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96 LINEAR EQUALIZATION 4.2 If we had r» to work with as well, what would be the partial zero-forcing linear equalization weight for r» so that ISI from So is canceled when detecting s 2 l 4.3 Consider the Alice and Bob example (n = 1, r 2 = —7). Suppose the noise power is σ 2 = 1 instead. a) What would the MMSE LE weights be? b) What is the value of z 2 and s 2 with MMSE LE? 4.4 Consider the Alice and Bob example (r\ — 1, r 2 = —7). Suppose the noise power is σ 2 = 1000 instead. a) What would the MMSE LE weights be? b) What is the value of z 2 and s 2 with MMSE LE? More details 4.5 In the dispersive scenario with c = —10 and d — 9, suppose the max-SINR weights for detecting s 2 are scaled by -10. a) Calculate the new output SINR. b) Did the SINR get better, worse, or stay the same? c) Can we still detect s 2 by taking the sign of ζ 2 Ί 4.6 A better approach to the partial ZF approach is to minimize ISI. a) If w 2 = —0.1, determine Wi to minimize ISI when detecting s 2 . b) What is the resulting SINR? c) Is the SINR bigger or smaller than the partial ZF SINR? d) Is the SINR bigger or smaller than the MMSE SINR? 4.7 In the dispersive scenario, consider detecting si using r\ and r 2 , setting w 2 = 1/c, and choosing w\ to minimize ISI in z 2 . a) Find the general expression for w\. b) As the noise power goes to 0, what happens to w\l c) Show that if the noise power is zero, the SINR is the same as the MMSE solution SINR. 4.8 In the dispersive scenario with c = —10 and d = 9, consider MMSE detection of s i using r\ and r 2 . a) Find the MMSE solution for wi and w 2 . b) For r\ = 1, r 2 — — 7, what is the value of the decision variable for βχ? c) What is the detected value for si? 4.9 Using the model for the received values for the general dispersive scenario, show that the average of r\ is c 2 + d 2 + σ 2 . 4.10 Consider the MIMO scenario in which c = 10, d = 7, e = 9, and / = 6. a) What are the MMSE weights for detecting s 2 l b) What is the output SINR? c) If ri = 9 and r 2 = 11, what is the detected value for s 2 ?
PROBLEMS 97 The math 4.11 Consider the general dispersive case, with channel coefficients c and d. a) Find the general expression for the MMSE weights for detecting s\ using r\ and r?. b) As the noise power goes to 0, what happens to »2? c) As the noise power becomes much larger than the ISI power d?, show that MMSE weights are proportional to the MF weights. 4.12 Show that the ML and MMSE weight solutions are equivalent (within a scaling factor). Use the following version of the matrix inversion lemma: [A + BCD]" 1 = A" 1 - A^BIC" 1 + DA^B^DA 1 . (4.136) 4.13 Using the model in (4.79), show that the SINR expressions in (4.97) and (4.99) are equivalent. You will need the matrix inversion lemma from the previous problem.
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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
Page 67 and 68: THE MATH 45 We can interpret f(t) a
Page 69 and 70: MORE MATH 47 10" 1 Odeg 90 deg X 18
Page 71 and 72: MORE MATH 49 With despread-level Ra
Page 73 and 74: MORE MATH 51 Figure 2.6 OFDM exampl
Page 75 and 76: MORE MATH 53 2.4.3 MF in colored no
Page 77 and 78: PROBLEMS 55 period) is examined in
Page 79 and 80: CHAPTER 3 ZERO-FORCING DECISION FEE
Page 81 and 82: MORE DETAILS 59 1/c r m —► H i
Page 83 and 84: MORE DETAILS 61 where «i = m - (d/
Page 85 and 86: 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: PROBLEMS 95 Another aspect of cocha
Page 122 and 123: 100 MMSE AND ML DECISION FEEDBACK E
Page 137 and 138: CHAPTER 6 MAXIMUM LIKELIHOOD SEQUEN
Page 139 and 140: MORE DETAILS 117 r 2 Ht, r, fcl + f
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
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PROBLEMS 147 the channel to minimum
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PROBLEMS 149 c) Suppose we know «o
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152 ADVANCED TOPICS detecting the w
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154 ADVANCED TOPICS Table 7.2 Examp
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156 ADVANCED TOPICS Let's look at t
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158 ADVANCED TOPICS Thus, for MAPSD
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160 ADVANCED TOPICS Table 7.6 Examp
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162 ADVANCED TOPICS If we assume no
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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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