Processamento Estatístico de Sinal

This class was not given due to the involvement of the instructor in the organization of the 2022 Workshop MAP-tele. Students were advised to attend the keynote speeches of the workshop.

Presentation of the curricular unit.
A review of the fundamentals of discrete-time signals and systems.
A review of the frequency-domain representation of signals and systems.
A review of the discrete-time Fourier Transform (DTFT).

The DTFT of the auto-correlation and of the cross-correlation.
A review of the Z-Transform: definition, region of convergence (RoC) and its properties, implications of stability and causality to the RoC, reference Z-Transform pairs, the inverse Z-Transform, properties of the Z-Transform.

The Z-Transform of the auto/cross-correlation.
Review of random signals and processes: basic concepts (ensemble averages, jointly distributed random variables, moments and joint moments, independent, uncorrelated and orthogonal random variables, Gaussian random variables). Stationary processes, autocorrelation and autocovariance matrices and their properties.
Eigenvalues and eigenvectors. Linear transformation of random vectors. Innovations representations of random vectors (eigenanalysis and the Karhunen-Loève transform).

First quiz on discrete-time signals and systems, the Fourier Transform, and the Z-Transform.
Ergodicity, white noise, and the power spectrum. The PDF of the sum of independent random vectors. Illustrative examples.
Filtering discrete-time random processes. The cross-correlation between output and input of an LSI system. The deterministic auto-correlation of an LSI system and the relation between the auto-correlation between its output and its input.

Second quiz on random processes.
Spectral factorization of the power spectrum. The Wold decomposition theorem.
The Yule-Walker equations in the analysis of discrete-time random processes. Special types of random processes: autoregressive moving-average processes (ARMA).

Special types of random processes: autoregressive processes (AR), and moving-average processes (MA). Harmonic processes.
Presentation of the home assignment to be developed in groups of two students.

Introdução aos Processos Estocásticos e Filtro de Kalman. - Introdução. - Observadores determinísticos - Estimação de estado em sistemas com perturbações determinísticas: Modelo interno - Modelização de perturbações como processos estocásticos ergódicos.

Introdução aos Processos Estocáticos e Filtro de Kalman (conclusão) - Modelos de estado de sistemas determinísticos-estocásticos - Estimação de estado de sistemas determinístico-estocásticos com observadores de Luenberger. - Matrizes (semi)definidas positivas e (semi)definidas negativas. - Previsor de Kalman. - Filtro de Kalman.

Simulação dum motor eletrico perturbado por ruído e estimação do seu estado utilizando um filtro de Kalman.

Dimensiomento dum filtro de Kalman para estimar a posição dum bola de basquetbol a saltar no solo. Determinação da posição da bola utilizando o filtro de Kalman.

Estimação de da densidade espetral: Métodos não paramétricos: - O periodograma: Definição, valor esperado e variância - Média de periodogramas

Conclusão da aula anterior.