In our previous posts, we established that signals are vectors living in an inner product space and explored how signal analysis is fundamentally about orthogonal decomposition of signals. In this article, we'll see how these abstract mathematical concepts come alive in one of the most ubiquitous technologies around

Note before we start: Don’t be intimidated by the math in this article! If you’ve been following my previous work, you’ll find it not only manageable but also genuinely enjoyable 😊. Orthogonal Decompositions Building on our foundation: In our three previous articles (Signals as Vectors, L² Norms &

Building on our previous discussions of L² norms and inner products, let’s dive into the mathematical foundation that makes signal processing actually work — and discover why Hilbert spaces are the perfect playground for Digital Signal Processing. Note: Before we start our discussion, you may have to read my previous

Following up on our previous exploration of why signals are vectors… This is where of the dark matter of digital signal processing starts — the invisible mathematical foundation that underlies everything you do, yet most practitioners never truly understand it. Hilbert spaces aren’t just abstract math. They’re the core

The Thing They Never Explained in Engineering School In my second year of engineering, during Digital Signal Processing class, the professor suddenly announced: “Signals are vectors.” A few confused faces around the room. Someone brave enough asked: “How?” The professor rattled off: “Hilbert spaces, inner products, linear operators, basis functions…