Unit SIGNAL THEORY
- Course
- Computer science and electronic engineering
- Study-unit Code
- 70000009
- Curriculum
- In all curricula
- Teacher
- Paolo Banelli
- Teachers
-
- Paolo Banelli
- Hours
- 81 ore - Paolo Banelli
- CFU
- 9
- Course Regulation
- Coorte 2021
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Ingegneria delle telecomunicazioni
- Academic discipline
- ING-INF/03
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- ITALIAN
- Contents
- Continuous-time signals and spectral analysis: Fourier series and transform- Continuous signals in linear and non-linear systems.Signal sampling, discrete-time systems and discrete spectral analysis Spectral analysis of random processes ans series in linear and non-linear systems. Using Matlab for digital signal processing.
- Reference texts
- SUGGESTED BOOK - M. Luise, G. Vitetta, Teoria dei Segnali, Casa Editrice: McGraw-Hill, III ed., 2009 AUXILIARY BOOKS - Verrazani, Corsini, Teoria dei Segnali: Segnali Determinati, Casa Ed.: ETS, Pisa, 1995. - A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw Hill, New York City, 1965 (I ed) -2002 (IV ed).
- Educational objectives
- To understand the meaning of spectral content associated with continuous and discrete signals, both deterministic and random, described (when possible) by single-valued functions (typically in the time-domain). To understand the concept of signal processing by a device (linear and non-linear) that is capable to modify the signal shape in time and its spectral content: continuous and discrete filters and frequency-domain system response.First practical implementation of digital signal processing algorithms and filter design, by using Matlab.
- Prerequisites
- Mandatory: Mathematical Analisis I e II (functions, limits, derivatives, integrals, series, function series expansions)Mandatory: Linear Algebra I ( vectors, spaces, cartesian products, orthogonality,)Higly Recommended: Foundations of Probability Theory (probability of events, random variables, marginal, joint, and conditional probability density functions, transformation of random variables, random process definition)Useful: Linear Algebra II (spectral decomposition of symmetric (Hermitian) matrices, eigenvalues and eigenvectors)
- Teaching methods
- The class is given with frontal oral tuitions for a total of 81 hours (45 minute/hour).
About 20 hours are dedicated to solutions of exercises with variable difficulty, which are spread across all the course.About 12 hours are dedicated to verifying by Matlab and audio processing boards, practical computer aided implementation of some of the spectral analysis and filter design techniques that are taught during the course. - Other information
- STATISTICAL ANALYSYS on 9 Exam sessions from June 13, 2016 to June 12, 2017Number of Attempts = 231Number of Individuals = 138Number of Positive scores = 86 (46 with 1 attempt, 28 with 2 att., 8 with 3 att., 2 with 4 att.., 2 with 5 att..) Mean Score = 22.8/30 Stand. Dev.= 3.4 SCORES INSTOGRAM (18 – 30): 12-5-6-12-8-6-11-8-5-4-2-4-3
- Learning verification modality
- - a 2 hours written test with the solution of 2 exercises (open answers)- a short report of a Matlab implementation of a digital signal processing algorithm. The algorithm/application can freely be chosen by the student, an must be sent by e.mail to the teacher, together with the source code, some days before the oral session.- an oral session of about 30-45 minutes.The 2 exercises of the written test, which are open answers and do not include the part of digital signal processing, aim to verify the student capability to analyse interconnected systems that process signals (either deterministic or random), both in the time and the frequency domain, including basic operations such as linear time-invariant filtering, instantaneous non-linearity, modulation and demodulation, sampling and reconstruction of analog signals.Only the candidates with a score at least equal to 18/30 to the written test will be admitted to the oral session.The oral exam always begins with a question on digital signal processing and one or two successive questions on analogic signals and random processes.The goal is to verify the student level of knowledge and comprehension of the methodological and theoretical content of the class, and particularly to verify the capability of the student of autonomous elaboration of the concepts.During the oral exam, the capability of autonomous exposition of the concepts and the language appropriateness will be also evaluated. The final score is generally obtained as the average of the two scores (from 1 to 30) obtained at the two tests (written and oral), corrected by plus/minus 1 or 2 depending on the quality and oral discussion of the Matlab-based report, which however is not mandatoryto be presented. The instructor may decide to waive, even partially, the oral test based on kind of errors of the written test and its score, and it will be the student to decide if taking or not the oral test.
- Extended program
- Concept of signal (continuous, discrete, periodic). Energy, Power, mean value. Operations with signals. Elementary signals.Fourier series expansion of periodic signals: convergence, orthogonality, spectrum lines (amplitude and phase), link between power and Fourier series coefficients. Continuous spectrum for aperiodic signals. Continuous Fourier Transform (CTF): existence criteria and proofs of the most important properties. Parseval's Theorem. CFT computation of important signals. CTF computation by the derivative method. Linear and non-linear systems: causality, time invariance. Linear time-invariant (LTI) systems: impulse response, convolution integral, transfer function, BIBO stability, causality. Cascade and parallel of LTI systems. Examples: RC (CR) filters, ideal integrator, differentiator, two-path wireless channel model, square law, modulus law. Definition and properties of energy (and power) cross-correlation integral. Energy and power autocorrelation function, Energy spectral density (ESD) and power spectral density (PSD). Proofs of the Wiener-Kintchin theorem for ESD and PSD. Autocorrelation, ESD, PSD at the input and the output of a LTI system. PSD and power autocorrelation function of periodic signals. Signal sampling: the Nyquist-Shannon sampling theorem (proof for energy signals), reconstruction, aliasing and anti-aliasing filtering. Non ideal sampling and reconstruction (real sampling, natural sampling, sample&hold, linear interpolation, finite samples number).Spectral analysis of discrete-time signals (sequences): links with the continuous-time signal samples, Discrete Time Fourier Transform (DTFT) and connections with the CFT. Inverse-DTFT.Bilateral Zeta transform (ZT) and connections with DTFT. Inverse ZT and residue theorem. Connections with the Laplace transform (LT) for sampled sequences. DTFT and ZT main properties and transform of important sequences (Kroneker delta, constant, monolateral decaying exponential, complex exponential). Discrete-time LTI systems: discrete convolution and transfer function H(z). H(z) region of convergence and stability. Discrete-time LTI system implementation: single pole and single zero, generalization for rational H(z) and difference equations, FIR and IIR systems, poles-zeros and partial-fraction (PF) expansions of H(z). Finite-length sequences obtained by sampling a continuous-time signal: the Discrete Fourier Transform (DFT) and the CFT sampling of the sampled signal. Fundamental property of the DFT: the circular convolution. DFT spectral resolution of a sampled signal: time-domain zero-padding. Frequency-domain zero-padding and time-domain interpolation. Digital filer design from analogue filters: impulse response invariance (IRI) theorem, and Bilinear Transform Method (BTM) and consequences on the associated EFP of TL and TZ.Computer aided design: analogue RC filter transformation in a digital filter by the IRI and the BTM. Vector-matrix representation of: discrete convolution (linear and circular) and DFT.Diagonalization of circulant matrices by DFT: concept of eigen-vectors of circulant systems and similarities with eigen-functions of time-continuous convolutive linear systems.Resume of: continuous and discrete random variables (definition, CDF, pdf, mean value, mean squared value, variance),.Transformations Y=g(X) (pdf, fundamental theorem of mean value), joint random variables (CDF, marginal, joint and conditional pdf, mixed momentsi, crosscorrelation coefficient, independence) Resume of: statistic autocorrelation of a Random Process (RP), stationary and ciclostationary RPs, Guassian and armonic RPs. Definition of PSD for a RP an the statistical average od the PSDs of each signal realization for stationary and non-stationary RPs (proof of the Wiener-Kintchin theorem for RPs). Suma and product of RPs and associated PSD.RPs in LTI systems: mean value, statistical autocorrelation and PSD at the input and the output.Gaussian RPs in linear and non-linear systems (marginal and joint pdf of the input/output). The special RP of the Pulse Amplitude Modulation (PAM): spots on its use in digital communication systems, autocorrelation and PSD with correlated and uncorrelated data symbols, Sampling of a RP: autocorrelation function od the sampled RP and covariance matrix for a samples vector. Discrete -time RP filtering, connections between mean values and covariance matrices of the input and output vectors, decorrelation of statistically correlated samples.White noise and “colouring” by filtering.Discrete-time equivalent for discrete-time RPs (Series) and FIR filters. Vector-Matrix notation Y = Hx for white x. Covariance matrix of y and colour introduced by H.Covariances and mean vectors of Y = Hz, and link by H. Decorrelation of y = Hx, and more generally of a random vector with assigned covariance Matrix: EVD of Y and relationship with SVD of H.MATLAB for digital signal processing - Short introduction to Matlab: generation of signal vectors, deterministic and random. Energy and Power computation. Graphical representation by Matlab commands (plot, stem, subplot, etc.)-- Matlab training: samplng and aliasing, spectral estimation by DFT and time-domain windowing. Application to audio signal.- Matlab training: moving average filter, FIR filter (fir1), and Matlab command “filter”. Equiripple Parks-McLellan filter design by Matlab. Short introduction to FDA-Tool in Matlab. Application to audio and music signal filtering, equalizer design by DFT processing. Joint Overlap&Add and DFT processing for digital signal filtering.- Matlab training: “simple” file audio compression by DFT and DCT processing, based on spectrogram considerations..