The performance analysis of differential orthogonal space time block codes

THE PERFORMANCE ANALYSIS OF DIFFERENTIAL ORTHOGONAL SPACE-TIME BLOCK CODES SOH THIAN PING (B.Eng(Hons.),NUS) A THESIS SUBMITTED IN FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2006 Table of Contents Table of Contents ii Statement v Summary vi Acknowledgments ix List of Tables xi List of Figures xii Notation xv Commonly used symbols xvii Abbreviations xx Introduction 1.1 Space-Time Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . 12 MIMO System Model and Differential STBC with unknown CSI 14 2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Differential Unitary Space-Time Modulation . . . . . . . . . . . . 20 Performance Analysis of DOSTBC over static Non-identical Fading Channels 27 3.1 BEP Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ii 3.2 3.3 Numerical Results and Discussions . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 51 Performance Analysis of DOSTBC over block-wise constant Fading Channels 4.1 BEP Analysis for the iid Rayleigh Channels . . . . . . . . . . . . 4.1.1 Numerical Results and Discussions . . . . . . . . . . . . . 4.2 BEP Analysis for the Rician Channels . . . . . . . . . . . . . . . 4.2.1 Numerical Results and Discussions . . . . . . . . . . . . . 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 58 61 74 81 85 Performance Analysis of DOSTBC with generalized receive selection combining 86 5.1 Generalized Selection Combining based on max S+N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Generalized Selection Combining based on max SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3 Generalized Selection Combining based on max output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 Numerical results and Discussions . . . . . . . . . . . . . . . . . . 109 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Performance comparison of DOSTBC with new DUSTM constellations 6.1 Search criteria for DUSTM . . . . . . . . . . 6.2 New Constellations for DUSTM . . . . . . . 6.3 Simulation Results . . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . DUSTM and some 116 . . . . . . . . . . . . 119 . . . . . . . . . . . . 121 . . . . . . . . . . . . 125 . . . . . . . . . . . . 131 Conclusions and future works 134 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 ∞ A Evaluation of the Integral γ L exp(−kγ)p (γ; {·}) dγ A.1 Nakagami-m and Rayleigh fading channel . . . . . . A.2 Nakagami-q fading channel . . . . . . . . . . . . . . . A.3 Rician fading channel . . . . . . . . . . . . . . . . . . A.4 Log-normal shadowing . . . . . . . . . . . . . . . . . A.5 Nakagami-m lognormal fading channel . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 140 140 141 142 142 A.6 A.7 A.8 A.9 Rice-lognormal fading channel . . . . . . . Chun Loo’s Land Mobile Satellite Channel Generalized Rice-lognormal fading channel Generalized Marcum-Q function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 143 144 145 B Conditional Multi-variate Distributions of Complex normal variables 148 Bibliography 150 C Publications 161 C.1 Journal Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C.2 Conference Papers . . . . . . . . . . . . . . . . . . . . . . . . . . 161 iv Statement I state that the thesis is my own work, and it has not been submitted for another degree. v Summary The future wireless communications systems should be able to offer high speed Internet access and multimedia communications. One major technological breakthrough that will support the high data rate, required by such applications, is the uses of multiple transmit and receive antennas. A system with multiple transmit and receive antennas is often referred to as a multiple-input multiple-output (MIMO) system. Various approaches to design codes, known as space-time codes (STC), to be used in MIMO system have been proposed in recent years. Some approaches are based on the receivers having some or complete knowledge of the channels; some are based on both the receivers and the transmitters having some or complete knowledge of the channels while others may employ a differential scheme whereby the channels are completely unknown. The differential scheme is apt in situation when it is not feasible to estimate the channel parameters, due to the complexity involved and/or when the channel characteristics are changing too rapidly. The thesis deals with the error performance analysis of one such differential scheme, which is the differential orthogonal space time block codes (DOSTBC) utilizing M-ary phase shift keying. DOSTBC are able to achieve the maximum diversity order inherent in a MIMO system with a simple linear symbol by symbol vi maximum likelihood decoder. This is in contrast to other STC schemes which require joint detection of the transmitted symbols and are thus more complex. The bit error probabilities (BEP) of DOSTBC, with M ∈ {2, 4, 8, 16}, for the static and independent generalized channels are obtained exclusively here. In the independent generalized channels model, the channels observed by different transmit-receive antennas pairs are independent and they may exhibit different statistical properties. The Rayleigh, Nakagami-q, Nakagami-m, Rician , lognormal shadowed, composite Nakagami-m lognormal, composite Rice-lognormal, the Chun Loo’s land mobile satellite and a generalized Rice lognormal channels are considered here. The static channels are assumed where the channel parameters are treated as constants during two transmission block duration. In the situation when the channels are block-wise static where the channel parameters remain constants within a transmission block duration but fluctuate from block to block, the BEP of DOSTBC are derived for independent and identically distributed (iid) Rayleigh channels and for iid Rician channels. For the case of Rician channels, the power of the line of sight components between different transmit-receive antenna pairs are non-identical and the BEP are only derived for M = 2. For the more general situation when the channels are assumed symbol-wise static, an approximated BEP is proposed. The BEP of DOSTBC are also derived for the iid block-wise static Rayleigh channels when suboptimal diversity techniques, namely antenna selection combining schemes, are used. Finally, we utilized the error performance results derived for DOSTBC and benchmarked them against the performance of existing and newly proposed unitary codes designed for differential unitary space-time modulation. Although, the more general unitary codes may outperform the orthogonal codes in term of error vii performance, this is achieved at the expense of having a more complex detecting algorithm. This trade-off has to be taken in account in system designs. viii Acknowledgments I would like to thank my thesis adviser, A/Professor Ng Chun Sum, and coadviser, Professor Kam Pooi Yuen, for their guidance and advice. Despite their busy workload, they patiently went through all my ideas and gave me the benefit of their insights. They also painstakingly read and edited my manuscript and brought my attention to the areas where I had not sufficiently clarified or explained. Several of my ideas were sharpened and in some cases radically changed by this ongoing process of intellectual exchange. Without their help, my work would have been much lesser than it is now. I am forever in debt to my parents for their great support, encouragement and love. Finally I would like to express my most heartfelt gratitude to my wife, Peng. Her support has been instrumental to the success of this work. To my son, Zack, and daughter, Jann, whom have been inspirational. Rather than to use the usual words to thank them, as is customary with most acknowledgments, I prefer to sing: “This is it, oh, I finally found someone. Someone to share my life. ix I finally found the one, to be with every night. ’Cause whatever I do, it’s just got to be you. My life has just begun, I finally found someone .” -Bryan Adams. x 147 Here, ξ1 = a21 + b21 + 2a1 b1 a21 + b21 2a1 b1 −1 (A.9.11) and 2k + a21 + b21 + ξ2 = 2a1 b1 2k + a21 + b21 2a1 b1 − 1. (A.9.12) For the case when b1 > a1 = 0, we have [35] l−1 Ql (0, b1 ) = n=0 n! b21 n l−1 I(l, a1 , b1 ; L, k) = n=0 exp − n! b21 b21 . Thus, I(l, 0, b1 ; L, k) can be shown to be n Γ(L + + n) , b1 > a1 = 0. (L + 12 /2 + k)L+1+n (A.9.13) Using the fact that Qm (a1 , 0) = 1, we have I(l, a1 , b1 ; L, k) = Γ(L + 1) , a1 > b1 = 0. k L+1 (A.9.14) Appendix B Conditional Multi-variate Distributions of Complex normal variables ˜ = y ˜ n (k − 1) ∼ CN (0, Rpp) and q ˜ = y ˜ n (k) ∼ CN (0, Rqq ). We have Let p from [72], ˜ ˜ = E[˜ η q|˜ p] = Rqp R−1 pp p (B.0.1) Σ = var[˜ q|˜ p] = Rqq − Rqp R−1 pp Rpq , (B.0.2) and 148 149 ˜ and q ˜ . It is easy to obtain where Rpq is the covariance matrix between p Rpp = Rqq = (2R(0)Es + N0 )I (B.0.3) ˜ Rqp = 2R(1)Es M(k). (B.0.4) and Finally, we obtain ˜= η ρ[1]¯ γsym ˜ M(k)˜rn (k − 1) γ¯sym + (B.0.5) and ρ(1)2 N0 γ¯sym I. γsym + 1)I − Σ = N0 (¯ γ¯sym + (B.0.6) Bibliography [1] M. Mouly and M. B. Pautet, “The GSM System for Mobile Communications”, Telecom Publishing, Olympia, WA, 1992. [2] I. E. Telatar, “Capacity of Multi-Antenna Gaussian Channels,”, AT&T Bell lab, Internal Tech. Memo, June 1995. [3] G. J. Foschini and M. J. 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Kam, “Unitary Signal Constellations for Differential Space-Time Modulation with more that Two Transmit Antenna”, in Proc. ICCS 2004 , Singapore, Session 2P-04-06, pp. 291-295. 5. T. P. Soh, P. Y. Kam and C. S. Ng, “Bit Error Probability For Orthogonal Space-Time Block Codes with Differential Detection”, in Proc. ICCS 2004, Session 2P-04-07, Singapore, pp. 296-300. 6. T. P. Soh, C. S. Ng and P. Y. Kam, “On Designing Full Rank Space Time Block Codes with High Code Rate”, in Proc. ICICS-PCM 2003, Singapore, Session 2A6.6, pp. 869-873. [...]... training The spacetime coding techniques when the channel propagation coeffients are known at the receivers include space- time block codes (STBC) [8] [9] [10] , space- time trellis codes (STTC) [7] [11] [12], space- time concatenated codes (which include space- time turbo codes, serial concatenated codes, etc) and the layered spacetime (LST) codes -which include Bell Laboratories layered space- time scheme... Hochward [4] spearheaded the development of STC and escalated the amount of research activities in this area of research 4 1.1 Space- Time Codes The fundamental principle of STC is illustrated in Figure 1.1 From an input stream of information bits, the input symbol is encoded by the space time encoder into NT code vectors of time length L forming a L × NT code matrix The row of the matrix will be transmitted... Notation a the scalar a a ˜ the complex number a ˜ a the vector a ˜ a ˜ the complex vector a A ˜ A the scalar matrix A ˜ the complex matrix A (·)∗ the complex conjugate T the transpose of A H j the conjugate transpose of A √ the imaginary unit (j = −1) Re {˜} , aR a ˜ the real part of a ˜ Im {˜} , aI a ˜ the imaginary part of a ˜ E {·} , (·) the expectation operator |A| the determinant of the matrix... Beside space diversity, there are other techniques for diversity These techniques are polarization diversity, frequency diversity and time diversity The main aim of these techniques is to provide the receiver(s) with various versions or replicas of the message signals through different dimensions (space, time or frequency) It is shown that by combining the effect of space and time diversity, via the use of. .. Organization of the thesis The thesis is organized as follows The system model and the description of DUSTM, which include DOSTBC, will be presented in chapter 2 The derivation of the BEP for DOSTBC for the generalized channels, will be presented in chapter 3 In chapter 4, we derived the BEP for the DOSTBC for the block- wise static channels The Rayleigh and the Rician channels will be considered For the symbol-wise... orthonormal with respect to time are transmitted among the different 8 transmit antennas The message information is carried in the subspace spanned by the orthonormal columns (the direction) of the unitary matrix, i.e., we can interpret the complex signal constellations as vector function of time and the message information is carried in the directions of these vectors To extend the coding scheme to a continuously... combining OFDM Orthogonal frequency division multiplexing OSTBC Orthogonal space- time block code /codes/ coding pdf probability density function PWEP Pairwise error probability RLN Rice Lognormal S+N Signal-plus-noise SIMO Single-input multiple output SISO Single-input single output SNR Signal-to-noise ratio STBC Space time block code /codes STC Space- time code /codes STTC Space- time trellis code /codes USTM... expectation operator |A| the determinant of the matrix A (A) (A) A the Frobenius norm or Euclidean norm of the matrix A Tr {A} the trace of the matrix A rank {A} the rank of the matrix A ⊗ the Kronecker product I the identity matrix 0 the zeros matrix xv p(x), f (x) the probability density function (pdf) of x P (E) the probability of an event E x ∼ N (μ, Λ) x is real-valued Gaussian random variable... transmitted simultaneously over the NT numbers of transmit antennas The entries of the vector are complex baseband representation of the symbols to be modulated and transmitted on the transmitter antenna We could, thus, view the transmission of STC as matrix transmission Transmit Antenna 1 Receive Antenna 1 Space Time Encoder x11 x1NT xL1 xLNT Time Space Information Source Space Time Receiver : : : Transmit... (especially so when the transmit and/or the receive antennas are spaced far apart) Hence, it is of practical interest to obtain the theoretical BEP for the non-identical channels case We will consider the BEP of DOSTBC for the non-identical channels here Our results can be viewed as a generalization of the BEP of the conventional DPSK for the non-identical channels [34] [35]to the MIMO system However, . THE PERFORMANCE ANALYSIS OF DIFFERENTIAL ORTHOGONAL SPACE- TIME BLOCK CODES SOH THIAN PING (B.Eng(Hons.),NUS) A THESIS SUBMITTED IN FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF. Euclidean norm of the matrix A Tr {A} the trace of the matrix A rank {A} the rank of the matrix A ⊗ the Kronecker product I the identity matrix 0 the zeros matrix xv p(x),f(x) the probability. transpose of A j the imaginary unit (j = √ −1) Re {˜a}, ˜a R the real part of ˜a Im {˜a}, ˜a I the imaginary part of ˜a E {·}, (·) the expectation operator |A| the determinant of the matrix A A the