Course objectives
This course is designed for engineering students who have already received a basic training in complex calculus, linear algebra and probability theory. It offers the theoretical foundations to information theory and communication theory. Both deterministic and stochastic signal representation and processing are addressed. The course
describes the time-frequency representation for continuous-time signals. Operations such as sampling, quantization and modulation are outlined in detail.
Teaching modalities
Individual work is complemented by roughly 108 hours of lectures and problem solving. Non mandatory lab work is also made available to facilitate the understanding of the course programme. Lecturing and problem solving activities are taught in Italian.
Prerequisites
Linear Algebra, Calculus I, Elements of probability and statistics
Evaluation methodology
The final evaluation of the Signals & Systems course is assessed through an oral exam. Prior to being admitted to the oral exam, the student must take a written test. During the semester, other types of continuous evaluation may take place through homework and/or lab activities, which would ease the final examination.
Detailed program
[lecture/problem solving]
Detailed program - part 1
Introduction [4/2]
Communication system; information and noise; signal; energy; power; periodicity.
Signal classification and elementary operations [6/4]
Signal classification (origin, energy, shape, frequency, time); elementary operations: sum, product, shift, time-inversion, scale change; decomposition in odd and even parts. Elementary signals: rectangle, triangle, gaussian, sinusoidal signals, unit step, sign function, sinc function. Elements of distribution theory; Dirac impulse (definition and properties); Dirac train.
System classification [6/4]
Linearity, causality, memory, shift-invariance, stability. Input-output relation of linear shift invariant (LSI) systems: LSI impulse response; LSI causality and stability. Linear convolution: definition, properties, geometrical interpretation. LSI eigenfunction. LSI frequency response.
Frequency representation of signals [12/8]
Revision of complex number. Fourier transform of continuous-time signals: definition and inversion formula. Spectrum amplitude and phase spectra. Fourier transform properties. Bandwidth. Parseval identities. Fourier transform convergence. Gibbs phenomenon. Periodic signal spectrum. Fourier series: definition and properties. Asymptotic behavior and convergence of Fourier series expansion. (Energy/Power) spectral density of a signal. Unilateral frequency representations.
Vectorial representation of signals [6/4]
Revision of linear algebra. Signal space, signal distance, signal norm, inner product between signals. Schwarz inequality. Orthogonal and biorthogonal basis. Generalized Parseval identities. Least square approximation of signals. Complete basis examples (Walsh transform, shifted sinc families, ...). Auto-/Cross-correlation functions for energy/power/periodic signals. Convolution versus correlation. Normalized and circular convolution
.
Detailed program - part 2
Introduction to stochastic processes [12/8]
Revision of probability theory: axiomatic definition, events, incompatibility, independence, conditional events, Total probability theorem, Bayes theorem. Random variables: pdf, distribution function, expectation (mean, variance). Join random variables: joint pdf, joint distribution, joint moments (correlation); independence of random variables; examples. Functions of random variables. Conditional random variables: conditional pdf, conditional expected values. Repeated trials. Law of large numbers. Poisson distribution. Central Limit Theorem. Stochastic process. Stationarity and ergodicity. Auto-correlation of a random process (definition and examples, periodogram method). Power spectral density of a random process. Wiener-Khintchine theorem. Sample processes and their statistical modeling (random phase sinusoid, gaussian, PAM, ...).
Noise (white, colored, narrow-band...).
Analog signal processing [6/4]
Phase and group delays; Complex envelop representation of a narrow band signal; Hilbert transform; Analytic signal. Parametric systems. Non linear systems. Ideal transmission system. Harmonic and cross-modulation distortion measures. Linear and non linear processing of arandom process. Memoryless non linear processing of a random process. Linear processing of a random process. Stationary process filtering. Sum and multiplication of random processes.
Fundaments of analog and digital modulation [6/4]
Amplitude modulation (AM, DSB, SSB, VSB). Coherent demodulation and envelop demodulation. Quadrature amplitude modulation. FDM. Frequency modulation systems. Modulated signal spectrum. Demodulation with discrimination (PLL idea). Inter-symbol interference. In-band digital transmission. PAM. PAM spectrum. Main PAM codes
(Manchester, AMI, ...).
Digital representation of an analog signal [6/4]
Analog signal sampling; frequency aliasing; reconstruction by interpolation and extrapolation (ZOH, linear interpolation, ideal interpolation). Quantization; quantization noise: statistical properties. Real A/D and D/A conversions; limit cycle of an A/D converter.
Examples of real systems using the described system components.
References
1. R. Leonardi, & P. Migliorati, Esercizi di Teoria dei Segnali, vol. 1, 360 pages, ISBN: 978-88-7488-401-8. Editor: Società Editrice Esculapio (Bologna, Italy), 3rd edition, Feb. 2011.
2. A.V. Oppenheim, A. Willsky, & I. Young. Signals and Systems, Prentice-Hall.
3. A. Papoulis. Probability, Random Variables and Stochastic Processes, Mc Graw-Hill.
4. C. M. Monti, & G. Pierobon. Esercitazioni di Teoria della Probabilità, vol. 1-3, Lib. Prog., Padova.
5. G. Proakis, D. Manolakis. Digital Signal Processing, Prentice-Hall.
6. F. de Coulon, Signal Theory and Processing, Artech House.
7. S. Bellini. Elementi di Teoria dei Segnali, CLUP.
8. S. Haykin. Communication Systems, Wiley.
9. A.B. Carlson, Communication Systems, McGraw Hill.
10. N.S. Jayant & P. Noll, Digital Coding of Waveforms, Prentice-Hall.