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168 ADVANCED TOPICS to achieve spatial multiplexing (MIMO). Because subcarriers are orthogonal, we can use the simple MIMO model introduced in Chapter 1, in which the decision variables for a particular subcarrier collected over multiple receive antennas can be modeled as r f= Hs + n. (7.53) Unlike Chapter 1, we will assume complex-valued quantities. Let's consider MAPSD assuming no prior information about the two symbols. For symbol si, we need to form the MAP symbol metrics specified in (7.20). For the case of only two M-ary symbols drawn from a set S — {Sj\j = 1... M}, this becomes M P{ 8l =Sj\r} = ^Pr{r|e 1 =S j) e 2 = Sfc} (7.54) Assuming n is zero-mean Gaussian with covariance matrix C n , then fc=l Pr{r| Sl = S jt s 2 = S fc } = N) exp {(r - Hs^C^r - H Sj -. fc )} , (7.55) where Sjj. — [Sj Sk\ T ■ To compute soft information for a particular bit í>¿ included in s\, we divide the set of possible symbol values S into two subsets: the set B+ corresponding to bi = +1 and the set B¿~ corresponding to 6» = — 1. The log-likelihood ratio for 6¿ can then be determined using LLRi = log I 53 P{ Sl = Sjlr} j - log j ^ P{ Sl = Sj\r) J . (7.56) 7.6 THE LITERATURE 7.6.1 MAP symbol detection The relation between MAPSD and minimizing symbol error rate is pointed out in [Abe68j. A structure for MAPSD is given in [Gon68]. Fixed-lag (finite decision depth) MAPSD forms are developed in [Gon68, Abe70), and a parallel architecture is given in [Erf94]. Early work on MAPSD is surveyed in [Bel79]. Like MLSD, MAPSD can be formulated using an MF front end [Col05]. The BCJR algorithm is described in [Bah74]. It can be viewed as a special case of Pearl's belief-propagation algorithm [Mce98]. Like the Viterbi algorithm, reduced-complexity versions of the BCJR, algorithm exist. The log-MAX approximate form that uses a forward and backward Viterbi algorithm is developed in [Vit98]. Reduced-state approximations have been proposed [ColOl, Vit07] as well as only considering a subset of symbols, treating the others as colored noise [P0088]. 7.6.2 Soft information In wireless communications, typical FEC codes are convolutional codes (proposed in [Eli55] according to [Vit67]), turbo codes [Ber96], and, more recently, low-density
PROBLEMS 169 parity-check (LDPC) codes [Gal62]. Error detection is usually performed with a Cyclic Redundancy Code (CRC). For MLSD, a soft output Viterbi algorithm (SOVA) provides soft information. The two main approaches are given in [Bat87] and [Hag89]. These two can be made equivalent by adding a certain term to the latter [Fos98]. Performance of various soft information generation approaches are compared in [Han96j. With approximate MLSD approaches, paths corresponding to all possible bit values may not be present. One solution is to set the soft value to some maximum value [Nas96]. The term dual maxima is introduced in [Vit98]. Kaiman filtering can be used as a form of equalization for estimating soft bit values [Thi97]. 7.6.3 Joint demodulation and decoding The classic reference for turbo equalization is [Dou95]. Turbo equalization for differential modulation can be found in [Hoe99, Nar99]. Channel estimation can also be included in the turbo equalization process [Ger97]. Linear turbo equalization is described in [LaoOl, Tüc02a, Tüc02b]. There is some evidence that using unadjusted feedback from the decoder can improve performance [Vog05]. So far we have considered structure provided by encoding. There may be further structure in the information bits as well. Using this information at the receiver is referred to as joint source/channel decoding [Hag95], and interesting performance gains are possible [Fin02]. Joint demodulation and decoding of CDMA signals is discussed in [Gia96]. Turbo equalization has been extended to joint detection of cochannel signals (multiuser detection) [Moh98, Ree98, Wan99]. PROBLEMS The idea 7.1 Consider the Alice and Bob example and Table 6.1. a) Find the two symbol metrics for si b) What is the MAP symbol estimate for si? c) Does it agree with the MLSD estimate? 7.2 Consider the Alice and Bob example and Table 6.1. a) Suppose the noise power is 0.01 instead of 100. Recompute the symbol metrics. b) If you observed a numerical issue, what was it? If not, try using a simple calculator. 7.3 Consider the Alice and Bob example. Suppose instead that z\ = —4 and zi = 3. a) What are the detected symbol values? b) Do the detected symbol values correspond to a valid message?
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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
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
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
Page 188 and 189: 166 ADVANCED TOPICS Notice we used
Page 192 and 193: 170 ADVANCED TOPICS c) What are the
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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