Fourier Series Approximation of a Square Wave


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1 OpenSaxCNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSaxCNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier series o approximae a square wave, as opposed o he sinusoidal waves seen previously. he Fourier series represenaion of a signal, as expressed for a square wave by s ( = a + ( 2πk a k cos + k= ( 2πk b k sin k= ( says ha he lef and righ sides are "equal". We need o invesigae equaliy hrough an example. Example Version 2.6: Jun 4, 29 2: pm 5 hp://creaivecommons.org/licenses/by/. hp://cnx.org/conen/m4/2.6/
2 OpenSaxCNX module: m4 2 Fourier series approximaion of a square wave K=  K=5  K=  K=49  Figure : Fourier series approximaion o sq (. he number of erms in he Fourier sum is indicaed in each plo, and he square wave is shown as a dashed line over wo periods. Le's nd he specrum of he square wave sq (. he expressions for he Fourier coeciens have he common form a k b k = 2 2 cos ( 2πk sin ( 2πk d 2 he cosine coeciens a k are all zero, and he sine coeciens are 4 πk if k is odd b k = if k is even (3 2 cos ( 2πk sin ( 2πk d (2 hus, he Fourier series for he square wave is sq ( = k {,3,... } ( 4 2πk πk sin (4 hp://cnx.org/conen/m4/2.6/
3 OpenSaxCNX module: m4 3 As we see in Figure (Fourier series approximaion of a square wave, he Fourier series requires many more erms o provide he same qualiy of approximaion as we found wih he halfwave recied sinusoid. We can verify ha more erms are needed by considering he power specrum and he approximaion error shown in Figure 2 (Power specrum and rms error. P s (k Power specrum and rms error k Relaive rms error K Figure 2: he upper plo shows he power specrum of he square wave, and he lower plo he rms error of he nielengh Fourier series approximaion o he square wave. he aserisk denoes he rms error when he number of erms K in he Fourier series equals 99. his dierence beween he wo Fourier series resuls because he halfwave recied sinusoid's Fourier coeciens are proporional o k while hose of he square wave are proporional o 2 k. In shor, he square wave's coeciens decay more slowly wih increasing frequency. Said anoher way, he squarewave's specrum conains more power a higher frequencies han does he halfwaverecied sinusoid. Exercise (Soluion on p. 6. Calculae he harmonic disorion for he square wave. hp://cnx.org/conen/m4/2.6/
4 OpenSaxCNX module: m4 4 Figure 3: Fourier series approximaion o sq (. he number of erms in he Fourier sum is indicaed in each plo, and he square wave is shown as a dashed line over wo periods. When comparing he square wave o is Fourier series represenaion i is no clear ha he wo are equal. he fac ha he square wave's Fourier series requires more erms for a given represenaion accuracy is no imporan. However, close inspecion of Figure 3 does reveal a poenial issue: Does he Fourier series really equal he square wave a all values of? In paricular, a each sepchange in he square wave, he Fourier series exhibis a peak followed by rapid oscillaions. As more erms are added o he series, he oscillaions seem o become more rapid and smaller, bu he peaks are no decreasing. Consider his mahemaical quesion inuiively: Can a disconinuous funcion, like he square wave, be expressed as a sum, even an innie one, of coninuous ones? One should a leas be suspicious, and in fac, i can' be hus expressed. his issue brough Fourier much criicism from he French Academy of Science (Laplace, Legendre, and Lagrange comprised he review commiee for several years afer is presenaion on 87. I was no resolved for also a cenury, and is resoluion is ineresing and imporan o undersand from a pracical viewpoin. hp://wwwgroups.dcs.sand.ac.uk/ hisory/mahemaicians/fourier.hml hp://cnx.org/conen/m4/2.6/
5 OpenSaxCNX module: m4 5 he exraneous peaks in he square wave's Fourier series never disappear; hey are ermed Gibb's phenomenon afer he American physicis Josiah Willard Gibbs. hey occur whenever he signal is disconinuous, and will always be presen whenever he signal has jumps. Le's reurn o he quesion of equaliy; how can he equal sign in he deniion of he Fourier series be jusied? he parial answer is ha poinwiseeach and every value of equaliy is no guaraneed. Wha mahemaicians laer in he nineeenh cenury showed was ha he rms error of he Fourier series was always zero. limi rms (ɛ K = (5 K Wha his means is ha he dierence beween an acual signal and is Fourier series represenaion may no be zero, bu he square of his quaniy has zero inegral! I is hrough he eyes of he rms value ha we dene equaliy: wo signals s (, s 2 ( are said o be equal in he mean square if rms (s s 2 =. hese signals are said o be equal poinwise if s ( = s 2 ( for all values of. For Fourier series, Gibb's phenomenon peaks have nie heigh and zero widh: he error diers from zero only a isolaed poins whenever he periodic signal conains disconinuiiesand equals abou 9% of he size of he disconinuiy. he value of a funcion a a nie se of poins does no aec is inegral. his eec underlies he reason why dening he value of a disconinuous funcion, like we refrained from doing in dening he sep funcion 2, a is disconinuiy is meaningless. Whaever you pick for a value has no pracical relevance for eiher he signal's specrum or for how a sysem responds o he signal. he Fourier series value "a" he disconinuiy is he average of he values on eiher side of he jump. 2 "Elemenal Signals": Secion Uni Sep <hp://cnx.org/conen/m4/laes/#sepdef> hp://cnx.org/conen/m4/2.6/
6 OpenSaxCNX module: m4 6 Soluions o Exercises in his Module Soluion o Exercise (p. 3 oal harmonic disorion in he square wave is ( 4 π 2 = 2%. hp://cnx.org/conen/m4/2.6/
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