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116 MAXIMUM LIKELIHOOD SEQUENCE DETECTION Mathematically, we form a metric M, = (r 1 -f 1 ) 2 + (r 2 -f 2 ) 2 + ..., (6.1) where the subscript q indicates a particular symbol sequence hypothesis. This metric is referred to as the Euclidean distance metric, because it is related to the distance between two points in geometry. Let's try it for the Alice and Bob example. We form predicted values using h = -lOtt+Qfl, (6.2) f 2 = -1092+9(7!. (6.3) For example, consider the hypothesis g () = +1, q\ = — 1, and q 2 = +1 (an incorrect hypothesis). Then h = -10(-1) +9(+l) = 19 (6.4) f 2 = -l0(+l)+9(-l) = -19. (6.5) Recall that r\ = 1 and r 2 — —7, so that the metric would be M q = (1 - 19) 2 + (-7 - (-19)) 2 = 468. (6.6) We would then need to consider all other hypothetical symbol combinations. All combinations and their associated metrics are shown in Table 6.1. Observe that the metric is minimized for q n — +1, q x = +1 and qi = +1. Thus, the detected symbol values would be s\ = +1 and s 2 = +1· Table 6.1 Example of sequence metrics Index Hypothesis Metric 9o 9i 92 ~1 +1 +1 +1 40 2 +1 +1 -1 680 3 +1 -1 +1 468 4 +1 -1 -1 388 5 -1 +1 +1 436 6 -1 +1 -1 1076 7 -1 -1 +1 144 8 -1 -1 -1 64 This approach is referred to as Maximum Likelihood Sequence Detection (MLSD). The "sequence detection" part of the name comes from the fact that we detect the entire symbol sequence together, rather than detecting symbols one at a time. The "maximum likelihood" part comes from the fact that the metric is related to the likelihood of noise values taking on certain values. Specifically, the log of the likelihood that a noise value n\ equals the number 7 is related to —7 2 or —49. The closer the number is to zero, the more likely the noise value. Thus, the highest log-likelihood value is —0 2 or 0. A block diagram for forming an MLSD metric is given in Fig. 6.1. Generation of the predicted received values is shown in Fig. 6.2. In our approach above, instead of maximizing the likelihood, we minimize the negative of the log-likelihood. For example, instead of maximizing — {r\ — f\) 2 , we minimize the Euclidean distance (r\ - r\) 2 .
MORE DETAILS 117 r 2 Ht, r, fcl + ft ► + w C) 2 —► + 4 C) 2 ir —► Figure 6.1 MLSD block diagram. 6.2 MORE DETAILS For the general dispersive case, we have cq m +dq m -i, (6.7) and, for ΛΓ Γ received values, we obtain Ma Nr-l (6.8) Suppose we know s () = +1. We can think of this as a single possibility. When we receive r\, there are now two possibilities, corresponding to q\ = +1 and q\ = —1. When we receive r 2 there are now four possibilities corresponding to the four possible combinations of q\ and q-¿. We can represent this growing number of possibilities with a tree, as shown in Fig. 6.3 (the tree is laying on its side). Each value of M q corresponds to a different "path" from the base of the tree to the top. Notice that paths share segments in the tree. This suggests a sharing of computations. Specifically, we start by considering Γχ, which introduces two branches corresponding to
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Channel Equalization for Wireless C
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To my colleagues at Ericsson
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CONTENTS List of Figures List of Ta
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CONTENTS XI 4.2 4.3 4.4 4.5 4.6 4.1
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CONTENTS XIII 8 Practical Considera
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xvi LIST OF FIGURES 2.2 Matched fil
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XviÜ LIST OF FIGURES 6.18 Scatter
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XX Prologue Alice was nervous. Woul
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XXÜ PREFACE is introduced as neede
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ACRONYMS AC A/D ADC AMLD AMPS ASK A
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ML MLD MLPD MLSD MLSE MRC MSE MSB O
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2 INTRODUCTION Table 1.1 Possible m
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4 INTRODUCTION s o S 1 8 2 S 3 S 0
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6 INTRODUCTION A block diagram of t
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8 INTRODUCTION Figure 1.7 QPSK. tra
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10 INTRODUCTION phases (phase modul
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12 INTRODUCTION rollo« 0.22 0.8 Ί
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14 INTRODUCTION is the "channel" re
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16 INTRODUCTION 1.4 MORE MATH In th
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18 INTRODUCTION For K = 1 (order 0)
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20 INTRODUCTION The K = 4 main bloc
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22 INTRODUCTION A sample of the noi
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24 INTRODUCTION The receiver may ha
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26 INTRODUCTION 1.5.1 Reference sys
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28 INTRODUCTION a) What most likely
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CHAPTER 2 MATCHED FILTERING In this
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MORE DETAILS 33 2.2 MORE DETAILS So
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THE MATH 35 Now compare the output
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THE MATH 37 The correlation in (2.2
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THE MATH 39 correlation function f(
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THE MATH 41 Thus, in the complex pl
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THE MATH 43 to the medium response
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THE MATH 45 We can interpret f(t) a
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MORE MATH 47 10" 1 Odeg 90 deg X 18
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MORE MATH 49 With despread-level Ra
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MORE MATH 51 Figure 2.6 OFDM exampl
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MORE MATH 53 2.4.3 MF in colored no
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PROBLEMS 55 period) is examined in
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CHAPTER 3 ZERO-FORCING DECISION FEE
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MORE DETAILS 59 1/c r m —► H i
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MORE DETAILS 61 where «i = m - (d/
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MORE MATH 63 Because there is no IS
Page 87 and 88: MORE MATH 65 In the later chapter o
Page 89 and 90: PROBLEMS 67 b) What is the detected
Page 91 and 92: CHAPTER 4 LINEAR EQUALIZATION With
Page 93 and 94: THE IDEA 71 estimate of symbol s 2
Page 95 and 96: THE IDEA 73 Notice that £2 (MSE) d
Page 97 and 98: MORE DETAILS 75 The first term is t
Page 99 and 100: MORE DETAILS 77 To obtain the unity
Page 101 and 102: THE MATH 79 and SINR becomes w T h
Page 103 and 104: THE MATH 81 To obtain the MMSE solu
Page 105 and 106: THE MATH 83 where *m n = ν ^ ^ ^ '
Page 107 and 108: THE MATH 85 power is much larger th
Page 109 and 110: MORE MATH 87 IQ' 1 OC 111 ω 10 2 1
Page 111 and 112: MORE MATH 89 values of h k m (f) fr
Page 113 and 114: MORE MATH 91 where C(d(),d 0 ) . C(
Page 115 and 116: AN EXAMPLE 93 between z and s. This
Page 117 and 118: PROBLEMS 95 Another aspect of cocha
Page 119: PROBLEMS 97 The math 4.11 Consider
Page 122 and 123: 100 MMSE AND ML DECISION FEEDBACK E
Page 137: CHAPTER 6 MAXIMUM LIKELIHOOD SEQUEN
Page 141 and 142: MORE DETAILS 119 Figure 6.3 MLSD tr
Page 143 and 144: THE MATH 121 legs of the journey is
Page 145 and 146: THE MATH 123 Figure 6.8 MLSD trelli
Page 147 and 148: THE MATH 125 We would then identify
Page 149 and 150: THE MATH 127 Expanding the square,
Page 151 and 152: THE MATH 129 Consider the matched f
Page 153 and 154: THE MATH 131 6.3.5.1 M-algorithm Wi
Page 155 and 156: THE MATH 133 results, DFE and MLSD
Page 157 and 158: THE MATH 135 IQ" 1 oc LU CD 10 2 10
Page 159 and 160: THE MATH 137 sometimes inaccurate w
Page 161 and 162: MORE MATH 139 POWER CONTROL effecti
Page 163 and 164: MORE MATH 141 15 POWER CONTROL 10
Page 165 and 166: MORE MATH 143 6.4.3 More approximat
Page 167 and 168: THE LITERATURE 145 EDGE is an evolu
Page 169 and 170: PROBLEMS 147 the channel to minimum
Page 171: PROBLEMS 149 c) Suppose we know «o
Page 174 and 175: 152 ADVANCED TOPICS detecting the w
Page 176 and 177: 154 ADVANCED TOPICS Table 7.2 Examp
Page 178 and 179: 156 ADVANCED TOPICS Let's look at t
Page 180 and 181: 158 ADVANCED TOPICS Thus, for MAPSD
Page 182 and 183: 160 ADVANCED TOPICS Table 7.6 Examp
Page 184 and 185: 162 ADVANCED TOPICS If we assume no
Page 186 and 187: 164 ADVANCED TOPICS Now we can focu
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166 ADVANCED TOPICS Notice we used
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168 ADVANCED TOPICS to achieve spat
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170 ADVANCED TOPICS c) What are the
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CHAPTER 8 PRACTICAL CONSIDERATIONS
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MORE DETAILS 175 8.2 MORE DETAILS I
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MORE DETAILS 177 One extreme is to
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THE MATH 179 8.3.1 Time-invariant c
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THE MATH 181 With recursive channel
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MORE PRACTICAL ASPECTS 183 timing).
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THE LITERATURE 185 8.6 THE LITERATU
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PROBLEMS 187 a) Draw the new receiv
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APPENDIX A SIMULATION NOTES Perform
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FADING CHANNELS 191 where JA = A 2
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APPENDIX B NOTATION Channel Equaliz
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NOTATION 195 Variable Meaning p(t)
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198 REFERENCES [Ari98] [Ari99] [Ari
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200 REFERENCES [Bra91] W. R. Braun
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202 REFERENCES [Div98] [Dou95] [Due
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204 REFERENCES [Gon68] R. A. Gonsal
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206 REFERENCES [Kaa94] [Kai60] V.-P
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208 REFERENCES [ManOl] [Mar73] [Mar
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210 REFERENCES [Pic95] [P0088] [Pri
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212 REFERENCES [Sto93] [Sto96] [Sui
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214 REFERENCES [Wan06a] T. Wang, J.
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