Fourier Transforms

Fourier Transforms

And Convolutions for the Experimentalist

1st Edition - January 1, 1961

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  • Author: R.C. Jennison
  • eBook ISBN: 9781483155401

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Description

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.

Table of Contents


  • 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

Product details

  • No. of pages: 128
  • Language: English
  • Copyright: © Pergamon 1961
  • Published: January 1, 1961
  • Imprint: Pergamon
  • eBook ISBN: 9781483155401

About the Author

R.C. Jennison

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