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