## EE3-08 Advanced Signal Processing

Lecturer(s): Prof Danilo Mandic

Aims:
To introduce students to the fundamentals of statistical signal processing, with particular emphasis upon classical and modern estimation theory, parametric and nonparametric modelling, time series analysis, least squares methods, and basics of adaptive signal processing.

To give students practical experience of utilising statistical signal processing on real world multimedia signals, such as their own speech and video recordings through the provision of structured coursework assignments based upon using MATLAB.

Learning Outcomes:
At the end of the course students should be able to:
Analyse discrete time random signals based upon their statistical properties, such as their probability density functions, degree of statistical stationarity and ergodicity, correlatedness and coupling.

Calculate the first, second, and higher statistical moments of discrete time random signals, namely their mean, correlation and covariance functions, and use these for analysis of real world signals.

Familiarise with a concept of an estimator of an unknown general scalar or vector parameter.

Based upon the concept of an estimator, produce estimates of unknown random variables.

Understand the importance of the the bias and variance of an estimator.

Become familiar with parametric stochastic models, such as the Autoregressive (AR), moving average (MA), and their combination (ARMA).

Learn about the Yule-Walker (Normal) equations and the importance of first and second order moments in statistical modelling of time series.

Derive stability and invertibility conditions for linear stochastic models.

Model real world signals using ARMA model theory, such as financial time series.

Understand the bias-variance dilemma in estimation theory.

Use Cramer Rao theory to obtain the Minimum Variance Unbiased Estimator (MVUE), that is, a theoretic bound on the performance of any estimator, and evaluate its variance with the Fisher information matrix.

Based on the Cramer Rao theory, derive more practical special cases such as the Best Linear Unbiased Estimator (BLUE).

Understand the the BLUE and other estimators can be seen as constrained optimisation problems.

Formulate the maximum likelihood and Bayesian estimator and compare their performance with standard estimators.

Formulate the Least-Squares estimation problem and apply it to a range of real world applications. Derive the block and sequential form of the least squares estimator.

Understand the principle of orthogonality within least squares estimation and the implications on practical estimators.

Become familiarised with the concept of Wiener filterining, and understand the difference between block and sequential stochastic models.

Learng the concept of steepest descent and benefits and drawbacks of approximate sequential models.

Derive the Least Mean Square (LMS) adaptive filtering algorithm for prediction and identification of real world signals.

Understand the operation of LMS on nonstationary signals and signals with large dynamical range.

Apply statistical signal processing in noise cancellation and signal enhancement applications such as acoustic echo control and foetal hearbeat monitoring.

Syllabus:
Discrete random signals; statistical stationarity, strict sense and wide sense. Averages; mean, correlations and covariances. Bias-Variance dilemma. Curse of dimensionality. Linear stochastic models. ARMA modelling. Stability of linear stochastic models. Introduction to statistical estimation theory. Properties of estimators; bias and variance. Role of Cramer Rao lower bound. Minimum variance unbiased estimator. Best linear unbiased estimator (BLUE) and maximum likelihood estimation. Maximul likelihood estimator. Bayesian estimation. Least square estimation: orthogonality principle, block and sequential forms. Wiener filtering, adaptive filtering and signal modelling. Concept of an artificial neuron. Applications: time series modelling (financial, biomedical), acoustic echo cancellation and signal enhancement, inverse system modelling and denoising.

Assessment:
80% coursework, 20% class test in late Spring Term

Coursework contribution: 80%

Term: Spring

Closed or Open Book (end of year exam): Closed

Coursework Requirement
To be announced

Oral Exam Required (as final assessment): no

Prerequisite: None required

Course Homepage: unavailable