Signal theory
Course objectives
The course of signal theory intends to provide the learner with the bases for calculating the probabilities and the frequency analysis of certain and random signals, together with their practical application in the context of filtering, numerical transmission and analog modulation techniques. Specific - Specifically, after passing the exam, the learner will have acquired the knowledge and understanding of the aspects reported in the general part, - including their application to the contexts of a telecommunications system. - The learner will therefore have acquired the skills necessary for the frequency analysis of certain and random signals, and their application in the field of digital base-band transmission techniques and analog modulation techniques, becoming able to evaluate the quality of a telecommunication system in terms of the relative signal to noise ratio, and of the possible worsening introduced by the devices used and by the transmission medium adopted. - Passing the exam test attests the learner's achievement of critical skills and judgment regarding the performance of a telecommunication system, and the development of the examination paper allows to evaluate his ability to communicate what he has learned. - Being a second year course, it makes use of the skills acquired in the context of the basic teachings previously given, grafting on these a new common basis of skills that the subsequent teachings can take advantage of. For this reason the contribution given by the course to the learner's ability to continue the study in an autonomous way is considered adequate
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FABRIZIO SANTI
Lecturers' profile
Program - Frequency - Exams
Course program
Part I – Deterministic Signals
- Introductory concepts – Introduction to the concept of a signal; classification of signals; basic properties (mean value, power, energy); elementary operations.
- Discrete representation of signals – Inner product; signal space; projection theorem; unilateral Fourier series; bilateral Fourier series; calculation of coefficients and their properties.
- Fourier transform – Existence criteria; signal bandwidth; properties of the transform (linearity, value at the origin, symmetry, scaling duality, time and frequency shifts, differentiation and integration properties); Gaussian signal; uncertainty principle for energy signals; Parseval’s theorem.
- Generalized Fourier transforms – Definition and properties of the Dirac delta (area, multiplication, sampling); Fourier transforms of constant, step, and sign functions; Fourier transform of periodic signals.
- Signal transmission through systems – System properties (linearity, time-invariance, causality, stability); impulse response and step response; convolution integral; convolution theorem; transfer function; eigenfunctions of LTI systems; high-pass and low-pass filters.
- Correlation and spectrum – Properties of correlation for energy and power signals; correlation properties in the Fourier domain; autocorrelation function and its properties; energy and power spectral densities; Wiener’s theorems; generalized harmonic analysis.
- Modulated signals – Representation of band-shifted signals (analytic signal, complex envelope, low-frequency analog components); Hilbert transform; sinusoidal carrier modulation; double-sideband amplitude modulation; single-sideband amplitude modulation; angular modulation (overview).
- Signal sampling – Sampling theorem; aliasing; interpolation formula; anti-aliasing filter.
Part II – Random Signals and Elements of Digital Telecommunications
- Probability theory review – Continuous and discrete random variables; distribution function; probability density; central and non-central moments; characteristic function; functions of random variables.
- Multidimensional random variables – Two-dimensional random variables; joint and marginal distribution and density functions; mixed central and non-central moments; statistical correlation; covariance; correlation coefficient; jointly Gaussian random variables; conditional probability densities; operations on pairs of random variables; linear combination of Gaussians; M-variate Gaussians; central limit theorem.
- Random processes – Definition of a random process; properties of stationarity and ergodicity; moments of a random process; Gaussian process; harmonic process; power spectral density; cyclostationarity.
- Digital transmissions – Pulse amplitude modulation; PAM waveform (expected value, autocorrelation, and power spectral density).
- Random signal processing – Modulation of a random process; transmission of random processes through LTI systems; sampling of a random process; thermal noise.
Prerequisites
Basic knowledge of mathematical analysis, linear algebra, and probability calculus.
Frequency
Attendance is not mandatory but strongly recommended.
Exam mode
Written exam consisting of exercises and theoretical questions.
Lesson mode
Lectures and classroom exercises.
FABRIZIO SANTI
Lecturers' profile
Program - Frequency - Exams
Course program
Part I – Deterministic Signals
- Introductory concepts – Introduction to the concept of a signal; classification of signals; basic properties (mean value, power, energy); elementary operations.
- Discrete representation of signals – Inner product; signal space; projection theorem; unilateral Fourier series; bilateral Fourier series; calculation of coefficients and their properties.
- Fourier transform – Existence criteria; signal bandwidth; properties of the transform (linearity, value at the origin, symmetry, scaling duality, time and frequency shifts, differentiation and integration properties); Gaussian signal; uncertainty principle for energy signals; Parseval’s theorem.
- Generalized Fourier transforms – Definition and properties of the Dirac delta (area, multiplication, sampling); Fourier transforms of constant, step, and sign functions; Fourier transform of periodic signals.
- Signal transmission through systems – System properties (linearity, time-invariance, causality, stability); impulse response and step response; convolution integral; convolution theorem; transfer function; eigenfunctions of LTI systems; high-pass and low-pass filters.
- Correlation and spectrum – Properties of correlation for energy and power signals; correlation properties in the Fourier domain; autocorrelation function and its properties; energy and power spectral densities; Wiener’s theorems; generalized harmonic analysis.
- Modulated signals – Representation of band-shifted signals (analytic signal, complex envelope, low-frequency analog components); Hilbert transform; sinusoidal carrier modulation; double-sideband amplitude modulation; single-sideband amplitude modulation; angular modulation (overview).
- Signal sampling – Sampling theorem; aliasing; interpolation formula; anti-aliasing filter.
Part II – Random Signals and Elements of Digital Telecommunications
- Probability theory review – Continuous and discrete random variables; distribution function; probability density; central and non-central moments; characteristic function; functions of random variables.
- Multidimensional random variables – Two-dimensional random variables; joint and marginal distribution and density functions; mixed central and non-central moments; statistical correlation; covariance; correlation coefficient; jointly Gaussian random variables; conditional probability densities; operations on pairs of random variables; linear combination of Gaussians; M-variate Gaussians; central limit theorem.
- Random processes – Definition of a random process; properties of stationarity and ergodicity; moments of a random process; Gaussian process; harmonic process; power spectral density; cyclostationarity.
- Digital transmissions – Pulse amplitude modulation; PAM waveform (expected value, autocorrelation, and power spectral density).
- Random signal processing – Modulation of a random process; transmission of random processes through LTI systems; sampling of a random process; thermal noise.
Prerequisites
Basic knowledge of mathematical analysis, linear algebra, and probability calculus.
Frequency
Attendance is not mandatory but strongly recommended.
Exam mode
Written exam consisting of exercises and theoretical questions.
Lesson mode
Lectures and classroom exercises.
- Lesson code10596204
- Academic year2025/2026
- CourseInformation Engineering
- CurriculumGestionale
- Year2nd year
- Semester2nd semester
- SSDING-INF/03
- CFU6