## Continuous-time signals and systems

### Teaching unit

### Description

Orthogonal expansions of signals. Periodic signals: Fourier series. Properties of Fourier series. Parseval's theorem. Non-periodic signals: continuous-time Fourier transform. Properties of the Fourier transform. Basic Fourier transform pairs.

Representation of signals in terms of impulses. The unit impulse response. The convolution integral. Properties of the convolution. Methods for calculating convolutions. Convolution of periodic signals. Causality, stability and invertibility of continuous-time LTI systems. Characterization of LTI systems: time and frequency domains.

The Laplace transform: definition and properties. Region of convergence for Laplace transforms. The inverse Laplace transform. Transfer function of a LTI system. Analysis and characterization of LTI systems using the Laplace transform. LTI systems described by differential equations.

### Objectives

To develop competency in the Fourier analysis for continuous-time signals (Fourier series and Fourier transform).

To develop skills in analyzing continuous-time LTI systems using the convolution integral.

To understand the basic concepts of the Laplace transform, and develop the ability to analyze LTI systems using the Laplace transform.

### Subjects

#### Lecture topics

- Fourier analysis for continuous-time signals. Objectives. Orthogonal expansions of signals. Fourier series of periodic signals. Continuous-time Fourier transform.
- Continuous-time LTI systems. Objectives of the unit. Representation of signals in terms of impulses. The unit impulse response. The convolution integral. Causality, stability and invertibility of continuous-time LTI systems. Characterization of LTI systems.
- The Laplace transform. Introduction. The Laplace transform. The region of convergence for Laplace transforms. The inverse Laplace transform. Properties of the Laplace transform. Analysis and characterization of LTI systems using the Laplace transform.

#### Laboratory topics

- Fourier analysis of periodic signals.
- Fourier analysis of non-periodic signals.
- The convolution integral and properties of continuous-time LTI systems