Planeamento
Aulas de Problemas
Discrete-time signals
Exercises from problem series 1.
Discrete-time systems and Fourier transform
Exercises from problem series 2.
z transform
Exercises from problem series 3.
Discrete Fourier transform
Exercises from problem series 4.
Least squares method
Exercises from problem series 5.
Random signals and parameter estimation
Exercises from problem series 6.
Maximum likelihood method and Crámer-Rao lower bound
Exercises from problem series 7.
Aulas Teóricas
Course presentation
Motivation. Course overview. Syllabus. Grading. Lab registration.
Discrete-time signals
Basic signals. Periodicity. Representation of signals as a sum of impulses.
Discrete-time systems
Discrete-time systems. Properties: linearity, time invariance, causality, stability. Linear and time invariant (LTI) systems. Convolution sum.
Fourier transform and sampling
Fourier transform of discrete-time signals. Periodicity. Properties. Sampling of continuous-time signals. Sampling theorem. Aliasing.
z transform
Definition of z transform. Region of convergence. Relationship with the Fourier transform. Examples.
z transform
Properties of the ROC. Properties of the z transform. Convolution property. Exercises.
z transform
Inverse transform. Inverse transform of rational functions with simple poles. Long division of polynomials. Partial fraction decomposition.
z transform
Application to LTI systems. Transfer function. Poles and zeros. Causality and stability. Difference equations.
Discrete Fourier transform
Finite transforms. Exponential base. The discrete Fourier transform and its inverse. Examples.
Discrete Fourier transform
Properties. Exercises.
Discrete Fourier transform
Circular convolution of finite signals. Circular convolution property.
Discrete Fourier transform
Exercises.
Discrete Fourier transform
Linear filtering with DFT. Overlap Add method.
Linear filtering
Types of filters. Ideal filters. Filter specification. IIR and FIR filters.
Linear filtering
Canonic form I and II. Design of FIR filters using windows.
Least squares method
Signal model. Sum of squared errors (SSE) criterion. Least squares problem. Matrix notation for linear models (in the parameters)
Least squares method
Exercises.
Finite random signals
Continuous and discrete random variables. Probability function and probability density function. Random vectors. Probability distribution of a random vector. Independent random variables. Second order description: mean vector and covariance matrix. Properties of the covariance matrix. Covariance of independent random variables. Linear transformation. Multivariate normal distribution.
Parameter estimation
Estimator of a parameter. Bias and covariance. Bayesian estimation vs classic estimation.
Maximum likelihood method
Likelihood function and log-likelihood function. Maximum likelihood method. Examples.
Maximum likelihood method
Examples
Maximum likelihood method
Properties. Examples.
Crámer-Rao lower bound
Crámer-Rao lower bound (scalar case). Examples. Crámer-Rao lower bound (vector case). Examples.
Bayesian estimation
Prior information. The a posteriori distribution. Examples.
Bayesian estimation
Computational difficulties. The exponential family. Examples.
Bayesian estimation
The Bayes classifier. Examples.