Let f(t) be a real or complex-valued function on the real line R (or on the integers) and square integrable. Think of f(t) as the value of a signal at time t. We want to analyse this signal in ways other than the time-value form t --> f(t) given to us. In particular we will analyse the signal in terms of frequency components and various combinations of time and frequency components. Once we have analysed the signal we may want to alter some of the component parts to eliminate some undesirable features or to compress the signal for more efficient transmission and storage. Finally, we will reconstitute the signal from its component parts.
The three steps are:
Analysis. Decompose the signal into basic components. We will think of the signal space as a vector space and break it up into a sum of subspaces, each of which captures a special feature of a signal.
Processing.Modify some of the basic components of the signal that were obtained through the analysis, or correlate with other information.
We will look at several methods for analysis:
Almost all of these methods are based on the decomposition of the Hilbert space of square integrable functions into orthogonal subspaces. Group representation theory lies at the core of the applied topics, particularly in its relationship to multi-scale analysis and self-similarity, and I will develop a bit of group theory as needed.
- Fourier series
- The Fourier integral
- Discrete Fourier transforms
- Windowed Fourier transforms
- Continuous wavelet transforms
- Discrete wavelet transforms (e.g., Daubechies wavelets)
- Ambiguity functions (signal correlation)
- Fractal transforms
TRANSFORM |
GROUP |
Fourier series | circle group |
Fourier integral | line group |
windowed Fourier transform | Heisenberg group |
discrete windowed Fourier transform | lattice subgroup |
continuous wavelet transform | ax+b group |
discrete wavelet transform | discrete semisubgroup of ax+b |
discrete Fourier transform (DFT) | cyclic group C_n |
(narrow band) RADAR ambiguity function | Heisenberg group |
(wide band) RADAR ambiguity function | ax+b group |
fractal transform | semigroup of "words" |
spherical harmonics |
rotation group |
Syllabus:
O. Motivation through signal processing, analysis and synthesis
I. Vector Spaces with Inner Product
Metrics and norms. Completeness
Hilbert spaces
Orthogonal projections
L² and l². The Lebesgue integral. Lebesgue
measure. Dominated convergence theorem.
Operators on Hilbert spaces. Riesz representation theorem. Contraction mappings.
II. Brief review of definitions and convergence results for Fourier series and integrals
IV. Radar Ambiguity Functions
V. Bases and Frames
VI. Windowed Fourier Transforms
VII. The Continuous Wavelet transform
VIII. Multiresolution Analysis and the Discrete Wavelet Transform
Haar wavelets as motivation,
Scaling functions, The dilation equation, The wavelet equation
Cascade Algorithm
Daubechies wavelets
Scaling Function by recursion, Evaluation at dyadic points
Infinite product formula for the scaling function
Accuracy of approximation, Convergence
Smoothness of scaling functions and wavelets
IX. Fractal Imaging
The Hausdorff metric.
Iterated fractal transforms
VIII. Other Topics
Biorthogonal Filters and Wavelets
Multifilters and Multiwavelets
IX. Applications of Wavelets
Denoising, compression, image processing, etc.
Applications to PDEs and numerical simulation.
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MATLAB demos and examples.
For course credit, there will be homework exercises.