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Artículo: AMZ-B0FB45K6KT

Discrete Fourier Transform in Linear Algebra: Data Science and Discrete Fourier Transform

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Paperback

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0.33 kg

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This book is designed to teach the Discrete Fourier Transform (DFT) from the perspective of linear algebra. Traditionally, we first learn the continuous Fourier transform in its integral form, and then proceed to the DFT through discretization in the time or spatial domain. However, in practice, Fourier transforms are almost always computed using DFTs on computers. Rarely is the integral form used directly. This led to the motivation behind this book: Why not begin with the DFT?The essence of the Fourier transform lies in basis transformation. Yet, when undergraduate students encounter Fourier analysis through its integral formulation, grasping this essence can be difficult. In recent years, many students have struggled with integration and are discouraged from engaging with Fourier transforms at all. Few students understand basis transformations in function spaces. On the other hand, linear algebra is generally easier for undergraduates to grasp, and with the growing interest in data science, more students are proactively learning linear algebra. Basis transformation in vector spaces is relatively easy to understand. The DFT is, in fact, a basis transformation in vector spaces, and is deeply connected with data science and deep learning. Learning the DFT is a fast track to a deeper understanding of convolutional neural networks (CNNs).This book is intended for readers who have completed an introductory course in linear algebra—first-year university students, graduate students, and professionals looking to revisit Fourier analysis. It is written to be accessible to second-year undergraduates as well. While the text contains a large number of equations to ensure mathematical rigor, readers are encouraged to skip over the detailed derivations on a first reading and focus on understanding the core ideas of the DFT. Once the essential concept of the DFT is understood, the book proceeds to discuss the continuous Fourier transform from the viewpoint of function bases and orthogonal function systems.This book also examines vector bases from two perspectives: "model-driven bases" and "data-driven bases." A typical example of the former is the Discrete Fourier Transform, while Principal Component Analysis (PCA) represents the latter. In data science and deep learning, data-driven bases are often used. However, in cases where only a small amount of data is available or the nature of the data is known in advance, model-driven bases are frequently more effective. This book also explores conditions under which the two approaches coincide—namely, when the DFT and PCA yield the same basis. This correspondence is not only theoretically meaningful but also practically important. In addition, the book discusses the Discrete Cosine Transform (DCT), which is used in JPEG image compression, and presents it as a real-valued analogue of the DFT.

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