Signal theory A - L

Signal Theory

The course aims to provide students with the basics of probability theory, signals theory and random
signals theory.
In relation to the Dublin Descriptors 1 (Knowledge and understanding) and 2 (Applying knowledge and
understanding), the course aims to provide students with a general understanding of simple problems
described with probabilistic methods. Furthermore, students will be allowed to understand how to
characterize deterministic signals with suitable mathematical tools. Finally, from the combination of the
tools and approaches described above, students will come to understand the concept of random
processes and their characteristics, thus applying the acquired knowledge to the solution of real
engineering problems.
In relation to the Dublin Descriptors 3 (Making judgemements), 4 (Communication skills) and 5 (Learning
skills), the aim of the teaching is that students acquire the ability to analyze and understand the
characteristics of deterministic and random signals. The students will be able to deepen what they have
learned in the course, and use the basic knowledge as a starting point for subsequent studies.
Furthermore, upon passing the exam, students will acquire the ability to mathematically formalize the
results of transformations of linear systems on determined and random signals with the ability to
communicate the acquired knowledge to others in a clear and complete way. Finally, students will
understand and will be able to formalize the transformations made by the basic components of a
communication system by applying the above knowledge to the solution of real problems. The student
will therefore become independent from the teacher, acquiring the ability to refine and deepen their
knowledge in an autonomous and original way. Upon completion of the course the students must gain
independent and critical investigation skills as well as ability to formalize through statistical methods
some real problems (also through the help of numerous exercises carried out during the course), as well
as the ability to discuss and present the results of such studies. Finally, at the end of the course, the
students will be able to continue independently their study of other engineering disciplines with the
ability to appropriately use statistical tools.

The course consists of lectures and exercises both on the blackboard and the computer. In case of COVID
emergency, lectures will be provided through an appropriate computer platform.
The final exam could be eventually done remotely in case of COVID contingency.
The theorethical lessons are taught by the teacher while the exercises are both done by the teacher and
the students who are invited to perform, with the support of the teacher, the tests.  Finally, seminars are usually scheduled at the end of the course in which the application
of signal theory and spectral investigation to the modulation and filtering of signals with laboratory
equipment (oscilloscope filters, modulators / demodulators) is demonstrated.
Should teaching be carried out  in mixed or only remote mode, it could be needed to introduce some
necessary variations as compared to what has been foreseen and reported in the syllabus.

Part 1. Probability Theory

*Random experiment; probability, *Bayes theorem; *total probability theorem; *Random variables, *probability density function and cumulative distribution; *transformation of a random variable; indexes of a distribution; *Gaussian random variable, other relevant random variables (e.g. uniform random variable, Poisson random variable, Bernoulli random variable, exponential random variable), *central limit theorem.

Part 2. Analysis of Periodic and Continuous Signals

*Definitions and examples of signals; elementary properties of signals; *Harmonic analysis of periodic signals; *amplitude and phase spectra of signals and their properties; synthesis of a signal from a finite number of harmonics.* Fourier integral;* Fourier transform; Fourier transform theorems (linearity, duality, delay, scale change, *modulation, derivation, integration, product, convolution); *Dirac delta generalized function and its transformation; *Periodicization of a signal and Poisson formulas; *Sampling theorem.

Part 3. Linear and stationary systems

*Definition of "system" and transformation of a signal through a system; properties of one-dimensional systems; *characterization and analysis of stationary linear systems (impulse response and frequency response);  decibels;* cascading and parallel systems; Ideal filters ( low pass, high pass, band pass, delete band; real filters; signal bandwidth; distortion due to the filtering process; *Parseval theorem and spectral energy density;*spectral power density; *autocorrelation function; Wiener-Khintchine theorem.

Part 4. Elementary transformations of random signals

*Random processes; *parametric random processes; *Stationary processes; Filtering of a stationary random process; spectral power density of a stationary random process; *White noise and thermal noise; *Ergodicity of a process.

*minimum required arguments

1) Marco Luise, Giorgio Vitetta: Teoria dei Segnali, Mc Graw Hill
2) Leon Couch: Fondamenti di Telecomunicazioni, VII Ed. Pearson, Prentice Hall