Signals and systems

The course Signals and Systems covers the basics of signal and system analysis, adapted to the knowledge and demands of engineers, and includes the mathematical skills for handling matrices and matrix equations.

Topics that are covered in the signals part include convolution, Fourier series, Fourier transform, Discrete Fourier Transform and the Sampling Theorem Students learn that the representation of signals in frequency domain is just as important as the time domain representations they already know. In the systems part the focus lies on dynamical systems that convert input signals into output signals as is essential in many engineering applications and forms the basis of controller design. Different representations of systems are discussed including state-space models, transfer functions and frequency response functions (FRF), which will be connected to Fourier series, Fourier transform, Laplace transform of input and output signals.

The course begins with an introduction on the relevance of signal analysis and the use of frequency domain characterizations. The course starts by introducing some basic concepts of signals including convolution. Then it focuses on the decomposition of periodic signals into harmonic components via the Fourier series. The concepts are extended to general aperiodic signals, leading to the Fourier Transform. Additional topics such as the Discrete Fourier Transform and the Sampling Theorem are covered to enable students to use these concepts in real-life applications. This is the main focus of the project.

Based on this knowledge the step is then made towards system representations and analysis. Input/output (I/O) measurements form the basic starting point to obtain the Frequency Response Function (FRF) of a system. The FRF as a system representation is then connected to alternative system descriptions. The connection between FRF and state-space models via the transfer function is discussed in detail. The use of state-space models and transfer functions motivates a deeper understanding of mathematical analysis of matrix equations. This leads to a discussion of many important concepts including rank, eigenvalues, and eigenvectors of matrices, characteristic equation, the concept of basis (L5c) in R^n and coordinate transformations, similarity of matrices, diagonalization and (real) Jordan form. The connections between these mathematical notions and signal and system analysis will be clarified through various examples.

For Student Mobility Alliance students: This course requires knowledge of: This course requires knowledge of: - complex numbers, - differentiation; - integration; - solving systems of linear equations, i.e., solve Ax = b; - inverse of a matrix, determinant of a matrix; - linear combination of vectors; and - first-order differential equations. The lectures will mostly be delivered in a hybrid format allowing you to follow online. The course assessment will be through a project and an exam with weights for the final grade of 25% and 75%, respectively. The project also requires group work. The final exam is a written on campus exam

General learning objective
Building knowledge and understanding of signals and systems in both time and frequency domain, and the operationalization of this knowledge, and understanding the first principles of system analysis as well as the ability to apply the necessary mathematical concepts and tools.

Learning objectives

• To compute the convolution of two signals
• To determine the response of an LTI system to sinusoids and exponentials
• To understand and compute the Fourier Series
• To understand and compute the Fourier Transform of simple and elaborate signals using the properties of the Fourier Transform
• To understand the Discrete Fourier Transform and apply it to a real-world problem
• To write a linear time-invariant (LTI) system as a state-space model consisting of a set of first order differential equations in vector/matrix form
• To compute different representations of an LTI system: FRF, transfer functions/matrices and state-space
• To compute eigenvalues and eigenvectors of square matrices and the poles of transfer functions/matrices
• To perform matrix calculations including rank, null spaces, image and various transformations
• To compute matrix diagonalization by coordinate transformations and to calculate canonical forms of matrices
• To compute the responses of LTI systems in time and frequency domain.
• To compute the steady-state output response of an LTI system given a sinusoidal input from the Bode plot.
• To compute and design the time-domain response of an LTI system given its Bode plot.

You must meet the following requirements

• Completed none of the course modules listed below
• Vector Analysis (32VAN)
• Linear Algebra and Applications (2DBI00)
• Signals and systems (4CB00)
• Systems (5ESB0)

• Via Canvas
• Signals and Systems (ISBN 9781292025902)
• Linear Algebra and Its Applications (ISBN 9780321982384)
• Feedback Control of Dynamic Systems (ISBN 978-0691140216)
• Linear Systems Theory (ISBN 978-0691140216)