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An Introduction to Fourier Analysis Fourier Series, Partial Differential Equations and Fourier Transforms

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An Introduction to Fourier Analysis Fourier Series, Partial Differential Equations and Fourier Transforms An Introduction to Fourier Analysis Fourier Series, Partial Differential Equations and Fourier Transforms Notes prepared for MA3139 Arthur L. Schoenstadt Department of Applied Mathematics Naval Postgraduate School Code MA/Zh Monterey, California 93943 August 18, 2005 c 1992 - Professor Arthur L. Schoenstadt 1 Contents 1 Infinite Sequences, Infinite Series and Improper Integrals 1 1.1Introduction.................................... 1 1.2FunctionsandSequences............................. 2 1.3Limits....................................... 5 1.4TheOrderNotation................................ 8 1.5 Infinite Series . ................................ 11 1.6ConvergenceTests................................ 13 1.7ErrorEstimates.................................. 15 1.8SequencesofFunctions.............................. 18 2 Fourier Series 25 2.1Introduction.................................... 25 2.2DerivationoftheFourierSeriesCoefficients.................. 26 2.3OddandEvenFunctions............................. 35 2.4ConvergencePropertiesofFourierSeries.................... 40 2.5InterpretationoftheFourierCoefficients.................... 48 2.6TheComplexFormoftheFourierSeries.................... 53 2.7FourierSeriesandOrdinaryDifferentialEquations............... 56 2.8FourierSeriesandDigitalDataTransmission.................. 60 3 The One-Dimensional Wave Equation 70 3.1Introduction.................................... 70 3.2TheOne-DimensionalWaveEquation...................... 70 3.3 Boundary Conditions ............................... 76 3.4InitialConditions................................. 82 3.5IntroductiontotheSolutionoftheWaveEquation.............. 82 3.6TheFixedEndConditionString......................... 85 3.7TheFreeEndConditionsProblem........................ 97 3.8TheMixedEndConditionsProblem...................... 106 3.9GeneralizationsontheMethodofSeparationofVariables........... 117 3.10 Sturm-Liouville Theory .............................. 120 3.11TheFrequencyDomainInterpretationoftheWaveEquation......... 132 3.12TheD’AlembertSolutionoftheWaveEquation................ 137 3.13 The Effect of Boundary Conditions . ..................... 141 4 The Two-Dimensional Wave Equation 147 4.1Introduction.................................... 147 4.2TheRigidEdgeProblem............................. 148 4.3FrequencyDomainAnalysis........................... 154 4.4TimeDomainAnalysis.............................. 158 4.5TheWaveEquationinCircularRegions.................... 159 4.6SymmetricVibrationsoftheCircularDrum.................. 163 i 4.7FrequncyDomainAnalysisoftheCircularDrum................ 170 4.8TimeDomainAnalysisoftheCircularMembrane............... 171 5 Introduction to the Fourier Transform 178 5.1PeriodicandAperiodicFunctions........................ 178 5.2 RepresentationofAperiodicFunctions .................... 179 5.3TheFourierTransformandInverseTransform................. 182 5.4ExamplesofFourierTransformsandTheirGraphicalRepresentation.... 185 5.5SpecialComputationalCasesoftheFourierTransform............ 189 5.6RelationsBetweentheTransformandInverseTransform........... 192 5.7 General Properties of the Fourier Transform - Linearity, Shifting and Scaling 195 5.8TheFourierTransformofDerivativesandIntegrals.............. 199 5.9TheFourierTransformoftheImpulseFunctionandItsImplications..... 203 5.10FurtherExtensionsoftheFourierTransform.................. 209 6 Applications of the Fourier Transform 215 6.1Introduction.................................... 215 6.2ConvolutionandFourierTransforms...................... 215 6.3 Linear,Shift-InvariantSystems......................... 223 6.4 DeterminingaSystem’sImpulseResponseandTransferFunction...... 228 6.5ApplicationsofConvolution-SignalProcessingandFilters.......... 234 6.6 Applications of Convolution - Amplitude Modulation and Frequency Division Multiplexing ................................... 237 6.7 TheD’AlembertSolutionRevisited ...................... 242 6.8 DispersiveWaves ................................ 245 6.9 Correlation.................................... 248 6.10Summary..................................... 249 7 Appendix A - Bessel’s Equation 251 7.1Bessel’sEquation................................. 251 7.2PropertiesofBesselFunctions.......................... 253 7.3VariantsofBessel’sEquation.......................... 256 ii List of Figures 1 Zeno’sParadox.................................. 1 2 The“BlackBox”Function............................ 2 3 The Natural Logarithm Function (ln( )) .................... 3 4 GraphofaSequence............................... 4 5 SamplingofaContinuousSignal......................... 4 6 ThePictorialConceptofaLimit........................ 6 7 Pictorial Concept of a Limit at Infinity ..................... 6 8 EstimatingtheErrorofaPartialSum..................... 17 9 ASequenceofFunctions............................. 19 10 AGeneralPeriodicFunction........................... 26 11 A Piecewise Continuous Function in the Example ............... 29 12 ConvergenceofthePartialSumsofaFourierSeries.............. 31 13 SymmetriesinSineandCosine......................... 35 14 IntegralsofEvenandOddFunctions...................... 36 15 f(x)=x, − 3 <x<3............................. 37 16 SpectrumofaSignal............................... 49 17 ATypicalPeriodicFunction........................... 56 18 SquareWave................................... 57 19 ATransmittedDigitalSignal.......................... 60 20 ASimpleCircuit................................. 61 21 APeriodicDigitalTestSignal.......................... 61 22 UndistortedandDistortedSignals........................ 65 23 FirstSampleOutput............................... 67 24 SecondSampleOutput.............................. 68 25 AnElasticString................................. 71 26 ASmallSegmentoftheString.......................... 72 27 FreeEndConditions............................... 77 28 MixedEndConditions.............................. 79 29 The Initial Displacement - u(x, 0)........................ 93 30 TheFreeEndConditionsProblem........................ 98 31 The Initial Displacement f(x).......................... 103 32 TheMixedEndConditionProblem....................... 107 2πct 33 A2(x)cos L .................................. 133 34 VariousModesofVibration........................... 134 35 TheMovingFunction............................... 139 36 Constructing the D’Alembert Solution in the Unbounded Region ....... 141 37 The D’Alembert Solution With Boundary Conditions . .......... 144 38 Boundary Reflections via The D’Alembert Solution .............. 145 39 ModesofAVibratingRectangle......................... 155 40 ContourLinesforModesofaVibratingRectangle............... 156 41 TheSpectrumoftheRectangularDrum.................... 157 42 ATravelingPlaneWave............................. 159 iii 43 Two Plane Waves Traveling in the Directions kˆ and k˜ ............. 160 44 The Ordinary Bessel Functions J0(r)andY0(r) ................ 166 45 ModesoftheCircularMembrane........................ 171 46 SpectrumoftheCircularMembrane....................... 172 47 ApproximationoftheDefiniteIntegral..................... 180 48 TheSquarePulse................................. 186 49 TheFourierTransformoftheSquarePulse................... 187 50 The Function h(t)givenby5.4.16........................ 188 51 AlternativeGraphicalDescriptionsoftheFourierTransform......... 189 52 The Function e−|t| ................................. 191 53 The Relationship of h(t)andh(−f)....................... 194 54 The Relationship of h(t)andh(at) ....................... 196 1 f 55 The Relationship of H(f)and H .................... 198 a a 56 The Relationship of h(t)andh(t − b)...................... 199 57 AFourierTransformComputedUsingtheDerivativeRule.......... 201 58 The Graphical Interpretations of δ(t) ...................... 204 59 The Transform Pair for F [δ(t)]=1....................... 205 60 The Transform Pair for F [cos(2πf0t)] ..................... 207 61 The Transform Pair for F [sin(2πf0t)]...................... 207 62 TheTransformPairforaPeriodicFunction.................. 208 63 The Transform Pair for the Function sgn(t) .................. 210 64 TheTransformPairfortheUnitStepFunction................ 211 65 The Relation of h(t) as a Function of t and h(t − τ) as a Function of τ ... 216 66 TheGraphicalDescriptionofaConvolution.................. 218 67 The Graph of a g(t) ∗ h(t)fromtheExample ................. 219 68 TheGraphicalDescriptionofaSecondConvolution.............. 220 69 The Graphical Description of a System Output in the Transform Domain . 228 70 ASampleRCCircuit............................... 229 71 AnExampleImpulseResponseandTransferFunction............. 230 72 An Example Input and Output for an RC Circuit............... 234 73 TransferFunctionsforIdealFilters....................... 235 74 Real Filters With Their Impulse Responses and Transfer Functions. Top show RC filter (lo-pass),middle is RC filter (high-pass), and bottom is LRC filter (band-pass).................................... 236 75 AmplitudeModulation-TheTimeDomainView............... 237 76 AmplitudeModulation-TheFrequencyDomainView............ 238 77 SingleSidebandModulation-TheFrequencyDomainView.......... 239 78 Frequency Division Multiplexing - The Time and Frequency Domain Views . 240 79 RecoveringaFrequencyDivisionMultiplexedSignal.............. 241 80 SolutionsofaDispersiveWaveEquationatDifferentTimes......... 247 81 The Bessel Functions Jn (a) and Yn (b)..................... 254 iv MA 3139 Fourier Analysis and Partial Differential Equations Introduction These notes
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