Portale di Ateneo - En.Unibs.it

Introduction.

Communication system; information and noise; signal; energy; power; periodicity.
Signal classification and elementary operations.

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.
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.
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.
Revision of linear algebra. Signal space, signal distance, signal norm, inner product between signals. Schwarz inequality. Orthogonal and bi-orthogonal basis. Generalized Parseval identities. Least square approximation of signals. Auto-/Cross-correlation functions for energy/power/periodic signals. Convolution versus correlation. Normalized and circular convolution.

Digital representation of an analog signal
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.

Discrete signals.
Basic notions and classification.
Energy/poewr/mean value of a sequence.
Causal, anti-causal and non-causal sequences.

Elementary sequences: step, impulse, rect, sinc, sinusoidal sequences etc.
Elementary operations on sequences: sum, difference, product, decimation and interpolation.

Dicrete processign systems. Classification: linearity, memory, causality, stability, shift-invariance.

LSI systems. input/output relation, causality and stability.
IR and IIR systems. Parallel and series concatenation of LSI systems.

Convolution of sequences. Linear and circular convolution.

Frequency analysis of discrete signals, DTFT and DFT.

Eigenfunctions of LSI systems and
fequency response of a LSI system.