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

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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.

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Data Availability

The FDM MATLAB code is publicly available for download at https://www.researchgate.net/publication/319877224_MATLABCodeOfFDMforImageDecomposition, and A MATLAB code for the proposed analytic image and 2D Hilbert transform computation by 2D-FFT is included in Appendix of the paper itself.

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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.

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Authors and Affiliations

  1. Department of ECE, School of Engineering and Applied Sciences, Bennett University, Greater Noida, India

    Pushpendra Singh

  2. Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India

    Shiv Dutt Joshi

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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.

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Correspondence to Pushpendra Singh.

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Appendices

1  An Analytic Image Computation Using the 2D-DFT

In this appendix, we derive analytic image representation by 2D-DFT as follows. Let g[m, n] be a real-valued function, then the 2D-DFT of g[m, n] is defined as

$$\begin{aligned} G[k,l]=\frac{1}{MN}\sum _{m=0}^{M-1} \sum _{n=0}^{N-1} g[m,n] e^{-j(\frac{km}{M}+\frac{ln}{N})} \end{aligned}$$
(40)

and 2D-IDFT is defined as

$$\begin{aligned} g[m,n]=\sum _{k=0}^{M-1} \sum _{l=0}^{N-1} G[k,l] e^{j(\frac{km}{M}+\frac{ln}{N})}. \end{aligned}$$
(41)

In (40), for odd numbers (\(M=N\)), there is only one real term G[0, 0] and \((MN-1)/2\) terms are complex conjugate of the rest \((MN-1)/2\) terms; and for even numbers (\(M=N\)), there are only four real terms G[0, 0], G[0, N / 2], G[M / 2, 0], G[M / 2, N / 2] and \((MN-4)/2\) terms are complex conjugate of the rest \((MN-4)/2\) terms. From the above discussions and using the conjugate symmetry of 2D-DFT, we obtain 2D-AS for odd numbers (\(M=N\)) as

$$\begin{aligned} z_{14}[m,n]= & {} 2\sum _{k=0}^{(M-1)/2} \sum _{l=0}^{{(N-1)}/{2}} G[k,l] e^{j\left( \frac{km}{M}+\frac{ln}{N}\right) } \nonumber \\&+ 2\sum _{k=(M+1)/2}^{(M-1)} \sum _{l=1}^{{(N-1)}/{2}} G[k,l] e^{j(\frac{km}{M}+\frac{ln}{N})}-G(0,0) \end{aligned}$$
(42)

and for even numbers (\(M=N\)) as

$$\begin{aligned} z_{14}[m,n]= & {} 2\sum _{k=0}^{M/2} \sum _{l=0}^{{N}/{2}} G[k,l] e^{j\left( \frac{km}{M}+\frac{ln}{N}\right) }+2\sum _{k=M/2+1}^{M-1} \sum _{l=1}^{{N}/{2}-1} G[k,l] e^{j\left( \frac{km}{M}+\frac{ln}{N}\right) }\nonumber \\&-G[0,0] -G[0,N/2]-G[M/2,0]-G[M/2,N/2], \end{aligned}$$
(43)

where real part of AS is original signal (i.e., \(g[m,n]=Re\{z_{14}[m,n]\}\)) and imaginary part of AS is the HT of original signal (i.e., \(\hat{g}[m,n]=Im\{z_{14}[m,n]\}\)). This 2D-AS has been obtained by considering the first and fourth quadrants of 2D DFT. These 2D-AS (analytic image) computation and 2D-HT can be easily implemented with 2D-FFT algorithms, e.g., A MATLAB implementation is presented in Algorithm 1.

figure a
figure b

2  MD-FS, MD-HT and MD-AS Representation

The 2D-FS discussion presented in Sect. 2 can be easily extended for MD-FS as follows

$$\begin{aligned} g(x_1,\ldots ,x_M)&=\sum _{k_M=-\infty }^{\infty } \ldots \sum _{k_{3}=-\infty }^{\infty } \sum _{k_{2}=0}^{\infty } \sum _{k_{1}=0}^{\infty } \big [a_{k_1,\ldots ,k_M} \cos (k_1\omega _1x_1+\ldots +k_M\omega _Mx_M)\nonumber \\&\qquad +\,b_{k_1,\ldots ,k_M} \sin (k_1\omega _1x_1+\ldots +k_M\omega _Mx_M)\big ]\nonumber \\&\qquad +\,\sum _{k_M=-\infty }^{\infty } \ldots \sum _{k_{3}=-\infty }^{\infty } \sum _{k_{2}=-\infty }^{-1} \sum _{k_{1}=1}^{\infty } \big [a_{k_1,\ldots ,k_M} \cos (k_1\omega _1x_1+\ldots +k_M\omega _Mx_M)\nonumber \\&\qquad +\,b_{k_1,\ldots ,k_M} \sin (k_1\omega _1x_1+\ldots +k_M\omega _Mx_M)\big ], \end{aligned}$$
(44)
$$\begin{aligned} g(x_1,\ldots ,x_M)= & {} \sum _{k_M=-\infty }^{\infty } \ldots \sum _{k_{2}=-\infty }^{\infty } \sum _{k_{1}=-\infty }^{\infty } c_{k_1,\ldots ,k_M} \exp [j(k_1\omega _1x_1+\ldots +k_M\omega _Mx_M)],\nonumber \\ c_{k_1,\ldots ,k_M}= & {} \frac{1}{T_1\ldots T_M} \int _{-\frac{T_M}{2}}^{\frac{T_M}{2}} \ldots \int _{-\frac{T_1}{2}}^{\frac{T_1}{2}} g(x_1,\ldots ,x_M)\nonumber \\&\quad \exp [-j(k_1\omega _1x_1+\ldots +k_M\omega _Mx_M)] \,\mathrm {d}x_1 \ldots \,\mathrm {d}x_M, \end{aligned}$$
(45)

where \(c_{k_1,\ldots ,k_M}=(a_{k_1,\ldots ,k_M}-jb_{k_1,\ldots ,k_M})/2\). We can easily obtain MD-HT by replacing \(\cos \) with \(\sin \) and \(\sin \) with \(-\cos \) in (44) and obtain AS such that real part of AS is original signal, AS has positive resultant frequency \(\omega =|\varvec{\omega }|=\sqrt{(k_1\omega _1)^2+\ldots +(k_M\omega _M)^2}\) and its FFV can be written as \(\varvec{\omega }=\begin{bmatrix}k_1\omega _1&\ldots&k_M\omega _M \end{bmatrix}^{T}\). The prowess and efficacy of the Fourier theory can be realized from the fact that the HT and analytic representation of a signal are, inherently, present in the Fourier representation.

Observation We can also use (44) for \((M-1)\)D space-time series \((x_1,\ldots ,x_{(M-1)},t)\) analysis, e.g., 2D wave equation, \(g(x,y,t)=\cos (k_1\omega _1 t-k_2\omega _2 x-k_3\omega _3 y)\), where \(k_1\omega _1=\frac{2\pi k_1}{T_1}=2\pi k_1f_1=2\pi f\), wave vector (or FFV) \(\varvec{\omega }=\begin{bmatrix}k_2\omega _2&k_3\omega _3 \end{bmatrix}^{T}\), wave number (or spatial frequency) \(|\varvec{\omega }|=\sqrt{(k_2\omega _2)^2+(k_3\omega _3)^2}=\frac{2\pi }{\lambda }\) and phase velocity \(v_p=\frac{k_1\omega _1}{|\varvec{\omega }|}=f\lambda \).

For non-periodic MD signal, \(g(x_1,\ldots ,x_M)\), the MD-FT and MD inverse FT (MD-IFT) are defined as

$$\begin{aligned} \begin{aligned}&C[f_1,\ldots ,f_M]\\&\quad =\int _{-\infty }^{\infty } \ldots \int _{-\infty }^{\infty } g(x_1,\ldots ,x_M) \exp [-j(\omega _1x_1+\ldots + \omega _Mx_M)] \,\mathrm {d}x_1 \ldots \,\mathrm {d}x_M,\\&g(x_1,\ldots ,x_M)\\&\quad =\int _{-\infty }^{\infty } \ldots \int _{-\infty }^{\infty } C[f_1,\ldots ,f_M] \exp [j(\omega _1x_1+\ldots +\omega _Mx_M)]\,\mathrm {d}f_1 \ldots \,\mathrm {d}f_M, \end{aligned} \end{aligned}$$
(46)

respectively. We define MD-AS, for real-valued signal \(g(x_1,\ldots ,x_M)\), as

$$\begin{aligned}&z(x_1,\ldots ,x_M)\nonumber \\&\quad =2\int _{-\infty }^{\infty } \ldots \int _{-\infty }^{\infty } \int _{0}^{\infty } C[f_1,\ldots ,f_M] \exp [j(\omega _1x_1+\ldots +\omega _Mx_M)]\,\mathrm {d}f_1 \ldots \,\mathrm {d}f_M, \end{aligned}$$
(47)

where its real part is original signal and imaginary part is HT of original signal.

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Singh, P., Joshi, S.D. Some Studies on Multidimensional Fourier Theory for Hilbert Transform, Analytic Signal and AM–FM Representation. Circuits Syst Signal Process 38, 5623–5650 (2019). https://doi.org/10.1007/s00034-019-01133-x

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