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.

Program

Unit 0. Review of Signals and Systems in the Time-Domain

Unit 1. Fourier Transform: continuous-time signals

  • Periodic signals: Fourier series representation

  • The continuous-time Fourier transform and its properties

  • Analysis of linear time-invariant systems

  • Applications: Filtering and systems described by linear differential equations

Unit 2. Fourier Transform: discrete-time signals

  • Discrete-time complex exponentials

  • Fourier series representation of discrete-time periodic signals

  • The Fourier transform of discrete-time sequences and comparison with continuous-time

Unit 3. Sampling in the time-domain

  • The sampling theorem and optimal signal reconstruction

  • Discrete-time processing of continuous-time signals

  • Decimation and interpolation

Unit 4. Discrete Fourier Transform (DFT)

  • Finite-length signals and periodic signals: DFT

  • Connection between Discrete Fourier Transform and the Fourier Transform

  • Efficient implementation and applications

Unit 5. The z-transform

  • Definition and connection with Fourier Transform

  • The region of convergence and its properties: zero-pole diagrams

  • Analysis and characterization of unstable LTI systems

Course material

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

  • Class notes (in Spanish). Available online soon.