• Blended Learning & Flipped Classroom

Estimation theory, statistical signal processing and time-data analytics are becoming very common theoretical tools needed in several interdisciplinary areas of engineering such as audio/video, digital communications, vibration analysis, array-processing, imaging and remote sensing, GPS and navigation systems, just to mention some.

The teaching activity aims to balance the fundamentals of discrete-time signals and the estimation theory with a set of engineering use-cases. Goal is to strength the capability of student to have a pragmatic, but still rigorous, way to solve stochastic signal related problems. More specifically, students will learn to approach an engineering problem that involves stochastic signals, formalize it into a mathematical model where all the unknowns are stochastic and/or deterministic variables, and find the most appropriate solving method always validated by the knowledge of the statistical bounds in idealistic connditions.

Goal is to handle with well practice algebra for signal processing and estimation theory, fundamentals of the estimation theory (BLUE, MLE, CRB, MMSE, MAP), parameter tracking and Kalman filtering, and adaptive LMS/RLS filtering. Spectral analysis (AR/MA/ARMA) and high-resolution methods for line spectra and array processing. Detection theory, pattern and feature detection/classification, and supervised/unsupervised classification methods.

The theory on statistical signal processing (60hours) are interleaved with numerical exercising (30hours) on simple settings to consolidate the concepts. More realistic engineering problems are dealt during the 1Cr of Innovative Teaching: practical problems are first discussed in class providing real data, then (typically for 2-3 days) students can prepare for solving the assigned tasks by using Matlab in collaborative Labs where students are randomly grouped in 3-4 to compare and improve their solutions under the supervision of a tutor. The grouping within the same class enables students to experience intra/inter-group dynamics with other unknown and randomly chosen students, this small-scale simulation of their professional life aims to better refine their mutual professional relationships.

Lectures and exercise sessions will allow the student to:

• Consolidate the theoretical tools to handle with finite-length single/multiple signals
• Have solid knowledge of the estimation theory for statistical signal processing
• Know what method to choose depending on statistical properties of stochastic part.
• Handle with adaptive time-varying systems and design Bayesian parameters tracking
• Estimate spectral properties from subset of stochastic samples
• Handle multiple signals from arrays, and extract the relevant parameters
• Extract a pattern from data in supervised/unsupervised forms
• Design the Matlab flowchart for some essential kernels such as estimation of frequency and delays, extraction signals from noise by filtering and by convolution/deconvolution matrixes.

 The project laboratory part of innovative-teaching will allow the student to:

• Analyze the requirements and write the algorithm flow
• Write the Matlab-code that solves the required problem
• Evaluate multiple solutions for the same problem and develop a criticism.
• Write a short document structured as a technical report
• Work in team with other students on the same working-group and learn to interact each-other

 Overall the teaching will give the student at the end of the course and after passing the exam the capability to design estimators and processing methods for limited-length data, to evaluate the reliability and accuracy of the algorithms, and tools for manipulating and classifying multiple-dimensions discrete signals. At the end of the course and after passing the exam, the student will be able to solve problems involving stochastic stationary or time-varying signals, and to evaluate the cost/benefit tradeoff with a deep knowledge of the corresponding statistical bounds.

Outcomes of the students:

• Knowledge and understanding:  students will learn how to manipulate signals using algebraic tools; apply the estimation theory and bounds to signals related problems; manipulate multidimensional signals for estimation and decision.
• Applying knowledge and undestanding: students will be able to define a statistical model for signals' related problems and their solution with fundamental estimation theory methods; solve a broad range of statistical signals' related engineering problems.
• Making judgements: students will be able to evaluate the fundamental limits on statistical signal processing applications even in presence of uncomplete/inaccurate conditions; tailor and adapt the statistical methods to complex problems. 
• Communication: student will learn how to describe a method and sinthetize the limits and results; write a technical report.

The course focuses on statistical signal processing and covers the following topics:

• Review of basics: matrix and linear algebra; quadratic and constrained optimization problems.
• Introduction to the estimation problem and models: definitions, performance, sufficient statistics, linear and non-linear models.
• Estimators: minimum variance unbiased estimation (MVUE), best linear unbiased estimation (BLUE), maximum likelihood estimation (MLE), least squares method. Cramer Rao lower bound.
• Bayesian estimators: a-posteriori estimation (MAP, MMSE and LMMSE); Wiener filter; linear prediction and Yule-Walker equations.
• Adaptive filters: LMS, RLS methods, convergence analysis and step-size selection.
• Spectral analysis: sample autocorrelation and power spectrum; non-parametric method (periodogram); parametric methods (MA, AR, ARMA models, and line spectra).
• Bayesian tracking: dynamic model and Kalman filter; examples of positioning.
• Array processing and direction of arrivals (DOA), beamforming methods and multichannel systems.
• Pattern and sequence recognition: supervised and unsupervised classification, classification of signals in noise, linear discriminant and support vectors, clustering methods.
• Montecarlo simulation and numerical analysis.

Knowledge of linear algebra and matrix manipulations obtained from the teaching of “Geometria e Algebra Lineare”. Basics on stochastic variables, distributions, moments,  statistical independence that are obtained from the teaching of “Teoria dei Fenomeni Aleatori” or “Probabilita’ e Statistica” or “Fondamenti di segnali e trasmissione”.

Knowledge of Fourier transform and its properties from the teaching of “Fondamenti di segnali e trasmissione” or “Analisi Funzionale e trasformate”. Knowledge of Z transform and its properties is recommended, but it is not mandatory as the essential properties for linear time invariant system are part of the teaching topic and are reviewed anyway.

The exam is in written form with at least one small project complete of the reporting activity and, optionally, the oral exam. More specifically, the written exam lasts 2.5h and it covers the entire theory part of the program with simple exercises to spot the degree of understanding of the concepts.  Exercises are more than necessary to pass the exam, The sum of total marks allotted for all the exercises in written exam usually ranges in 42-45, so each student can choose what exercises to do and how much in depth according to her/his skill. Each exercise is in increasing-difficulty, and marks are ranked accordingly. Marks of the entire written exam saturate to 24. During written students cannot use books but a set of formulas chosen by themselves. For computations, students can use computers only with Matlab.

 In parallel, the students can develop at least one of the 3 projects (called Homeworks, or Hw, within the teaching jargon as they can be freely carried out at home): Hw#1, Hw#2, Hw#3. During the teaching semester, the Hws’ text and data are released to the students so that they can start solving Hws at home while studying theory and exercising. Solving the Hw during the teaching semester is recommended as let the students take benefit of the Project Laboratory as Hws have similar structure except should be completed independently on their own. Hws are of incremental complexity, with increasing marks 1,2,3, so that max Hw marks is 1+2+3=6. Hw#1 is mandatory and there is no restrictions on when to solve the Hws, but students can deliver the Hw-reports only after passing the written exam >17.

For students that wish to have to oral exam, the written exam should be passed with unsaturated mark >27, with at least Hw#1. Oral exam is a discussion that aims to evaluate the maturity and the knowledge of the topics when solving some use cases; during oral the student can use the textbook and notes. Notice that 30Lode is only by Oral exam.

There is an intermediate written exam, typically the end of October, covering half of the program. The second half can be completed in any written exam dates till the end of the winter session of exams. The written exam can be repeated anytime without penalty.

Notes/slides on the book can be downloaded from the folder of the course shared with students during the semester