Handbook of Fourier analysis & its applications:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2009
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXVI, 772 S. Ill., graph. Darst. |
| ISBN: | 9780195335927 |
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| 008 | 090313s2009 xxuad|| |||| 00||| eng d | ||
| 010 | |a 2008019802 | ||
| 020 | |a 9780195335927 |c cloth : alk. paper |9 978-0-19-533592-7 | ||
| 035 | |a (OCoLC)600908878 | ||
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| 040 | |a DE-604 |b ger |e aacr | ||
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| 084 | |a SK 450 |0 (DE-625)143240: |2 rvk | ||
| 100 | 1 | |a Marks, Robert J. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Handbook of Fourier analysis & its applications |c Robert J. Marks II |
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2009 | |
| 300 | |a XXVI, 772 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Fourier analysis | |
| 650 | 0 | 7 | |a Harmonische Analyse |0 (DE-588)4023453-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Fourier-Reihe |0 (DE-588)4155109-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Fourier-Transformation |0 (DE-588)4018014-1 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Fourier-Reihe |0 (DE-588)4155109-6 |D s |
| 689 | 1 | |5 DE-604 | |
| 689 | 2 | 0 | |a Fourier-Transformation |0 (DE-588)4018014-1 |D s |
| 689 | 2 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017171809&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017171809 | ||
Record in the Search Index
| DE-BY-862_location | 2000 |
|---|---|
| DE-BY-FWS_call_number | 2000/SK 450 M346 |
| DE-BY-FWS_katkey | 707631 |
| DE-BY-FWS_media_number | 083000520849 |
| _version_ | 1824555531786780672 |
| adam_text | Contents
Preface
vi
Acronym List
ix
Notation
xi
1
Introduction
3
1.1
Ubiquitous Fourier Analysis
3
1.2
Jean
Baptiste
Joseph Fourier
4
1.3
This Book
6
1.3.1
Surveying the Contents
6
2
Fundamentals of Fourier Analysis
10
2.1
Introduction
10
2.2
Signal Classes
11
2.3
The Fourier Transform
13
2.3.1
The Continuous Time Fourier Transform
14
2.3.2
The Fourier Series
14
2.3.2.1
Parseval s Theorem for the Fourier Series
16
2.3.2.2
Convergence
16
2.3.3
Relationship Between Fourier and Laplace Transforms
18
2.3.4
Some Continuous Time Fourier Transform Theorems
18
2.3.4.1
The Derivative Theorem
18
2.3.4.2
The Convolution Theorem
19
2.3.4.3
The Inversion Theorem
19
2.3.4.4
The Duality Theorem
19
2.3.4.5
The Scaling Theorem
20
2.3.5
Some Continuous Time Fourier Transform Pairs
20
2.3.5.1
The Dirac Delta Function
20
2.3.5.2
Trigonometric Functions
21
2.3.5.3
The Sine and Related Functions
27
2.3.5.4
The Rectangle Function
28
2.3.5.5
The
Signum
Function
29
2.3.5.6
The Gaussian
30
2.3.5.7
The Comb Function
31
2.3.5.8
The Triangle Function
32
2.3.5.9
The Array Function
32
2.3.5.10
The Gamma Function^
35
2.3.5.11
The Beta Function*
36
xv
xvi CONTENTS
2.3.5.12 Bessel
Functions4
37
2.3.5.13 The Sine Integral 39
2.3.5.14
Orthogonal Polynomials 1
40
2.3.6
Other Properties of the Continuous Time Fourier Transform
45
2.3.6.1
The Signal Integral Property
45
2.3.6.2
Conjugate Symmetry of a Real Signal s Spectrum
45
2.3.6.3
The Hubert Transform
46
2.3.6.4
Hubert Transform Relationships Within a Causal Signal s Spectrum
46
2.3.6.5
The Power Theorem
47
2.3.6.6
The
Poisson
Sum Formula
47
2.4
Orthogonal Basis Functions
48
2.4.1
Parseval s Theorem for an
Orthonormal
Basis
48
2.4.2
Examples1
49
2.4.2.1
The Cardinal Series1
49
2.4.2.2
The Fourier Series1
49
2.4.2.3
Prolate Spheroidal Wave Functions1
49
2.4.2.4
Complex Walsh Functions5
50
2.4.2.5
Orthogonal Polynomials1
50
2.4.2.6
Fourier Transforms of
Orthonormal
Functions are Orthonormal1
50
2.5
The Discrete Time Fourier Transform
50
2.5.1
Relation of the DFT to the Continuous Time Fourier Transform
50
2.6
The Discrete Fourier Transform
52
2.6.1
Circular Convolution
53
2.6.2
Relation to the Continuous Time Fourier Transform
54
2.6.3
DFT Leakage
55
2.7
Related Transforms
56
2.7.1
The Cosine Transform1^
56
2.7.2
The Sine TransfoW
57
2.7.3
The Hartley Transform
57
2.7.4
The
z
Transforms
58
2.8
Exercises
60
2.9
Solutions for Selected Chapter
2
Exercises
72
3
Fourier Analysis in Systems Theory
104
3.1
Introduction
104
3.2
System Classes
104
3.2.1
System Types
104
3.2.1.1
Homogeneous Systems
104
3.2.1.2
Additive Systems
105
3.2.1.3
Linear Systems
105
3.2.1.4
Time-Invariant Systems
106
3.2.1.5
LTI Systems
106
3.2.1.6
Causal Systems
107
3.2.1.7
Memoryless Systems
107
3.2.1.8
Stable Systems
107
3.2.1.9
Invertible Systems
107
3.2.2
Example Systems
107
3.2.2.1
The Magnifier
108
3.2.2.2
The Fourier Transformer
109
3.2.2.3
Convolution
110
CONTENTS xvii
3.3 System Characterization 112
3.3.1 Linear System Characterization 112
3.3.1.1
Continuous Time
Systems 112
3.3.1.2
Discrete Time
Systems 113
3.3.2
Causal
Linear Systems 113
3.3.3 Linear Time Invariant (LTI) Systems 114
3.3.3.1
Convolution
Algebra 115
3.3.3.2
Convolution Mechanics
117
3.3.3.3
Circular Convolution Mechanics
118
3.3.3.4
Step and Ramp Response Characterization
118
3.3.3.5
Characterizing an LTI System
120
3.3.3.6
Filters
120
3.4
Amplitude Modulation
122
3.4.1
Coherent Demodulation
122
3.4.1.1
Loss of Coherence and Fading
124
3.4.2
Envelope Demodulation
124
3.5
Goertzel s Algorithm for Computing the DFT
125
3.6
Fractional Fourier Transforms
126
3.6.1
Periodicity of the Fourier Transform Operator
126
3.6.2
Fractional Fourier Transform Criteria
127
3.6.3
The Weighted Fractional Fourier Transform
128
3.7
Approximating a Linear System with LTI Systems
129
3.7.1
Examples of the Piecewise Invariant Approximation
130
3.7.1.1
The Magnifier
130
3.7.1.2
The Fourier Transformer
131
3.8
Exercises
132
3.9
Solutions for Selected Chapter
3
Exercises
139
Fourier Transforms in Probability, Random Variables and
Stochastic Processes
151
4.1
Introduction
151
4.2
Random Variables
151
4.2.1
Probability Density Functions, Expectation, and Characteristic Functions
152
4.2.1.1
Discrete Random Variables
152
4.2.1.2
Moments from the Characteristic Functions
154
4.2.1.3
Chebyshev s Inequality
157
4.2.2
Example Probability Functions, Characteristic Functions, Means
and Variances
158
4.2.2.1
The Uniform Random Variable
159
4.2.2.2
The Triangle Random Variable
159
4.2.2.3
The Gaussian Random Variable
159
4.2.2.4
The Exponential Random Variable
160
4.2.2.5
The Laplace Random Variable
161
4.2.2.6
The Hyperbolic Secant Random Variable
161
4.2.2.7
The Gamma Random Variable
162
4.2.2.8
The Beta Random Variable
164
4.2.2.9
The Cauchy Random Variable
166
4.2.2.10
The Deterministic Random Variable
168
4.2.2.11
The Bernoulli Random Variable
169
4.2.2.12
The Discrete Uniform Random Variable
169
xviii CONTENTS
4.2.2.13 The
Binomial
Random Variable 170
4.2.2.14 The
Poisson
Random Variable 171
4.2.3
Distributions of Sums of Random
Variables 173
4.2.3.1
The Sum of Independent Random Variables
174
4.2.3.2
Distributions Closed Under Independent Summation
174
4.2.3.3
The Sum of i.i.d. Random Variables
176
4.2.3.4
The Average of i.i.d. Random Variables
176
4.2.4
The Law of Large Numbers
177
4.2.4.1
Stochastic Resonance
178
4.2.5
The Central Limit Theorem
179
4.2.5.1
The Central Limit Theorem Applied to Randomly Generated Probability
Density Functions
187
4.2.5.2
Random Variables That Are Asymptotically Gaussian
189
4.3
Uncertainty in Wave Equations
190
4.4
Stochastic Processes
193
4.4.1
First and Second Order Statistics
194
4.4.1.1
Stationary Processes
195
4.4.2
Power Spectral Density
196
4.4.3
Some Stationary Noise Models
197
4.4.3.1
Stationary Continuous White Noise
197
4.4.3.2
Stationary Discrete White Noise
198
4.4.3.3
Laplace Autocorrelation
198
4.4.3.4
Ringing Laplace Autocorrelation
198
4.4.4
Linear Systems with Stationary Stochastic Inputs
198
4.4.5
Properties of the Power Spectral Density
200
4.4.5.1
Second Moment
200
4.4.5.2
Realness
200
4.4.5.3
Nonnegativity
200
4.5
Exercises
201
4.6
Solutions for Selected Chapter
4
Exercises
204
The Sampling Theorem
217
5.1
Introduction
217
5.1.1
The Cardinal Series
217
5.1.2
History
218
5.2
Interpretation
220
5.3
Proofs
220
5.3.1
Using Comb Functions
221
5.3.1.1
Aliasing
222
5.3.1.2
The Resulting Cardinal Series
222
5.3.2
Fourier Series Proof
222
5.3.3
Papoulis Proof
223
5.4
Properties
223
5.4.1
Convergence
223
5.4.1.1
For Finite Energy Signals
224
5.4.1.2
For Bandlimited Functions with Finite Area Spectra
224
5.4.2
Trapezoidal Integration
225
5.4.2.1
Of Bandlimited Functions
225
5.4.2.2
Of Linear Integral Transforms
226
CONTENTS xix
5.4.2.3
Derivation of the Low Passed Kernel
227
5.4.2.4
Parseval s Theorem for the Cardinal Series
231
5.4.3
The Time-Bandwidth Product
231
5.5
Application to Spectra Containing Distributions
232
5.6
Application to Bandlimited Stochastic Processes
233
5.7
Exercises
234
5.8
Solutions for Selected Chapter
5
Exercises
236
6
Generalizations of the Sampling Theorem
242
6.1
Introduction
242
6.2
Generalized Interpolation Functions
242
6.2.1
Oversampling
243
6.2.2
Restoration of Lost Samples
243
6.2.2.1
Sample Dependency
243
6.2.2.2
Restoring a Single Lost Sample
243
6.2.2.3
Restoring
M
Lost Samples
245
6.2.2.4
Direct Interpolation from
M
Lost Samples
246
6.2.2.5
Relaxed Interpolation Formulae
247
6.2.3
Criteria for Generalized Interpolation Functions
248
6.2.3.1
Interpolation Functions
248
6.2.4
Reconstruction from a Filtered Signal s Samples
250
6.2.4.1
Restoration of Samples from an Integrating Detector
250
6.3
Papoulis Generalization
252
6.3.1
Derivation
252
6.3.2
Interpolation Function Computation
256
6.3.3
Example Applications
257
6.3.3.1
Recurrent
Nonuniform
Sampling
257
6.3.3.2
Interlaced Signal-Derivative Sampling
258
6.3.3.3
Higher Order Derivative Sampling
259
6.3.3.4
Effects of Oversampling in Papoulis Generalization
261
6.4
Derivative Interpolation
261
6.4.1
Properties of the Derivative Kernel
262
6.5
A Relation Between the Taylor and Cardinal Series
265
6.6
Sampling Trigonometric Polynomials
266
6.7
Sampling Theory for Bandpass Functions
267
6.7.1
Heterodyned Sampling
268
6.7.2
Direct Bandpass Sampling
270
6.8
A Summary of Sampling Theorems for Directly Sampled Signals
271
6.9
Lagrangian Interpolation
272
6.10
Kramer s Generalization
273
6.11
Exercises
274
6.12
Solutions for Selected Chapter
6
Exercises
278
7
Noise and Error Effects
288
7.1
Introduction
288
7.2
Effects of Additive Data Noise
288
7.2.1
On Cardinal Series Interpolation
288
7.2.1.1
Interpolation Noise Level
289
7.2.1.2
Effects of Oversampling and Filtering
290
XX
CONTENTS
7.2.2 Interpolation
Noise Variance for Directly Sampled Signals
293
7.2.2.1
Interpolation with Lost Samples
294
7.2.2.2
Bandpass Functions
301
7.2.3
On Papoulis Generalization
302
7.2.3.1
Examples
304
7.2.3.2
Notes
304
7.2.4
On Derivative Interpolation
306
7.2.5
A Lower Bound on the NINV
308
7.2.5.1
Examples
309
7.3
Jitter
313
7.4
Filtered Cardinal Series Interpolation
314
7.4.1
Unbiased Interpolation from Jittered Samples
314
7.4.2
Effects of Jitter In Stochastic Bandlimited Signal Interpolation
316
7.4.2.1
NINV of Unbiased Restoration
318
7.4.2.2
Examples
318
7.5
Truncation Error
320
7.5.1
An Error Bound
320
7.6
Exercises
322
7.7
Solutions for Selected Chapter
7
Exercises
324
8
Multidimensional Signal Analysis
326
8.1
Introduction
326
8.2
Notation
327
8.3
Visualizing Higher Dimensions
328
8.3.1 N
Dimensional
Tic Tac Toe
328
8.3.2
Vectorization
329
8.4
Continuous Time Multidimensional Fourier Analysis
330
8.4.1
Linearity
332
8.4.2
The Shift Theorem
332
8.4.3
Multidimensional Convolution
332
8.4.3.1
The Mechanics of Two Dimensional Convolution
333
8.4.4
Separability
334
8.4.5
Rotation, Scale and Transposition
336
8.4.5.1
Transposition
337
8.4.5.2
Scale
337
8.4.5.3
Rotation
337
8.4.5.4
Sequential Combination of Operations
338
8.4.5.5
Rotation in Higher Dimensions
339
8.4.5.6
Effects of Rotation, Scale and Transposition on
Fourier Transformation
340
8.4.6
Fourier Transformation of Circularly Symmetric Functions
341
8.4.6.1
The Two Dimensional Fourier Transform of a Circle
343
8.4.6.2
Generalization to Higher Dimensions
343
8.4.6.3
Polar Representation of Two Dimensional Functions that are
Not Circularly Symmetric
344
8.5
Characterization of Signals from their
Tomographie
Projections
345
8.5.1
The Abel Transform and Its Inverse
345
8.5.1.1
The Abel Transform in Higher Dimensions
347
8.5.2
The Central Slice Theorem
348
8.5.3
The Radon Transform and Its Inverse
349
CONTENTS xxi
8.6
Fourier
Series
352
8.6.1
Multidimensional
Periodicity
352
8.6.2
The Multidimensional Fourier Series Expansion
356
8.6.3
Multidimensional Discrete Fourier Transforms
359
8.6.3.1
Evaluation
360
8.6.3.2
The Role of Phase in Image Characterization
360
8.7
Discrete Cosine Transform-Based Image Coding
360
8.7.1
DCT Basis Functions
362
8.7.2
The DCT in Image Compression
362
8.8
McClellan Transformation for Filter Design
366
8.8.1
Modular Implementation of the McClellan Transform
372
8.8.2
Implementation Issues
373
8.9
The Multidimensional Sampling Theorem
373
8.9.1
The Nyquist Density
376
8.9.1.1
Circular and Rectangular Spectral Support
377
8.9.1.2
Sampling Density Comparisons
379
8.9.2
Generalized Interpolation Functions
380
8.9.2.1
Tightening the Integration Region
380
8.9.2.2
Allowing Slower Roll Off
381
8.10
Restoring Lost Samples
382
8.10.1
Restoration Formulae
382
8.10.1.1
Lost Sample Restoration Theorem
382
8.10.1.2
Restoration of a Single Lost Sample
383
8.10.2
Noise Sensitivity
383
8.10.2.1
Filtering
385
8.10.2.2
Deleting Samples from Optical Images
385
8.11
Periodic Sample Decimation and Restoration
387
8.11.1
Preliminaries
387
8.11.2
First Order Decimated Sample Restoration
390
8.11.3
Sampling Below the Nyquist Density
392
8.11.3.1
The Square Doughnut
392
8.11.3.2
Sub-Nyquist Sampling of Optical Images
393
8.11.4
Higher Order Decimation
393
8.12
Raster Sampling
395
8.12.1
Bandwidth Equivalence of Line Samples
397
8.13
Exercises
398
8.14
Solutions for Selected Chapter
8
Exercises
404
9
Time-Frequency Representations
411
9.1
Introduction
411
9.2
Short Time Fourier Transforms and Spectrograms
413
9.3
Filter Banks
413
9.3.1
Commonly Used Windows
415
9.3.2
Spectrograms
415
9.3.3
The Mechanics of Short Time Fourier Transformation
416
9.3.3.1
The Time Resolution Versus Frequency Resolution Trade Off
416
9.3.4
Computational Architectures
417
9.3.4.1
Modulated Inputs
418
9.3.4.2
Window Design Using Truncated IIR Filters
419
9.3.4.3
Modulated Windows
423
χχϋ
CONTENTS
9.4
Generalized Time-Frequency Representations
424
9.4.1
GTFR Mechanics
425
9.4.2
Kernel Properties
426
9.4.3
Marginals
427
9.4.3.1
Kernel Constraints
428
9.4.4
Example GTFR s
431
9.4.4.1
The Spectrogram
431
9.4.4.2
The Wigner Distribution
434
9.4.4.3
Kernel Synthesis Using Spectrogram Superposition
434
9.4.4.4
The Cone Shaped Kernel
434
9.4.4.5
Cone Kernel GTFR Implementation Using Short Time Fourier
Transforms
435
9.4.4.6
Cone Kernel GTFR Implementation for Real Signals
438
9.4.4.7
Kernel Synthesis Using POCS
438
9.5
Exercises
439
9.6
Solutions for Selected Chapter
9
Exercises
442
10
Signal Recovery
447
10.1
Introduction
447
10.2
Continuous Sampling
447
10.3
Interpolation From Periodic Continuous Samples
450
10.3.1
The Restoration Algorithm
451
10.3.1.1
Trigonometric Polynomials
453
10.3.1.2
Noise Sensitivity
456
10.3.2
Observations
461
10.3.2.1
Comparison with the NINV of the Cardinal Series
461
10.3.2.2
In the Limit as an Extrapolation Algorithm
462
10.3.2.3
Application to Interval Interpolation
463
10.4
Interpolation of Discrete Periodic Nonuniform Decimation
463
10.4.1
Problem Description
466
10.4.1.1
Degree of Aliasing
467
10.4.1.2
Interpolation
467
10.4.1.3
The [N,P] Matrix
470
10.4.2
The Periodic Functions,
4>м(у)
470
10.4.3
Quadrature Version
472
10.5
Prolate Spheroidal Wave Functions
473
10.5.1
Properties
473
10.5.2
Application to Extrapolation
475
10.5.3
Application to Interval Interpolation
476
10.6
The Papoulis-Gerchberg Algorithm
477
10.6.1
The Basic Algorithm
477
10.6.1.1
An Alternate Derivation of the PGA Using Operators
479
10.6.1.2
Application to Interpolation
480
10.6.2
Proof of the PGA using PSWFs
480
10.6.3
Remarks
482
10.7
Exercises
482
10.8
Solutions for Selected Chapter
10
Exercises
486
Π
Signal and Image Synthesis: Alternating Projections Onto Convex Sets
495
11.1
Introduction
495
CONTENTS xxiii
11.2 Geometical POCS 496
11.2.1
Geometrical
Convex
Sets 496
11.2.2
Projecting onto a Convex Set
497
11.2.3
POCS
498
11.3
Convex Sets of Signals
501
11.3.1
The Hubert Space
502
11.3.
11.3.
11.3.
11.3.
11.3.
.1
Convex Sets
503
.2
Subspaces
503
.3
Null Spaces
503
.4
Linear Varieties
504
.5
Cones
504
11.3.1.6
Convex Hulls and Convex Cone Hulls
504
11.3.2
Some Commonly Used Convex Sets of Signals
504
11.3.2.1
Matrix equations
505
11.3.2.2
Bandiimited Signal
507
11.3.2.3
Duration Limited Signals
507
11.3.2.4
Real Transform
Positivity
508
11.3.2.5
Constant Area Signals
508
11.3.2.6
Bounded Energy Signals
511
11.3.2.7
Constant Phase Signals
511
11.3.2.8
Bounded Signals
512
11.3.2.9
Signals With Identical Middles
513
11.4
Example Applications of POCS
514
11.4.1 Von
Neumann s Alternating Projection Theorem
514
11.4.2
Solution of Simultaneous Equations
514
11.4.3
The Papoulis-Gerchberg Algorithm
514
11.4.4
Howard s Minimum-Negativity-Constraint Algorithm
516
11.4.5
Associative Memory
516
11.4.5.1
Template Matching
517
11.4.5.2
POCS Based Associative Memories
517
11.4.5.3
POCS Associative Memory Examples
517
11.4.5.4
POCS Associative Memory Convergence
521
11.4.5.5
POCS Associative Memory Capacity
526
11.4.5.6
Heteroassociative Memory
527
11.4.6
Recovery of Lost Image Blocks
527
11.4.7
Subpixel
Resolution
532
11.4.7.1
Formulation as a Set of Linear Equations
532
11.4.7.2
Subpixel
Resolution using POCS
533
11.4.8
Reconstruction of Images from
Tomographie
Projections
533
11.4.9
Correcting Quantization Error for Oversampled Bandiimited Signals
535
11.4.10
Kernel Synthesis for GTFR s
537
11.4.10.1
GTFR Constraints as Convex Sets
537
11.4.10.2
GTFR Kernel Synthesis Using POCS
543
11.4.11
Application to
Conformai
Radiotherapy
545
11.4.11.1
Convex Constraint Sets
547
11.4.11.2
Example Applications
557
11.5
Generalizations
558
11.5.1
Iteration Relaxation
558
11.5.2
Contractive and
Nonexpansive
Operators
558
11.5.2.1
Contractive and
Nonexpansive
Functions
559
xxiv CONTENTS
11.6
Exercises
563
11.7 Solutions
for Selected Chapter
11
Exercises
567
12
Mathematical Morphology and Fourier Analysis on Time Scales
570
12.1
Introduction
570
12.2
Mathematical Morphology Fundamentals
570
12.2.1
Minkowski Arithmetic
570
12.2.2
Relation of Convolution Support to the Operation of Dilation
572
12.2.3
Other Morphological Operations
573
12.2.4
Minkowski Algebra
576
12.3
Fourier and Signal Analysis on Time Scales
583
12.3.1
Background
586
12.3.1.1
The Hilger Derivative
586
12.3.1.2
Hilger Integration
587
12.3.2
Fourier Transforms on a Time Scale
587
12.3.3
The Minkowski Sum of Time Scales
590
12.3.4
Convolution on a Time Scale
592
12.3.4.1
Convolution on Discrete Time Scales
592
12.3.4.2
Time Scales in Deconvolution
594
12.3.5
Additively Idempotent Time Scales
595
12.3.5.1
АГГЅ
Hulls
5%
12.3.5.2
АГГЅ
Examples
596
12.3.5.3
АГГЅ
Conservation
597
12.3.5.4
Asymptotic Graininess of At,
597
12.3.6
Discrete Convolution of AITS
601
12.3.6.1
Applications
601
12.3.7
Multidimensional AITS Time Scales
604
12.4
Exercises
608
12.5
Solutions for Selected Chapter
12
Exercises
609
13
Applications
610
13.1
The Wave Equation, Its Fourier Solution and Harmony in Western Music
610
13.1.1
The Wave Equation
610
13.1.2
The Fourier Series Solution
612
13.1.3
The Fourier Series and Western Harmony
613
13.1.4
Pythagorean Harmony
614
13.1.4.1
Melodies of Harmonics: Bugle Tunes
616
13.1.4.2
Pythagorean and Tempered String
Vibrations
617
13.1.5
Harmonics Expansions Produce Major Chords and
the Major Scale
618
13.1.5.1
Subharmonics Produce Minor Keys
620
13.1.5.2
Combining the Harmonic and Subharmonic Expansions
Approximates the Tempered Chromatic Scale
621
13.1.6
Fret Calibration
621
13.1.6.1
Comments
623
13.2
Fourier Transforms in Optics and Wave Propagation
623
13.2.1
Scalar Model for Wave Propagation
624
13.2.1.1
The Wave Equation
624
13.2.1.2
Solutions to the Wave Equation
626
13.2.2
The Angular Spectrum
627
CONTENTS xxv
13.2.2.1 Plane
Waves as Frequencies
628
13.2.2.2
Propagation of the Angular Spectrum
629
13.2.2.3
Evanescent Waves
630
13.2.3
Rayleigh-Sommerfield Diffraction
630
13.2.3.1
The Diffraction Integral
633
13.2.3.2
The Fresnel Diffraction Integral
633
13.2.3.3
The
Fraunhofer
Approximation
634
13.2.4
A One Lens System
638
13.2.4.1
Analysis
638
13.2.4.2
Imaging System
640
13.2.4.3
Fourier Transformation
641
13.2.4.4
Implementation of the PGA Using Optics
642
13.2.5
Beamforming
643
13.2.5.1
Apodization
644
13.2.5.2
Beam Steering
644
13.3
Heisenberg s Uncertainty Principle
645
13.4
Elementary Deterministic Finance
646
13.4.1
Some Preliminary Math
647
13.4.2
Compound Interest on a One Time Deposit
647
13.4.2.1
Yield Increases with Frequency of Compounding
648
13.4.2.2
Continuous Compounding
649
13.4.2.3
Different Rates and Compounding Periods
-
Same Yield
649
13.4.2.4
The
Extrema
of Yield
650
13.4.2.5
Effect of Annual Taxes
650
13.4.2.6
Effect of Inflation
651
13.4.3
Compound Interest With Constant Periodic Deposits
652
13.4.3.1
Continuous Time Solution
653
13.4.3.2
Be a Millionaire
653
13.4.3.3
Starting With a Nest Egg
654
13.4.4
Loan and Mortgage Payments
655
13.4.4.1
Monthly Payment
655
13.4.4.2
Amortization
656
13.4.4.3
Monthly Payment Over Long Periods of Time
656
13.5
Exercises
656
13.6
Solutions for Selected Chapter
13
Exercises
658
14
Appendices
660
14.1
Schwarz s Inequality
660
14.2
Leibniz s Rule
661
14.3
A Useful Limit
661
14.4
Series
662
14.4.1
Binomial Series
662
14.4.2
Geometric Series
662
14.4.2.1
Derivation
623
14.4.2.2
Trigonometric Geometric Series
663
14.5
Ill-Conditioned Matrices
663
14.6
Other Commonly Used Random Variables
664
14.6.1
The Pareto Random Variable
665
14.6.2
The Weibull Random Variable
666
14.6.3
The Chi Random Variable
667
xxvi CONTENTS
14.6.4
The Noncentral Chi-Squared
Random Variable
667
14.6.5
The Half Normal Random Variable
668
14.6.6
The Rayleigh Random Variable
668
14.6.7
The Maxwell Random Variable
669
14.6.8
The Log Random Variable
669
14.6.9
The
Von
Mises
Variable
670
14.6.10
The Uniform Product Variable
671
14.6.11
The Uniform Ratio Variable
671
14.6.12
The Logistic Random Variable
672
14.6.13
The Gibrat Random Variable
673
14.6.14
The
F
Random Variable
673
14.6.15
The
Noncentral F
Random Variable
673
14.6.16
The Fisher-Tippett Random Variable
675
14.6.17
The Gumbel Random Variable
675
14.6.18
The Student s
t
Random Variable
676
14.6.19
The
Noncentral
Student s
t
Random Variable
676
14.6.20
The Rice Random Variable
677
14.6.21
The Planck s Radiation Random Variable
677
14.6.22
The Generalized Gaussian Random Variable
677
14.6.23
The Generalized Cauchy Random Variable
678
15
References
680
Index
745
|
| any_adam_object | 1 |
| author | Marks, Robert J. |
| author_facet | Marks, Robert J. |
| author_role | aut |
| author_sort | Marks, Robert J. |
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| building | Verbundindex |
| bvnumber | BV035367873 |
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| callnumber-label | QA403 |
| callnumber-raw | QA403.5 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 450 |
| ctrlnum | (OCoLC)600908878 (DE-599)BVBBV035367873 |
| dewey-full | 515/.2433 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.2433 |
| dewey-search | 515/.2433 |
| dewey-sort | 3515 42433 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035367873 |
| illustrated | Illustrated |
| indexdate | 2025-02-20T07:09:09Z |
| institution | BVB |
| isbn | 9780195335927 |
| language | English |
| lccn | 2008019802 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017171809 |
| oclc_num | 600908878 |
| open_access_boolean | |
| owner | DE-20 DE-19 DE-BY-UBM DE-634 DE-703 DE-11 DE-824 DE-862 DE-BY-FWS |
| owner_facet | DE-20 DE-19 DE-BY-UBM DE-634 DE-703 DE-11 DE-824 DE-862 DE-BY-FWS |
| physical | XXVI, 772 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| spellingShingle | Marks, Robert J. Handbook of Fourier analysis & its applications Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd Fourier-Reihe (DE-588)4155109-6 gnd Fourier-Transformation (DE-588)4018014-1 gnd |
| subject_GND | (DE-588)4023453-8 (DE-588)4155109-6 (DE-588)4018014-1 |
| title | Handbook of Fourier analysis & its applications |
| title_auth | Handbook of Fourier analysis & its applications |
| title_exact_search | Handbook of Fourier analysis & its applications |
| title_full | Handbook of Fourier analysis & its applications Robert J. Marks II |
| title_fullStr | Handbook of Fourier analysis & its applications Robert J. Marks II |
| title_full_unstemmed | Handbook of Fourier analysis & its applications Robert J. Marks II |
| title_short | Handbook of Fourier analysis & its applications |
| title_sort | handbook of fourier analysis its applications |
| topic | Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd Fourier-Reihe (DE-588)4155109-6 gnd Fourier-Transformation (DE-588)4018014-1 gnd |
| topic_facet | Fourier analysis Harmonische Analyse Fourier-Reihe Fourier-Transformation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017171809&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT marksrobertj handbookoffourieranalysisitsapplications |
Indexes
Sonderstandort Fakultät
| Call Number: |
2000 SK 450 M346 |
|---|---|
| Copy 1 | nicht ausleihbar Checked out – Due: 31.12.2099 Recall This |