Consider the random variable
where the are independent, exponentially distributed random variables with unequal positive means The exponential probability density func- tion of is
A straightforward calculation yields the characteristic function
Since Z is the sum of independent random variables, (D-43) implies that its characteristic function is
To derive the probability density function of Z, (D-7) is applied after first expanding the right-hand side of (D-44) in a partial-fraction expansion. The result is
where
and A direct integration and algebra yields the probability distribution function
Equations (D-45) and (D-12) give
When the are equal so that then
Therefore, the probability density function of Z is
which is a special case of the gamma density function. Successive integration by parts yields
From (D-49) and (D-12), the mean and variance of Z are found to be
Index
Acquisition, 183–208
consecutive-count strategy, 187 lock mode, 187
matched-filter, 184–185, 192, 214–
221
multiple-dwell, 186 parallel array, 183 sequential detection, 208 sequential estimation, 183 serial-search, 185–208, 221–226 single-dwell, 186
up-down strategy, 187 verification mode, 187 Acquisition correlator, 201–208 Acquisition time, 188
Adaptive ACM filter, 123–125 Adaptive filters, 413–415 Adjacent splatter ratio, 143 Analytic signal, 418
Aperiodic autocorrelation, 101 Approximate conditional mean(ACM)
filter, 122–123 Area-mean power, 231 Attenuation power law, 231 Autocorrelation, 421 Autoregressive process, 121 Average autocorrelation, 68 Average power spectral density, 68 BCJR algorithm, 45
Beta function, 272 Block code, 1–27
BCH, 3, 10 cyclic, 3, 9 extended, 4 Golay , 4 Hamming , 4
linear, 3
maximum-distance-separable, 5 perfect, 3
Reed-Solomon, 5 repetition, 3 systematic, 5
Bluetooth, 173, 174, 176, 362 Burst communications, 184 Cell, 185
Cellular network, 326–349, 366–368, 372–382
Channel
frequency response, 246 impulse response, 245–247 Channelization code, 327
Channelized radiometer, 401–407 Characteristic function, 431 Characteristic polynomial, 303 Chase algorithm, 15, 51
Chebyshev’s inequality, 410–411 Chernoff bound, 25–27
Chi-square distribution, 431–433 Chip waveform, 56
Circular state diagram, 197 Circularly symmetric process, 429 Code rate, 4
Code tracking, 183, 209–214, 226 delay-locked loop, 210–212 early-late-gate loop, 226 tau-dither loop, 212–214 Code-aided methods, 125
Code-division multiple access(CDMA) definition, 293
Code-shift keying(CSK), 106–108 Coding gain, 16
Coherence bandwidth, 245
Coherence time, 238
Complementary error function, 12 Complex envelope, 418
Complex-valued quaternary sequence, 302–306
Concatenated code, 40–42
Confluent hypergeometric function, 434
Constraint length, 28
Continuous-phase modulation continuous-phase frequency-shift
keying(CPFSK), 144 Continuous-phase modulation(CPM),
143–150
Gaussian MSK(GMSK), 145 minimum-shift keying(MSK), 144 Convex function, 409
Convolutional code, 27–37 catastrophic, 32 costraint length, 28 generating function, 36 generators, 29
linear, 28
minimum free distance, 30 punctured, 34
sequential decoding, 30 state, 29
systematic, 28 trellis diagram, 29 Viterbi decoder, 30 Convolver, 103
Cross-correlation aperiodic, 302
continuous-time partial, 302 parameter, 148
periodic, 297 Cycle swallower, 175 Decimation, 299
Decision-directed demodulator, 112 Decoder
bounded-distance, 2 complete, 2
erasing, 6 incomplete, 2 reproducing, 6
sequential, 30 Viterbi, 30 Decoding
errors-and-erasures, 13 hard-decision, 6 soft-decision, 12 Dehopping, 131 Delay spread, 243 Despreading, 57 Deviation ratio, 143
Differential phase-shift keying(DPSK), 108, 146–148
Direct-conversion receiver, 426–429 Diversity, 247
frequency, 247 path, 275 polarization, 247 spatial, 373 time, 247 Divider, 171–173
dual-modulus, 172 Doppler
factor, 346 shift, 233
spectrum, 238, 247 spread, 238
Double-dwell system, 191–193 Double-mix-divide system, 166 Downlink, 327
Downlink capacity, 379 DS/CDMA, 294–361 Duplexing, 328 Duty factor, 362 Dwell interval, 131
Energy detector, see Radiometer Equal-gain combining, 261–269 Erasure, 12
Error probability channel-symbol, 6 decoded-symbol, 9 decoding failure, 7 information-bit, 8 information-symbol, 8, 9 undetected, 7
word, 6
Error rate
decoded-symbol, 9 information-symbol, 9 Euler function, 73
Fading, 232–245 fast, 238 slow, 238 Fading rate, 240–241 False alarm, 394
rate, 395
Fast frequency hopping, 132 Feedback shift register, 60 FH/CDMA, 362–382
Fourth-generation cellular systems, 324
Fractional power, 144 Frequency channel, 129 Frequency discriminator, 149 Frequency synthesizer, 166–176
digital, 167–170 direct, 166–167 fractional-N, 175–176 indirect, 170–176 multiple-loop, 173–174 Frequency-hopping pattern, 129 Galois field, 62
Gamma function, 432 Gaussian approximation
improved, 313 standard, 313
Gaussian interference, 83–86
Global System for Mobile(GSM), 146, 328
Gold sequence, 299–300 Hadamard matrix, 296 Hamming bound, 3 Hamming distance, 1
Hard-decision decoding, 6–12 Hilbert transform, 417 Hop duration, 129 Hop interval, 129 Hop rate, 129 Hopping band,129 Hopset, 129
Hybrid systems, 151–152
Ideal detection, 387–390, 398–401 Incomplete beta function, 403 Incomplete gamma function, 394 Intercell interference factor, 332 Interference canceller, 358–361
multistage, 360 parallel, 360 successive, 358 Interleaving, 39–40
block, 39
convolutional, 40 helical, 40
odd-even separation, 44 pseudorandom, 40 S-random, 40 IS-95, 317, 328, 344, 345 Isotropic scattering, 238 Jensen’s inequality, 409–410 Kalman-Bucy filter, 119, 121 Kasami sequence, 300–301 Key, 77, 131
Least-mean-square(LMS) algorithm, 115, 415
Likelihood function, 13 Linear span, 131 Local-mean power, 231 Lock detector, 193
Lognormal distribution, 232 Low probability of interception, 387 MAP algorithm, 45
Marcum Q-function, 218 generalized, 394 Matched filter, 100–112
bandpass, 101 convolver, 103
SAW transversal filter, 102 Matched-filter acquisition, 184–185,
192, 214–221 Maximal sequence, 65–74
preferred pair, 299
Maximal-ratio combining, 251–261
Message privacy, 56 Metric, 13, 18–24
AGC, 96 correlation, 351
maximum-likelihood, 95 Rayleigh, 20, 285 self-normalization, 140 variable-gain, 137 white-noise, 97 Minimum distance, 2 Minimum free distance, 30 Modified Bessel function, 20, 432 Moment generating function, 25 Mother code, 296
Moving-window detection, 405 MSK, see continuous-phase modu-
lation
Multicarrier direct-sequence system, 318–321
Multicarrier DS/CDMA system, 324–
325 Multipath, 232
diffuse components, 236 intensity profile, 246 intensity vector, 322
resolvable components, 244, 275 specular components, 236 unresolvable components, 234 Multiple access, 293
Multiple frequency-shift keying(MFSK), 21–24, 134–142
Multiuser detector, 349–361 adaptive, 358
decorrelating, 352–356 for frequency hopping, 360 interference canceller, 358–360 minimum mean-square error(MMSE),
356–358 optimum, 350–352 Nakagami density, 236
Narrowband interference, 113–125 Near-far problem, 327
Network capacity, 317
Noncoherent combining loss, 136 Noncoherent correlator, 183
Nonlinear filter, 119–125 Nonlinear generator, 75–77
OFDM, see orthogonal frequency- division multiplexing One-coincidence sequence, 373 Optimal array, 247–251
Orthogonal frequency-division mul- tiplexing(OFDM), 325 Orthogonal variable-spreading-factor
codes, 296 Outage, 333
Output threshold test, 157 Packing density, 10
Partial-band interference, 152–160 Peer-to-peer communications, 328 Peer-to-peer network, 366–372 Penalty time, 189, 193–194 Periodic autocorrelation, 65–69 Phase stripper, 252
Poisson sum formula, 425 Polynomial, 70–74
characteristic, 70 generating function, 70 irreducible, 71
primitive, 72
Power control, 328–329, 336–339, 343–
349
closed-loop, 328 open-loop, 328
Power spectral density, 421, 423–
424
Probability densities, see Probabil- ity distributions
Probability distributions, 431–437 central chi-square, 433 chi-square, 431 exponential, 436 lognormal, 232
noncentral chi-square, 431 Rayleigh, 435
Rice, 434
Processing gain, 56, 77 overall, 321 Product code, 49
Pseudonoise sequence, 68
Psi function, 330
Pulsed interference, 91–99 q-ary symmetric channel, 11 Quaternary system, 86–91
balanced, 88 dual, 86
Radiometer, 97, 390–398, 401–407 Rake receiver, 275–281, 322–324
fingers, 277
Random binary sequence, 58–60 autocorrelation, 60
Ratio threshold test, 158 Rayleigh distribution, 435–436 Rayleigh metric, 20, 266, 285 Recirculation loop, 109–111 Reed-Solomon code, 154–160 Rewinding time, 188
Rice distribution, 434–435 Ricean fading, 236
Riemann zeta function, 330 Sampling theorems, 424–426 SAW elastic convolver, 103–105 SAW transversal filter, 102 Scrambling code, 327 Search strategy
broken-center Z, 187 equiexpanding, 195 expanding-window, 195 nonuniform alternating, 196 uniform, 187
uniform alternating, 196 Z, 194
Selection diversity, 270–274 generalized, 279 Self-interference, 203
Separated orthogonal signals, 374 Serial-search acquisition, 185–208, 221–
226 Shadowing, 231
factor, 232
Shift-register sequence, 60–77 linear, 60
maximal, 65
Side information, 136, 156 Turbo code, 42–52, 161–165 Triangular function, 60
161
Trellis-coded modulation, 37–38, 51, Transmission security, 131
119
Transform-domain processor, 117–
Tone interference, 80–83 117
Time-domain adaptive filter, 114–
Time of day (TOD), 131 350
Third-generation cellular systems, 345, Test symbols, 156
Switching time, 131
Switch-and-stay combining, 274 Step size, 186
method, 415 Steepest descent
Spreading waveform, 56, 58–77 short, 74
long , 74
linear complexity, 75 Spreading sequence, 56, 58–77 Spreading factor, 296
Spectral splatter, 142 Spectral notching, 131 Spatial reliability, 370 Spatial diversity, 241–245
algorithm
Soft-in soft-out algorithm, see SISO 141
Soft-decision decoding, 12–16, 136–
Slow frequency hopping, 133 SOVA, 47
max-log-MAP, 47 log-MAP, 47 SISO algorithm, 47 Singleton bound, 5 Single-dwell system, 192
321–324
Single-carrier direct-sequence system, Sinc function, 392
Signum function, 417 Signature sequences, 294