# SIGNAL THEORY

## LEARNING OUTCOMES OF THE COURSE UNIT

Goal of the course is to provide the student with 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.

The course is divided into three parts:

- First part: Basics of probability theory and random variables with engineering applications

- Second part: Deterministic signals and their transformation through systems.

- Third part (synthesis of the first two parts): Stochastic processes, random signals and filtering.

## 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.

## 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-Hill Europe, 2002

## ASSESSMENT METHODS AND CRITERIA

Written and oral exam.

## FURTHER INFORMATIONS

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

lea.unipr.it