MATH 464 (Transform Methods for Scientists and Engineers)
| DESCRIPTION |
Fourier transform, Fourier series, discrete and fast Fourier transform
(DFT and FFT). Laplace transform. Poisson summation, and sampling.
Optional Topics: Distributions and operational calculus,
PDEs, Wavelet transform, Radon transform and Applications such as
Imaging, Speech Processing, PDEs of Mathematical Physics,
Communications, Inverse Problems.
|
| PREREQUISITES |
Math 246 and a 400-level mathematics
or electrical
engineering course, perhaps taken concurrently.
|
| TOPICS |
The course components above are now described more fully.
Fourier Transform Component
Algebraic properties of the Fourier
transform: convolution, modulation, and translation.
Analytic properties of the Fourier transform:
Riemann-Lebesgue Lemma, transforms of derivatives, and
derivatives of transforms.
Inversion theory: Approximate
identities,
L1 inversion, Jordan's theorem, and examples.
The L2 theory: Parseval's
formula, Plancherel's theorem, and examples.
Fourier series component
Representation theory:
Dirichlet's
theorem and examples.
Differentiation and integration
of
Fourier series.
The L1 and L2 theories.
Absolutely convergent Fourier series and
Wiener's
inversion theorem.
Gibbs phenomenon.
Laplace transform component
Review of complex variables.
Algebraic properties of the Laplace transform.
Analytic properties of the Laplace transform:
regions of convergence, transforms of derivatives, and derivatives of
transforms.
Representation and inversion theory of the
Laplace transform.
Evaluation of the complex inversion formula
by residues.
Differential equations component
Applications of Fourier
transforms,
Fourier series, and Laplace transforms to ODE's and PDE's. These
include recent applications in signal processing, classical
applicsations
in mathematical physics, initial and boundary value problems, Bessel
functions,
etc.
Distribution theory component
Motivation, definitions, elementary results,
and examples.
Fourier transforms of distributions.
Convolution equations.
Linear translation invariant systems.
Operational calculus.
DFT and FFT component
Definition and properties of the DFT.
Description of the FFT algorithm, and
examples.
Applications with MATLAB.
Signal processing component
Poisson summation and applications.
The classical sampling theorem.
Uncertainty principle and entropy
inequalities.
Temporal and spectral widths.
Power spectrum: definitions, estimation,
calculations,
and examples.
Maximum entropy and linear prediction.
Wavelet theory and MATLAB component
Shannon wavelets and the classical sampling
theorem.
Multiresolution and analysis wavelet
orthonormal
bases.
Quadrature mirror filters and perfect
reconstruction
filter banks.
Multidimensional results.
Wavelet packets.
Applications with the MATLAB Wavelet Toolbox.
Other transforms component
hankel, Hilbert, Mellin, and Radon transforms.
|
| TEXT |
Text(s)
typically used in this course.
Additional Information on texts:
Harmonic analysis and applications - by J.J.
Benedetto.
The Fourier transform and its applications,
by R.N.
Bracewell
MATLAB Wavelet Toolbox.
Fast Fourier transforms, by J.S. Walker
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