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The Fourier Transform (FT) is an essential mathematical tool in signal processing, transforming a time-domain signal into its frequency-domain representation. This transformation elucidates the relationship between time and frequency domains through several properties, each revealing unique aspects of signal behavior.
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The property of Accumulation in signal processing is derived by analyzing the accumulated sum of a discrete-time signal and using the time-shifting property to determine its z-transform. This principle reveals that the z-transform of the summed signal is related to the z-transform of the original signal by a multiplicative factor.
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The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...
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In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
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On Generalizations of the Nonwindowed Scattering Transform.

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Summary

This study generalizes wavelet scattering transforms using continuous wavelet transforms and nonlinearities. The research provides mathematical norms and proves operators are well-defined and Lipschitz continuous, extending to rotation-invariant and equivariant operators.

Keywords:
Deformation StabilityWavelet Scattering TransformWavelets

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Area of Science:

  • Mathematics
  • Signal Processing
  • Harmonic Analysis

Background:

  • Wavelet scattering transforms are powerful tools for analyzing signals.
  • Existing methods have limitations in handling certain transformations like rotations and diffeomorphisms.
  • Generalizing these transforms is crucial for broader applications in machine learning and data analysis.

Purpose of the Study:

  • To generalize finite depth wavelet scattering transforms.
  • To provide rigorous mathematical formulations and proofs for these generalized transforms.
  • To extend the framework to handle rotational symmetries and actions of diffeomorphisms.

Main Methods:

  • Formulating wavelet scattering transforms as norms of cascaded continuous wavelet transforms and nonlinearities.
  • Developing and proving norms for the resulting operators.
  • Demonstrating Lipschitz continuity with respect to diffeomorphisms in specific settings.
  • Constructing rotation-invariant and rotation-equivariant operators.

Main Results:

  • Generalization of finite depth wavelet scattering transforms.
  • Well-definedness and Lipschitz continuity of the generalized operators under specific conditions.
  • Development of operators invariant and equivariant to rotations.
  • Provides a theoretical foundation for applying scattering transforms in more complex scenarios.

Conclusions:

  • The generalized wavelet scattering transforms offer enhanced capabilities for signal analysis.
  • The theoretical framework supports the development of more robust and versatile signal processing tools.
  • This work paves the way for improved feature extraction in the presence of geometric transformations.