## Learning objectives

Knowledge and understanding:

The main goal of this course is to provide the students with A refresher (with a reinforcement tailored towards the upcoming courses) of the key concepts in linear algebra, probability theory and system analysis they should have encountered in their Bachelor studies.

The refreshed mathematical tools will allow them to manage both deterministic and stochastic signals both in continuous- and discrete-time, as well as their linear transformations.

Applying knowledge and understanding:

The abilities to apply the acquired knowledge and understanding are:

PART 1:

- to understand and apply the basic concepts of linear vector spaces and their basic operations.

- to manipulate Hermitian, unitary, and projection matrices. To perform a spectral or singular value matrix decomposition

- to apply basic probabilty theory to solve practical problems

PART 2:

- to apply Fourier and Z transform techniques to solve linear filtering problems

- to calculate the spectral properties of filtered stochastic processes

- to use the complex equivalent lowpass representation of real bandpass stochastic processes

PART 3:

- to provide students with the tools do describe and solve communications and signal processing problems in the MATLAB environment.

## Prerequisites

A very basic knowledge of linear spaces, matrix operations, basic probability and basic transform theory applied to linear systems analysis is assumed. Basics of communications and signal processing.

Prior knowledge of a programming language (recommended).

## Course unit content

PART 1 (FOGGI)

Basics of linear spaces

matrix theory, eigenvalues, eigenvectors, spectral decomposition, singular value decomposition

Introduction to probability theory. Basic concepts: conditioning, total probability, Bayes law

Scalar and muti-dimensional random variables

stochastic processes.

PART 2 (PIEMONTESE)

Wrapup of basic complex-number calculus

Continuous Fourier transform and its properties

Sampling and aliasing

Discrete Fourier transforms and its properties

Two-sided Z transform and its properties

Spectral analysis of stochastic processes: the Wiener Khinchin theorem.

Passband signals: lowpass equivalent.

PART 3 (UGOLINI)

Introduction to the MATLAB environment.

Solution of algebraic problems with MATLAB.

Modeling and solution of signal processing and communication problems.

## Full programme

PART 1 (FOGGI)

Basics of linear spaces

matrix theory, eigenvalues, eigenvectors, spectral decomposition, singular value decomposition

Introduction to probability theory. Basic concepts: conditioning, total probability, Bayes law

Scalar and muti-dimensional random variables

stochastic processes.

PART 2 (PIEMONTESE)

Wrapup of basic complex-number calculus

Continuous Fourier transform and its properties

Sampling and aliasing

Discrete Fourier transforms and its properties

Two-sided Z transform and its properties

Spectral analysis of stochastic processes: the Wiener Khinchin theorem.

Passband signals: lowpass equivalent.

PART 3 (UGOLINI)

Introduction to the MATLAB environment (4 hours)

- Variables: scalars, vectors, matrices

- Basic operations and plotting

- Linear algebra

- Scripts and functions

Probability (2 hours)

Signals and systems (8 hours)

- Definitions of signals

- Analog and discrete-time signals

- Operations on signals

- Frequency-domain representation

- Sampling and reconstruction

Transmission on the AWGN channel (2 hours)

Fourier transform and spectral analysis (6 hours)

## Bibliography

TEXTBOOKS

[1] A. B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication. McGraw-Hill, 1986.

[2] A. Papoulis, Probability, Random Variables and Stochastic Processes. New York, NY: McGraw-Hill, 1991.

[3] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. New Jersey: Prentice Hall, 2nd ed., 1999.

Notes and programs developed during the lectures.

## Teaching methods

Lectures and exercises (approximately with a ratio 80%-20%).

Laboratory lectures. The students are required to install the MATLAB software before the beginning of the course.

## Assessment methods and criteria

UNIT 1 exam will be written and simultaneous for both parts.

UNIT 2 exam will be held in a computer lab, and will consist in the solution of a telecommunications or signal processing problem using MATLAB.

## Other information

1) Course structure

(each lecture 2 hours)

UNIT 1 part 1 will engage two lectures per week for the first six weeks, then UNIT 2 will start with two lectures per week for the following six weeks; UNIT 1 part 2 will engage one lecture per week across the whole semester.

This scheme will allow coordination of the 3-part contents.

A calendar of lectures for all 3 parts will be posted on the web-learning (Elly) site.

2) Office hours:

You will be able to meet your teachers either physically at their office or remotely via Teams after making an appointment by email. Check the individual units for their official meeting days.

3) teaching material:

It will be posted on the (unique) Elly course website.