SE 379. Fourier Analysis, Transforms and Applications
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INTRODUCTION: Who was Fourier?

PRELIMINARIES:

  1. Hilbert Spaces: Orthonormal Bases, Operators on Hilbert Spaces. The spaces L2[0,1], L2(z). Examples of Orthonormal Bases: the Fourier series, the Hear System.
  2. Cesaro Summability, Plancherel theorem, computing Fourier Coefficients
  3. Fourier Transforms, Placherel and Inversion Theorems (d) Schwarz Class Functions and other dense subspaces. Sobolev Spaces - different norms.
  4. Fast Fourier Transform.
  5. Poisson summation Formula.
  6. Shannon sampling Theorem.                                                         [20]

The 'global' character of the Fourier Transform. Time, frequency, and Scale. Localization. The Gabor Transform and the Short time Fourier Transform. Scaling Functions and Wavelets. The multiresolution concept. Discrete and continuous wavelet transforms. Filter Banks and trees of filter banks.                                     [ 7 ]

Other orthogonal transforms and fast transform computation. The computational complexity of a transform. Symmetries in the Fourier Transform, and their exploitation: the Fast Fourier transform.
Associated real transforms: the cosine and sine transforms and their fast computation. Dyadic transforms: the Walsh Hadamard transform and its fast computation. The fast wavelet transform.                 [ 4 ]

Spectrum Estimation: the problem of estimation of spectra from finite-duration observations. The periodogram. Frequency resolution. Nonparametric estimation: Bartlett, Welch and Blackman-Tukey estimates.
Performance of nonparametric estimation techniques. Parametric estimation: AR, MA and ARMA modeling, maximum entropy estimation. Minimum variance spectral estimation.                        [ 9 ]

REFERENCES

Proposed by:
Dr. K S Venkatesh, Department of Electrical Engg.
Dr. S. Madan, Department of Mathematics

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