A Unified Vector Space Approach to Teaching the Fourier Transform

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 exponential 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 development (the DFT) that supports all four transforms with a common foundation.