Course No. 389.166

2019W, VU, 3.0h, 4.5EC

**General Information**

**Lecturers:**

**Semester hours per week:**

3 (approx. 2 for the lectures, 1 for the exercises)

**Language:**

English

**Time and place:**

Th. 14:00 – 15:30 EI3A

Fr. 08:45 – 10:30 EI4

The first lecture of the winter semester 2019: Thursday, 3.10.2019, 14:00 – 15:30, room EI 3A

**Oral exam**:

The final grade consists of 39 points (maximum) from the exercises and 67 points from the oral exam. You need at least 18 points from the exercises to register for the oral exam. Please register for the oral exam via TISS. New exam dates will be offered upon request via email to sp1exercise(at)nt.tuwien.ac.at.

**Content**

**0. Introduction**

**1. Basics:**

Notation – vector, matrix, models of linear systems, state-space descriptions, Fourier-, Laplace- and z-transform, sampling theorems.

**2. Vector spaces and linear algebra:**

Metrical spaces, groups, topological concepts, supremum and infimum, series, Cauchy series, vector spaces, linear combinations, linear independence, basis and dimension norms and normed vector spaces, inner vector products and inner product spaces, induced norms and Cauchy-Schwarz inequality, orthogonality, Hilbert and Banach spaces.

**3. Representation and approximation in vector spaces**:

Approximation problem in the Hilbert space, orthogonality principle, minimizing via the gradient method, least square filtering, linear regression, signal transformation and generalized Fourier series, examples for orthogonal functions, wavelets.

**4. Linear operators:**

Linear functionals, norms on operators, orthogonal subspaces, nullspace and range, projections, adjoint operators, matrix rank, inverse and condition number, matrix decompositions, subspace methods: Pisarenko, MUSIC, ESPRIT, singular value decomposition.

**5. Matrix computation (Kronecker products):**

Kronecker products and sums, DFT, FFT, Hadamard transformations, special forms of the FFT, Split-radix FFT, overlap add and save methods, circulant matrices, examples to OFDM, vec-operator, big data, asymptotic equivalence of Toeplitz and circulant matrices.

**Course Material**

__Slides to download__:

- Chapter 0: Introduction
- Chapter 1 (with resume): Basics
- Chapter 2: Signal Spaces
- Chapter 3: Representation and Approximation in Vector Spaces
- Chapter 4: Linear Operators
- Chapter 5: Matrix Computation

__Lecture notes__:

The lecture notes are available at the TU Wien Copy Center in Freihaus. It consists of the script and a collection of slides for the remaining chapters.

__Additional literature__:

T.Moon, W.Stirling: “Mathematical Methods for Signal Processing,” Prentice Hall

(There are several copies available at the main library.)

As a reference for discrete signals and systems and the z transform, we recommend:

M. Vetterli, J. Kovacevic, and V. K. Goyal: *Signal Processing: Foundations* available as **free PDF** here: http://www.fourierandwavelets.org

__A paper for discussion__:

E. Y. Sidky, X. Pan: “Recovering a Compactly Supported Function from Knowledge of its Hilbert Transform on a Finite Interval”, IEEE Signal Processing Letters, vol. 12, No. 2, Feb. 2005. (.pdf)

** Presentation slides by Stefan Müller-Stach (in German)**:

“Tarnkappen und mathematische Räume” (.pdf)

**Exercises**

The exercises are managed over the **TUWEL course** for the Signal Processing 1. Please enroll for this course. For **QUESTIONS **regarding the exercises, please schedule a date via e-mail (sp1exercise(at)nt.tuwien.ac.at) or phone call. Otherwise we can’t guarantee to have enough time for your concerns.

__Procedure:__

1. You can get at most 35 points from the exercise section. These points will directly go to your final grade

- 20 from the exercises,
- 15 from a written exam.

2. Each exercise consists of analytical problems and Python problems.

- The analytical part has to be done individually. Discussion is encouraged, but you are supposed to write your solution alone without checking out one from another.
- The Python exercises can be done in groups. No group more than three is allowed.

3. For each exercise you show on the blackboard, you earn an extra point. You can gain at most 4 points from presenting at the blackboard.

4. You need all together at least 18 points to be eligible for the oral exam.

5. If we catch you copying results from another student, you lose all your previous points.

**Exercise Hand-in:**

Solutions must be handed in at least **48 hours** before the exercise lecture.

1. The analytical part can be handed in either by paper or via upload in TUWEL. Solutions have to be readable in general. **Only machine-written solutions are accepted via TUWEL upload.**

**by paper**: to the box on the 5^{th}floor or personally to one of the tutors. Please write the problem number clearly and make sure**your name and registration number are on every page**.**via TUWEL**: only**machine-written**solutions are accepted (a single pdf file). If drawn figures are necessary for the exercises, you may append hand-drawn figures to the pdf file. The requirements for the printout are the same as for the paper-wise submission.

2. Python code is only accepted when valid .py file is uploaded

- At the top of the .py file, write
**the names and registration numbers of all group members**. - Pay attention to a clear structure of the code and provide explanatory comments.
- Make the py-file executable such that all required results and comments are produced or hand in Python simulation results together with analytical part.
- If the Python problem contains an analytical part, hand it in with your solution of the analytical problems (the analytical part has to be handed in by each student).

3. If the exercises are delivered **after the deadline**, you will **lose at least half of the points, provided we still have the time to correct your exercises. Otherwise you will not get any points**.

**Written Exam:**

**Midterm 2019**: 18. Dec. 2019, 15:00-18:00, EI9 (Hlawka HS)- You are permitted to use the following items during the SP1 written exam:

– Lecture notes

– Additional literature (formulary, etc.)

– A simple calculator (no notebooks/tablets/smart-phones) - Elaborated exercises or other electronic equipment are not permitted.
**NOTE: There will be NO alternative date !**

**Mathematical Basics**

In addition to the problems of the exercises, we provide a set of “basic exercises” as self-test. These exercises should be solvable by all students of the 7th semester.