Part 2: Systems are maps

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[This is Part II of my earlier post]

If there is one sentence that I think is most enlightning and important to understand coming out of a signal processing course, it’s

Fourier transforms are useful to signal processing because complex exponentials are eigenfunctions of Linear Time Invariant systems and fourier transforms transform signal vectors into a complex exponential basis, diagonalising convolutions.

My hope in this post, is to help you understand that sentence. Along the way, with some patience, we’ll talk about complex exponentials, eigenvectors and eigenfunctions, and hopefully build some geometric intuition for what frequency transforms are doing and why they’re useful. The broader goal is to recognise that not only are signals vectors, but our ‘systems’ are maps, like functions or matrices that map from one vector to another, and how we can use that to simplify understanding systems.