# SIGNAL THEORY - channel 2

ING-INF/03 - 9 CFU - 2° Semester

LAURA GALLUCCIO

## Learning Objectives

The course aims to provide students with the basics of probability theory, signals theory and random signals theory.

Concerning the Dublin Descriptors No. 1 and 2, the course aims to provide students with a general understanding of simple problems described through probabilistic methods. Furthermore, students will be allowed to understand how to characterize certain signals with appropriate mathematical tools. Finally, from the combination of the tools and approaches described above, students will come to understand the concept of real random process and their characteristics.

Concerning the Dublin Descriptors No. 3, 4 and 5, students will gain the ability to analyze and understand the characteristics of deterministic and random signals. Furthermore, students will have the ability to mathematically formalize the results of transformations of linear systems on deterministic and random signals. Finally, the students will understand and will be able to formalize the transformations operated by the basic components of a communication system.

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.

## Course Structure

The course consists of lectures and exercises both on the blackboard and the computer.

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. In addition, there are lessons in which it is shown how to use software tools es. Mathworks Matlab, for the solution of signal theory problems. Finally, seminars are also 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.

## Detailed Course Content

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 transformtheorems (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.

## Textbook Information

1) Marco Luise, Giorgio Vitetta: Teoria dei Segnali, Mc Graw Hill

2) Leon Couch: Fondamenti di Telecomunicazioni, VII Ed. Pearson, Prentice Hall

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