The course aims to present the basic knowledge of signal theory and analysis. It provides the fundamental methodological tools for the description and analysis of continuous-time and discrete-time, both deterministic and random signals.

Deterministic, continuous-time signals and systems and their analysis in the frequency domain.
Deterministic, discrete-time signals and systems.
Basic concepts of probability theory.
Continuous and discrete-time random signals.
Selection and application of appropriate methods for signal analysis.

Basic mathematical analysis (integrals, derivatives), linear algebra (vectors, matrices, scalar product, norm).
In-depth knowledge of complex numbers and trigonometry.

Definition and classification of signals: continuous / discrete time, deterministic / random
Energy and mean power
Fourier Series and Transform
LTI systems, impulse response and transfer function
Energy spectral density and autocorrelation function
Periodic signals and related power spectral density
Power spectral density of non-periodic signals
Sampling Theorem
Discrete-time signals; z-transform, discrete time Fourier transform (DTFT)
LTI discrete-time systems
FIR and IIR filters
Introduction to random variables
Random processes: definitions
Stationary processes: autocorrelation and power spectral density
Ergodic processes

Classification of signals: continuous-time or discrete-time, deterministic or random

Dirac delta

Linear time invariant systems: definition, impulse response, convolution (continuous and discrete time)

Deterministic and continuous-time systems and signals:
- LTI systems: Fourier transform (eigenfunction of convolution), transfer function
- Signals: Fourier transform and its properties, evaluation of the main Fourier transforms
- Finite-energy signals: energy, energy spectrum, autocorrelation function
- Periodic signals: period and power, Fourier series, Fourier transform, power spectrum, autocorrelation function


Sampling theorem

Deterministic and discrete-time systems and signals:
- LTI systems: Z transform and inverse transform, transfer function, poles and zeroes
- Signals: Z transform of the main signals
- Periodic signals: DFT and IDFT, circular convolution
- Use of DFT/IDFT to evaluate the output of an LTI system for a non-periodic input

Relationship between the Fourier transform of a time-continuous signal and the Z transform of its sampled version.

Probability thoery:
- Axioms
- Conditional probability, Bayes rule, statistical independence
- Random variables: probability distribution and density
- Couple of random variables: joint probability distribution and density
- Mean, mean square value, variance, standard deviation
- Gaussian and uniform probability denisty functions and distributions
- Central limit theorem

Continuous-time random processes:
- Examples of random processes
- First order probability density function for a random process
- Autocorrelation and power spectrum for a stationary and ergodic process
- Filtered stationary and ergodic processes
- White Gaussian noise

The course consists of lectures and exercises. The classroom exercises consist in the solution of exercises related to the program carried out in class.

- B. P. Lahti, R. A. Green, Essentials of Digital Signal Processing, Cambridge University Press, 2014
- (alternatively) Lo Presti and F. Neri, The analysis of the signals, CLUT, 1992.
In addition, teachers will provide handouts and exercises.

L. Lo Presti, F. Neri, 'L'Analisi dei Segnali', CLUT.
L. Lo Presti, F. Neri, 'Introduzione ai processi casuali', CLUT.
Soved exercises and exams (download from didattica portal)

Written exam: two exercises, each one encompassing a number of questions, to be carried out in extended form. The exercises will be related to any part of the program, and may also include theoretical questions.

Written exam with theory questions and problems to be solved. Closed book exam (apart from tables of Fourier and Z transforms). Oral exam if requested by the professor.