  In this paper, we propose the Fourier frequency vector (FFV), inherently,
associated with multidimensional Fourier transform. With the help of FFV, we
are able to provide physical meaning of so called negative frequencies in
multidimensional Fourier transform (MDFT), which in turn provide
multidimensional spatial and space-time series analysis. The complex
exponential representation of sinusoidal function always yields two
frequencies, negative frequency corresponding to positive frequency and vice
versa, in the multidimensional Fourier spectrum. Thus, using the MDFT, we
propose multidimensional Hilbert transform (MDHT) and associated
multidimensional analytic signal (MDAS) with following properties: (a) the
extra and redundant positive, negative, or both frequencies, introduced due to
complex exponential representation of multidimensional Fourier spectrum, are
suppressed, (b) real part of MDAS is original signal, (c) real and imaginary
part of MDAS are orthogonal, and (d) the magnitude envelope of a original
signal is obtained as the magnitude of its associated MDAS, which is the
instantaneous amplitude of the MDAS. The proposed MDHT and associated DMAS are
generalization of the 1D HT and AS, respectively. We also provide the
decomposition of an image into the AM-FM image model by the Fourier method and
obtain explicit expression for the analytic image computation by 2DDFT.
