Course detail

# Selected Lectures on Mathematics

The proposed course is a compulsory basic theoretical course profiling the basis, which focuses on the use of selected mathematical methods in the field of space technology. It provides not only a theoretical apparatus for the design and implementation of space applications, but also their practical verification in simulations. The Matlab programming environment will be used for this purpose.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Learning outcomes of the course unit

After completing the course, students should be able to independently solve problems associated with mathematical modeling, verification and testing of designs for space applications.

Prerequisites

Mathematical subjects and knowledge , including the basics of statistics and signal processing, are required.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the way of teaching and are described in Article 7 of the BUT Rules for Studies and Examinations.
Teaching methods include lectures and computer exercises. Student develops individual tasks during computer exercises.

Assesment methods and criteria linked to learning outcomes

The conditions for the successful completion of the course are specified in the annually updated guarantor's order. In general:
- obtaining credit on the basis of a test for 16 points and active participation in exercises for 14 points (max. 30 points, min. 15 points),
- Written parts of the final exam (max 70 points, minimum 35 points)

Course curriculum

Lectures:
1. Vector algebra and analysis
2. Differential geometry
3. Differential calculus of a function of two or more variables (including extrema)
4. Integral calculus of functions of two and more variables (double, triple integrals; use in geometry and physics)
5. Transformation: Z-transformation, KLT, SVD, FFT.
6. Relationship of impulse char, and LTI transfer functions. FIR filters
7. Basics of probability and statistics. Random variable. Moment characteristics.
8. Theory of estimation in general: BLUE, ML, LS. Estimation quality criteria.
9. Theory of estimates and testing (point and interval estimation, testing of moment characteristics).
10. Reliability of systems.
11. Random processes. Stationary, ergodic.
12. Spectral analysis of stochastic signals. Autocorrelation.
13. Detection of signals hidden in noise.

Exercises
1. Vector algebra and analysis
2. Examples from the field of differential geometry
3. Differential calculus of a function of two or more variables (including extrema)
4. Integral number of functions of two or more variables
5. Modeling and use of KLT, SVD, FFT transformations in Matlab.
6. Design of filters and modeling of the relationship between impulse response and transfer function of the system.
7. Test or individual work
8. Modeling of a random variable and calculation of their characteristics.
9. Work with estimates and measurement of their quality.
10. Hypothesis testing: simulation, numerical analysis and testing in Matlab.
11. Simulation of random processes.
12. Spectral analysis of stochastic signals. Autocorrelation.
13. Detection and testing of signals hidden in noise. ROC curve.

Work placements

Not applicable.

Aims

The aim of the course is to present to students a specialized mathematical-statistical apparatus, which is important for understanding and interconnecting the principles of electrical and mechanical systems and practical verification of acquired skills.

Specification of controlled education, way of implementation and compensation for absences

The definition of the controlled education and the way of its implementation are stipulated by the updated guarantor's annual regulation.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

MOON,T., STIRLING, W. Mathematical Methods and Algorithms for Signal Processing, Prentice Hall, New Jersey 2000, pp. 978. ISBN 0-201 -361 86-8 (EN)
KAZ, S. Fundamentals of Statistical signal processing. Estimation Theorz. Prentice Hall, New Jersey, 2011, 595 pp. ISBN 0-13-345711-7 (EN)

JAN, J. Digital Signal Filtering, Analysis and Restoration. volume 44. volume 44. London: The Institution of Electrical Engineers, 2000. 407 s. ISBN: 0-85296-760- 8.
REKTORYS, K. Přehled užité matematiky I, II, Prometheus, Praha 2002, 720 s., ISBN80-7196-181-7 (EN)
JAN, J. Číslicová filtrace, analýza a restaurace signálů. vědecké monografie. vědecké monografie. Brno: VUTIUM Brno, 2002. 427 s. ISBN: 80-214-1558- 4.

Classification of course in study plans

• Programme MPA-SAP Master's, 1. year of study, summer semester, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer