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502 Chapter 12. Fast Fourier Transform (a) (b) (c) h(t) − 1 2∆ H( f ) 0 H( f ) 0 T ∆ aliased Fourier transform 1 2∆ t f true Fourier transform Figure 12.1.1. The continuous function shown in (a) is nonzero only for a finite interval of time T . It follows that its Fourier transform, whose modulus is shown schematically in (b), is not bandwidth limited but has finite amplitude for all frequencies. If the original function is sampled with a sampling interval ∆, as in (a), then the Fourier transform (c) is defined only between plus and minus the Nyquist critical frequency. Power outside that range is folded over or “aliased” into the range. The effect can be eliminated only by low-pass filtering the original function before sampling. so that the sampling interval is ∆. To make things simpler, let us also suppose that N is even. If the function h(t) is nonzero only in a finite interval of time, then that whole interval of time is supposed to be contained in the range of the N points given. Alternatively, if the function h(t) goes on forever, then the sampled points are supposed to be at least “typical” of what h(t) looks like at all other times. With N numbers of input, we will evidently be able to produce no more than N independent numbers of output. So, instead of trying to estimate the Fourier transform H(f) at all values of f in the range −fc to fc, let us seek estimates only at the discrete values fn ≡ n N , n = − N∆ 2 ,...,N (12.1.5) 2 The extreme values of n in (12.1.5) correspond exactly to the lower and upper limits of the Nyquist critical frequency range. If you are really on the ball, you will have noticed that there are N +1, not N, values of n in (12.1.5); it will turn out that the two extreme values of n are not independent (in fact they are equal), but all the others are. This reduces the count to N. f Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by <strong>Numerical</strong> Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order <strong>Numerical</strong> Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).
12.1 Fourier Transform of Discretely Sampled Data 503 The remaining step is to approximate the integral in (12.0.1) by a discrete sum: H(fn) = � ∞ −∞ h(t)e 2πifnt dt ≈ N−1 � k=0 � hk e 2πifntk N−1 ∆=∆ k=0 hk e 2πikn/N (12.1.6) Here equations (12.1.4) and (12.1.5) have been used in the final equality. The final summation in equation (12.1.6) is called the discrete Fourier transform of the N points hk. Let us denote it by Hn, Hn ≡ N−1 � k=0 hk e 2πikn/N (12.1.7) The discrete Fourier transform maps N complex numbers (the h k’s) into N complex numbers (the Hn’s). It does not depend on any dimensional parameter, such as the time scale ∆. The relation (12.1.6) between the discrete Fourier transform of a set of numbers and their continuous Fourier transform when they are viewed as samples of a continuous function sampled at an interval ∆ can be rewritten as H(fn) ≈ ∆Hn (12.1.8) where fn is given by (12.1.5). Up to now we have taken the view that the index n in (12.1.7) varies from −N/2 to N/2 (cf. 12.1.5). You can easily see, however, that (12.1.7) is periodic in n, with period N. Therefore, H−n = HN−n n =1, 2,.... With this conversion in mind, one generally lets the n in Hn vary from 0 to N − 1 (one complete period). Then n and k (in hk) vary exactly over the same range, so the mapping of N numbers into N numbers is manifest. When this convention is followed, you must remember that zero frequency corresponds to n =0, positive frequencies 0
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Numerical Recipes in C The Art of S
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Contents Preface to the Second Edit
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Contents vii 7.2 Transformation Met
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Contents ix 15.6 Confidence Limits
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Preface to the Second Edition Our a
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Preface to the Second Edition xiii
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Preface to the First Edition xv lin
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Types of License Offered License In
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Computer Programs by Chapter and Se
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Chapter 1. Preliminaries 1.0 Introd
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1.0 Introduction 3 Tested Machines
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1.1 Program Organization and Contro
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true 1.1 Program Organization and C
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} 1.2 Some C Conventions for Scient
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1.2 Some C Conventions for Scientif
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(a) **m (b) **m 1.2 Some C Conventi
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1.3 Error, Accuracy, and Stability
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2.0 Introduction 33 Accumulated rou
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2.0 Introduction 35 between singula
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2.1 Gauss-Jordan Elimination 37 Eli
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2.1 Gauss-Jordan Elimination 39 pre
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2.2 Gaussian Elimination with Backs
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2.3 LU Decomposition and Its Applic
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a b c d 2.3 LU Decomposition and It
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2.4 Tridiagonal and Band Diagonal S
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2.6 Singular Value Decomposition 59
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2.7 Sparse Linear Systems 2.7 Spars
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2.7 Sparse Linear Systems 77 Finall
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2.7 Sparse Linear Systems 79 In row
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2.7 Sparse Linear Systems 83 Matrix
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in terms of which 2.11 Is Matrix In
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Chapter 3. Interpolation and Extrap
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(a) (b) 3.0 Introduction 107 Figure
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3.1 Polynomial Interpolation and Ex
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3.2 Rational Function Interpolation
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3.3 Cubic Spline Interpolation 113
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3.4 How to Search an Ordered Table
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3.6 Interpolation in Two or More Di
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3.6 Interpolation in Two or More Di
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Chapter 4. Integration of Functions
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4.1 Classical Formulas for Equally
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#define FUNC(x) ((*func)(x)) 4.2 El
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4.2 Elementary Algorithms 139 trapz
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4.4 Improper Integrals 141 which co
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4.4 Improper Integrals 143 The rout
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4.4 Improper Integrals 145 If you n
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4.5 Gaussian Quadratures and Orthog
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Chapter 5. Evaluation of Functions
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5.1 Series and Their Convergence 16
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5.2 Evaluation of Continued Fractio
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5.3 Polynomials and Rational Functi
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5.4 Complex Arithmetic 177 Actually
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5.6 Quadratic and Cubic Equations 1
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5.7 Numerical Derivatives 187 With
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Chebyshev polynomials 1 .5 0 −.5
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6.1 Gamma, Beta, and Related Functi
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incomplete gamma function P(a,x) 1.
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6.3 Exponential Integrals 223 where
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6.4 Incomplete Beta Function 227 Th
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Bessel functions 1 .5 0 −.5 −1
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modified Bessel functions 4 3 2 1 6
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6.11 Elliptic Integrals and Jacobia
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#define TINY 1.0e-25 #define BIG 4.
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6.11 Elliptic Integrals and Jacobia
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#include "complex.h" #define ONE Co
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7.1 Uniform Deviates 275 As for ref
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7.1 Uniform Deviates 277 This corre
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#define IA 16807 #define IM 2147483
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RAN 3 1 7.1 Uniform Deviates 281 iy
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7.1 Uniform Deviates 285 Constants
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7.2 Transformation Method: Exponent
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first random deviate in 7.3 Rejecti
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7.4 Generation of Random Bits 297 s
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8.1 Straight Insertion and Shell’
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9.0 Introduction 349 good first gue
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(a) (b) (c) (d) 9.1 Bracketing and
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Bisection Method 9.1 Bracketing and
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Eigenvalue Methods 9.5 Roots of Pol
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X1 A B C 10.0 Introduction 395 D Fi
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18.1 Fredholm Equations of the Seco
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#include "nrutil.h" 18.1 Fredholm E
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18.2 Volterra Equations 795 in §18
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18.3 Integral Equations with Singul
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f (x) 1 .5 0 −.5 0 18.3 Integral
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18.4 Inverse Problems and the Use o
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18.4 Inverse Problems and the Use o
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18.5 Linear Regularization Methods
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18.6 Backus-Gilbert Method 815 nece
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18.6 Backus-Gilbert Method 817 Subs
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18.7 Maximum Entropy Image Restorat
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Chapter 19. Partial Differential Eq
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19.0 Introduction 829 possible stab
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y L ∆ y1 19.0 Introduction 831 A
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19.0 Introduction 833 So-called rap
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19.1 Flux-Conservative Initial Valu
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t or n 19.1 Flux-Conservative Initi
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19.2 Diffusive Initial Value Proble
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19.2 Diffusive Initial Value Proble
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and the heuristic stability criteri
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19.3 Initial Value Problems in Mult
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19.3 Initial Value Problems in Mult
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19.4 Fourier and Cyclic Reduction M
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FACR Method 19.5 Relaxation Methods
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19.5 Relaxation Methods for Boundar
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19.6 Multigrid Methods for Boundary
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S S S E γ = 1 19.6 Multigrid Metho
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} 19.6 Multigrid Methods for Bounda
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#include 19.6 Multigrid Methods fo
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Chapter 20. Less-Numerical Algorith
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20.1 Diagnosing Machine Parameters
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20.1 Diagnosing Machine Parameters
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(a) G(i) (b) MSB 4 3 2 1 0 LSB i MS
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20.3 Cyclic Redundancy and Other Ch
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} 20.4 Huffman Coding and Compressi
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20.4 Huffman Coding and Compression
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} 20.4 Huffman Coding and Compressi
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} } } 20.4 Huffman Coding and Compr
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1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
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20.5 Arithmetic Coding 913 Individu
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20.6 Arithmetic at Arbitrary Precis
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} for (j=n;j>=1;j--) { ireg=u[j]+HI
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#include "nrutil.h" #define RX 256.
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} 20.6 Arithmetic at Arbitrary Prec
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