The two courses are related to each other and will be given in parallel. However, it is also possible to attend only one of the courses. Course 1 assumes some prior knowledge in elementary linear algebra and will use some MATLAB. Course 2 develops the mathematical foundations of signal analysis based on a „simplified theory of distributions“, which was developed based on modern developments in time-frequency analysis by using the Segal algebra . Necessary concepts and notions from functional analysis (including references) will be recalled (as we go) during the lecture. So prior knowledge of functional analysis is not a prerequisite for successfully participating in this course.

 The two courses are related to each other in such a way that the first course will illustrate the basic principles of Fourier analysis and Harmonic analysis, respectively, including the short-time Fourier transform and Gabor expansion of functions. The second course on the other hand, will discuss in detail the mathematical tools (based on generalized functions, explained in a mathematically exact but also intuitive way) which are necessary to understand the continuous and non-periodic case and will apply these concepts to practical problems. The second course will demonstrate that it is possible to derive the fundamentals of signal and system theory (approximately up to the convolution theorem and Shannon’s sampling theorem) in a mathematically precise and consistent way, without using divergent integrals or groundless exchanges of limiting processes, without Lebesgue integrals, and without using Schwartz‘s theory of tempered distributions.

Both courses can be attended independently. Advanced students may skip Course 1. However, it might be helpful to look at the course material and the motivating explanation, since it deviates considerably from the usual approach found in standard textbooks. Younger students may attend the first course and then take a look at the beginning of the second course to get an idea how the approach can be extended towards distribution theory.

Lecture notes are available for both courses, partially based on existing material, but which will be adapted to the actual courses. MATLAB materials are provided and explained, especially for Course 1, but also for Course 2 by way of illustration.

www.univie.ac.at/nuhag-php/home/skripten.php

www.univie.ac.at/nuhag-php/login/skripten/data/AngAnal15Skript.pdf

An extensive reference list will be given and some of the original works can be downloaded at

www.nuhag.eu/bibtex

There will be an individual oral exam at the end of the semester. Preparation of materials by the students (e.g. MATLAB code or short summary reports on parts of the subject which can be used by other students) will be evaluated as a partial exam result and can improve the mark of the oral exam. However, this part is not obligatory and it is not necessary for a positive evaluation. It should only encourage the students to reflect and apply the material in the course.

Harmonic Analysis provides a way to better understand the analogy between translation invariant linear systems in a discrete or continuous, periodic or non-periodic setting. In each case, there is some underlying group structure, which in turn provides the concept of convolution or Fourier transform and thus provides a complete description of such systems using the concept of impulse response and transfer function.

The discrete-periodic setting can be handled properly within MATLAB, because it deals with signals living on the group  of unit roots of order  (sitting on the unit circle in the complex domain) and because it is a finite dimensional problem which can be completely described in the context of linear algebra. Moreover, polynomials can be used to explain properties of both the convolution and the Fourier transform.-

This course provides the proper understanding of the setting and the connection between mathematical description and MATLAB code (used for demonstration and simulation). A number of applications will be provided. In addition, applications related to image analysis will be given, where the pure frequencies are replaced by plane waves.

 In the second part of the course, the short-time Fourier transform and its sampled version, the so-called Gabor transform and its properties, are studied. These tools are at the basis for MP3 compression algorithms, but they can also be used to design time-variant filters, or to realize slowly-varying systems as Gabor multipliers.  

All the subjects of this discrete setting can also be properly described in the continuous setting, using distributions, as will be done in the second course. The connection between the two settings is beyond the scope of these courses and the subject of current research.

Harmonic Analysis is the mathematical discipline which allows us to take a unifying view of various types of signals, independent of the fact whether these are continuous or discrete, periodic or non-periodic signals. For each setting, there is a unique Fourier transform, which turns the convolution (slightly different in each setting) into pointwise multiplication. Translation invariant systems are convolution operators with some impulse response, whose Fourier transform are the transfer functions of the system.

The treatment of such questions (up to the  Shannon sampling theorem, which is a consequence of Poisson’s formula) often involves the Dirac Delta-“function” or divergent integrals, while the more strict mathematical description of Fourier Analysis puts considerable emphasis on the proper definition of the Lebesgue integral or the convergence of Fourier series.

 The course offered will describe a novel approach, allowing a mathematically concise description of these concepts, which are highly relevant for engineering applications. It is based on the so-called Banach-Gelfand Triple consisting of a Banach space of test functions, the so-called Segal algebra , the Hilbert space  and the dual space , which contains most of the generalized functions (i.e. distributions) needed for engineering applications. The analogy to the threefold number systems,  (rationals), inside  (real numbers) and  (complex numbers) will help in understanding why these three layers are necessary. Operations like Fourier inversion are simple (even via ordinary Riemann integrals) at the level of test functions (just like the simple rule  for rationals), while the properties of the Hilbert space allow us to describe the fact that the Fourier transform is unitary, meaning energy preserving. Finally, the outer layer allows us to take the Fourier transform even for pure frequencies, to talk about Dirac distributions and to avoid divergent integrals, but requires us to give up the idea of pointwise values so much emphasized in the usual mathematical setting.

The course will start with a short introduction to the basic principles of functional analysis, which can be seen as a continuation of linear algebra to vector spaces of signals which are not finite-dimensional anymore, so cannot be simply described by a finite basis. We continue by introducing the space of test functions and its many good properties, which will then allow us to work on a solid mathematical platform using the functionals (the elements of the space ) to perform the actually interesting and otherwise only vaguely described objects. There is still a Fourier transform for these objects, and it is characterized by the “natural fact” that it maps pure frequencies to Dirac distributions (sitting at the right frequency point).  

Depending on the amount of explanation needed (based on the feedback of the audience, also in the exercises), either the foundations will be described in more detail, or preferably, after having set the stage, some applications (e.g. related to the treatment of slowly varying channels) will be given.

The exercises for this course will be given by the lecturer (Prof. Hans G. Feichtinger) and will require continuous participation in the course and completed homework, showing that the participants have a working knowledge of the material presented. There will be a number of exercises to be done by everyone, but if the size of the audience permits, there will also be more personalized assignments, depending on the needs and the backgrounds of the participants.