Introduction

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.

Synthesis. Reconstitute the signal from its (altered) component parts. An important requirement we will make  is perfect  reconstruction. If we don't alter the component parts, we want the synthesised signal to agree exactly with the original signal.
 

We will look at several methods for analysis:

  1. Fourier series
  2. The Fourier integral
  3. Discrete Fourier transforms
  4. Windowed Fourier transforms
  5. Continuous wavelet transforms
  6. Discrete wavelet transforms (e.g., Daubechies wavelets)
  7. Ambiguity functions (signal correlation)
  8. Fractal transforms
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.
 
 
Associations between transforms and groups
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.