# SIGNAL THEORY

## Learning outcomes of the course unit

This course aims at introducing and developing the concepts of deterministic signal and random signal, as models for physical systems of interest in the engineering contexts (electronics, computer science, telecommunications and related subjects). Signal transfomrations are introduced, as models of various systems encountered in several areas of ICT (amplifiers, filters, transmission lines, modulators, samplers, etc.).

At the end of the course, the Student should:

- know the techniques for the analysis of signals in the frequency domain;

- being able to apply these techniques to the filtering an sampling of an analog signal;

- understand and being able to model problems in the context of probability theory;

- personally manage simple probabilistic models, and be able to compare them critically, with the tools of random variables;

- know and be able to communicate concepts related to random signal (mean parameters; power spectral density, stationarity,).

- be able to face the face and deepen autonomously the case-study of specific random processes (harmonic process, PAM signal, etc.).

___1) Knowledge and understanding.___

The students learn the concepts and the mathematical tools necessary to manipulate both deterministic and random signals (stochastic processes). Signals are treated as mathematical models of physical signals, in particular those found in telecommunications, electrical engineering and computer science. Transformation of signals are also studied as models of physical systems like amplifiers, filters, transmission lines, modulators, samplers, etc.

___2) Applying knowledge and understanding.___

The students learn to apply the acquired knowledge to the modeling, analysis and design of the main systems encountered in Electrical Engineering and Information and Communication Technology (ICT) such as: amplifiers, filters, transmission lines, modulators, samplers etc.

## Prerequisites

Knowledge of the fundamentals of Claculus, with specific reference to the algebra of complex numbers and their representation in the exponential form (and related Euler's forulae).

"Mathematical analysis 1" and "Geometry" (suggested)

## Course contents summary

The course is divided in three parts:

- in the first part, we introduce deterministic signals and their analysis, both in the time- and in the frequency-domain. We shall analyze the trasformations that signals undergo through linear systems and through the sampling process;

- the second part is an introduction to the fundamentals of the theory of probability and random variables, with applications to engineering problems;

- the third part sums up the first two in the concept of stochastic process, as a model to study random signals and their transformations throgh systems.

--- Probability theory and random variables

Elements of set theory. The axioms of probability. Elements of combinatorics. Conditional probability, total probability and Bayes' formula. Repeated trials. Random Variables: definitions, distribution function, probability density function, discrete and continuous random variables. Transformations of a random variables and the fundamental theorem. Expected value and moments. Continuous and mixed forms of Bayes' formula and total probability theorem. Two random variables: joint distribution and probability density functions, conditional expected values, functions of two r.v. Vectors of random variables and Gaussian vectors.

--- Signals and systems

Definition of signal. Finite power and finite energy signals. Basic signals and transformations. The Dirac delta function. Systems an their properties: time- invariant, linear, memoryless, causal and stable systems. Linear time invariant (LTI) systems: impulse response and its use, convolution, stable and causal LTI systems, cascade and parallel of LTI systems. The complex exponential. Response of LTI systems to complex exponentials (eigenfunctions) and to sinusoids. Frequency response of LTI systems. Fourier Series representation of periodic signals. The Fourier Transform (FT) of non periodic signals: properties of the Fourier Transform, basic Fourier Transform and Fourier Series pairs. Signals through LTI systems (filtering), ideal filters and real filters, distortionless systems and distortions. Power spectral density.

--- Stochastic processes

Definitions. Distribution function and probability density function of stochastic processes. Mean, variance, autocorrelation and autocovariance. Stationary processes: strict-sense and wide-sense stationary processes. Power spectral density and its properties. The white noise ad other basic processes. Filtering of stationary processes. Gaussian processes and their filtering. Ergodic processes.

## Course contents

--- Probability theory and random variables

Elements of set theory. The axioms of probability. Elements of combinatorics. Conditional probability, total probability and Bayes' formula. Repeated trials. Random Variables: definitions, distribution function, probability density function, discrete and continuous random variables. Transformations of a random variables and the fundamental theorem. Expected value and moments. Continuous and mixed forms of Bayes' formula and total probability theorem. Two random variables: joint distribution and probability density functions, conditional expected values, functions of two r.v. Vectors of random variables and Gaussian vectors.

--- Signals and systems

Definition of signal. Finite power and finite energy signals. Basic signals and transformations. The Dirac delta function. Systems an their properties: time- invariant, linear, memoryless, causal and stable systems. Linear time invariant (LTI) systems: impulse response and its use, convolution, stable and causal LTI systems, cascade and parallel of LTI systems. The complex exponential. Response of LTI systems to complex exponentials (eigenfunctions) and to sinusoids. Frequency response of LTI systems. Fourier Series representation of periodic signals. The Fourier Transform (FT) of non periodic signals: properties of the Fourier Transform, basic Fourier Transform and Fourier Series pairs. Signals through LTI systems (filtering), ideal filters and real filters, distortionless systems and distortions. Power spectral density. Sampling. Relationship between Fourier and Lapalce Transforms.

--- Stochastic processes

Definitions. Distribution function and probability density function of stochastic processes. Mean, variance, autocorrelation and autocovariance. Stationary processes: strict-sense and wide-sense stationary processes. Power spectral density and its properties. The white noise ad other basic processes. Filtering of stationary processes. Gaussian processes and their filtering. Ergodic processes.

## Recommended readings

- A. Vannucci, "Segnali analogici e sistemi ineari", Pitagora Editrice, Bologna, 2003, ISBN: 88-371-1416-8.

- A. Bononi, G. Ferrari, "Introduzione a Teoria della probabilità e variabili aleatorie con applicazioni all'ingegneria e alle scienze", Soc. Editrice Esculapio, Bologna, aprile 2008, ISBN: 978-88-7488-257-1.

- A. Vannucci, "Esercizi d'esame di Teoria dei Segnali", Pitagora Editrice, Bologna, 2018. (in print)

--- R.E. Ziemer, "Elements of Engineering Probability and Statistics", Prentice Hall,

1996

--- B. Carlson, "Communication Systems", Mcgraw Hill Higher Education, 2009

(This textbook is used in other courses also)

--- A. Papoulis "Probability Random Variables and Stochastic Processes", McGraw-Europe, 2002

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## Teaching methods

This is a course of 9 credits (CFU), for which 72 hours of Class Lecturesare foreseen, to build up the understanding and critical processing of the subject matter.

Applications and exercises related to the presented topics are regularly proposed (via traditional or flipped classroom techniques) to develop application skills. Other assignments are given through the web learning platform Elly.

When foreseen by the long-term teaching plan of this University, a Tutor shall give guided exercises.

Teaching methods shall be adapted to the possible presence (detected according to laws and guidelines) of students with special needs and or disabilities.

Classroom teaching. In-class problem solving.

Homeworks assigned weekly.

## Assessment methods and criteria

The overall assessment of the learning outcomes foresees two tests:

1) a structured written test with 3 open questions, where the student shall demonstrate his/her ability to:

- analyze an analog signal and its possible transformations, in the time and frequency domain;

- model and solve a problem of probabiility theory, that possibly employs random variables;

- analyze a random signal and its possible filtering, by evaluating its moments and/or power spectrum

The duration of the written test is 2hours. The test is evaluated on a 0-30 scale, plus possible honors, in the case of top grades in all items together with an appropriateness of language.

2) an oral test, consisting of a critical discussion about the topics developed during the course. The student shall demonstrate proper knowledge and ability to illustrate the topic, with sufficient accuracy in the language.

Students can optionally take two intermediate written tests- to take place halfway during the course and at the end of the course, respectively - that shall replace the overall assessment procedure above.

The first test is a sequence of 10 closed multiple choice questions, followed by two open questions, similar to those illustrated at point 1) above.

The second test is is ismply made up of two open questions, similar to those illustrated at point 1) above.

In case the student takes both intermediate tests, the overall evaluation shall be assessed without an oral test.ue prove, senza necessità di una prova orale successiva.

Assessment methods shall be adapted to the possible presence (detected according to laws and guidelines) of students with special needs and or disabilities.

The final exam is a written test and an oral exam, unless specified further in relation to the first call of the first session. The written test is aimed at verifying students' ability to apply theory to solve problems and exercises. The oral test is aimed at verifying the understanding of the theoretical aspects of the subject and the ability to correctly apply various parts of the theory to the solution of problems and exercises. The written test consists of four or five problems or exercises on various parts of the program to be solved in about three hours. The maximum grade for the written test is 30/30. To be admitted to the oral exam a minimum grade of 16/30 in the written test is required. The final examination grade is based on an overall evaluation of the written and oral parts. The oral examination must be taken in the same session of the written test even if in a different call.

As an alternative way to take the exam, valid only for the first call of the first session, it is possible to take the exam by completing two written partial tests, without oral exam. If the overall grade of the parital tests is not less tha 18/10, an optional oral test can be taken in the first call.

## Other informations

Information to students and various documents are provided through the platform:

elly.dii.unipr.it