ONE ASSUMPTION widely made in performance analysis

Transcription

1 282 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007 Analysis of Differential Orthogonal Space Time Block Codes Over Semi-Identical MIMO Fading Channels Meixia Tao, Member, IEEE, and Pooi Yuen Kam, Senior Member, IEEE Abstract We study the performance of differential orthogonal space time block codes (OSTBC) over independent and semi-identically distributed block Rayleigh fading channels. In this semiidentical fading model, the channel gains from different transmit antennas to a common receive antenna are identically distributed, but the gains associated with different receive antennas are nonidentically distributed. Arbitrary fluctuation rates of the fading processes from one transmission block to another are considered. We first derive the optimal symbol-by-symbol differential detector, and show that the conventional differential detector is suboptimal. We then derive expressions of exact bit-error probabilities (BEPs) for both the optimal and suboptimal detectors. The results are applicable for any number of receive antennas, and any number of transmit antennas for which OSTBCs exist. For two transmit antennas, explicit and closed-form BEP expressions are obtained. For an arbitrary number of transmit antennas, a Chernoff bound on the BEP for the optimal detector is also derived. Our results show that the semi-identical channel statistics degrade the error performance of differential OSTBC, compared with the identical case. Also, the proposed optimal detector substantially outperforms the conventional detector when the channel fluctuates rapidly. But in near-static fading channels, the two detectors have similar performances. Index Terms Differential detector, independent and nonidentical channels, Rayleigh fading, space time block codes (STBC), transmit diversity. I. INTRODUCTION ONE ASSUMPTION widely made in performance analysis for diversity-combining techniques is that the diversity branches are statistically identical. In real propagation environments, however, this assumption may not hold. For instance, the environmental scattering on the propagation paths observed by different branches may be different. This may incur shadowing effects and, consequently, unequal path losses on different branches. In multi-input multi-output (MIMO) systems, this is especially common when the antenna spacing is relatively large (compared with the carrier wavelength) to ensure low correlation. The performance of diversity reception over nonidentically distributed branches in single-input multi-output (SIMO) systems has been studied in, for example, [1] and [2]. The goal Paper approved by A. Anastasopoulos, the Editor for Wireless Communications of the IEEE Communications Society. Manuscript received June 13, 2005; revised February 20, This paper was presented in part at the IEEE International Conference on Communications, Istanbul, Turkey, June 2006, and in part at the IEEE Vehicular Technology Conference, Dallas, TX, September The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore ( mxtao@nus.edu.sg; elekampy@nus.edu.sg). Digital Object Identifier /TCOMM of this paper is to investigate the effects of nonidentical distributions on the performance of joint diversity transmission and reception in MIMO systems. Of the existing transmit-diversity techniques, orthogonal space time block codes (OSTBC) [3] [6] are most popular because of their elegant code structure and low decoding complexity. The performance of OSTBC is known to be identical to that of maximum ratio combing in SIMO channels (apart from being scaled down by the number of transmit antennas and coding rate) when channel state information (CSI) is available at the receiver. When CSI is unavailable, OSTBC can be used in a differential encoding manner and still possess low decoding complexity [7] [9]. Existing work on performance analysis of differential OSTBC can be found in [10] [12], and they all assumed identical channel distributions. Our first objective is to model the independent and nonidentically distributed (i.n.i.d.) MIMO fading channels, with focus on the semi-identical distribution. The semi-identical distribution case refers to the situations the path gains from different transmit antennas to a common receive antenna are identically distributed, but the gains associated with different receive antennas are nonidentically distributed. Such situations would most likely occur in the uplink of a cellular system, the antennas on the base station are mounted several wavelengths apart from one another, as the antennas on a mobile handset are inside a small antenna panel. In ground-to-air communications, the semi-identical scenario may also occur, since the receiving paths of some antennas are likely to be obstructed by the aircraft body during maneuvers, but the paths of other receive antennas may be clear. This independent and semi-identically distributed (i.s.i.d.) fading model is also appropriate for the general i.n.i.d. fading channels, given that proper decomposition on the long-term channel-parameter matrix in conjunction with power allocation, asproposed in this paper, is applied. To the bestofour knowledge, the modeling of nonidentical and semi-identical distributions for MIMO channels have not been addressed by previous works. Our next objective is to analyze the performance of differential OSTBC under i.s.i.d. and block-wise time-varying Rayleigh fading channels. In the block-wise time-varying fading model considered, the channel coefficients are assumed to keep constant over one STBC transmission, but fluctuate from one block to another at an arbitrary fluctuation rate. We first show that the conventional symbol-by-symbol differential detector described in [9] is suboptimal, and that the optimal symbol-by-symbol differential detector involves weighting the output from each receive antenna according to its channel statistical information /$ IEEE

2 TAO AND KAM: ANALYSIS OF DIFFERENTIAL ORTHOGONAL SPACE TIME BLOCK CODES 283 These findings are similar to the ones shown in [1] for differential phase-shift keying (DPSK) over nonidentical SIMO channels. We then derive exact bit-error probability (BEP) expressions for both the optimal and suboptimal detectors. These results are for any number of receive and transmit antennas OSTBCs exist. In the special case with two transmit antennas, explicit and closed-form expressions are obtained. From the expressions, the behavior of the exact error probability as a function of the total transmit signal-to-noise ratio (SNR), block fading rates, and the unequal characteristics of the semi-identical channels can be obtained. Simple and more insightful upper bounds for arbitrary numbers of transmit antennas are also derived. The rest of this paper is organized as follows. In Section II, we introduce the nonidentical and semi-identical MIMO channel models. In Section III, we present the optimal and suboptimal detectors of differential OSTBC. The derivations of exact BEP and their bounds are provided in Section IV. Numerical results and discussions are presented in Section V. Finally, we conclude this paper in Section VI. Throughout this paper, matrices (vectors) are denoted using boldface upper (lower) case letters. and stand for the zero matrix and the identity matrix, the dimensions may be dropped if there is no confusion. stands for expectation over the random variables within the brackets. Superscripts,, and denote transpose, conjugate, and conjugate transpose, respectively. II. CHANNEL MODEL We consider a point-to-point wireless communication link equipped with transmit antennas and receive antennas. The channel is assumed to be frequency-nonselective Rayleigh fading and block-wise time-varying, with each block consisting of symbol intervals. It is also assumed that the channel coefficients between any transmit receive antenna pairs are independent. Let denote the channel gain matrix during the th transmission block. The th entry is the path gain from the th transmit antenna to the th receive antenna. Each sequence is modeled as samples of a complex, zero-mean, Gaussian random process having autocorrelation function. Here, the factor of 2 arises from the assumption that the real and imaginary parts of each path gain are independent and identically distributed (i.i.d.). In general, is not necessarily the same for all and. It takes into account the nonidentical distribution of the channel gains between different transmit receive antenna pairs. The fading correlation coefficient across two adjacent blocks, called the block correlation coefficient, is given by, and is a measure of the fluctuation rate of the channel fading process. In practice, the Doppler frequency shift experienced by the communication links due to relative movement of the sender and receiver would likely be the same for all transmit receive antenna pairs. As a result, will likely be the same for all s and s. Nevertheless, we assume that they are not the same, unless otherwise stated, for the sake of generality. In this paper, we consider only the semi-identical channels with and, for all. That is, the path gains from different transmit antennas to a common receive antenna are identically distributed. This channel model is general in the sense that nonidentical channels can also be treated as the semi-identical ones under certain conditions. Specifically, we define an diagonal matrix whose th diagonal entry is, for and. The matrix is essentially the covariance matrix of the vector,, formed by stacking the columns of under one another. If, for all and, and can be decoupled as, stands for the Kronecker product, and and are and diagonal matrices, respectively, then the statistical properties of are identical to those of the matrix.we denote the statistical equivalence by In (1),,, and the matrix contains i.i.d. entries, each of which is modeled as samples of a complex, zero-mean, Gaussian random process having autocorrelation function, with and. Therefore, a power-allocation matrix can be applied at the transmitter such that the effective channel gain matrix becomes semi-identical. All the results obtained in this paper can then be used directly in the general nonidentical channels. It is also assumed that the channel is corrupted with additive white Gaussian noise (AWGN). The complex baseband channel input output relationship can be modeled as,, and are the transmitted signal matrix, received signal matrix, and noise matrix, respectively, during the th block. satisfies the average power constraint. The entries of are i.i.d., each with mean zero and variance per dimension. is the average total energy transmitted on all antennas. III. OPTIMAL AND SUBOPTIMAL DIFFERENTIAL DETECTION OF OSTBC In this section, we first review differential transmission of OSTBC, and then propose the optimal symbol-by-symbol differential detector over i.s.i.d. MIMO channels. We show that the conventional differential detector described in [9] is suboptimal, and that the optimal detector should involve weighting the output from each receive antenna according to its channel statistical information. In the differential transmission of a square OSTBC codeword, the block length equals the number of transmit antennas, i.e.,. Let denote a set of information symbols to be transmitted in the th block. They are complex scalars chosen from phase-shift keying (PSK) constellations with, and encoded in an OSTBC codeword as (1) (2)

3 284 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007 and are the encoding matrices. The set of encoding matrices are linked to the theory of amicable orthogonal designs [6], [13], [14], satisfying the following conditions: natural logarithm of (6) and ignoring the terms that are independent of, we obtain the ML decision rule for as The OSTBC is completely determined by the encoding matrices and, as each individual codeword is determined by the choice of the phases. The data matrix is differentially encoded in the transmitted signal matrix as Since is a unitary matrix, is also unitary. We consider the maximum-likelihood (ML) detection of based on two consecutive received matrices and.wedefine and, is the received signal vector on the th antenna during the th block, for. Conditioned on and, the vectors s can be shown to be independent for different s, and each has a complex Gaussian distribution with mean zero and covariance matrix In (5), the block index in is omitted for notational brevity. The conditional probability density function (PDF) of can thus be written as It is seen that is independent of. Thus, the ML detection of at block does not depend on the signal transmitted in the previous block. Using the matrix formulas [15] we obtain and (3) (4) (5) (6) (7) (8) (9) represents the real part, and the weighting coeffi- is given by cient (10) Here, we assume that the second-order channel statistics, s and s, vary much more slowly than the instantaneous channel coefficients, and hence, can be estimated at the receiver. Now, using the inherent linear structure of the data matrix by design in (2), the above matrix-by-matrix detector reduces to a symbol-by-symbol detector given by (11) is the contribution to the decision variable from the th receive antenna. The contributions from different receive antennas are weighted by the s. Note that the optimal weights in (10) for the semi-identical MIMO channels considered here coincide with the result in [1] for SIMO channels. A detailed discussion on the weights will be provided in Section V. When the second-order channel statistics are unknown at the receiver, one would let the contributions from different receive antennas be weighted equally in the detector. Thus, by letting for all, (11) reduces to (12) which agrees with the decision rule in [9] and [11] for identically distributed channels. Hence, the traditional differential detector is suboptimal in the i.s.i.d. case. IV. PERFORMANCE ANALYSIS In this section, we derive exact BEP expressions for both the optimal and suboptimal detectors. The results are for any number of receive and transmit antennas for which OSTBC exist. In the case with two transmit antennas, explicit and closedform expressions are obtained. An insightful upper bound for the optimal detector for an arbitrary number of transmit antennas is also derived. The analysis of BEP for detecting symbol involves the computation of the following probability [16]: In both (8) and (9), SNR per symbol interval on the is the average received th receive antenna. Taking the (13)

4 TAO AND KAM: ANALYSIS OF DIFFERENTIAL ORTHOGONAL SPACE TIME BLOCK CODES 285 (14) We investigate below the eigenvalues of, for. Let undergo a similitude transformation producing is the decision phasor for, and is some angle. For ease of presentation, we shall use (14) as a unified expression for both optimal and suboptimal detection, in which the weights are given in (10) for the optimal detector, and are the same and equal to one for the suboptimal detector. It will be shown later that the above probability is the same for all indices, and thus the BEP conditioned on corresponds also to the overall average BEP. For binary (B)PSK and quaternary (Q)PSK (with Gray mapping) constellations, the exact BEPs are given using the half-plane decision-region approach [16], respectively, by (20) is a unitary matrix. Then the eigenvalues of are the same as those of. Here we choose Substituting (18), (5), and (21) into (20) yields (21) (15) (16) For 8PSK (with Gray mapping), the BEP is well approximated by [17] The rest of this section is devoted to the evaluation of under various scenarios. (17) A. Exact for Optimal and Suboptimal Detection The decision phasor in (14) can be written as the Hermitian quadratic form of Gaussian vectors in which Now we are ready to evaluate the eigenvalues of The eigenvalues of are the solutions of From (2) and (3), it can be shown that We then have (22) directly. (23) (24) is a Hermitian matrix given by Moreover, it can be shown that the two matrices and commute. Thus, (23) readily becomes (18) Applying the cumulative distribution function of a Hermitian quadratic form [18], we have In obtaining (25), the matrix formulas (7) and (25) (19) denotes the residue of at, is the set of nonzero distinct eigenvalues of the matrix with multiplicities, and are also used. Finally, solving the quadratic equation (25) produces the two distinct eigenvalues of as shown in (26) at the bottom of the page, for. Since the matrix has full rank, the multiplicity of each of the eigenvalues is.we choose and. The evaluation of the probability can thus be accomplished by computing the residues in (19) at, for. For a general formula of computing these residues, readers are referred to [19, D3]. (26)

5 286 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007 In what follows, we consider the special case, which is Alamouti s STBC [3]. According to the residue equation of a function at a pole of multiplicity we can have is the parameter to be optimized, and s for and are given in (26). The tightest bound on can be obtained by selecting the value of that minimizes the right-hand side (RHS) of the inequality (28). This is equivalent to selecting to maximize. The value of that maximizes the th factor is determined by solving the equation, which gives the result (27) Substituting (10) and (26) into (29), we obtain (29) (30) Equation (27) is a closed-form expression and depends explicitly on the average received SNR on each antenna and the block correlation coefficients. In concluding this subsection, we have several remarks. First, the channel weights are not specified throughout the derivation in this subsection. Thus, the expressions of in (19) for a general and (27) for are for both the optimal and suboptimal detectors. Specifically, will be replaced by (10) for the optimal detector, and be equal to one for the suboptimal detector. Second, in computing (27), we assumed that the eigenvalues are distinct for different s. Therefore, such explicit and closed-form expressions cannot be easily extended to the i.i.d. channels for all. Moreover, the results in both (19) and (27) may not be well-suited for numerical evaluation as approaches, because computing the sum of the residues tends to be numerically unstable in this case. Finally, the exact BEP expressions using BPSK and QPSK constellations can be obtained by substituting (19) or (27) into (15) and (16), respectively. An approximate BEP expression using the 8PSK constellation can also be obtained by substituting (19) or (27) into (17). B. Upper Bound on for Optimal Detection Based on the eigenvalues obtained in Section IV-A, an upper bound on for optimal detection can also be obtained, from which the asymptotic behavior of the BEP can be predicted. Applying the well-known Cherenoff bound inequality [20] to (13) and using the characteristic function for a Hermitian quadratic form of Gaussian vectors, we have This value is independent of the index and, hence, also maximizes. Therefore, substituting (30) into (28), we have (31) The RHS of (31) is a closed-form expression and depends explicitly on the average received SNR on each antenna and the block-correlation coefficients. Accordingly, the Chernoff bounds on the BEP using BPSK and QPSK modulation can be found by letting and in (31), respectively, with. We now study the asymptotic behavior of the Chernoff bound for high SNRs. We first consider a static channel with, for all. By letting, the inequality (31) can be approximated by (32) is the geometric mean of, given by. Recall that for a set of real positive values, the geometric mean is always upper bounded by the arithmetic mean. Hence, when the total received SNR over all receive antennas, defined as, isfixed, the semi-identical channel will degrade the BEP performance, compared with the identical channel. Next, we consider a time-varying channel with for all. By letting again, the RHS of the inequality (31) approaches a constant given by (28) in which This constant term is obviously upper bounded by

6 TAO AND KAM: ANALYSIS OF DIFFERENTIAL ORTHOGONAL SPACE TIME BLOCK CODES 287, and equality holds when is the same for all. This result indicates that the asymptotic irreducible error floor of differential OSTBC with optimal detection, despite its dependence on the different fluctuation rates, is independent of the unequal power distribution on different receive antennas. In the case, it can be expected that the error floors of the semi-identical and identical channels are the same. Note that the above Chernoff bound is only for the optimal detector. For the suboptimal detector, the value of that maximizes the th factor in with is dependent on the index. Therefore, the optimal can only be found by evaluating the derivative of, which can be very difficult for large and. C. An Alternative for Optimal Detection In this subsection, we present an alternative expression for for the optimal detector, which is more suitable for numerical evaluation when the channels are nearly identical. From this expression, the same Chernoff bound as obtained in Section IV-B can also be found. Given, the decision phasor in (14) can be shown to be conditionally Gaussian distributed with mean We define. It is easy to see that each is distributed according to a chi-square distribution with degrees of freedom. The characteristic function of is given by [20] Averaging (36) over the distribution of yields. Using the alternative representation of the -function [21] we can evaluate as (37) (33) and variance (34) stands for the squared Frobenius norm. The proof is outlined in the Appendix. The conditional probability that can thus be expressed using the -function [20] as (35) Substituting the optimal weights (10) into both and,we rewrite (35) as (38) Equation (38) involves only a single integral over finite limits, and hence, can be numerically computed easily. Using the alternative expression for in (38), we can now compute the exact BEP for BPSK and QPSK, and the approximate BEP for 8PSK, as in (15), (16), and (17), respectively. These results are for an arbitrary number of transmit antennas for which an OSTBC exists. They are also applicable for the i.i.d. channels by letting and for all. An upper bound on the probability can be easily obtained by letting in (38). Doing this and using the expression for in (37) leads to (36) (39) This bound differs from the bound obtained previously in (31) by a factor of only. V. NUMERICAL RESULTS AND DISCUSSIONS We first study the behavior of the optimal weighting coefficients in optimal detection as a function of various system parameters. We then proceed with the BEP versus SNR plots using

7 288 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007 Fig. 1. Normalized optimal weighting coefficients for N = 2 at = 0:1 ( : =10% :90%). Fig. 3. Analytical and simulated BEP comparison for M = P = N = 2 with QPSK at =0:1( : =10% :90%). Fig. 2. Normalized optimal weighting coefficients for N =2at =0:49( : =49% :51%). Fig. 4. Analytical and simulated BEP comparison for M = P = N = 2 with QPSK at =0:49 ( : =49% :51%). both analytical results and simulation results. Several interesting observations shall be made. In all our figures, the total average received SNR per bit over receive antennas is defined as, is the transmission rate in bits per channel use. When QPSK modulation is used, we have for the differential OSTBC with, and for the one with and [9]. The fluctuation rates of the fading processes on all receive antennas are assumed to be the same, i.e., for all. Figs. 1 and 2 show the values of the weights as a function of (with and QPSK) computed using (10) for receive antennas. The quantity denotes the fraction of the total average received signal energy from receive antenna 1 (and hence is from receive antenna 2). For instance, we have % % when, and % % when. The values of the weights are normalized with respect to the sum.wefirst observe that the relative importance of each receive antenna depends not only on the SNR fraction at that antenna, but also on the operating total average SNR per bit. In particular, at low SNR ( 10 db), the receive antenna contributing more received power has a higher weight, as at high SNR ( 15 db at ), the receive antenna contributing less received power has a higher weight. It is also observed that when the block correlation coefficient approaches unity, the two antennas tend to be weighted equally at high SNR. This implies that the optimal and suboptimal differential detectors have similar performance at high SNR when the channel approaches static fading. In Figs. 3 5, we plot the BEP results for receive antennas obtained from both theoretical analysis and computer simulation. Figs. 3 and 4 show the results for the differential OSTBC with and QPSK modulation at

8 TAO AND KAM: ANALYSIS OF DIFFERENTIAL ORTHOGONAL SPACE TIME BLOCK CODES 289 Fig. 5. Analytical and simulated BEP comparison for M = 4, P = 3, and N =2with QPSK at =0:1( : =10% :90%). Fig. 6. Analytical BEP comparison for M = P = 2and N = 3with QPSK at : : =10% :30%:60%. 0.1 and 0.49, respectively. Fig. 5 illustrates the results for the scheme with,, and QPSK at. Though block-wise time-varying fading is assumed in our theoretical analysis, we use the more realistic symbol-wise time-varying fading model in our simulations. The time variation is given by the Jakes model with normalized autocorrelation function, is the zeroth-order Bessel function of the first kind, is the normalized Doppler frequency, and is the index of symbol intervals. To determine the block correlation coefficient for computing the weights (10) used in the optimal detector, we take the average over two adjacent transmission blocks as. The same is also adopted in the theoretical analysis. The upper bound for the optimal detector is plotted using (39). The exact BEP for i.i.d. channels, from [11], is plotted for comparison. From these figures, several observations can be made. First, it is seen that for both two and four transmit antennas, the analytical BEP derived assuming block-wise time-varying fading matches very well with the simulated BEP over symbol-wise time-varying fading channels. This observation shows that the block-wise fading model with block correlation coefficient determined above serves as a good model for practical time-varying channels in performance analysis. Second, the nonidentical channel gain distribution among the receive antennas degrades the BEP performance. For instance, in Fig. 3, when the total average SNR per bit is 20 db and the normalized Doppler frequency is 0.015, the analytical BEP is equal to for the i.i.d. channel ( 20 db), but is only for the i.s.i.d. channel ( 13 db and db) with optimal detection. Third, the irreducible error floor of i.s.i.d. channels as when the optimal detector is used approaches that of i.i.d. channels. These two observations agree with the asymptotic behavior of the upper bound we obtained in Section IV-B. Last, when the normalized Doppler frequency is large, the optimal detector can substantially improve the BEP performance, compared with the suboptimal detector, especially in the regime of high SNR. For example, in Fig. 5, the error floor of the optimal detector is one order of magnitude lower than that of the suboptimal detector. On the other hand, when is small, the performances of the optimal and suboptimal detectors are almost identical, regardless of the unequal received powers among the receive antennas. This agrees with our observation from Figs. 1 and 2. Finally, in Fig. 6, we present the analytical BEP results for the code with using receive antennas, the average received signal energy distribution among the three receive antennas is set to % % %. Similar observations can be made. VI. CONCLUSION This paper presents the analysis of BEPs for differential OSTBCs over semi-identical MIMO Rayleigh fading channels. We take into account both unequal received SNR distribution among the receive antennas and arbitrary fluctuation rates of the block-wise time-varying channel fading processes. We show that the optimal symbol-by-symbol differential detector involves weighting the output from each receive antenna according to its channel statistical information. The exact BEP expressions for any number of transmit antennas for which OSTBCs exist are derived in the form of residues, from which the exact and closed-form expressions for two transmit antennas are derived. An alternative expression of exact BEP for the optimal detector is also derived, which is well-suited for numerical evaluation. Our results reveal that the nonidentical channel statistics degrade the error performance, compared with the identical case. We also find that the proposed optimal detector outperforms the conventional suboptimal detector when the channel fluctuates rapidly, but they perform similarly in near-static fading channels. Though they are derived for semi-identical MIMO channels, our results can be applied to nonidentical MIMO channels under certain conditions.

9 290 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007 We rewrite in (14) as APPENDIX The correlation between and is is given in (22). It follows from (5) that the joint PDF of and is complex Gaussian with mean zero and covariance matrix Since the real and imaginary parts of are uncorrelated and both have the same covariance matrix, it follows that. Hence,. and can be obtained using (41). To summarize, the conditional variance of is computed as (34). REFERENCES Therefore, the conditional distribution complex Gaussian with mean and covariance matrix is also After manipulation, the mean vector and covariance matrix can be simplified as (40) (41) respectively. Thus, is also conditionally Gaussian distributed. Using (40), we obtain the conditional mean of as Since satisfies (24), reduces to (33). The conditional variance of is computed as [1] H. Fu and P. Y. Kam, Performance of optimum and suboptimum combining diversity reception for binary DPSK over independent, nonidentical Rayleigh fading channels, in Proc. IEEE Int. Conf. Commun., Jun. 2005, vol. 4, pp [2] M. Z. Win and J. H. Winters, Analysis of hybrid selection/maximalratio combining of diversity branches with unequal SNR in Rayleigh fading, in Proc. IEEE VTC, Fall, 1999, vol. 1, pp [3] S. M. Alamouti, A simple transmit diversity technique for wireless communication, IEEE J. Sel. Areas Commun., vol. 16, no. 3, pp , Oct [4] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory, vol. 45, no. 5, pp , Jul [5] G. Ganesan and P. Stoica, Space-time diversity using orthogonal and amicable orthogonal designs, in Proc. Int. Conf. Acoust., Speech, Signal Process., 2000, vol. 5, pp [6] G. Ganesan and P. Stoica, Space-time block codes: A maximum SNR approach, IEEE Trans. Inf. Theory, vol. 47, no. 5, pp , May [7] V. Tarokh and H. Jafarkhani, A differential detection scheme for transmit diversity, IEEE J. Sel. Areas Commun., vol. 18, no. 7, pp , Jul [8] M. Tao and R. S. Cheng, Differential space-time block codes, in Proc. IEEE Global Telecommun. Conf., 2001, vol. 2, pp [9] G. Ganesan and P. Stoica, Differential detection based on space-time block codes, Wireless Pers. Commun., vol. 21, pp , [10] C. Gao and A. M. Haimovich, BEP analysis of MPSK space-time block codes with differential detection, IEEE Commun. Lett., vol. 7, no. 7, pp , Jul [11] T. P. Soh, P. Y. Kam, and C. S. Ng, Bit error probability for orthogonal space-time block codes with differential detection, IEEE Trans. Commun., vol. 53, no. 11, pp , Nov [12] E. Chiavaccini and G. M. Vitetta, Further results on differential space-time modulation, IEEE Trans. Commun., vol. 51, no. 7, pp , Jul [13] Z. Chen, G. Zhu, J. Shen, and Y. Liu, Differential space-time block codes from amicable orthogonal designs, in Proc. IEEE Wireless Commun. Netw., Mar. 2003, vol. 2, pp [14] A. V. Geramita and J. Seberry, Orthogonal Designs, Quadratic Forms and Hadamard Matrices. New York: Marcel Dekker, 1978, vol. 43, Lecture Notes in Pure and Appl. Math.. [15] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge Univ. Press, [16] P. Y. Kam, Bit error probabilities of MDPSK over the nonselective Rayleigh fading channel with diversity reception, IEEE Trans. Commun., vol. 39, no. 2, pp , Feb [17] Y. Ma and Q. T. Zhang, Accurate evaluation of MDPSK with noncoherent diversity, IEEE Trans. Commun., vol. 50, no. 7, pp , Jul [18] M. Brehler and M. K. Varanasi, Asymptotic error probability analysis of quadratic receivers in Rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers, IEEE Trans. Inf. Theory, vol. 47, no. 6, pp , Sep [19] G. M. Vitetta, U. Mengali, and D. P. Taylor, Optimal noncoherent detection FSK signals transmitted over linearly time-selective Rayleigh fading channels, IEEE Trans. Commun., vol. 45, no. 11, pp , Nov

10 TAO AND KAM: ANALYSIS OF DIFFERENTIAL ORTHOGONAL SPACE TIME BLOCK CODES 291 [20] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw- Hill, [21] M. K. Simon and M.-S. Alouini, A unified approach to the performance analysis of digital communication over generalized fading channels, Proc. IEEE, vol. 86, no. 9, pp , Sep Meixia Tao (S 00-M 04) received the B.S. degree in electronic engineering from Fudan University, Shanghai, China, in 1999, and the Ph.D. degree in electrical and electronic engineering from Hong Kong University of Science and Technology, Hong Kong, in From August 2003 to August 2004, she was a Member of Professional Staff in the Wireless Access Group, Hong Kong Applied Science and Technology Research Institute Co. Ltd., she worked on the design of wireless local area networks. Since August 2004, she has been with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, she is currently an Assistant Professor. Her research interests include multiple-antenna techniques, coding and modulation, dynamic resource allocation in wireless networks, cooperative communications, and cross-layer design and optimization. Pooi Yuen Kam (M 83 SM 87) was born in Ipoh, Malaysia in 1951, and educated at the Massachusetts Institute of Technology, Cambridge, he obtained the S.B., S.M., and Ph.D. degrees in electrical engineering in 1972, 1973, and 1976, respectively. From 1976 to 1978, he was a Member of the Technical Staff at the Bell Telephone Laboratories, Holmdel, NJ, he was engaged in packet network studies. Since 1978, he has been with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, he is now a Professor. His research interests are in detection and estimation theory, and its applications to digital communications and coding. He spent the sabbatical year 1987 to 1988 at the Tokyo Institute of Technology, Tokyo, Japan, under the sponsorship of the Hitachi Scholarship Foundation. In the summer of 2006, he was invited to the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada, as the David Bensted Fellow. Dr. Kam is a member of Eta Kappa Nu, Tau Beta Pi, and Sigma Xi. Since 1996, he has been the Editor for Modulation and Detection for Wireless Systems of the IEEE TRANSACTIONS ON COMMUNICATIONS. He won the Best Paper Award at IEEE VTC 2004 Fall, Los Angeles, CA.