Signal theory

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

In relation to the Dublin Descriptors 1 (Knowledge and understanding) and 2 (Applying knowledge and understanding), the course aims to provide students with a general understanding of simple problems described with probabilistic methods. Furthermore, students will be allowed to understand how to characterize deterministic signals with suitable mathematical tools. Finally, from the combination of the tools and approaches described above, students will come to understand the concept of random processes and their characteristics, thus applying the acquired knowledge to the solution of real engineering problems.

In relation to the Dublin Descriptors 3 (Making judgemements), 4 (Communication skills) and 5 (Learning skills), the aim of the teaching is that students acquire the ability to analyze and understand the characteristics of deterministic and random signals. The students will be able to deepen what they have learned in the course, and use the basic knowledge as a starting point for subsequent studies. Furthermore, upon passing the exam, students will acquire the ability to mathematically formalize the results of transformations of linear systems on determined and random signals with the ability to communicate the acquired knowledge to others in a clear and complete way. Finally, students will understand and will be able to formalize the transformations made by the basic components of a communication system by applying the above knowledge to the solution of real problems. The student will therefore become independent from the teacher, acquiring the ability to refine and deepen their knowledge in an autonomous and original way. 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.

The course consists of lectures and exercises both on the blackboard and the computer. In case of COVID emergency, lectures will be provided through an appropriate digital platform.

If possible, attendance in presence is strongly recommended to simplify and optimize learning assessment.

Learning assessment may also be carried out on line, should the conditions require it.

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 usually 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.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus. 

Ability to solve integrals, derivatives and inequalities, knowledge of complex numbers, knowledge of elementary electrical circuits, e.g.  RC  and RLC circuits.

Students are required (not mandatory) to self-assess their preparation  at the beginning of the course.

Attendance is not mandatory, although it is strongly recommended to take the exam (see indication below). Statistically, students who do not attend the course have greater difficulty in passing the exam. Furthermore, if the pandemic conditions allow it, physical attendance at the lessons is recommended. It has been observed that this represents a powerful tool for simplifying the study for the student and for clarifying and consolidating skills with consequent greater speed and excellent performance in passing the exam.

The student who intends to take the test in progress must accumulate at least 70% of attendance in the lessons prior to the test.

The learning assessment may also be carried out electronically, if the conditions require it.

Part 1. Probability Theory: Reference book 1)

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 Reference books 1) and 2)

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 Reference book 1)

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); 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 Reference books 1) and 2)

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.
 

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

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

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) referring teacher within their department.

Examples of exercises Available on Studium

Examples of oral questions:

PROBABILITY THEORY AND RANDOM VARIABLES

• Random experiment
• Concept of probability
• Bayes theorem;
  
• Total probability theorem;
• Random variables, probability density function and cumulative distribution, characteristic indices;
• Transformation of a random variable;
• Characteristic indices of a distribution;
• Uniform random variable
• Gaussian random variable
• Exponential random variable
• Poisson random variable
• Bernoulli random variable
• Central limit theorem
• Pairs of random variables
• Correlation and independence between random variables
• Jointly Gaussian random variables
• Central limit theorem

 SIGNALS

• Definition and examples of determined and random signals
• Elementary properties of signals
• Harmonic analysis of periodic signals
• Amplitude and phase spectra and their properties
• Even, odd signals
• Synthesis of a signal starting from a limited number of harmonics
• Fourier transform of a determined signal
• Properties of the Fourier transform and theorems (linearity, duality, delay, change of scale, modulation, derivation, integration, product, convolution
• Fourier transform of the generalized impulsive function Dirac delta and notable transforms (step function and sign function); Periodicization and Poisson formulas;
• Sampling theorem
• Dimensionality theorem
• Concept of “system” and transformation of a signal
• properties of one-dimensional systems
• characterization and analysis of stationary linear systems (impulse response and frequency response);
• decibel
• cascade and parallel systems
• Ideal low-pass, high-pass, band-pass, band-stop filters; real filters
• bandwidth of a signal and a system;
• notes on distortions introduced by filters;
• Parseval's theorem and energy spectral density;
• power spectral density;
• autocorrelation function;
• Wiener-Khintchine theorem;
• Orthogonal expansion theorem
• Fourier series expansion for periodic signals

RANDOM PROCESSES

• Continuous-time random processes
• Parametric random processes
• Harmonic process
• First and second order statistical indices of a random process
• Stationarity in the narrow sense and in the broad sense
• Energy and power of a process
• Power spectral density and autocorrelation function of a stationary process
• Correlation between random processes
• White noise and continuous-time Gaussian random processes
• Thermal noise
• Ergodicity
• Filtering of a stationary random process in the broad sense