## Fundamentals of Wavelet and Time- Frequency Analysis (Exercise)

Lecturer (assistant) Volker Pohl 0000003213 Exercise 2 SWS Wintersemester 2019/20 German See TUMonline See TUMonline

### Objectives

At the end of this module, the students have a profound theoretical understanding of methods and principles of wavelet and time-frequency analysis. They know the pros and cons of the different signal representations and transforms. They know how to use the learned methods to analyse and synthesise signals, and they can construct appropriated wavelets and Gabor systems. Moreover, the students are familiar with the basic applications where the learned methods are applied and they can implement basic algorithms in MATLAB.

### Description

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.

### Prerequisites

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.

### Teaching and learning methods

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.

### Examination

There will be an oral or written exam (dependent on the number of participants). In this exam, the students have to solve problems using the learned methods from the lecture and exercise. Solving these problems includes calculations, mathematical manipulations and short sketches of proofs. Up to 20% of the exam might consist of multiple-choice questions. No helping material is allowed during the exam.

### Recommended literature

S. Mallat, “A Wavelet Tour of Signal Processing", 2nd. Ed., Academic Press, 1999. G. Strang, T. Nguyen, “Wavelets and Filter Banks", Wellesley-Cambridge Press, 1997. D. F. Walnut, “An Introduction to Wavelet Analysis”, Springer 2002.