FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

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1 FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in many areas o engineering, and mos o you will discuss he mehod again in your second year mahemaics unis We consider here Fourier series expansions o periodic uncions, ie uncions which repea hemselves exacly a regular inervals wo examples are shown below Figure (a Figure (b De A uncion is periodic o period ( > i and only i ( + =( or all hereore he period is deined as he ime inerval required or one complee lucuaion Hence ( =cos is periodic wih period π since ( +π=cos(+π=cos=( or all NB I is periodic wih period, hen clearly rom he graphs, or rom repeaed use o he deiniion, is also periodic wih periods,3, you should choose he minimum period o he uncion o be is period Ex Deermine wheher he ollowing uncions are periodic and, i so, deermine he periods:- (i ( = sin(, (ii ( =cos( 3, (iii ( =cos+ sin( (i ( = sin( is periodic wih period π since ( + π =sin(+π=sin(+π=sin=( or all (ii ( =cos( 3 is periodic wih period π/ 3 since ( + π ( ( 3 =cos + π =cos( 3+π=cos( 3=( or all 3 3

2 (iii (more complicaed cos has periods π, 4π, 6π, sin has periods π, π, 3π, and clearly he minimum period or he sum o hese quaniies is he smalles number ha appears in boh liss o periods In his example clearly =π he uncion ( =cos+cos( would be even more diicul, since cos has periods π, 4π, 6π, cos( π has periods, 4π, 6π, Since he muliplicaive acors o π in he periods or cos are whole numbers bu he corresponding acors or cos( are irraional (always involving here is no number ha appears in boh liss Hence he uncion ( =cos+cos( is NO periodic (i never repeas isel, despie is comparaively simple orm More generally, we can say ha he sum o wo or more cosine waves will be periodic only when he raios o all pairs o periods orm raional numbers (raios o inegers Sine wave Le us recall deiniions linked o a sine wave sin φ φ/ω Figure Consider ( = sin(ω + φ = sin(πν + φ, shown above, where is ampliude, ω is he circular (or angular requency in radians/uni ime, ν is requency in cycles/uni ime (Herz φ is phase angle wih respec o ime origin in radians he period o above sine wave is /ν =π/ω seconds A posiive phase angle φ shis waveorm o he le (a lead and a negaive phase angle moves waveorm o he righ (a lag π/ω Whole range Fourier series Fourier analysis decomposes a complicaed periodic wave shape ino a sum o sine and cosine waves Suppose ( = ( ( ] πn πn a + a n cos + b n sin, ( n= where ( is a periodic uncion o period and a n and b n are he (consan Fourier coeiciens o (, deined by a n = +/ ( πn (cos d, n =,,, ( /

3 b n = +/ / ( πn (sin d, n =,,3 (3 Equaions (, ( and (3 appear on he Formula Shee (NB a and a n (or n> are deined by he same inegral (, bu a mus always be calculaed separaely since orm o inegral changes when n = Equaions ( (3 also appear in he Daa Book bu in a slighly dieren orm since ha book assumes he uncion has period L (ge amiliar wih he equaions you mus use Is he orm ( reasonable? Does he ininie sum converge? Well, given a periodic uncion ( i we calculae is Fourier coeiciens using he inegrals ( and (3, and hen wrie ( a + n= a n cos ( πn + b n sin ( ] πn, (4 where he series on he RHS o (4 is called he whole-range Fourier series o (, i can be proved ha he ininie series in (4 does converge, under cerain condiions, or a wide variey o uncions Suicien condiions or convergence, known as he Dirichle condiions, are saed below I ( is a periodic uncion wih period and (i is piecewise coninuous in he inerval / <<+/ (ie i is bounded in he inerval and is coninuous in he inerval excep a a inie number o poins, (ii has only a inie number o maxima and minima in he inerval, ] hen he Fourier series converges a a poin o he value lim ( + lim ( + he value wrien above represens he average o he le and righ hand limis o a (see Figure 3 Obviously, a all poins a which he uncion ( is coninuous, he le and righ-hand limis o are he same, so he Fourier series converges o ( value o Fourier series a Figure 3 NB he symbol is used in equaion (4 o show ha ( is no necessarily equal o he series on he righ (as above discussion shows Ex Find he Fourier series o he uncion deined by {, ( = < +, < (+=( 3

4 For his uncion (shown in Figure 4 Figure 4 a = +/ / = ] a n = = +/ / (cos(d = + ] = ( +/ / ( (d = + ( { πn (cos d = { sin(πn/ (πn/ ] + { ( ( cos / } ] sin(πn/ (πn/ ( d + / =; ( πn d + +/ / } (+ d = { sin ( sin(n + sin(πn sin } = ( sin(πn + sin(πn = ; πn πn b n = +/ ( { πn (sin d = ( πn ( sin d + / / { cos(πn/ = ] + cos(πn/ ] } (πn/ (πn/ / ( } πn (+ cos d ( } πn (+ sin d = {cos cos(n cos(πn+cos}= { cos(πn} = πn πn πn { ( n } In he calculaion o b n he resul cos(nπ =( n has been used, since cos(nπ alernaes beween and + or all ineger n When n is even, ( n = + and hence b n = When n is odd hen ( n =, and b n = 4 ( ( = nπ nπ hus b = 4 π, b =, b 3 = 4 3π, b 4 =, and i ω =π/ we obain ( 4 π sin(ω + 3 sin(3ω + ] 5 sin(5ω + 4

5 Parial sums De he mh parial sum S m ( is he sum o a Fourier series up o he erms in sin(πm/ and cos(πm/ m For he Fourier series in Ex i ollows ha S m ( = b n sin(nω soha n= S (= 4 π sin ω S 3 (= 4 sin ω + ] π 3 sin 3ω S 5 ( = 4 sin ω + π 3 sin 3ω + ] 5 sin 5ω A =, a poin o disconinuiy o he original uncion, all erms in he calculaed Fourier series are zero so he oal sum is also zero his illusraes he resul ha he Fourier series is convergen a =o ] lim ( + lim ( = + ( ] = + 3 Even and odd uncions In Ex he Fourier coeiciens were calculaed direcly rom he inegral deiniions In ha example, and many ohers, calculaions can be simpliied by using properies o he uncion ( Recall a uncion ( iseven i ( =(; a uncion ( isodd i ( = ( For example, ( =cosis even since ( =cos( =cos=(, ( =sinis odd since ( = sin( = sin = ( Noe ha he produc o wo even uncions and g is even, since ( g( =(g(; he produc o wo odd uncions and g is even, since ( g( =( ( ( g( = ( g( bu he produc o an odd uncion wih an even uncion g is odd, since ( g( =( ( g( = (g( he crucial resuls concern he inegrals o odd and even uncions From he deiniions o hese uncions, or rom heir graphs, i is easily seen ha i ( isodd hen i ( iseven hen a a a a ( d =; ( d = hese inegral resuls are illusraed below: a (d a A A a A a A a Figure 5(a Figure 5(b 5

6 Figure 4 shows ha he uncion in Ex is odd, and so use o he above inegral resuls would have simpliied he calculaion o he a n and b n, in an analogous way o heir use in he example below Ex 3 Deermine he Fourier series expansion o he uncion ( deined by: i </, ( = + i / <π/, and ( +π=( or all i π/ <π, ( is shown on Figure 6 clearly uncion is even wih period =π(or L =π, i you are using he Daa Book 3π Figure 6 π π 3π Using he general orm or he Fourier coeiciens, wih =π(or L = π, b n = π +π ( nπ (sin d = π π +π ( sin(n d Since ( is even and sin(n is odd, he produc o hese uncions is odd and hence he inegral is zero, ie b n = For he coeicien a, a = +π ( cos(d = +π ( d, π π and since ( is even i is possible o rewrie he laer o give urning o he coeiciens a n, a = { +π ( d = +π/ π π = { } ] π/ + ] π π/ = π π a n = π +π (cos } π d + ( d π/ { π + ( ( nπ d = π π +π ( π } = (cos(n d Boh ( and cos(n are even, so he produc is even and hence a n = { +π (cos(n d = +π/ } π cos(n d + ( cos(n d π π π/ { sin = ] π/ ] } π n sin n π n n π/ = { ( nπ ( nπ } sin sin sin(nπ+sin = 4 ( nπ πn πn sin 6

7 Now ( nπ sin = { i n is even, + i n =,5,9, i n =3,7,, so i n is even, 4 a n = πn i n =,5,9, 4 πn i n =3,7,, and, combining ogeher hese resuls, gives he inal soluion ( ( nπ a n cos = {cos cos(3+ cos(5 } π π n= Graphs o sums o various numbers o non-zero erms in his soluion (ie he parial sums S n are shown on he nex page he igures show ha he series becomes closer o he given uncion ( as more erms are included, bu here will always remain considerable error in he viciniy o a disconinuiy no maer how many erms you use In rying o achieve he ininie slope a he disconinuiy, he parial sum acually overshoos hese addiional peaks become o vanishingly small widh as he number o erms increases (see he igures, bu he overshoo isel does no reduce o zero his resul is known as Gibbs phenomenon Inhecaseo he square wave in Ex 3 he smalles magniude o he overshoo is abou 9% Jusiicaion o ormulae in Fourier series Finally we jusiy he Fourier series ormulae he proos involve orhogonaliy relaions which will now be saed and proved: / ( ( πm πn Resul cos sin d =, or all m and n (5 / ( ( πm πn (5 holds because cos is even, sin is odd, hence he produc is odd and he above inegral is zero / ( ( πm πn Resul cos cos d =, or all m n (6 / On he LHS o (6 boh cosine uncions are even, so he produc is even he inegral on he LHS can hereore be expressed / ( ( πm πn cos cos d On using he rigonomeric ideniies on he Formula shee, or in he Daa Book, he above inegral becomes / ( ( πm πn / { ( ( } π(m + n π(m n cos cos d = cos +cos d ] / sin(π(m + n/ sin(π(m n/ = + π(m + n/ π(m n/ =, since sin(m nπ = sin(m + nπ = Using a very similar mehod (which is no given here i can be shown ha Resul 3 / / ( πm sin sin ( πn d =, or all m n (7 7

8 Figures or his page o come 8

9 he case m = n or he inegrals in (6 and (7 need o be reaed separaely: Resul 4 / / ( πm cos d =, m (8 o prove he laer / / ( πm cos d = = / + ( πm cos d = sin (4πm/ 4πm/ Using an analogous argumen i can be shown ha Resul 5 / / ] / / d =, ( +cos ( 4πm d since sin πm =sin= ( πm sin d =, m (9 he relaions (5 o (9 can be used o jusiy he expressions or he Fourier coeiciens as ollows he general ormulae or Fourier series were saed earlier, equaions ( o (4 Muliply boh sides o ( by he quaniy cos(πm/ and inegrae rom /o /ogive / ( πm (cos d = / ( a πm / / cos d { / ( ( }] πm πn + a n cos cos d n= / { / ( ( }] πm πn + b n cos sin d n= / ( Relaion (5 implies ha he hird inegral on he RHS o ( is always zero When m = hen he irs inegral on RHS o ( is, bu he second inegral = / Hence a = (d / When m, he irs inegral on RHS o ( is zero bu, using (6 and (8, he second inegral is non-zero only when m = n I ollows ha a m = / ( πm (cos d / In a similar way, on muliplying ( by sin(πm/ and inegraing, we can show b m = / ( πm (sin d / Dirichle condiions are needed o ensure he convergence o he various inegrals, and he legiimacy o inerchanging he orders o inegraion and summaion which has been used in wriing down (] rec/ls 9