Course: 2020/2021

Linear Systems

(15372)

Students are expected to have completed

'Calculus II' and 'Circuits and Systems'

ObjectivesFurther information on this link

The goal of the course is to provide the students with the theoretical and methodological knowledge necessary to work with continuous and discrete-time signals and LTI (linear and time-invariant) systems in the frequency domain.
Upon successful completion of the course a student will meet the following ABET Program Outcomes (PO): a, b, e, k.
1. GENERAL/TRANSVERSAL COMPETENCES:
1.1. Individual-work skills (PO: a, b, e, k)
1.2. Capacity for analysis and synthesis (PO: b, e).
1.3. Ability to apply theoretical concepts to practice (PO: a, b, e, k)
1.4. Skills related to group work, collaboration and coordination with other students (PO: a, e, k)
2. SPECIFIC COMPETENCES:
2.1. Theoretical knowledge of signals and systems representation in the frequency domain (PO: a, b, e, k)
2.2. Capacity for analyzing signals and systems in the frequency domain, with emphasis in applications related to Communications (PO: a, b, e, k)
2.3. Use of fundamental tools for the analysis of signals and systems in the frequency domain, with emphasis in Communications (PO: b, e, k)

Description of contents: programme

Unit 0. Review of Signals and Systems in the Time-Domain
Unit 1. Fourier Transform of continuous-time signals
1.1. Fourier series representation of continuous-time periodic signals: analysis and synthesis equations
1.2. Convergence
1.3. Properties of continuous-time Fourier series
1.4. The continuous-time Fourier transform of aperiodic signals
1.5. The continuous-time Fourier transform of periodic signals
1.6. Properties of the continuous-time Fourier transform
1.7. Applications: Filtering and frequency response of systems described by linear constant-coefficient differential equations
Unit 2. The discrete-time Fourier transform
2.1. Fourier series representation of discrete-time periodic signals: analysis and synthesis equations
2.2. Properties of the discrete-time Fourier series. Comparison with continuous-time
2.3. The discrete-time Fourier transform for non-periodic signals
2.4. The discrete-time Fourier transform for periodic signals
2.5. Properties of the continuous-time Fourier transform. Parseval's Theorem. Duality
2.6. Filtering and frequency response of systems characterized by linear constant-coefficient difference equations
Unit 3. Sampling in the time-domain
3.1. The sampling theorem
3.2. Ideal reconstruction
3.3. Discrete-time processing of continuous-time signals
3.4. Decimation and interpolation
Unit 4. Discrete Fourier Transform (DFT)
4.1. Sampling of the Fourier transform
4.2. Discrete Fourier Transform
4.3. Properties
4.4. Applications
Unit 5. The z-transform
5.1. The z-transform
5.2. The region of convergence. Properties
5.3. The inverse z-transform
5.4. Properties of the z-transform
5.5. Evaluation of the frequency response from the pole-zero diagram
5.6. Analysis and characterization of LTI systems using the z-transform
5.7. Block diagram representation

Learning activities and methodology

The course comprises four types of activity: lectures, problem solving sessions, group working sessions and laboratory practice.
LECTURES (3 ECTS)
Lectures provide an oveview of the main mathematical and analytical tools for analysis of signals and systems in the frequency domain mainly using the board and aided by slides and other audiovisual media for the illustration of certain topics. Recommended readings and self-evaluation quizzes are provided for homework. (PO: a)
PROBLEM SOLVING SESSIONS (2 ECTS)
Students are provided with problem sets for each of the units of the program together with the answers (but not the solving procedures). These are designed to probe a thorough understanding of fundamental concepts and to encourage practice on algebraic manipulations. The instructor solves on the board a selection of the problems allowing students self-evaluation by comparison with their answers. During these sessions students are encouraged to ask questions and suggest alternative answers (PO: a, e and k).
LABORATORY EXERCISES (1 ECTS)
Laboratory exercises using MATLAB are designed for applying the mathematical tools presented in the lecture. The students learn to model and simulate signals and systems, and to interpret data from their computational work. The degree of freedom is increased from the first towards the fourth session, progressing from mere demonstrations to more open problems. (PO: a, b and k)

Assessment System

- % end-of-term-examination 50
- % of continuous assessment (assigments, laboratory, practicals...) 50

Basic Bibliography

- Alan V. Oppenheim, Alan S. Willsky, with S. Hamid. Signals and Systems. 2nd edition. Prentice Hall. 1996