This appendix contains several tables pertinent to the material contained in this book. The tables are
1. The GaussianQ-Function 2. Trigonometric Identities 3. Series Expansions 4. Integrals
5. Fourier Transform Pairs 6. Fourier Transform Theorems
n G.1 THE GAUSSIAN Q-FUNCTION
In this appendix we examine the Gaussian Q-function in more detail and discuss several approximations to theQ-function.1The Gaussian probability density function of unit variance and zero mean is
Zðxị ẳ 1 ffiffiffiffiffiffi
p2pex2=2 ðG:1ị
and the corresponding cumulative distribution function is Pðxị ẳ
ðx
ƠZðtịdt ðG:2ị The GaussianQ-function is defined as2
Qðxị ẳ1Pðxị ẳ ð¥
x
Zðtịdt ðG:3ị
An asymptotic expansion forQðxị, valid for largex, is Qðxị ẳZðxị
x 1 1
x2 ỵ 1ð3ị
x4 ỵ ð1ịn1ð3ị ð2n1ị x2n
ỵRn ðG:4ị
1The information given in this appendix is extracted from Abramowitz and Stegun, (1972) (originally published in 1964 as part of the National Bureau of Standards Applied Mathematics Series 55).
2Forx<0;Qðxị ẳ1Qðjxjị.
719
where the remainder is given by
Rnẳ ð1ịnỵ11ð3ị ð2n ỵ1ị ð¥
x
Zðtị
t2nỵ2 dt ðG:5ị which is less in absolute value than the first neglected term. Forx 3, less than 10% error results if only the first term in (G.4) is used to approximate the GaussianQ-function.
A finite-limit integral for theQ-function, which is convenient for numerical integration, is3
Qðxị ẳ 1 p
ðp=2 0
exp
x2 2 sin2f
df; x 0
1 1 p
ðp=2 0
exp
x2 2 sin2f
df; x<0 8>
>>
<
>>
>:
ðG:6ị
The error function can be related to the GaussianQ-function by erfðxị/ 2
ffiffiffiffip p ðx
0
et2dtẳ12Qð ffiffiffi 2
p xị ðG:7ị
The complementary error function is defined as erfcxẳ1erfxso that Qðxị ẳ1
2erfc x ffiffiffi2 p
ðG:8ị which is convenient for computing values using MATLAB since erfc is a subprogram in MATLAB but theQ-function is not (unless you have a Communications Toolbox).
Table G.1 A Short Table ofQ-Function Values
x Q(x) x Q(x) x Q(x)
0 0.5 1.5 0.066807 3.0 0.0013499
0.1 0.46017 1.6 0.054799 3.1 0.00096760
0.2 0.42074 1.7 0.044565 3.2 0.00068714
0.3 0.38209 1.8 0.035930 3.3 0.00048342
0.4 0.34458 1.9 0.028717 3.4 0.00033693
0.5 0.30854 2.0 0.022750 3.5 0.00023263
0.6 0.27425 2.1 0.017864 3.6 0.00015911
0.7 0.24196 2.2 0.013903 3.7 0.00010780
0.8 0.21186 2.3 0.010724 3.8 7:2348105
0.9 0.18406 2.4 0.0081975 3.9 4:8096105
1.0 0.15866 2.5 0.0062097 4.0 3:1671105
1.1 0.13567 2.6 0.0046612 4.1 2:0658105
1.2 0.11507 2.7 0.0034670 4.2 1:3346105
1.3 0.096800 2.8 0.0025551 4.3 8:5399106
1.4 0.080757 2.9 0.0018658 4.4 5:4125106
3J. W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations.IEEE MILCOM91 Conference Record., Boston, MA, 25.5.1– 25.5.5, November 1991.
M. K. Simon and D. Divsalar, Some new twists to problems involving the Gaussian probability integral.IEEE Transactions on Communication,46: 200–210, February 1998.
A short table of values forQðxịis given in Table G.1. Note that values ofQðxịforx<0 can be found from the table by using the relationship
Qðxị ẳ1Qðxị ðG:9ị
For example, from Table G.1,Qð0:1ị ẳ1Qð0:1ị ẳ10:46017ẳ0:53983.
n G.2 TRIGONOMETRIC IDENTITIES
cosuẳ eju ỵeju 2 sinu ẳ ejueju
2j cos2uỵsin2uẳ1
cos2usin2u ẳcosð2uị 2 sinucosu ẳsinð2uị
cosucosv ẳ1
2cosðuvị ỵ 1
2cosðu ỵvị sinucosvẳ 1
2sinðuvị ỵ 1
2sinðuỵvị sinusinvẳ 1
2cosðuvị 1
2cosðuỵvị cosðuvị ẳcosucosvsinusinv sinðuvị ẳ sin u cosvcosusinv
cos2u ẳ 1 2 þ 1
2cosð2uị cos2nu ẳ 1
22n X
n1 kẳ0
2 2n k
!
cosẵ2ðnkịu ỵ 2n n !
( )
; na positive integer
cos2n1u ẳ 1 22n2
X
n1 kẳ0
2n1 k
!
cosð2n2k1ịu
( )
sin2uẳ 1 21
2cosð2uị sin2nuẳ 1
22n X
n1 kẳ0
ð1ịnk2 2n k
!
cosẵ2ðnkịu ỵ 2n n !
( )
sin2n1u ẳ 1 22n2
X
n1 kẳ0
ð1ịnỵk1 2n1 k
!
sinð2n2k1ịu
" #
n G.3 SERIES EXPANSIONS
ðuỵvịnẳXn
kẳ0
n k
unkvk; n
k
ẳ n!
ðnkị!k!
Lettingu ẳ1 andvẳx; wherejxj1 results in the approximations:
ð1 ỵxịn ffi1 ỵnx; ð1xịn ffi1nx; ð1 ỵxị1=2ffi1ỵ 1 2x loga uẳlogeulogae; logeuẳlnuẳlogealogau
euẳXƠ
kẳ0
uk
k!ffi1 þu; juj1 lnð1 ỵuị ffiu; juj1
sinuẳXƠ
kẳ0
ð1ịk u2kỵ1
ð2k ỵ1ị!ffiuu3
3!; juj1 cosu ẳXƠ
kẳ0
ð1ịk u2k
ð2kị!ffi1u2
2!; juj1 tanuẳu ỵ 1
3u3 þ 2
15u5 þ
Jnðuị ffi
(
un
2nn! 1 u2
22ðn ỵ1ị ỵ u4
224ðnỵ1ịðnỵ2ị
;juj1
ffiffiffiffiffiffi 2 pu r
cos
unp
2 p
2
;juj1
I0ðuị ffi
(
1þ u2 22 þ u4
24 þ ffieu2=4; 0u1 eu
ffiffiffiffiffiffiffiffiffi
p2pu; u1
n G.4 INTEGRALS
G.4.1 Indefinite
ésinðaxịdxẳ 1
acosðaxị
écosðaxịdxẳ1 asinðaxị
Ð
sin2ðaxịdxẳx 2 1
4asinð2axị
Ð
cos2ðaxịdxẳx 2 þ 1
4asinð2axị
Ð
xsinðaxịdxẳa2ẵsinðaxị axcosðaxị
Ð
xcosðaxịdxẳa2ẵcosðaxị ỵ axsinðaxị
Ð
xmsinx dxẳ xmcosxỵm
Ð
xm1cosx dx
Ð
xmcosx dxẳxmsinxm
Ð
xm1sinx dx
Ð
expðaxịdxẳa1expðaxị
Ð
xmexpðaxịdxẳa1xmexpðaxị a1m
Ð
xm1expðaxịdx
Ð
expðaxịsinðbxịdxẳ ða2 ỵb2ị1expðaxịẵasinðbxị bcosðbxị
Ð
expðaxịcosðbxịdxẳ ða2 ỵb2ị1expðaxịẵacosðbxị ỵbsinðbxị
G.4.2 Definite ð¥
0
xm1
1 ỵxndxẳ p=n
sinðmp=nị; n>m>0
Ðp
0sin2ðnxịdxẳép
0cos2ðnxịdxẳp
2; nan integer
Ðp
0sinðmxịsinðnxịdxẳép
0cosðmxịcosðnxịdx ẳ 0; m6ẳn;mandninteger
Ðp
0sinðmxịcosðnxịdxẳ
( 2m
m2n2; mþnodd
0; mþneven
Ð¥
0xa1cosbx dxẳGðaị ba cos pa
2
; 0<jaj<1; b>0
Ð¥
0xa1sinbx dxẳGðaị ba sin pa
2
; 0<jaj<1; b>0
Ð¥
0xnexpðaxịdxẳ n!
anþ1; n an integer and a>0
Ð¥
0expða2x2ịdxẳ ffiffiffiffip p
Ð¥ 2jaj
0x2nexpða2x2ịdxẳ1 ð3ị ð5ị ð2n1ịpffiffiffiffip
2nþ1a2nþ1 ; a>0
Ð¥
0expðaxịcosðbxịdxẳ a
a2 þb2; a>0
Ð¥
0expðaxịsinðbxịdxẳ b
a2 þb2; a>0
Ð¥
0expða2x2ịcosðbxịdxẳ ffiffiffiffip p
2a exp b2 4a2
Ð¥
0xexpðax2ịIkðbxịdxẳ 1
2aexp b2 4a
; a>0
ð¥ 0
cosðaxị b2 ỵx2dxẳ p
2bexpðabị; a>0; b>0 ð¥
0
xsinðaxị b2 ỵx2 dxẳp
2expðabị; a>0; b>0 ð¥
0
sincðxịdxẳ ð¥
0
sinc2ðxịdxẳ1 2
n G.5 FOURIER TRANSFORM PAIRS
Signal Fourier transform
Pðt=tị ẳ 1; jtj t
0; otherwise2 (
tsincðftị ẳtsinðpftị pft
2Wsincð2Wtị P f
2W
Lðt=tị ẳ 1jtj
t; jtj t 0; otherwise 8<
: tsinc2ðftị
Wsinc2ðWtị L f
W
expðatịuðtị; a>0 1
ðaỵj2pfị
texpðatịuðtị; a>0 1
ðaỵj2pfị2
expðajtjị; a>0 2a
a2ỵ ð2pfị2 exp p t
t
2
texpẵ pðtfị2
dðtị 1
1 dðfị
cosð2pf0tị 1
2dðff0ị ỵ1
2dðf ỵf0ị
sinð2pf0tị 1
2jdðff0ị 1
2jdðf ỵf0ị
uðtị 1
j2pf þ1 2dðfị 1
ðptị jsgnf;sgnf ẳ
1; f>0 1; f <0 P¥
mẳ ƠdðtmTsị fsPƠ
nẳ Ơdðfn fsị; fsẳ 1 Ts
n G.6 FOURIER TRANSFORM THEOREMS
Name
Time domain operation (signals assumed real)
Frequency domain operation Superposition a1x1ðtị ỵa2x2ðtị a1X1ðfị ỵa2X2ðfị
Time delay xðtt0ị Xðfịexpðj2pt0fị
Scale change xðatị jaj1X f
a
Time reversal xðtị Xðfị ẳXðfị
Duality Xðtị xðfị
Frequency translation xðtịexpðj2pf0tị Xðff0ị
Modulation xðtịcosð2pf0tị 1
2Xðff0ị ỵ1
2Xðf ỵf0ị
Convolution* x1ðtịx2ðtị X1ðfịX2ðfị
Multiplication x1ðtịx2ðtị X1ðfịX2ðfị
Differentiation dnxðtị
dtn ðj2pfịnXðfị
Integration Ðt
Ơxðlịdl Xðfị
j2pf þ1 2Xð0ịdðfị
x1ðtịx2ðtị/éƠ
Ơx1ðlịx2ðtlịdl:
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Poor, H. V., 598 Prabha, V. K., 486, 492
Proakis, J. G., 240, 331, 437, 439, 453, 549
Pursley, M. B., 517 Rappaport, S. S., 520 Rappaport, T. S., 538, 543 Rice, S. O., 1, 496 Ross, S., 293 Rowe, H. E., 492 Salehi, M., 240, 331 Sari, H., 526
Scheibe, E. H., 688, 689, 698 Schilling, D. L., 202, 321, 378 Sharf, L., 598
Scholtz, R. A., 186, 504, 511 Serpedin, E., 503
Seshadri, N., 543 Shanai, 437
Shannon, C. E., 1, 675 Siebert, W., 59
Simon, M. K., 274, 440, 502-504, 581, 601, 720
Simpson, R. S., 380 Sklar, B., 668, 675 Skolnik, M. J., 510 Stegun, I., 78, 288, 572, 719 Stiffler, J. J., 501, 502, 504 Stuber, G. L., 538 Sundberg, C. E., 668 Tarokh, V., 543 Taub, H., 202, 371, 378 Taylor, D. P., 16 Taylor, F. J., 70 Thitimajshima, P., 15 Thomas, J. A., 672
729
Torrieri, D. J., 639 Tranter, W. H., 16, 202, 378 Treichler, J. R., 447 Tse, D., 538
Ungerboeck, G., 668, 670, 671, 672 Van der Ziel, A., 698
Van Trees, H. L., 407, 555, 562, 563, 581, 598
Verdu, S., 15, 518, 543 Viswanath, P., 538 Viterbi, A. J., 672 Walpole, 293 Wang, Z., 522 Weaver, W., 675 Weinstein, S. B., 525 Whalen, A. D., 598 Wicker, S. B., 675
Wiener, N., 1 Williams, A. B., 70 Winters, J. H., 12 Wintz, P. A., 504 Woodward, P. M., 14
Wozencraft, J. M., 14, 471, 571, 598 Zhuang, W., 538
Ziemer, R. E., 58, 92, 99, 240, 508, 549, 656, 668, 675, 698
Absorption, 10–11
Adaptive delta modulation, 189 Adaptive equalization, 449 Adaptive filter, 14
Adjacent channel interference, 497 Administrative Radio Conference, 7 Advanced Mobile Phone System, 537 Advanced Technology Satellite, 11 Aliasing, 80
Alphabet, 620
Amplitude density spectrum, 38 Amplitude distortion, 64 Amplitude jitter, 233 Amplitude modulation (AM)
coherent detection, 115 defined, 115
detection gain, 348
effect of interference on, 159–162 effect of noise on, 357–353 efficiency, 117
envelope detection of, 115–117 index, 115
optimal performance of, 667 square law detection of, 204 Amplitude response function, 60 Amplitude-shift keying (ASK), 238, 385,
392, 404 Amplitude spectrum, 38 Analog baseband system, 342 Analog pulse modulation, 182–186 Analog signal, 4
Analog-to-digital conversion (see also Pulse-code modulation), 384 Analytic signal, 85
Angle modulation (see alsoFrequency modulation)
bandwidth of signal, 147–152 demodulation of, 154–159 deviation ratio, 148 effect of noise on, 357–363 frequency deviation, 136 frequency deviation constant, 137 index, 141
interference in, 162–167 narrowband modulation, 138, 149 narrowband-to-wideband conversion,
139, 152–154 phase deviation, 136 phase deviation constant, 137 power in signal, 147–152 spectrum with sinusoidal signal,
141–147
wideband modulation, 149 Antenna coverage, 528–530 Antenna gain, 533 Antipodal signals, 399
Aperiodic signal, 18
A posterioriprobability, 14, 563 Apparent carrier, 468
Arithmetical average, 268 Asynchronous system, 385 Atmospheric attenuation, 10 Atmospheric noise, 5 Attenuator noise, 694 Autocorrelation function
deterministic signals, 52 properties, 53, 313 random signals, 305 random pulse train, 314 Available power, 685 Average cost, 557 Average information, 608 Average power, 23 AWGN model, 342 Balanced discriminator, 158 Bandlimited channels, 426–431 Bandlimited white noise, 313 Bandpass limiter, 156 Bandpass signals, 87–89 Bandpass systems, 89–91 Bandwidth
bit-rate, 389 efficiency, 491
efficient modulation, 688–672 expansion factor, 666 limited operation, 626 noise-equivalent, 322–325 relation to risetime, 75–78 Barker sequence, 510
Baseband data transmission, 210, 386–391 Basis set
complete, 27 defined, 25 normalized, 26 orthonormal, 26 Basis vector, 25, 564 Bayes detection, 554–564 Bayes estimation, 554, 585–589 Bayes’ rule, 248, 253
Bent-pipe system, 526, 532–535 Bessel filter, 72
Bessel functions, table of, 142 Bessel polynomial, 72 BIBO stability, 58
Binary random waveform, 314–316 Binary system, 385
Binary unit, 385 Binit, 387
Binomial coefficient, 280 Binomial distribution, 280, 282 Binomial theorem, 281
Biphase-shift keying (BPSK), 405–407 Bit, 211, 387, 607
Bit-rate bandwidth, 389 Bit synchronization, 387 Boltzmann’s constant, 341 Burst-error-correcting code, 657 Butterworth filter, 71–73, 324 Capacity limits, 517 Carrier frequency, 111 Carrier nulls, 144 Carrier reinsertion, 125
Carrier synchronization, 167, 499–502 Carson’s rule, 149
Causal system, 59
Cellular mobile radio, 537–546 Central-limit theorem, 284 Channel
Bandlimited, 422–432 binary erasure, 676 binary symmetric, 615 capacity, 613–617 characteristics, 5–14 continuous, 624 defined, 5
electromagnetic wave, 7–11 fading, 6, 424, 542, 582 feedback, 661–665
guided electromagnetic wave, 11 matrix, 610
measurement, 689–691 memoryless, 609 models, 609–612 multipath, 431–437 noiseless, 614 optical, 12
representation of, 609–612 satellite, 611, 526–537, 695–698 slowly fading, 582
transition probability, 609 transmission, 6–12 types of, 6–12 Channel capacity
binary symmetric channel, 615 continuous channel, 624 defined, 613
noiseless channel, 614 Characteristic function, 275 Chebyshev filter, 72 Chebyshev inequality, 289 Chebyshev polynomial, 72 Chip period, 515
Cochannel interference, 540
Code division multiple access (CDMA), 517
Code synchronization, 520
731
Coding definitions
alphabet, 620 block codes, 626–646 constraint span, 647 efficiency, 620 error vector, 631 generator matrix, 633 Hamming distance, 627 Hamming weight, 627 instantaneous codes, 620 nonblock codes, 620 noninstantaneous codes, 620 parity-check matrix, 631 space-time, 543 syndrome, 632 perfect code, 639 systematic code, 631 word length, 620 for error control
BCH codes, 637–638 block codes, 626–646
burst-error correcting codes, 657 code rate, 627
convolutional codes, 647–657 cyclic codes, 635
Golay codes, 636 group codes, 634 Hamming codes, 634–635 interleaved codes, 657 linear codes, 634 repetition codes, 629 single parity-check codes, 628,
630–635
structure of parity-check codes, 628–634
trellis-coded modulation, 668–671 turbo code, 659
Viterbi decoding (Viterbi algorithm), 650–657
source encoding described, 384, 617 Huffman, 623 Shannon-Fano, 622
Coherent demodulation, 114, 385, 500 Communication system, 3
Communication theory, 13 Commutator, 195 Companding, 375 Compound event, 246
Complementary error function, 289 Complex envelope, 87
Compressor, 376
Conditional expectation, 272 Conditional entropy, 612 Conditional mean, 587 Conditional probability, 247 Conditional probability density, 260 Conditional risk, 587
Consistent estimate, 592 Constraint span, 647
Continuous-phase modulation (CPM), 668–672
CONUS, 529 Convolution, 40
Convolutional code, 647–657 Convolution theorem, 44 Correlation, 309, 398 Correlation coefficient, 278 Correlation detection, 401 Correlation receiver, 400
Cost of making a decision, 557 Costas phase-lock loop
for carrier synchronization, 438 demodulation of DSB, 114 Courier satellite, 526 Covariance, 278, 304 Cramer-Rao inequality
Cross-correlation function, 316–317 Cross-power, 262
Cross-power spectral density, 316–317 Crosstalk, 195
Cumulative distribution function, 254–256 Cycle-slipping phenomenon, 177 Cyclic codes, 635
Cyclic prefix, 525
Cyclostationary process, 314 Data transmission
Baseband, 210–237, 386–391 with modulation, 391–426 Data vector, 574
Decimation in time, 92 Decision feedback, 449 Decision rule, 577
De-emphasis (seePre-emphasis) Delay distortion, 64
Delay spread, 524, 542 Delta function, 21 Delta modulation, 187–190 Demod/remod system, 535–537 Demodulation phase errors, 353–357 Detection, statistical
Bayes detection, 555–559
maximuma posterioridetection, 563 minimum probability of error detection,
562–563
Neyman-Pearson detection, 562 Detection gain
in AM, 358 defined, 345 in DSB, 345 optimal, 666 in SSB, 347
Differential encoding, 409
Differential phase-shift keying (DPSK), 409–417
Differentiation theorem, 43 Diffuse multipath, 6
Digtal audio broadcasting, 522 Digital modulation
amplitude-shift keying (ASK), 238, 385, 392, 403
biphase-shift keying (BPSK), 405–407 differential phase-shift keying (DPSK),
409–417, 485–486
frequency-shift keying (FSK), 238, ,385, 392, 407, 468, 480–485
M-ary PAM, 418
minimum-shift keying (MSK), 465–471 noncoherent FSK, 417
offset quadriphase-shift keying (OQPSK), 464
phase-shift keying (PSK), 238, 385, 392, 404
quadriphase-shift keying (QPSK), 385, 474–478
staggered QPSK, 464 Digital signal, 4
Digital subscriber lines, 522 Digital telephone system, 197 Digital–to-analog conversion, 384
Dimensionality theorem, 571 Direct sequence (DS) spread-spectrum,
512–519 Dirichlet conditions, 28 Discrete Fourier transform, 91–95 Discriminator, 154
Disjoint sets, 246 Distortion
amplitude, 64 harmonic, 67, 108 intermodulation, 67 nonlinear, 64, 67 phase (delay), 64
Distortionless transmission, 64 Diversity transmission, 439, 585 Dot product, 564
Double-sideband modulation (DSB) coherent demodulation of, 112 defined, 112
detection gain, 345
effect of interference on, 159–160 effect of noise on, 343–345 optimal performance of, 667 Duality theorem, 43
Earth stations, 530–532 Echo I, 526
Effective carrier, 161
Effective noise temperature, 691 Effective radiated power, 696 Efficient estimate, 592 Electromagnetic spectrum, 8 Electromagnetic-wave propagation
channels, 7–11 Energy, 23
Energy spectral density, 39 Ensemble, 303
Entropy, 608, 621 Envelope, 76, 85 Envelope detection
of AM signals, 115 of FSK signals, 419–423
Envelope-phase representation of noise, 325
Equal gain combining, 439 Equalization
Adaptive, 14 decision-directed, 449 filter, 184, 436
minimum mean-square error, 446–450 transversal implementation, 229 zero-forcing, 442–445 Equivalent noise temperature, 691 Ergodic process, 304, 306 Error correcting codes (seeCoding) Error-detection feedback, 661–665 Error function, 289
Error probability (seespecific system) Estimation
applications
estimation of signal phase, 594–596 pulse amplitude modulation, 593–594 based on multiple oberservations,
589–591 Bayes, 586–588 conditional mean, 587 conditional risk, 587 cost function, 586 Cramer–Rao inequality, 591 Efficient, 592
likelihood equations, 589
likelihood function, 589
maximuma posteriori (MAP), 587 maximum likelihood, 588–589, 592 multiple observations, 589–591 rule, 586
theory, 585–592 unbiased, 591 Euler’s theorem, 18 Event, 246 Excess phase, 468 Expander, 376 Expectation, 269 Extended source, 618, 621 Eye diagrams, 232–234 Fading, 437–443, 582 Fading margin, 458 False alarm, 559
Fast Fourier transform, 91–95
Fast frequency-shift keying (FFSK), 468 Fast hop, 519
Federal Communications Commission (FCC), 9
Feedback channels, 661–665 Feedback demodulators
Costas phase-lock loop, 180 phase-lock loop, 167–180 Filter
adaptive, 14 Bessel, 71–73
Butterworth, 71–73, 324 Chebyshev, 71–73 de-emphasis, 167, 362 equalization, 226–231 ideal, 68–70
intermediate-frequency, 134 matched, 14, 394–402 postdetection, 343 predetection, 343, 362 pre-emphasis, 167 radio frequency, 134 reconstruction, 80 transversal, 229 Weiner, 14 whitening, 402
Filtered Gaussian process, 320 Fixed system, 57
Fourier coefficients, 26, 567–568 Fourier series
complex exponential, 28 generalized, 25–27 symmetry properties, 29, 30 trigonometric, 30
Fourier transforms
amplitude and phase spectra, 37 defined, 37
discrete, 91–95 fast, 92 inverse, 31 periodic signals, 50 symmetry properties, 38 table of, 724
theorems, 41–50, 725 Frame, 504
Free distance, 655–671 Free-space loss, 696
Free-space propagation, 695–698 Frequency bands, 8, 9
Frequency deviation, 137, 148 Frequency diversity, 439 Frequency divider, 182
Frequency division multiplexing, 192
Frequency hopped (FH) spread-spectrum, 519
Frequency modulation bandwidth of signal, 147–150 Carson’s rule, 149
de-emphasis, 166 demodulation of
noiseless, 154–159, 167–180 in the presence of noise, 360–362 deviation constant, 137
deviation ratio, 148 discriminator, 154
effect of interference on, 162–167 effect of noise on, 360–362 index, 145
indirect, 153
narrowband modulation, 138–140 narrowband-to-wideband conversion,
139
optimal performance of, 667 power in signal, 147–152 pre-emphasis in, 166
spectrum with sinusoidal modulation, 141–147
stereophonic broadcasting, 193 threshold effects, 162, 363–371, 373 Frequency multiplier, 181
Frequency reuse, 538 Frequency-shift keying (FSK)
Coherent, 407, 480 M-ary, 480
Noncoherent, 481– 485 Frequency translation, 133–136 Frequency translation theorem, 43 Friis’ formula, 692
Fundamental period, 18
Fundamental theorem of information theory, 624
Gamma function, 290 Gaussian process, 304 Gaussian Q-function, 288 Generalized Fourier series, 25–28 Generator matrix, 633
Geometric distribution, 284 Geostationary satellite, 528 Global positioning system, 510 Globalstar system, 528 Global system for mobile, 537 Golay code, 636
Gram–Schmidt procedure, 569 Gray code, 419
Ground-wave propagation, 8 Group codes, 634
Group delay, 65 Guard band. 531 Guard time, 532
Guided electromagnetic-wave channels, 11 Halfwave symmetry, 30
Hamming codes, 634–635 Hamming distance, 505, 627 Hamming weight, 628 Handoff, 538 Harmonic term, 31 Hartley, 607
Hermite functions, 569 High-side tuning, 135 Hilbert transforms
analytic signals, 85 defined, 82 properties, 83
History of communications, 2–3 Huffman code, 623
Hybrid spread spectrum, 548 Ideal filters, 68–70
Ideal sampling waveform, 78 Ignition noise, 6
Image frequency, 134 Impulse function, 23 Impulse noise, 6 Impulse response
ideal filters, 69 of linear system, 57
Indirect frequency modulation, 153 Information, 607
Information feedback, 661 Information rate, 617
Information theory, 15, 606–624 Instantaneous sampling, 78 Intangible economy, 1 Integrals (table of), 722–724 Integral-squared error, 26
Integrate-and-dump detector, 386–387, 401
Integration theorem, 44 Intelsat, 526
Interference
adjacent channel, 497 in angle modulation, 162–167 in linear modulation, 159–162 intersymbol, 211, 220–222, 402, 413 multipath, 431–437
Interleaved codes, 657 Intermodulation distortion, 67
International Telecommunications Union (ITU), 7
Intersatellite communications
Intersymbol interference, 211, 220–222, 402, 413
Ionosphere, 8 Iridium system, 528 Isotropic radiation, 695 Jacobian, 266 Joint entropy, 612 Joint event, 246 Joint probability
cumulative distribution function, 259 density function, 259, 303
matrix, 610 Kraft inequality, 678 Kronecker delta, 26, 494, 568 Laplace approximation, 282 Laser, 11
Last mile problem, 12, 522 Legendre functions, 569 Likelihood function, 589 Likelihood ratio, 558 Limiter, 156 Line codes, 211–220 Linear modulation
amplitude modulation, 115–120 double-sideband modulation, 112–136 interference in, 159–162
single-sideband modulation, 121–129 vertigial-sideband modulation, 129–133 Linear systems
amplitude response, 60 causal, 58–59 definition of, 56