Aims

The aims of the course are to:

• Study more advanced probability theory, leading into random process theory.
• Study random process theory and focus on discrete time methods.
• Introduce inferential methodology, including maximum likelihood and Bayesian procedures, and examples drawn from signal processing. Objectives

Objectives

As specific objectives, by the end of the course students should be able to:

• By the end of the course students should be familiar with the fundamental concepts of statistical signal processing, including random processes, probability, estimation and inference.

Content

Lectures 1-8: Advanced Probability and Random Processes

• Probability and random variables

  • Sample space, events, probability measure, axioms.

  • Conditional probability, probability chain rule, independence, Bayes rule.

  • Random variables (discrete and continuous), probability mass function (pmf), probability density function (pdf), cumulative distribution function, transformation of random variables.

  • Bivariate: conditional pmf, conditional pdf, expectation, conditional expectation.

  • Multivariates: marginals, Gaussian (properties), characteristic function, change of variables (Jacobian.)

• Random processes

  • Definition of a random process, finite order densities.

  • Markov chains.

  • Auto-correlation functions.

  • Stationarity–strict sense, wide sense. Examples: iid process, random-phase sinusoid.

  • Ergodicity, Central limit theorem.

  • Spectral density.

  • Response of linear systems to stochastic inputs – time and frequency domain.

  • Time series models: AR, MA, ARMA

Lectures 9-16: Detection, Estimation and Inference

• Basic linear estimation theory: BLUE, MMSE, bias, variance

• Wiener filters

• Matched filters

• Least squares, maximum likelihood, Bayesian inference.

• The ML/Bayesian linear Gaussian model

• Maximum likelihood and Bayesian estimation

• Example inference models: frequency estimation, AR model, Estimation of parameters for discrete Markov chain.

This syllabus contributes to the following areas of the UK-SPEC standard:

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Last modified: 23/09/2019 14:25

Aims

The aims of the course are to:

• Study more advanced probability theory, leading into random process theory.
• Study random process theory and focus on discrete time methods.
• Introduce inferential methodology, including maximum likelihood and Bayesian procedures, and examples drawn from signal processing. Objectives

Objectives

As specific objectives, by the end of the course students should be able to:

• By the end of the course students should be familiar with the fundamental concepts of statistical signal processing, including random processes, probability, estimation and inference.

Content

Lectures 1-8: Advanced Probability and Random Processes

• Probability and random variables

  • Sample space, events, probability measure, axioms.

  • Conditional probability, probability chain rule, independence, Bayes rule.

  • Random variables (discrete and continuous), probability mass function (pmf), probability density function (pdf), cumulative distribution function, transformation of random variables.

  • Bivariate: conditional pmf, conditional pdf, expectation, conditional expectation.

  • Multivariates: marginals, Gaussian (properties), characteristic function, change of variables (Jacobian.)

• Random processes

  • Definition of a random process, finite order densities.

  • Markov chains.

  • Auto-correlation functions.

  • Stationarity–strict sense, wide sense. Examples: iid process, random-phase sinusoid.

  • Ergodicity, Central limit theorem.

  • Spectral density.

  • Response of linear systems to stochastic inputs – time and frequency domain.

  • Time series models: AR, MA, ARMA

Lectures 9-16: Detection, Estimation and Inference

• Basic linear estimation theory: BLUE, MMSE, bias, variance

• Wiener filters

• Matched filters

• Least squares, maximum likelihood, Bayesian inference.

• The ML/Bayesian linear Gaussian model

• Maximum likelihood and Bayesian estimation

• Example inference models: frequency estimation, AR model, Estimation of parameters for discrete Markov chain.

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 05/01/2024 12:25

Aims

The aims of the course are to:

• Cover three basic topics in signals and systems which provide the basis for further topics in signal processing, communications, control and related subjects.
• Introduce the z-transform, which is the generalisation of the Laplace transform to discrete time systems.
• Introduce digital filtering.
• Introduce stochastic processes.

Objectives

As specific objectives, by the end of the course students should be able to:

• Be familiar with the theory and application of the z-transform.
• Analyse the stability of discrete-time systems
• Understand the use of correlation and spectral density functions.
• Analyse the behaviour of linear systems with random inputs.

Content

Enabling theory, application and design, Dr T. O’Leary and Dr F. Forni

Introduction to signals and systems, discrete time signals and systems, Z-transform (5L – O’Leary)

• Discrete signals and systems, LTI systems, convolution. 
• z-transform and solution of linear difference equations
• System analysis in the z-domain. 
• Impulse and frequency responses.

Applications & digital filtering (8L – Forni)

• Design and properties of digital feedback systems. Nyquist stability criterion. 
• Design and properties of Digital Filters, FIR and IIR
• Analysis of systems with discrete/continuous interfaces.
• DTFT/DFT and links to z-transforms 
• The Fast Fourier Transform (FFT)
• Windowed spectral analysis of data 
• Introduction to 2D filtering, image analysis

Introduction to random processes and linear systems (3L – O’Leary)

• Continuous time random processes, correlation functions, spectral density.
• Response of continuous time linear systems to random excitation.

This syllabus contributes to the following areas of the UK-SPEC standard:

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Last modified: 16/05/2018 13:31

Aims

The aims of the course are to:

• Provide an understanding of a range of management science modelling methods involving randomness, such as statistics, decision analysis, behavioral factors, portfolio management, process analysis, queueing theory, forecasting, and regression.
• For each of the modelling areas, students will become familiar with the types of situations in which the method is useful.

Objectives

As specific objectives, by the end of the course students should be able to:

• Understand basic concepts of probability and the rationale behind statistical reasoning.
• Be able to calculate statistical measures like mean and variance, and interpret these in realistic situations.
• Use confidence intervals to quantify risk.
• Conduct hypothesis testing.
• Be able to understand decision trees and how to apply them in decision making.
• Identify and manage the bottleneck in a serial process, calculate the throughput of the entire system and utilisation at each step.
• Understand and use simple formulas for queues in which arrivals occur as a Poisson process.
• Understand the role of behavioral biases in decision making.
• Forecast data using short range extrapolative techniques such as exponential smoothing.
• Know how to take account of seasonality when forecasting.
• Apply regression techniques to estimate the way in which two variables are related.
• Be able to understand investment strategies for portfolios.
• Be able to incorporate risk into investment and decision making.

Content

"There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. These are things we don't know we don't know."

- Donald Rumsfeld

Note: The content covered across all lectures and example papers will be as listed below. However, elements of the content may be re-sequenced to achieve a better flow. 

Mathematical Analysis of Deterministic and Stochastic Processes (4L)

• Process Analysis: Identify and manage the bottleneck in a serial process, calculate the throughput of the entire system and utilisation at each step, evaluate the impact of improvements to different steps in a process.
• Queueing theory: Poisson arrival processes, classification of queueing systems, steady state, performance measures, Little's formula, benefits and limitations of queueing theory.

Regression Analysis and Forecasting (4L)

• Simple linear regression analysis, least squares estimates, significance of  regression, multiple regression, multi-collinearity.
• Different methods for forecasting: moving average, exponential smoothing, modelling seasonality and trends.

Inventory Management (2L)

• Basic concepts in inventory management: inventory management under stochastic demand.

Portfolio Management (2L)

• Basic portfolio concepts
• Risk and expected return on a portfolio, and the efficient frontier.

Decision Analysis (4L)

• Events and decisions, decision trees, expected monetary value, sensitivity analysis, expected value of perfect information, expected value of sample information.
• Behavioural Factors in Decision Making

This syllabus contributes to the following areas of the UK-SPEC standard:

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Last modified: 30/10/2023 14:58