11/16/99 (T.F. Weiss) Lecture #18: Continuous time periodic signals. Outline: Fourier series of periodic functions. Motivation:

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1 /6/99 F Weiss) Lecure #8: Coninuous ime periodic signals Moivaion: Represenaion o coninuous ime, periodic signals in he requency domain Periodic signals occur requenly moion o planes and heir saellies, vibraion o oscillaors, elecric power disribuion, beaing o he hear, vibraion o vocal chords, ec Ouline: Fourier series o periodic uncions Examples o Fourier series periodic impulse rain Fourier ransorms o periodic uncions relaion o Fourier series Conclusions 2 Fourier series o a periodic uncion Periodic ime uncion x) is a periodic ime uncion wih period x) Such a periodic uncion can be expanded in an ininie series o exponenial ime uncions called he Fourier series, x) n X[n]e j2πn/ 3 Fourier series coeiciens he coeiciens o he Fourier series can be ound as ollows /2 /2 x)e j2πn/ d /2 X[k]e j2πk/ e j2πn/ d, /2 k /2 ) /2 /2 x)e j2πn/ d X[k] k /2 ej2πk n)/ d he inegral can be evaluaed as ollows { /2 i k n, /2 ej2πk n)/ d i k n he se o exponenial ime uncions are said o be an orhonormal basis he coeiciens are X[n] /2 /2 x)e j2πn/ d 4

2 Deiniion o line specra, harmonics he undamenal requency o / he Fourier series coeiciens ploed as a uncion o n or n o is called a Fourier specrum X[n] DC n o 2 o 3 o 4 o Fundamenal 2nd harmonic 3rd harmonic 4h harmonic Harmonic # Frequency Examples o Fourier series o periodic ime uncions Periodic impulse rain he periodic impulse rain is an imporan periodic ime uncion and we derive is Fourier series coeiciens s) s) δ n ) n he Fourier series coeiciens are ound as ollows S[n] /2 /2 s)e j2πn/ d, /2 δ n )e j2πn/ d, /2 n /2 /2 δ)e j2πn/ d 6 5 Periodic impulse rain, con d he Fourier series coeiciens are S[n] he ime uncion and specrum are shown below s) S[n] / 23n o summarize, he periodic impulse rain can be represened by is Fourier series, s) δ n ) e j2πn/ n n Periodic impulse rain, con d he Fourier series o he periodic impulse rain is s) δ n ) e j2πn/ n n I is no obvious ha he wo expressions are equal o invesigae his, we deine he parial sum o he Fourier series, s N ), s N ) N e j2πn/, n N and invesigae is behavior as N 7 8

3 Periodic impulse rain, con d he parial sum o he Fourier series is s N ) N e j2πn/ N e j2π/ ) n n N n N We can use he summaion ormula or a inie geomeric series Lecure ) o sum his series, e j2π/ ) N e j2π/ ) N+ s N ) e j2π/, s N ) e j2n+)π/ e j2n+)π/ e jπ/ e jπ/, s N ) sin2n +)π/ sin π/ 9 Periodic impulse rain, con d 2 s N ) N 2 N 5 N s N ) sin2n +)π/ sin π/ Noe ha his uncion is periodic wih period, and 2N + s N n ) he irs zero o s N ) isa 2N + hus, as N, each lobe ges larger and narrower o deermine i each lobe acs as an impulse, we need o ind is area Periodic impulse rain, con d he area o each period o s N ) is simply Area /2 /2 s N N) d e j2πn/ d /2 /2 n N N /2 n N /2 ej2πn/ d he inegral is zero excep when n when i equals so ha Area hus, each lobe o s N ) becomes: all, heigh is s N n )2N +)/ ; narrow, widh is 2/2N + ); and is area is hus, he parial sum approaches an ininie impulse rain o uni area, s) δ n ) e j2πn/ n n Fourier ransorm o a periodic uncion Fourier ransorm o a periodic impulse rain We have wo expressions or a periodic impulse rain, s) δ n ) e j2πn/ n n he Fourier ransorm o each expression is S) e j2πn δ n ) n n hereore, he Fourier ransorm o a periodic impulse rain in ime is a periodic impulse rain in requency s) F S) / 2

4 Relaion o Fourier series specrum o Fourier ransorm o a periodic impulse rain s) S[n] / 23n S) / Fourier ransorm o an arbirary periodic uncion Represenaion o a periodic uncion An arbirary periodic uncion can be generaed by convolving a pulse, x ), ha represens one period o he periodic uncion wih a periodic impulse rain, s) n δ n ), x) x ) s) x ) δ n ) x n ) n n x ) s) x) hereore, he Fourier ransorm o he periodic impulse rain has an impulse a he requency o each Fourier series componen and he area o he impulse equals he Fourier series coeicien 3 4 Fourier ransorm o a periodic uncion he Fourier ransorm o he periodic uncion is x) x ) s) F X ) X ) S) X ) X ) δ n ) X n δ n n x ) s) x) X ) S) X ) n ) 5 Fourier ransorm o a periodic uncion, con d An imporan conclusion is ha he Fourier ransorm o a periodic uncion consiss o impulses in requency a muliples o he undamenal requency hus, periodic coninuous ime uncions can be represened by a counably ininie number o complex exponenials 6

5 Fourier series coeiciens he Fourier ransorm o he periodic uncion is X ) X n δ n ) n Recall ha /2 X ) x )e j2π d /2 hereore, X n /2 x )e j2πn/ d X[n] /2 hereore, or an arbirary periodic coninuous ime uncion, he Fourier ransorm consiss o impulses locaed a he harmonic requencies) whose areas are he Fourier series coeiciens Fourier series o a square wave generaion o he square wave We will ind he Fourier series o a square wave by inding he Fourier ransorm o one period x ) 4 4 x) x) x ) s) 7 8 Fourier series o a square wave Fourier ransorm o one period o he square wave x ) x ) F X ) X ) 2 /2 sinπ/2) π/2 Fourier series o a square wave Fourier ransorm o square wave o obain he Fourier ransorm o he square wave, we ake he Fourier ransorm o one period o he square wave and muliply i by he Fourier ransorm o he periodic impulse rain X ) X ) S) sinπ/2) 2 π/2 n sinnπ/2) δ n ) n 2 nπ/2 δ n ), X ) 4 / X )

6 wo-minue miniquiz soluion wo-minue miniquiz problem Problem 8- Fourier series o he square wave a) Deermine he Fourier series coeiciens o he square wave Problem 8- Fourier series o he square wave a) Since he Fourier ransorm o he square wave is sinnπ/2) X ) δ n ), n 2 nπ/2 he Fourier series coeiciens are sinnπ/2) b) From he Fourier series coeiciens deermine he average value o he square wave b) X[] X[n] 2 nπ/2 x) /2 x) d /2 /2 / Fourier series o a square wave synhesis o a square wave We can synhesize he square wave by adding complex exponenials weighed by heir Fourier series coeiciens, x) X[n]e j2πn/ sinnπ/2) e j2πn/ n n 2 nπ/2 Noe ha he coeiciens are even uncions o n Hence, we can rewrie he series as ollows x) 2 + sinnπ/2) e j2πn/ + e j2πn/ ), n 2 nπ/2 x) sinnπ/2) 2 + cos2πn/ ) n nπ/2 Noe ha all he even harmonics o x) are zero excep or he erm or n 2 Fourier series o a square wave synhesis o a square wave, con d o invesigae he synhesis o he square wave, we consider he parial sum o he Fourier series, x N ) N sinnπ/2) 2 + cos2πn/ ) n nπ/ / N N N3 N5 N7 N9 N N3 22

7 Fourier series o a square wave synhesis o a square wave, con d Demo illusraing he Gibbs phenomenon Fourier series o a square wave Gibbs phenomenon As harmonics are added o synhesize he square wave, he parial sum o he Fourier series converges o he square wave everywhere excep near he disconinuiy where he parial sum akes on he value o /2 here are oscillaions on eiher side o he disconinuiy whose maximum over and undershoo approach 9% o he disconinuiy independen o N We can inerpre he Gibbs phenomenon by examining he parial sum o he Fourier series as a ilering problem Gibbs phenomenon inerpreaion as an ideal LPF o a square wave runcaion o he Fourier series o a square wave can be inerpreed in he requency domain as passing he square wave hrough an ideal lowpass iler ha runcaes he specrum ha can be inerpreed in he ime domain as he convoluion o he square wave wih a sinc uncion he response o he ideal lowpass iler o he disconinuiy o he square wave gives rise o he oscillaions X ) H) X N ) x) h) x N ) 25 Gibbs phenomenon sep response o an ideal LPF he Gibbs phenomenon can be invesigaed urher by examining he sep response o an ideal lowpass x) H) x) x w ) h) W H) x w ) Hence, sin2πw) x w ) u) h) hτ) dτ where h) 2W 2πW W 26

8 Gibbs phenomenon sep response o an ideal LPF, con d We evaluae he inegral sin2πwτ) x w ) 2W dτ, 2πWτ by changing variables y 2πW which yields 2πW sin y x w ) dy y his is closely relaed o a abulaed uncion called he sine inegral uncion Si) deined as sin y Si) dy y Gibbs phenomenon sep response o an ideal LPF, con d he sine inegral uncion, sin y Si) dy, y is ploed below 5 π/2 sin π/ Si) Gibbs phenomenon sep response o an ideal LPF, con d We express he sep wih he runcaed specrum in erms o he sine inegral uncion as x w ) 2 + π Si2πW), Conclusions ime Funcion Fourier ransorm Fourier Series X ) x) x) X ) X[n] which is shown ploed below or several values o W n 5 5 W W2 5 W3 W F aperiodic, con- Aperiodic, coninuous uncions o ime inuous uncions o requency Periodic, coninuous uncions o ime F impulses whose areas are he Fourier series coeiciens and locaed a discree requencies 29 3

9 Hisorical perspecive Jean Bapise Joseph Fourier ) Joseph Fourier, coninued Born o humble origins his aher was he own ailor, he was orphaned a age 9, and raised by a neighbor Joseph Fourier was born in Auxerre, France on March 2, 768 and died in Paris on May 4, 83 his is one o wo porrais ha has survived 3 Wen o miliary school and discovered mahemaics a age 3 augh mahemaics, rheoric, philosophy, and hisory in a Benedicine school in his home own Became acive in local poliics and social causes His poliical honesy go him in ho waer he was arresed irs by he Robespierre regime and laer by he enemy pos-robespierre regime He was jus barely saved rom he guilloine wice 32 Joseph Fourier, coninued Joseph Fourier, coninued augh a he Ecole Polyechnique where he developed an excellen repuaion as a lecurer and began publishing mahemaical research He augh wih Lagrange and Monge wo eminen mahemaicians When Napoleon Bonapare was pu in charge o a French expediion o Egyp, Fourier was chosen as a scieniic member He held poliical and adminisraive posiions in Egyp and proved o be an able adminisraor When he French occupaion o Egyp ended, Fourier reurned o eaching mahemaics in France In 82, Napoleon appoined Fourier as he preec o Isère, a French deparmen whose cener was Grenoble his was an imporan adminisraive/poliical posiion somewha akin o he governor o a US sae Fourier always hoped o complee his asks and reurn o a scholarly lie, bu he coninued o ake on imporan adminisraive poss hroughou his lie His mahemaical and scieniic sudies were largely par-ime eors 33 34

10 Joseph Fourier, coninued In 87, Fourier submied a paper on he use o rigonomeric series o solve problems in hea conducion o he Insiue o France I was reviewed by our amous French mahemaicians Lagrange, Laplace, Lacroix, and Monge hree o he our reerees voed o accep he paper, Lagrange was opposed Lagrange had been on one side o a raging conroversy on he represenaion o uncions by rigonomeric series Lagrange did no believe ha arbirary uncions could be expanded in rigonomeric series as Fourier s paper claimed Fourier s original paper was never published, bu in 822 he published his work in a book he Analyical heory o Hea in 822 Alhough he did no originae rigonomeric series, nor did he deermine he precise condiions or heir validiy, he did use hem o solve problems in hea conducion hus illusraing heir uiliy 35 Joseph Fourier, coninued Fourier s work and ha o hose ha ollowed him have had a proound eec on science and mahemaics Fourier ransorms are common ools in many dieren ields o science Fourier Fourier ransormed he image o Fourier le) is 582 by 582 pixels he logarihm o he magniude o he Fourier ransorm o his image is ploed on he righ