## 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 tosolve linear filtering problems

- to calculate the spectral properties of filtered

stochastic processes

- to use the complex equivalent lowpass

representation of real bandpass stochastic

processes

## Prerequisites

A very basic knowledge of linear spaces, matrix

operations and basic probability is

assumed.

## 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.

## 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.

## 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.

## Teaching methods

Lectures and exercises (approximately with a ratio

80%-20%)

## Assessment methods and criteria

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

## 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.

2) Office Hours

FOGGI:

Wednesday 15:00-17:00 (Scientific Complex, Building 2, floor 2).

PIEMONTESE:

Wednesday 14:00-16:00 (Scientific Complex, Building 2, floor 2).

You can also meet your instructor on Teams after making an appointment by email.

3) teaching material:It will be posted on the (unique) Elly course website.