# Linear Systems

## Course Description

• ### Course Name

Linear Systems

• ### Area of Study

Systems Engineering

• ### Language Level

Taught In English

• ### Prerequisites

Calculus II, Circuits and Systems

### Hours & Credits

• ECTS Credits

6
• Recommended U.S. Semester Credits
3
• Recommended U.S. Quarter Units
4
• ### Overview

Linear Systems
Course Number: 214 - 13326
ECTS credits: 6
YEAR 2/ Lower Division

PREREQUISITES/STUDENTS ARE EXPECTED TO HAVE COMPLETED:
Calculus II, and Circuits and Systems

COMPETENCES AND SKILLS THAT WILL BE ACQUIRED AND LEARNING RESULTS:

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:

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 ACTIVITES AND METHODOLOGY:

The course comprises three types of activity: lectures, problem solving sessions and laboratory practice.

LECTURES (3,5 ECTS)
Lectures provide an overview of the main mathematical and methodological 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 (1,5 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 work in teams, 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:

Assessment is broken up into the following evaluation procedures:

1. Continuous evaluation: Block exams plus laboratory questionnaires. (PO: a, b, e and k)

2. Final exam: covering all the topics of the program. (PO: a, e and k)

A minimum grade of 4 (over 10) will be required in the final exam to pass the course.

Extraordinary exam: Assessment can be fitted to continuous assessment process (with the same porcentages as in ordinary exam) or with a final exame with the 100% of qualification

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).

### Course Disclaimer

Courses and course hours of instruction are subject to change.

ECTS (European Credit Transfer and Accumulation System) credits are converted to semester credits/quarter units differently among U.S. universities. Students should confirm the conversion scale used at their home university when determining credit transfer.