Portale di Ateneo - En.Unibs.it

Continuous time signals. Elementary signals (rect, tri, sinc, ecc.) and basic operations (horizontal flip, shift, ecc.). Main properties of signals (energy, power, periodicity, etc.)

Geometric representation of signals.
Review of linear spaces and inner products. Linear space of signals. Subspaces of energy and power signals with their inner products.
Orthogonality of signals, Pythagorean theorem.
Cauchy-Schwarz inequality, cosine theorem. Orthonormal basis, expansion of a signal over an orthonormal basis and preservation of the euclidean norm and of the inner product.
Projection of a signal into a subspace. Normal equations. Gram-Schmidt orthonormalization.

Correlations of signals. Cross- and self-correlation for energy signals. Normalized and circular cross- and self-correlation. Definitions and properties.

Dirac's delta function, definition and main properties.

Continuous time systems.
Definition, classification: linearity, causality, memory, stability, time-invariance.

Linear, time-invariant systems.
Input-output relation, impulse response. Causality and stability of an LTI system.
Linear convolution, definition, graphical interpretation and properties.
Convolutions with delta functions. Connection between convolution and cross-correlation. Circular convolution.

Frequency analysis.
Eigenfunction of an LTI. Frequency response of an LTI system.
Fourier transform of a signal: definition, inverse transform and existence. Spectrum of a signal (amplitude and phase). Examples.
Properties of the Fourier transform (duality of the inversion formula, linearity, hermitian symmetry, evenness/oddness, spectrum of a real signal, Parseval identity, shift, scale change, modulation, multiplication, linear convolution, derivative).

Fourier transforms of elementary signals (sign, Dirac's delta, step, sinusoids, rect, tri etc.)
Notion of signal and filter bandwidth. Low-pass, high-pass and band-pass filters.
Convergence of the Fourier transform, Gibbs phenomenon.
Proprietà di convergenza della T.F. Fenomeni di Gibbs.

Fourier Series.
Definition, examples, properties.

Energy and power spectral densities.

Principles of Analog-to-Digital and Digital-to-Analog conversion.
Examples of expansion of a band-limited signal over a basis of sinc functions.
Sampling of a signal, aliasing, sampling theorem.
Interpolation of a sampled signal (zero-order hold, linear and ideal interpolator)
Real sampling, anti-aliasing filter.
Quantization, definition and examples. Quantization noise, model for many quantization levels.
Signal-to-quantization noise ratio.

Discrete Time signals and systess.
Sampling of sinusoids, interpretation of aliasing, normalized frequency.
Discrete sinusoids, discrete complex exponential functions.
Elementary signals and operations.
Discrete-time systems, classification, LTI systems, examples.
Impulse response and convolution. FIR and IIR systems.
Systems described by difference equations.
Convolutional Networks (CNN)
Frequency response of a discrete time LTI system.
Discrete-Time Fourier Transform (DTFT), definition and connection with Fourier series.
Inversion formula for the DTFT.
Discrete Fourier Transform (DFT) by sampling of the DTFT.
Inversion formula (IDFT) and intepretations. Properties of the DFT.
Indirect convolution with the DFT.