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fourier analysis: signals and filters 235(10.8) determines the values⇒b n = 2 Ty(t)= 2 π∫ +T/2−T/2tdt sin nωtT/2 = ω π[sin ωt − 1 2 sin 2ωt + 1 3∫ +π/ω−π/ωdt sin nωt ωtπ = 2nπ (−1)n+1 ,sin 3ωt −···]. (10.14)10.2.2 Example 2: Half-wave FunctionThe half-wave function⎧⎨sin ωt, for 0
236 chapter 1010.4 Fourier Transforms (Theory)Although a Fourier series is the right tool for approximating or analyzing periodicfunctions, the Fourier transform or integral is the right tool for nonperiodic functions.We convert from series to transform by imagining a system described by a continuumof “fundamental” frequencies. We thereby deal with wave packets containingcontinuous rather than discrete frequencies. 2 While the difference between seriesand transform methods may appear clear mathematically, when we approximatethe Fourier integral as a finite sum, the two become equivalent.By analogy with (10.7), we now imagine our function or signal y(t) expressed interms of a continuous series of harmonics:y(t)=∫ +∞−∞dω Y (ω) eiωt√2π, (10.17)where for compactness we use a complex exponential function. 3 The expansionamplitude Y (ω) is analogous to the Fourier coefficients (a n ,b n ) and is called theFourier transform of y(t). The integral (10.17) is the inverse transform since it convertsthe transform to the signal. The Fourier transform converts y(t) to its transform Y (ω):Y (ω)=∫ +∞−∞dt e−iωt√2πy(t). (10.18)The 1/ √ 2π factor in both these integrals is a common normalization in quantummechanics but maybe not in engineering where only a single 1/2π factor is used.Likewise, the signs in the exponents are also conventions that do not matter as longas you maintain consistency.If y(t) is the measured response of a system (signal) as a function of time, thenY (ω) is the spectral function that measures the amount of frequency ω present inthe signal. [However, some experiments may measure Y (ω) directly, in which casean inverse transform is needed to obtain y(t).] In many cases Y (ω) is a complexfunction with positive and negative values and with significant variation in magnitude.Accordingly, it is customary to eliminate some of the complexity of Y (ω)by making a semilog plot of the squared modulus |Y (ω)| 2 versus ω. This is called a2 We follow convention and consider time t the function’s variable and frequency ω thetransform’s variable. Nonetheless, these can be reversed or other variables such as positionx and wave vector k may also be used.3 Recollect the principle of linear superposition and that exp(iωt) = cos ωt + i sin ωt. Thismeans that the real part of y gives the cosine series, and the imaginary part the sine series.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 236
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viiicontents2.3 Experimental Error
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xcontents6.3 Experimentation 1356.4
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xxivprefacewhich includes understan
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2 chapter 1PhysicsCPFigure 1.1 A re
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4 chapter 1Scientific LibrariesPerf
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6 chapter 1C DWe have tried to make
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10 chapter 11.4.1 Structured Progra
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20 chapter 1TABLE 1.4The IEEE 754 S
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24 chapter 1TABLE 1.6Representation
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26 chapter 1The computer fetches th
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28 chapter 1Your problem is to use
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2Errors & Uncertainties in Computat
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32 chapter 2purposes, let us consid
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34 chapter 2can be avoided by combi
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42 chapter 2To see if these assumpt
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44 chapter 2computation. Accordingl
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48 chapter 3to plot for x and y, we
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50 chapter 3Sample PtPlot Data file
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52 chapter 3TABLE 3.1Text Files Gra
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54 chapter 3Figure 3.4 Left: A plot
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56 chapter 3TABLE 3.2Grace Menu and
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58 chapter 33.4.1 Gnuplot Input Dat
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60 chapter 3gnuplot> set output "pl
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62 chapter 3gnuplot> set terminal e
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66 chapter 33.6 Texturing and 3-D I
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68 chapter 4R R 2RL L 2LC C 2CFigur
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80 chapter 41 / Z0.51400R0400R80021
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84 chapter 42. Define a daughter cl
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88 chapter 4qFigure 4.7 Left: The t
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90 chapter 4with properties differi
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96 chapter 46. Change the mass of t
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98 chapter 4Check that all the plot
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100 chapter 43. You should see now
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102 chapter 44.9.11 Complex Object
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104 chapter 4✞/ / KomplexTest : t
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106 chapter 4motion in other direct
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108 chapter 4✝protected double y(
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110 chapter 5Mathematically, the li
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112 chapter 5The linear congruent m
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114 chapter 5TABLE 5.1A Table of a
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116 chapter 55.3 Unit II. Monte Car
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118 chapter 5300y0-10-20R2001000 20
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120 chapter 54100,000log[N(t)]10,00
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122 chapter 5✞/ / Decay . java :
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124 chapter 6f(x)a x i x i+1 x i+2
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126 chapter 6f(x)f(x)parabola 1para
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128 chapter 6evaluate the function
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130 chapter 6⇒ N =( ) 2/91 1(ɛ m
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132 chapter 63. [−∞→∞], sca
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134 chapter 6}public static double
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136 chapter 6the integral of f(x)=1
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138 chapter 6f(x)< f(x) >xFigure 6.
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140 chapter 61. Conduct 16 trials a
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142 chapter 6acceptrejectFigure 6.6
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144 chapter 6The crux of this techn
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7Differentiation & SearchingIn this
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148 chapter 7the forward-difference
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150 chapter 7algorithm (7.7) is O(h
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152 chapter 7algorithms in which de
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154 chapter 7we know a zero occurs.
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156 chapter 7Figure 7.3 Two example
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8Solving Systems of Equationswith M
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160 chapter 8the spheres, and the d
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162 chapter 8We now have a solvable
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164 chapter 8of (8.19) by A −1 :x
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166 chapter 8Row MajorColumn Majora
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168 chapter 8having different varia
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170 chapter 8Matrix objects, add an
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172 chapter 8✞/∗ JamaFit : JAMA
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174 chapter 8( ) α β3. Consider t
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176 chapter 8discarding some inform
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178 chapter 8Lagrange interpolation
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180 chapter 8Figure 8.3 Three fits
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182 chapter 8if you have the abilit
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288 chapter 113. Reproduce the scal
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290 chapter 12discrete decay law,
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14High-Performance Computing Hardwa
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354 chapter 14A(1)A(2)A(3)CPUA(N)M(
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356 chapter 14ADBCFigure 14.4 Multi
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358 chapter 14as Fortran or C, tran
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360 chapter 14TABLE 14.2Computation
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362 chapter 14The processors in a p
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364 chapter 14Figure 14.7 Two views
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366 chapter 14Speedup8Amdahl's Lawp
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368 chapter 1414.11 Parallelization
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370 chapter 14The problem affects p
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372 chapter 14• A race condition
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374 chapter 14yet in order to obtai
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376 chapter 14slightly faster or sm
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378 chapter 14You see (Listing 14.1
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386 chapter 14as high-performance c
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16Simulating Matter withMolecular D
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430 chapter 162 3 2 3 2 31 4 1 4 1
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17PDEs for Electrostatics & Heat Fl
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θpde waves: string, quantum packet
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pde waves: string, quantum packet,
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solitons & computational fluid dyna
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✞solitons & computational fluid d
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integral equations in quantum mecha
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✞integral equations in quantum me
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Appendix A: Glossaryabsolute value
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glossary 557control character — A
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glossary 559log in (on) — To sign
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glossary 561supercomputer — The c
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installing ptplot & java developer
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industrial-strength data visualizat
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Appendix D: An MPI TutorialIn this
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an mpi tutorial 595• Open a shell
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an mpi tutorial 597of all the compu
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an mpi tutorial 599TABLE D.1Some Co
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an mpi tutorial 601jobs to MPI. 2 A
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an mpi tutorial 603✞> cd $HOME Do
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an mpi tutorial 605✞/ / MPIhello
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an mpi tutorial 607Argument Namemsg
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an mpi tutorial 609D.3.4 Broadcast
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an mpi tutorial 611D.4 Parallel Tun
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an mpi tutorial 613✝}MPI_Recv ( &
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an mpi tutorial 615✞/ / Code list
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an mpi tutorial 617-1) does not sen
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an mpi tutorial 619D.6.1 Nonblockin
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an mpi tutorial 621D.9 List of MPI
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Appendix E: Calling LAPACK from CCa
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calling LAPACK from C 625E.2 Compil
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software on the CD 627JavaCodes Con
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software on the CD 629JavaCodes Con
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software on the CD 631Applets Direc
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software on the CD 633Fortran77code
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Appendix G: Compression via DWTwith
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compression via DWT with thresholdi
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compression via DWT with thresholdi
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BIBLIOGRAPHY[Abar 93] Abarbanel, H.
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ibliography 643[Erco] Ercolessi, F.
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ibliography 645[L&F 93] Landau, R.
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ibliography 647[P&W 91] Pinson, L.
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ibliography 649[VdeV 94] van de Vel
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IndexAbstract data structures,70Abs
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index 653drand, 114Driving force, 2
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index 655Libraries, see Subroutines
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index 657structured, 10for virtual
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