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Differential amplitude/phase space-time modulation Zi, Juan 2003

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Differential Amplitude/Phase Space-Time Modulation Juan Zi B. Eng., Huazhong University of Science and Technology Hankou Branch, 1994  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF  Master of Applied Science in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard  The University of British Columbia December 2003 © Juan Zi, 2003  Library Authorization  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Name of Author (please print)  Date  Title of Thesis: /\Aooiu(cx4->  Degree:  ft.flU.  Year:  2*ew>3  Abstract In this thesis, a differential amplitude/phase space-time modulation ( D A P S T M ) is proposed for multiple transmit antenna wireless systems over flat Rayleigh fading channels. Two conventional noncoherent detection schemes, namely, simply heuristic (SH) differential detection (DD) and maximum likelihood (ML) D D are presented. Furthermore, two improved noncoherent detection schemes, multiple-symbol detection (MSD) and decisionfeedback D D (DF-DD) with lower decoding complexity are derived. By taking the dependencies among the received symbols into account, M S D and D F - D D can reduce the error floor of M L - D D . The pairwise error probability (PEP) based on SH-DD, and an approximation of the bit error rate (BER) based on the union bound, are derived. Analytical considerations agree well with the simulation results. Compared with the known differential unitary space-time modulation (DUSTM), D A P S T M can be said to generalize the diagonal structure from phase signals to a combination of phase signals and amplitude signals. This generalization potentially allows the spectral efficiency to be increased by carrying information, not only in phases, but also in amplitudes. D A P S T M is not as power efficient as space-time codes with differential amplitude/phase shift keying ( S T C - D A P S K ) , which is based on Alamouti's orthogonal space-time code (OSTC), when two transmit antennas are employed. However, D A P S T M allows easy implementation at the transmitter, due to the group property of its constellation under matrix  ii  Ill  multiplication. D A P S T M can be employed for an arbitrary number of transmit antennas while keeping full diversity and full rate. It is also suitable for exploiting time diversity when only one transmit antenna is used i n the system.  In contrast, S T C - D A P S K can  only achieve full diversity and full rate for two transmit antennas and can not exploit time diversity, due to its nondiagonal structure.  Contents Abstract  ii  List of Tables  vii  List of Figures  ix  Acknowledgments  x  1 Introduction  1  1.1  Diversity  1  1.2  Motivation and Objectives  2  1.3  Contributions  3  1.4  Thesis Outline  4  2 Background and Related Work 2.1  2.2  2.3  6  System Model  6  2.1.1  Space Diversity System Model  6  2.1.2  Time Diversity System Model  7  Channel Model  10  2.2.1  Flat Rayleigh Fading Ghannel Model  10  2.2.2  MIMO Channel Model  12  Differential Modulation and Detection  14  iv  CONTENTS  •  3  4  5  6  v  2.3.1  MDPSK  14  2.3.2  MDAPSK  16  2.3.3  Differential Unitary Space-Time Modulation  20  2.3.4  Space-Time Codes with D A P S K  "...  23  Differential Amplitude and Phase Space-Time Modulation  27  3.1  DAPSTM  27  3.2  Conventional Differential Detection  31  3.2.1  Simple Heuristic Differential Detection  32  3.2.2  Maximum Likelihood Differential Detection  33  3.3  Multiple-Symbol Detection  38  3.4  Decision-Feedback Differential Detection  41  3.5  Modifications for Time Diversity  43  Performance Analysis  44  4.1  Pairwise Error Probability  44  4.2  Approximation for Bit Error Rate  48  Simulation Results  49  5.1  Numerical Results Compared with Simulations  49  5.2  Constellations Design  52  5.3  Space Diversity  55  5.4  Time Diversity  61  Conclusions and Recommendations  64  6.1  Conclusions  64  6.2  Recommendations for Future Work  65  Glossary  69  CONTENTS  vi  Bibliography  70  List of Tables 2.1  D U S T M parameters for N  = 2  2.2  D U S T M parameter for NT = 3  22  5.1  D A P S T M parameters for N  = 2 by minmax P E P  54  5.2  D A P S T M parameters fox N  = 2 and N  55  T  T  T  21  T  vii  = 3  List of Figures 2.1  Space diversity system model  7  2.2  Time diversity system model  8  2.3  N  x Ni rectangular interleaver  9  2.4  Generic model for a MIMO channel  12  2.5  Structure of the differential encoder for M D P S K modulation  14  2.6  Block diagram of the conventional differential detector for M D P S K  15  2.7  Structure of the differential encoder for M D A P S K modulation  16  2.8  Signal constellation for 16DAPSK  2.9  Block diagram of the conventional differential detector for M D A P S K  B  , .  18 19  2.10 Structure of the differential encoder for D U S T M  20  2.11 Structure of the differential encoder for S T C - D A P S K  23  2.12 STC-24APSK signal constellation  24  3.1  Structure of the differential encoder for D A P S T M  28  5.1  Comparison of simulation vs. numerical results of P E P . N  = 2, R = 1.5  T  bits/(channel use) (RP — 1 bit/(channel use), RA = 0.5 bit/(channel use)), BT f  5.2  = 0.001, p = 2.1, u = l, 0  Ul  Comparison of simulation vs.  = 1, 0Q = 0, and 0 = 1 are valid 1  numerical result of B E R . N  T  50  = 2, R = 4  bits/(channel use) (RP = 3 bits/(channel use), RA = 1 bit/(channel use)), BT f  = 0.001, p = 1.4, UQ = 1, ui = 15, #o = 0, and d = 1 are valid x  viii  51  LIST  OF FIGURES  5.3  ix  B E R vs. 101og (£ /iV ) for D A P S T M and D U S T M with R = 3 bits/(channel 10  6  0  use) and R = 4 bits/(channel use). N  T  = 2, N  R  = 1 and B T = 0.001 are F  valid. D A P S T M is detected by SH-DD and M L - D D ; D U S T M is only detected by M L - D D 5.4  56  B E R vs. 10log (E /N ) W  B  for  0  M S D and D F - D D with (a) R = 3 bits/(channel  use), B T = 0.005, (b) R = 4 bits/(channel use), B T = 0.005, (c) R = 5 F  F  bits/(channel use), B T F  0.0025. N  = 2, N  T  5.5  = 0.0035, (d) R = 6 bits/(channel use), B T F  = 1 is valid  R  57  B E R vs. 101og (£ /Ar ) for D A P S T M with N 10  N  R  =  6  0  T  = I, N  T  = 2, and N  T  = 3.  = 1, B T = 0.001, R = 4 bits/(channel use) (RA = 1 bit/(channel use), F  RP = 3 bits/(channel use)), p = 1.4, u\ = 23, Ip = 3 are used 5.6  B E R vs. 101og (£ /W ) 10  N  T  5.7  = 3. N  R  6  f o r  0  58  D A P S T M and D U S T M with (a) A ^ = 2, (b) r  = 1, B T = 0.001 are valid  59  F  B E R vs. 101og (£ /A o) for D A P S T M vs. S T C - D A P S K . N /  10  6  T  = 2, N  R  = 1,  and B T = 0.001 are valid  61  F  5.8  B E R vs. 101og (i5&/7Vo) for space diversity with N 10  T  = 2 and time diversity  with NB = 2. (a) R = 4 bits/(channel use), (b) R = 5 bits/(channel use), (c) R = 6 bits/(channel use) are valid 5.9  B E R vs. 101og (/^b/A ) for time diversity with N  62  /  10  0  with NT — 3. R — 4 bits/(channel use) is valid  B  = 3 and space diversity 63  Acknowledgments  I would like to express my heartfelt gratitude to my supervisor, Dr. Robert Schober, for his continuous guidance and insight throughout this thesis. His enthusiasm and devotion have always inspired me during hard times. I would like to thank Dr. Victor C M . Leung for his constant encouragement from the beginning of my graduate study at U B C . I would also like to thank all of my friends and fellow students, who have made my past years of study and work at U B C productive, cheerful and enjoyable. Last but not least, I would like to thank my family, for their never ending love and unconditional support.  Chapter 1 Introduction 1.1  Diversity  In wireless communication systems, fading [1] is a major obstacle to the efficient and reliable transmission of data. T o mitigate the adverse effects of fading, it is desirable to provide the receiver w i t h more than one replica of the transmitted signal to improve error-rate performance. T h i s so-called diversity is based upon the observation that i f the same information signal is received redundantly, over two or more independent fading channels ("diversity branches"), the probability that all the signals will fade simultaneously is reduced [2]. Space diversity and time diversity are two widely-applied diversity techniques.  Space diversity (antenna diversity) is an attractive technique for achieving a .diversity advantage, since it does not incur any bandwidth expansion.  T h e basic idea here is to  employ two or more transmit antennas and/or receive antennas i n order to receive uncorrected signals. It is a practical technique for reducing the effects of fading i n most scattering environments [2].  For time diversity, the same information-bearing signal is repeatedly transmitted i n different time slots. If the time spacing equals or exceeds the coherence time of the channel, the multiple repetitions of the signal undergo nearly independent fading, thereby achieving 1  1.2 Motivation and Objectives  2  diversity [1].  1.2  Motivation and Objectives  For single antenna wireless communication systems, differential encoding is employed for phase shift keying (PSK) when the phase shift introduced by the channel is unknown at the receiver [3]. In this so-called differential phase-shift keying (DPSK) scheme, the absolute phase of the received signal cannot be exploited, and the information is mapped to the phase difference between two successively transmitted symbols. To achieve a higher bandwidth efficiency, differential amplitude/phase-shift keying (DAPSK) modulation, see e.g.  [4], is  introduced. To make use of the diversity of multiple transmit antennas, differential unitary space-time modulation (DUSTM) [5], [6], and differential space-time block coding (DSTBC) [7], are proposed.  These noncoherent transmission schemes avoid the need for channel  estimation but often pay a price in performance. As a special case, diagonal signals are introduced for D U S T M , where at any given time only one antenna is active. The diagonal signal constellations form groups under matrix multiplication, allowing easy implementation at the transmitter while providing full diversity and full rate.  D S T B C is more power  efficient than D U S T M , but in general, it can only achieve full diversity and full rate with two transmit antennas.  In [5]-[7], signal constellations consisting of unitary matrices are  employed. D U S T M and D S T B C can be seen to be natural extensions of standard D P S K as used in single-antenna unknown-channel systems to multiple antenna transmission systems. To take advantage of the "amplitude", space-time codes with D A P S K (STC-DAPSK) [8] are introduced to improve the bandwidth efficiency of D S T B C [7] for two transmit antenna systems.  1.3 Contributions  1.3  3  Contributions  In this thesis, we propose a differential amplitude/phase space-time modulation (DAPSTM) scheme for multiple antenna wireless communication systems in flat Rayleigh fading channels. In our proposed D A P S T M scheme, diagonal signals are employed not only for phase but also for amplitude modulation. The total differential signal is the product of the amplitude signal and the phase signal. The D A P S T M signal keeps the group properties of DUSTM.  Two corresponding differential detection (DD) schemes, simple heuristic differential detection (SH-DD) and maximum likelihood D D (ML-DD) are derived for detection of D A P S T M . Two improved noncoherent receivers are also considered. The first one is the multiple-symbol detection (MSD) receiver. Because of the high computational complexity of MSD, a low-complexity decision-feedback D D (DF-DD) receiver is also derived. We also optimize the parameters in our signal design to achieve higher performance.  Performance analyses of the proposed D A P S T M for Rayleigh fading with uncorrelated diversity branches are presented.  In particular, we calculate pairwise error probability  (PEP) based on SH-DD. The analysis results agree well with the simulation results. Also, an approximation for the bit error rate (BER) is derived by the union bound.  Compared with D U S T M , the proposed constellations do not necessarily have a constant amplitude. In other words, we extend the differential phase modulation to combined differential amplitude and phase modulation in multiple-antenna systems. Since we take advantage of the amplitude information, D A P S T M can achieve a higher bandwidth efficiency than D U S T M .  D A P S T M is not as power efficient as the nondiagonal S T C - D A P S K [8] when two trans-  1.4 Thesis Outline  mit antennas are employed.  4  But D A P S T M can be easily extended to more than two  transmit antennas, while keeping full diversity and the full transmission rate. For D A P S T M and fast fading, higher performance can be achieved if time diversity is exploited instead of space diversity. S T C - D A P S K cannot be applied to exploit time diversity due to its nondiagonal structure.  Space and time diversity are closely related, since in both schemes a number of symbols are jointly modulated to obtain a multi-dimensional hypersymbol. All the features that work for space diversity also work for time diversity if signals are of a diagonal structure. But these two diversity techniques have different diversity characteristics when the channel varies slowly. Time diversity is less effective, because a very long interleaver is necessary to obtain sufficient diversity gain. On the other hand, space diversity requires multiple antennas at the transmitter, which increases implementation costs.  1.4  Thesis Outline  The outline of this thesis is as follows:  Chapter 2 reviews some basic concepts, such as the M-ary DPSK (MDPSK) and M ary D A P S K (MDAPSK) modulation schemes, differential encoding, the employed channel models, D U S T M and S T C - D A P S K , and conventional coherent and noncoherent receivers.  chapter 3 discusses D A P S T M with M L - D D , MSD, and D F - D D receivers and introduces time diversity.  In chapter 4, we derive an exact, closed form expression for P E P and an approximation for B E R .  1.4 Thesis Outline  5  Chapter 5 presents constellation designs and simulation results for space diversity and time diversity of D A P S T M , and compares the simulation results for space diversity with those for D U S T M and S T C - D A P S K .  Chapter 6 summarizes the main contributions and conclusions of this thesis and gives recommendations for future work.  In this thesis, script English letters denote sets, bold upper case letters denote matrices, bold lower case letters denote vectors, and lower case letters denote scalars.  Chapter 2 Background and Related Work 2.1 2.1.1  System M o d e l Space Diversity System Model  Fig. 2.1 depicts the discrete-time equivalent baseband model of a single-user communication system with JVy transmit antennas and N  R  receive antennas. The discrete message source  first emits L binary bits b[i], b[i] € {0, l}, i = 0 , 1 , . . . , L — 1, in each time interval i V . As r  usual, the bits at different time instances are assumed to be independent and identically distributed (i.i.d.).  The mapper then modulates these L bits into a space-time matrix  symbol V[k) of size A^ x N T  T  according to some mapping rules. After that, the transmit  matrix S[k] is obtained by differentially encoding V[k]. The symbol s [N k  + K] in row  K and column m of the matrix S[k] is transmitted at time N k + K, 0<K<N  — 1, by the  M  T  T  T  mth antenna, 0<m<NT — 1.  At the receiver, r [N k N  receive antenna, 0<n<N  T  R  + n] is received at time N k + K, 0<K<N T  — 1. The N N T  R  T  — 1 by the nth  received signals constitute the matrix R[k] of  size NT x N , which is processed by a noncoherent receiver. This processing combines D D R  6  2.1 System Model  Discrete Message Source  7  fc[/]e{0,l}  V[k]  Mapper  S[k]  Differential Encoder  J L  H[k]  N[k]  Channel  Message Sink  b[i]z{0A}  DeMapper  V[k]  R[k]  Differential Detector  Noncoherent Receiver  Figure 2.1: Space diversity system model.  and demapping, and determines the estimates b[i] of the transmitted binary symbols b[i]. Finally, the estimates b[i] are passed to the message sink. A more detailed description of the different parts of Fig. 2.1 is given in the following sections.  2.1.2  Time Diversity System Model  The discrete-time equivalent complex-baseband model of the time diversity transmission system is sketched in Fig. 2.2. There are only one transmit antenna and N  R  tennas in the system. To achieve N  B  transmit diversity branches, an N  receive an-  x A / rectangular 7  B  interleaver (cf. Fig. 2.3) is used to space the successive signals sufficiently far apart through the channel.  8  2.1 System Model  Discrete Message Source  v, e {0,1}  {0,1}  Message Sink  V[k]  Mapper  *H  DeMapper  Differential Encoder  V[k]  Differential Detector  ^  S[k]  •  R[k] ^  Rectangular Interleaver  b[k]  Rectangular Deinterleaver  Noncoherent Receiver  Figure 2.2: T i m e diversity system model.  First, the L x Nj i.i.d. binary symbols b[i], b[i] £ { 0 , l } , i = 0 , l , . . . , i V / x L - l , are mapped into Ni diagonal matrices V[k], 0 < k < Afj — 1, of size N  B  x A^.  E a c h V[k] is  differentially encoded to obtain one transmission matrix S[k]. T h e main diagonal elements of S[k], s [k], 0 < K < N K  B  -1, 0 < k < N - 1 are written into the kth, 0 < k < R  of the interleaver's memory. After interleaving, a block of NBNJ  symbols is read out from  the interleaver, column by column, to form Ni vectors b[k] of size N . B  the vector b[k], b [k], 0 < n < N K  0 < K< N  B  B  - 1 , row  T h e /cth element of  — 1, 0 < k < Ni — 1, is transmitted at time N k B  - 1, 0 < k < AT/ - 1. b [k] = s '[k'}, 0 < K' < N K  K  B  + K,  — I, 0 < k' < Ni — 1 a n d  A ^ A ; + K = K'NI + k'. W i t h this arrangement, the successive symbols of S[k] are spaced B  N  B  symbols apart i n b[k], so that they encounter independent fading.  2.1 System Model  9  *,[0]  s [0] 2  *«,-..[!]  * [1] 2  fe [0]i,[0]fo [0]-0  s [N,-l]  -1]  s [N,-i]  s,[N,-l]  0  2  2  Figure 2.3: A ^ X AT/ rectangular interleaver.  Each of the N  receive antennas collects the Nj x N  B  B  received signals and feeds them  in its own deinterleaver. Each deinterleaver performs the reverse operation on each block of Ni x N  B  received symbols to recover their original sequence order and reads out N  symbols  B  each time. At the output of the corresponding deinterleaver of the nth, 0<n<N  — 1, receive  R  antenna, the received signal at time N k + K, 0<K<N B  r [N k n  where 0 < K < N  B  —  1, is r [N k n  + K],  B  + K] = s [k]h [N K + k] + rt [N K + k], K  n  I  n  - 1, 0 < k < Ni - 1. The received N  B  B  with elementsr [N k+n}, 0<K,<N n  — 1, 0<k<Nj  B  B  B  — 1, 0<k<Nr  —  (2.1)  I  x A^j-matrix R[k], 0<k<Nj - 1,  l-, is processed by a noncoherent receiver  to determine the estimates v[i] of the transmitted binary symbols v[i], similar to space diversity. In particular, for given k and n, the fading gains h [NiK + k], 0 < K < N n  B  — 1,  should be approximately uncorrected to ensure full diversity for the jointly modulated symbols s [k], 0 < K < N K  B  — 1.  All concepts proposed for the space diversity system using diagonal signals can also be applied directly to the time diversity system. Therefore, in the following, we concentrate on the space diversity aspect, and discuss time diversity only when necessary. Note that in space diversity NT diversity branches are achieved by A ^ transmit antennas, and in time  2.2 Channel Model  10  diversity, where just one transmit antenna: exists, N  B  diversity branches are achieved by  the interleaver.  2.2 2.2.1  Channel Model Flat Rayleigh Fading Channel Model  This thesis considers a stationary slowly time-varying, frequency non-selective, Rayleigh fading channel along, with additive white Gaussian noise (AWGN) distortion. We assume that the fading processes of the diversity branches are statistically independent. As usual, channel state and carrier phase are expected to be constant over a block of at least NT symbol periods, and perfect symbol synchronization is presumed. The received signal at each receive antenna is a noisy superposition of independent, Rayleigh faded, transmitted signals. At time Nrk+K, 0<K,<N  T  — 1, the equivalent complex-  baseband received signal r [Nxk + K] at the nth, 0<n<NR — 1, receive antenna can be n  modelled as NT-1  r [N k + «] = J2 m[N k + K\h [N k s  n  T  T  mn  T  + ] + nnWrk  K  + K],  (2.2)  rn—O  where the gain'./i [A xA; + K] models the zero mean complex flat fading channel from the /  mn  rath transmit antenna to the nth receive antenna, and n [NTk + K] is an independent n  zero-mean complex white Gaussian noise process with two sided power spectral density u\. A l l the fading processes are assumed to have identical statistical properties and are correlated in time. The fading autocorrelation function (ACF)  iphh[ ] m  is [9]  2.2 Channel Model  11  <Pkh[>] = £ h* [N k  + K}h [N k  J  mn  T  mn  + K + A]  T  (2.3)  where £ { • } and ( • )* denote expectation and complex conjugation, respectively, and the following definitions are used.  • oj refers to the variance of the fading processing,  ^2 A o-j-E  c  h [N k mn  T  + K]  For a simple representation, we can normalize h [NTk mn  (2.4) + K] properly to ensure that  the variance of the received noise-free signal is equal to 1.  Since the transmitted  symbols satisfy N -1 T  s  \  £  we get,  (2.5)  \Sm[N k + n} \ = T  '  'm=0.  = 1.  Jo(-) refers to the zeroth order Bessel function of the first kind.  Mx)  = ^f  e  dn,  JxsHv)  x e  (2.6)  Bf refers to one-sided bandwidth of the underlying continuous-time fading processes. T (scalar) refers to the symbol duration.  The white Gaussian noise processes of different receive antennas have equal variance  n [N k n  T  +K  (2.7).  2.2 Channel  12  Model  The mean S N R per receive antenna is SNR = crj/a^ = l/cr?. Since the path gains are assumed to be constant during the transmission of one symbol matrix V[k] and vary from one block to the next, h [N k mn  + K] can be represented as  T  hmniNrk]. Hence, the r [Nrk + K] can also be obtained by n  N -1 T  r [N k n  T  + K] =  m[N k  + n]h [N k]  s  T  mn  T  + n [N k n  T  + K].  (2.8)  m=0  2.2.2  M I M O Channel Model  h [N k] n  \  T  KM;® h jN k]\ N  T  /K J.N k] T  T  \  .  Figure 2.4: Generic model for a M I M O channel.  Fig. 2.4 represents the basic layout of a multiple-input multiple-output (MIMO) channel model with N  T  size NT x N  R  transmit antennas and N. receive antennas. The channel matrix H[k] of R  consists of the fading gains for the different diversity branch pairs,  13  2.2 Channel Model  h [N k}  h [N k]  h _i[N k]  •h [Nfk]  h [N k]  hi ^i[N k]  00  T  10  h - [N k] NT  where h [NTk], mn  10  T  01  0NR  T  n  NR  T  h _ [N k] NT  n  T  T  ••• ' ^ - 1 ^ . - 1  T  (2.9)  [AVA;]  0 < m < NT•— 1, 0 < n < NR — 1, is the complex fading gain between  the mth transmit antenna and the nth receive antenna in kth NT period. The output is related to the input and the channel parameters by the following equation: (2.10)  R[k] = S[k]H[k] + N[k], where  R[k] is an NT x NR received signal matrix at the kth block with (K, n)th entry r [NTk n  + K] representing the received data at the nth, 0 < n < NR — 1, receive  antenna at time A^/c + K, 0 < K < NT — 1,  r [N k] 0  r [N k 0  R[k]  + N  T  rx -i[N k]  T  + 1]  T  r [N k 0  n[N k]  T  T  ri[N k  + l]  T  - 1] ri[N k  H  + A ^ - 1]  T  T  r -i[N k+l] NR  T  • • • r -i[N k NR  + N  T  T  - 1] J (2.11)  • S[k] is an NT X NT transmitted signal matrix at the fcth block with (K, m)th entry Sm[NTk + K] representing the transmitted symbols from mth, 0 < m < N  T  — 1,  transmit antenna at time AT^A; + K,  s [N k} 0  S[k]  T  s [N k 0  T  + l],  so[N k + N -l] T  T  .  s [iV /c] x  [N k  Sl  T  NT  + l]  s{[N k + N T  s -i[N k]  T  T  -.1]  T  s ^[N k+l] NT  T  ••• s -i[N k NT  T  +N  T  - 1] J (2.12)  14  2.3 Differential Modulation and Detection  x N R noise matrix at the kth block with (re, n)th .entry n [N k + K]  • N[k] is an N  T  n  T  representing A W G N at time Nrk + K,  n [N k} n [N k + 1] 0  0  N[k] =  2.3  T  T  n -i[N k] n .i[N k+  T  T  n [N k + N 0  ni[N k] ni[N k + 1]  T  NR  T  - 1] n [N k + N -1] t  T  T  NR  T  ••• n -i[N k Nrt  1]  T  T  +N  T  - I]  (2.13)  Differential Modulation and Detection  Coherent modulation schemes require CSI at the receiver. However, in some situations, e.g., fast fading, reliable channel estimation is difficult. In these cases, differential modulation is usually used in wireless communication systems. In this section, first, two differential modulation schemes with one transmit antenna and one receive antenna, namely, M D P S K and M D A P S K , are reviewed. Then, D S T M and S T C - D A P S K are presented for more transmit antennas.  2.3.1  MDPSK  v[k] o  s[k] •o  Figure 2.5: Structure of the differential encoder for M D P S K modulation.  M D P S K , the differential form of M-ary P S K (MPSK), is a very important modulation  2.3 Differential Modulation and Detection  15  scheme for single antenna wireless communication systems. The structure of the differential encoder for M D P S K is depicted in Fig. 2.5. In this scheme, notation ?;[£;].and s[k] are used instead of V[k] and S[k], which only have one element in them, respectively. First, the information bits are mapped into the phase difference symbol v[k]. The transmitted symbol at time k is generated from v[k] via differential encoding:  (2.14)  s[k] = v[k]-.s[k - I}.  The phase difference symbols v[k] are drawn from an M-ary alphabet  (2.15)  In conventional differential detectors for M D P S K , cf., Fig. 2.6, the estimated transmitted M D P S K symbol v[k] is determined from the decision variable [3]  (2.16)  d[k] = r[k]-r*[k - 1].  r[k] &  d[k]  o  Figure 2.6: Block diagram of the conventional differential detector for M D P S K .  For this, the complex plane is divided into M sectors, corresponding to the M possible values of v[k]. The sector into which cZ[A;] falls determines the value of v[k]. The decision also can be expressed as  2.3 Differential  Modulation  and  16  Detection  [A;] = argmax |&{r[fc]r*[fc - l]w*[A:]} j ,  where  (2.17)  • } denotes the real part of a complex number and v[k] <G A (M) V  denotes a trial  symbol.  2.3.2  MDAPSK  Another more bandwidth-efficient modulation scheme for single antenna wireless communication systems is M D A P S K modulation [4]. M D A P S K , where both amplitude and phase are differentially encoded, may be viewed as a combination of differential amplitude-shift keying (DASK) and D P S K , which are independent of each other (cf. Fig. 2.7). a[k]  s[k]  NT  —o  T  [k]  "JUL  P  NT T  Figure 2.7: Structure of the differential encoder for M D A P S K modulation.  The M D A P S K symbol is given by s[k] = a[A:]p[A:], where the absolute amplitude symbol a[k] is a positive real number, and the absolute phase symbol has magnitude one. For convenience, an appropriate normalization is employed to ensure  s{k)\  \=£{a [k} 2  (2.18)  2.3 Differential Modulation and Detection  17  The absolute amplitude at time k, which is taken from the alphabet  • A = <j>,ai,--- , a _ i Y eN,Y > 2J,  (2.19)  y  where N is the natural number set. a[k] is differentially encoded by  a[k] = v {k]-a[k - 1],  (2.20)  a  where v [k] is the amplitude difference symbols. The alphabet A a  Va  of the amplitude differ-  ence symbols v [k] depends on A a  a  Similar to M D P S K , the absolute phase at time k is also obtained from differentially encoding the phase difference symbols v [k], p  p[k] = v [k]-p[k - 1}.  (2.21)  p  The absolute phase symbol is drawn from the alphabet,  A (M/Y) P  4 j e ^ ' | /G{0,1,--- ,M/Y-  And the alphabet of the phase difference symbols A (M/Y) Vp  1}|.  (2.22)  is identical to  A (M/Y). P  From another point of view, we can see the M D A P S K symbol v[k] as the product of the amplitude difference symbol v [k] and the phase difference symbol v [k], a  p  v[k] = v [k]v [k}. a  p  (2.23)  16DAPSK ( y = 2) modulation, which is also referred as 16-star quadrature-amplitude modulation (QAM) [4], is a popular example of M D A P S K modulation. The single constellation for 16DAPSK is shown in Fig. 2.8. It consists of 2DASK with absolute amplitude  2.3 Differential Modulation and Detection  18  Figure 2.8: Signal constellation for 16DAPSK.  {r ,r } L  H  and 8DPSK with  > [k]e{e^ \l  e  l  p  The amplitude ratio p =  TH/TL  >  {0,1,-••,7}}.  (2.24)  1 is the most significant parameter of 16DAPSK. The  absolute amplitude can be expressed by p as  (2.25)  l + p  2  2p  2  (2.26)  l + p  2  where the normalization  z{Ak]} = \(rl + r ) = 2  H  has been taken into account.  (2.27)  One out of four bits is carried by 2DAPSK. This one bit  is mapped to amplitude difference symbol v [k]. The bit "0" causes no amplitude change a  2.3 Differential Modulation and Detection  19  [a[k] = a[k — 1]) and the bit "1" causes an amplitude change (a[k) =^ a[k — 1]). The remaining 3 bits are carried by 8DPSK. They are Gray mapped onto the phase difference symbol v [k], according to this rule: (000)H->1, (00l)t->e / , ( O l l ) ^ ' ^ , jn 4  7  p  ( H O ) ^ e ^ , (lll)^e- / , j3n 4  (101)^e- / , jn 2  (100)^ -^/ ,  2  (010)^e ' ^ , J  3  /4  • •  4  e  In the first stage of the receiver, differential decoding involves as shown in Fig. 2.9. The decision variable d[k]  is generated.  A  r[k] r[k - 1]  (2.28)  Amplitude and phase difference symbols are estimated separately in the  receiver, since the amplitude and phase modulation are independent. d[k]  r[k]  m  Figure 2.9: Block diagram of the conventional differential detector for M D A P S K .  In 16DAPSK, the phase decision rule is the same as that employed for conventional D D of 8DPSK. For the amplitude decision of 16DAPSK, the arithmetic mean is used for decision thresholds [4]:  7o 7i  1 2(l + l / p ) ' 1 2(1 +p)"  (2.29) (2.30)  The above decision thresholds are not optimum. For a given channel and a given ring ratio, we can optimize 70 and 71 together with ring ratio p by numerical evaluation of the  20  2.3 Differential Modulation and Detection  analytical expression for the B E R provided i n [10]. So, the estimated amplitude difference symbol becomes  f 9: \m\>iv Aa[k]=l  1  :  7o<ffl|<7i,  (2-31)  [ 1/p, \d[k}\ <.  7o  2.3.3  Differential Unitary Space-Time Modulation  In the case of multiple transmit antennas, D U S T M is proposed by Hochwald et a l . [5] and Hughes [6], independently. In D U S T M , cf. F i g . 2.10, the transmitted signal m a t r i x at each  V[k]  S[k]  <8>  o  NT T  Figure 2.10: Structure of the differential encoder for D U S T M .  time block is the product of the previous transmitted m a t r i x and the current unitary data matrix:  S[k] = V[k]S[k-l].  (2.32)  T h e transmitted matrix symbol  S[k} = dmg{s [N k},s [N k Q  T  1  T  +!],••• ,s ^[N k NT  T  +N  T  where diag{ • } denotes a diagonal matrix. >S[0] = I J V , where J J V t  identity matrix.  T :  - 1]},  (2.33)  stands for NT X N  T  2.3 Differential Modulation and Detection  21  The constellations for unitary space-time modulated signals proposed in [5] and [6] form groups under matrix multiplication, thus simplifying the differential encoding process. In particular, the diagonal cyclic group constellations proposed in [5] are systematically designed for an arbitrary number of transmit antennas while providing full rate. The unitary data matrix  2„  V[k] = d i a g j e ^ " , e -r \ •/• • , e ~? , 0  j2  u  where 0 < l[k] < L — 1, L = 2  j2  N t R  Um  1i] l  k  (2.34)  with data rate R. For diagonal signals only one antenna  is active at any time. The transmitted phase-shifted symbols V[k] are potentially different for each antenna due to different values of u . The coefficients UQ. = 1, u , 1 < m < NT — 1 m  m  can be obtained by an exhaustive computer search to maximize the diversity product  C=  Y[sin(Tru l[k]/L)  min  where, u  m  (2.35)  m  l[k]€{l,-,L-1}  ™  —1  G { l , 3 , ••• ,L/2] and l[k] G { l , 3 , ••• ,L/2 — l } . The search results of u , m  1 < m < N — 1, of NT = 2 and NT = 3 are given in Tables 2.1 and 2.2, respectively. In T  order to yield a low B E R , the distance between the two nearest neighbors I, is determined using the method in [11] to generate a Gray mapping for a given signal constellation. The Table 2.1: D U S T M parameters for JV = 2. r  R=3  R=4  R=5  u  t  I  u'i  I  19  1  75 *  1*  19*  99  75  .. 27*  R=6 I.  449*  2*  Ui  I  1557  171  1731*  7*  2.3 Differential Modulation and Detection  22  optimum value of I is also shown in Tables 2.1 and 2.2. The values with " * " are actually used for the simulations in Chapter 5. Table 2.2: D U S T M parameter for N  T  = 3.  R=4  R=3  I  R=5  u  I  2  119  221  185  1735  889  14  139  157  75  1737*  961*  158*  185  75  1  u  I  11375*  3*  2  6921*  In D D at the receiver, the channel H[k] is unknown and assumed to be approximately constant during two adjacent blocks. The decision is based on the observation of R[k — 1] and R[k],  R[k]  4  R[k - 1] R[k]  (2.36)  In this case, R[k] can be expressed as  R[k] = S[k]H[k] + N[k],  (2.37)  where  S[k)  N[k]  S[k - 1] S[k]  N[k - 1] N[k]  (2.38)  (2.39)  2.3 Differential Modulation and Detection  23  It is shown in [11] that the probability density function (pdf) of R[k] conditioned on S[k], p(R[k]\S[k]), is independent of S[k - 1]. By maximizing p (R[k}\S[k\), the decoding metric of V[k] is  V[k] = argmin R[k] - V[k]R[k - 1]  (2.40)  V[k] where || • || denotes the Frobenius norm. V[k] is the estimated symbol, and V[k] is the trial symbol. The decoding metric can be either of quadratic form, or correlation form, or of minimum distance form, which are all equivalent.  2.3.4  Space-Time Codes with D A P S K  In D S T M and D S T B C , all the codes have the same norm. This implies that these spacetime signal constellations are extensions of PSK, which is of limited bandwidth efficiency. To achieve a high bandwidth efficiency, Xia [8] proposes S T C - D A P S K for Alamouti's orthogonal space-time code (OSTC) [12] with two transmit antennas, cf. Fig. 2.11.  y [*l fl  a[k]  S[k]  VJk]  P[k]  Figure 2.11: Structure of the differential encoder for S T C - D A P S K .  2.3 Differential Modulation and Detection  24  For S T C - ( M i + M ) D A P S K , let R = l o g M i , R = l o g M . R + R 2  1  2  2  2  2  x  2  bits are mapped  to one Alamouti codeword matrix [12],  v [2k]  v^[2k]  v [2k]  -v* [2k}  P0  V [k] = p  n  where v [2k] e A , Pi  Mi  (2.41)  po  i e {0, l}, is the M;PSK signal,  | ^ ' | z e  { 0 , 1 , 2 , . . .  ,M,-l}|.  (2.42)  The [Ri + R + l)th bit is introduced to decide whether there is an amplitude change in the 2  two successive transmit symbols. The bandwidth efficiency is [Ri + R +1) /2 bits/(channel 2  use).  Figure 2.12: STC-24APSK signal constellation.  Specifically, in STC-24APSK, there are two independent 8PSK and 16PSK constellations and one 2ASK, with p = r /rL, H  as shown in Fig. 2.12. v [2k] and i> [2/c] are picked po  Pl  up from the sets A& and Aie, respectively In 2ASK, the (4 + 3 + l)th bit is mapped to v [k] = p l ] , where Aa[k] G { 0 , ± l } . Aa  a  fc  By using differential encoding , cf. Fig. 2.10, we  2.3 Differential Modulation and Detection  25  have  (2.43)  S[k] = a[k]P[k], a[k] = a[k — 1] • v [k], a  P[k] = where P[0] = IN , a[0] = r . T  L  P[k-l]-V [k], p  (2.44) (2.45)  In each two time steps, we first decide upon the amplitude  ring, and then upon the codeword matrix P[k]. In particular, if the r# ring is selected, then the signal constellation is drawn from 16PSK. Otherwise, an 8PSK signal is used. In the next step, the amplitude may change and a different signal alphabet is used. On average, there are a total of 4 + 3 + 1 bits carried in 2 time slots. Hence, the bandwidth efficiency is 4 bits/(channel use).  The single receive antenna case is considered in [8]. Here, the received signals in the 2NT time intervals form the following 2 x 1 vector:  r[k] =  r [2k] 0  r [2A; + l]  (2.46)  0  Thus, we can get the receive signal  r[k] = S[k]h[k] + n[k],  (2.47)  where  h[k]  n[k] =  h [2k] 0  hiftk]  n [2k} 0  n [2k + 1] 0  (2.48)  (2.49)  2.3 Differential Modulation and Detection  26  There are two steps in the differential decoding. The first step is to detect the value of the (Rx + R + l)th bit, "0" or "1", by 2  Aa\k] = arg  min  ^Aa[k]  r[k]  r[k - 1]  (2.50)  Aa[fe]e{0,±l}  If Aa[k] = 0, the (R\ + R + l)th bit is 0. Otherwise, it is equal to 1. After deciding on 2  the bit to be carried by 2ASK, symbols v [2A;], v [2/c] are detected by Po  v [2k] = arg  max  P0  Pl  M{ (r* [2k]r [2{k - 1)] + n[2k}rl[2(k - 1)]  v [2k] },  (2.51)  m frS[2fc]ri[2(fc - 1)] + n[2A;]r^2(A; - 1)]) v [2k] \.  (2.52)  0  0  Po  v [2k]€A PO  v [2k] = arg  Mo  max  Pl  v [2k\€A vl  Ml  Pl  IV  /  I  Since Alamouti's O S T C is orthogonal, the decoding complexity of S T C - D A P S K remains similar to that of a single antenna system.  Chapter 3 Differential Amplitude and Phase Space-Time Modulation In this chapter, we propose a D A P S T M scheme based on diagonal signals, originally proposed for D U S T M [5], [6]. For D A P S T M , two differential detectors, SH-DD and M L - D D , are investigated. A n improved noncoherent receiver, MSD for D A P S T M is also introduced. To reduce the computation complexity of the MSD, the decoding metric is simplified to D F - D D . In SH-DD, we assume the channel is unchanged in the two successive symbols. In M L - D D , M S D and D F - D D , the channel is assumed to be constant in one symbol interval, but the second order statistics of fading and noise are yet to be known.  3.1  DAPSTM  Fig. 3.1 shows the block diagram of the proposed D A P S T M transmission scheme. consider a transmission scheme comprising N  T  transmit antennas and N  R  We  receive antennas.  The transmitted signals are organized in the square matrix S[k], k — 0,1, • • • , with element  SmWrk + /«] in row K, 0 < n < NT — 1, and column rn, 0 < m < N — 1. The transmitted T  signal matrix at k  th  time block  27  3.1  DAPSTM  28  S[k] = A[k]P[k],  (3-1)  consists of an amplitude matrix symbol A[k] and an unitary phase matrix symbol P[k]. The data rate R = RA + Rp, where RA and Rp are the rate of the bits mapped to phase and amplitude symbols, respectively.  A[k]  S[k]  NT T  V [k]  P[k]  P  NT T  Figure 3.1: Structure of the differential encoder for D A P S T M .  The phase matrix symbol P[k] is obtained through differential encoding from P[k — 1]  and Vp[fc]: P[k] =  For Vp[4  V [k]P[k-l].  (3.2)  P  we restrict the phase signals to constellations whose elements form a group  under matrix multiplication; that is, the possible values for Vp[fc] and P[k] belong to a finite set with LP = 2  NtRp  elements [13]. In particular, we are interested in the diagonal  constellations where the phase difference matrix symbols VV[/c] = VA/ [fc] are drawn from P  the set [5] [6],  A = {V  = diag <j e^ , U0  Alp  e^TP  Alpe  jo, 1 • • • L P - 1 | | .  (3.3)  3.1 DAPSTM  29  where j = y/—l denotes the imaginary unit, the Alp[k] refers to the information-carrying < LP — 1, 0 < m < N — 1 is a  phase difference symbols and the coefficient u , 0 < u m  m  T  design parameter. Note that the matrix V [k] and P[k] are uniquely associated with the P  symbol A/p[/c] and lp[k] by  .  V [k]  = V  P  ,  P[k]  (3.4)  Alp[k] P  = V  \  (3.5)  lp[k P  respectively, where Vp = Vp[l], lp[k] represents the absolute phase symbols.  Through  differential encoding, we have  p[k] = V [k]P[k P  =  v  A p  - 1]  'p[ ]+'p[ f c  f c _ 1  ]  (3  The amplitude matrix symbol A[k] is obtained by differentially encoding from VA[/C] and A[k — 1],  A[k] = V [k]A[k-l}.  (3.7)  A  The possible values for V^[fc] belong to a finite set Av with LA = 2  NtRa  A  elements [13].  The alphabet A A of the absolute amplitude matrix symbols A[k] depends on Ay . V \k\ A  A  and A[k] are also diagonal matrices, and given by  V [k] A  A[k]  = p  I ,  (3.8)  Ala[k]  =  (  NT  Ji g|p( a  A2a  [ l+' [ - l+ o)modLA fe  a  fc  1  0  p(Ala[k]+la[k—l]+6 )mod m  p(Ala[k]+la[k—l]+9i)mod  LA  LA  ^(AJa[fc]+/a[fc-l]-|-0 v -i)mod L A | ^ | - Q J J  T  ^3  3.1  DAPSTM  30  In the above equations, the following definition is used. A[0] = QIN , T  with scalar Q > 0.  The information-carrying amplitude difference symbols Ala[k] belong to {0,1, • • • , LA — l}. The la[k] represents the absolute amplitude symbols. The design parameter 6 , 0 < 8 m  m  <  N — 1, determines the initial amplitude value for each antenna. Another important design T  parameter p > 0 stands for the magnitude ratio between two amplitudes of the neighboring symbol matrices. Since the A[k] and P[k] are diagonal matrices, we can define them as  Po[N k}  0  T  0  P[k] =  [N k  Pl  0  + l]  T  0  0  0  a [N k]  0  0  0  a^Nrk + l]  0  0  0  0  A[k} =  T  p -![N k NT  (3.10)  + N - 1] J  T  T  aN^Nrk  (3.11) + Nr - 1] J  Using Eq. (3.1), the transmitted signal matrix S[k] also becomes a diagonal matrix:  s [N k] 0  S[k] =  0  T  0  s^Nrk + 1]  0  0  0 0  s ^[N k NT  T  (3.12)  +N  - 1]  T  where  S [N k m  with 0 < m < N  T  T  + m]=  — 1. s [N k m  T  0TP rn(^lv[k}+lp\k-l\) u  e  .  (Ala[k]+la[k-l}+e )modLA^  gp  m  (3.13)  + m] is transmitted by the rath antenna, 0 < ra < NT — 1  at time NTk + m. We transmit over NT antennas in NT time steps to achieve full rate. The  3.2 Conventional Differential Detection  31  transmitted symbols are normalized to  (  J V - l r  \s [N k m  + m]  T  (3.14)  = 1,  K m=0  which means  (3.15)  7^ ((P )° + (P ) + (P ) + • • • + (P )^- ) = i , 2  2  1  2  0  2  1  and hence we have,  1A Q =  (3.16)  lLA-1  £  (p ) 2m  m=0  For given N  T  and data rate R, the design parameters p, u  m  and 9  m  are optimized to  achieve best performance, cf. Section 5.2.  3.2  Conventional Differential Detection  In conventional D D , two consecutively received signal matrices, R[k] and R[k — 1], are stacked to form a matrix R[k] with 2NT rows:  R[k] =  R[k - 1] R[k]  (3.17)  By using Eqs. (2.9), (2.10), (2.13), (3.12), R[k] can be expressed as  R[k] = S[k]H[k] + N[k],  with the definitions  (3.18)  3.2 Conventional Differential Detection  32  S[k] =  H[k]  S[k-1]  O  0  S[k]  (3.19)  NT  N T  H[k - 1]  =  (3.20)  H[k]  N[k - 1]  N[k] =  (3.21)  N[k]  By substituting Eqs. (3.2), (3.7) into Eq. (3.1), we can get  S[k] = V [k]A[k A  - l]V [k]P[k P  - 1].  -  (3.22)  Since A[k — 1] and Vp[fc] commute, by denning V[k] = V [k]V p[k], we can then write A  S[k] as  S[k] = V [k]V [k]A[k A  P  - l]P[k - 1],  = V[k]S[k-l],  (3.23)  which is desirable for D D . In the following, we derive two conventional noncoherent decoding methods, namely SH-DD and M L - D D (in the sense of maximum likelihood over two consecutive received signal matrices).  3.2.1  Simple Heuristic Differential Detection  SH-DD is a suboptimum means of detecting D A P S T M , based on the minimum distance between the received signal matrix and the product of the current information conveying matrix V[k] and the previously received signal matrix. The estimated signal matrix is given by  3.2 Conventional Differential Detection  33  V[k] = argmin  R[k] - V[k]R[k - 1]  (3.24)  V[k]  Trial matrix' V[k] is an NT X N  diagonal matrix,  T  v [N k} 0  0  T  0  V[k] =  vi[N k T  0  0 0  . + 1]  0  VNT-ilNrk  (3.25) + Nx - 1]  where Vm[N k T  + m] = ip j  W  .  UmAip  e  (3.26)  Ua[k]^  p  with 0 < m < NT — 1. The decision rule can be written simply as  {  N -1 R  N  T  -1  5^  \rn[N k + m] T  -v [N k m  T  + m]r [N (k n  T  - 1) + m]\  },  (3.27)  where Ala[k] and Alp[k] denote the estimated amplitude and phase difference symbols, respectively, while Al~a[k] e {0,1, • • • , LA - l} and Al~p[k] G {0,1, • • • , LP — l} present the corresponding trial symbols, respectively.  3.2.2  Maximum Likelihood Differential Detection  Since h [k], 0 < m < N mn  T  — 1, 0 <n < N  R  — 1, and n [k], 0 < n < N n  R  — 1 are zero-mean  complex Gaussian random processes, the pdf of R[k] conditioned on S[k] is given by [11]  expf - tr  (^[QC^lQRlk])  p(fl[A;]|5[fc]) =  (3.28)  N  R  ^7r^det{c [fc]} fl  3.2 Conventional Differential Detection  34  Here N = 2 stands for the observation window size, [ • j  denotes Hermitian transposition,  and det{ • } and tr{ • } denote the determinant and the trace of a matrix, respectively. C [k] R  denotes the conditional covariance matrix,  C [k] R  =S^R[k]R [k]\s[k}^  (3.29)  H  Because of the mutual independence oiH[k] and N[k], and the uncorrelatedness of  n [N k+ n  T  m) in space and time, C [k] can be rewritten as R  C [k] = S[k}£ JH[k]H [k]}  S [k] + £ ^N[k]N [k]  H  H  R  S[k]C S [k]  +  H  H  H  Na I ,  (3.30)  2  R  n  2NT  with  C ±s{&[k]H [k]}. H  (3.31)  H  Due to the spatial uncorrelatedness of h [-], CH can be rewritten as mn  C  H  =  N (c ®I ), R  h  2NT  (3.32)  where <g> denotes the Kronecker product [14] and Ch is defined as  C  h  4  <Phh[-N ] T  ip [0] hh  1  Vhh[N ]  Vhh[N ]  1  T  T  -  (3.33)  ifhh[- ] is given by Eq. (2.3). By substituting S[k] with Eq. (3.1), and taking into account that S[k], P[k], and A[k] are diagonal matrices,.we obtain  3.2 Conventional Differential Detection  C [k]  35  = P[k}A[k]C A [k]P [k] H  R  =  P[k] (A[k]C A [k]  =  P[k]M[k]P [k],  No I  +  H  H  2  R  N 2NT  P [k)  +NaJ )  H  2  H  R  H  2NT  (3.34)  H  where  P[k-  P[k} =  (3.35)  P[k]  T  A[k-  A[k]  M[k]  1]  Ojv  1]  ON  (3.36)  T  A[k]  = A[k]C A [k]  NOI  +  H  H  2  R  2NT  A [k - 1] + O I  = N,R  HH  Nt  ip [N ]A[k]A[k HH  T  A [k] +  - 1]  T  - 1]  <p [N ]A[k]A[k  2  2  OI 2  2  (3.37)  NT  By using the matrix formula  det  A  B  C  D  CA^B},  | = det{A}det JZ> -  the determinant of iW[A;] is T  = N 2 N R  ( 1 1 l[{o*  n n  N -1  det{M[k}}  JJ  m=0  L  + a [N (k-Z)  + m]}  2  T  £=0  1  N -l T  =  (  n m=0  1  n{^+(P I,  2 A H K ]  )^ [^(fc-i)+m]} 2  £=0  a [ i V ( ^ - l ) + m] 2  T  n=0  (3.38)  36  3.2 Conventional Differential Detection (  N -1 T  N  R  N  IJ  T  1  n{^+p  \  I  m=0  fc-l]+0 )mod LA m  2 ( 5 A M f c i + M f c  £=0  1  LA  (3.39)  Now, the determinant of C [k] can be calculated to R  = det{p[fc]}det{M[fc]}det  det|cR[A;]|  =  {P^[A;]}  det{M[fc]}.  (3.40)  From Eqs. (3.39), (3.40), we observe that det{Cp[&;]} depends on Ala[k] and the previous absolute amplitude symbol la[k — 1]. We also obtain  -c [k]=p[k]{-M[k]y p [k] i  1  H  R  = P[k}T[k]P [k],  (3.41)  H  with the definition T[k]  =  -M~ [k] l  too  0  0  tn  •••  0  0  •••  0  tjv _i./v -l  0  •••  0  tN 0 T  0  0  tjV +ll T  •••  0  •  -  r  0  T  0  T  t  1Nr  0 tw N T  0  t2N -lN -i  0  0Nr  T  0  -  t  0  T  0  •••  0  •••  0  0  •' •  tN +lN T  •••  T  ' •'  t^ -i2N -l T  0  •••  U  T  0  t N -12N -l 2  T  T  (3.42)  3.2 Conventional Differential Detection  37  where _  _|_ p2(Ala[k]+la[k-l]+e )mod m  LA  (3.43)  . ^ 2 _|_ p2{la[k-l]+9m)mod LA^j a  t(N +m)(N +m) T  (3.44)  —  T  ^ ^ ^p{Ala{k\+2la[k-\\+2e )raod hh  T  LA  m  (3.45)  With  = fl  1?  {°n  €=0  tice that  + p  A  l  a  ^  +  l  ^  a  k  - ^  +  e  ^  m  o  d  -ip [N ] 2  }  LA  hh  T  "  fl  p2{ZAla[k}+la\k-l}+e ) od m  m  LA_  N  q  _  £=0  £ (jv +m) m  ^  2  T  =  t  (  A  r  T  +  m  )  m  ,  for 0 < m < N  T  - 1.  Also we have the relation [11] 2vr Pm[N {k - 1) + m}p* [N k + m] = exp j — u A / p [ A ; ] LP T  m  T  (3.46)  m  So we can further simplify the numerator of Eq. (3.28)  -tr =  [R [k]C^[k]R[k] H  -tr  (R [k]P[k)N P [k)R[k}) H  H  T  N -x  N ~i  T  i  R  r  - E E E 771=0  71=0  f=0  CN ~1 T  +  23? \J2  TTI=0  I,  N -1 R  E  71=0  *m(^  r + m )  e ' - ^ ^ W • r*[AT /c + m}r [N (k r  n  T  - 1) +  m  ]  (3.47)  Maximizing p(il[fc]|S[A;]) is equivalent to maximizing ln(p(J?[A;]|S[A;])), where ln( • ) refers to the natural logarithm. Therefore, we can get the decision rule  3.3 Multiple-Symbol  Detection  38  Ala[k], Alp[k] ( 1  N -1 T  =  arg  max ^  i l l {n + p a  Ala[fc],Alp[k] ^  ^  ^ * ^ -  2  l ( ^ [ A T y ] JJ  -  ^ ™ *  1  L A  }  p K '" [ ]+Hfc-l]+^m)mod LA 2  A  a  f c  5=0 N -1N -1 T  I f  R  + Y Y Y  m=0 n=0 5=0  {  2  \ HtNT+mXtNr+m)  I  n[N (k  r  T  \  - 1 + f) + m] >  J  JV -1N —1 T  ^  ^  R  ^  t iv m (  T + m )  e(^  U m A r p [ f c l  ) • r*[N k + m]r [N (k T  n  T  m=0 n=0  -l)  + m} \ I,  J J  (3.48)  where A/a[/c] e {0,1, • • • , LA - l } and A/p[fc] G {0,1, • • • , L P - l } . This decision rule depends on la[k — 1], which is estimated from the previous estimated amplitude difference symbol as  la[k-l]  =  (Y [x\)  =  Ala[k - 1] + la[k - 2],  Ala  modLA  X=0  (3.49)  with Ala[0] = 0.  3.3  Multiple-Symbol Detection  The observation window of M L - D D can be generalized from 2 to N, N > 2, consecutive received signal matrices, which leads to multiple-symbol detection (MSD) [15]. M S D makes a joint decision on N — 1 symbols based on N received matrices. These N consecutive received matrices are stacked to form a matrix R[k] with NNT rows to obtain  3.3 Multiple-Symbol  Detection  39  R[k-N+ R[k]  =  R[k  1]  — N + 2]  (3.50)  R[k]  For this, R[k] can be expressed as  R[k] = S[k]H[k] + N[k],  (3.51)  with the definitions S[k-N+1] S[k]  0  =  N  T  0  N  S{k-N  T  + 2]  O  O  H[k H[k]  N[k]  =  =  (3.52) S[k)  Nt  J  N + 1]  —  H[k - N + 2]  (3.53)  JV[fc - JV + 1] JV[fc - TV + 2]  (3.54)  N[k] The amplitude matrix becomes  A[k - N + 1] A[k] =  0  N  T  ON  T  with the elements of A [k — £] given by  O  0]sr  NT  A[k-N  T  + 2]  (3.55) Q  N  T  A[k]  3.3 Multiple-Symbol  a[N (k-£)  Detection  + m]  T  Ala[k-v]+la[k-N+l]+O  mod LA  m  = p\"=«  0<£,<N-2,  (3  The channel covariance matrix  C  =  H  N (c ®I ) R  h  NNT  (3  :  with  C  h  <Phh[0] Whh[-N )  T  <Phh[0]  T  A  L y [-{N hh  <p [(N - 1)N ] <p [(N - 2)N ]  Vhh[N )  hh  T  hh  T  - l)N ]  (3  fhh[0]  T  The matrix T[k] can be represented as - l  4  T[k]  {-(A[k)C A [k].+  N all )}  H  H  too  R  NNT  •••  0  to(N-l)N  0  0  •••  0  0  0  0  0  T  tN „-1N -1 NT  0  T  0  t(N-l)N 0 T  0  T  T  NNr  0  T  0  0  t NN -1N -1  0  T  t(N-l)N (N-l)N  0  0  t^ ^i _i  0  T  £jVJV - I . / V J V - I T  T  (3  Also we have the relation [11]  ^ JJ  2  p [N (k m  T  -  + m]p* [N (k - &) + m] = m  T  ( 2TT exp (j—u Alp[k m  \  - u]) .  (3  3.4 Decision-Feedback Differential Detection  41  When H[k] is unknown, the pdf of R[k] conditioned on S[k] is given by Eq. (3.28) with N > 2. Following the same step as M L - D D for N = 2, we get the decision rule of MSD  (Ala[k],Alp[k]j, = arg  f  • • • , (Ala[k -N  5S[*D-  (Afa[k},Aip[k}) (Al~a[k-N+2},A.fp[k-N+2}) N i < T--1i N - 1 T  R  N-l N-1  + EE E m=0  f  n=0 5=0 N  T  - l N  R  d  e  t  {  W  A  T  - l N - l  N *>I } R  N-1  NNT  5  2  2  H ^ r  +  m  + m]r [N (k-£ )  1  +  W  H  J  1=  n  A  ^  E E EE  r* [N (k-Z )  »  ,  K m=0 n=0 5 0 5 =5i+l  •  C  + 2], Alp[k — N + 2]  n  T  m  , N  JI e ( ^ - ^ * - ] )  i/=5i T  +  m  )  +m ] ^ ,  2  (3.61)  where {Ala[k], • • • , AZa[fc-A +2]} e {0,1, • • • , L.4-1} and {Al~p[k}, • • • , Al~p[k-N+2}} r  e  {0,1, • • • , LP — l}. This decision rule depends on la[k — N + 1], which is estimated from the previous estimated amplitude symbols in the same way as for M L - D D . M S D takes the dependencies among the received symbols into account and can reduce the irreducible error floor of M L - D D . Unfortunately, the above MSD decision rule requires the calculation of 2 ( ~ ^/(N R(N NtR  n  1  T  — 1)) metrics per bit decision, i.e., its computational complexity is  exponential in the number of transmit antennas N , data rate R, and observation window T  size N. A n alternative scheme with similar performance to MSD, but with a complexity almost independent of N, is subsequently derived.  3.4  Decision-Feedback Differential Detection  As mentioned in the previous section, we can reduce computational complexity by introducing decision-feedback [11], i.e., the Ala[k — v] and Alp[k — v\ are replaced by the previously  3.4 Decision-Feedback Differential Detection  42  decided symbols Ala[k - v] and Alp[k - u], 1 < v < N - 2, in Eq. (3.61). Therefore the decision is made only on Ala[k) and Alp[k}.  (Ala[k], Alp[k]) =  arg  \ detjA[k]C A [k)  max  H  H  A/o[fc],A/p[fc] I N -1  N -1  T  +  +  r  jV-1  E  5=0 I  E  L m=0  NNT  J  \ k^T+m^N +m)\r [N (k  E n=0  E  2  R  (  Yl  \  NOI \  ^  m=0  m  +  T  e  ^  U m 4  "' ' [  n  T  ~ 1 + 0 + ™] f  •<[iV fc + m ] r  l )  T  refin  [ A ^ ( f c - l ) + m]l  L  -  (3.62)  JJ  n=0  with the reference phase signal  r et,n[ T(k  - l )  E  kiiNT+m^NT+m)  N  T  E  +  m  JJ exp f J —  U Alp[k m  - v] J • T [N (k n  T  - £>) +  77l].  (3.63)  The (iV A; + m, iV A; + m)th element of A[k] is T  r  ( £  a[N k + m] T  ia[fc-t/]+Za[fc-iV+l]+0 +A/a[A:] (mod LA m  = p\"=<  /  .  ;  (3.64)  and the (N (k — £) + m, N (k — £) + ra)th element of A [A;] is T  T  N-2 „ JJ la[k-u]+la[k-N+l]+e  \ mod LA  m  a[N (k-£)+m]  ' = p\-e  T  J  ,  (3.65)  with 1 < £ < N — 1. The trial symbols Ala[k] e {0,1, • • • , LA - 1} and A/p[A;] € {0,1,--- , L P - 1 } . Now, only 2^ ^/(N R) N  T  metrics per bit decision have to be calculated,  3.5 Modifications for Time Diversity  43  i.e., the computational complexity is only exponential in N - and R. For the special case T  N = 2, both the MSD (cf. Eq. (3.61)) and the D F - D D (cf. Eq. (3.62)) decision rules are identical to the decision rule for M L - D D given in Eq. (3.48).  3.5  Modifications for Time Diversity  For D A P S T M to exploit space diversity, each transmit antenna is active only in every N th T  symbol interval, therefore; the effective fading bandwidth relevant for the receiver is  N BfT, T  instead of BfT, which has a negative influence on receiver performance [16]. In  time diversity, the effective fading bandwidth is BfT, independent of N . B  Time diversity  leads to a lower error floor. On the other hand, since an interleaver is used for time diversity, a transmission delay of N xNj B  is introduced in space diversity  is introduced, whereas only a delay of NT symbol periods  Chapter 4 Performance Analysis In this chapter, the performance of D A P S T M is analyzed for flat Rayleigh fading channels without spatial correlation. First, the P E P is evaluated based on SH-DD, since the calculation of P E P for MI-DD is too involved, if not impossible. A n approximation for B E R is obtained from the weighted PEPs.  4.1  Pairwise Error Probability  The P E P P (a, (3) is the probability of detecting V[k] = Vp[k] (Alp[k] = Alp [k], Ala[k] = e  Ala [k]),  p  when V[k] = V [k]  p  (Alp[k] = Alp [k],  a  transmitted. Here Ala [k], Ala [k\ a  Ala[k] = Ala [k]),  a  p  a  G {0,1, • • • LA-l}  V [k]  and Alp [k],Alpp[k] a  a  ^ V [k], is p  G {0,1, • • • LP-  1}. Using Eq. (3.27), it becomes a straightforward task to show that the P E P of D A P S T M with SH-DD can be expressed as  P (a,p) e  =  Pr<  R[k] - V [k]R[k p  - 1]  < \\R[k] - V [k]R[k a  44  - 1]  4.1 Pairwise Error Probability  N -1  N -1  R  T  Pr<{ E  Y  ri=0  \rn[N k + m} - v [N k T  <  n  - 1) + m]  T  T  Y  Y  n=0 • N -1N -1  + m] - v [N k  r  T  am  + m]r [N (k  T  n  T  - 1) + m]  m=0 ,  T  n=0  \ n[N k  m=0 ^  I  +  =  + m]r [N (k  T  N -1  R  R  0m  m=0  N -l  L  45  /  2  2  | r „ [ A V ( f c - l ) + m] ( ^ J i V V / c + m]  2  -  u [A^A; + m] Qm  \1  H<0  Pr {A(a,/3) < 0 .  where v [N k m  T  + m] =  (4.1)  ^ mAi u  e  P  . Aia[k]_  [k}  N -1N -1 R  A(a, /3)  H  p  ^ E E i< n [ ^ m=0  \vp [ Tk  x [k] mn  y [k-l] mn  = v [N k 0m  n  4  - 1] + x [ A : ] C ^ ^ J / c - 1]  f e [ f c  m n  Vmn[  k  ~ tyrant ~ 1]B„  v [N k am  am  +m  T  + m] - v [N k  T  = r [N k  m  + m]  N  m  m  A(a,f3) is defined as  e  ^  +  C  r  (  T  n=0  e  (4.3)  + m],  T  (4.4)  + m],  T  (4.5) (4.6)  r [ J V ( f c - l ) + m]. n  (4.2)  r  Using vector notation, A(a,f3) can be written as quadratic form  A(a,/?)  = =  with  xCy H  H  9 Fg, H  +yCx H  H  +  y By H  (4.7)  4.1 Pairwise Error Probability  r  A  g  =  T  TV  [x  (4.8)  y  ON N T  C  R  C X  A  0  H  (4.9)  B  r [N k]  =  46  r [N k  T  Q  + 1]  T  •••  r [N k 0  T  + N - 1]  r^Nrk]  T  -iT  (4.10) •••  r „ [N k NR  1  + N -l}\  T  ,  T  A  y  r [N {k  =  0  r [N (k-l)] n[N (k-i)] 0  r [N {k  T  0  T  T  B  =  I  C  =  I  N R  - 1) + N  T  T  - 1] (4.11)  + NT-l]  <g> diag | B ,  #i,  Bjv _i}, r  (4.12)  <g>diag{c ,  Ci,  CJV -I}.  (4.13)  0  0  N R  - 1) + 1} rx^lNTik-y  T  Since A(a,/3) is a quadratic form of Gaussian random variables, the two-sided Laplace transform  $A(a,/3)  ( ) of its pdf can be expressed as [16] s  $A(a,/,)(s) •loo  exp ^ - sg  H  (V  -1  (4.14) det(/  + ^ggF s  2N N T  R  where the definitions  £  =  ^{o},  (4.15)  4  -$)"},  are used. Since vector <7 has zero mean, g = 02jv jv xi) where r  0 vector.  $A(a,/3)( ) S  H  (4.16)  0 JV AT XI 2  T  R  is a 2N N T  R  x 1 all  can be simplified to  $A(a,/?)(s)  =  7  x-.  det I N N 2  T  R  + S^ggF )  (4.17)  4.1 Pairwise Error  Probability  47  We can calculate P ( a, (3^ from e  ( '' )  ft  for 0 < 7 <  9?{si},  where  Si  a  3  =  2^i,_  J O O  refers to the pole of  * ' M W  <&A(a,/3)  (4.18)  7  (s) which has the minimum positive  real part.  The integral in Eq. (4.18) can be calculated in closed form using the residues theorem [16]  P (a,/?)=c  Y  Residues ^  A ( Q ] / g )  (s)/s),  (4.19)  R H poles  where the summation is taken over all residues corresponding to poles located in the righthand (RH) side of the complex s-plane. Eq. (4.19) constitutes a very general method of calculating P (a,/3). We can easily get P (a,0) when the poles are simple. But it is very e  e  difficult to calculate the residues if <&A(S)/S has multiple poles or essential singularities. To avoid this problem, some approximations, such as the Chernoff bound [17] can be used to simplify the calculation. The bound takes the simple form  P (a,/?) < m i n $ e  o  i  A ( a  ,  / 3 )  (7).  (4.20)  The Chernoff bound is a general technique for upperbounding the PEP. But in most cases, it fails to provide a tight enough bound for error probability. Therefore, we favor a numerical calculation approach, which is based on Gauss-Chebyshev quadrature rules [17]:  Pe{<X,P)  =J7~Y1 G  k=l  ( { A(a,/3)(7 + i7^)} K  $  +^^{^A(a,/3)(7 + J7Tfc)}) •  '  + P 7 V  G  ,  (4.21)  4.2 Approximation for Bit Error Rate  48  where N is even, and 0 < 7 < 3?{si} and r = tan((2A: - l)n/(2N )) G  E  NG  k  vanishes for N  G  are valid. The error  G  —> 00. In practice, relatively small values for N  G  chosen properly [17]. For the numerical results in this section, N  G  can be used if 7 is  = 128 and 7 = !R{si}  are adopted.  4.2  Approximation for B i t Error Rate  The exact calculation of the B E R is quite involved. A simple approximation of the B E R is given by the union bound [18]  a=lp^a  where N{, — 2 Nt<  \  ra+rp  and Vp is detected.  and n(a,P)  b  denotes the number of bit errors if V  a  is transmitted  Pb is an upper bound for the achievable B E R for space-time coded  transmission. However, for moderate-to-large signal-to-noise ratios (SNR), the union bound becomes tight and is a very good approximation for the achievable B E R .  Chapter 5 Simulation Results  In this chapter, simulation and numerical results are presented and discussed.  We have  simulated a system as described in Chapter 3, i.e., independent flat Rayleigh fading channels are assumed between any pair of transmit and receive antennas. We restrict ourselves to consider only one receive antenna (N  = 1), since the main focus of this work is on transmit  R  (modulation) diversity. In the simulation, Eb/N  = l/(a R) 2  0  is used as the channel SNR,  where Eb is the total energy per bit used in the transmission. For the numerical evaluation, the results given in Chapter 4 are used.  5.1  Numerical Results Compared with Simulations  First, we compare the simulation and numerical results of PEP. For simulation SH-DD is applied, and for the numerical expression of P E P Eq. (4.21) is used. B E R vs. lOlogjoC-Eft/JVo) for N  T  = 2. R = 1.5 bits/(channel use) (RP = 1 bit/(channel  use), RA = 0.5 bit/(channel use)), B T f  0i = 1 are adopted.  Fig. 5.1 shows the  = 0.001, p = 2.1, u  0  = 1, m = 1, 0 = 0, and  So, Ala e {0, l} and Alp e {0,1,2,3}.  probability of detecting Vp, when V  a  O  V  a  —> Vp denotes the  is transmitted. Two particular pairs of P E P are  49  5.1 Numerical Results Compared with Simulations  + —B— -o-X-©-Q-  Simulation Numerical Simulation Simulation Numerical Simulation Numerical Simulation Numerical  (la=1 lp=3) -> (la=1 lp=3) -> (la=1 lp=3) -> (la=0 lp=3) -> (la=0 lp=3) -> (la=1 lp=0) -> (la=l lp=0) -> (la=1 lp=3) -> (la=1 lp=3) ->  10"  50  (la=0 lp=3) (la=0 lp=3) (la=0 lp=3) No Error Propagation (la=1 lp=3) (la=1 lp=3) (la=1 lp=3) (la=1 lp=3) (la=1 lp=0) (la=1 lp=0)  10  15  20  25  30  10log(E /N ) b  0  Figure 5.1: Comparison of simulation vs. numerical results of P E P . NT = 2, R = 1.5 bits/(channel use) (RP = 1 bit/(channel use), RA = 0.5 bit/(channel use)), B T = 0.001, p = 2.1, u = 1, ui = 1, = 0, and 6i = l,are valid. f  0  considered. The first one is the P E P between V i ( A / a = 1, Alp = 3) and V (Ala 3  03  = 0,  Alp = 3), where only an amplitude change occurs, and the other is the P E P between Vi3(Ala  = 1, Alp = 3) and V (Ala w  We observe that P E P ( V i , V 3  1 0  = 1, Alp = 0), where only a phase change occurs.  ) = P E P ( V , V^),  of the P E P match well with simulations.  1 0  and in this case, the numerical results  When the amplitude change happens, both  numerical and simulation results of P E P are not symmetric, i.e., P E P ( V , V03) ^ P E P 1 3  ( ^ 0 3 , ^ 1 3 ) in either numerical or simulation results. For numerical results, we assume the previous absolute amplitude symbols are known at the receiver, and no error propagation  5.1 Numerical Results Compared with Simulations  51  is considered, while error propagation actually occurs in the simulation. The numerical result for PEP(V~ , V 0 3 ) is lower than the simulation result. These two results match 13  remarkably well when we cancel the effect of the error propagation in simulation.  DC HI CQ  15  20  10log(Eb/N0)  Figure 5.2: Comparison of simulation vs. numerical result of B E R . NT = 2, R = 4 bits/(channel use) (RP = 3 bits/(channel use), RA = 1 bit/(channel use)), BfT = 0.001, p — 1.4, UQ = 1, ui = 15, OQ = 0, and 0\ — \ are valid.  Fig. 5.2 shows B E R vs. 10\og (E /No) w  Eq. (4.22).  for simulation and numerical evaluation using  b  Again, SH-DD is assumed.  N  T  = 2, R = 4 bits/(channel use) (RP  bits/(channel use), RA = 1 bit/(channel use)) and BfT  = 3  = 0.001 are adopted, p = 1.4,  UQ = 1, ui = 15, Oo = 0, and 0i = 1 optimized in Section 5.2 are valid. To make this comparison conclusive, we assume all the decisions of the previous absolute amplitude symbols are correct, i.e., no error propagation occurs in the simulation. It can be observed  5.2 Constellations Design  52  that the gap between the numerical result of the union bound and the simulation result gets smaller with increasing  E  B  / N  0  Thus, the numerical result becomes tight and is a good  .  approximation for the achievable B E R for moderate-to-large  5.2  E  B  /N  0  ranges.  Constellations Design  As mentioned in Section 3.1, the B E R performance depends on three parameters in our code designs, magnitude ratio p, p > 0, the initial amplitude value for each antenna 6 , M  0 < m < NT — 1, and coefficients u , m  0 < m < NT — 1, for the phase signal. It is necessary  to find the optimum combination of these three parameters to get the best performance. For a very simple case, N  T  RA  = 2, R = 1.5 bits/(channel use) (RP = 1 bit/(channel use),  = 0.5 bit/(channel use)), 0 and 6\ can be taken from the set {0,l}, and u , u\ is 0  0  drawn from the set {0,1,2,3}. We find the optimum values, u  0  = 1, u\ = 1 according  to the search method in [5], p = 2.1 and 9 ^ #i (9 = 0, #i = 1 or 6 — 1,0\ = 0) by 0  0  0  simulation. In more complicated cases, such as with multiple antennas or high data rates, there are many possible combinations of p, 6 , u . M  u, m  m  To optimize the parameters p, 0 , M  we need to calculate and compare the B E R for different combinations of parameters.  However, it is too difficult to take into account all the PEPs, which are used to calculate BER. Consequently, we turn to a suboptimum way, in which we use the maximum P E P as the metric. The P E P (cf. Eq. (4.1)) is derived for SH-DD, and the parameters resulting from the optimization also give good performance for M L - D D . Our goal is to find p, 9 , M  and u  m  satisfying  arg ;  (5.1)  mm p>0  0<«o,",^N -I<LP—1 O<0CV,0JV -I<£A-1 T  T  where (cf. Eq. (4.21))  5.2 Constellations Design  P e  (  a , / ?  )  =  W $ A ( a , / 3 ) ( 7 + J77-fc)} + r 9 { ^ A ( a , « ( 7 + J77jfc)} )•  AT £ G  53  (5-2)  f c  fc=i ^  '  Because each antenna is statistically.equivalent to the others, we may impose the ordering UQ <  « l  <  • • • <  ItjVr-l,  00  <  01  <  • • • <  07V -1T  It is impossible to get an explicit solution for the procedure described above; therefore, we resort to exhaustive computer searches.  The search space can be reduced using the  following rules:  1. u = l , u e {3,5,..., LP/2 - l}, where 1 < m < N 0  m  T  - 1 [5].  2. We restrict p > 1. If p < 1, it is effectively equivalent to another p > 1. 3. The phase difference symbol Alp[k] may be taken from the set {l, 2, • • • , LP/2 — l} [5], and the amplitude difference symbol Ala[k] e {0,1, 2, • • • , LA — l}. 4. We assume 6 = 0, 0  0AT -I = T  LA  — 1 and try to arrange the rest of the 0i, 0 , 2  • • • , 8N -2 t  equally spaced between 0 and LA — 1 to maximize the distance between any two elements in a diagonal amplitude matrix symbol. For the special case of i V = 2, r  RA = 1 bit/(channel use), we have 0 = 0, Q\ = 3. For the case of N O  T  = 3, RA = 1  bit/(channel use), we assign 0 = 0, Q\ = 3 and 0 = 7. O  2  Using the proposed procedure, we find the optimum values of the parameters for a given E /N b  0  and R = 3 bits/(channel use), R = 4 bits/(channel use) when N  T  = 2, cf. Table  5.1. In order to yield a low B E R for a given signal constellation, a kind of Gray mapping is desirable for the phase signal Alp[k] and the amplitude signal Ala[k], respectively. For this, the nearest neighbors of the phase signal for each matrix symbol are first determined, according to the minmax value of P E P (cf. Eq. (5.1)), which can be calculated by Eq. (4.21).  5.2 Constellations Design  54  Table 5.1: D A P S T M parameters for N  T  = 2 by minmax P E P .  R = 3(RP = 2,RA = 1) E /N (dB) 0  P  20  R = 4 (RP = 3, RA = 1)  Ml  la  lp  P  1.5  5  1  3  1.4  23  1  3  30  1.5  3  1  1  1.4  15  1  4  200  1.9  7  1  2  1.5  27  1  1  b  la  Ul  Assume each phase signal has two nearest neighbors with the same distance lp, and lp for all the phase signals are identical; that is the nearest neighbor symbols for any given Alp[k], 0 < Alp[k] < LP — 1, can be expressed as (Alp[k] ± lp)mod LP, and lp is unique. In this case, a Gray mapping may be constructed by assigning bit patterns to the phase signal in such a way that Alp[k] and its neighboring signals differ by only one bit. Otherwise, there may be constellations for which no Gray mapping exists. We managed to find lp for N  T  = 2 and R = 3,4 bits/(channel use) as given in Table 5.1.  However, following  the same procedure as for Alp[k], we could not express the nearest neighbors of Ala[k] as (Ala[k] ± /a)mod LA with a unique distance la. Here, we simply use Gray mapping with la = 1 for the amplitude signal mapping. Simulation results show that for A ^ = 3, RA = 1 bit/(channel use) this Gray mapping for the amplitude signal yields the best performance. Further, while NT = 2 and RA = 1 bit/(channel use), different mapping methods yield a similar performance.  Notice that the differential transmission schemes proposed in [5] and [6] appear as special cases of our scheme for RA = 0. Therefore, it appears to be reasonable to use optimum values of u , m  signal design.  lp for D U S T M (cf. Table 2.1, Table 2.2) as the suboptimum values in our This then provides an alternative means to the parameter optimization  method described above. By this means, we may lose a little performance quality, but the computational complexity is significantly reduced. Again, we assign 6 = 0 and 6i = 3 for 0  5.3 Space Diversity  55  the case of NT = 2 and RA = 1 bit/(channel use). Further we assign 9Q = 0, 9\ = 3, and #2 = 7 in the case of NT = 3 and RA — 1 bit/(channel use).  Once 9 , u m  m  and Ip ave  obtained, the magnitude ratio p, p > 1, can be found by  p = argmin< max P (a,0) >.  (5.3)  e  p>l  [<*,/3,a//3  J  Table 5.2 lists the optimized parameters using the two described methods. For NT = 2, R — 3,4bits/(channel use), the parameters are obtained by reduced-space-exhaustive-searching when Eb/No — 20 dB. For the remaining cases, we borrow the optimal parameters u  m  and  la from D U S T M and then find p, according to Eq. (5.3). We use the parameters in Table 5.2 for our simulations and analytical calculations in the following. Table 5.2: D A P S T M parameters for NT — 2 and N  N - 3  Ar =2 T  p  Ui  = 3.  r  T  fp  P  R=3 (RP = 2 , RA = 1)  1.5  5  3  R=4 (RP = 3, RA = 1 )  1.4  23  3  1.2  119  221  185  R=5 (RP =  1.2  75  1  1.2  1735  889  14  1.2  449  2  4  t  RA = 1 )  R=6 (RP = 5 , RA =  5.3  1 )  Space Diversity  In this section, we simulate the system with BfT = 0.001, without making special claims. As mentioned in the last section, the parameters for D A P S T M are adopted from Table 5.2. For D U S T M , the parameters with " * " in Tables 2.1 and 2.2 are drawn for simulation.  Fig. 5.3 shows the simulation results of D A P S T M and D U S T M with N  T  = 2 for R = 3  5.3 Space Diversity  56  bits/(channel use) and R = 4 bits/(channel use). D A P S T M is detected by SH-DD and M L DD, while D U S T M is only detected by M L - D D . At the rate of R = 3 bits/(channel use), the performance of D A P S T M with SH-DD is worse than that of D U S T M , while D A P S T M with M L - D D performs slightly better than D U S T M . When R = 4 bits/(channel use), D A P S T M with both SH-DD and M L - D D yields better performance than D U S T M . D A P S T M with M L - D D outperforms D U S T M by 3.3 dB at a B E R = lCT . For D A P S T M , M L - D D always 3  outperforms SH-DD; therefore, we use M L - D D in our simulation for D A P S T M .  0  5  10  15 20 10log(E /N ) b  25  30  35  0  Figure 5.3: B E R vs. 10\og (E /N ) for D A P S T M and D U S T M with R = 3 bits/(channel use) and R = 4 bits/(channel use). N = 2, N = 1 and BfT = 0.001 are valid. D A P S T M is detected by SH-DD and M L - D D ; D U S T M is only detected by M L - D D . 10  b  0  T  R  Fig. 5.4 shows the simulation results of M S D and D F - D D for D A P S T M with N  T  = 2  5.3 Space Diversity  57  at four different data rates (R = 3,4,5,6 bits/(channel use)). The results of M S D and D F - D D (N — 3) are compared with those of M L - D D (N = 2) at different fading velocities, including (a) B T f  = 0.005, (b) B T f  = 0.005, (c) B T f  = 0.0035, and (d) B T f  = 0.0025.  It is shown in Fig. 5.4(a) that D A P S T M detected by D F - D D and M S D performs almost equally well and can take full advantage of the enhanced diversity provided by multiple transmit antennas even for fast fading. All the figures in Fig. 5.4 show that D F - D D with observation window N = 3 can yield a significant gain over conventional D D at high Also in the E /N b  0 (C)  Q  E /N . b  0  range of interest an error floor can be avoided, even though both yield  10  20 10log(E /N ) b  30  40  0  0  (  d  )  10  20 10log(E /N ) b  30  40  0  Figure 5.4: B E R vs. 101og (£ /A o) for M S D and D F - D D with (a) R=-3 bits/(channel use), B T = 0.005, (b) R = 4 bits/(channel use), BfT = 0.005, (c) R = 5 bits/(channel use), B T = 0.0035, (d) R = 6 bits/(channel use), B T = 0.0025. N = 2, N = 1 is valid. 7  10  6  f  f  f  almost the same performance at lower Eb/N s, 0  T  R  E^/NQ < 30 dB. It is worth mentioning  5.3 Space Diversity  58  that the computational complexity of M S D is much higher than that of D F - D D . These observations are also in accordance with the results for D U S T M , cf.[ll].  In order to illustrate the diversity gain of D A P S T M for different numbers of transmit antennas, Fig. 5.5 shows the performance for R = 4 bits/(channel use) (RA = 1 bit/(channel use), RP = 3 bits/(channel use)) with NT = 1, NT = 2, and N  T  Eb/N  0  = 3. We observe that if  is low, D A P S T M with three transmit antennas yields the worst performance of the E^/NQ,  three schemes, while a single antenna has the best performance. With increasing  -| Q~  s  I  0  i  i  i  —  10  5  15  i  i  20  i  25  i  30  35  I 40  10log(E /N ) b  0  Figure 5.5: B E R vs. 1 0 1 o g ( £ / N ) for D A P S T M with N = 1, N = 2, and N = 3. N = 1, B T = 0.001, R = 4 bits/(channel use) (RA = 1 bit/(channel use), RP = 3 bits/(channel use)), p = 1.4, U\ = 23, Ip = 3 are used. ,  10  R  6  0  T  T  T  f  D A P S T M with multiple antennas starts to benefit from the diversity gain. The performance  5.3 Space Diversity  59  for two transmit antennas surpasses that for the single antenna at E^/NQ = 18 dB, while D A P S T M with three antennas yields the best performance at E^/NQ = 23 dB. Obviously, at high Eb/No B E R decreases as the number of transmit antenna increases. D A P S T M is especially effective at high E /N . b  This is consistent with the results in [5], where unitary  0  space-time signals are especially effective at high  Eb/N s. 0  Fig. 5.6 compares the performance of D A P S T M and D U S T M with various rates for NT = 2,3. It is observed that D A P S T M outperforms D U S T M when R > 3 bits/(channel use). The gap between D A P S T M and D U S T M increases with increasing the data rate and  0 (  a  10  20  30  lOlog(E /N )  )  b  0  40  0 ( b  )  10  20  30  40  l0log(E /N ) b  0  Figure 5.6: B E R vs. 101og (£ /JVo) for D A P S T M and D U S T M with (a) NT = 2, (b) N = 3. N = 1, B T = 0.001 are valid. 10  T  R  6  f  the number of transmit antennas. Specifically for NT — 2, we compare the two schemes  5.3 Space Diversity  60  from R = 4 bits/(channel use) to R = 6 bits/(channel use).  At a B E R = I O , D A P - 2  S T M outperforms D U S T M by 3 dB for R = 4 bits/(channel use) to 10.4 dB for R = 6 bits/(channel use). For NT = 3, when the data rate increases from R = 4 bits/(channel use) to R = 5 bits/(channel use), D A P S T M only degrades slightly, while D U S T M degrades by over 10 dB at a B E R = 1 0 . Since all signals in D U S T M are restricted to diagonal unitary -3  matrices, with increasing data rates, the minimum distance between two arbitrary signals becomes smaller, which leads to poorer performance. With D A P S T M we take advantage of amplitude so that the signals are not restricted to unitary matrices. Its minimum signal distance, which determines the B E R , is larger than that of D U S T M , given a fixed data rate. This is the major difference between our scheme and D U S T M , which allows us to increase the spectral efficiency by carrying information, not only on the phase, but also on the amplitude of a data matrix. The advantage of D A P S T M over D U S T M is more pronounced at higher data rates.  Fig. 5.7 compares the performance of D A P S T M and S T C - D A P S K with N  T  = 2. For  96STC-DAPSK with R = 6 bits/(channel use), two independent 64PSK and 32PSK constellations are used and the amplitude ratio p = 1.2.  For 48STC-DAPSK with R = 5  bits/(channel use), two independent 32PSK and 16PSK constellations are used and p = 1.4. For S T C - D A P S K with R = 4 bits/(channel use), two independent 16PSK and 8PSK constellations are adopted and p = 1.5.  Since our proposed D A P S M scheme is based on a  diagonal signal, it is not as power efficient as a nondiagonal signal constellation. We can see that the performance of D A P S T M is worse than S T C - D A P S K when NT = 2. However, the gap between D A P S T M and S T C - D A P S K becomes smaller with increasing the data rate. At B E R = 1 0 , the gap decreases from 2.0 dB for R = 4 bits/(channel use) to 1.4 dB for -3  R = 6 bits/(channel use). We expect that the gap to be narrower and for D A P S T M to surpass S T C - D A P S K at a certain higher data rate. Since S T C - D A P S K extends Alamouti's O S T C [12] from phase modulation to combined differential amplitude and phase modula-  5.4 Time Diversity  61  tion, we conjecture that S T C - D A P S K cannot achieve full diversity and full rate when more than two transmit antennas are employed. D A P S T M can achieve full diversity and full rate for an arbitrary number of transmit antennas. As shown subsequently, D A P S T M can be used to exploit time diversity when only one transmit antenna is employed. S T C - D A P S K , however, cannot due to its nondiagonal structure. 10  10  •A•A-6" -O-6-Q-  DAPSTM 96STC-APSK DAPSTM 48STC-APSK DAPSTM 24STC-APSK  10  R=6 R=6 R=5 R=5 R=4 R=4  10  15  20  25 b  Figure 5.7: B E R vs. 101og (£ //Vo) BfT = 0.001 are valid. 10  5.4  6  30  35  40  10log(E /N )  f o r  0  D A P S T M vs. S T C - D A P S K . N  T  = 2, N  R  = 1, and  Time Diversity  In this section, inter leaver length Ni = 250 is adopted to achieve an uncorrelated diversity branch in time diversity (TD) and M L - D D is applied for all the simulations. In Fig. 5.8,  5.4 Time Diversity  62  two different diversity schemes with NT = 2 for space diversity (SD), and NT = 1 and N  B  = 2 for T D are compared for independent diversity branches. Fig.5.8, (a), (b), and  (c) shows B E R vs. 10\og (E /N ) B  10  for R = 4, 5,6 bits/(channel use), with different fading  0  velocities, respectively. Clearly the major difference between SD and T D is the error floor. T D leads to a lower error floor than SD. For R = 4 bits/(channel), BfT = 0.005, SD yields  0 (  a  )  10  20 30 10log(E /N ) b  40  0  0  (  b  10  20 30 lOlogfE^/N,,)  )  40  0 (  c  10 20 30 10log(E /N )  )  b  40  0  Figure 5.8: B E R vs. lOlog (E /N ) for space diversity with N = 2 and time diversity with NB = 2. (a) R = 4 bits/(channel use), (b) R = 5 bits/(channel use), (c) R = 6 bits/(channel use) are valid. 10  B  0  an error floor of B E R = 7.7 x I O  T  - 4  at Eb/N  0  = 40 dB, while T D significantly decreases the  error floor to 7 x 10~ . The same applies to R = 5 and 6 bits/(channel use). The effective 5  fading bandwidth in T D is BfT,  which is lower than A ^ B / T in SD [16]. Since the error  floor increases with increasing fading bandwidth, a better performance can be achieved if  5.4 Time Diversity  63  T D is exploited instead of SD, especially for fast fadings.  10log(E /N ) b  0  Figure 5.9: B E R vs. 10\og (E /N ) for time diversity with NB — 3 and space diversity with N = 3. R — 4 bits/(channel use) is valid. 10  B  0  T  Similar observations as can be made from Fig. 5.8 can be made from Fig. 5.9, which is valid for NT = 3 and NB = 3. The performances of SD and T D with R = 4 bits/(channel use) are illustrated. SD exhibits error floors of B E R = 0.18 for B T f  = 0.02 and at B E R  = 3.3 x 10~ for BfT = 0.005. T D performs better than SD, resulting in lower error floors 3  of B E R = 8 x 10" and B E R = I O 3  - 5  for B T f  = 0.02 and B T f  = 0.005, respectively  Chapter 6 Conclusions and Recommendations 6.1  Conclusions  We have presented a differential amplitude/phase space-time modulation scheme based on diagonal signals over fiat Rayleigh fading channels when neither the transmitter nor the receiver has access to CSI. The proposed D A P S T M extends unitary space-time modulation and combines phase and amplitude modulation while keeping the group property. D A P S T M keeps the diagonal structure in both spatial and temporal dimensions and achieves full diversity and full rate.  We analyze the performance of D A P S T M with SH-DD. The performance of calculation results are consistent with the simulation results.  Compared, with the existing D U S T M proposed by Hochwald and Sweldens [5], we relax the restriction of differential phase modulation for data symbols to achieve significant improvement in error performance for high data rates.  We use the maximized P E P to  optimize the parameter in our signal design.  Compared with the S T C - D A P S K [8] proposed by Xia, we extend the S T C - D A P S K from  64  6.2 Recommendations for Future Work  65  two transmit antennas to any number of transmit antennas, while keeping full diversity and full rate to achieve high bandwidth efficiency Because of the diagonal structure of both phase and amplitude, D A P S T M can be used to exploit the pure time diversity while only one transmit antenna is employed. S T C - D A P S K , on the other hand, can achieve full diversity and full rate only when two transmit antennas are employed, and can only be used in space diversity. However, D A P S T M is not as power efficient as S T C - D A P S K when two transmit antennas are employed. Two corresponding DD schemes, SH-DD and M L - D D are derived for a conventional noncoherent receiver. To further reduce the loss of performance for fast fading, two improved noncoherent D D schemes, i.e., M S D and D F - D D , with lower complexity, are investigated in this thesis. .  6.2  •  .  .  '  " '  Recommendations for Future Work  The D A P S T M constellation design and validation have been presented in this thesis. However, there is still much work to be done before perfection is achieved. Recommendations for possible future work are as follows:  • Further reducing complexity of M L - D D . • Optimizing design parameters p, u  m  and 6  m  with lower complexity to achieve better  performance. • Combining of space and time diversity to yield high performance for a wide range of fading velocities.  Glossary Operators argmaxj • }  argument maximizing the expression in brackets  argminj • }  argument minimizing the expression in brackets  diag{ • }  diagonal matrix with diagonal entries of vector argument  <?{ • }  expectation  jftja;}, Sja;}  real and imaginary part of x  P{ •}  error probability  [•]  Hermitian transpose  e  H  [ •]  T  (• )* II  transpose complex conjugate  112  II • ||  the Frobenius norm  | •|  magnitude of a complex number  Sets A  signal alphabet  Z  integer numbers  N  natural numbers  K  real numbers  66  GLOSSARY  R  +  67  positive real numbers  Constants j  imaginary unit: j  TT  the number pi TT = 3.14159265358979  e  Euler number e = 2.718281828  I  n x n identity matrix  2  n  O  n  O xi n  = —1  n x n zero matrix n-dimensional all zero row vector  Other Functions det{ • }  determinant of a matrix  tr{ • }  trace of a matrix  exp( • )  exponential function  Jo ( •)  zeroth order Bessel function of the first kind  ln( • )  logarithm to base e  l°Sio(')  logarithm to base 10  tan( •)  tangent function  Acronyms ACF  Autocorrelation Function  AWGN  Additive White Gaussian Noise  BER  Bit Error Rate  CSI  Channel State Information  DASK  Differential Amplitude Shift Keying  GLOSSARY  68  DAPSK  Differential Amplitude/Phase Shift Keying  DAPSTM  Differential Amplitude/Phase Space-Time Modulation  DD  Differential Detection  DF-DD  Decision-Feedback Differential Detection  DPSK  Differential Phase Shift Keying  DUSTM  Differential Unitary Space-Time Modulation  DSTBC  Differential Space-Time Block Coding  PEP  Pairwise Error Probability  MIMO  Multiple-Input Multiple-Output  ML  Maximum Likelihood  MSD  Multiple-Symbol Detection  i.i.d.  Independent and Identically Distributed  OSTC  Orthogonal Space-Time Code  SH  Simple Heuristic  SNR  Signal-to-Noise Ratio  STC-DAPSK  Differential Amplitude/Phase Shift Keying  QAM  Quadrature Amplitude Modulation  Bibliography [1] B. Sklar. Rayleigh fading channels in mobile digital communication systems.  IEEE  Commun. Magazine, Jul. 1997. [2] S. Stein. Fading channel issues in system engineering. IEEE J. Select. Areas Commun., SAC-5(No.2), Feb. 1987. [3] J. G . Proakis. Digital Communications. McGraw-Hill, New York, 3rd edition, 1995. [4] W . T . Webb, L. Hanzo, and R. Steele. Bandwidth efficient Q A M schemes for Rayleigh fading channels.  Commun. Speech and Vision, IEE Proceedings I, 138:169-175, Jun.  1991. [5] B. M . Hochwald and W. Sweldens. Differential unitary space-time modulation.  IEEE  Trans. Commun., 48:2041-2052, Dec. 2000. [6] B. L . Hughes.  Differential space-time modulation.  IEEE  Trans. Inform. Theory,  46:2567-2578, Nov. 2000. [7] V . Tarokh and H. 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