# Multidimensional Hartley transforms

Article Abstract:

The study of dispersion in fast Hartley transforms for two and three dimensions requires a frequency analysis be done on each of many three-dimensional harmonics. This requires four-dimensional (space-time) Hartley transforms to be calculated with the least computational effort possible. Running fast Fourier transforms on complex data for each dimension successively generates multidimensional Fourier transforming. Using 'cas' equals cos plus sin as the kernel in Hartley transforms, the same process generates a decomposition into products of cas-functions, one factor for each dimension. Earlier methods can be improved upon by building the Hartley wave up form four one-dimensional transform products rather than eight as may have been presumed. Going from three to four using the technique and then the step shown before requires two plus two extra adds per item.

Publication Name: Proceedings of the IEEE

Subject: Electronics

ISSN: 0018-9219

Year: 1987

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# Multidimensional complex signals with single-orthant spectra

Article Abstract:

Signal and systems theories frequently employ analytic complex signals with real and imaginary parts forming a pair of Hilbert transforms and with one-sided Fourier spectra. The concept of analytic signal is extended to multidimensional signals. The signal parts are a linear combination of the original signal and its Hilbert transforms. The Fourier image exists in a single orthant of the multidimensional frequency space. The orthant is a half-axis in a single dimension, a quadrant in two dimensions, an octant in three dimensions and so forth. Multidimensional complex signals make possible the formulation of a modulation theory of a multidimensional harmonic carrier. A single orthant spectrum is defined as a convolution of the real signal with the multidimensional complex delta distribution.

Publication Name: Proceedings of the IEEE

Subject: Electronics

ISSN: 0018-9219

Year: 1992

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