I SignalsandSystems VijayK.Madisetti GeorgiaInstituteofTechnology DouglasB.Williams GeorgiaInstituteofTechnology 1FourierSeries,FourierTransforms,andtheDFT W.KennethJenkins Introduction • FourierSeriesRepresentationofContinuousTimePeriodicSignals • TheClassical FourierTransformforContinuousTimeSignals • TheDiscreteTimeFourierTransform • The DiscreteFourierTransform • FamilyTreeofFourierTransforms • SelectedApplicationsofFourier Methods • Summary 2OrdinaryLinearDifferentialandDifferenceEquations B.P.Lathi DifferentialEquations • DifferenceEquations 3FiniteWordlengthEffects BruceW.Bomar Introduction • NumberRepresentation • Fixed-PointQuantizationErrors • Floating-PointQuan- tizationErrors • RoundoffNoise • LimitCycles • OverflowOscillations • CoefficientQuantization Error • RealizationConsiderations T HESTUDYOF“SIGNALSANDSYSTEMS”hasformedacornerstoneforthedevelopmentof digitalsignalprocessingandiscrucialforallofthetopicsdiscussedinthisHandbook.While thereaderisassumedtobefamiliarwiththebasicsofsignalsandsystems,asmallportionis reviewedinthischapterwithanemphasisonthetransitionfromcontinuoustimetodiscretetime. Thereaderwishingmorebackgroundmayfindinitanyofthemanyfinetextbooksinthisarea,for example[1]-[6]. Inthechapter“FourierSeries,FourierTransforms,andtheDFT”byW.KennethJenkins,many importantFouriertransformconceptsincontinuousanddiscretetimearepresented.Thediscrete Fouriertransform(DFT),whichformsthebackboneofmoderndigitalsignalprocessingasitsmost commonsignalanalysistool,isalsodescribed,togetherwithanintroductiontothefastFourier transformalgorithms. In“OrdinaryLinearDifferentialandDifferenceEquations”,theauthor,B.P.Lathi,presentsa detailedtutorialofdifferentialanddifferenceequationsandtheirsolutions.Becausetheseequations arethemostcommonstructuresforbothimplementingandmodellingsystems,thisbackgroundis necessaryfortheunderstandingofmanyofthelatertopicsinthisHandbook.Ofparticularinterest areanumberofsolvedexamplesthatillustratethesolutionstotheseformulations. c 1999byCRCPressLLC While most software based on workstations and PCs is executed in single or double precision arithmetic, practical realizations for some high throughput DSP applications must be implemented in fixed point arithmetic. These low cost implementations are still of interest to a wide community in the consumer electronics arena. The chapter “Finite Wordlength Effects” by Bruce W. Bomar describes basic number representations, fixed and floating point errors, roundoff noise, and practical considerations for realizations of digital signal processing applications, with a special emphasis on filtering. References [1] Jackson, L.B., Signals, Systems, and Transforms, Addison-Wesley, Reading, MA, 1991. [2] Kamen, E.W. and Heck, B.S., Fundamentals of Signals and Systems Using MATLAB , Prentice-Hall, Upper Saddle River, NJ, 1997. [3] Oppenheim, A.V. and Willsky, A.S., with Nawab, S.H., Signals and Systems , 2nd Ed., Prentice-Hall, Upper Saddle River, NJ, 1997. [4] Strum, R.D.andKirk,D.E., ContemporaryLinearSystems Using MATLAB ,PWSPublishing,Boston, MA, 1994. [5] Proakis, J.G. and Manolakis, D.G., Introduction to Digital Signal Processing , Macmillan, New York; Collier Macmillan, London, 1988. [6] Oppenheim, A.V. and Schafer, R.W., Discrete Time Signal Processing , Prentice-Hall, Englewood Cliffs, NJ, 1989. c 1999 by CRC Press LLC