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                                                                                                                                                                          N I V E
                          FOURIER BOOKLET                                                                                                                             U              -1R
                                                                                                                                                                  E                         S I
                                                                                                                                                               T H                              T Y
                          School of Physics                                                                                                                 O                                       H
                                                                                                                                                              F                                  G
                                                                                                                                                                   E D                        R
                                                                                                                                                                           I   N B U
                                                              TheFourierTransform
                                                          (Whatyouneedtoknow)
                               Mathematical Background for:
                                                               Senior Honours                   ModernOptics
                                                               Senior Honours                   Digital Image Analysis
                                                               Senior Honours                   Optical Laboratory Projects
                                                               MSc                              Theory of Image Processing
                                                                                        Session: 2007-2008
                                                                                             Version: 3.1.1
                           School of Physics                                               Fourier Transform                            Revised: 10 September 2007
                         FOURIER BOOKLET                                                                                                                                    -1
                         Contents
                         1 Introduction                                                                                                                                       2
                               1.1      Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                  2
                         2 TheFourierTransform                                                                                                                                3
                               2.1      Properties of the Fourier Transform                          .  . . . . . . . . . . . . . . . . . . . . . .                           4
                               2.2      TwoDimensionalFourier Transform . . . . . . . . . . . . . . . . . . . . . . .                                                         5
                               2.3      TheThree-Dimensional Fourier Transform . . . . . . . . . . . . . . . . . . . .                                                        6
                         3 DiracDeltaFunction                                                                                                                                 7
                               3.1      Properties of the Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . .                                                    8
                               3.2      TheInfinite Comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                     9
                         4 SymmetryConditions                                                                                                                               10
                               4.1      One-DimensionalSymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                       11
                               4.2      Two-DimensionalSymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                       12
                         5 ConvolutionofTwoFunctions                                                                                                                        13
                               5.1      Simple Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                 14
                               5.2      TwoDimensionalConvolution . . . . . . . . . . . . . . . . . . . . . . . . . .                                                       14
                         6 Correlation of Two Functions                                                                                                                     15
                               6.1      Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                 16
                         7 Questions                                                                                                                                        17
                               7.1      Thesinc() function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                  17
                               7.2      Rectangular Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                  18
                               7.3      Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                 19
                               7.4      Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                               21
                               7.5      Delta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                 22
                               7.6      Sines and Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                 23
                               7.7      CombFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                    24
                               7.8      Convolution Theorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                    25
                               7.9      Correlation Theorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                  27
                               7.10 Auto-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                      28
                          School of Physics                                            Fourier Transform                         Revised: 10 September 2007
                -2                                                                        FOURIER BOOKLET
                1 Introduction
                Fourier Transform theory is essential to many areas of physics including acoustics and signal
                processing, optics and image processing, solid state physics, scattering theory, and the more
                generally, in the solution of differential equations in applications as diverse as weather model-
                ingtoquantumfieldcalculations. TheFourierTransformcaneitherbeconsideredasexpansion
                in terms of an orthogonal bases set (sine and cosine), or a shift of space from real space to re-
                ciprocal space. Actually these two concepts are mathematically identical although they are
                often used in very different physical situations.
                Theaimofthisbookletistocover the Fourier Theory required primarily for the
                   • Junior Honours course OPTICS.
                   • Senior Honours course MODERN OPTICS1 and DIGITAL IMAGE ANALYSIS
                   • Geoscience MSc course THEORY OF IMAGE PROCESSING.
                It also contains examples from acoustics and solid state physics so should be generally useful
                for these courses. The mathematical results presented in this booklet will be used in the above
                courses and they are expected to be known.
                There are a selection of tutorial style questions with full solutions at the back of the booklet.
                These contain a range of examples and mathematical proofs, some of which are fairly difficult,
                particularly the parts in italic. The mathematical proofs are not in themselves an examinal part
                of the lecture courses, but the results and techniques employed are.
                Further details of Fourier Transforms can be found in “Introduction to the Fourier Transform
                and its Applications” by Bracewell and “Mathematical Methods for Physics and Engineering”
                by Riley, Hobson & Bence.
                1.1   Notation
                Unlike many mathematical field of science, Fourier Transform theory does not have a well
                defined set of standard notations. The notation maintained throughout will be:
                                           x,y → RealSpaceco-ordinates
                                           u,v → FrequencySpaceco-ordinates
                and lower case functions (eg f(x)), being a real space function and upper case functions (eg
                F(u)), being the corresponding Fourier transform, thus:
                                                   F(u) = F {f(x)}
                                                    f(x)  = F−1{F(u)}
                where F {} is the Fourier Transform operator.
                                                       √
                The character ı will be used to denote   −1, it should be noted that this character differs from
                the conventional i (or j). This slightly odd convention and is to avoid confusion when the
                digital version of the Fourier Transform is discussed in some courses since then i and j will be
                used as summation variables.
                   1not offered in 2006/2007session.
                Revised: 10 September 2007             Fourier Transform                      School of Physics
                FOURIER BOOKLET                                                                                 -3
                                               1                            sinc(x)
                                              0.8
                                              0.6
                                              0.4
                                              0.2
                                               0
                                             -0.2
                                             -0.4
                                               -10       -5        0        5        10
                                                  Figure 1: The sinc() function.
                Twospecial functions will also be employed, these being sinc() defined2 as,
                                                         sinc(x) = sin(x)                                      (1)
                                                                      x
                giving sinc(0) = 13 and sinc(x0) = 0 at x0 = ±π, ±2π,..., as shown in figure 1. The top hat
                function Π(x), is given by,
                                                  Π(x) = 1         for |x| ≤ 1/2                               (2)
                                                         = 0       else
                being a function of unit height and width centered about x = 0, and is shown in figure 2
                                              1.2
                                               1
                                              0.8
                                              0.6
                                              0.4
                                              0.2
                                               0-2  -1.5 -1  -0.5  0   0.5  1   1.5   2
                                                  Figure 2: The Π(x) function
                2 TheFourierTransform
                Thedefinitionofaonedimensionalcontinuousfunction,denotedby f(x),theFouriertransform
                is defined by:
                                               F(u)=Z ∞ f(x)exp(−ı2πux)dx                                      (3)
                                                         −∞
                   2The sinc() function is sometimes defined with a “stray” 2π, this has the same shape and mathematical prop-
                erties.
                   3See question 1
                 School of Physics                       Fourier Transform          Revised: 10 September 2007
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...N i v e fourier booklet u r s t h y school of physics o f g d b thefouriertransform whatyouneedtoknow mathematical background for senior honours modernoptics digital image analysis optical laboratory projects msc theory processing session version transform revised september contents introduction notation properties the twodimensionalfourier thethree dimensional diracdeltafunction dirac delta function theinnite comb symmetryconditions one dimensionalsymmetry two convolutionoftwofunctions simple twodimensionalconvolution correlation functions autocorrelation questions thesinc rectangular aperture gaussians differentials sines and cosines combfunction convolution theorm auto is essential to many areas including acoustics signal optics solid state scattering more generally in solution differential equations applications as diverse weather model ingtoquantumeldcalculations thefouriertransformcaneitherbeconsideredasexpansion terms an orthogonal bases set sine cosine or a shift space from rea...

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