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62 ZERO-FORCING DECISION FEEDBACK EQUALIZATION 3.3 THE MATH We start this section by determining when a zero-forcing solution is possible. The basic equations for ZF DFE are then given. A full zero-forcing solution is only possible in a few special cases. This is because the pulse shape is typically bandlimited, making it nonzero for many symbol periods in both the future and the past. Even if the pulse shape is root-Nyquist and partial MF is employed, ISI from future symbol periods occurs if fractionally spaced equalization is employed or the medium path delays are fractionally spaced. As we will consider the more general case in Chapter 5, we will focus on a special case for which full ZF is possible. Specifically, we will assume the following. 1. The channel can be modeled with symbol-spaced paths (τ( = ÍT). 2. Root-Nyquist pulse shaping is used at the transmitter, and partial MF is used at the receiver (R p (qT) = 6(q)). 3. The sampling phase is aligned with the first tap delay of the channel (i n = TO). 4. Symbol-spaced sampling is used (T s = T). With these simplifying assumptions, the received samples (r m = v(mT)) can be modeled as L-l r m = \/E~s Σ 9e s m-e + u m , (3.31) where u m is zero-mean, complex Gaussian r.v. with variance No. The traditional block diagram for DFE is given in Fig. 3.3. The received signal is processed by a feedforward filter (FFF). The output of a feedback filter (FBF) is then subtracted, removing ISI from past symbol periods. The result is a decision variable, which is used by a decision device or detector (DET) to determine a detected symbol value. We have divided the FFF into two filters: a front-end filter matched to the pulse shape (partial MF or PMF) and a forward filter (FF). Also, because the FF is linear, the DFE can be formulated with the FBF being applied prior to the FF, as shown in Fig. 3.4. This formulation is convenient as it decouples the forward and backward filter designs under the design assumptions to be used. The FBF subtracts the influence of past symbols on r m , giving which can be modeled as L-l y m = r m -^2 ges(m - (.), (3.32) 1=1 L-l Vm h V~Ës9oSm + Σ 9e\s{m - i) - s(m - ί)\ + u m . (3.33) e=i We will assume that the past symbol decisions are correct, simplifying (3.33) to Um \= \/E s gns m + u m . (3.34)
MORE MATH 63 Because there is no ISI and u m is complex Gaussian, we can use MLD to obtain s m = detect(j/ m , \fW s g 0 ). (3.35) ! FFF PMF —* ΓΓ + * DET 1 1 FBF 4 Figure 3.3 Traditional DFE. PMF — ► + ► FF —► DET 1 L FBF 4 Figure 3.4 Alternative DFE. 3.3.1 Performance results We will defer ZF DFE results until Chapter 5. When available, ZF DFE results will be labeled Minimum ISI (MISI), which is a broader category of DFE that does not necessarily force ISI to zero. 3.4 MORE MATH We start by considering when a ZF strategy is possible in our extended system model. The introduction of multiple receive antennas increases the opportunities for a ZF strategy. Two scenarios are then explored in more detail. For a truly ZF solution, we need to be able to avoid or cancel ISI from future symbol blocks as well as symbols within the current symbol block. We will make similar assumptions as in the previous section: chip-spaced paths, root-Nyquist chip pulse shaping, ideal sampling, and chip-spaced samples. For CDM with orthogonal codes, having a one-path channel would keep symbols orthogonal, but such a channel model is usually not reasonable. With multiple paths, ZF is possible if we only use the received samples that depend on the present and past symbol period symbols. However, this is a heavy cost in signal energy. If
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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
Page 34 and 35: 12 INTRODUCTION rollo« 0.22 0.8 Ί
Page 36 and 37: 14 INTRODUCTION is the "channel" re
Page 38 and 39: 16 INTRODUCTION 1.4 MORE MATH In th
Page 40 and 41: 18 INTRODUCTION For K = 1 (order 0)
Page 42 and 43: 20 INTRODUCTION The K = 4 main bloc
Page 44 and 45: 22 INTRODUCTION A sample of the noi
Page 46 and 47: 24 INTRODUCTION The receiver may ha
Page 48 and 49: 26 INTRODUCTION 1.5.1 Reference sys
Page 50 and 51: 28 INTRODUCTION a) What most likely
Page 53 and 54: CHAPTER 2 MATCHED FILTERING In this
Page 55 and 56: MORE DETAILS 33 2.2 MORE DETAILS So
Page 57 and 58: THE MATH 35 Now compare the output
Page 59 and 60: 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: MORE DETAILS 61 where «i = m - (d/
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 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
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112 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
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MORE DETAILS 119 Figure 6.3 MLSD tr
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THE MATH 121 legs of the journey is
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THE MATH 123 Figure 6.8 MLSD trelli
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THE MATH 125 We would then identify
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THE MATH 127 Expanding the square,
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THE MATH 129 Consider the matched f
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THE MATH 131 6.3.5.1 M-algorithm Wi
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THE MATH 133 results, DFE and MLSD
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THE MATH 135 IQ" 1 oc LU CD 10 2 10
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THE MATH 137 sometimes inaccurate w
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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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