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160 ADVANCED TOPICS Table 7.6 Example of message metrics for (7.4) Hamming code Message Information Bits Check Bits Message Metric 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 +1 +1 +1 +1 +1 +1 +1 -1 +1 +1 -1 +1 + 1 +1 -1 -1 + 1 -1 +1 +1 +1 -1 +1 -1 +1 -1 -1 +1 +1 -1 -1 -1 -1 +1 +1 +1 -1 +1 +1 -1 -1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 -1 -1 +1 -1 -1 -1 -1 4-1 4-1 4-1 -1 -1 -1 4-1 -1 -1 -1 4-1 4-1 -1 4-1 -1 4-1 -1 4-1 -1 -1 4-1 4-1 4-1 -1 -1 -1 4-1 4-1 4-1 -1 -1 4-1 -1 4-1 -1 4-1 4-1 -1 -1 -1 4-1 4-1 4-1 4-1 4-1 -1 -1 -1 13 -3 -11 25 23 -1 19 7 -7 -19 1 -23 -25 11 3 -13 that exit the coding process. Thus, if there are 400 information bits and we encode them into 600 modem bits, we are using a rate 2/3 code. In general, lower rate codes are more powerful, being able to correct more errors. 7.2.3 Joint demodulation and decoding Let's consider joint demodulation and decoding for the case of MLSD and the (7,4) Hamming code. In Chapter 6, we introduced the notion of MLSD as a tree search, using the tree in Fig. 6.5. The branches of the tree corresponded to all possible symbol sequences. When we including the fact that the symbol sequences were generated with a code, we can immediately prune the tree and drop any branches that do not correspond to a valid codeword. For the (7,4) Hamming code, 7 bits are transmitted. With conventional MLSD, there would be 2 7 = 128 possible sequences. With joint demodulation and decoding, there would be only 2 4 = 16 possible sequences. We would form metrics for each possible sequence using a tree with 16 final branches. 7.3 THE MATH 7.3.1 MAP symbol detection MAPSD tries to find the symbol value that maximizes the conditional symbol likelihood, conditioned on the received signal. How does this differ from MLSD? There are two main differences. First, MLSD tries to find a sequence value, not a symbol value. Second, MLSD tries to maximize the received signal likelihood, not a symbol likelihood. The implication of first difference is that MAPSD will focus on each symbol separately. As a result, it will minimize symbol error rate, as opposed to sequence error rate. Keep in mind that when detecting a particular symbol, MAPSD will
THE MATH 161 still take advantage of the entire received signal and the fact that other, discrete symbols are being sent. The implication of the second difference is that MAPSD will allow us to introduce prior information about the symbol likelihoods. Mathematically, MAPSD gives s(m) — arg max Pr{s(m) = q(m)\r(t) Vi}, (7-16) q(m,)eS where we use Pr{·} to denote likelihood, which can be either a discrete probability (sum to one) or a PDF value (integrates to one). Thus, we find the hypothetical symbol value q(m) that is most likely, given the received signal. Applying Bayes' rule, we can rewrite this as = arg max P r {r(«)Vtk(m) = g(m)}l>rMm) = g(m)} V ' g(m)es Pr{r(í)Ví} v ; Each of the three terms on the r.h.s. is discussed separately. It helps to start with the second term in the numerator. This term is the a priori or prior information regarding the symbol. If we knew that the possible symbol values are not equi-likely, we would introduce that information here. The first term in the numerator looks like the MLSD criterion, except the likelihood of the received signal is conditioned on a single symbol, rather than a sequence. Thus, this first term is really an ML symbol detection metric. We will see later that this term can also include prior information, prior information about other symbols besides s(m). The term in the denominator is the likelihood of the received signal. As it is an unconditional likelihood, it will be the same for each value of q(m). Thus, we can ignore this term when performing the maximization operation. We can rewrite the first term in the numerator in a way that relates it to sequence detection. Let s m denote the subsequence of s that excludes s(m), and let q m denote a hypothetical value for s m . Also, let S^'~ x denote the set of all possible subsequences for s m . Then, we can write Pr{r(i) Vt|« = q(m)} = £ Pr{r(i) Vt|«(m) = (?(m), s m = q m } xPr{s m = q m }. (7.18) Using this result and dropping the denominator term gives the MAP symbol metric P{s = q(m)\r(t) Vi} = £ Pr{r(t) Vt|e(m) = q(m),s m = q} m xPr{s(m) = 9(m)}Pr}s m = q m }. (7.19) Observe that the metric consists of the sum of a product of three terms. The first term is the MLSD metric for a particular sequence. However, it is only evaluated for sequences for which s(m) = q(m). The second and third terms give prior sequence likelihood information. Here is where prior information about other symbols can be introduced.
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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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Page 132 and 133: 110 MMSE AND ML DECISION FEEDBACK E
Page 134 and 135: 112 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
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 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 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
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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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