Signals, Systems and Processes
Summary
These notes constitute elements regarding the mathematical representation of time series. They have been used for teaching second year Master students at the Neural Information Processing program of the Graduate Training Center of Neuroscience of Tübingen, Germany. This program is oriented towards students that want to approach Neuroscience from a computational perspective. Given the variety of backgrounds, and the diversity of aspects of time series computational neuroscientists may encounter, lectures where aimed at:
- Introduce and learn to manipulate the mathematical representations of signals (convolution, difference equations, z-transform, point-processes);
- Introduce key principles behind classical signal processing tools: filtering, spectral analysis, wavelets;
- Understand the capabilities and limitations of time series analysis approaches and how to use them safely.
To achieve this purpose, I have attempted to gather in a coherent way a broad range of topics that are typically taught in different fields. Notably, I tried to put in coherence the discrete and continuous time perspectives on signals and systems, but also link deterministic and stochastic aspects, such that the later can be explained without resorting to heavy formalism. Mathematical refreshers are introduced in order to allow students with a B.Sc. from various scientific fields (Physics, Computer Science, Biochemistry, Mathematics) to follow the technical developments without referring to other textbooks.
I am very grateful to Kaidi Shao for helping me giving these lectures. Constructive feedback on these notes is of course welcome.