Known in the signal processing literature mainly as a computational technique, the Hartley transform has proven to possess attributes in the physical world of experiment, especially in optics and microwaves. The Fourier transform, which has constituted the standard approach to spectral analysis, is understood both in the domain of numerical analysis and in the world of natural phenomena, gaining its central significance from the ubiquity of sinusoidal natural behavior; it also gains appeal from the convenience offered by complex variable analysis, which is a basic part of technical education. Computers, however, prefer real numbers, an attribute that first opened a niche for the Hartley transform. As applications have multiplied, and others open up, it will be helpful to understand the Hartley idea from more than one point of view. Several topics, including multidimensional transforms, the complex generalization, and microwave phase measurement by amplitude measurement only, are discussed with the intention of throwing light on the numerical and physical relationships between the Hartley and Fourier transforms.>

Many well-known theorems for the two-dimensional (2D) Hartley transform are encapsulated in the affine theorem which is presented here and illustrated by example. The mode of presentation permits ready visualization of the effect on the spectral makeup when an object is subjected to a particular distortion as specified by the six disposable parameters of the affine coordinate transformation and of the effects of progressive distortion. By contrast, 2D Fourier transform pairs for images do not lend themselves to visualization in the same way. Affine transformations possess considerable flexibility and generality of a useful kind that arises from the property that a single 2D spectral component transforms into another single component. >

The author describes how Maxwell's equations are indebted to observations of Mars, Jupiter, and Saturn. The electromagnetic relations were derived in terms of a medium with 'small but real density' and with the capability of yielding elastically, but the mechanical origin to which the curl and div equations trace back is a surprise to many engineering students. Maxwell developed his ability to deal with waves by starting his career with a four-year study of the rings of Saturn, which constitute a rare, elastic medium subject to internal inverse-square forces. Applying the laws of motion, Maxwell deduced that electricity and magnetism should combine to support what he called an electromagnetic wave, whose velocity in that medium he could calculate. Meanwhile, observations of Jupiter's satellites revealed that light has a finite velocity. Knowing this value, Maxwell was able to propose that light itself might be electromagnetic. These foundations of modern electrical engineering thus rest firmly on astronomical observation of Mars, Jupiter, and Saturn. Astronomy has continued to contribute to many branches of electrical engineering.>

A three-dimensional (3-D) Discrete Fourier Transform (DFT) algorithm for real data using the one-dimensional Fast Hartley Transform (FHT) is introduced. It requires the same number of one-dimensional transforms as a direct FFT approach but is simpler and retains the speed advantage that is characteristic of the Hartley approach. The method utilizes a decomposition of the cas function kernel of the Hartley transform to obtain a temporary transform, which is then corrected by some additions to yield the 3-D DFT. A Fortran subroutine is available on request.

The fast Hartley transform algorithm introduced in 1984 offers an alternative to the fast Fourier transform, with the advantages of not requiring complex arithmetic or a sign change of i to distinguish inverse transformation from direct. A two-dimensional extension is described that speeds up Fourier transformation of real digital images.