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Sampling Trajectories for the Short-Time Fourier Transform.

Michael Speckbacher1

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

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|October 24, 2022
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Summary

This study explores stable reconstruction of the short-time Fourier transform (STFT) from trajectory samples. Researchers found that specific spiraling curves enable stable STFT reconstruction using Hermite function-based windows.

Keywords:
Gabor analysisMobile samplingPolyanalytic functionsSampling theory

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Area of Science:

  • Signal Processing
  • Harmonic Analysis
  • Time-Frequency Analysis

Background:

  • The Short-Time Fourier Transform (STFT) is crucial for analyzing non-stationary signals.
  • Stable reconstruction of STFT from discrete samples is a challenging problem.
  • Understanding the relationship between sampling trajectories and signal reconstruction is vital.

Purpose of the Study:

  • To investigate the stable reconstruction of STFT from trajectory samples.
  • To analyze the impact of trajectory density on reconstruction properties.
  • To identify specific trajectories, like spiraling curves, that facilitate stable STFT reconstruction.

Main Methods:

  • Analysis of sampling properties on general trajectories.
  • Focus on the specific case of spiraling curves.
  • Utilizing window functions as linear combinations of Hermite functions.

Main Results:

  • Established a connection between trajectory density and STFT reconstruction stability.
  • Characterized sampling and uniqueness properties for spiraling curves.
  • Demonstrated stable STFT reconstruction from samples on specific spiraling curves using Hermite function windows.

Conclusions:

  • The choice of sampling trajectory significantly impacts STFT reconstruction.
  • Spiraling curves offer a promising framework for stable STFT reconstruction.
  • Hermite function windows are effective for stable STFT reconstruction on these specific curves.