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26 INTRODUCTION 1.5.1 Reference system and channel models In later chapters, we will use simulation to compare different equalization approaches for a TDM system. Notes on how these simulations were performed are given in Appendix A. Most results will be for QPSK. The pulse shape is root-raised cosine with rolloff (ß) 0.22 (22% excess bandwidth). The following channel models will be used. TwoTS Dispersive medium with two, nonfading symbol-spaced paths with relative powers 0 and — 1 dB (sum of path energies normalized to unity) and angles 0 and 90 degrees. TwoFS Dispersive medium with two, nonfading half-symbol-spaced paths with relative powers 0 and — 1 dB (sum of path energies normalized to unity) and angles 0 and 90 degrees. TwoTSfade Similar to the TwoTS channel, except that each path experiences independent, Rayleigh fading, i.e., each path is a complex Gaussian random variable. The variances of the random variables are set so that £{g H g} = 1}, and the relative average powers are 0 and —1 dB. 1.6 THE LITERATURE The general system model and its notation are based on [Wan06b, Ful09]. Real and complex Gaussian random variables are addressed in a number of places, including [Wha71]. Digital communications background material, including modulation, channel modeling, and performance analysis, can be found in [Pro89, ProOl]. The notion of Nyquist rate for distortionless transmission is developed in [Nyq28]. Nyquist rate is the result of the fact that if one is given bandwidth B and time duration T, there are 2TW independent dimensions or degrees of freedom [Nyq28, Sha49]. Sending more symbols than independent dimensions leads to ISI. The notion of channel capacity is developed in [Sha48, Sha49]. Cellular communications is described in [Lee95, Rap96]. Background material on OFDM and CDMA can be found in [Sch05, Dah08]. For OFDM, use of the FFT can found in [Wei71], and application to mobile radio communications is discussed in [Cim85]. Using a cyclic prefix in CDM systems is considered in [Bau02]. Information on the discrete Fourier transform can be found in standard signal processing textbooks, such as [Rob87]. While modeling the channel as linear is fairly general, the assumption of Rayleigh or Rice fading is particular to wireless communications. Accurate modeling is important, because equalization design is usually targeted to particular scenarios for which reliable communications is desirable. In the literature, channel modeling information can be found for • wireless (radio) communications [Tur72, Suz77], • wireline communications (twisted pair) [Fis95],
PROBLEMS 27 • optical communications over fiber [Aza02], • underwater acoustic communications [Sin09], • underwater optical communications [Jar08], and • magnetic recording [Kum94, Pro98]. OFDM equalization when the delay spread exceeds the length of the cyclic prefix is considered in [Van98]. We will not consider it further, though results for the CDM case are applicable by redefining the spreading sequences. Equalization when the channel varies within a symbol period is considered in [Jeo99, Wan06a]. We will not consider it further. PROBLEMS The idea 1.1 Suppose a transmitted symbol, either +1 or —1, passes through a channel, which multiplies the symbol by —10 and introduces a very small amount of noise. Suppose the received value is 8. a) What most likely is the transmitted symbol? b) What most likely is the noise value? c) What is the other possible noise value? 1.2 Suppose a transmitted symbol, either +1 or —1, passes through a channel, which multiplies the current symbol by 1, adds the previous symbol multiplied by 2, and introduces a very small amount of noise. Suppose you know that the previous symbol is —1 and the current received value is —1. What most likely is the current symbol? 1.3 Suppose a transmitted symbol s, either +1 or —1, passes through a channel which scales the symbol by 5 and adds —10 and introduces a very small amount of noise. Suppose the current received value is —3. What most likely is the current symbol? 1.4 Suppose a transmitted symbol s, either +1 or —1, passes through a nonlinear channel, which produces 20s 2 + 10s, and introduces a very small amount of noise. Suppose the current received value is +9. What most likely is the current symbol? More details 1.5 Suppose we have the MIMO scenario in which c = 1, d Also, suppose the two received values are n = —1.2 and r 2 a) What most likely was the symbol s\ ? b) What most likely was the symbol S2? 0, e = 0, and/= 1. -0.8. 1.6 Suppose we have the MIMO scenario in which c = 1, d = 0, e = 2, and / = 1. Also, suppose the two received values are r\ = —1.2 and r 2 = —0.8.
Page 2 and 3: Channel Equalization for Wireless C
Page 4 and 5: To my colleagues at Ericsson
Page 6 and 7: CONTENTS List of Figures List of Ta
Page 8 and 9: CONTENTS XI 4.2 4.3 4.4 4.5 4.6 4.1
Page 10 and 11: CONTENTS XIII 8 Practical Considera
Page 12 and 13: xvi LIST OF FIGURES 2.2 Matched fil
Page 14 and 15: XviÜ LIST OF FIGURES 6.18 Scatter
Page 16 and 17: XX Prologue Alice was nervous. Woul
Page 18 and 19: XXÜ PREFACE is introduced as neede
Page 20 and 21: ACRONYMS AC A/D ADC AMLD AMPS ASK A
Page 22 and 23: ML MLD MLPD MLSD MLSE MRC MSE MSB O
Page 24 and 25: 2 INTRODUCTION Table 1.1 Possible m
Page 26 and 27: 4 INTRODUCTION s o S 1 8 2 S 3 S 0
Page 28 and 29: 6 INTRODUCTION A block diagram of t
Page 30 and 31: 8 INTRODUCTION Figure 1.7 QPSK. tra
Page 32 and 33: 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 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 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
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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
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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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