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Linear Algebra and Adaptive Filters (Forår 2008)

Kursuskode : ELAAF-U01
ECTS Point : 7,5 Status : Tilvalg
Revideret : 01/10 2007 Oprettet : 15/03 2002
Placering : 5-7 semester Timer pr. uge : 4
Længde : 1 semester Undervisningssprog : Ikke valgt
Målsætning : Purpose:
The purpose of the course is twofold. It offers
1)
A solid introduction to linear algebra where the material has been selected to present the theoretical basis of adaptive filters in particular - as well as a range of other technical fields of work in general.
2)
An introduction to the design of adaptive filters which is an important area in modern signal processing: The design of hearing aids, echo cancelling in mobile telephone headsets, the filtering of cockpit or traffic noise in the telecommunication with pilots or policeofficers on motorcycles are a few examples.

The course focuses equally on the mathematical theory and on it’s practical application. The students will be required to make extensive use of related software as well as develop their own.

Learning outcomes:
In the practical context of a project where the task is the design and analysis of adaptive filters the student must demonstrate the ability to
• define and describe the fundamental problems and concepts introduced in the course – using proper mathematical notation when appropriate,
• define and describe the fundamental methods for solution introduced in the course – using proper mathematical notation when appropriate,
• define and describe a feasible algorithm based on a mathematical method for the design of an adaptive filter - allowing subsequent implementation in MATLAB,
• define and motivate a strategy for the verification and testing of a proposed implementation of an algorithm for the design
• apply MATLAB in the development of own software for implementation of the design,
• apply a defined strategy for the verification and testing of the implementation of an algorithm for the design,
• analyze and discuss the relationship between problem, solution method and computed results,
• document the analysis of a designed adaptive filter and the mathematical model behind it.
Hovedindhold : - - -Basic matrix concepts, the four related vector spaces, rank.
- Vector spaces, linear combinations, linear dependence and independence, orthogonal projections, vector norms, orthogonalization.
- Square systems of linear equations, Gauss elimination, partial pivoting, LU- and Cholesky factorizations.
- The SVD (Singular Values Decomposition) of a matrix and it’s applications
- Overdetermined systems, normal equations and least squares solution.
- Eigenvalues- and vectors, diagonalization, quadratic forms, SVD.
- Multivariable functions, partial derivatives, gradients, directional derivatives, Taylor approximations, quadratic forms, deterministic optimization techniques (briefly).
- Random signals and their statistics, Wiener filters, linear prediction, LMS (least mean sqares) adaptive filters, RMS (recursive least mean squares) adaptive filters.

Time allocation approximately: (Pure) linear algebra 60%, adaptive filters 40%.
Undervisningsform : Classroom lectures. Projects and group work on the basis of exercises, assignments and projects using MATLAB on a PC.
Krævede forudsætninger : DSM2 (MAT2), a knowledge of probability theory and random signals on an introductory level, solid experience in the use of MATLAB
Anbefalede forudsætninger : - DSM3 or DSM4 (MAT4)
Relationer : -
Prøveform : Se under bemærkninger
Censur : Ekstern
Bedømmelse : 7-trinsskala
Bemærkninger : The exam is oral, individual, and based on a project assignment. The subject of the project will be the design, implementation, and analysis of adaptive filters. The assignment is handed out 4 weeks before the examination and may be solved individually or in groups as decided by the students. The response report is handed in and evaluated by the examiner and the external censor prior to the examination. The oral examination will mainly focus on the project, but will also include a discussion of (pure) linear algebra subjects not covered explicitly in the project.

- A grade from the 7-step scale is given in agreement with the following main departmental guidelines:
2 for the accomplishment of the task to
define and describe
7 for the additional accomplishment of the
task to explain and apply.
12 for the additional accomplishment of the
task to discuss and analyze

- Two project assignments are given during the semester. Responses to both must be handed in and accepted in order to achieve admission to the exam. The project work is done in groups and will require development of a MATLAB-program for the solution and analysis of a practical problem. The response will be a report presenting the documentation of the following working process:
• A preliminary study of relevant theory.
• Problem statement and delimitation.
• Statement and analysis of the method for solution.
• Implementation plan and development of algorithms for solution, i.e. the logic of
• The development of MATLAB-code for the solution and analysis of the problem.
• Verification, test, and experimental analysis of the implemented solution strategy.
The following two projects are given during the semester:
• The use of the SVD (Singular Value Decomposition) of a matrix for data compression in image processing.
• The design, implementation, and analysis of adaptive filters by simulation of the Wiener solution and use of the LMS (Least Mean Squares) and RLS (Recursive Least Squares) designs.
Undervisningsmateriale : - Notes written particularly for this course by members of the EIT-staff.
(The main reason for using notes is that the course would otherwise require two textbooks: One on Linear Algebra and another on Adaptive filters – which would be costly. However, many excellent books are available and recommendations are given on request.)
The latest version of MATLAB.
Ansvarlig underviser : Hans Pedersen , hchpe@dtu.dk