Unit name | Mathematics for Signal Processing and Communications |
---|---|

Unit code | EENGM0014 |

Credit points | 20 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 4 (weeks 1-24) |

Unit director | Professor. Pitas |

Open unit status | Not open |

Pre-requisites |
Admission to PhD in Communications programme |

Co-requisites |
None |

School/department | Department of Electrical & Electronic Engineering |

Faculty | Faculty of Engineering |

While mathematics is core to Communications and Signal Processing, many students enter the discipline at postgraduate level lacking many of the fundamental skills and knowledge that enables them to maximise their research potential or contribution. The aim of this unit is to fill this void, initially targeted at students on the Centre for Doctoral Training in Communications. Target topics would be: a) Linear algebra, vector analysis, b) probability theory and random processes, c) graph theory d) optimization methods that form a good part of the mathematical foundations of communications systems, multimedia communication/processing, signals and systems. The basic mathematical essentials cover the following topics which will be addressed in this unit:

**Revisiting Linear Algebra and Vector Analysis**: The properties and manipulation/decomposition of a matrix, linear transformations, orthogonality and bases, systems of equations and solutions, Eigen values and Eigen vectors, principle component analysis QR and single value decomposition.

**Probability and Random Processes**: Randomness is inherent to physical systems, and plays a particular important role in communications and networking, grounded in Information Theory, a probabilistic theory that describes the fundamental limits on communication imposed by random noise in communication and storage media, as well as implications for secure communications and network design and deployment. This part will cover probability axioms, conditional probabilities and independence, Bayes theorem, distribution types, random variables, random vectors, probability density functions, correlation and covariance; random processes, stationarity, ergodicity, power spectral density, Gaussian random processes, sampling and reconstruction, power spectral density, modelling of data sets, least squares and maximum likelihoods, nonlinear models, confidence limits, robust estimation, polynomial interpolation and extrapolation, cubic spines, dimensionality aspects.

**Graph Theory: **Graph theory has emerged in recent years as an important topic in communications engineering and signal processing, relevant to a wide range of topics such as network analysis and design and cross media analytics: Types of graphs, properties, paths and circuits, trees, topologies, graph clustering, linear algebra in graph analysis.

**Optimisation: **The design or efficient operation of most signal processing and communication systems requires some form of off-line or on-line optimisation, usually incorporating numerous constraints. This part will introduce optimisation theory covering: unconstrained optimisation, iterative methods, roots of equations, multidimensional optimisation, conjugate gradients and quasi Newton methods, constrained optimisation- linear, integer, simplex and dynamic programming, annealing and evolutionary methods.

This unit aims to provide students with an appreciation of such issues and to enable them to apply the relevant mathematical skills in the context of Engineering Communication systems.

At the end of the course the student should be able to:

This unit aims to provide students with an appreciation of the mathematical background of communications and signal processing and enable them to apply the relevant mathematical skills in the context of their studies with applications ranging from communication systems to multimedia compression/communications to biomedical signal analysis, compression, storage and communication. At the end of the course the student should be able to:

- Master Linear Algebra and vector analysis.
- Work on applications in speech, biomedical/financial signal and image analysis/communications.
- Master probability theory: and Random Processes: Probability axioms, conditional probabilities and independence, Bayes theorem, distribution types, random variables, random vectors, probability density functions, correlation and covariance.
- Use random processes: stationarity, ergodicity, power spectral density, Gaussian random processes, sampling and reconstruction, power spectral density.
- Perform data modelling interpolation and extrapolation: least squares and maximum likelihoods, nonlinear models, confidence limits, robust estimation, polynomial interpolation and extrapolation, cubic spines, dimensionality aspects.
- Perform graph analysis: finding graph paths and circuits, trees, topologies, perform graph clustering, linear algebra in graph analysis (spectral analysis).

Use Optimisation in communication/analysis problems: unconstrained optimisation, iterative methods, roots of equations, multidimensional optimisation, conjugate gradients and quasi Newton methods, constrained optimisation- linear, integer, simplex and dynamic programming, annealing and evolutionary methods.

Lectures, Software simulation tutorials

2 hour written examination (80%)

Software Simulation based coursework assignment (20%) :Development of MATLAB or C/C++ code for analysing e.g., speech, biomedical/financial signals and images complemented by an essay/documentation.

Papoulis, Athanasios, and S. Unnikrishna Pillai, “Probability, random variables, and stochastic processes”, McGraw-Hill, 2002.

2. S. M. Kay, “Intuitive Probability and Random Processes Using MATLAB”, New York, N.Y.: Springer 2006.

3. Strang, Gilbert. "Introduction to Linear Algebra", Wellesley-Cambridge Press; 4 edition (10 Feb. 2009).

4. Fletcher, Roger, “Practical methods of optimization”, John Wiley & Sons, 2013.

5. D.B. West, “Introduction to graph theory”, Pearson, 2015.

6. I. Pitas (editor), “Graph-based social media analysis”, CRC Press, 2015.