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90 LINEAR EQUALIZATION where emj(n) can be interpreted as a chip estimate during symbol period m» for transmitter i () . We can stack these estimates into an N c x 1 vector β^" = [e ( m,}(0) ... e {^{N c - l)] r , which is obtained by where e = W H v, (4.122) W = C" 1^"' (4.123) F(¿„) = [f(¿n)( 0 ) ... f( ¿ ")(iV c -l)l . (4.124) We can interpret the operation in (4.122) as a matrix equalizer (matrix multiply), in which the nth column of W is a weight vector used to obtain an estimate of the nth chip transmitted from transmitter ¿ ( > during symbol period mo. While the elements of F**"' do not depend on fco or mo, the elements of C„ do. As a result, the entries in W will be code-specific , a function of the spreading codes used during symbol period mo. We can trade performance for reduced complexity by approximating C„ with its average, averaged over the possible spreading codes. Then W would be the same for each symbol period, requiring fewer weight computations. However, even with code-averaging, we would still need a separate weight vector for each chip period. We can reduce complexity further by constraining the equalizer to use a sliding window of receive samples when computing different chip estimates. Such a form of equalization is called transversal equalization. Specifically, when forming em„ (no), we use v(m 0 7' + n„T c + djT s ), j = 0, ...,J-1, (4.125) where dj is a relative processing delay, relative to both the symbol period and the chip period within the symbol. Complexity is reduced because with code averaging, the weight vector is the same for each chip period no and each symbol period mo. Before performing code averaging, let's look at the code-specific transversal solution. To obtain the weight vector for forming eS„(no), we can use the analysis above, assume Aj = n t) T c + d/F a , (4.126) and examine the noth column of W in (4.123) (denoted w). From (4.123), C v (m 0 )w = ¥*\n 0 T c ). (4.127) where C„(mo) is a JN r x JN r matrix of the form given in (4.111) and (4.113) and f (i ">(n7' c ) is defined in (4.120). From (4.120) and (4.126), we see that f' io >(n 0 ï' c ) is really independent of no and can be denoted f**"'. The elements of f' 1 "' are code-independent, but the elements of C„(mo) are not. To obtain a code-averaged solution, we average C„(mo) over the spreading codes and use the result in (4.127) to solve for the weights. Thus, C„w = f (i,l) , (4.128)
MORE MATH 91 where C(d(),d 0 ) . C(dj_i,do) C(d 0 ,dj-i) C(dj_i,dj_i) . (4.129) C v (dj 1 ,dj 2 ) ^E^C^id^d^+NoiRpUi »=i L-l ί,-1 Σ Σ 8Ίβ2 x
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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
Page 61 and 62: THE MATH 39 correlation function f(
Page 63 and 64: THE MATH 41 Thus, in the complex pl
Page 65 and 66: 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: MORE MATH 89 values of h k m (f) fr
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 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
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MORE MATH 141 15 POWER CONTROL 10
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MORE MATH 143 6.4.3 More approximat
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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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