Fourier transforms are eternal. They have not changed their nature since the last edition ten years ago: but the intervening time has allowed the author to correct errors in the text and to expand it slightly to cover some other interesting applications. The van Cittert–Zernike theorem makes a belated appearance, for example, and there are hints of some aspects of radio aerial design as interesting applications.

I also take the opportunity to thank many people who have offered criticism, often anonymously and therefore frankly, which has (usually) been acted upon and which, I hope, has improved the appeal both of the writing and of the contents.

History

Fourier transformation is formally an analytic process which uses integral calculus. In experimental physics and engineering, however, the integrand may be a set of experimental data, and the integration is necessarily done artificially. Since a separate integration is needed to give each point of the transformed function, the process would become exceedingly tedious if it were to be attempted manually, and many ingenious devices have been invented for performing Fourier transforms mechanically, electrically, acoustically and optically. These are all now part of history since the arrival of the digital computer and more particularly since the discovery – or invention – of the ‘fast Fourier transform’ algorithm or FFT as it is generally called. Using this algorithm, the data are put (‘read’) into a file (or ‘array’, depending on the computer jargon in use), the transform is carried out, and the array then contains the points of the transformed function. It can be achieved by a software program, or by a purpose-built integrated circuit. It can be done very quickly so that vibration sensitive instruments with Fourier transformers attached can be used for tuning pianos and motor engines, for aircraft and submarine detection and so on. It must not be forgotten that the ear is Nature's own Fourier transformer, and, as used by an expert piano-tuner, for example, is probably the equal of any electronic simulator in the 20–20 000-Hz range.

Fraunhofer diffraction

The application of Fourier theory to Fraunhofer diffraction problems, and to interference phenomena generally, was hardly recognized before the late 1950s. Consequently, only textbooks written since then mention the technique. Diffraction theory, of which interference is only a special case, derives from Huygens' principle: that every point on a wavefront which has come from a source can be regarded as a secondary source; and that all the wave fronts from all these secondary sources combine and interfere to form a new wavefront.

Some precision can be added by using calculus. In Fig. 3.1, suppose that at O there is a source of ‘strength’ q, defined by the fact that at A, a distance r from O, there is a ‘field’, E, of strength E = q/r. Huygens' principle is now as follows:

In elementary Fraunhofer diffraction theory we simplify. We assume the following.

• That only two dimensions need be considered. All apertures bounding the transparent part of the surface S are rectangular and of length unity perpendicular to the plane of the diagram.

• […]

This edition follows much advice and constructive criticism which the author has received from all quarters of the globe, in consequence of which various typos and misprints have been corrected and some ambiguous statements and anfractuosities have been replaced by more clear and direct derivations. Chapter 7 has been largely rewritten to demonstrate the way in which Fourier transforms are used in CAT scanning, an application of more than usual ingenuity and importance: but overall this edition represents a renewed effort to rescue Fourier transforms from the clutches of the pure mathematicians and present them as a working tool to the horny-handed toilers who strive in the fields of electronic engineering and experimental physics.

Showing a Fourier transform to a physics student generally produces the same reaction as showing a crucifix to Count Dracula. This may be because the subject tends to be taught by theorists who themselves use Fourier methods to solve otherwise intractable differential equations. The result is often a heavy load of mathematical analysis.

This need not be so. Engineers and practical physicists use Fourier theory in quite another way: to treat experimental data, to extract information from noisy signals, to design electrical filters, to ‘clean’ TV pictures and for many similar practical tasks. The transforms are done digitally and there is a minimum of mathematics involved.

The chief tools of the trade are the theorems in Chapter 2, and an easy familiarity with these is the way to mastery of the subject. In spite of the forest of integration signs throughout the book there is in fact very little integration done and most of that is at high-school level. There are one or two excursions in places to show the breadth of power that the method can give. These are not pursued to any length but are intended to whet the appetite of those who want to follow more theoretical paths.

The book is deliberately incomplete. Many topics are missing and there is no attempt to explain everything: but I have left, here and there, what I hope are tempting clues to stimulate the reader into looking further; and of course, there is a bibliography at the end.