# Some Studies on Multidimensional Fourier Theory for Hilbert Transform, Analytic Signal and AM–FM Representation

## Abstract

In this paper, we propose the notion of Fourier frequency vector (FFV) which is inherently associated with the multidimensional (MD) Fourier representation (FR) of a signal. The proposed FFV provides physical meaning to the so-called negative frequencies in the MD-FR that in turn yields MD spatial and MD space-time series analysis. The one-dimensional Hilbert transform (1D-HT) and associated 1D analytic signal (1D-AS) of an 1D signal are well established; however, their true generalization to an MD signal, which possess all the properties of 1D case, are not available in the literature. To achieve this, we observe that in MD-FR the complex exponential representation of a sinusoidal function always yields two frequencies, namely negative frequency corresponding to positive frequency and vice versa. Thus, using the MD-FR, we propose MD-HT and associated MD analytic signal (AS) as a true generalization of the 1D-HT and 1D-AS, respectively, and obtain an explicit expression for the analytic image computation by 2D discrete Fourier transform (2D-DFT). We also extend the Fourier decomposition method for 2D signals that decomposes an image into a set of amplitude-modulated and frequency-modulated (AM–FM) image components. We finally propose a single-orthant Fourier transform (FT) of real MD signals which computes FT in the first orthant, and values in rest of the orthants are obtained by simple conjugation defined in this study.

## Keywords

Fourier representation (FR) and Fourier frequency vector Hilbert transform (HT) and analytic signal (AS) Single-orthant Fourier transform (SOFT) Fourier decomposition method (FDM) Linearly independent non-orthogonal yet energy-preserving (LINOEP) vectors## Notes

### Acknowledgements

Authors would like to thank the editors and anonymous reviewers for their constructive and thorough comments and suggestions which improved the presentation of manuscript.

### Author Contributions

P. Singh conceived and designed the study, carried out the simulation work, participated in data analysis, and drafted the manuscript; S. D. Joshi discussed and checked the mathematical analyses, coordinated the study and helped in drafting the manuscript. All authors commented and gave final approval for publication.

### Funding

There is no funding to support this research.

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Permission to Carry out Fieldwork

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## Supplementary material

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