# MA3K9 Mathematics of Digital Signal Processing

Lecturer: Randa Herzallah

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 lectures

Assessment: 80% fusion of a report and take home exam and 20% assignments

Formal registration prerequisites: None

Assumed knowledge:

• MA106 Linear Algebra: A foundational understanding of vectors, matrices, and linear transformations is essential. These concepts are fundamental to many DSP techniques and algorithms

Useful background:

• MA124 Mathematics by Computer: Our tutorials will leverage Python to explore DSP concepts. While we will offer coding guidance, a basic knowledge of Python will enhance your comprehension of the implementations and aid in evaluations
• MA251 Algebra I: Advanced Linear Algebra: Grasping advanced linear algebra is crucial for understanding multi-dimensional DSP techniques. Many DSP algorithms and transformations originate from these linear algebra concepts
• MA269 Asymptotics and Integral Transforms: This module provides insights into integral transforms, vital for signal analysis in the frequency domain. Asymptotic analysis is key for assessing signal behaviour at boundary conditions

Synergies: This module pairs well with:

• MA2K4 Numerical Methods and Computing: Digital Signal Processing often necessitates the use of numerical methods for signal analysis and filtering. This module introduces computational techniques directly applicable to DSP, enhancing both the efficiency and precision of signal processing algorithms
• MA398 Matrix Analysis and Algorithms: Given that many DSP techniques revolve around matrix operations and transformations, a profound understanding of matrix analysis can significantly enhance the optimization and execution of DSP algorithms. This module delves into advanced matrix computations, essential for multi-dimensional signal processing

Aims: Digital signal processing lies at the core of many new and emerging areas of science and technology including telecommunication, biomedical applications, image processing and recognition, and digital media. This module is an introduction to the basic mathematical tools that are required to present, analyse, understand and design digital signal processing systems. This includes the basics of sampling of continuous time signals, digital filters, digital spectral analysis and digital multirate signal processing.

Content: Digital signal processing is based on many, core disciplines covering many theoretical and application fields. In the field of Mathematics, calculus, probability statistics, deterministic and stochastic processes, and numerical analysis are all core disciplines for digital signal processing. It also lies at the core of many new and emerging areas of science and technology, including network theory and interconnected systems, signals and systems, cybernetics, communication theory, control theory, and fault diagnosis. In addition, digital signal processing forms the basic foundation for the exciting fields of artificial intelligence, pattern processing and analysis, and neural networks. This module offers a comprehensive introduction to the fundamental aspects of DSP, reinforcing understanding through practical applications using programming languages like Python.

Main Topics:

1. Data Acquisition and Sampling: Understanding discrete sequences and systems, diving into the essence of sampling, addressing aliasing, quantization, and signal reconstruction

2. Time Domain Methods: Delving into techniques like correlation and various forms of convolution

3. Frequency Domain Methods: Introducing forward and inverse z-transforms, as well as discrete Fourier transforms

4. Digital Filter Design: Exploring finite and infinite impulse response filters and the realization of digital filters

5. Foundations of Multirate Signal Processing: While advanced topics in multirate signal processing might be touched upon, the primary focus will be on its foundational principles

Objectives: By the end of the module, students should be able to:

• Describe the terminology and concepts of core methods and techniques of digital signal processing
• Design and develop digital signal processing systems and applications
• Formulate and code DSP algorithms to simulate and implement digital signal processing algorithms
• Analyse and explain the behaviour of digital systems
• Design and implement various digital filters including FIR and IIR filters

• Ifeachor, Emmanuel C, Jervis, Barrie W. Digital Signal Processing: A Practical Approach. Prentice Hall, ISBN: 0201596199
• Proakis, John G, Manolakis, Dimitris G. Digital Signal Processing. Pearson Prentice Hall, ISBN: 0131873741
• Porat, Boaz. A Course in Digital Signal Processing. John Wiley, ISBN: 0471149616
• Paulo S. R. Diniz, Eduardo A. B da Silva, and Sergio L. Netto. Digital Signal Processing: System Analysis and Design. ISBN: 0521781752