Inform. Theory, vol. 44, pp , Mar

10 2964 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 11, NOVEMBER 2003 [6] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block coding from orthogonal designs, IEEE Trans. Inform. Theory, vol. 45, pp , July [7] S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Select. Areas Commun., vol. 16, pp , Oct [8] B. Hochwald and T. Marzetta, Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading, IEEE Trans. Inform. Theory, vol. 46, pp , Mar [9] V. Tarokh and H. Jafarkhani, A differential detection scheme for transmit diversity, IEEE J. Select. Areas Commun., vol. 18, pp , July [10] B. M. Hochwald and W. Sweldens, Differential unitary space time modulation, IEEE Trans. Commun., vol. 48, pp , Dec [11] B. L. Hughes, Differential space time modulation, IEEE Trans. Inform. Theory, vol. 46, pp , Nov [12] T. Marzetta and B. Hochwald, Capacity of a mobile multiple-antenna communication link in rayleigh flat fading, IEEE Trans. Inform. Theory, vol. 45, pp , Jan [13] Z. Liu, G. B. Giannakis, and L. Hughes, Double differential space-time block coding for time-selective fading channels, IEEE Trans. Commun., vol. 49, pp , Sept [14] H. Jafarkhani and V. Tarokh, Multiple transmit antenna differential detection from generalized orthogonal designs, IEEE Trans. Inform. Theory, vol. 47, pp , Sept [15] J. Liu, J. Li, H. Li, and E. G. Larsson, Differential space code modulation for interference suppression, IEEE Trans. Signal Processing, pp , Aug [16] S. Wilson, J. Freebersyser, and C. Marshall, Multi-symbol detection of M-DPSK, in Proc. IEEE GLOBECOM, Nov. 1989, pp [17] J. G. Proakis, Digital Communications. New York: McGraw-Hill, [18] X. Xia, Differentially en/decoded orthogonal space time block codes with APSK signals, IEEE Commun. Lett., vol. 6, pp , Apr [19] W. Jakes, Microwave Mobile Communications, 2nd ed. New York: IEEE Press, [20] L. L. Scharf, Statistical Signal Processing-Detection, Estimation, and Time Series Analysis. Reading, MA: Addison-Wesley, [21] V. Tarokh, A. F. Naguib, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless communication: Performance criteria in the presence of channel estimation errors, mobility, and multiple paths, IEEE Trans. Commun., vol. 47, pp , Feb Chan-Soo Hwang (M 99) received the B.S. and M.S. degree in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejon, Korea, in 1997 and 1999, respectively. He joined Samsung Advanced Institute of Technology (SAIT), Kyungkido, Korea in He participated in the standardization of VDSL in ITU from 1999 to Since 2001, he has done research on next-generation wireless communication systems. His primary research interests include space-time codes, MIMO, and multicarrier modulation. Seung Hoon Nam received the B.S. and M.S. degrees from Seoul National University, Seoul, Korea, in 2000 and 2002, respectively. He joined Samsung Advanced Institute of Technology (SAIT), Kyungkido, Korea in 2002 and was involved in the next-generation wireless communication systems project. His primary research interests include space-time codes and MIMO. Jaehak Chung (SM 01) received the B.S. and M.S. degree from Yonsei University, Seoul, Korea, in 1988 and 1990, respectively, both in electrical engineering, and the Ph.D. degree in 2000 from the University of Texas at Austin. He is currently a member of technical staff at Samsung Advanced Institute of Technology (SAIT), Kyungkido, Korea. His research interests include next-generation wireless communications such as space-time codes, MIMO, and multicarrier modulations. Vahid Tarokh (M 97) received the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in From 1995 to 1996, he was a postdoctorate fellow at the University of Illinois, Urbana-Champaign. He joined AT&T Labs-Research, Red Bank, NJ, in 1996, where he was (in chronological order) a Senior and Principal Technical Staff Member and the Head of the Department of Wireless Communications and Signal Processing. In September 2000, he joined Departemtn of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, in 2000 as an Associate Professor, where he worked until June Since July 2002, he has been the a Gordon McKay Professor of electrical engineering at the Division of Engineering and Applied Sciences, Harvard University, Cambridge, where he is also a holder of a Hammond Vinton Hayes Senior Research Fellowship. Dr. Tarokh received the 1995 Governor General of Canada s Academic Gold Medal, the 1999 IEEE Information Theory Society Prize Paper Award (jointly with Seshadri and Calderbank), and the 2001 Alan T. Waterman Award, and was selected as one of the 100 young inventors of the year by the Technology Review Magazine in He also holds a number of honorary degrees.