ECE Course Outline

ECE6500

Fourier Techniques and Signal Analysis (3-0-3)

Prerequisites
None
Corequisites
None
Catalog Description
Introduction to the use of Fourier Methods for analysis of signals.
Textbook(s)
Bracewell, The Fourier Transform and Its Application (3rd edition), McGraw Hill, 1999. ISBN 9780073039381 (required)

Topical Outline
1.   Groundwork (Fourier's integral theorem, oddness and evenness, complex 
conjugates, sine and cosine transforms, transforms in the limit)
2.   Convolution (skill with graphical techniques, serial products, the 
autocorrelation function, cross-correlation, energy spectrum)
3.   Notation for Useful Functions (rectangle, triangle, Gaussian, etc.)
4.   The Impulse Symbol (models, sifting property, sampling comb, null 
functions, particularly well-behaved functions, generalized functions)
5.   Basic Theorems (brief review of basic theorems of Fourier transform 
theory)
6.   The Two Domains (definite integral, moments, mean-square abscissa, 
variance, smoothness and compactness, width measures, upper limits to 
ordinate and slope, Schwarz's inequality, uncertainty relation, 
central-limit theorem)
7.   Sampling and Series (sampling theorem and variations, interpolation, 
undersampling, ordinate and slope sampling, interlaced sampling, sampling 
in the presence of noise, relation between Fourier series and Fourier 
integral transform)
8.   The Discrete Fourier Transform (relationship to continuous-domain 
Fourier transform, cyclic convolution, application to real-world signals)
9.   Dynamic Spectra and Wavelets (short-time amplitude spectra, Gabor 
time-frequency analysis, time-frequency energy density function, Wigner 
distribution, ambiguity function, wavelets and wavelet decomposition)
10.  Relatives of the Fourier transform (Laplace, Hankel, Mellin, Abel, Z, 
Hartley, and Hilbert transforms, analytic and complex signal 
representations)
11.  Two-dimensional Signals (2-D Fourier transform, polar coordinates 
transform, 2-D sampling and replication, impulses in two dimensions)
12.  Applications (antennas and optics, television image formation, 
statistics and noise waveforms, heat conduction and diffusion)