Space Time Wireless Channels

Preface......Page 2
1.1.1 Early Years of Radio......Page 4
Figure 1.2. Some important milestones in radio communications.......Page 5
Figure 1.3. Fading for mobile communications causes sporadic moments of poor signal-to-interference+noise ratio (SINR) levels.......Page 6
Figure 1.4. Small-scale fading for a mobile receiver in a multipath environment.......Page 7
Fade Margin......Page 8
1.1.5 Channels with Multiple Dependencies......Page 9
Figure 1.6. In contrast to the idealized additive white Gaussian noise (AWGN) channel, the true wireless radio channel has numerous dependencies.......Page 10
1.2.2 Channel Primacy in Communications......Page 11
1.2.3 Wasted Space......Page 12
Figure 1.9. In a multipath channel, a single-port radio wastes much of the impinging signal power.......Page 13
1.3.3 Smart Antennas......Page 14
1.4 About This Book......Page 15
1.4.2 Contents......Page 16
1.4.3 Features of This Book......Page 17
2.1.1 Signal Spectrum......Page 18
2.1.2 Signal Modulation......Page 19
2.1.3 Inverse Modulation......Page 21
Example 2.1: Amplitude and Phase......Page 22
2.1.4 The Baseband Channel......Page 23
2.1.5 Time-Invariant Versus Time-Varying Channels......Page 24
Figure 2.5. Spectral diagram of baseband and passband signals and channel.......Page 25
2.1.6 Detection Terminology......Page 26
2.2.2 Temporal Coherence......Page 27
Figure 2.6. A time-varying channel.......Page 28
Figure 2.8. A frequency-varying channel.......Page 29
2.2.4 Spatial Coherence......Page 30
Large-Scale Versus Small-Scale Fading......Page 31
2.3.1 Spectral Domain Representations......Page 32
Table 2.2. Fourier Transform Definitions for Each Channel Dependency......Page 33
Theorem 2.1: Space–Time Transmission......Page 34
2.3.3 Time-Invariant Channel Transmission......Page 35
2.3.4 Mobile Receiver Transmission......Page 36
Problems......Page 37
3.1.1 The Meaning of Correlation......Page 40
3.1.2 Autocorrelation Relationships......Page 41
3.1.3 Autocovariance......Page 42
3.2 Power Spectral Density (PSD)......Page 43
Theorem 3.1: Uncorrelated Spectrum for WSS......Page 44
3.2.2 The Wiener-Khintchine Theorem......Page 45
3.2.3 Statistics with Three-Dimensional Space......Page 46
3.3 Joint Statistics......Page 48
3.3.1 Joint Autocorrelation and Spectrum......Page 49
Example 3.1: WSS-US Autocorrelation......Page 50
Figure 3.2. Autocorrelation function and PSD relationships for time and frequency.......Page 51
Figure 3.3. Autocorrelation function and PSD relationships for space and frequency.......Page 52
Figure 3.4. Autocorrelation function and power spectrum relationships for space, time, and frequency.......Page 53
Table 3.2. How to Remove a Dependency in a Random......Page 54
Example 3.2: Exponential Delay Spectrum......Page 55
3.4.2 RMS Doppler Spread......Page 56
3.4.3 RMS Wavenumber Spread......Page 57
Example 3.4: Omnidirectional Wavenumber Spectrum......Page 58
Example 3.5: Coherence Time Duality......Page 59
Figure 3.5. Time-varying, Rayleigh-distributed stochastic processes with different second-order statistics.......Page 60
3.4.6 Fundamental Spectral Spread Theorem......Page 62
Problems......Page 64
4.1.1 Electromagnetic Fields and Received Signals......Page 68
Figure 4.1. An antenna maps the complex electric field vector, 
, to a scalar baseband channel voltage, 
.......Page 69
4.1.2 The Maxwellian Basis......Page 70
Theorem 4.1: Wavevector Criterion for free-space......Page 71
4.1.3 Homogeneous Plane Waves......Page 73
4.1.4 Inhomogeneous Plane Waves......Page 74
4.1.5 Homogeneous Versus Inhomogeneous Plane Waves......Page 75
Figure 4.3. Rules of thumb for homogeneous and inhomogeneous plane wave propagation.......Page 76
An Analogy From Circuit Theory......Page 77
Figure 4.4. A linear circuit contains capacitors, inductors, resistors, and an AC source.......Page 78
Analogy to Free-Space Plane Waves......Page 79
4.2.2 Scatterer Proximity......Page 80
4.2.3 A Wideband Plane Wave......Page 81
Figure 4.7. The basic quantities of time-harmonic wave propagation through a scattering environment.......Page 82
Adding More Bandwidth......Page 83
4.2.4 The Bandwidth-Distance Threshold......Page 84
Table 4.1. Maximum Size of a Local Area (only) According to the Bandwidth-Distance Threshold for Example Wireless Applications......Page 86
4.3.2 Nonspecular Wave Component......Page 87
4.3.4 Reduced Wave Grouping......Page 88
4.4.1 Stochastic Model......Page 89
4.4.3 Other Random Quantities......Page 90
Uncorrelated Phases......Page 91
Figure 4.9. Venn diagram of various local area channel models.......Page 92
4.4.5 Fourier Transforms......Page 93
Theorem 4.2: U-SLAC Wide-Sense Stationarity......Page 94
Theorem 4.3: WSS Heterogeneous Scattering......Page 95
Diffuse Components......Page 96
Figure 4.10. Different delay spectra using the standard power spectrum, 
and the integrated power spectrum,......Page 97
Problems......Page 98
5.1.1 Average Versus Received Power......Page 103
Theorem 5.1: First-Order Stationarity......Page 104
5.1.3 Mean U-SLAC Power......Page 106
5.1.5 Ergodicity......Page 107
Theorem 5.3: Power Ergodicity......Page 108
5.2.1 Notes and Concepts......Page 110
5.2.2 Characteristic Functions......Page 111
5.2.4 Diffuse, Nonspecular Characteristic Function......Page 112
5.3.1 The One-Wave PDF......Page 115
Table 5.1. Summary of Envelope PDFs in Different Fading Environments......Page 116
Figure 5.3. Two-wave PDF and CDF with varying ? [Dur02].......Page 117
5.3.3 The Three-Wave PDF......Page 118
Figure 5.4. Three-wave CDF and PDF for four cases [Dur02].......Page 120
5.3.4 The Rayleigh PDF......Page 121
5.3.5 The Rician PDF......Page 122
Figure 5.6. Rician PDF and CDF as the dominant multipath component increases 
[Dur02].......Page 123
5.4.1 Approximate Representation......Page 125
Table 5.2. Exact Coefficients for the First Five Orders of the Approximate TWDP Fading PDF......Page 126
5.4.2 Graphical Analysis......Page 127
5.4.3 Rayleigh and Rician Approximations......Page 128
Figure 5.9. TWDP PDF and CDF for K = 3 dB [Dur02]......Page 129
Figure 5.10. TWDP PDF and CDF for K = 6 dB [Dur02]......Page 130
Figure 5.11. TWDP PDF and CDF for K = 10 dB [Dur02]......Page 131
5.5 Chapter Summary......Page 133
Problems......Page 135
5.A Envelope Characteristic Functions......Page 137
6.1.1 Scalar Collapse of Position Vectors......Page 140
6.1.2 Scalar Collapse of Wavevectors......Page 141
6.2.1 Definition of the Angle Spectrum......Page 144
6.2.2 Mapping Angles to Wavenumbers......Page 147
6.2.3 From-the-Horizon Propagation......Page 148
Figure 6.4. Multipath power is mapped from the angle spectrum, p(?), to the wavenumber spectrum, 
, as a function of its angle-of-arrival [Dur00a].......Page 149
Figure 6.5. Autocorrelation and spectrum relationships for the space-varying channel.......Page 150
Figure 6.6. Examples of multipath angular spread values, ?.......Page 153
Azimuthal Direction of Maximum Fading......Page 154
6.3.3 Comparison to Omnidirectional Propagation......Page 155
Figure 6.8. A graphical summary of shape factor behavior.......Page 156
Figure 6.9. Overview of multipath azimuth spectrum studied in this section [Dur00b].......Page 157
6.4.1 Two-Wave Channel Model......Page 158
6.4.2 Sector Channel Model......Page 159
Figure 6.11. Multipath sector propagation model [Dur00b].......Page 160
6.4.3 Double-Sector Channel Model......Page 161
6.4.4 Rician Channel Model......Page 162
6.5 Chapter Summary......Page 163
Problems......Page 164
7.1.1 Level-Crossing Rate......Page 170
Figure 7.1. Envelope process with level crossings, fade durations, and threshold level ?rms.......Page 171
7.1.2 Average Fade Duration......Page 172
7.1.4 Level Crossing in Space......Page 173
7.2 Envelope Unit Autocovariance......Page 175
7.2.1 Temporal Unit Autocovariance......Page 176
7.2.2 Frequency Unit Autocovariance......Page 177
7.2.3 Spatial Unit Autocovariance......Page 178
7.2.4 Joint Unit Autocovariance......Page 180
7.3.1 Classical Models......Page 181
7.3.2 Channel Model Solutions......Page 182
Table 7.1. Second-Order Statistics for Rayleigh Envelopes in Time, Frequency, and Space......Page 183
Figure 7.4. Comparison of Clarke theoretical and approximate envelope autocovariance functions for Ez-case.......Page 185
Figure 7.6. Comparison of Clarke theoretical and approximate envelope autocovariance functions for Hy-case.......Page 186
7.4.1 Discrete Wideband Channels......Page 187
Figure 7.7. A discrete, tap-delay line channel model with additive white Gaussian noise.......Page 188
7.4.2 Time-Varying Wideband Channels......Page 189
Figure 7.8. An example of how the complex, baseband radio channel is sampled in time and delay to form a matrix.......Page 190
7.4.3 Discrete Transmission......Page 191
7.4.4 Notes on Temporal Modeling......Page 192
7.4.5 Rician Fading in Time-Varying Channels......Page 193
Figure 7.10. Computer-generated example of a wideband temporal channel 
with Rician K = 10 dB.......Page 194
Problems......Page 195
7.A Approximate Spatial Autocovariance......Page 198
7.B Classical Envelope Autocovariance......Page 200
7.C Rician Mean Approximation......Page 201
Figure 7.11. Comparison of the exact Rician mean and the algebraic approximation.......Page 202
8.1.1 The Role of Diversity......Page 203
8.1.2 Antenna Diversity......Page 204
8.1.4 Diversity Failure......Page 205
8.2 Combining Techniques......Page 206
8.2.1 Gain Combining......Page 207
8.2.2 Signal Envelope for Gain Combining......Page 210
8.2.3 Switch Combining......Page 211
Figure 8.3. A comparison of outputs from EGC, MRC, and pure selection diversity-combining techniques.......Page 212
Table 8.2. Overview of Diversity-Combining Techniques......Page 213
Table 8.3. BER Expressions for Various Modulation Types[*]......Page 214
8.3.4 Empirical BER and Capacity......Page 216
8.3.5 Diversity Gain for Multiple Branches......Page 218
Example 8.2: Diversity Gain for QPSK......Page 219
Figure 8.7. CDF of Shannon capacity for EGC diversity using two correlated Rayleigh branches, each with average SINR = 10 dB.......Page 220
Figure 8.8. CDF of Shannon capacity for pure selection diversity using two correlated Rayleigh branches, each with average SINR = 10 dB.......Page 221
8.3.7 Illustration of Unequal Branch Power on Diversity......Page 222
Figure 8.9. CDF of Shannon capacity for MRC diversity using two Rayleigh-fading branches with unequal average signal strength.......Page 223
Figure 8.11. CDF of Shannon capacity for pure selection diversity using two Rayleigh-fading branches with unequal average signal strength.......Page 224
Problems......Page 225
Figure 9.1. A SISO antenna configuration.......Page 228
Figure 9.2. A SIMO antenna configuration.......Page 229
9.1.3 Multiple-Input, Single-Output (MISO)......Page 230
9.1.4 Multiple-Input, Multiple-Output (MIMO)......Page 231
9.2.1 MIMO Channel Matrix......Page 232
Figure 9.5. Block diagram of the different channels that exist in a MIMO radio link.......Page 233
At the Receiver......Page 235
9.2.3 Separate Channels......Page 237
Figure 9.7. Physically, MIMO spatial coding sends different symbol streams to and from different directions in space.......Page 239
9.2.4 Formal Capacity Expressions......Page 240
Absence of Multipath......Page 242
Figure 9.10. A keyhole channel creates a bottleneck for the MIMO system.......Page 244
9.3.1 Practical Signal Extraction......Page 245
Figure 9.11. A practical M = 4, N = 4 MIMO transmission system [Fos96].......Page 246
9.3.3 Subtraction of Interference......Page 248
9.4 Space–Time Block Coding......Page 250
9.4.1 MISO Revisited......Page 251
Example 9.1: Capacity of MISO Block Code......Page 252
Figure 9.12. Space–time block codes allow the rec......Page 254
Problems......Page 256
Chapter 10. Array Design in Multipath......Page 259
10.1.2 Approximate Autocovariance......Page 260
Figure 10.1. When the roles of transmitter and receiver are reversed, angles-of-arrival become angles-of-departure.......Page 261
10.1.3 Forbidden Zones of Correlation......Page 262
Figure 10.4. Antenna elements must be placed so that they do not intrude into the forbidden zone (dotted ellipse) of another element [Dur01c].......Page 263
Figure 10.5. The three cases of forbidden zones when coupling is considered.......Page 264
Figure 10.6. Two design philosophies when designing systems in which the orientation of propagation with respect to transmitter or receiver is unknown.......Page 265
Figure 10.7. The physics of local area propagation for transmitter and receiver antennas [Dur01b].......Page 266
10.2.2 Double Spatial Channel Correlation......Page 267
Figure 10.8. Top view of the antenna configurations and multipath angles-of-departure (from transmitter) and angles-of-arrival (to receiver) [Dur01b].......Page 269
10.3 Example System......Page 270
Figure 10.11. Multipath angle spectrum models for the base station and user terminals.......Page 271
Figure 10.12. Three elements in an equilateral triangle configuration.......Page 272
10.4.2 Description of Peer-to-Peer Measurement Technique......Page 273
Figure 10.14. In a local area, power delay profiles are measured along two orthogonal linear tracks using an omnidirectional antenna (a) and also by spatially averaging angular sweeps with a directional antenna (b) [Dur03].......Page 274
Figure 10.17. A local area angle–delay spectrum a......Page 275
Table 10.1. Dispersion Statistics Calculated from Track Measurements [Dur03]......Page 276
Table 10.2. Spatial Multipath Parameters Calculated from Spatially Averaged Azimuthal Sweeps of a Horn Antenna [Dur03]......Page 277
Figure 10.18. The trend between multipath angular spread, ?, and RMS delay spread, ??, for indoor and outdoor receiver locations [Dur03].......Page 278
Problems......Page 279
10.A.1 Noncoherent Channel Measurements......Page 282
10.A.2 Power Spectra......Page 283
10.A.4 Angle-of-Arrival Parameters......Page 284
Table A.1. Properties of the Delta Function, ?(t)......Page 285
Figure A.1. A graph of the communications sinc function.......Page 286
Table A.2. Common Examples of Singularity Functions......Page 287
A.4 Bessel Functions......Page 288
Table A.3. Useful Bessel Function Properties......Page 289
A.5 Complete Elliptic Integral Functions......Page 290
Table A.4. Tabulated Values for the Bessel Function, J0(x), and Modified Bessel Function, I0(x).......Page 291
Table A.5. Tabulated Values for the Complete Elliptic Integrals......Page 293
Figure A.4. A graph of the Q-function.......Page 294
Table A.6. Tabulated Values for the Q-function, Q(x)......Page 295
Example B.1: Frequency Domain to Time Domain......Page 297
Table B.1. Elementary Fourier Transform Pairs......Page 299
Table B.2. Advanced Fourier Transform Pairs......Page 301
Table B.3. Properties of the Fourier Transform......Page 302
B.4 Space–Wavenumber Transforms......Page 303
Table B.6. Steps for Calculating a Space–Wavenumb......Page 304
Table B.7. Useful Trigonometric Identities......Page 305
C.1.1 Random Variables......Page 307
Figure C.1. The time-varying random process, 
, has a stochastic ensemble consisting of many individual realizations.......Page 308
C.2.1 Definitions......Page 309
C.2.2 Joint Distributions......Page 311
C.2.3 Computing Statistics......Page 312
C.3.1 Functions With Inverses......Page 313
C.3.2 Multiroot Functions......Page 314
Example C.2: Sinusoidal Mapping......Page 315
D.1 Mathematical Symbols and Conventions......Page 317
Bibliography......Page 320