Syllabus

• Week 1

  Signals in the time-domain

  • Introduction

    We begin by introducing the meaning of "signal" and "system," and provide a brief overview of the contents of the course

  • Aperiodic signals

    Some fundamental measures are introduced for continuous-time as well as discrete-time aperiodic signals, namely: area, mean value, energy, and power

  • Periodic signals

    We discuss the periodicity properties of continuous and dicrete-time periodic signals, and apply the measures of mean value and power to learn some fundamental properties of complex exponentials

  • Basic transformations and symmetries

    A few basic transformations, that is time-reversal, time-shift, and time-scale, are key to interpret and write signals. They further lead to the concept of symmetries in signals.

  • Periodic repetition and ideal impulses

    Periodic repetition connects the world of aperiodic signals to that of periodic signals, and is a key element here. Ideal impulses, instead, are a powerful tool to manage signal properties, as we will appreciate.

  • Wrap-up

    It is time to wrap-up your understanding of signals in the time domain! Take the test!

• Week 2

  Systems in the time-domain

  • Systems properties

    Systems can be classified on some fundamental aspects including their invertibility, memory properties, BIBO stability, linearity, or time-invariance. We will also learn what a connection in series or parallel is.

  • LTI systems

    Linearr time-invariant (LTI) systems are the classical systems of physics, and can be described through a so-called convolution operation that we will learn to interpret and manage.

  • Circular convolution and filters

    We generalize the convolution operator to periodic signals, and learn how to identify LTI system properties dirrectly from the system impulse response.

  • Introduction to MatLab

    MatLab is a numerical computer environment which allows matrix manipulations, plotting of functions, and implementation of algorithms. Here we give a brief introduction and discuss the key aspects related to plotting a signal.

  • Convolution in MatLab

    Convolution can be numerically evaluated in MatLab also for continuous-time signals. We investigate this important aspect related to our course.

  • Wrap-up

    It is time to wrap-up your understanding of systems in the time domain! Take the test!

• Week 3

  Fourier transforms

  • Fourier series

    The Fourier series is, hystorically, the first Fourier transform, and is able to represent a continuous-time periodic signal through a collection of coefficients. We will learn its definition and properties.

  • Discrete Fourier transform

    The discrete Fourier transform generalises the Fourier series to discrete-time periodic signals, the only class that is effectively manageable by a digital device.

  • Fourier transform

    The Fourier transform applies to continuous-time aperiodic signals. It is by far the most flexible and general definition, for which a vast range of results is available.

  • Relations among the transforms and filters

    Connecting Fourier transforms allows easy derivation of results in very many cases, and is therefore important knowledge. We will also learn to interpret continuous-time filters through Fourier lenses.

  • Wrap-up

    It is time to wrap-up your understanding of Fourier transforms! Take the test!

• Week 4

  Discrete-time Fourier transform and sampling theorem

  • Discrete-time Fourier transform

    The discrete-time Fourier transform applies to discrete-time aperiodic signals and has a form of a Fourier series where time and Fourier domain are swapped. It was originally introduced by Shannon in 1949.

  • Filters in discrete-time, and wrap up

    We learn to exploit the Fourier formalism to investigate discrete-time filters, with an additional complication that the Fourier domain is now periodic. Then, we wrap up on Fourier transforms to highlight their common structure.

  • Shannon's Sampling Theorem

    Shannon's Sampling Theorem is the basis for undestanding signal digitalisation, and provides a powerful method to reconstruct a continuous-time signal from its discrete-time samples by means of the so-called interpolation.

  • Fourier transform in MatLab

    Inspecting the Fourier domain in MatLab is a matter of understanding the relations among different transform and signal counterparts, i.e., continuous and discrete-time, and periodic. We will learn how this is achieved by the FFT.

  • Wrap-up

    It is time to wrap-up your understanding of discrete-time fourier transforms and the sampling theorem! Take the test!

• Week 5

  Laplace transforms

  • The Laplace transform

    The Laplace transform is a generalization of the Fourier concept that allows you to manage signals that do not necessarily decay over time. It is a more "mathematical" transformation of which we will learn some simple properties.

  • Solving differential equations

    Solving differential equations (e.g., in physics of motion) is a particularly easy task by use of the Laplace transform. Their generalization to discrete-time is obtained through the Z transform and difference equations.

  • Wrap-up

    It is time to wrap-up your understanding of Laplace transforms and differential equations! Take the test! And also take the test to complete the course!

Who will you learn with?