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28 INTRODUCTION a) What most likely was the symbol si? b) Assuming you detected s\ correctly, what most likely was the symbol S2? 1.7 Suppose we have the dispersive scenario in which c = 4, d = 2, Γ2 = 2.1, and ri is not available. a) Each symbol can take one of two values. For each of the four combinations of s 1 and S2, determine the corresponding noise value for n-¿. b) Which combination corresponds to thé smallest magnitude noise value? 1.8 Suppose we have the dispersive scenario in which c = —2 and d = 0. Suppose QPSK is sent and r x = -1.8 + j'2.3. a) What most likely is the I component of si ? b) What most likely is the Q component of si? 1.9 Sometimes we want to send two bits in one symbol period. One way to do this is to send one of four possible symbol values: —3, —1, +1 or +3. Consider the mapping 00 = —3, 01 = —1, 10 = +1 and 11 = +3. At the receiver, when mistakes are made due to noise, they typically involve mistaking a symbol for one if its nearest neighbors. For example, —3 is detected as —1, its nearest neighbor. a) When —3 is mistaken as —1, how many bit errors are made? b) When —1 is mistaken as +1, how many bit errors are made? c) What is the average signal power, assuming each symbol is equi-likely to occur? d) Suppose the symbol passes through a channel, which multiplies the symbol by —10 and introduces a very small amount of noise. Suppose the received value is —11. What most likely was the transmitted symbol? What were the transmitted bits? 1.10 Suppose we change the mapping to 00 = —3, 01 = — 1, 11 = +1 and 10 = +3, referred to as Gray-mapping. a) When —1 is mistaken as +1, how many bit errors are made? b) Is there a case where a nearest neighbor mistake causes two bit errors? The math 1.11 Consider a TDM transmitter using a root-Nyquist pulse shape. The signal passes through a single-path medium with delay τ 0 = 0. Suppose the receiver initially filters the received signal using v(qT) = f τ{τ)ρ*(τ — qT — to) dr. a) For ¿o = 0, how many symbols does v(qT) depend on? b) For to = T/2, how many symbols does v(qT) depend on? c) For to = T, how many symbols does v{qT) depend on? d) Suppose the medium consists of two paths, with path delays 0 and T seconds. Now how many symbols does v(qT) depend on for i 0 = 0? 1.12 Suppose the pulse shape is a rectangular pulse shape, so that p(t) is \/T on the interval [0,7') and zero otherwise. a) What is ñ p (r)? b) Is this pulse shape root-Nyquist?
PROBLEMS 29 c) Suppose the receiver initially filters the received signal using v(qT) = J^° τ{τ)ρ*{τ — qT) dr. How many symbols does v(qT) depend on? d) Suppose the receiver initially filters the received signal using v(qT) = /_oo r ( r )P*( T — ÇÏ' — 2/2) dr. How many symbols does v(qT) depend on? 1.13 Consider BPSK, in which a detect static can be modeled as z \= y/Ë^s + n, where s is +1 or —1 and n is a Gaussian random variable with zero mean and variance ./Vo/2. Derive (A.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 48 and 49: 26 INTRODUCTION 1.5.1 Reference sys
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
Page 99 and 100: 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 137 and 138:
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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