Index

A p/4-QPSK, 133–136 American Digital Cellular and Japanese time domain waveforms, 131, 132 Digital Cellular systems, 138 waveforms, 130 Amount of fading (AF) signal space, 129 cascaded fading channels, 233, 235 symbol error Gaussian pdf, 366 coherent MPSK, 137, 138 lognormal shadowing channel, 364, 366 equivalent bit error rate, 137 MRC diversity, 362 generic MPSK constellation, 136, 137 Nakagami fading, 211 k bits encoding/mapping, 136 Nakagami parameter, 364, 366 spectral efficiencies and SNR, 138, 139 Rayleigh fading, 200–201 , 13–15 Rician fading, 203, 204 Bit error rate (BER), 255, 256, 307 shadowing models, 216 BPSK, 256–259 short term faded Nakagami channel, 362 CDF, 259 SSC, 363 fading channel, 259 two channel selection combining moderately shadowed channel, 260, 262 diversity, 363 MRC, 448–450 Amplitude shift keying (ASK), 121–122 Nakagami channels Arnold and Strauss’s bivariate gamma average error probability, 442, 444, 445 distribution, 76 hypergeometric function, 443 MeijerG function, 443 pdf, 449 B Rayleigh channels, 441–442 Bayes theorem, 86–87 severely shadowed channel, 260, 261 BER. See Bit error rate shadowed fading channels Bessel function, 221, 303 average error rates, 446, 447 , 11–13 CCI effect, 446 Binary phase shift keying (BPSK) cochannels, 448 digital signal spectra, 127 N Nakagami interferers, 445 gray encoding, 129, 130 BPSK. See Binary phase shift keying M-ary signals, 128–129 MQAM, 139–141 null-to-null criterion, 127 C QPSK Cascaded channels, 306 modulator, 131, 132 Cascaded fading channels offset-QPSK scheme, 133, 134 AF, 233, 235 phase constellation, 131 average SNR, 233

P.M. Shankar, Fading and Shadowing in Wireless Systems, 455 DOI 10.1007/978-1-4614-0367-8, # Springer Science+Business Media, LLC 2012 456 Index

Cascaded fading channels (cont.) probability of miss, 91, 92 CDF, 235, 236 Q function, 92 density function, 233, 234 Rayleigh densities, 90 gamma-distribution, 232 Rician densities, 90 MeijerG function, 232–233 Differential phase shift keying (DPSK), 154 Nakagami-gamma model, 234 Digital frequency modulation (DFM) Cascaded Nakagami channels block diagram, 142, 143 AF, 395, 400, 401 CPFSK, 144 average error probabilities, 405, 406 frequency deviation, 143 BER, 401, 402 pulse shape, 143 CDFs, 396–401, 404, 405 Diversity techniques, 1 density function, SNR, 395 AF Matlab, 396 Gaussian pdf, 366 M-fold convolution, 403, 404 lognormal shadowing channel, 364, 365 MGF, 404, 405 MRC diversity, 363 MRC diversity, AF, 403–404 Nakagami parameter, 364, 366 multihop relayed communication short term faded Nakagami channel, 362 system, 395 SSC, 364 order statistics, 400 two channel selection combining outage probability, 401, 403, diversity, 363 405, 407 average error probability pdfs, SNR, 396, 398–400, 402 average error rate, 366–367 received signal power, 395 CBPSK, 367 threshold SNR, 404 CDF, 369 , 15–16 density functions, 373–377 CDFs. See Cumulative distribution functions dual branch correlated MRC, 372 Central limit theorem (CLT), 46–47 dual branch MRC, 375, 377–379 Characteristic function (CHF), 10–11, 57 dual branch SC, 371, 372, 378, 379 Chebyshev inequality, 94–95, 323 four-branch diversity, 371 Chi-squared distribution, 16–18 four channel MRC, 375, 378 Cochannel interference, 5 GSC algorithm, 371, 373 Coherent binary phase shift keying hybrid diversity approach, 375 (CBPSK), 367 Laplace transform, 367 Cumulative distribution functions (CDFs), 8–9, MGF, 367–368 207–208, 322, 330–331 MRC algorithm, 369 Nakagami channel, 368, 373 Nakagami-lognormal model, 373 D Nakagami-m channel, MRC, 370, 371 Decision theory and error rates Q function, 368 channel noise, 85–86 shadowed fading channels, 375, 377, Gaussian case 378, 379 Bayes theorem, 86–87 branch correlation effects binary channel, 86 correlated SC and SSC receiver, 337 conditional density functions, 87 exponential correlation, 334 conditional probability, 87 joint CDF, 339 error probability, 87, 89, 90 Marcum’s Q function, 337 false alarm probability, 89 modified Bessel function, 337, 339 hypothesis testing, 86, 88 MRC algorithm (see Maximal ratio likelihood ratio, 89 combining algorithm) probability of miss, 89 Nakagami parameter, 337, 338 non-Gaussian case pdf, SC algorithm, 339 false alarm probability, 91 cascaded Nakagami channels (see hypothesis testing, 91, 93 Cascaded Nakagami channels) Index 457

CDF, 325 short-term faded channel, 381 diversity receivers, 321 pdf, 314 EGC algorithm, 324 polarization diversity, 318–319 frequency diversity, 318 Rayleigh faded channel, 315 gamma shadowing, 362 received signal power, 315 generalized gamma and Weibull channels SC algorithm AF, 386 CDFs, 322 average error probability, 391, 392 SNR expression, 321–322 BER, dual diversity, 393, 394 SSC algorithm (see Switched and stay CDFs, 387–389, 391 combining algorithm) coherent BPSK modem, 389 signal processing methods, 313 complementary incomplete SNR, 316 gamma function, 390 space diversity, 317–318 detection scheme, 390 time diversity, 319 dual MRC, 392 types, 316 dual SC, BER, 394 “keyhole” scattering, 395 MGF, 392 E modulation type, 389 Equal gain combining (EGC) algorithm, 324 Nakagami-m distribution, 385 Ergodic channel capacity Nakagami pdf, 385 additive white Gaussian noise, 284, 285 SNR, pdfs, 385, 387, 388, 391 cascaded short-term fading channel, 287 GSC algorithm, 361 channel bandwidth, 285 average error probabilities, 408, 409 density function, generalized K average SNR, 357 distribution, 289–290 CDFs, 357–359, 408, 409 double Nakagami cascaded channels, 288 CDMA system, 353 MeijerG function, 285 densities and distributions, 359–360 Nakagami faded channel, 285, 286 gamma random variables, 359 Nakagami-Hoyt channels, 287, 288 joint pdf, 354–356 normalized average channel capacity, 285 Mc signals, 354 quadruple cascaded channels, 287, 289 Nakagami channels, 358, 360 Rayleigh fading, 286 Nakagami-m faded channels, 360, 361 Rician faded channel, 286, 287 normalized peak values, 360 shadowed fading channels, 290, 291 outage probabilities, 407, 410 triple cascaded channels, 287, 289 output SNR, pdfs, 408 , 18 RAKE reception, 353 , 18–20 Rayleigh channel, 354–356 three-branch diversity receiver, pdfs, 358, 359 F macrodiversity (see Macrodiversity F (Fisher-Snedecor) distribution, 20–21 techniques) Frequency diversity, 318 MRC algorithm (see Maximal ratio Frequency shift keying (FSK), 124–126, combining algorithm) 141–142 multipath diversity, 320 Nakagami-m distribution (see Nakagami-m distribution) G noise power, 315 , 21–23 outage probability Gaussian distribution, 33–35 CDF, 381 Gaussian function, 301 MRC algorithm, 382, 384 Gaussian minimum shift keying (GMSK), Nakagami fading channels, 382 146–149 SC algorithm, 382 Generalized Bessel K (GBK) distribution, 65 sensitivity, 381 Generalized gamma distribution, 25 shadowed fading channel, 383–385 definition, 22 458 Index

Generalized gamma distribution (cont.) gamma pdf, 344 generalized gamma pdf, 24 GK distribution, 344, 345 normalization factor, 25 joint pdf, 345–346 random variable, 24 MeijerG functions, 351 scaling factor, 25 MRC diversity, 348, 349 Stacy distribution, 24 MRC–SC, 345–347, 350–351 two-sided generalized gamma pdf, 26 pdfs, 352, 353 Generalized K (GK) distribution, 344, 345 shadowed fading channel, 349 Generalized selection combining (GSC) short-term fading effects, 343 algorithm, 361 SNR, SC, 349–350 average error probabilities, 408, 409 shadowing mitigation average SNR, 357 CDF, SC algorithm, 341 CDFs, 357–359, 408, 409 dual correlated Nakagami channels, CDMA system, 353 342 densities and distributions, 359–360 multiple base stations, 340 gamma random variables, 359 normalized Gaussian variable, 341 joint pdf, 354–356 SNR, 340, 342, 343 Mc signals, 354 three-base station arrangement, Nakagami channels, 358, 360 340, 341 Nakagami-m faded channels, 360, 361 Marcum’s Q functions, 3, 179–182 normalized peak values, 360 M-ary phase shift keying (MPSK). See Binary outage probabilities, 407, 410 phase shift keying output SNR, pdfs, 408 M-ary quadrature amplitude modulation RAKE reception, 353 (MQAM), 139–141 Rayleigh channel, 354–356 Matlab and Maple, 5 three-branch diversity receiver, pdfs, 358, 359 Maximal ratio combining (MRC) algorithm, GK distribution. See Generalized K 448–450 distribution Chebyshev inequality, 323 GMSK. See Gaussian minimum shift keying correlation coefficient correlated branch expression, 334 density functions, 335, 336 K SNR, fractional decline, 337 Kibble’s bivariate gamma distribution, 76 noise power, 323 coefficient, 10 processing algorithm, 322 signal power, 323 McKay’s bivariate gamma distribution, 76 L Meijer G function, 67, 68, 299–301, 304 , 27–28 MGF. See generating function Laplace transforms, 11, 307 Minimum shift keying (MSK) 8-Level phase shift keying (8PSK) modulator, 144, 145 demodulator, 158 power spectrum, 144 modulator, 158 waveform, 144–146 phase constellation, 156, 157 Modems phase encoding, 156, 157 bandwidth requirement, 174 waveform, 159 bit energy, 163 Lognormal distribution, 28–29 carrier regeneration and synchronization, 166–168 channel capacity, 175 M complementary error function, 176–179 Macrodiversity techniques correlator, 114 microdiversity systems cos() and sine() functions, 119 CDFs, 352 differentially encoded signals density functions, 348, 349 coherent vs. noncoherent modems, 156 Index 459

DPSK, 154 inter channel interference, 160, 162 error probability, 152 inverse fast Fourier transform approach, 8-level PSK (see 8-Level phase shift 162 keying) subcarrier frequencies, 160–161 modified Bessel function, 153 transmitter and receiver, 160, 161 noise , 153 optimum filter, impulse response, 113 noncoherent detection, 152 phase mismatch noncoherent receiver, 154, 155 angular mismatch, 168, 169 non-Gaussian statistics, 152 coherent QPSK, 172 phase encoding, 155 correlation function, 171 Rician statistics, 153 fixed phase mismatch, 168, 169 digital communication system, 110 jitter, 170 digital modulation techniques, 117–118 matched filter, 170 digital signal bandwidth, 164–166 normalized timing offset, 171–172 digital transmission, 110, 111 random phase mismatch, 168, 170 error function, 176 timing offset, 170 Euclidian distance, 120 power spectral density, 113 gamma functions, 177–178 probability of error, 119–121 Gaussian Q function, 112 probablity density function, 112 general nonlinear modulation schemes pulse shapes, 114–116 DFM (see Digital frequency Q functions, 176, 179 modulation) Shannon’s channel capacity theorem, 174 error rates, 151 signal-to-noise ratio, 163 FSK, 141–142 sin c pulse, 116 GMSK, 146–149 spectral and power efficiencies, 174 MSK (see Minimum shift keying) symbol energy, 163 orthogonal M-ary FSK, 147, 149–151 zero mean Gaussian white noise, 111 gray coding, 110 Moment generating function (MGF), 11, 367 inter symbol interference MQAM. See M-ary quadrature amplitude cosine pulse shape, 183–185 modulation eye pattern, 185, 186 MRC algorithm. See Maximal ratio combining impulse response, 182 algorithm roll-off factor, 183 MSK. See Minimum shift keying Kroenecker delta function, 118 Multipath diversity, 320 Marcum’s Q functions, 179–182 Multipath propagation phenomenon, 3 M-ary signaling, 110 M-ary transmission system, 118 matched filter, 114 N modulation techniques Nakagami channels, 426–427 ASK, 121–122 bit error rates BPSK (see Binary phase shift keying) average error probability, 442, 444, 445 FSK, 124–126 hypergeometric function, 443 PSK, 122–124 MeijerG function, 443 QAM, 125 pdf, 436 N-dimensional orthogonal space, 118 , 30–32 noise power, 163 Nakagami fading noise variance, 112 AF, 211 Nyquist criteria, 115, 117 CDF, 207–208 OFDM channel, 303 digital Fourier transforms principle, chi-square density function, 208–209 161, 162 clustering/bunching concept, 208 fast Fourier transform algorithm, 162 density moments, 207 frequency domain representations, 160 Gaussian densities, 209, 210 460 Index

Nakagami fading (cont.) correlation coefficient, 45 Nakagami-m pdf moments, 208, 209 covariance, 45 received signal envelope, 206–207 decision theory and error rates (see Rician envelope densities, 209–211 Decision theory and error rates) Nakagami-inverse Gaussian distribution, 226 , random variables, Nakagami-m distribution 44–45 CDF, 325, 328–330 independent random variables, 45 density functions, 328–329 joint densities, 43 EGC algorithm, 327 joint pdf, multiple random variable Matlab, 328 CDF differentiation, 56, 57 MRC algorithm, 325–327 diversity combining algorithm, 53 received signal power, 325 Jacobian, 51, 52 SC algorithm, 325, 326 Leibniz’s rule, 55 SNR improvement, 329 probability volume calculation, 55 statistical model, 324 region of interest, 53, 54 Nakagami-N-gamma channels, 4 two-scaled random variables, 52 Natural logarithm, 303 Laplace transforms, 11 Non-central chi-squared distribution, 32–33 marginal densities, 43–44 Nyquist criteria, 115, 117 moment generating functions, 11 order statistics bivariate correlated pdf, 83 O bivariate lognormal densities, 84 Orthogonal frequency division multiplexing bivariate Nakagami pdf, 82 (OFDM), 3 bivariate Rayleigh pdf, 82–83 digital Fourier transforms principle, 161, exponential pdf, 82 162 generalized Bernoulli trial, 81 fast Fourier transform algorithm, 162 joint CDF, 80 frequency domain representations, 160 lognormal random variables, 85 inter channel interference, 160, 162 Marcum Q function, 83 inverse fast Fourier transform approach, 162 selection combining algorithm, 79 subcarrier frequencies, 160–161 short-term fading, indoor wireless transmitter and receiver, 160, 161 channels, 84 orthogonality, 45 pdf, 10 P beta distribution, 11–13 pdf. See Probability density function binomial distribution, 13–15 Phase shift keying (PSK), 122–124 Cauchy distribution, 15–16 , 35 chi-squared distribution, 16–18 Polarization diversity, 318–319 Erlang distribution, 18 Probability and statistics concepts exponential distribution, 18–20 bivariate correlated distributions F distribution, 20–21 gamma pdf, 76–77 gamma distribution, 21–23 generalized gamma pdf, 77–78 Gaussian distribution, 33–35 Nakagami pdf, 75–76 generalized gamma distribution (see normal pdf, 73–75 Generalized gamma distribution) Rician distribution, 79 Laplace distribution, 27–28 Weibull pdf, 78 lognormal distribution, 28–29 CDF, 8–9 Nakagami distribution, 30–32 Chebyshev inequality, 94–95 non-central chi-squared distribution, Chernoff bound, 96–97 32–33 CHF, 10–11, 57 Poisson distribution, 35 CLT, 46–47 , 36–38 conditional densities, 44 rectangular/uniform distribution, 38–40 Index 461

student’s t distribution (see Student’s Nakagami distribution, 30–32 t distribution) Nakagami-Hoyt pdf, 282, 283 , 41–43 Nakagami-m pdf, 201 random variable transformations non-central chi-squared distribution, 32–33 Bessel function, 59, 63 Poisson distribution, 35 doubling formula, 62 Rayleigh channels, 434–435 gamma random variable sum, 62, 66 Rayleigh distribution, 36–38 Gaussian, chi-squared and student rectangular/uniform distribution, 38–40 t distributions relationship, 72–73 Rician fading, 206 GBK distribution, 65 second order statistics K distribution, 64 conditional pdf, 293 lognormal fading conditions, 68 envelope pdf, 296 marginal density function, 58 SNR, joint pdf, 292 Meijer’s G function, 67, 68 shadowed fading channels monotonic transformation property, 63 N interfering channels, 439 Nakagami-m distributed envelope probability density functions, 437, 438, values, 69 440, 441 phase statistics, 59 signal-to-CCI ratio, 437, 438 Rician factor, 60–61 SNR, density function, 436, 437, 439 SNR, 70, 71 student’s t distribution (see Student’s stochastic processes, 97 t distribution) autocovariance, 99 Weibull distribution, 41–43 ergodicity concept, 100 PSK. See Phase shift keying first order density function, 98 higher-order density functions, 99 power spectral density, 100 Q second order density function, 98 Quadrature amplitude modulation (QAM), 125 spectral density, 101 Quaternary phase shift keying (QPSK) thermal noise, 101 modulator, 131, 132 time domain function, 97 offset-QPSK scheme, 133, 134 wide stationary sense process, 100 phase constellation, 131 Y = g(X), pdf and CDF derivation p/4-QPSK, 133–136 filter, input-output relationship, 47, 48 time domain waveforms, 131, 132 many-to-one transformation, 49, 50 waveforms, 130 monotonic transformation, 47, 48 non-monotonic transformation, 48, 49 Z = X + Y, pdf, 50–51 R Probability density function (pdf), 10, 198 RAKE receiver, 320 beta distribution, 11–13 Rayleigh channels binomial distribution, 13–15 bit error rates, 421, 441–442 Cauchy distribution, 15–16 N cochannels, 419 chi-squared distribution, 16–18 outage probability, 418, 419 Erlang distribution, 18 CCI channels, 425, 426 exponential distribution, 18–20 desired signal power, 424, 425 F distribution, 20–21 multiple cochannels, 424 gamma distribution, 21–23 SIR/q, 422 Gaussian distribution, 33–35 Z0/ZT, 423 Gaussian pdf, 215 pdf, 434–435 generalized gamma distribution (see Poisson random variable, 424 Generalized gamma distribution) protection ratio, 420 Laplace distribution, 27–28 signal-to-interference, 420 lognormal distribution, 28–29 threshold and protection factor, 421 Nakagami channels, 436 Rayleigh distribution, 36–38 462 Index

Rayleigh fading MeijerG function and Bessel function, AF, 200–201 238, 240 amplitude, 197 Nakagami-N-gamma distribution, 237 exponential densities, 198, 199 outage probabilities and error rates, 241 Gaussian random variables, 198 shadowing, higher level, 240, 241 histogram, 198–200 conditional density function, 429 inphase and quadrature notation, 197 equivalent gamma model, 429 multipath phenomenon, 197 generalized gamma model, 223–225 Nakagami-m pdf, 201 Nakagami-gamma/generalized K models, pdf, 198 220–222 Rayleigh density function, 200 Nakagami-inverse-Gaussian model, Rayleigh-lognormal channel, 221 221–223 Rectangular/uniform distribution, 38–40 Nakagami-lognormal models Rician factor, 60–61 average power, 218 Rician fading density functions, 219 AF, 203, 204 vs. short term fading, 219, 220 average power, 203, 204 outage probability density function, 201, 202, 206 CDF, 271, 273 error function, 206 density function, 271, 272 Gaussian statistics, 204–205 multiple interfering shadowed fading Marcum Q function, 203 channels, 432, 433 mean and second moments, 202–203 Nakagami-cascaded gamma channel, phase, pdf, 206 268, 270 received power, 201 Nakagami inverse Gaussian channel, Rician factor, 202, 205 271 Nakagami-lognormal channel, 267–271 S Nakagami-m distributed cochannel, Selection combining (SC) algorithm, 429, 430 321–322 Nakagami parameter, 268, 269 Shadowed fading channels single interfering channel, 431, 432 AF, 225–226 pdf, 440, 441 BER, 255, 256 N interfering channels, 439 average error rates, 446, 447 signal-to-CCI ratio, 437, 438 BPSK, 256–259 SNR, density function, 436, 437, 439 CCI effect, 446 shadowing density, 429 CDF, 259 Shannon’s channel capacity theorem, 174 cochannels, 448 Short-term fading, 194 fading channel, 259 attenuation, 195 moderately shadowed channel, average power, gamma pdf moments, 211 260, 262 CDF, 213 Nakagami interferers, 445 gamma fading channel, 211, 214, 215 severely shadowed channel, 261 generalized gamma channel, 212–215 cascaded approach Nakagami fading AF, 242, 243 AF, 211 CDF, 242, 244, 245 CDF, 207–208 central limit theorem, 237 chi-square density function, 208–209 density function, 238, 240, 241 clustering/bunching concept, 208 digamma and trigamma function, 238 density moments, 207 gamma parameter, 239 Gaussian densities, 209, 210 gamma random variables, 238 Nakagami-m pdf moments, 208, 209 GK distribution, 236 received signal envelope, 206–207 inverse Gaussian distribution, 237 Rician envelope densities, 209–211 Index 463

Rayleigh fading vs. SC dual branch, density functions, AF, 200–201 331, 332 amplitude, 197 vs. SC SNR enhancement, 333, 334 exponential densities, 198, 199 Gaussian random variables, 198 histogram, 198–200 T inphase and quadrature notation, 197 Thermal noise, 101 multipath phenomenon, 197 Time diversity, 319 Nakagami-m pdf, 201 Time domain function, 97 pdf, 198 Rayleigh density function, 200 Rician fading W AF, 203, 204 Weibull distribution, 41–43 average power, 203, 204 Weibull fading model, 213–215 density function, 201, 202, 206 Wireless channels, 1, 2, 193 error function, 206 attenuation, 195 Gaussian statistics, 204–205 average probability of error Marcum Q function, 203 bit error rate (see Bit error rate) mean and second moments, 202–203 coherent BPSK modem, 433 phase, pdf, 206 probability density function (see received power, 201 Probability density function) Rician factor, 202, 205 cascaded models statistical characteristics, 196 cascaded fading channels (see Cascaded Weibull fading model, 213–215 fading channels) Signal transmission and deterioration, 1 cooperative diversity, 231 coefficient, 10 double Rician channels, 245–246 Space diversity, 317–318 multiple scattering, 231 SSC algorithm. See Switched and stay N*Weibull channels, 245 combining algorithm shadowed fading channels (see Stacy distribution, 24 Shadowed fading channels) Statistical decision theory, 3 composite model Statistical distributions, 2 cluster power, 227 Stochastic processes, 97 double Nakagami fading and autocovariance, 99 Nakagami-m fading, 228–229 ergodicity concept, 100 double Rayleigh and Rayleigh densities, first order density function, 98 228 higher-order density functions, 99 gamma–Weibull/Weibull–Weibull power spectral density, 100 channel, 230 second order density function, 98 “miniclusters,” 230 spectral density, 101 received signal power, 226–227 thermal noise, 101 scaling factors, 227 time domain function, 97 signal-to-noise ratio, 230–231 wide stationary sense process, 100 Doppler fading, 196 Student’s t distribution, 41 ergodic channel capacity cumulative distributions functions, 41, 42 additive white Gaussian noise, 284, 285 Gaussian and Cauchy distributions, 39 cascaded short-term fading channel, 287 moments of, 40 channel bandwidth, 285 random variable, 40 density function, generalized K Switched and stay combining (SSC) algorithm distribution, 289–290 CDFs, 330–331 double Nakagami cascaded channels, logic and switching circuits, 329 288 Nakagami parameter, SNR improvement, MeijerG function, 285 332, 333 Nakagami faded channel, 285, 286 464 Index

Wireless channels (cont.) long-term fading, 195 Nakagami-Hoyt channels, 287, 288 multipath transmission, 194 normalized average channel multiple scattering, 195 capacity, 285 outage probability quadruple cascaded channels, 287, 289 cascaded N*Nakagami channel, Rayleigh fading, 286 266, 267 Rician faded channel, 286, 287 CDF, 265 shadowed fading channels, 290, 291 generalized gamma fading channel, triple cascaded channels, 287, 289 264, 265 error rates Nakagami channels, 426–428 cascaded channels, 253, 255, 256 Nakagami-Hoyt channels, 283, 284 complimentary error function, 248 Nakagami-m faded channel, 263 generalized gamma channel, 252–254 Rayleigh channels (see Rayleigh ideal Gaussian channel, 247 channels) lognormal channel, 252, 253 shadowed fading channels (see lognormal random variable, 251–252 Shadowed fading channels) lognormal shadowing, 251 SNR, 262 Nakagami channels, 250–251 Weibull channels, 265, 266 Nakagami-Hoyt channels, 283, 284 second order statistics power penalty, 247, 249 average fade duration, 292, 294, 296 Rayleigh fading channel, 248 conditional pdf, 293 Rician channel, 251 envelope pdf, 296 shadowed fading channels (see level crossing rates, 292 Shadowed fading channels) maximum Doppler frequency shift, 293 signal-to-noise ratio, 249 Nakagami fading channel, 297 SNR correction factor, 252 Rayleigh faded channel, 292–293 Weibull channels, 253, 254 Rician channel, LCR, 295 fading channel, 304 shadowed fading channels, frequency selective fading channels, 196 297, 298 general fading models SNR, joint pdf, 292 AF, 283 temporal characteristics, 291 central limit theorem, 274 transmitter and receiver effects, cluster based scattering model, 274, 275 290–291 Gaussian random variables, 279 Weibull fading, 295–296 generalized Nakagami pdf, 282 shadowed fading channels, 196 (see also Z–m densities, 276 Shadowed fading channels) a–Z–m distributions, 273 shadowing models Hoyt fading channel, 278–279 AF, 216 k–m density, 280, 281 definition, 215 a–l–m distributions, 273 gamma distribution, 216 modified Bessel function, 277 Gaussian pdf, 215 Nakagami-Hoyt distribution, 276 lognormal density function, 216, 217 Nakagami-Hoyt pdf, 282, 283 shadowing parameter vs. gamma Nakagami parameter, 277, 278 parameter, 217, 218 , 279 short-term fading (see Short-term fading) total power, 280 variance ratio, inphase and quadrature unified model, 273 component, 275, 278