The elegant geometry of fourier analysis

We outline a method that brings the elegant, unifying geometry of orthogonal function expansions to the teaching of Fourier Analysis in our gateway course on Signals and Systems at UC Berkeley. Our approach starts with discrete-time periodic signals. Their straightforward representation as finite-dimensional Cartesian vectors provides a gentle ingress into the more abstract Euclidean vector spaces that inform the Fourier decompositions of richer signal types. As we describe how a signal fragments into its elemental frequencies, we are careful with the mathematics but we do not let rigor eclipse clarity; plausible reasoning often suffices. We sequence the topics and develop the theory to reduce algebraic clutter and promote geometric insight into the progressively nuanced world of frequency decompositions nestled in the beautiful heart of Fourier Analysis.