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56 MATCHED FILTERING 2.7 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. The noise power is σ 1 = 100. a) What is the value of the MF decision variable z\ for si? b) What is the SINK of zi? c) What is the value of the MF decision variable z 2 for s 2 ? d) What is the SINR of ¿ 2 ? 2.8 Suppose we have the dispersive scenario in which c = —2 and d = 1. Suppose QPSK is sent and r\ = — 3 + j and r 2 = 2 + j'3. a) What is the value for the MF decision variable for si? b) What most likely is the I component of «i? c) What most likely is the Q component of «i? The math 2.9 Consider a BPSK system with root-Nyquist pulse shaping. Suppose the medium consists of a single path with delay To = 0 and coefficient po- There is also AWGN with one-sided PSD 7V () . Assume the receiver uses a matched filter. a) Express the amplitude of the decision variable in terms of Ef„ N n , and p () . b) What is the variance of the noise on the decision variable? c) What is the output SNR? d) If po = 3 and Et/No = 0.1, what is the probability of a bit error in terms of the erfc function? If possible, evaluate this to get a numerical result as well. 2.10 Consider a BPSK system with root-Nyquist pulse shaping. Suppose the medium consists of a two paths with delays To = 0, τχ = T and coefficients po and Pi. There is also AWGN with one-sided PSD iVo. Assume the transmitter sends only one symbol and the receiver uses a matched filter. a) Express the amplitude of the decision variable in terms of £¡,, No, and po. b) What is the variance of the noise on the decision variable? c) What is the output SNR? d) If po = 3, pi = 2 and Eb/N tt = 0.1, what is the probability of a bit error in terms of the erfc function? If possible, evaluate this to get a numerical result as well. 2.11 Consider a BPSK system with root-Nyquist pulse shaping. Suppose the transmitter sends each symbol twice before sending the next symbol (i.e., s 2 = s\, Si = S;i, etc.). Suppose the medium consists of one path with delay To = 0 and coefficient p () . There is also AWGN with one-sided PSD ΛΌ. Assume the receiver uses a matched filter. a) Express the matched filter for si in terms of r(t), p(t), T, and pob) Express the amplitude of the decision variable in terms of Ef,, N», and poc) What is the variance of the noise on the decision variable? d) What is the output SNR?
CHAPTER 3 ZERO-FORCING DECISION FEEDBACK EQUALIZATION Decision feedback equalization (DFE) uses past symbol decisions (detected values) to remove ISI from previous symbols. In this chapter we will consider a zero-forcing strategy, in which ISI from future symbols is avoided. Assuming the detected values are correct, ISI from past symbols is also avoided. Thus, ISI is forced to zero. This is not necessarily the best strategy, but it is a useful starting point for understanding more sophisticated strategies. In Chapter 5, we will examine other strategies, including Minimum Mean-Square Estimation (MMSE) DFE and Maximum Likelihood (ML) DFE. 3.1 THE IDEA Let's assume we have a detected or known value for s () . Next we want to determine si using ri, which can be modeled as ri =-10si+9so + ni. (3.1) Since we have an idea of what SQ is, we can subtract its influence on rj, giving 0i = ri - 9So, (3-2) which, assuming So = so, can be modeled as i/i = —lOsi + rii- (3.3) Channel Equalization for Wireless Communications: From Concepts to Detailed 57 Mathematics, First Edition. Gregory E. Bottomley. © 2011 Institute of Electrical and Electronics Engineers, Inc. Published 2011 by John Wiley & Sons, Inc.
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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
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 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: PROBLEMS 55 period) is examined in
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 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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106 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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