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Fourier and Complex Analysis RUSSELLL.HERMAN ANINTRODUCTIONTO FOURIERANDCOMPLEXANALYSIS WITHAPPLICATIONSTOTHE SPECTRALANALYSISOFSIGNALS R. L. HERMAN - VERSION DATE: DECEMBER 20, 2014 Copyright © 2005-2014 by Russell L. Herman published by r. l. herman This text has been reformatted from the original using a modification of the Tufte-book documentclass in LATEX. See tufte-latex.googlecode.com. an introduction to fourier and complex analysis with applications to the spectral analysis of signals by Russell Herman is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. These notes have resided at http://people.uncw.edu/hermanr/mat367/fcabook since Spring 2005. Sixth printing, December 2014 Contents Introduction xv 1 Review of Sequences and Infinite Series 1 1.1 Sequences of Real Numbers . 2 1.2 Convergence of Sequences . 2 1.3 Limit Theorems . 3 1.4 Infinite Series . 4 1.5 Geometric Series . 5 1.6 Convergence Tests . 8 1.7 Sequences of Functions . 12 1.8 Infinite Series of Functions . 15 1.9 Special Series Expansions . 17 1.10 Power Series . 19 1.11 Binomial Series . 26 1.12 The Order of Sequences and Functions . 31 Problems . 34 2 Fourier Trigonometric Series 37 2.1 Introduction to Fourier Series . 37 2.2 Fourier Trigonometric Series . 40 2.3 Fourier Series over Other Intervals . 47 2.3.1 Fourier Series on [a, b] ................... 52 2.4 Sine and Cosine Series . 53 2.5 The Gibbs Phenomenon . 59 2.6 Multiple Fourier Series . 64 Problems . 67 3 Generalized Fourier Series and Function Spaces 73 3.1 Finite Dimensional Vector Spaces . 73 3.2 Function Spaces . 79 3.3 Classical Orthogonal Polynomials . 83 3.4 Fourier-Legendre Series . 86 3.4.1 Properties of Legendre Polynomials . 87 3.4.2 The Generating Function for Legendre Polynomials . 89 3.4.3 The Differential Equation for Legendre Polynomials . 94 3.4.4 Fourier-Legendre Series Examples . 95 iv 3.5 Gamma Function . 97 3.6 Fourier-Bessel Series . 98 3.7 Appendix: The Least Squares Approximation . 103 3.8 Appendix: Convergence of Trigonometric Fourier Series . 106 Problems . 112 4 Complex Analysis 117 4.1 Complex Numbers . 118 4.2 Complex Valued Functions . 121 4.2.1 Complex Domain Coloring . 124 4.3 Complex Differentiation . 127 4.4 Complex Integration . 131 4.4.1 Complex Path Integrals . 131 4.4.2 Cauchy’s Theorem . 134 4.4.3 Analytic Functions and Cauchy’s Integral Formula . 138 4.4.4 Laurent Series . 142 4.4.5 Singularities and The Residue Theorem . 145 4.4.6 Infinite Integrals . 153 4.4.7 Integration over Multivalued Functions . 159 4.4.8 Appendix: Jordan’s Lemma . 163 Problems . 164 5 Fourier and Laplace Transforms 169 5.1 Introduction . 169 5.2 Complex Exponential Fourier Series . 170 5.3 Exponential Fourier Transform . 172 5.4 The Dirac Delta Function . 176 5.5 Properties of the Fourier Transform . 179 5.5.1 Fourier Transform Examples . 181 5.6 The Convolution Operation . 186 5.6.1 Convolution Theorem for Fourier Transforms . 189 5.6.2 Application to Signal Analysis . 193 5.6.3 Parseval’s Equality . 195 5.7 The Laplace Transform . 196 5.7.1 Properties and Examples of Laplace Transforms . 198 5.8 Applications of Laplace Transforms . 203 5.8.1 Series Summation Using Laplace Transforms . 203 5.8.2 Solution of ODEs Using Laplace Transforms . 206 5.8.3 Step and Impulse Functions . 209 5.9 The Convolution Theorem . 214 5.10 The Inverse Laplace Transform . 217 5.11 Transforms and Partial Differential Equations . 220 5.11.1 Fourier Transform and the Heat Equation . 220 5.11.2 Laplace’s Equation on the Half Plane . 222 5.11.3 Heat Equation on Infinite Interval, Revisited . 224 5.11.4 Nonhomogeneous Heat Equation . 226 v Problems . 229 6 From Analog to Discrete Signals 235 6.1 Analog to Periodic Signals . 235 6.2 The Comb Function . 238 6.3 Discrete Signals . 242 6.3.1 Summary . 244 6.4 The Discrete (Trigonometric) Fourier Transform . 245 6.4.1 Discrete Trigonometric Series . 247 6.4.2 Discrete Orthogonality . 248 6.4.3 The Discrete Fourier Coefficients . 250 6.5 The Discrete Exponential Transform . 252 6.6 Applications . 255 6.7 Appendix . 257 6.7.1 MATLAB Implementation . 257 6.7.2 Matrix Operations for MATLAB . 261 6.7.3 FFT: The Fast Fourier Transform . 262 Problems . 266 7 Signal Analysis 267 7.1 Introduction . 267 7.2 Periodogram Examples . 268 7.3 Effects of Sampling . 273 7.4 Effect of Finite Record Length . 278 7.5 Aliasing . 283 7.6 The Shannon Sampling Theorem . 285 Problems . 288 Bibliography 289 Index 293 List of Figures 1 Plot of the second harmonic of a vibrating string. xv 2 Plot of an initial condition for a plucked string. xvi 1.1 Plot of the terms of the sequence an = n − 1, n = 1, 2, . , 10. 2 1 1.2 Plot of the terms of the sequence a = , for n = 1, 2, . , 10. 2 n 2n (−1)n 1.3 Plot of the terms of the sequence an = 2n , for n = 1, 2, . , 10. 2 (−1)n 1.4 Plot of an = 2n for n = 1 . 10 showing the tail. 3 (−1)n 1.5 Plot of an = 2n for n = 1 . 10 showing the tail. 3 1.6 Plot of s = n 1 for n = 1 . 10. 5 n ∑k=1 2k−1 k 1 1.7 Plot of the partial sums, sk = ∑n=1 n . 9 1 1 1.8 Plot of f (x) = x and boxes of height n and width 1. 9 n 1.9 For fn(x) = x we see how N depends on x and e. 13 1.10 Plot defining uniform convergence. 14 n 1.11 fn(x) = x does not converge uniformly. 14 2 1.12 fn(x) = cos(nx)/n converges uniformly. 15 1 2 1.13 Comparison of 1−x to 1 + x and 1 + x + x . 19 1 2 2 3 1.14 Comparison of 1−x to 1 + x + x and 1 + x + x + x . 19 1 20 n 1.15 Comparison of 1−x to ∑n=0 x . 20 2.1 Plots of y(t) = A sin(2p f t). 38 2.2 Problems can occur while plotting. 38 2.3 Superposition of several sinusoids. 39 2.4 Functions y(t) = 2 sin(4pt) and y(t) = 2 sin(4pt + 7p/8). 39 2.5 Plot of a function and its periodic extension. 41 2.6 Plot of discontinuous function in Example 2.3. 45 2.7 Transformation between intervals. 47 2.8 Area under an even function on a symmetric interval, [−a, a]. 49 2.9 Area under an odd function on a symmetric interval, [−a, a]. 49 2.10 Partial sums of the Fourier series for f (x) = jxj. 51 2.11 Plot of the first 10 terms of the Fourier series for f (x) = jxj 51 2.12 Plot of the first 200 terms of the Fourier series for f (x) = x 51 2.13 A sketch of a function and its various extensions. 54 2.14 The periodic extension of f (x) = x2 on [0, 1]. 57 2.15 The even periodic extension of f (x) = x2 on [0, 1]. 58 2.16 The odd periodic extension of f (x) = x2 on [0, 1]. 58 2.17 The Fourier representation of a step function, N = 10. 59 viii 2.18 The Fourier representation of a step function, N = 10. 60 2.19 The Fourier representation of a step function, N = 20. 60 2.20 The Fourier representation of a step function, N = 100. 61 2.21 The Fourier representation of a step function, N = 500. 61 2.22 The rectangular membrane of length L and width H. 64 3.1 Basis vectors a1, a2, and a3. 83 3.2 Vectors e1, a2, and e2. 84 3.3 A plot of vectors for determining e3. 84 3.4 Plots of the Legendre polynomials 88 3.5 Earth-moon system. 90 3.6 Fourier-Legendre series expansion of Heaviside function. 96 3.7 Plot of the Gamma function. 97 3.8 Plots of the Bessel functions J0(x), J1(x), J2(x), and J3(x). 99 3.9 Plots of the Neumann functions, N0(x),..., N3(x). 100 3.10 Fourier-Bessel series expansion for f (x) = 1 on 0 < x < 1. 103 3.11 Nth Dirichlet Kernel for N=25. 110 3.12 Nth Dirichlet Kernel for N=50. 110 3.13 Nth Dirichlet Kernel for N=100. 111 4.1 The Argand diagram for plotting complex numbers 118 4.2 Locating 1 + i in the complex z-plane. 119 4.3 Locating the cube roots of unity in the complex z-plane. 121 4.4 Defining a complex valued function on C 122 4.5 f (z) = z2 maps lines in the z-plane into parabolae in the w-plane. 123 4.6 f (z) = z2 maps a grid in the z-plane into the w-plane. 123 4.7 f (z) = ez maps the z-plane into the w-plane. 123 4.8 f (z) = ez maps the z-plane into the w-plane. 124 4.9 Domain coloring of the complex z-plane 124 4.10 Domain coloring for f (z) = z2. 125 4.11 Domain coloring for f (z) = 1/z(1 − z). 125 4.12 Domain coloring for f (z) = z. 125 4.13 Domain coloring for the function f (z) = z2. 125 4.14 Domain coloring for several functions. 126 4.15 Domain coloring for f (z) = z2 − 0.75 − 0.2i.
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