'Calculus II' and 'Circuits and Systems'

Linear systems, or systems defined by a linear operator, can be used to model many real-world systems, and find applications in control theory, signal processing, and telecommunication technologies, among other areas. The goal of this 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 both time and frequency domains. Upon attending this course students will acquire: - 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.

CB1: Students have demonstrated possession and understanding of knowledge in an area of study that builds on the foundation of general secondary education, and is usually at a level that, while relying on advanced textbooks, also includes some aspects that involve knowledge from the cutting edge of their field of study CB2: Students are able to apply their knowledge to their work or vocation in a professional manner and possess the competences usually demonstrated through the development and defence of arguments and problem solving within their field of study. CG3: Knowledge of basic and technological subject areas which enable acquisition of new methods and technologies, as well as endowing the technical engineer with the versatility necessary to adapt to any new situation. CG13: Understanding and command of basic concepts of linear systems and related functions and transformers. Electrical circuit theory, electronic circuits, physical principles of semiconductors and logic families, electronic and photonic devices, materials technology and their application in resolving problems characteristic of engineering. RA1: Knowledge and understanding of the general fundamentals of engineering, scientific and mathematical principles, as well as those of their branch or specialty, including some knowledge at the forefront of their field. RA5: Applications. Graduates will have the ability to apply their knowledge and understanding to solve problems, conduct research, and design engineering devices or processes. These skills include knowledge, use and limitations of materials, computer models, process engineering, equipment, practical work, technical literature and information sources. They must be aware  of all the implications of engineering practice: ethical, environmental, commercial and industrial.

Unit 0. Review of Signals and Systems in the Time-Domain Unit 1. Fourier Transform: continuous-time signals 1.1. Periodic signals: Fourier series representation 1.2. The continuous-time Fourier transform and its properties 1.3. Analysis of linear time-invariant systems 1.4. Applications: Filtering and systems described by linear differential equations Unit 2. Fourier Transform: discrete-time sequences 2.1. Discrete-time complex exponentials 2.2. Fourier series representation of discrete-time periodic signals 2.3. The Fourier transform of sequences and differences with continuous-time 2.4. Applications: Filtering. Systems characterized by linear difference equations Unit 3. Sampling in the time-domain 3.1. The sampling theorem and optimal signal reconstruction 3.2. Discrete-time processing of continuous-time signals 3.3. Decimation and interpolation Unit 4. Discrete Fourier Transform (DFT) 4.1. Finite-length signals and periodic signals: DFT 4.2. Connection between Discrete Fourier Transform and the Fourier Transform 4.3. Efficient implementation and applications Unit 5. The z-transform 5.1. Definition and connection with Fourier Transform 5.2. The region of convergence and its properties: zero-pole diagrams 5.3. Analysis and characterization of unstable LTI systems

The course comprises three types of activities: lectures, problem solving sessions, and laboratory sessions. LECTURES (3 ECTS) Lectures provide an overview 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 guided exercises to more open problems.