M8120 Spectral Analysis II

Faculty of Science
Spring 2025
Extent and Intensity
2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
doc. Phuoc Tai Nguyen, PhD (lecturer)
Guaranteed by
doc. RNDr. Martin Kolář, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 17. 2. to Sat 24. 5. Mon 13:00–14:50 M3,01023
  • Timetable of Seminar Groups:
M8120/01: Mon 17. 2. to Sat 24. 5. Mon 15:00–15:50 M3,01023, P. Nguyen
Prerequisites
M7120 Spectral Analysis I
Calculus of complex numbers, Vector and matrix calculus, Linear functional analysis, Basics of Fourier analysis of periodic and nonperiodic functions including convolution operators.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The course aims at providing several advanced methods of spectral analysis. The first part of the course is focused on different transforms, including Z-transform, Laplace transform, Fourier transform, and Radon transform, and their applications in solving difference and differential equations. The second part is devoted to multiresolution analysis and wavelet which have applications in signal processing. The last part covers spectral theory for linear operators which has applications in Quantum Mechanics.
Learning outcomes
At the end of the course students should be able to: - understand and explain discrete analogs to the relevant concepts and operations from Spectral Analysis I, - apply Z-transform, Fourier transform (discreet Fourier transform), Laplace transform and Radon transform to solve difference and differential equations, - master the main techniques and use of multiresolution analysis and wavelets, - understand the spectral theory for linear operators.
Syllabus
  • Z-transform: Definition and properties, Z-transform of element functions, the inverse Z-transform, applications to difference equations.
  • Fourier transform: Quick recall of Fourier transform, uncertainty principle, discreet Fourier transform, fast Fourier transform (FFT), an approximation to the Fourier transform.
  • Laplace transform: Definition and properties, Laplace transform of element functions, inverse Laplace transform, some applications.
  • Radon transform: Definition and properties, relation to Fourier transform.
  • Haar wavelets: introduction of scaling function and wavelet, properties, Haar decomposition and reconstruction algorithms.
  • Multiresolution analysis and wavelets: framework, the scaling relation, the associated wavelet and wavelet spaces, Haar wavelets, Daubechies wavelets.
  • Spectral theory: Spectrum of self-adjoint, compact operators.
Literature
    recommended literature
  • HOWELL, Kenneth B. Principles of Fourier Analysis. Boca Raton-London-New York-Washington: Chapman & Hall, 2001, 776 pp. Studies in Advanced Mathematics. ISBN 0-8493-8275-0. info
    not specified
  • BRIGHAM, E. Oran. Fast Fourier transform. Englewood Cliffs: Prentice Hall, 1974, 252 s. ISBN 0-13-307496-X. info
  • VAN LOAN, Charles. Computational frameworks for the fast fourier transform. Philadelphia: Society for Industrial and Applied Mathematics, 1992, 273 s. ISBN 0-89871-285-8. info
Teaching methods
Lectures, exercises, homeworks
Assessment methods
Exams: One midterm exam and one final exam (or presentation).
Language of instruction
English
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught once in two years.
Teacher's information
The lessons and exam will be in English.
The course is also listed under the following terms Spring 2011 - only for the accreditation, Spring 2001, Spring 2003, Spring 2005, Spring 2007, Spring 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Spring 2021, Spring 2023.
  • Enrolment Statistics (recent)
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