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170 ADVANCED TOPICS c) What are the three soft decision decoding metrics? d) What is the detected message number? 7.4 Consider the Alice and Bob example. Suppose instead that z\ = —3 and ¿2 = 4. a) What are the detected symbol values? b) Do the detected symbol values correspond to a valid message? c) What are the three soft decision decoding metrics? d) What is the detected message number? 7.5 Consider the simple parity check code in which s;¡ = s\S2- Suppose we decide to use this code for error correction in the Alice and Bob example. Suppose the signed soft values are —1,3, — 2. a) Using Table 1.1, create a table of valid messages including values for s\, «2, and su. Assume that s ( i = +1 is a known symbol. b) Using soft decision decoding, add to the table decoding metrics for each possible message, using ζχ, Z2, and z,-j. c) Identify the decoded message. 7.6 Consider the Alice and Bob example. Suppose instead that r\ = — 1 and Γ2 = 4. Assume s ( > = +1. a) Form a table of message metrics for only valid combinations of s\ and s 2 - b) Which metric is best? What is the detected sequence? 7.7 Consider the Alice and Bob example. Suppose instead that r\ = — 1 and Γ2 = 4. Assume s () = +1. Suppose a parity check symbol s 3 = s\s 2 was also transmitted and r.i = 2. a) Form a table of message metrics for only valid combinations of s\, s 2 , and s.-j. b) What is the detected message number? More details 7.8 Consider the Alice and Bob example and suppose we know that so = +1 because it is a pilot symbol. a) Find the two symbol metrics for s 2 using Table 6.1. b) Did the MAPSD detected value for s 2 change? c) Compute the ratio of the symbol metrics, with the larger one in the numerator. Do the same for the case when s» was not known. What happened to the ratio when we added the information that so = +1? 7.9 Consider the Alice and Bob example. Using the normalized sequence metrics in Table 7.4, compute the two symbol metrics for S2- 7.10 Consider the Alice and Bob example. Suppose instead that r\ = — 1 and r 2 = 4. Assume SQ = +1. Consider the MLSD tree, except only valid message branches are allowed. a) What are the two path metrics after processing Γχ?
PROBLEMS 171 b) What are the three path metrics after processing r 2 ? c) What is the detected message number? 7.11 Consider the MIMO scenario in which c = 10, d = 7, e = 9 and / = 6. The received values are r\ = 9 and r^ = 11. Assume only the messages in Table 1.1 are allowed. a) Form a table of metrics for only valid combinations of s\ and s?. b) What is the detected message number? The math 7.12 To derive the recursive formula for the beta values in the BCJR algorithm, a) How should the B¡ values be defined when using (7.26)? b) How should A and B be defined to use (7.22)? 7.13 Consider an arbitrary, 4-ary modulation. a) How many symbol likelihoods are there? b) With 2-ary modulation, it is possible to store one, signed soft value by forming the LLR. For 4-ary modulation, how could one store fewer symbol likelihoods? 7.14 Consider the general MIMO scenario (channel coefficients c, d, e, and /) with M-ary symbols. a) How many sequence metrics are there? b) How many symbol likelihoods total need to be computed for MAPSD of both symbols? c) How many sequence metrics are used to compute each symbol likelihood value? 7.15 Consider using the BCJR algorithm for processing received values of the form r m = cs m + eis m _i +n m with QPSK symbols. Suppose a finite set of symbols, «o through sç, are transmitted. No symbols are sent either before or after. Suppose So and s» are pilot symbols. a) What would be the first received value we would want to process? b) What would be the last received value we would want to process? 7.16 Consider using the BCJR. algorithm for processing received values of the form r m = cs m + ds m -\ + es m -i + n m with QPSK symbols. Suppose a finite set of symbols, so through sg are transmitted. No symbols are sent either before or after. Suppose so and so are pilot symbols. a) What would be the first received value we would want to process? b) What would be the last received value we would want to process?
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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
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MORE MATH 65 In the later chapter o
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PROBLEMS 67 b) What is the detected
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CHAPTER 4 LINEAR EQUALIZATION With
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THE IDEA 71 estimate of symbol s 2
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THE IDEA 73 Notice that £2 (MSE) d
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MORE DETAILS 75 The first term is t
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MORE DETAILS 77 To obtain the unity
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THE MATH 79 and SINR becomes w T h
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THE MATH 81 To obtain the MMSE solu
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THE MATH 83 where *m n = ν ^ ^ ^ '
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THE MATH 85 power is much larger th
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MORE MATH 87 IQ' 1 OC 111 ω 10 2 1
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MORE MATH 89 values of h k m (f) fr
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MORE MATH 91 where C(d(),d 0 ) . C(
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AN EXAMPLE 93 between z and s. This
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PROBLEMS 95 Another aspect of cocha
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PROBLEMS 97 The math 4.11 Consider
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100 MMSE AND ML DECISION FEEDBACK E
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CHAPTER 6 MAXIMUM LIKELIHOOD SEQUEN
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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
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
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Page 184 and 185: 162 ADVANCED TOPICS If we assume no
Page 186 and 187: 164 ADVANCED TOPICS Now we can focu
Page 188 and 189: 166 ADVANCED TOPICS Notice we used
Page 190 and 191: 168 ADVANCED TOPICS to achieve spat
Page 195 and 196: CHAPTER 8 PRACTICAL CONSIDERATIONS
Page 197 and 198: MORE DETAILS 175 8.2 MORE DETAILS I
Page 199 and 200: MORE DETAILS 177 One extreme is to
Page 201 and 202: THE MATH 179 8.3.1 Time-invariant c
Page 203 and 204: THE MATH 181 With recursive channel
Page 205 and 206: MORE PRACTICAL ASPECTS 183 timing).
Page 207 and 208: THE LITERATURE 185 8.6 THE LITERATU
Page 209 and 210: PROBLEMS 187 a) Draw the new receiv
Page 211 and 212: APPENDIX A SIMULATION NOTES Perform
Page 213 and 214: FADING CHANNELS 191 where JA = A 2
Page 215 and 216: APPENDIX B NOTATION Channel Equaliz
Page 217: NOTATION 195 Variable Meaning p(t)
Page 220 and 221: 198 REFERENCES [Ari98] [Ari99] [Ari
Page 222 and 223: 200 REFERENCES [Bra91] W. R. Braun
Page 224 and 225: 202 REFERENCES [Div98] [Dou95] [Due
Page 226 and 227: 204 REFERENCES [Gon68] R. A. Gonsal
Page 228 and 229: 206 REFERENCES [Kaa94] [Kai60] V.-P
Page 230 and 231: 208 REFERENCES [ManOl] [Mar73] [Mar
Page 232 and 233: 210 REFERENCES [Pic95] [P0088] [Pri
Page 234 and 235: 212 REFERENCES [Sto93] [Sto96] [Sui
Page 236 and 237: 214 REFERENCES [Wan06a] T. Wang, J.
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