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186 PRACTICAL CONSIDERATIONS a) Suppose channel estimates and a noise power estimate are used to determine weights. Is this direct or indirect adaptation? b) Suppose the LMS algorithm is used to adapt the weights. Is this direct or indirect adaptation? More details 8.6 Consider estimating three channel coefficients: c, d, and e such that r m = cs m + ds m -i + es m -2- Suppose We have four known symbols, «n through s ; ¡, and we plan on using Γ2 and r,¡ to estimate the channel coefficients. a) Give the equations for c, d, and ê in terms of ri and r%. b) What properties of the known symbols must be satisfied so that paths don't interfere with one another in channel estimation. c) Given So = +1 and s\ = —1, can these properties be met? If so, what values for S2 and s.-¡ work? 8.7 Consider MMSE DFE and determining the weights for the forward filter. a) Suppose channel estimates and a noise power estimate are used to form a data correlation matrix, which is then used to determine weights. Is this parametric or nonparametric? b) Suppose data correlations are estimated directly from the received samples. These data correlation estimates are used to determine weights. Is this parametric or nonparametric? 8.8 Suppose QPSK symbols are being sent through a single-path channel with coefficient c = 1. a) Copy the constellation diagram from Fig. 1.7. In practice, one would receive one of these four points plus noise. b) Add the lines y = x and y = —x to the diagram. If we think of the diagram as possible received values, these lines divide up the diagram into four regions, determining which symbol we would detect if the received value fell into that region. Notice that the constellation points are midway between two lines. c) Now, suppose there is an unknown phase error that rotates the signal 22.5 degrees counterclockwise. If there were no noise, would the receiver make an error? d) Now, suppose there is an unknown phase error that rotates the signal 50 degrees counter clockwise. If there were no noise, would the receiver make an error? e) Now, suppose there is an unknown phase error that rotates the signal 10 degrees counter clockwise. If 1 + j is transmitted and there is noise, the receiver will sometimes make mistakes. Which mistaken value will the receiver decide more often, 1 — j or — 1 + p. 8.9 Suppose QPSK symbols are being sent through a single-path channel with coefficient c = 1. Also, suppose there is I/Q imbalance, so that the real part is scaled by 0.8.
PROBLEMS 187 a) Draw the new received constellation points due to I/Q imbalance. b) If the symbol 1 + j is sent and the symbol is detected incorrectly, what is the most likely value of the incorrect symbol? c) With random symbols being sent, which bit will have more errors, the one on the I channel or the one on the Q channel? The math 8.10 Suppose QPSK symbols are being sent through a single-path channel with coefficient c = 1. Also, suppose there is I/Q imbalance, so that the real part is scaled by 0.8. a) Draw the new received constellation points due to I/Q imbalance. b) If the symbol 1 + j is sent and the symbol is detected incorrectly, what is the most likely value of the incorrect symbol? c) With random symbols being sent, which bit will have more errors, the one on the I channel or the one on the Q channel? 8.11 Suppose QPSK symbols are being sent through a single-path channel with coefficient c = 1. Also, suppose there is frequency offset of 25 degrees per symbol period. a) Assume the accumulated phase offset is 0 degrees at time m = 1. Draw the constellation points at that time and label them 1, 2, 3, and 4. These labels correspond to certain bit combinations. b) Draw the received constellation points at time m = 2. Label the four points. c) Assume the received values are not corrected, so that frequency offset must be handled by changing the possible symbol values. Give an equation for the possible symbol values as a function of m. 8.12 Suppose pilot symbols are used to estimate the channel response, which is then used to determine equalization parameters. Suppose the pilot symbols have a different energy per symbol than the traffic symbols. Consider the following equalization approaches: MF, ZF DFE, MISI LE, MMSE LE, MMSE DFE, MLSD. a) Which of these approaches require us to know the power relation between the pilot symbols and the traffic symbols. Assume all symbols are QPSK. b) What is your answer if the traffic symbols are 16-QAM? 8.13 Show that for QPSK symbols and tracking of a single channel coefficient, LMS channel tracking and exponential filtering are equivalent. What is the relationship between μ and a? 8.14 Research the Doppler effect and the equations that govern it. Show, mathematically, how the Doppler effect can be approximated by a frequency shift.
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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
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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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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 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: THE LITERATURE 185 8.6 THE LITERATU
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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