Course: 2024/2025

Linear Systems

(13326)

Requirements (Subjects that are assumed to be known)

Calculus II
Circuits and Systems

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:
- Theoretical knowledge of signals and systems representation in the frequency domain
- Capacity for analyzing signals and systems in the frequency domain, with emphasis in applications related to Communications
- Use of fundamental tools for the analysis of signals and systems in the frequency domain, with emphasis in Communications

Skills and learning outcomes

Description of contents: programme

BLOCK 0: Introduction
Unit 0. Review of Signals and Systems in the Time-Domain
BLOCK 1: Fourier Series Representation of Periodic Signals
Unit 1. Fourier Series Representation of Continuous-Time Periodic Signals
1.1. Introduction: Response of LTI Systems to Complex Exponentials
1.2. Fourier Series Representation of Continuous-Time Periodic Signals: Analysis and Synthesis Equations
1.3. Convergence
1.4. Properties of Continuous-Time Fourier Series. Examples
Unit 2. Fourier Series Representation of Discrete-Time Periodic Signals
2.1. Fourier Series Representation of Discrete-Time Periodic Signals: Analysis and Synthesis Equations
2.2. Properties of Discrete-Time Fourier Series. Comparison with the Continuous Case. Examples.
BLOCK 2. Fourier Transform
Unit 3. The Continuous-Time Fourier Transform
3.1. Introduction
3.2. The Continuous-Time Fourier Transform for Aperiodic Signals
3.3. The Continuous-Time Fourier Transform for Periodic Signals
3.4. Properties of the Continuous-Time Fourier Transform. Examples. Parseval¿s Theorem.
Unit 4. The Discrete-Time Fourier Transform
4.1. Introduction
4.2. The Discrete-Time Fourier Transform for Aperiodic Signals
4.3. The Discrete-Time Fourier Transform for Periodic Signals
4.4. Properties of the Continuous-Time Fourier Transform. Parseval¿s Theorem. Duality
Unit 5. Systems
5.1. Introduction
5.2. Frequency Response of Systems Characterized by Linear Constant-Coefficient Differential Equations
5.3. Frequency Response of Systems Characterized by Linear Constant-Coefficient Difference Equations
BLOCK 3. Sampling
Unit 6. Sampling in the Time-Domain
6.1. Introduction
6.2. The Sampling Theorem
6.3. Reconstruction of Continuous-Time Signals from Its Samples Using Interpolation
6.4. Discrete-Time Processing of Continuous-Time Signals
6.5. Decimation and Interpolation
Unit 7. Sampling in the Frequency-Domain: Discrete Fourier Transform
7.1. Introduction
7.2. Sampling of the Fourier Transform
7.3. Discrete Fourier Transform
7.4. Properties
7.5. Circular Convolution and Linear Convolution
BLOCK 4. The z-Transform
Unit 8. The z-Transform
8.1. Introduction
8.2. The z-Transform
8.3. The Region of Convergence. Properties
8.4. The Inverse z-Transform
8.5. Properties of the z-Transform
8.6. Evaluation of the Frequency Response from the Pole-Zero Plot
8.7. Analysis and Characterization of LTI Systems Using the z-Transform
8.8. 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.
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.
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.

Assessment System

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

Calendar of Continuous assessment

Extraordinary call: regulations

Basic Bibliography

- Alan V. Oppenheim, Ronald W. Schafer, with John R. Buck. Tratamiento de Señales en Tiempo Discreto. Prentice Hall; 2 edition (January 1, 2000).
- Alan V. Oppenheim, Alan S. Willsky, with S. Hamid. Signals and Systems. Prentice Hall; 2 edition (August 16, 1996).
- S. S. Soliman and M. D. Srinath. Continuous and Discrete Time Signals and Systems. Prentice Hall; 2 edition (January 1, 1998).

Additional Bibliography

- A. Bracewel. "The Fourier Transform and Its Applications". McGraw-Hill, New York, Second Edition. 1986.
- A. Papoulis. "Signal Analysis". McGraw-Hill, New York, 1977.
- A. Papoulis. "The Fourier Integral and Its Applications". McGraw-Hill, New York, 1962.
- J. G. Proakis and D. G. Manolakis. "Digital Signal Processing: Principles, Algorithms and Applications". Prentice-Hall, Upper Saddle River, NJ, 1996.
- J. R. Buck, M. M. Daniel and A. C. Singer. "Computer Explorations in Signals and Systems Using MATLAB". Prentice-Hall, Upper Saddle River, NJ, 1997.
- R. A. Gabel and R. A. Roberts. "Signals and Linear Systems". John Wiley & Sons, New York, NY, Third Edition, 1987.
- R. A. Roberts and D. T. Mullis. "Digital Signal Processing". Addison-Wesley, Reading, MA, 1987.

The course syllabus may change due academic events or other reasons.