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E-raamat: Mastering the Discrete Fourier Transform in One, Two or Several Dimensions: Pitfalls and Artifacts

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Mastering the Discrete Fourier Transform in One, Two or Several Dimensions: Pitfalls and ArtifactsSuurem pilt
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The discrete Fourier transform (DFT) is an extremely useful tool that finds application in many different disciplines. However, its use requires caution. The aim of this book is to explain the DFT and its various artifacts and pitfalls and to show how to avoid these (whenever possible), or at least how to recognize them in order to avoid misinterpretations. This concentrated treatment of the DFT artifacts and pitfalls in a single volume is, indeed, new, and it makes this book a valuable source of information for the widest possible range of DFT users. Special attention is given to the one and two dimensional cases due to their particular importance, but the discussion covers the general multidimensional case, too. The book favours a pictorial, intuitive approach which is supported by mathematics, and the discussion is accompanied by a large number of figures and illustrative examples, some of which are visually attractive and even spectacular.

Mastering the Discrete Fourier Transform in One, Two or Several Dimensions is intended for scientists, engineers, students and any readers who wish to widen their knowledge of the DFT and its practical use. This book will also be very useful for ‘naive’ users from various scientific or technical disciplines who have to use the DFT for their respective applications. The prerequisite mathematical background is limited to an elementary familiarity with calculus and with the continuous and discrete Fourier theory.

Arvustused

From the book reviews:

This book is about the discrete Fourier transform and its practical limitations, pitfalls and artifacts. It is addressed to a wide audience in various scientific and engineering branches, and because of that an intuitive approach supported by mathematics is preferred over a rigorous mathematical treatment. (Biljana Jolevska-Tuneska, Mathematical Reviews, May, 2014)

This book is for readers who have already taken an introductory course in the theory of discrete and continuous Fourier transforms, and now want to use their new knowledge in various science or engineering fields. Amidror (Swiss Federal Institute of Technology in Lausanne) illustrates some of the most frequent mistakes using several hundred figures, which are a central part of the books approach. it may be useful for professional audiences. Summing Up: Recommended. Professionals/practitioners. (M. Bona, Choice, Vol. 51 (9), May, 2014)

The current book aims to explain the DFT and the various ways one can get into trouble with it. More importantly, it also suggests how to avoid the pitfalls or recognize them and escape any consequent misinterpretation. it would be useful resource for practitioners and a valuable addition to libraries. It has a good bibliography and a very nice glossary of signal and image processing terms. (William J. Satzer, MAA Reviews, April, 2014)

The aim of this textbook is to explain the various artifacts and pitfalls of the DFT. This book is mainly written for people with a basic knowledge of DFT, but are not sufficiently familiar with practical limitations and artifacts of DFT. The textbook is written in a very informal style and contains numerous examples, figures, and tables. Without any doubt, this book will be a valuable source of information for a wide range of DFT users. (Manfred Tasche, zbMATH, Vol. 1277, 2014)

Preface xi
1 Introduction
1(14)
1.1 The discrete Fourier transform
1(1)
1.2 A brief historical background
2(2)
1.3 The scope of the present book
4(1)
1.4 Overview of the following chapters
5(2)
1.5 About the graphic presentation of sampled signals and discrete data
7(6)
1.5.5 Graphic presentations in the ID case
7(1)
1.5.5 Graphic presentations in the 2D case
8(5)
1.5.5 Graphic presentations in the MD case
13(1)
1.6 About the exercises and the internet site
13(2)
2 Background and basic notions
15(30)
2.1 Introduction
15(1)
2.2 The continuous Fourier transform: definitions and notations
15(2)
2.3 The discrete Fourier transform: definitions and notations
17(4)
2.4 Rules for deriving new Fourier transforms from already known ones
21(10)
2.4.4 Rules for the ID continuous Fourier transform
22(1)
2.4.4 Rules for the 2D continuous Fourier transform
23(2)
2.4.4 Rules for the MD continuous Fourier transform
25(1)
2.4.4 Rules for the ID discrete Fourier transform
26(2)
2.4.4 Rules for the 2D discrete Fourier transform
28(1)
2.4.4 Rules for the MD discrete Fourier transform
29(2)
2.5 Graphical development of the DFT - a three-stage process
31(5)
2.6 DFT as an approximation to the continuous Fourier transform
36(5)
2.7 The use of DFT in the case of periodic or almost-periodic functions
41(4)
Problems
42(3)
3 Data reorganizations for the DFT and the IDFT
45(24)
3.1 Introduction
45(1)
3.2 Reorganization of the output data of the DFT
45(2)
3.3 Reorganization of the input data of the DFT
47(4)
3.4 Data reorganizations in the case of IDFT
51(3)
3.5 Examples
54(6)
3.6 Discussion
60(9)
Problems
66(3)
4 True units along the axes when plotting the DFT
69(20)
4.1 Introduction
69(1)
4.2 True units for the input array
69(3)
4.3 True units for the output array
72(6)
4.3.3 The particular case of periodic functions
74(4)
4.4 True units for the DFT element values (heights along the vertical axis)
78(1)
4.5 Examples
79(10)
Problems
84(5)
5 Issues related to aliasing
89(54)
5.1 Introduction
89(1)
5.2 Aliasing in the one dimensional case
89(6)
5.3 Examples of aliasing in the one dimensional case
95(18)
5.4 Aliasing in two or more dimensions
113(1)
5.5 Examples of aliasing in the multidimensional case
114(18)
5.6 Discussion
132(1)
5.7 Signal-domain aliasing
133(10)
Problems
136(7)
6 Issues related to leakage
143(42)
6.1 Introduction
143(1)
6.2 Leakage in the one dimensional case
143(5)
6.3 Examples of leakage in the one dimensional case
148(10)
6.4 Errors due to signal-domain truncation
158(4)
6.5 Spectral impulses that fall between output array elements
162(3)
6.6 Leakage in two or more dimensions
165(9)
6.6.6 The case of 2D 1-fold periodic functions
170(2)
6.6.6 The case of 2D 2-fold periodic functions
172(1)
6.6.6 The general MD case
173(1)
6.7 Signal-domain leakage
174(11)
Problems
179(6)
7 Issues related to resolution and range
185(10)
7.1 Introduction
185(1)
7.2 The choice of the array size
185(1)
7.3 The choice of the sampling interval
186(1)
7.4 The choice of the sampling range
187(2)
7.5 The choice of the frequency step and of the frequency range
189(6)
Problems
192(3)
8 Miscellaneous issues
195(52)
8.1 Introduction
195(1)
8.2 Representation of discontinuities
195(7)
8.3 Phase related issues
202(1)
8.4 Symmetry related issues
203(7)
8.5 Jaggies on sharp edges as aliasing or reconstruction phenomena
210(13)
8.6 Sub-Nyquist artifacts
223(18)
8.7 Displaying considerations
241(1)
8.8 Numeric precision considerations
241(6)
Problems
242(5)
Appendices
A Impulses in the continuous and discrete worlds
247(18)
A.1 Introduction
247(1)
A.2 Continuous-world impulses vs. discrete-world impulses
247(1)
A. 3 Impulses in the spectral domain
248(11)
A.4 Impulses in the signal domain
259(6)
B Data extensions and their effects on the DFT results
265(8)
B.1 Introduction
265(1)
B.2 Method 1: Extending the input data by adding new values beyond the original range
265(1)
B.3 Method 2: Extending the input data by denser sampling within the original range
266(2)
B.4 Method 3: Extending the input data by adding zeroes after each value (zero packing)
268(1)
B.5 Method 4: Extending the input data by replicating it
269(1)
B.6 Method 5: Extending the input data by replicating each of its elements
269(1)
B.7 Method 6: Extending the input data by adding zeroes beyond its original range (zero padding)
270(2)
B.8 Conclusions
272(1)
C The roles of p and q and their interconnections
273(26)
C.1 Introduction
273(1)
C.2 The one dimensional case
273(7)
C.3 Generalization of p and q to the multidimensional case
280(19)
C.3.1 Multidimensional generalization in the continuous world
281(4)
C.3.2 Multidimensional generalization in the discrete world
285(14)
D Miscellaneous remarks and derivations
299(36)
D.1 The periodicity of the input and output arrays of the DFT
299(1)
D.2 Explanation of the element order in the DFT output array
300(2)
D.3 The beating effect in a sum of cosines with similar frequencies
302(4)
D.4 Convolutions with a sine function which have no effect
306(1)
D.5 The convolution of a square pulse with a sine function
307(3)
D.6 A more detailed discussion on Remark A.2 of Sec. A.3
310(4)
D.7 The effect of the sine lobes on its convolution with a sharp-edged function
314(1)
D.8 On the order of applying the sampling and truncation operations
315(2)
D.9 Relation of the DFT to the CFT and to Fourier series
317(15)
D.9.1 Examples illustrating the relation of the DFT to the CFT and to Fourier series
325(7)
D.10 The 2D spectrum of a rotated bar passing through the origin
332(3)
E Glossary of the main terms
335(18)
E.1 About the glossary
335(1)
E.2 Terms in the signal domain
336(5)
E.3 Terms in the spectral domain
341(4)
E.4 Miscellaneous terms
345(8)
List of the main relations 353(2)
List of notations and symbols 355(4)
List of abbreviations 359(2)
References 361(6)
Index 367
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