Fundamentals of Signal Processing in Metric Spaces with Lattice Properties: Algebraic Approach

453 Pages·2017·4.631 MB·English

by  Andrey Popoff

Fundamentals Signal
of
Processing Metric Spaces
in
Lattice Properties
with
A L G E B R A I C A P P R O A C H

Fundamentals Signal
of
Processing Metric Spaces
in
Lattice Properties
with
A L G E B R A I C A P P R O A C H
Andrey Popoff
Boca Raton London New York
CRC Press is an imprint of the
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© 2018 by Andrey Popoff
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Version Date: 20171004
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Contents
List of Figures ix
Preface xiii
Introduction xix
List of Abbreviations xxvii
Notation System xxix
1 General Ideas of Natural Science, Signal Theory, and Information
Theory 1
1.1 General System of Notions and Principles of Modern Research
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Information Theory and Natural Sciences . . . . . . . . . . . . . . 5
1.3 Overcoming Logical Difficulties . . . . . . . . . . . . . . . . . . . 10
2 Information Carrier Space Built upon Generalized Boolean
Algebra with a Measure 25
2.1 Information Carrier Space . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Geometrical Properties of Metric Space Built upon Generalized
Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Main Relationships between Elements of Metric Space Built
upon Generalized Boolean Algebra with a Measure . . . . . 29
2.2.2 Notion of Line in Metric Space Built upon Generalized
Boolean Algebra with a Measure . . . . . . . . . . . . . . . 32
2.2.3 Notions of Sheet and Plane in Metric Space Built
upon Generalized Boolean Algebra with a Measure . . . . . 41
2.2.4 Axiomatic System of Metric Space Built upon Generalized
Boolean Algebra with a Measure . . . . . . . . . . . . . . . 46
2.2.5 Metric and Trigonometrical Relationships in Space Built
upon Generalized Boolean Algebra with a Measure . . . . . 51
2.2.6 Properties of Metric Space with Normalized Metric Built
upon Generalized Boolean Algebra with a Measure . . . . . 54
2.3 Informational Properties of Information Carrier Space . . . . . . . 59
v
vi Contents
3 Informational Characteristics and Properties of Stochastic
Processes 75
3.1 Normalized Function of Statistical Interrelationship of Stochastic
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 Normalized Measure of Statistical Interrelationship of Stochastic
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Probabilistic Measure of Statistical Interrelationship of Stochastic
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Information Distribution Density of Stochastic Processes . . . . . 101
3.5 Informational Characteristics and Properties of Stochastic Processes 108
4 Signal Spaces with Lattice Properties 123
4.1 Physical and Informational Signal Spaces . . . . . . . . . . . . . . 123
4.2 Homomorphic Mappings in Signal Space Built upon Generalized
Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.2.1 Homomorphic Mappings of Continuous Signal Into a
Finite (Countable) Sample Set in Signal Space Built
upon Generalized Boolean Algebra with a Measure . . . . . 133
4.2.2 Theorems on Isomorphism in Signal Space Built upon
Generalized Boolean Algebra with a Measure . . . . . . . . 143
4.3 Features of Signal Interaction in Signal Spaces with Various
Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.3.1 Informational Paradox of Additive Signal Interaction in
Linear Signal Space: Notion of Ideal Signal Interaction . . . 147
4.3.2 InformationalRelationshipsCharacterizingSignalInteraction
in Signal Spaces with Various Algebraic Properties . . . . . 156
4.4 Metric and Informational Relationships between Signals Interacting
in Signal Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5 Communication Channel Capacity 173
5.1 Information Quantity Carried by Discrete and Continuous Signals 174
5.1.1 Information Quantity Carried by Binary Signals . . . . . . 174
5.1.2 Information Quantity Carried by m-ary Signals . . . . . . . 179
5.1.3 Information Quantity Carried by Continuous Signals . . . . 190
5.2 Capacity of Noiseless Communication Channels . . . . . . . . . . 196
5.2.1 Discrete Noiseless Channel Capacity . . . . . . . . . . . . . 199
5.2.2 Continuous Noiseless Channel Capacity . . . . . . . . . . . 200
5.2.3 Evaluation of Noiseless Channel Capacity . . . . . . . . . . 201
5.2.3.1 Evaluation of Capacity of Noiseless Channel
Matched with Stochastic Stationary Signal with
Uniform Information Distribution Density . . . . . 201
5.2.3.2 Evaluation of Capacity of Noiseless Channel
Matched with Stochastic Stationary Signal with
Laplacian Information Distribution Density . . . . 204
Contents vii
6 Quality Indices of Signal Processing in Metric Spaces with
L-group Properties 207
6.1 Formulation of Main Signal Processing Problems . . . . . . . . . 208
6.2 Quality Indices of Signal Filtering in Metric Spaces with L-group
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.3 Quality Indices of Unknown Nonrandom Parameter Estimation . 218
6.4 Quality Indices of Classification and Detection of Deterministic
Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.4.1 Quality Indices of Classification of Deterministic Signals in
Metric Spaces with L-group Properties . . . . . . . . . . . . 229
6.4.2 Quality Indices of Deterministic Signal Detection in Metric
Spaces with L-group Properties . . . . . . . . . . . . . . . 237
6.5 Capacity of Communication Channels Operating in Presence of
Interference (Noise) . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.5.1 Capacity of Continuous Communication Channels Operating
in Presence of Interference (Noise) in Metric Spaces with
L-group Properties . . . . . . . . . . . . . . . . . . . . . . . 244
6.5.2 Capacity of Discrete Communication Channels Operating
in Presence of Interference (Noise) in Metric Spaces with
L-group Properties . . . . . . . . . . . . . . . . . . . . . . . 247
6.6 Quality Indices of Resolution-Detection in Metric Spaces with
L-group Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.7 Quality Indices of Resolution-Estimation in Metric Spaces with
Lattice Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7 Synthesis and Analysis of Signal Processing Algorithms 267
7.1 Signal Spaces with Lattice Properties . . . . . . . . . . . . . . . . 269
7.2 EstimationofUnknownNonrandomParameterinSampleSpacewith
Lattice Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
7.2.1 Efficiency of Estimator λˆ in Sample Space with Lattice
n,∧
Properties with Respect to Estimator λˆ in Linear Sample
n,+
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
7.2.2 Quality Indices of Estimators in Metric Sample Space . . . 282
7.3 Extraction of Stochastic Signal in Metric Space with Lattice
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.3.1 Synthesis of Optimal Algorithm of Stochastic Signal
Extraction in Metric Space with L-group Properties . . . . 288
7.3.2 AnalysisofOptimalAlgorithmofSignalExtractioninMetric
Space with L-group Properties . . . . . . . . . . . . . . . . 294
7.3.3 Possibilities of Further Processing of Narrowband Stochastic
Signal Extracted on Basis of Optimal Filtering Algorithm in
Metric Space with L-group Properties . . . . . . . . . . . . 302
7.4 Signal Detection in Metric Space with Lattice Properties . . . . . 304
7.4.1 Deterministic Signal Detection in Presence of Interference
(Noise) in Metric Space with Lattice Properties . . . . . . 305
viii Contents
7.4.2 Detection of Harmonic Signal with Unknown Nonrandom
Amplitude and Initial Phase with Joint Estimation of Time
of Signal Arrival (Ending) in Presence of Interference (Noise)
in Metric Space with L-group Properties . . . . . . . . . . . 311
7.4.3 Detection of Harmonic Signal with Unknown Nonrandom
Amplitude and Initial Phase with Joint Estimation of
Amplitude, Initial Phase, and Time of Signal Arrival
(Ending) in Presence of Interference (Noise) in Metric Space
with L-group Properties . . . . . . . . . . . . . . . . . . . . 324
7.4.4 Features of Detection of Linear Frequency Modulated Signal
with Unknown Nonrandom Amplitude and Initial Phase
with Joint Estimation of Time of Signal Arrival (Ending) in
PresenceofInterference(Noise)inMetricSpacewithL-group
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
7.5 Classification of Deterministic Signals in Metric Space with Lattice
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
7.6 Resolution of Radio Frequency Pulses in Metric Space with Lattice
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
7.7 Methods of Mapping of Signal Spaces into Signal Space with Lattice
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
7.7.1 Method of Mapping of Linear Signal Space into Signal Space
with Lattice Properties . . . . . . . . . . . . . . . . . . . . 379
7.7.2 Method of Mapping of Signal Space with Semigroup
Properties into Signal Space with Lattice Properties . . . . 383
Conclusion 385
Bibliography 389
Index 403
List of Figures
I.1 Dependences Q(m2/m ) of some normalized signal processing
1 2
quality index Q on the ratio m2/m . . . . . . . . . . . . . . . xxii
1 2
I.2 Dependences Q(q2) of some normalized signal processing quality
index Q on the signal-to-noise ratio q2 . . . . . . . . . . . . . . xxii
2.2.1 Hexahedron built upon a set of all subsets of element A+B . 30
2.2.2 Tetrahedron built upon elements O,A,B,C . . . . . . . . . . . 30
2.2.3 Simplex Sx(A +A +A ) in metric space Ω . . . . . . . . . . 55
i j k
2.3.1 Information quantities between two sets A and B . . . . . . . . 61
(cid:83)
2.3.2 Elements of set A = A situated on n-dimensional sphere
α α
Sp(O,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.1 Metric relationships between samples of signals in metric signal
space (Γ,µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2.2 Metric relationships between samples of signals in metric space
(Γ, µ) elucidating Theorem 3.2.14 . . . . . . . . . . . . . . . . 97
3.5.1 IDDsi (t ,t),i (t ,t)ofsamplesξ(t ),ξ(t )ofstochasticprocess
ξ j ξ k j k
ξ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.5.2 IDD i (t ,t) and mutual IDD i (t0,t) of samples ξ(t ), η(t(cid:48)) of
ξ j ξη k j k
stochastic processes ξ(t), η(t(cid:48)) . . . . . . . . . . . . . . . . . . . 119
4.2.1 Graphs of dependences of ratio I (X(cid:48))/I(X) on discretization
∆
interval ∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.3.1 Dependences I (I ) of quantity of mutual information I on
bx ax bx
quantity of mutual information I . . . . . . . . . . . . . . . . 160
ax
5.1.1 NFSI ψ (τ) and normalized ACF r (τ) of stochastic process . 175
u u
5.1.2 IDD i (τ) of stochastic process . . . . . . . . . . . . . . . . . . 176
u
5.1.3 NFSI of multipositional sequence . . . . . . . . . . . . . . . . . 181
5.1.4 IDD of multipositional sequence . . . . . . . . . . . . . . . . . 181
6.2.1 Metric relationships between signals elucidating Theorem 6.2.1 213
6.2.2 Dependences of NMSIs ν (q), ν (q) on signal-to-noise ratio
ssˆ sx
q2 = S /N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
0 0
6.2.3 Dependences of metrics µ (q), µ (q) on signal-to-noise ratio
ssˆ sx
q2 = S /N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
0 0
6.3.1 CDFs of estimation (measurement) results X , X and esti-
∨,i ∧,i
mators λˆ , λˆ . . . . . . . . . . . . . . . . . . . . . . . . . 221
n,∧ n,∨
ix