Calculus I Calculus II Linear Algebra

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 time and frequency domain. Upon successful completion of the course a student will meet the following ABET Program Outcomes (PO): a, b, e, k. 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 OBJECTIVES: 2.1. Theoretical knowledge of signals and systems representation in the time domain (PO: a, b, e, k) 2.2. Theoretical knowledge of signals and systems representation in the frequency domain (PO: a, b, e, k) 2.3. Capacity for analyzing signals and systems in the frequency domain, with emphasis in applications related to Bioengineering (PO: a, b, e, k) 2.4. Use of fundamental tools for the analysis of signals and systems in the frequency domain, with emphasis in Bioengineering (PO: b, e, k)

Unit 1. Signals 1.1. Definition and introduction to biomedical signals 1.2. Properties of the signals: regularity, symmetry, etc. 1.3. Characterization of signals: energy and average power. 1.4. Basic operations with signals: time reversal, scaling, shifting. Unit 2. Systems 2.1. Properties of the systems: causality, stability, time invariance, linearity. 2.2. Linear Time-Invariant Systems (LTI). 2.3. Convolution. 2.4. Properties of LTI systems. Unit 3. Fourier Series Representation of Continuous-Time Periodic Signals and sequences 3.1. Introduction: Response of LTI Systems to Complex Exponentials. 3.2. Fourier Series Representation of Continuous-Time Periodic Signals: Analysis and Synthesis Equations. 3.3. Properties of Continuous-Time Fourier Series. Examples. 3.4. Fourier Series Representation of Discrete-Time Periodic Signals: Analysis and Synthesis Equations. 3.5. Properties of Discrete-Time Fourier Series and comparisons with the Continuous Case. Examples. Unit 4. The Continuous-Time Fourier Transform 4.1. The Continuous-Time Fourier Transform for Aperiodic Signals. 4.2. The Continuous-Time Fourier Transform for Periodic Signals. 4.3. Properties of the Continuous-Time Fourier Transform. Examples. Parseval's Theorem. 4.4. The Discrete-Time Fourier Transform. Properties. Unit 5. Sampling 5.1. The Sampling Theorem. 5.2. Reconstruction of Continuous-Time Signals from Its Samples Using Interpolation. 5.3. Discrete-Time Processing of Continuous-Time Signals. 5.4. Decimation and Interpolation. Unit 6. The Laplace transform and the z-Transform 6.1. The Laplace transform. 6.2. The z-Transform. 6.3. The Region of Convergence. Properties. 6.4. The Inverse Transforms. 6.5. Properties of the Transforms. 6.6. Evaluation of the Frequency Response from the Pole-Zero Plot. 6.7. Analysis and Characterization of LTI Systems Using transforms.

The course will be taught in three types of classes: theory, exercises and laboratory practice. THEORY (2.5 ECTS) The sessions will explain the basic fundamentals and analysis tools corresponding to the core of the course. Numerous examples of signals, systems, their properties and their behaviour, both in the time domain and in the frequency domain, will be given. For this purpose, a blackboard and audiovisual media (slides, video, ...) will be used. The main objective is that the student qualitatively understands the basic tools of linear systems. EXERCISES (2.5 ECTS) For the exercises class, students will be provided in advance with the corresponding statements. Likewise, the detailed solution of different proposed exercises will be provided so that the student acquires a practical knowledge of the subject. LABORATORIES (1 ECTS) The laboratories provide students with a hands-on experience to understand the fundamentals of signals, systems and, most particularly, signal analysis and signal processing. Students will also learn how to use Matlab for signal processing. Students should come prepared for the lab sessions. A paper on a particular application of signal processing in the area of Biomedicine will be proposed and carried out in groups. TUTORIALS Tutorials will be held weekly in several sessions spread throughout the week in order to give students more options to attend. Group tutorials will be held when required by the students, at the times scheduled for this purpose.