CONTINUOUS AND DISCRETE WAVELET TRANSFORMS

CONTINUOUS AND DISCRETE WAVELET TRANSFORMS

December 1989 | CHRISTOPHER E. HEIL† AND DAVID F. WALNUT†
This paper provides an expository survey of integral representations and discrete sum expansions of functions in \( L^2(\mathbf{R}) \) using coherent states. Two types of coherent states are considered: Weyl-Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, known as "wavelets," which arise from translations and dilations of a single function. The paper discusses how to represent any function in \( L^2(\mathbf{R}) \) as a sum or integral of these states. It surveys the literature, particularly the work of I. Daubechies, A. Grossmann, and J. Morlet, and includes some new results by the authors. The paper begins with an introduction that highlights the importance of representing signals in terms of their spectrum or Fourier transform, but notes that this representation is not always the most natural or useful. It introduces two methods for achieving time-dependent frequency analysis: the Gabor transform and the wavelet transform. The Gabor transform, named after D. Gabor, involves dividing a signal into short segments and computing the Fourier coefficients of each segment, providing a time-frequency localization technique. The wavelet transform, on the other hand, involves translations and dilations of a mother wavelet, offering a more localized representation in both time and frequency. The continuous versions of these transforms are discussed in detail, showing how they arise as representations of groups on \( L^2(\mathbf{R}) \). The Feichtinger-Gröchenig theory is outlined, demonstrating how any representation can give rise to a discrete transform. The discrete Gabor and wavelet transforms are then described, including the construction of frames and the use of approximate identities. The paper concludes with a summary of the key concepts and results, emphasizing the deep connection between the Gabor and wavelet transforms through the theory of group representations.This paper provides an expository survey of integral representations and discrete sum expansions of functions in \( L^2(\mathbf{R}) \) using coherent states. Two types of coherent states are considered: Weyl-Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, known as "wavelets," which arise from translations and dilations of a single function. The paper discusses how to represent any function in \( L^2(\mathbf{R}) \) as a sum or integral of these states. It surveys the literature, particularly the work of I. Daubechies, A. Grossmann, and J. Morlet, and includes some new results by the authors. The paper begins with an introduction that highlights the importance of representing signals in terms of their spectrum or Fourier transform, but notes that this representation is not always the most natural or useful. It introduces two methods for achieving time-dependent frequency analysis: the Gabor transform and the wavelet transform. The Gabor transform, named after D. Gabor, involves dividing a signal into short segments and computing the Fourier coefficients of each segment, providing a time-frequency localization technique. The wavelet transform, on the other hand, involves translations and dilations of a mother wavelet, offering a more localized representation in both time and frequency. The continuous versions of these transforms are discussed in detail, showing how they arise as representations of groups on \( L^2(\mathbf{R}) \). The Feichtinger-Gröchenig theory is outlined, demonstrating how any representation can give rise to a discrete transform. The discrete Gabor and wavelet transforms are then described, including the construction of frames and the use of approximate identities. The paper concludes with a summary of the key concepts and results, emphasizing the deep connection between the Gabor and wavelet transforms through the theory of group representations.
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