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:**

Due to COVID19 restrictions, the lectures and exercises are held online in WS2020.

For more information, enroll in the **TUWEL course**.

**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 oral 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, 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, the asymptotic equivalence of Toeplitz and circulant matrices.

**Course Material**

** Slides and lecture notes:
**You can download the slides and the lecture notes below. If you want a hard-copy of the lecture notes + slides, you can buy it at the Grafisches Zentrum in the Freihaus for about 25€. Should there be no more copies left, then please contact us and we will put some in production.

__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

**Exercises**

The exercises are managed over the **TUWEL course** for the Signal Processing 1. Please enroll in this course.

For **QUESTIONS **regarding the exercises, use the Lecture/Exercise Forum link provided in TUWEL.

__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.
- Python exercises can be done in groups. Note that the maximum size of the group is 3 people.

3. For each exercise you present in the Zoom meeting (see TUWEL), you earn an extra point. You can gain at most 4 points from presenting.

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.

6. If you handed in exercise solutions, you have to be present in the Zoom meeting for that exercise, to actually earn points.

You are expected to be able to present your solutions step by step, in the Zoom meeting, using the screen sharing option (see Zoom tutorial for screen sharing).

**Exercise Hand-in:**

Solutions must be** handed in before the provided deadline in TUWEL**. All solutions are to be uploaded in TUWEL (no hand in solutions this semester!).

You are allowed to upload** 3 files per each exercise**:

- One
**.pdf**file containing analytical solutions for examples 1 and 2. - One
**.pdf**file containing analytical solutions for examples 3 and 4. - One
**.py**file containing the Python implementation of example 4 (only one of the group members needs to upload this file)

Both machine- and hand-written analytical solutions are accepted, but only as .pdf files. But please note, **solutions have to be readable** in general, even after scanning!

Python code is only accepted when a 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 the analytical part (in the .pdf file for examples 3 and 4). - If the Python problem contains an analytical part, solve it yourself and hand it in with your solution of the analytical problems
**(the analytical part has to be handed in by each student individually**).

3. If the exercises are submitted **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: 17. December 2020 (**you will be notified here as soon as the time is set).- 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 midterm date!**

**Mathematical Basics**

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