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Fourier Transforms

And Convolutions for the Experimentalist

1st Edition - January 1, 1961

Author: R.C. Jennison

eBook ISBN:

9 7 8 - 1 - 4 8 3 1 - 5 5 4 0 - 1

Fourier Transforms and Convolutions for the Experimentalist provides the experimentalist with a guide to the principles and practical uses of the Fourier transformation. It aims… Read more

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Fourier Transforms and Convolutions for the Experimentalist provides the experimentalist with a guide to the principles and practical uses of the Fourier transformation. It aims to bridge the gap between the more abstract account of a purely mathematical approach and the rule of thumb calculation and intuition of the practical worker. The monograph springs from a lecture course which the author has given in recent years and for which he has drawn upon a number of sources, including a set of notes compiled by the late Dr. I. C. Browne from a series of lectures given by Mr. J . A. Ratcliffe of the Cavendish Laboratory. The book begins with an introduction to Fourier Transform. It provides a definition o Fourier Transform, describes its applications, and presents the formal mathematical statement of the transform. Separate chapters discuss the elementary transform, extended functions, and direct applications of Fourier transforms. The final two chapters deal with limitations, products, and convolutions; and the differentiation of Fourier transforms.

Preface

I Introduction

What is a Fourier Transform?

Typical Applications of the Transform

Formal Statement of the Fourier Transform

Convolutions

Notations

The Meaning of Negative Frequencies

Basic Formula

II The Elementary Transform

The Spatial Relationships

Reciprocity

Time and Frequency

The Delta Function

III Extended Functions: The Superposition or Summation Property

The Superposition of Elementary Functions

The Rectangular Function

Complex Distributions

A List of Common Fourier Pairs

IV The Direct Application of Fourier Transforms

Frequency and Time Relationships in Simple Circuits

One Dimensional Aerial Systems

Optical Shots and Gratings

An Optical Example — The Rayleigh Refractometer

Examples in Acoustics

Two Dimensional Fourier Transforms

The Numerical Evaluation of Fourier Transforms

V Limitations, Products and Convolutions

The Effect of Interposing Limits

Convolutions

Physical Interpretation of Convolutions

Some Examples of Convolutions

The Solution of Fourier Transforms by the Application of the Convolution Theorem

The Isosceles Triangle

The Doublet Pulse

Convolutions involving the Sine Integral Si(x)

Two Dimensional Convolutions

VI The Differentiation of Fourier Transforms

Differentiation and Repeated Differentiation

The Fourier Transform of a Step Function or Straight Edge

The Convolution — Differentiation Relationship

The Differential Operator h(x)

The Integral Operator

VII The Auto-Correlation Function and The Transfer Function of a System Linear in Intensity

The Auto-correlation Function

The Transfer Function of a System Linear in Intensity

Examples on Chapter VII:

1. The Telescope

2. Stellar Interferometers

Note on Fourier Synthesis of Apertures

Appendix I Analogue Computers of Fourier Transforms

(i) The Diffraction Computer

(ii) A Mechanical Computer

(iii) A Coherent Electronic Computer

(iv) A Coherent Electronic Computer with an Incoherent Source

Appendix II

Tables of J1X/X, Si(x), sin X/X

Index