NUMERICAL SIGNAL PROCESSING
*) Knowledge and understanding: Knowledge of the mathematical foundations for the representation, analysis and digital processing of signals. Knowledge of design methods of FIR and IIR filters. Knowledge of classical and Bayesian statistical estimation techniques.
*) Applied knowledge and understanding: analysis of a signal in the frequency domain, design of a digital filter, performing digital filtering of signals, statistical estimation of parameters from noisy data.
*) Making judgments: developing the ability to critically and synergistically use various tools for numerical processing and statistical estimate of signals. Knowing how to evaluate the design constraints of a processing system in terms of error, computational complexity and stability of the algorithm.
*) Communication skills: Ability to present the topics in a clear and rigorous technical-scientific point of view. Knowing how to submit an application solution in a simple and comprehensive.
*) Learning skills: knowing how to integrate knowledge from various sources for the purpose of deepening. Knowing how to use the concepts covered for applications other than those disclosed.
For the successful achievement of objectives, the knowledge of signals and systems theory is required.
Digital signal processing systems. Advantages of Digital over Analog signal processing. Continuous-Time and Discrete-Time signals. Continuous-Valued and Discrete-Valued signals. Deterministic and random signals. Sinusoidal signals. Harmonically related complex exponentials. Analog-to-Digital and Digita-to-Analog conversion. Sampling of analog signals. The Sampling Theorem. Quantization. Quantization of sinusoidal signals. Coding of quantized samples. Digital-to-Analog conversion. Some elementary discrete-time signals. Classification of discrete-time signals. Manipulations of discrete-time signals.
Discrete-Time systems. Input-Output description. Block diagram representation of discrete-time systems. Classification and interconnection of discrete-time systems. Analysis of discrete-time Linear Time-Invariant (LTI) systems. Response of LTI Systems: the Convolution Sum. Properties of convolution and the interconnection of LTI systems. Causal LTI systems. Stability. Systems with finite-duration and infinite-duration impulse response.
Discrete-Time systems described by difference equations. Recursive and nonrecursive discrete-time systems. LTI systems characterized by constant-coefficient difference equations. Solution of linear constant-coefficient difference equations. The ompulse response of an LTI recursive system. Implementation of discrete-time systems. Structures for the realization of LTI systems. Recursive and nonrecursive realizations of FIR systems. Crosscorrelation and autocorrelation sequences. Properties of the autocorrelation and crosscorrelation sequences. Correlation of periodic sequences. Computation of correlation sequences. Input-Output correlation sequences.
The z-Transform. The direct and inverse z-Transform. Properties of the z-Transform. Poles and Zeros. Pole location and time-domain behaviour for causal signals. The System Function of an LTI system.
The Discrete Fourier Transform (DFT). The DFT as a linear transformation.
Properties of the DFT. Frequency analysis of signals using the DFT.
Design of digital filters. Causality and its implications. Characteristics of practical frequency-selective filters. FIR and IIR filters Applications. Zeros of linear phase filters: COMB Filters. Notch Filters. Design of FIR filters. Symmetric and antisymmeiric FIR filters. Design of linear-phase FIR filters using Windows. Design of linear-phase FIR filters by the frequency-sampling method. Design of optimum equiripple linear-phase FIR filters. Design of IIR filters from analog filters. IIR filter design by approximation of derivatives.
Theory of statistical estimation. The mathematical problem of the estimate. Classical estimation theory. Biased estimators. Minimum variance estimators. MVU estimators. Extension to the vector case. Cramer-Rao Bounds. CRLB for signals in AWGN. Transformation of parameters. Extension to the vector case. CRLB for the general Gaussian case. Examples of Signal Processing. Linear models. Generalized linear models. Maximum likelihood estimators. Properties of ML estimators. MLE for transformation of parameters. Extension to the vector case. MLE numerical determination. Asymptotic properties of the ML estimators. Least squares. Linear case. Bayesian estimation. Bayesian cost. Bayesian risk. MMSE estimators. MAP estimators. Bayesian Gaussian case. Iterative methods. Global minimization. A priori information.
Classroom lectures and computer labs in Matlab.
Dimitris Manolakis, Vinay Ingle, ‘Applied Digital Signal Processing: Teory and Practice’, Cambridge University Press, 2011.
S. M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory (Vol. 1). Prentice Hall.
The objective of the examination test is to verify the level of achievement of the training objectives indicated above.
The examination is divided into 2 parts which take place on different days.
• A computer-based practical test in Matlab that aims to evaluate the student's ability to solve problems encountered during laboratory exercises; to pass the test he must acquire at least 16 points out of 30. The test time is 2 hours; books and notes cannot be consulted.
• An oral test in which the ability to connect and compare different aspects dealt with during the course will be evaluated; to pass the test it is necessary to acquire at least 18 points out of 30.
The final score is the average of the 2 scores. If the total score is less than 18, all two tests must be repeated.