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4 INTRODUCTION s o S 1 8 2 S 3 S 0 S 1 S 2 S 3 Γ 1 r 2 r 3 Figure 1.3 Received signal example. While we don't know the noise values, we do know that they are usually small. In fact, physics tells us that the likelihood of noise taking on a particular value is given by the histogram in Fig. 1.4. Such noise is called Gaussian, named after the scientist Gauss. The average noise value is 0. The average of the square of a noise value is denoted σ 2 (the average of n\ or n 2 ). We call the average of the square energy or power (energy per sample). We will assume we know this power. If needed, it would be estimated in practice. One more assumption regarding the noise terms. We will assume different noise values are unrelated (uncorrelated). Thus, knowing m would tell us nothing about n^. 1.2 MORE DETAILS How well an equalizer performs depends on how large the noise power is, relative to the signal power. A useful measure of this is the signal-to-noise ratio (SNR). It is defined as the ratio of signal power (S) to noise power (N), i.e., S/N. If we are told that the noise power is σ 2 = 100, we just need to figure out the signal power S. We can use the model for Ti in (1.1) to determine S. The input signal power S is the average of the signal component (—10s2 + 9si) 2 , averaged over the possible values of s\ and Si. This turns out to be 181, which can be computed one of two ways. One way is to consider all possible combinations of s\ and «2- For example, the combination s\ — +1 and S2 = +1 gives a signal term of —10(+1) +9(+l) = —1 which has power (—l) 2 = 1. Assuming all combinations are possible 1 , the average power becomes S = (l/4)[(-l) 2 + (-19) 2 + (19) 2 + l 2 ] = 181. (1.3) Another way to compute S is to use the fact that si and S2 are assumed to be unrelated. When two terms are unrelated, their powers add. The power in — lOsi 1 This is not quite true, because one combination does not occur according to Table 1.1. However, for most practical systems, this aspect can be ignored.
MORE DETAILS 5 0.5 0.4 I 0.2 0.3 0.1 0 - 4 - 3 - 2 - 1 0 1 2 3 4 noise values Figure 1.4 Noise histogram for noise power σ 2 = 1. is the average of [(-10)(+1)] 2 and [(—10)(—l)] 2 , which is 100. We could have used the property that the average of cs is c 2 times the average of s 2 . The power in 9s i is 81, so the total signal power is 181. Thus, the input SNR is SNR = 181/100= 1.81. (1.4) It is common to express SNR in units of decibels, abbreviated dB. These units are obtained by taking the base 10 logarithm and then multiplying by 10. Thus, the SNR of 1.81 becomes 101og 10 (1.81) = 2.6 dB. We will be interested in two extremes: low input SNR and high input SNR. When input SNR is low, performance is limited by noise. When input SNR is high, performance is limited by ISI. 1.2.1 General dispersive and MIMO scenarios In general, we can write the received values in terms of channel coefficients c and d, keeping in mind that we know the values for c and d. Thus, for the dispersive scenario, we have r m = cs m + ds m _i + n m ; m =1,2, etc., (1.5) where the noise power is σ 2 . The corresponding SNR is SNR = (c 2 + d 2 )/a 2 . (1.6)
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 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
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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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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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