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Classical Fourier analysis represents signals as a superposition of trigonometric functions. Such a representation is especially suited for describing stationary properties of signals. For non-stationary processes or signals, methods of wavelet and time-frequency analysis are often the better choice. These Methods represent the signals as a superposition of building blocks, which are obtained from a “mother-signal” by scalings, shifts, and modulations.
Therefore, these signal representations are usually more flexible and can better be better adapted to particular applications. However, the construction of “good” mother signals is generally a challenging and non-trivial process. This module introduces the principles and methods of Wavelet and time-frequency analysis. Both, time-continuous and time-discrete methods are considered in the lecture. The lecture covers in particular the following topics: Haar systems and bases; Discrete Haar transform (DHT); multiresolution analysis; discrete wavelet transform (DWT), construction of wavelet bases; spline bases; time-frequency representations; short-time Fourier transform (STFT); Gabor frames. From the practical side, applications from image processing, channel estimation and radar will be discussed.

Knowledge in linear algebra, analysis, and MATLAB. It is recommended that the students have already attended the following modules: Analysis 1-3, Linear algebra, signal representations.

Presentation of the main content on the blackboard. Applying the learned theory to concrete Problems by solving exercises or/and working on small projects during the exercises.