Fourier Analysis of analog deterministic signals.
Definition and analysis of simple LTI systems.
Complex envelope of band-pass signals.
Sampling of band-limited signals. Aliasing.
Definition of random process, stationary random processes, Gaussian random processes.
Random processes through LTI systems.
White noise, Statistical characteristics of narrow-band Gaussian noise.
Monica Gherardelli, Mario Fossi
"Appunti di Teoria dei Segnali",
Simon Haykin, Michael Moher: "Introduzione alle telecomunicazioni analogiche e digitali"
Casa Editrice Ambrosiana
J.G Proakis, M. Salehi::"Communication Systems Engineering",
Prentice Hall International Editions
A. Papoulis: "Probability, Random variables and Stochastic processes",
Ed. Mc Graw-Hill (3* edizione).
The course provides the basic methodological tools for
description, analysis and modeling of deterministic signals and
random signals. The course introduces the study of simple linear systems and their
behavior in the presence of input signals; it allows to know and
interpret the effects of the sampling of continuous signals.
Limits, series, integrals. Linear Algebra. Complex Algebra. Trigonometry.
Analytic Geometry. Probability Theory.
Further information is available at the website e-l.unifi.it
Type of Assessment
The final exam consists of two parts: an intermediate written test and an oral test.
The intermediate written test involves the solution of three exercises aimed at evaluating the student's ability to perform Fourier transforms of signals in low and high frequency ranges and to calculate the output of simple LTI systems with different inputs, both in the time domain and in the frequency.
The oral exam is aimed at verifying the learning of:
- techniques for modeling of deterministic and random signals;
- methods for the classification of simple linear systems and for the study of their behavior in the presence of different inputs;
- sampling techniques of time-continuous signals and their effects.
Definitions of information, signal and communication system.
Classifications of signals - Deterministic and random signals; energy signals and power signals; analog signals, discrete-time signals and digital signals; periodic signals, no-periodic signals and cyclic signals. Examples.
Fourier analysis - Complex exponential Fourier series of periodic and energy signals. Fourier transform; graphic evaluation of convolution integral; autocorrelation, cross-correlation and Parseval theorem. Dirac Delta: definition and properties. Fourier transforms of generalized functions: Dirac delta, complex exponential function, signum function, unit step function, periodic functions, Dirac Comb. Definition of bandwidth of a signal. Autocorrelation and Spectral power density of power signals.
Linear transformations of analog signals – Classification of electrical systems: linear systems, Time-invariant systems, casual systems, stable systems, active and passive systems. Analytical characterization of LTI systems. Physical reliability. LTI system analysis in the frequency domain: frequency response or transfer function of a system, frequency response theorem, I/O relation, relation between I/O power spectral densities. Distortionless transmission: linear distortions, amplitude and phase distortions. Power gain of a LTI system. Filtering systems: band-pass filters, low-pass filters, definition of filter bandwidth.
Complex envelope of a band-pass signal - Hilbert transform. Complex envelope of finite energy signal. Canonical representation of band-pass signals. Examples.
Sampling of signals – Sampling theorem of finite energy and band-limited signals: Fourier transform of the sampled signal, Nyquist rate, analog signal reconstruction by mean of interpolation formula. Aliasing. Sampling by switching circuit. Sample-hold sampling. Sampling of band-pass signals. Examples.
Random vectors - Covariance matrix, Joint distribution function and joint probability density function. Gaussian vectors, linear transformations of n jointly Gaussian random variables.
Random processes – Definition. Nth order distribution function of a process, Nth order probability density function of a process. Multi-dimensional processes. Complex processes. Mean, autocorrelation and auto-covariance functions. Cross-correlation and cross-covariance functions of two processes. Uncorrelated processes, orthogonal processes, statistically independent processes. Gaussian processes. Stationary processes: stationarity in the strict sense and in the wide sense, joint stationarity. Autocorrelation and power spectral density of stationary processes. Cross correlation and cross spectrum of stationary processes. Linear transformation of random processes. Ergodic processes.
Noise – White noise , white gaussian noise through a low-pass filter: ideal and real cases (RC filter). Noise equivalent bandwidth and decorrelation time. Matched filter. Statistical characteristics of Gaussian narrow-band: properties of the in-phase and quadrature components, properties of the envelope and phase.