Signal Processing

In previous parts we have already discussed how changing a coordinate system or basis (Definition 1.8) can simplify expression or computation, see 3  Eigendecomposition and 4  Singular Value Decomposition, among others.

One of the most fundamental coordinate transformations was introduced by J.-B. Joseph Fourier in the early 1800s. While investigating heat, he discovered that sine and cosine functions with increasing frequency form an orthogonal basis (Definition 1.6 & Definition 1.9). In fact, the sine and cosine functions form an eigenbasis for the heat equation \[ \frac{\partial u}{\partial t} = \Delta u \] and solving it becomes trivial once you determine the corresponding eigenvalues that are connected to the geometry, amplitudes, and boundary conditions.

In the 200+ years since, this discovery has not only founded new corners of mathematics but also allows via the fast fourier transform or FFT the real-time image and audio compression that make our global communication networks work.

In the same area, the related wavelets where developed for advanced signal processing and compression.

In this section we are going to discuss basics of signal processing in terms of these and other signal transformations, see

• Fourier Transform in 8  Fourier Transform
• Wavelet Transform in 10  Wavelet Transform
• Two-Dimensional Transforms 11  Two-Dimensional Transform

Parts of this section are based on (Brunton and Kutz 2022, chap. 2).

Brunton, Steven L., and J. Nathan Kutz. 2022. Data-Driven Science and Engineering - Machine Learning, Dynamical Systems, and Control. 2nd ed. Cambridge: Cambridge University Press.