# SIGNAL THEORY

## Learning outcomes of the course unit

___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

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

## Course contents summary

--- 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

--- 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

Classroom teaching. In-class problem solving.

Homeworks assigned weekly.

## Assessment methods and criteria

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