Fourier Analysis for Beginners Indiana University School of Optometry Coursenotes for V791: Quantitative Methods for Vision Research (Sixth edition) © L.N. Thibos (1989, 1993, 2000, 2003, 2012, 2014) Table of Contents Preface.............................................................................................................................v Chapter 1: Mathematical Preliminaries .....................................................................1 1.A Introduction ...............................................................................................1 1.B Review of some useful concepts of geometry and algebra. ................3 Scalar arithmetic....................................................................................3 Vector arithmetic...................................................................................4 Vector multiplication............................................................................6 Vector length..........................................................................................8 Summary. ...............................................................................................9 1.C Review of phasors and complex numbers.............................................9 Phasor length, the magnitude of complex numbers, and Euler's formula. .....................................................................................10 Multiplying complex numbers ...........................................................12 Statistics of complex numbers.............................................................13 1.D Terminology summary. ...........................................................................13 Chapter 2: Sinusoids, Phasors, and Matrices ..........................................................15 2.A Phasor representation of sinusoidal waveforms..................................15 2.B Matrix algebra. ...........................................................................................16 Rotation matrices. .................................................................................18 Basis vectors...........................................................................................20 Orthogonal decomposition..................................................................20 Chapter 3: Fourier Analysis of Discrete Functions .................................................23 3.A Introduction. ..............................................................................................23 3.B A Function Sampled at 1 point...............................................................24 3.C A Function Sampled at 2 points.............................................................25 3.D Fourier Analysis is a Linear Transformation.......................................26 3.E Fourier Analysis is a Change in Basis Vectors. ....................................27 3.F A Function Sampled at 3 points..............................................................29 3.G A Function Sampled at D points............................................................32 3.H Tidying Up................................................................................................34 3.I Parseval's Theorem....................................................................................36 3.J A Statistical Connection............................................................................39 3.K Image Contrast and Compound Gratings............................................41 3.L Fourier Descriptors of the Shape of a Closed Curve ...........................43 Chapter 4: The Frequency Domain............................................................................47 4.A Spectral Analysis.......................................................................................47 4.B Physical Units.............................................................................................48 4.C Cartesian vs. Polar Form. .........................................................................50 4.D Complex Form of Spectral Analysis.......................................................51 4.E Complex Fourier Coefficients..................................................................53 4.F Relationship between Complex and Trigonometric Fourier Coefficients.........................................................................................................55 4.G Discrete Fourier Transforms in Two or More Dimensions.................58 4.H Matlab's Implementation of the DFT ....................................................59 4.I Parseval's Theorem, Revisited .................................................................60 Chapter 5: Continuous Functions..............................................................................61 5.A Introduction. ..............................................................................................58 5.B Inner products and orthogonality...........................................................63 5.C Symmetry. ..................................................................................................65 5.D Complex-valued functions. .....................................................................67 Chapter 6: Fourier Analysis of Continuous Functions...........................................69 6.A Introduction. ..............................................................................................69 6.B The Fourier Model.....................................................................................69 6.C Practicalities of Obtaining the Fourier Coefficients. ............................71 6.D Theorems....................................................................................................73 1. Linearity .............................................................................................73 2. Shift theorem......................................................................................73 3. Scaling theorem.................................................................................75 4. Differentiation theorem....................................................................76 5. Integration theorem ..........................................................................77 6.E Non-sinusoidal basis functions ...............................................................79 Chapter 7: Sampling Theory.......................................................................................81 7.A Introduction. ..............................................................................................81 7.B The Sampling Theorem.............................................................................81 7.C Aliasing.......................................................................................................83 7.D Parseval's Theorem...................................................................................84 7.E Truncation Errors. .....................................................................................85 7.F Truncated Fourier Series & Regression Theory. ...................................86 Chapter 8: Statistical Description of Fourier Coefficients......................................89 8.A Introduction. ..............................................................................................89 8.B Statistical Assumptions.............................................................................90 8.C Mean and Variance of Fourier Coefficients for Noisy Signals. ..........92 8.D Distribution of Fourier Coefficients for Noisy Signals........................94 8.E Distribution of Fourier Coefficients for Random Signals....................97 8.F Signal Averaging. ......................................................................................98 Chapter 9: Hypothesis Testing for Fourier Coefficients.........................................101 9.A Introduction. ..............................................................................................101 9.B Regression analysis. ..................................................................................101 9.C Band-limited signals. ................................................................................104 9.D Confidence intervals.................................................................................105 9.E Multivariate statistical analysis of Fourier coefficients........................107 Chapter 10: Directional Data Analysis......................................................................109 10.A Introduction. ............................................................................................109 10.B Determination of mean direction and concentration. ........................109 10.C Hypothesis testing. .................................................................................110 10.D Grouped data...........................................................................................110 10.D The Fourier connection. .........................................................................112 10.E Higher harmonics....................................................................................113 Chapter 11: The Fourier Transform...........................................................................115 11.A Introduction. ............................................................................................115 11.B The Inverse Cosine and Sine Transforms.............................................115 11.C The Forward Cosine and Sine Transforms..........................................117 11.D Discrete