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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 February 23, 2006 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 4.7FrequencyDomainAnalysisoftheCircularDrum............... 170 4.8TimeDomainAnalysisoftheCircularMembrane............... 171 5 Introduction to the Fourier Transform 177 5.1PeriodicandAperiodicFunctions........................ 177 5.2 RepresentationofAperiodicFunctions .................... 178 5.3TheFourierTransformandInverseTransform................. 181 5.4ExamplesofFourierTransformsandTheirGraphicalRepresentation.... 184 5.5SpecialComputationalCasesoftheFourierTransform............ 188 5.6RelationsBetweentheTransformandInverseTransform........... 191 5.7 General Properties of the Fourier Transform - Linearity, Shifting and Scaling 194 5.8TheFourierTransformofDerivativesandIntegrals.............. 198 5.9TheFourierTransformoftheImpulseFunctionandItsImplications..... 202 5.10FurtherExtensionsoftheFourierTransform.................. 208 6 Applications of the Fourier Transform 214 6.1Introduction.................................... 214 6.2ConvolutionandFourierTransforms...................... 214 6.3 Linear,Shift-InvariantSystems......................... 222 6.4 DeterminingaSystem’sImpulseResponseandTransferFunction...... 227 6.5ApplicationsofConvolution-SignalProcessingandFilters.......... 233 6.6 Applications of Convolution - Amplitude Modulation and Frequency Division Multiplexing ................................... 236 6.7 TheD’AlembertSolutionRevisited ...................... 241 6.8 DispersiveWaves ................................ 244 6.9 Correlation.................................... 247 6.10Summary..................................... 248 7 Appendix A - Bessel’s Equation 250 7.1Bessel’sEquation................................. 250 7.2PropertiesofBesselFunctions.......................... 252 7.3VariantsofBessel’sEquation.......................... 257 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, top left the initial solu- tion, top right the solution at time t = 2, below that the solution at times t =4andt = 6 and at the bottom, the solution at t =8 ........... 145 39 ModesofAVibratingRectangle......................... 155 40 ContourLinesforModesofaVibratingRectangle............... 156 iii 41 TheSpectrumoftheRectangularDrum.................... 157 42 ATravelingPlaneWave............................. 159 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 Spectrum of the Circular Membrane. Horizontal axis refers to the zeros of J0(ξnL). The numbers next to each vertical line measure the distance to the nextzero...................................... 172 47 ApproximationoftheDefiniteIntegral..................... 179 48 TheSquarePulse................................. 185 49 TheFourierTransformoftheSquarePulse................... 186 50 The Function h(t)givenby5.4.16........................ 187 51 AlternativeGraphicalDescriptionsoftheFourierTransform......... 188 52 The Function e−|t| ................................. 190 53 The Relationship of h(t)andh(−f)....................... 193 54 The Relationship of h(t)andh(at) ....................... 195 1 f 55 The Relationship of H(f)and H .................... 197 a a 56 The Relationship of h(t)andh(t − b)...................... 198 57 AFourierTransformComputedUsingtheDerivativeRule.......... 200 58 The Graphical Interpretations of δ(t) ...................... 203 59 The Transform Pair for F [δ(t)]=1....................... 204 60 The Transform Pair for F [cos(2πf0t)] ..................... 206 61 The Transform Pair for F [sin(2πf0t)]...................... 206 62 TheTransformPairforaPeriodicFunction.................. 207 63 The Transform Pair for the Function sgn(t) .................. 209 64 TheTransformPairfortheUnitStepFunction................ 210 65 The Relation of h(t) as a Function of t and h(t − τ) as a Function of τ ... 215 66 TheGraphicalDescriptionofaConvolution.................. 217 67 The Graph of a g(t) ∗ h(t)fromtheExample ................. 218 68 TheGraphicalDescriptionofaSecondConvolution.............. 219 69 The Graphical Description of a System Output in the Transform Domain . 227 70 ASampleRCCircuit............................... 228 71 AnExampleImpulseResponseandTransferFunction............. 229 72 An Example Input and Output for an RC Circuit............... 233 73 TransferFunctionsforIdealFilters....................... 234 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).................................... 235 75 AmplitudeModulation-TheTimeDomainView............... 236 76 AmplitudeModulation-TheFrequencyDomainView............ 237 77 SingleSidebandModulation-TheFrequencyDomainView.......... 238 78 Frequency Division Multiplexing - The Time and Frequency Domain
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