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Ash R.B. — Information theory
Ash R.B. — Information theory

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Название: Information theory

Автор: Ash R.B.

Аннотация:

Analysis of channel models and proof of coding theorems; study of specific coding systems; and study of statistical properties of information sources. Sixty problems, with solutions. Advanced undergraduate to graduate level.
Excellent introduction treats three major areas: analysis of channel models and proof of coding theorems; study of specific coding systems; and study of statistical properties of information sources. Appendix summarizes Hilbert space background and results from the theory of stochastic processes. Advanced undergraduate to graduate level. Bibliography.


Язык: en

Рубрика: Computer science/Теория информации/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1965

Количество страниц: 339

Добавлена в каталог: 10.11.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$l_2$ space      250 257 262
$\pi$-sequence      24
Abramson code      166
Algebra, of binary polynomials      147
Alphabet of a code      27
Alphabet of a source      172 185
Alphabet, of a channel      46 215
Ambiguous sequence      30
Arithmetic and geometric means, inequality of      25
Arzela — Ascoli theorem      270
Asymptotic equipartition property (AEP)      197 223
Band-limited functions      256 ff.
Bessel’s inequality      269
Binomial distribution      3 14 15
Binomial distribution, estimate of “tail” of      114
bits      11 84
Bose — Chaudhuri — Hocquenghem codes      156 ff.
Capacity of a binary symmetric channel      54
Capacity of a channel      1 3 50
Capacity of a discrete memoryless channel      50
Capacity of a finite state regular channel      220 228
Capacity of a general discrete channel      223
Capacity of a symmetric channel      53
Capacity of a time-continuous Gaussian channel      251 252
Capacity of a time-discrete Gaussian channel      234
Capacity of a time-discrete, amplitude-continuous channel      234
Capacity, calculation of      53 ff.
Capacity, X -capacity      224
Cascade combination, of discrete memoryless channels      85
Cayley — Hamilton theorem      139
Channel(s)      1
Channel(s), alphabet of      46 215
Channel(s), band-limited      256 ff.
Channel(s), binary erasure      86
Channel(s), binary symmetric      53 54 87
Channel(s), burst noise      213
Channel(s), capacity of      see “Capacity”
Channel(s), cascade combination of      85
Channel(s), code for      see “Code”
Channel(s), compound      228
Channel(s), connection of a source to      214 216
Channel(s), continuous      230 ff.
Channel(s), deterministic      51
Channel(s), discrete      47
Channel(s), discrete, memoryless      47
Channel(s), discrete, with finite memory      227
Channel(s), discrete, with memory      211 ff.
Channel(s), finite state      215 ff.
Channel(s), finite state, regular      216
Channel(s), information processed by      50 219
Channel(s), input n-sequence for      65
Channel(s), lossless      50
Channel(s), matrix of      49 212—215
Channel(s), matrix of, source-channel matrix      215—217
Channel(s), noiseless      27 51
Channel(s), output n-sequence for      65
Channel(s), product of      85
Channel(s), states of      46 215 230
Channel(s), sum of      85
Channel(s), symmetric      52
Channel(s), time-continuous Gaussian      251
Channel(s), time-discrete Gaussian      231
Channel(s), time-discrete Gaussian with average power limitation      231
Channel(s), time-discrete, amplitude-continuous      230
Channel(s), trap door      211
Channel(s), useless (zero-capacity)      52
Chebyshev’s inequality      15
Code      4
Code, (n, k)      99
Code, Abramson      166
Code, alphabet      27
Code, block      4 39 63
Code, Bose — Chaudhuri — Hocquenghem      156
Code, characters      27
Code, check digits of      93
Code, close-packed      132
Code, cyclic      see “Cyclic code”
Code, error correcting      87 ff.
Code, Fire      167
Code, for a binary symmetric channel      87 ff.
Code, for a discrete memoryless channel      65 66
Code, for a finite state regular channel      220
Code, for a general discrete channel      223
Code, for a noiseless channel      27
Code, for a time-continuous Gaussian channel      251
Code, for a time-continuous Gaussian channel, explicit construction of      256
Code, for a time-discrete, amplitude continuous channel      230
Code, geometric interpretation of      90
Code, group      93 99
Code, group, coset of      100
Code, Huffman      42
Code, information digits of      93
Code, instantaneous      see “Instantaneous code”
Code, lossless      132
Code, nonbinary      126
Code, optimal, absolutely      38
Code, optimal, absolutely, instantaneous      41
Code, parity check      see “Parity check code”
Code, random      66 67 74 110
Code, single-error correcting      105
Code, single-error correcting, cyclic      161
Code, uniquely decipherable      28 29 35 40
Code, word      4 27 63 65
Coding theorem      see “Fundamental theorem of information theory”
Communication entropy      24
Communication system      1
Conditional uncertainty      19 219 229 238 241
Convexity, of information and uncertainty      54 81 242
Convexity, of the logarithm      17
Corrector      94
Correlation detector      254
Coset leader ( = correctible error pattern)      103
Coset, of a group code      100
Coset, relation to correctible error patterns      102 103
Covariance function, of a random process      250 256 275
Critical rate, for the binary symmetric channel      117
Cycle set, of a binary matrix      136 143 144
Cycle set, of a feedback shift register      136
Cyclic code      137
Cyclic code, application to burst-error correction      164
Cyclic code, decoding of      161
Cyclic code, generation by a feedback shift register      137 151
Cyclic code, generator polynomial of      148
Cyclic code, properties of      147 ff.
Cyclic code, set of generators for      149
Cyclic code, single-error correcting      161
Decision scheme      1 4 60 65 220 230 251
Decision scheme, ideal observer      see “Ideal observer decision scheme”
Decision scheme, maximum likelihood      62
Decision scheme, minimum distance      see “Distance
Decision scheme, randomized      84
Decision scheme, with uniform error bound      62 66
Decoder      see “Decision scheme”
Decoding sets      65
Dini’s theorem      274
Distance (between code words)      87 ff. 126
Distance (between code words), minimum distance decoder      88 126 254
Distance (between code words), minimum distance decoder, for a parity check code      95
Distance (between code words), minimum distance principle      87 88
Distance (between code words), minimum, bounds on      129
Distance (between code words), relation to error correction      89
Effective procedure, for determining existence of steady state probabilities in a finite Markov chain      180 218
Effective procedure, for determining whether a finite state channel is regular      218
Eigenvalues, eigenfunctions, and eigenvectors      210 250 264
Encoder      1 4
entropy      24
Ergodic information source      see “Source”
Error bounds (exponential)      77 ff. 227 260
Error bounds (exponential), for general binary codes      113 ff.
Error bounds (exponential), for nonbinary codes      127
Error patterns      95
Error patterns, correctible, relation to cosets      102 103
Fano’s inequality      80 243
Feedback shift register      134 ff.
Feedback shift register, characteristic matrix of      135
Feedback shift register, cycle set of      136
Feedback shift register, generation of a cyclic code by      137 151
Feedback shift register, state (content) of      135
Field      126 142
Field, Galois (finite)      142
Fire code      167
Fourier transforms      251 257 258 281
Fourier transforms, convolution      282
Fourier transforms, Parseval relation      258 282
Fundamental lemma (in the proof of the fundamental theorem)      68 86 232
Fundamental theorem of information theory      1
Fundamental theorem of information theory, for the discrete memoryless channel      66 307
Fundamental theorem of information theory, for the discrete memoryless channel, algorithmic proof      68
Fundamental theorem of information theory, for the discrete memoryless channel, random coding proof      74
Fundamental theorem of information theory, for the discrete memoryless channel, Shannon’s original proof      66
Fundamental theorem of information theory, for the finite state regular channel      222
Fundamental theorem of information theory, for the general discrete channel      223
Fundamental theorem of information theory, for the time-continuous Gaussian channel      252
Fundamental theorem of information theory, for the time-discrete Gaussian channel      234
Fundamental theorem of information theory, strong converse to      see “Strong converse”
Fundamental theorem of information theory, weak converse to      see “Weak converse”
Generator matrix, of a cyclic code      149
Generator matrix, of a parity check code      132
Generator polynomial (of a cyclic code)      148
Generator polynomial (of a cyclic code), relation to the minimal polynomial of a matrix      150
Generators, set of, for a cyclic code      149
Generators, set of, for a parity check code      110
Greatest common divisor, of two polynomials      141
Group      96
Group code      93 99
Group code, coset of      100
Group, abelian (commutative)      96 141 147
Grouping axiom, for the uncertainty function      8 80 81
Grouping axiom, generalized      26
Hamming bound, for general binary codes      90 133
Hamming bound, for parity check codes      107
Hamming bound, strengthened      127
Hamming code      130 133
Hamming distance      see “Distance”
Hilbert space      250 262
Hilbert space, Bessel’s inequality      269
Hilbert space, inner product      262
Hilbert space, norm      262
Hilbert space, orthogonal complement of a subspace      268
Hilbert space, orthonormal family      266
Hilbert space, parallelogram law      265
Hilbert space, space spanned by a subset      267
Huffman code      42
Ideal observer decision scheme      61 73
Ideal observer decision scheme, for the binary symmetric channel      88
Ideal observer decision scheme, for the q-ary symmetric channel      126
Ideal observer decision scheme, for the time-discrete Gaussian channel      254
Ideal, in the algebra of binary polynomials      149
Independent random variables      14 39 195
Independent random variables, as a Markov information source      185
Independent random variables, uncertainty of      18 19 21 239 241 242
information      4
Information, convexity of      54 242
Information, conveyed about one random variable by another      22 23 239 242
Information, conveyed about one stationary sequence of random variables by another      219
Information, processed by a channel      50 219
Information, source of      see “Source”
Instantaneous code      28 39
Instantaneous code, conditions for existence of      33
Instantaneous code, optimal      41
Instantaneous code, relation to sequence of “yes or no” questions      40
Integral operator      269 ff.
Integral operator, associated with a covariance function      250 282
Integration, of second order random processes      275 ff.
Irreducible polynomial      140 146
Joint uncertainty      18—21 238—240
Karhunen — Loeve theorem      250 277
Karhunen — Loeve theorem for Brownian motion      279
Karhunen — Loeve theorem for Gaussian processes      279
Language, uncertainty of      206
Linear operator (on a Hilbert space)      262 ff.
Linear operator (on a Hilbert space), bilinear form of      265
Linear operator (on a Hilbert space), compact (completely continuous)      263
Linear operator (on a Hilbert space), continuous (bounded)      262
Linear operator (on a Hilbert space), integral operator      269 ff.
Linear operator (on a Hilbert space), integral operator, associated with a covariance function      250 282
Linear operator (on a Hilbert space), quadratic form of      265
Linear operator (on a Hilbert space), symmetric (self-adjoint)      265
Markov chain(s) (finite)      171
Markov chain(s) (finite), essential sets of      180 183
Markov chain(s) (finite), indecomposable      181 184
Markov chain(s) (finite), periodic and aperiodic states of      183 184
Markov chain(s) (finite), regular      185
Markov chain(s) (finite), states of      171
Markov chain(s) (finite), stationary distribution of      174 181 184
Markov chain(s) (finite), steady state probabilities of      174 176
Markov chain(s) (finite), steady state probabilities of, effective determination of existence of      180 218
Markov chain(s) (finite), theory of      172 ff.
Markov chain(s) (finite), transient states of      180 183
Markov chain(s) (finite), transition matrix (Markov matrix, stochastic matrix) of      171
Markov information source      see “Source”
Markov property      171
Matrix      138 ff.
Matrix, binary      138
Matrix, binary, characteristic polynomial of      138
Matrix, binary, cycle set of      136 143 144
Matrix, binary, minimal polynomial of      see “Minimal polynomial”
Matrix, binary, period of      143
Matrix, binary, polynomial in      140
Matrix, binary, powers of, and matric polynomials      145
Matrix, binary, with maximal period      144
Matrix, channel      49 212—215
Matrix, connection, of a unifilar Markov source      209
Matrix, doubly stochastic      26
Matrix, Markov (stochastic, transition)      171
Matrix, parity check      93 106
Matrix, source-channel      215—217
Maximal period      144
Maximum likelihood decision scheme      62
Mercer’s theorem      272
Message      1 27 63
Minimal polynomial of a matrix      139
Minimal polynomial of a matrix, irreducible      140 141 143 144
Minimal polynomial of a matrix, relation to generator polynomial of a cyclic code      150
Minimum distance decoder      see “Distance”
Noise      1
Noiseless coding      27 ff.
Noiseless coding, theorem      13 37
Nonnegative definite function      272
Order of a unifilar Markov source      189 ff.
Parity check code(s)      91 ff. 126
Parity check code(s), adequacy of      110 ff.
Parity check code(s), bounds on error-correcting ability of      105 ff.
Parity check code(s), generator matrix of      132
Parity check code(s), set of generators for      110
Parity check matrix      93 106
Period of a matrix      143
Period of a matrix, maximal      144
Polynomial(s), algebra of      147
Polynomial(s), characteristic      138
Polynomial(s), generator      see “Generator polynomial”
Polynomial(s), in a binary matrix      140
Polynomial(s), irreducible      140 146
Polynomial(s), minimal      see “Minimal polynomial”
Polynomial(s), representation of, by a binary sequence      147
Prefix      28
Probability of error      65
Probability of error, average      65 73
Probability of error, exponential bounds on      see “Error bounds”
Probability of error, for a group code      104
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