# Harmonic Analysis: Theory and Applications in Advanced Signal Processing

Dr. Erwin Riegler, Prof. Dr. Helmut Bölcskei

##### Offered in:

- Doctoral and Post-Doctoral Studies: Department of Information Technology and Electrical Engineering
- Electrical Engineering and Information Technology Master: Recommended Subjects (Empfohlene Fächer)
- Mathematics Master: Selection: Further Realms (Auswahl: Weitere Gebiete)
- Physics Master: General Electives (Allgemeine Wahlfächer)
- Computational Science and Engineering Master: Electives (Wahlfächer)

Lecture: | Thursday 10:15-12:00, HG F 26.5 |

The first lecture takes place on Thursday, February 23, 2017, 08:15-10:00. | |

Discussion session: | Thursday 08:15-10:00, HG F 26.5 |

The first discussion session takes place on Thursday, February 23, 2017, 10:15-12:00. | |

Instructor: | Dr. Erwin Riegler, Prof. Dr. Helmut Bölcskei |

Teaching assistant: | Dmytro Perekrestenko |

Office Hours: | Thursday, 15:00-16:00, ETF E 119 (Dmytro Perekrestenko) |

Lecture Notes: | Lecture notes and problem sets with documented solutions are available. |

Credits: | 6 ECTS credits |

Note: This class will be taught in English. To get credits, you
need to pass an oral exam, which will be in German (unless you wish to take it in English, of course).
The exam is also required for doctoral students to get credits for the
course.

##### News

We will post important announcements, links, and other information here in the course of the semester, so please check back often!

##### Course Info

This course is an introduction to the field of applied harmonic analysis with emphasis on applications in signal processing such as transform coding, inverse problems, imaging, signal recovery, and inpainting. We will consider theoretical, applied, and algorithmic aspects.

The outline of the course is as follows.

###### Frame theory:

Frames in finite-dimensional spaces, frames for Hilbert spaces, sampling theorems as frame expansions###### Spectrum-blind sampling:

Sampling of multi-band signals with known support set, density results by Beurling and Landau, unknown support sets, multi-coset sampling, the modulated wideband converter, reconstruction algorithms###### Sparse signals and compressed sensing:

Uncertainty principles, recovery of sparse signals with unknown support set, recovery of sparsely corrupted signals, orthogonal matching pursuit, basis pursuit, the multiple measurement vector problem###### High-dimensional data and dimension reduction:

Random projections, the Johnson-Lindenstrauss Lemma, the Restricted Isometry Property, concentration inequalities, covering numbers, Kashin widths##### Prerequisites

The course is heavy on linear algebra, operator theory, and functional
analysis. A solid background in these areas is beneficial. We will, however,
try to bring everybody on the same page in terms of the
mathematical background required, mostly through reviews of the
mathematical basics in the discussion sessions. Moreover, the lecture notes
contain detailed material on the advanced mathematical concepts used in the course. If you are unsure about the prerequisites, please contact
Dmytro Perekrestenko or Erwin Riegler.

##### Homework Assignments

There will be 6 homework assignments. Every other week a new assignment will be handed out. You can hand in your solutions and get feedback from us, but it is not mandatory to turn in solutions. Complete solutions to the homework assignments will be posted on the course web page.

##### Homework Problem Sets

##### Lecture Notes

We will use the following book chapter as material for the first few lectures and will send you, by e-mail, the material for the remaining lectures as we go along.

- A
short course on frame theory

V. I. Morgenshtern and H. Bölcskei, Chapter in "Mathematical Foundations for Signal Processing, Communications, and Networking", CRC Press, pp. 737-789, 2012.

##### Handouts

##### Exam

There is an oral exam (in German, unless you wish to take it in English, of course).

##### Recommended Reading

If you want to go into more depth or if you need additional background material, please check out these books:- S. Mallat, "A wavelet tour of signal processing: The sparse way", 3rd ed., Elsevier, 2009
- I. Daubechies, "Ten lectures on wavelets", SIAM, 1992
- O. Christensen, "An introduction to frames and Riesz bases", Birkhäuser, 2003
- K. Gröchenig, "Foundations of time-frequency analysis", Springer, 2001
- M. Elad, "Sparse and redundant representations -- From theory to applications in signal and image processing", Springer, 2010