D1 - Knowledge and understanding. The student will acquire knowledge of the main characteristics of the signals at both discrete and continuous time, both deterministic and random. D2 - Applying knowledge and understanding. Students will acquire familiarity with the analysis and elaboration of signals and systems, both by analytical techniques and by calculation and simulation programs. D3 - Making judgements. The student should be able to identify the most appropriate techniques for analyzing and processing the signals with which he / she must work. D4 - Communication. The student should be able to illustrate the characteristics of the signals, and to expose the techniques for their elaboration. D5 - Lifelong learning skills. The student should be able to extend the concepts expressed to types of signals not addressed in the course, such as images.

Knowledge of the basics of mathematical analysis (especially of complex numbers and their properties) and of theory of probability.

COMPLEX NUMBERS
Property. Exercises
LINEAR SYSTEMS
Time domain analysis.
Impulsive response.
LINEAR AND TIME INVARIANT SYSTEMS (LTI).
Impulse response of an LTI system.
Integral and convolution sum.
LINEAR AND TIME INVARIANT SYSTEMS IN THE FREQUENCY DOMAIN
Continuous time systems: Fourier transform.
Discrete time systems: Discrete Time Fourier Transform, Discrete Fourier Transform.
SAMPLING OF A CONTINUOUS TIME SIGNAL.
Sampling theorem.
Reconstruction of a sampled signal.
BAND PASS SIGNALS AND SYSTEMS.
Hilbert transform, analytic signal, complex envelope.
Sampling a Band Pass signal.
Z TRANSFORM.
Transformation and anti-transformation operations. Regions of convergence.
Difference equations.
Unilateral transform.
PROBABILITY
Properties, exercises
Random variables.
RANDOM PROCESSES AND NOISE.
Distribution functions and probability density of nth order.
Gaussian processes.
Stationary, regular, ergodic processes.
Study in the frequency domain: Wiener-Kintchine theorem (statement only).
Random processes through linear and time-invariant systems.

A.V.Oppenheim, A.S.Willsky: “Signals and Systems”, Prentice-Hall Int. Claudio Prati: “Segnali e Sistemi per le Telecomunicazioni”, McGraw-Hill S. Haykin, M. Moher: “ Introduzione alle telecomunicazioni analogiche e digitali”, R. D. Yates, D. J. Goodman:” Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers”, John Wiley.

COMPLEX NUMBERS
Property. Exercises.

INTRODUCTION. Communication Systems. Information, Source, Destination, Transmitter, Receiver. Signal Classification.

SIGNALS AND SYSTEMS. Continuous time and discrete time signals (sequences). Elementary operation on the signals: transformation of the independent variable. Elementary signals: complex sinusoidal and exponential signal, impulse function (delta-dirac) and unit pulse, unit step. Continuous time systems and discrete time systems. Interconnection of systems. Systems properties: systems with and without memory, causality, stability, linearity, invariance over time.

LINEAR TIME-INVARIANT SYSTEMS (LTI). Representation of time-discrete signals in terms of unitary pulses. Representation of continuous time signals in terms of impulse functions. Impulsive response of an LTI system. Discrete time LTI systems: sum of convolution. Continuous time LTI systems: convolution integral. Properties of LTI systems: systems with and without memory, causality and stability.
Response of LTI systems to complex exponential signal. Representation of periodic signals in terms of complex sinusoidal and exponential signals. Fourier series for continuous-time, time-discrete periodic signals. Fourier transform. Spectrum of a signal. Fourier transform properties: transformation linearity, symmetry property, time translation and frequency translation, derivation and integration (summation), Parseval relationships, convolution, periodic convolution, modulation properties. The discrete Fourier transform for discrete time signals. Frequency response of an LTI system and its properties.

SAMPLING. Theoretically limited band signals; Practically limited bandwidth signals. Sampling theorem. Reconstruction of a sampled signal as a filtering operation. Aliasing. Conversion of a continuous time signal into a discrete time signal. Relationships between the frequency response of a discrete time LTI system and its equivalent continuous time.

PASS BAND SIGNALS AND SYSTEMS. Hilbert's transformation of a continuous-time signal and its properties. Analytical signal associated with a real signal. Narrow bandwidth signals, complex envelope and natural envelope, phase and quadrature components.

Z. TRANSFORMATION Definition of the Z transform for a discrete time signal. Uniform circle on the complex plane. Zeros and poles, convergence region. Properties of the convergence region. Reverse Z transformation: general formula, development in partial fractions, development in series of powers. Z transformation properties: linearity, translation in time and frequency, convolution, etc. Use of Z transform in LTI system analysis.

PROBABILITY
Properties, exercises
Random variables.

RANDOM PROCESSES AND NOISE. Distribution functions and probability density of n-th order. Gaussian processes. Statistical description of random processes. Algebraic moments of order n; Average value, average quadratic value, central moments, average power of a random process. Mixed algebraic moments, autocorrelation. Joint description of two random processes, mutual correlation. Stationary processes in the narrow and broad sense, cycling processes. Time averages, time autocorrelation function. Ergodic processes. Frequency description of random processes. Spectral Power Density, Wiener-Kintchine Theorem (only enunciated). Joint processes through LTI systems.

Lectures and exercitations.

The didactic material available on the site summarizes all the topics of the course, but many comments, examples and demonstrations that complete the exam program can only be found in classroom lessons.

The teaching material is available at http://moodle2.units.it and on the Team of the course. To get a copy of the lesson slides, the student must enroll in the course. The material on the site includes slides presented, lesson exercises, previous written tests, course schedule, exam calendar.

The exam is divided into a written and an oral exam. During the course there are 2 written tests that can be supported in replacement of the examination. The written exam consists of basic exercises on topics related to Mathematical Analysis, Probability Calculation and Signal Theory. The oral test consists of specific theories of Signal Theory, as well as in verifying the concepts learned. To support the oral test the student must have supported the written statement with a positive result (15/30). Or both the preliminary written tests with an individual rating of not less than 12/30 and a mean rating of not less than 15/30. Who attended both preliminary tests with no evaluation less than 15/30 in each test can record the grade without taking the oral exam. Who has taken a single preliminary test only in which the rating is no less than 12/30, may solve in the written test the exercises of the part not included in the preliminary one. Will have to take anyway the oral exam, always with the rules illustrated above. Oral examination can be made in any test of any session of any academic year, or by appointment. Written exam can be repeated in any test of any session of any academic year. However, when a student submits to the written test, the result obtained in any previous one is canceled.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs)