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A Unified Vector Space Approach to Teaching the Fourier Transform Andrew Sterian Grand Valley State University

Abstract

This paper is concerned with the approach to teaching an introductory course on Fourier theory for engineers. Commonly called Signals and Systems (or some variation), this course generally introduces four transforms: the Fourier Transform, the Fourier Series, the Discrete-Time Fourier Transform (DTFT), and the Discrete Fourier Transform (DFT). Our concern is that the method and order of topic presentation (as reflected in popular textbooks) creates unnecessary difficulties for students. We propose spending less time on the transforms themselves and more time at the beginning of the course in presenting a finite-dimensional vector space framework. The DFT then becomes a natural application of this framework: the projection of a signal onto a complex expo- nential basis. The remaining three transforms follow with the same interpretation, differing only in the domain of application. Thus, students are presented with a rigorous but tractable develop- ment (the DFT) that supports all four transforms with a common foundation.

1. Introduction

Electrical engineering curricula traditionally include an introductory course on signals and sys- tems as a foundation for subsequent courses in communications, control, and signal processing. A large component of this course (in addition to LTI systems, convolution, the impulse response, etc.) is the presentation of the Fourier Transform in its four different forms (Fourier Transform, DFT, DTFT, and Fourier Series). Students often have difficulty with this material because it is highly abstract and because the four transforms are tantalizingly similar yet have subtly different properties and domains of application. These differences and similarities are sources of confusion because students are usually not given an underlying foundation to bind the transforms together. In addition, students are expected to accept on faith that the various transform definitions are cor- rect. Computer demonstrations of harmonic sinusoids summing together to eventually describe a square wave are reassuring and compelling, but there is still a residual feeling that there is magic involved.

Our approach to teaching the Fourier transforms is founded upon basic concepts of abstract linear

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Sterian, A. (2000, June), A Unified Vector Space Approach To Teaching The Fourier Transform Paper presented at 2000 Annual Conference, St. Louis, Missouri. 10.18260/1-2--8793