23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes


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1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals involving sinusoids. We hen assume ha if f() is a periodic funcion, of period, hen he Fourier series expansion akes he form: f() = a + (a n cos n + b n sin n) Our main purpose here is o show how he consans in his expansion, a n (for n =,,, 3... and b n (for n =,, 3,... ), may be deermined for any given funcion f(). Prerequisies Before saring his Secion you should... Learning Oucomes On compleion you should be able o... know wha a periodic funcion is be able o inegrae funcions involving sinusoids have knowledge of inegraion by pars calculae Fourier coefficiens of a funcion of period calculae Fourier coefficiens of a funcion of general period HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 9
2 . Inroducion We recall firs a simple rigonomeric ideniy: cos = + cos or equivalenly cos = + cos () Equaion can be inerpreed as a simple finie Fourier series represenaion of he periodic funcion f() = cos which has period. We noe ha he Fourier series represenaion conains a consan erm and a period erm. A more complicaed rigonomeric ideniy is sin 4 = 3 8 cos + cos 4 () 8 which again can be considered as a finie Fourier series represenaion. (Do no worry if you are unfamiliar wih he resul ().) Noe ha he funcion f() = sin 4 (which has period ) is being wrien in erms of a consan funcion, a funcion of period or frequency (he firs harmonic ) and a funcion of period or frequency (he second harmonic ). The reason for he consan erm in boh () and () is ha each of he funcions cos and sin 4 is nonnegaive and hence each mus have a posiive average value. Any sinusoid of he form cos n or sin n has, by symmery, zero average value. Therefore, so would a Fourier series conaining only such erms. A consan erm can herefore be expeced o arise in he Fourier series of a funcion which has a nonzero average value.. Funcions of period We now discuss how o represen periodic nonsinusoidal funcions f() of period in erms of sinusoids, i.e. how o obain Fourier series represenaions. As already discussed we expec such Fourier series o conain harmonics of frequency n (n =,, 3,...) and, if he periodic funcion has a nonzero average value, a consan erm. Thus we seek a Fourier series represenaion of he general form f() = a + a cos + a cos b sin + b sin +... The reason for labelling he consan erm as a will be discussed laer. The ampliudes a, a,... b, b,... of he sinusoids are called Fourier coefficiens. Obaining he Fourier coefficiens for a given periodic funcion f() is our main ask and is referred o as Fourier Analysis. Before embarking on such an analysis i is insrucive o esablish, a leas qualiaively, he plausibiliy of approximaing a funcion by a few erms of is Fourier series. HELM (8): Workbook 3: Fourier Series
3 Task Consider he square wave of period one period of which is shown in Figure. 4 (a) Wrie down he analyic descripion of his funcion, (b) Sae wheher you expec he Fourier series of his funcion o conain a consan erm, (c) Lis any oher possible feaures of he Fourier series ha you migh expec from he graph of he squarewave funcion. Your soluion (a) We have f() = f( + ) = f() 4 < < < <, < < (b) The Fourier series will conain a consan erm since he square wave here is nonnegaive and canno herefore have a zero average value. This consan erm is ofen referred o as he d.c. (direc curren) erm by engineers. (c) Since he square wave is an even funcion (i.e. he graph has symmery abou he y axis) hen is Fourier series will conain cosine erms bu no sine erms because only he cosines are even funcions. (Well done if you spoed his a his early sage!) HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series
4 I is possible o show, and we will do so laer, ha he Fourier series represenaion of his square wave is + 8 {cos 3 cos cos 5 7 } cos i.e. he Fourier coefficiens are a =, a = 8, a =, a 3 = 8 3, a 4 =, a 5 = 8 5,... Noe, as well as he presence of he consan erm and of he cosine (bu no sine) erms, ha only odd harmonics are presen i.e. sinusoids of period, 3, 5,,... or of frequency, 3, 5, 7,... 7 imes he fundamenal frequency. We now show in Figure 8 graphs of (i) he square wave (ii) he firs wo erms of he Fourier series represening he square wave (iii) he firs hree erms of he Fourier series represening he square wave (iv) he firs four erms of he Fourier series represening he square wave (v) he firs five erms of he Fourier series represening he square wave Noe: We show he graphs for < < only since he square wave and is Fourier series are even. (i) 4 (ii) (iii) + 8 cos + 8 (cos cos 3 ) (iv) + 8 (v) (cos 3 cos cos 5 ) (cos 3 cos cos 5 cos 7 ) Figure 8 HELM (8): Workbook 3: Fourier Series
5 We can clearly see from Figure 8 ha as he number of erms is increased he graph of he Fourier series gradually approaches ha of he original square wave  he ripples increase in number bu decrease in ampliude. (The behaviour near he disconinuiy, a =, is slighly more complicaed and i is possible o show ha however many erms are aken in he Fourier series, some overshoo will always occur. This effec, which we do no discuss furher, is known as he Gibbs Phenomenon.) Orhogonaliy properies of sinusoids As saed earlier, a periodic funcion f() wih period has a Fourier series represenaion f() = a + a cos + a cos b sin + b sin +..., = a + (a n cos n + b n sin n) (3) To deermine he Fourier coefficiens a n, b n and he consan erm a use has o be made of cerain inegrals involving sinusoids, he inegrals being over a range α, α +, where α is any number. (We will normally choose α =.) Task Find sin n d and cos n d where n is an ineger. Your soluion In fac boh inegrals are zero for [ sin n d = ] n cos n = ( cos n + cos n) = n n (4) cos n d = [ ] sin n = n (5) n As special cases, if n = he firs inegral is zero and he second inegral has value. N.B. Any inegraion range α, α +, would give hese same (zero) answers. These inegrals enable us o calculae he consan erm in he Fourier series (3) as in he following ask. HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 3
6 Task Inegrae boh sides of (3) from o and use he resuls from he previous Task. Hence obain an expression for a. Your soluion We ge for he lefhand side f()d (whose value clearly depends on he funcion f()). Inegraing he righhand side erm by erm we ge { } a d + a n cos n d + b n sin n d = (using he inegrals (4) and (5) shown above). Thus we ge f() d = (a ) or a = [ ] a + { + } f() d (6) Key Poin The consan erm in a rigonomeric Fourier series for a funcion of period is a = f() d = average value of f() over period. 4 HELM (8): Workbook 3: Fourier Series
7 This resul ies in wih our earlier discussion on he significance of he consan erm. Clearly a signal whose average value is zero will have no consan erm in is Fourier series. The following square wave (Figure 9) is an example. f() Figure 9 We now obain furher inegrals, known as orhogonaliy properies, which enable us o find he remaining Fourier coefficiens i.e. he ampliudes a n and b n (n =,, 3,...) of he sinusoids. Task Using he sandard rigonomeric ideniy ha sin n cos m {sin(n + m) + sin(n m)} evaluae sin n cos m d where n and m are any inegers. Your soluion We ge sin n cos m d = { sin(n + m) d + using he resuls (4) and (5) since n + m and n m are also inegers. This resul holds for any inerval of. } sin(n m) d = { + } = HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 5
8 Key Poin Orhogonaliy Relaion For any inegers m, n, including he case m = n, sin n cos m d = We shall use his resul shorly bu need a few more inegrals firs. Consider nex cos n cos m d where m and n are inegers. Using anoher rigonomeric ideniy we have, for he case n m, cos n cos m d = {cos(n + m) + cos(n m)}d = { + } = using he inegrals (4) and (5). For he case n = m we mus ge a nonzero answer since cos n is nonnegaive. In his case: cos n d = = ( + cos n) d [ + sin n n ] = ( provided n ) For he case n = m = we have cos n cos m d = Task Proceeding in a similar way o he above, evaluae sin n sin m d for inegers m and n. Again consider separaely he hree cases: (a) n m, (b) n = m and (c) n = m =. 6 HELM (8): Workbook 3: Fourier Series
9 Your soluion (a) Using he ideniy sin n sin m {cos(n m) cos(n + m)} and inegraing he righhand side erms, we ge, using (4) and (5) sin n sin m d = n, m inegers n m (b) Using he ideniy cos θ = sin θ wih θ = n gives for n = m (c) When n = m =, sin n d = sin n sin m d =. We summarise hese resuls in he following Key Poin: ( cos n)d = For inegers n, m Key Poin 3 sin n cos m d = cos n cos m d = sin n sin m d = n m n = m n = m = { n m, n = m = n = m All hese resuls hold for any inegraion range of widh. HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 7
10 3. Calculaion of Fourier coefficiens Consider he Fourier series for a funcion f() of period : f() = a + (a n cos n + b n sin n) (7) To obain he coefficiens a n (n =,, 3,...), we muliply boh sides by cos m where m is some posiive ineger and inegrae boh sides from o. For he lefhand side we obain f() cos m d For he righhand side we obain a cos m d + {a n cos n cos m d + b n The firs inegral is zero using (5). } sin n cos m d Using he orhogonaliy relaions all he inegrals in he summaion give zero excep for he case n = m when, from Key Poin 3 cos m d = Hence f() cos m d = a m from which he coefficien a m can be obained. Rewriing m as n we ge a n = f() cos n d for n =,, 3,... (8) Using (6), we see he formula also works for n = (bu we mus remember ha he consan erm is a.) From (8) a n = average value of f() cos n over one period. 8 HELM (8): Workbook 3: Fourier Series
11 Task By muliplying (7) by sin m obain an expression for he Fourier Sine coefficiens b n, n =,, 3,... Your soluion A similar calculaion o ha performed o find he a n gives f() sin m d = a sin m d + { a n cos n sin m d + All erms on he righhand side inegrae o zero excep for he case n = m where b m sin m d = b m Relabelling m as n gives b n = } b n sin n sin m d f() sin n d n =,, 3,... (9) (There is no Fourier coefficien b.) Clearly b n = average value of f() sin n over one period. HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 9
12 Key Poin 4 A funcion f() wih period has a Fourier series f() = a + (a n cos n + b n sin n) The Fourier coefficiens are a n = b n = f() cos n d n =,,,... f() sin n d n =,,... In he inegrals any convenien inegraion range exending over an inerval of may be used. 4. Examples of Fourier series We shall obain he Fourier series of he halfrecified square wave shown in Figure. f() period We have Figure f() = f( + ) = f() { < < < < The calculaion of he Fourier coefficiens is merely sraighforward inegraion using he resuls already obained: a n = f() cos n d in general. Hence, for our square wave a n = () cos n d = [ sin n n ] = provided n HELM (8): Workbook 3: Fourier Series
13 Bu a = () d = so he consan erm is a =. (The square wave akes on values and over equal lengh inervals of so value.) Similarly b n = () sin n d = Some care is needed now! b n = ( cos n) n Bu cos n = + n =, 4, 6,..., [ ] cos n n is clearly he mean b n = n =, 4, 6,... However, cos n = n =, 3, 5,... b n = ( ( )) = n n i.e. b =, b 3 = 3, b 5 = 5,... Hence he required Fourier series is n =, 3, 5,... f() = a + (a n cos n + b n sin n) in general f() = + (sin + 3 sin sin ) in his case Noe ha he Fourier series for his paricular form of he square wave conains a consan erm and odd harmonic sine erms. We already know why he consan erm arises (because of he nonzero mean value of he funcions) and will explain laer why he presence of any odd harmonic sine erms could have been prediced wihou inegraion. The Fourier series we have found can be wrien in summaion noaion in various ways: + (n odd) Fourier series as sin n or, since n is odd, we may wrie n = k k =,,... and wrie he n + k= sin(k ) (k ) HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series
14 Task Obain he Fourier series of he square wave one period of which is shown: 4 Your soluion HELM (8): Workbook 3: Fourier Series
15 We have, since he funcion is nonzero only for < <, a = 4 d = 4 a = is he consan erm as we would expec. Also a n = = 4 n { sin 4 cos n d = 4 ( n } [ sin n n ( sin n )) ] = 8 n sin ( n I follows from a knowledge of he sine funcion ha n =, 4, 6,... ) n =,, 3,... Also a n = b n = 8 n 8 n n =, 5, 9,... n = 3, 7,,... 4 sin n d = 4 [ ] cos n n = 4 { ( n cos n Hence, he required Fourier series is f() = + 8 (cos 3 cos cos 5 7 ) cos ) ( cos n )} = which, like he previous square wave, conains a consan erm and odd harmonics, bu in his case odd harmonic cosine erms raher han sine. You may recall ha his paricular square wave was used earlier and we have already skeched he form of he Fourier series for, 3, 4 and 5 erms in Figure 8. Clearly, in finding he Fourier series of square waves, he inegraion is paricularly simple because f() akes on piecewise consan values. For oher funcions, such as sawooh waves his will no be he case. Before we ackle such funcions however we shall generalise our formulae for he Fourier coefficiens a n, b n o he case of a periodic funcion of arbirary period, raher han confining ourselves o period. HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 3
16 5. Fourier series for funcions of general period This is a sraighforward exension of he period case ha we have already discussed. Using x (insead of ) emporarily as he variable. We have seen ha a periodic funcion f(x) has a Fourier series f(x) = a + (a n cos nx + b n sin nx) wih a n = f(x) cos nx dx n =,,,... b n = f(x) sin nx dx n =,,... Suppose we now change he variable o where x = T. Thus x = corresponds o = T/ and x = corresponds o = T/. Hence regarded as a funcion of, we have a funcion wih period T. Making he subsiuion x =, and hence dx = a n = T b n = T T T T T T T d, in he expressions for a n and b n we obain ( ) n f() cos d T n =,,... ( ) n f() sin d T n =,... These inegrals give he Fourier coefficiens for a funcion of period T whose Fourier series is f() = a + { ( ) ( )} n n a n cos + b n sin T T Various oher noaions are commonly used in his case e.g. i is someimes convenien o wrie he period T = l. (This is paricularly useful when Fourier series arise in he soluion of parial differenial equaions.) Anoher alernaive is o use he angular frequency ω and pu T = /ω. Task Wrie down he form of he Fourier series and expressions for he coefficiens if (a) T = l (b) T = /ω. Your soluion 4 HELM (8): Workbook 3: Fourier Series
17 (a) f() = a + { ( ) ( )} n n a n cos + b n sin wih a n = l ( ) n f() cos d l l l l l and similarly for b n. (b) f() = a + {a n cos(nω) + b n sin(nω)} wih a n = ω ω f() cos(nω) d ω and similarly for b n. You should noe ha, as usual, any convenien inegraion range of lengh T (or l or ω used in evaluaing a n and b n. ) can be Example Find he Fourier series of he funcion shown in Figure which is a sawooh wave wih alernae porions removed. f() Figure Soluion Here he period T = l = 4 so l =. The Fourier series will have he form f() = a + { ( ) ( )} n n a n cos + b n sin The coefficiens a n are given by a n = ( ) n f() cos d where { < < f() = < < Hence a n = ( ) n cos d. f( + 4) = f() HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 5
18 Soluion (cond.) The inegraion is readily performed using inegraion by pars: ( ) n cos d = = = [ 4 n n sin [ cos ( )] n ( )] n 4 (cos n ). n Hence, since a n = cos( n )d n =, 4, 6,... a n = 4 n =, 3, 5,... n The consan erm is a where a = Similarly b n = ( ) n sin d where ( ) n sin d = [ n cos ( )] n d =. n + n The second inegral gives zero. Hence b n = n =, 4, 6,... n cos n = n + n =, 3, 5,... n ( ) n sin d n ( ) n cos d. Hence, using all hese resuls for he Fourier coefficiens, he required Fourier series is f() = 4 ( ) {cos + ( ) 3 9 cos + ( ) } 5 5 cos { ( ) sin ( ) sin + ( ) } 3 3 sin... Noice ha because he Fourier coefficiens depend on n (raher han as was he case for n he square wave) he sinusoidal componens in he Fourier series have quie rapidly decreasing ampliudes. We would herefore expec o be able o approximae he original sawooh funcion using only a quie small number of erms in he series. 6 HELM (8): Workbook 3: Fourier Series
19 Task Obain he Fourier series of he funcion f() = < < f( + ) = f() f() Firs wrie ou he form of he Fourier series in his case: Your soluion Since T = l = and since he funcion has a nonzero average value, he form of he Fourier series is a + {a n (cos n) + b n sin(n)} Now wrie ou inegral expressions for a n and b n. Will here be a consan erm in he Fourier series? Your soluion Because he funcion is nonnegaive here will be a consan erm. Since T = l = hen l = and we have a n = b n = cos(n) d n =,,,... sin(n) d n =,,... The consan erm will be a where a = d. HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 7
20 Now evaluae he inegrals. Try o spo he value of he inegral for b n so as o avoid inegraion. Noe ha he inegrand is an even funcions for a n and an odd funcon for b n. Your soluion The inegral for b n is zero for all n because he inegrand is an odd funcion of. Since he inegrand is even in he inegrals for a n we can wrie a n = cos n d n =,,,... The consan erm will be a o where a = d = 3. For n =,, 3,... we mus inegrae by pars (wice) { [ ] a n = n sin(n) } sin(n) d n { [ = 4 ] n n cos(n) + } cos(n) d. n The inegral in he second erm gives zero so a n = 4 cos n. n Now wriing ou he final form of he Fourier series we have f() = cos n cos(n) = n { cos() + 4 cos() 9 } cos(3) HELM (8): Workbook 3: Fourier Series
21 Exercises For each of he following periodic signals skech he given funcion over a few periods find he rigonomeric Fourier coefficiens wrie ou he firs few erms of he Fourier series. < < /. f() = f( + ) = f() / < < square wave. f() = < < f( + ) = f() T/ < < 3. f() = f( + T ) = f() < < T/ < < 4. f() = f( + ) = f() < < T/ < < 5. f() = A sin T < < T/ s { cos 3 cos 3 }... f( + T ) = f() cos { sin sin 3 sin 5 sin { } cos cos 3 cos 4 cos { 4 sin ω + 3 sin 3 ω + 5 } sin 5ω +... where ω = /T. { 6 cos + A + A sin ω A square wave halfwave recifier sin 6 6 } +... } cos cos { ( 4 ) sin ( sin ) sin 3 4 } 3 3 sin { } cos ω cos 4ω + ()(3) (3)(5) +... HELM (8): Secion 3.: Represening Periodic Funcions by Fourier Series 9
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