An Introduction to Fourier Analysis - BGU

1 An Introduction to Fourier AnalysisFourier Series, Partial Differential Equationsand Fourier TransformsNotes prepared for MA3139 Arthur L. SchoenstadtDepartment of Applied MathematicsNaval Postgraduate SchoolCode MA/ZhMonterey, California 93943 August 18, 2005c 1992 - Professor Arthur L. Schoenstadt1 Contents1 Infinite Sequences, Infinite Series and Improper Limits .. Infinite Series .. ConvergenceTests .. 182 Fourier DerivationoftheFourierSeriesCoefficients .. ConvergencePropertiesofFourierSeries .. InterpretationoftheFourierCoefficients .. TheComplexFormoftheFourierSeries .. 603 The One-Dimensional Wave Boundary IntroductiontotheSolutionoftheWaveEquati on.

2 TheMixedEndConditionsProblem .. Sturm-Liouville .. AlembertSolutionoftheWaveEquation .. The Effect of Boundary Conditions .. 1414 The Two-Dimensional Wave FrequencyDomainAnalysis .. TheWaveEquationinCircularRegions .. SymmetricVibrationsoftheCircularDrum .. TimeDomainAnalysisoftheCircularMembrane .. 1715 Introduction to the Fourier PeriodicandAperiodicFunctions .. RepresentationofAperiodicFunctions .. TheFourierTransformandInverseTransform .. ExamplesofFourierTransformsandTheirGraph icalRepresentation .. SpecialComputationalCasesoftheFourierTra nsform .. RelationsBetweentheTransformandInverseTr ansform.

3 General Properties of the Fourier Transform - Linearity, Shifting and Scaling TheFourierTransformofDerivativesandInteg rals .. 2096 Applications of the Fourier ConvolutionandFourierTransforms .. Linear,Shift-InvariantSystems .. DeterminingaSystem Applications of Convolution - Amplitude Modulation and Frequency DivisionMultiplexing .. TheD AlembertSolutionRevisited .. DispersiveWaves .. Correlation .. 2497 Appendix A - Bessel s Bessel sEquation .. VariantsofBessel sEquation .. 256iiList of Figures1 Zeno sParadox ..12 The BlackBox Natural Logarithm Function (ln( ))..34 GraphofaSequence ..45 ThePictorialConceptofaLimit.

4 67 Pictorial Concept of a Limit at EstimatingtheErrorofaPartialSum .. 179 ASequenceofFunctions .. 1910 2611 A Piecewise Continuous Function in the 2912 ConvergenceofthePartialSumsofaFourierSer ies .. 3113 SymmetriesinSineandCosine .. 3514 IntegralsofEvenandOddFunctions .. 3615f(x)=x, 3<x<3 .. 3716 SpectrumofaSignal .. 4917 5618 SquareWave .. 5719 ATransmittedDigitalSignal .. 6020 ASimpleCircuit .. 6121 APeriodicDigitalTestSignal .. 6122 6523 FirstSampleOutput .. 6724 6825 AnElasticString .. 7126 7227 FreeEndConditions .. 7728 MixedEndConditions .. 7929 The Initial Displacement -u(x,0) .. 9330 9831 The Initial Displacementf(x).

5 10332 10733A2(x)cos 2 ctL .. 13334 VariousModesofVibration .. 13435 13936 Constructing the D Alembert Solution in the Unbounded 14137 The D Alembert Solution With Boundary Conditions .. 14438 Boundary Reflections via The D Alembert 14539 15540 15641 TheSpectrumoftheRectangularDrum .. 15742 ATravelingPlaneWave .. 159iii43 Two Plane Waves Traveling in the Directions kand 16044 The Ordinary Bessel FunctionsJ0(r)andY0(r) .. 16645 ModesoftheCircularMembrane .. 17146 17247 ApproximationoftheDefiniteIntegral .. 18048 18649 18750 The Functionh(t) 18851 AlternativeGraphicalDescriptionsoftheFou rierTransform .. 18952 The Functione |t|.

6 19153 The Relationship ofh(t)andh( f).. 19454 The Relationship ofh(t)andh(at) .. 19655 The Relationship ofH(f)and1aH fa .. 19856 The Relationship ofh(t)andh(t b).. 19957 AFourierTransformComputedUsingtheDerivat iveRule .. 20158 The Graphical Interpretations of (t) .. 20459 The Transform Pair forF[ (t)]= 20560 The Transform Pair forF[cos(2 f0t)] .. 20761 The Transform Pair forF[sin(2 f0t)].. 20762 TheTransformPairforaPeriodicFunction .. 20863 The Transform Pair for the Function sgn(t) .. 21064 TheTransformPairfortheUnitStepFunction .. 21165 The Relation ofh(t) as a Function oftandh(t ) as a Function of .. 21666 TheGraphicalDescriptionofaConvolution.

7 21867 The Graph of ag(t) h(t)fromtheExample .. 21968 TheGraphicalDescriptionofaSecondConvolut ion .. 22069 The Graphical Description of a System Output in the Transform Domain. 22870 22971 23072 An Example Input and Output for 23473 TransferFunctionsforIdealFilters .. 23574 Real Filters With Their Impulse Responses and Transfer Functions. Top showRCfilter (lo-pass),middle isRCfilter (high-pass), and bottom isLRCfilter(band-pass) .. 23675 AmplitudeModulation-TheTimeDomainView .. 23776 AmplitudeModulation-TheFrequencyDomainVi ew .. 23877 23978 Frequency Division Multiplexing - The Time and Frequency Domain Views . 24079 24180 SolutionsofaDispersiveWaveEquationatDiff erentTimes.

8 24781 The Bessel FunctionsJn(a) andYn(b).. 254ivMA 3139 Fourier Analysis and Partial Differential EquationsIntroductionThese notes are, at least indirectly, about the human eye and the human ear, and abouta philosophy of physical phenomena. (Now don t go asking for your money back yet! Thisreally will be a mathematics - not an anatomy or philosophy - text. We shall, however,develop the fundamental ideas of a branch ofmathematics that can be used to interpretmuch of the way these two intriguing human sensory organs function. In terms of actualanatomical descriptions we shall need no more than a simple high school-level concept ofthese organs.)The branch of mathematics we will consider is calledFourier Analysis , after the Frenchmathematician Jean Baptiste Joseph Fourier1(1768-1830), whose treatise on heat flow firstintroduced most of these concepts.

9 Today, Fourier Analysis is, among other things, perhapsthe single most important mathematical tool used in what we callsignal the fundamental procedure by which complex physical signals may be decom-posed into simpler ones and, conversely, by which complicated signals may be created outof simpler building blocks. Mathematically, Fourier Analysis has spawned some of the mostfundamental developments in our understanding of infinite series and function approxima-tion - developments which are, unfortunately, much beyond the scope of these notes. Equallyimportant, Fourier Analysis is the tool with which many of the everyday phenomena - theperceived differences in sound between violins and drums, sonic booms, and the mixing ofcolors - can be better understood.

10 As we shall come to see, Fourier Analysis does this by es-tablishing a simultaneous dual view ofphenomena - what we shall come to call thefrequencydomainand thetime we shall also come to argue later, what we shall call the time and frequency domainsimmediately relate to the ways in which the human ear and eye interpret stimuli. The ear, forexample, responds to minute variations in atmospheric pressure. These cause the ear drumto vibrate and, the various nerves in the inner ear then convert these vibrations into whatthe brain interprets as sounds. In the eye, by contrast, electromagnetic waves fall on therods and cones in the back of the eyeball, and are converted into what the brain interpretsas colors.