# Biomedical Engineering (MEng)

## Signals and Control

### Module aims

To introduce the foundations of signal processing and linear systems analysis, with a clear focus on achieving an intuitive understanding of the main mathematical descriptions that are used, and the reasons for their use.  Significant time is also spent on explaining the applications of signal processing, in biomedical applications, medical imaging, and elsewhere.

To give the mathematical basis of linear control systems analysis, which is used in many fields of (bio)engineering. Examples from bioengineering and robotics are also provided. This part of the course is composed of lectures, tutorials and Matlab practicals. The concepts are typically introduced in the lectures, while solving the exercices in the tutorials and practicals is necessary to fully grasp the concepts and become able to use the learned techniques.

### Learning outcomes

Learning Outcomes - Knowledge and Understanding

Transform-domain representation of time-varying signals, in particular in terms of harmonics. Understanding and appreciation of applications of signal processing in biomedical engineering - this includes in imaging techniques, and describing physiological signals. An appreciation of the power of systems-level approaches (particularly LTI systems) to engineering problems. An appreciation of the range of applications of Fourier and Laplace transforms in engineering.

Dynamic systems analysis. Design of control systems.

Learning Outcomes - Intellectual Skills

Appreciation of when to use DFT/analytic techniques to tackle particular problems; understanding of how to predict outputs of LTI systems to arbitrary inputs. An understanding of how to model or represent a signal using either mathematical expressions or simulations in Matlab, or some other programming language.

Dynamic systems analysis. Design of control systems.

Learning Outcomes - Practical Skills

Sketching semi-analytically defined signals; performing graphical and numeric convolution. Use of Matlab to develop simulations of systems, generating plots, analysing signals quantitatively and with the use of computational methods.

Tracing of Bode plots, design of linear controller with time and frequency methods. Use of Matlab to design and analyse controllers.

Learning Outcomes - Transferable Skills

Programming for solving engineering problems.  Quantitative descriptions of signals and signal properties.

Problem solving for engineering problems.

### Module syllabus

Definition of Signals & Systems: inputs and outputs. Deterministic and random/stochastic signal, continuous and discrete time signals, periodic signals; Examples in bioengineering; Typical signals such as sinusoidal, exponential, step, impulse, ramp. Simple operations on signals: time translation, dilation/contraction; scaling and addition, convolution, products and correlation between two signals, inner products, and norms. Euler formula, roots of unity; Signals in vector spaces with inner products; Fourier Representations:  Fourier integral representation, Fourier forward and inverse transforms, duality between time and frequency domains. Examples of Fourier analysis in bioengineering. Properties of Fourier transforms: Parseval relation, convolution property, continuous/discrete duality. Relationship to Fourier Series. Discrete time Fourier representations. Sampling and digital signals: sampling theorem, Nyquist frequency, continuous signal reconstruction, discrete sampling, decimation, the numerical Fourier transform. LTI systems: linear and nonlinear, time varying and time invariant systems; representation of linear time invariant systems by convolution and transform-domain operations.

Control: Introduction to LTI systems, Laplace Transform. Description of LTI systems. Block diagrams. Poles & Stability. Time reponse analysis. PID control. Frequency domain analysis. State-space systems.

### Pre-requisites

The Department of Bioengineering's Mathematics 1 course is a prerequisite. A working knowledge of Matlab is also required. A clear understanding of BE2-HMATH2 Mathematics II helps in this course.

### Teaching methods

Lectures: 31 hours
Study groups: 17 hours
Labs: 15 hours

### Supplementary

• #### Signals and systems / Alan V. Oppenheim, Alan S. Willsky, with S. Hamad Nawab.

Oppenheim, Alan V.

2nd ed., Prentice Hall

### Core

• #### Modern control engineering

Ogata, Katsuhiko.

5th ed., International ed., Boston, Mass. ; London : Pearson