Chapter 4Basic IdeaTaylor SeriesSquare WaveSlide 5Periodic SignalsFourier SeriesFourier Series CoefficientsEuler’s RelationshipExamplesDifferent Forms of Fourier SeriesSlide 12Slide 13ExampleSlide 15Practical ApplicationSlide 17DemoSlide 19Use the Fourier Series Table (Table 4.3)Fourier Series - AppletUsing Fourier Series TableSlide 23Fourier Series TransformationSlide 25Slide 26Slide 27Slide 28Slide 29Fourier Series and Frequency SpectraSchaum’s Outline ProblemsChapter 4Fourier Series & TransformsBasic IdeanotesTaylor Series•Complex signals are often broken into simple pieces •Signal requirements –Can be expressed into simpler problems –The first few terms can approximate the signal•Example: The Taylor series of a real or complex function ƒ(x) is the power series •http://upload.wikimedia.org/wikipedia/commons/6/62/Exp_series.gifSquare WaveS(t)=sin(2ft)S(t)=1/3[sin(23f)t)]S(t)= 4/{sin(2ft) +1/3[sin(2f)t)]}oddKkkftAts/1)2sin(4)(Fourier ExpansionSquare WaveoddKkkftAts/1)2sin(4)(Frequency Components of Square WaveK=1,3,5 K=1,3,5, 7K=1,3,5, 7, 9, …..Fourier ExpansionPeriodic Signals•A Periodic signal/function can be approximated by a sum (possibly infinite) sinusoidal signals. •Consider a periodic signal with period T•A periodic signal can be Real or Complex•The fundamental frequency: o•Example: ootjoTAetxComplexttxaltxnTtxPeriodico/2)()cos()(Re)()(Fourier Series•We can represent all periodic signals as harmonic series of the form–Ck are the Fourier Series Coefficients; k is real –k=0 gives the DC signal –k=+/-1 indicates the fundamental frequency or the first harmonic 0 –|k|>=2 harmonicsFourier Series Coefficients•Fourier Series Pair•We have•For k=0, we can obtain the DC value which is the average value of x(t) over one periodkoojkktjkktjkkkkeCCeCeCCC&**Series of complex numbers Defined over a period of x(t)Euler’s Relationship–Review Euler formulas notesExamples•Find Fourier Series Coefficients for•Find Fourier Series Coefficients for•Find Fourier Series Coefficients for•Find Fourier Series Coefficients for)cos()( ttxo)sin()( ttxo)4/2cos()( ttxnotesC1=1/2; C-1=1/2; No DCC1=1/2j; C-1=-1/2j; No DC)(sin)(2ttxoDifferent Forms of Fourier Series•Fourier Series Representation has three different forms Also: HarmonicAlso: Complex Exp.oddKkkftAts/1)2sin(4)(Which one is this?What is the DC component?What is the expression for Fourier Series CoefficientsExamples•Find Fourier Series Coefficients for•Find Fourier Series Coefficients forRemember:ExamplesFind the Complex Exponential Fourier Series Coefficientsttttxoooo3s in4)302cos(5cos310)( notestextbookExample•Find the average power of x(t) using Complex Exponential Fourier Series – assuming x(t) is periodic dttxPoT2)(2****2)()()()()(kkkkkTtkjkktkjkkTCCCdttxtxPeCtxeCtxdttxPooooThis is called the Parseval’s IdentityExample•Consider the following periodic square wave•Express x(t) as a piecewise function•Find the Exponential Fourier Series of representations of x(t)•Find the Combined Trigonometric Fourier Series of representations of x(t)•Plot Ck as a function of kVTo/2-VTonotesX(t))sin(4)90cos(42)(/1/12//tkkVtkkVeekVtxooddkooddktkjjoddko2|Ck||4V/||4V/5||4V/3|0 300Use aLow Pass Filter to pick any tone you want!!Practical Application•Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies?Practical Application•Using a XTL oscillator which produces positive 1Vp-p how can you generate a sinusoidal waveforms with different frequencies? )sin()( tkBtyoSquare Signal@ woFilter @ [ko]Level ShifterSinusoidal waveform1To/2ToX(t)0.5To/2ToX(t)-0.5@ [kwo]kwo)sin(42)(/12//tkkVeekVtxooddktkjjoddkoB changes depending on k valueDemoCk corresponds to frequency componentsIn the signal.Example•Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. Sinc FunctionOnly a functionof freq.1Note: sinc (infinity)  1 & Max value of sinc(x)1/xNote: First zero occurs at Sinc (+/-pi)Use the Fourier Series Table (Table 4.3)•Consider the following periodic square wave•Find the Exponential Fourier Series of representations of x(t)•X0V)sin(4)90cos(42)(/1/12//tkkVtkkVeekVtxooddkooddktkjjoddkoVTo/2-VToX(t)2|Ck||4V/||4V/5||4V/3|0 3 0  0tkjoddkoekVjtx/20)(Fourier Series - ApplettkjoddkoekVjtx/20)(http://www.falstad.com/fourier/Using Fourier Series Table•Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. (Rectangular wave) X01C0=T/ToT/2=T1T=2T1Ck=T/T0 sinc (Tkw0/2)tkjkooooookoekTcTTkTcTTTkcTTC)(sin2)(sin22sin1111Same as beforeNote: sinc (infinity)  1 & Max value of sinc(x)1/xUsing Fourier Series Table•Express the Fourier Series for a triangular waveform?• Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. ToXoFourier Series Transformation•Express the Fourier Series for a triangular waveform?• Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal. ToXoToXo/2-Xo/2From the table:Fourier Series Transformation•Express the Fourier Series for a triangular waveform?• Express the Fourier Series for a triangular waveform that is amplitude shifted down by –X0/2 ? Plot the signal.  tkjoddkoooekXXtx/222)(ToXo  tkjoddkotkjoddkooooooekXekXXXtxXty/2/22222)(2)(ToXo/2-Xo/2Only DC value changed!From the table:Fourier Series Transformation•Express the Fourier Series for a sawtooth waveform?•Express the Fourier Series for this sawtooth waveform?ToXoToXoFrom the table:-31Fourier Series Transformation•Express the