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amidror i mastering discrete fourier transform several dimensions 2013

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Textbook in PDF format The continuous Fourier transform (CFT) and its discrete counterpart, the discrete Fourier transform (DFT), are probably among the most widely used mathematical tools in science and engineering. Their fields of application are so diverse and numerous that it would be practically impossible to list them exhaustively in a short paragraph. It is not surprising, therefore, that almost anyone studying or working in a scientific or technological discipline had or will have the opportunity to use them one day or another. The present book originates from the author’s own experience as a frequent user of the DFT. Although several excellent textbooks already exist for teaching the DFT and its applications, it turns out that many practical issues of great importance to the users can still hardly be found in the existing literature. Most of the existing textbooks do not explain in sufficient detail the various pitfalls and artifacts that very often await users of the DFT, while in other books these topics are not easy to find, being dispersed between several sections and chapters. The situation in the multidimensional (MD) case is even worse, since many books defer this subject to a later chapter and only treat it there briefly as a straightforward extension of the one-dimensional case, without considering the particularities of the DFT pitfalls and artifacts in the MD case. Moreover, one can rarely find in the literature a detailed discussion on practical questions such as how to place the true unit values along the axes when plotting the DFT results, or similar practical issues that are often acquired only through the user’s own experience, sometimes simply by trial and error. The aim of the present book is, therefore, to close these gaps in the existing literature. This book is not intended to be a self-contained textbook for teaching the basics of the Fourier theory, a subject which is already covered by many excellent books. It rather intends to assist users who already have a basic knowledge of the DFT, but are not sufficiently familiar with its practical limitations, artifacts and pitfalls. The present book can be used, therefore, as a useful additional reference for students and researchers in various branches of science and engineering who have to work on applications of DFT. For this reason we have preferred throughout this book an informal style based on a pictorial, intuitive approach over a rigorous mathematical treatment, and we have intentionally avoided a purist’s approach based on theorems and proofs. And just like the author’s previous books on the theory of the moiré phenomenon, this volume, too, contains a large number of illustrative examples and figures, some of which are visually fascinating and even spectacular. The material covered in this book includes a wide range of subjects, ranging from the aliasing and leakage phenomena to less widely known topics such as symmetry related issues and the DFT artifacts that may result thereof, or the influence of the various possible types of data extension on the DFT results. It is our aim in this book to attract the reader’s attention to the many possible pitfalls and sources of error in the use of DFT, including potential errors due to the different CFT or DFT definitions being used in different books and software packages, the need for input and output data reorganizations, etc. And most importantly, we show the users how to correctly interpret the DFT results they obtain, and how to distinguish between true spectral contents and the various artifacts that are only due to DFT. This book is intended for students, scientists and engineers wishing to widen their knowledge of the DFT and its practical use. In particular, it will be very useful for naive users from various scientific or technical disciplines who have to use the DFT for their respective applications. The reader will find in this book not only a theoretical explanation of the DFT artifacts and pitfalls in question, but also practical recipes accompanied by many examples for the correct use of DFT in one or several dimensions. The prerequisite mathematical background is limited to an elementary familiarity with calculus and with the continuous and discrete Fourier theory. Background and basic notions . Data reorganizations for the DFT and the IDFT True units along the axes when plotting the DFT Issues related to aliasing Issues related to leakage Issues related to resolution and range . Miscellaneous issues A. Impulses in the continuous and discrete worlds B. Data extensions and their effects on the DFT results C. The roles of p and q and their interconnections D. Miscellaneous remarks and derivations E. Glossary of the main terms

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amidror i mastering discrete fourier transform several dimensions 2013

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