What is the Fourier Transform of a Spatial Point Process?

This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier transform, and what are its spectral moments, second we calculate fou...

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Bibliographic Details
Published in:IEEE transactions on information theory 2023-08, Vol.69 (8), p.1-1
Main Authors: Rajala, Tuomas A., Olhede, Sofia C., Grainger, Jake P., Murrell, David J.
Format: Article
Language:English
Subjects:
Debiased periodogram
Discrete Fourier transforms
Fourier transforms
Questions
Smoothing methods
Spatial point processes
Spectral analysis
Spectral density function
Spectrum analysis
Stochastic processes
Tapering
Time series analysis
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Summary:This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier transform, and what are its spectral moments, second we calculate fourth order moments of the Fourier transform using Campbell's theorem. Third we determine how to implement tapering, an important component for spectral analysis of other stochastic processes. Fourth we answer the question of how to produce an isotropic representation of the Fourier transform of the process. This determines the basic spectral properties of an observed spatial point process.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2023.3269514