Fourier Techniques and Applications

1. Front Matter

   Pages i-viii

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2. The Life of Joseph Fourier

   1. The Life of Joseph Fourier

      • R. N. Bracewell

      Pages 1-4

3. Applied Fourier Analysis

   1. Front Matter

      Pages 5-5

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   2. Linear Systems, Filters and Convolution Theorems

      • J. W. Sanders

      Pages 7-24

   3. Uncertainty Principles and Sampling Theorems

      • John F. Price

      Pages 25-44

   4. Fourier Techniques in Two Dimensions

      • R. N. Bracewell

      Pages 45-71

   5. Some Mathematical Methods for Spectrum Estimation

      • John J. Benedetto

      Pages 73-100

   6. Some Practical and Statistical Aspects of Filtering and Spectrum Estimation

      • T. P. Speed

      Pages 101-118

4. Fourier Techniques and Applications

   1. Front Matter

      Pages 119-119

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   2. Two-Dimensional Phase Restoration

      • R. H. T. Bates, W. R. Fright

      Pages 121-148

   3. Fourier Uniqueness Criteria And Spectrum Estimation Theorems

      • John J. Benedetto

      Pages 149-170

   4. Harmonic Synthesis — Theoretical Bounds

      • Gavin Brown

      Pages 171-181

   5. Seismic Image Processing for Petroleum Exploration

      • P. W. Buchen

      Pages 183-183

   6. Fourier Theory in Modern Imaging

      • T. W. Cole

      Pages 185-199

   7. An Overview of Time and Frequency Limiting

      • H. J. Landau

      Pages 201-220

   8. From Rainbows to Rings: A History of the Idea of the Spectrum

      • D. E. Taylor

      Pages 221-224

5. Back Matter

   Pages 225-231

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The first systematic methods of Fourier analysis date from the early eighteenth century with the work of Joseph Fourier on the problem of the flow of heat. (A brief history is contained in the first paper.) Given the initial tempera­ ture at all points of a region, the problem was to determine the changes in the temperature distribution over time. Understanding and predicting these changes was important in such areas as the handling of metals and the determination of geological and atmospheric temperatures. Briefly, Fourier noticed that the solution of the heat diffusion problem was simple if the initial temperature dis­ tribution was sinusoidal. He then asserted that any distri­ bution can be decomposed into a sum of sinusoids, these being the harmonics of the original function. This meant that the general solution could now be obtained by summing the solu­ tions of the component sinusoidal problems. This remarkable ability of the series of sinusoids to describe all "reasonable" functions, the sine qua non of Fourier analysis and synthesis, has led to the routine use of the methods originating with Fourier in a great diversity of areas - astrophysics, computing, economics, electrical engineering, geophysics, information theory, medical engineering, optics, petroleum and mineral exploration, quan­ tum physics and spectroscopy, to name a few.

• Diffusion
• calculus
• distribution
• function
• metals
• processing
• spectroscopy
• theorem

• School of Mathematics, University of New South Wales, Kensington, Australia

  John F. Price

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