# Signal theory: Part 1

## Table of Contents

## News

*optional homework*- Given data \(\big(w(1),\ldots,w(T)\big)\) that is generated by a linear static system \(\mathcal{B}\), explain how to find a kernel representation ker\((R)\) of \(\mathcal{B}\).
- Test your solution on the data \(W = \begin{bmatrix} w(1) & \cdots & w(T) \end{bmatrix}\) saved in the mat-file W.mat. Compare your answer with the "true" data generating system \(\mathcal{B} = \text{ker}(\begin{bmatrix} 1 & 2 & 3 \end{bmatrix})\).

## General information about the course

**Lecturers:**- Ivan Markovsky (part 1, weeks 1, 2, 3, 4)
- Leo Van Biesen (part 2, weeks 4, 5, 6, 8)

**Classes:**- Tuesday, 10:00–12:00
- Tuesday, 13:00–15:00
- Friday, 10:00–12:00

**Evaluation:**- 100% final exam

## Learning outcomes

### Specific

- In-depth knowledge of linear systems theory and its application for signal processing.
- Representations of linear time-invariant systems.
- Converting one representation to another.
- Realization theory (Kung's method).
- Least-squares estimation (Kalman filtering).
- System identification (subspace and optimization methods).

- Can formulate a precise mathematical problem from a given engineering specification.
- Can solve signal processing problems by converting them to already solved problems.
- Can solve signal processing problems numerically using Matlab/Octave.
- Can present solution of problems in a clear, well structured way.

### General

(Extracted from the learning outcomes of the masters program in electronics and information technology engineering.)

- In-depth knowledge and understanding of exact sciences with the specificity of their application to engineering.
- Can reformulate complex engineering problems in order to solve them (simplifying assumptions, reducing complexity).
- Can present and defend results in a scientifically sound way, using contemporary communication tools, for a national as well as for an international professional or lay audience.
- Can collaborate in a (multidisciplinary) team.
- Has a creative, problem-solving, result-driven and evidence-based attitude, aiming at innovation and applicability in industry and society.
- Has a critical attitude towards one’s own results and those of others.
- Has an active knowledge of the theory and applications of information and communication technology.
- Has a profound knowledge of modelling and control.

### Part 1 topics

- Behavioral approach to signals and systems
- Representations of linear time-invariant systems
- Random signals and least-squares estimation

## Slides

### Short MATLAB demos of what the theory does in practice

## Additional materials for Part 1

- Lecture notes written by Leo Van Biesen
- References
*Signals and systems:*A. Oppenheim and A. Willsky, Signals and Systems*Linear algebra:*G. Strang, Linear Algebra and Its Applications*System theory:*D. Luenberger, Introduction to Dynamical Systems: Theory, Models and Applications*Behavioral approach:*J.-W. Polderman and J. Willems, Introduction to Mathematical Systems Theory*Realization theory:*Sections 2.2 and 3.1 from I. Markovsky, Exact and approximate modeling in the behavioral setting and Sections 6.5–8 from E. Sontag, Mathematical control theory: Deterministic finite dimensional systems