23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

Transcription

1 Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl easier o obain as, if he signal is even onl cosines are involved whereas if he signal is odd hen onl sines are involved. We also show ha if a signal reverses afer half a period hen he Fourier series will onl conain odd harmonics Prerequisies Before saring his Secion ou should... Learning Oucomes Afer compleing his Secion ou should be able o... 1 know how o obain a Fourier series be able o inegrae funcions involving sinusoids 3 have knowledge of inegraion b pars deermine if a funcion is even or odd or neiher easil calculae Fourier coefficiens of even or odd funcions

2 1. Even and Odd Funcions We have shown in he previous Secion how o calculae, b inegraion, he coefficiens a n (n =, 1,, 3,...) and b n (n =1,, 3,...)inaFourier Series. Clearl his is a somewha edious process and i is advanageous if we can obain as much informaion as possible wihou recourse o inegraion. Referring back o he examples in he previous Secion we showed ha he square wave (one period of which shown in Figure 1(a)) had a Fourier Series conaining a consan erm and cosine erms (i.e. all he Fourier coefficiens b n were zero) while he funcion shown in Figure 1(b) had a more complicaed Fourier Series conaining boh cosine and sine erms as well as a consan. 4 Figure 1(a) f() Figure 1(b) Conras he smmer or oherwise of hese wo funcions. Your soluion The square wave in Figure 1(a) has a graph which is smmerical abou he xis and is called an even funcion. The sawooh shown in Figure 1(b) has no paricular smmer. In general a funcion is called even if is graph is unchanged under reflecion in he xis. This is equivalen o f( ) =f() for all Obvious examples of even funcions are, 4,, cos, cos, sin, cos n. A funcion is said o be odd if is graph is smmerical abou he origin. This is equivalen o he condiion f( ) = f() HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions

3 Figure shows an example of an odd funcion. f() Figure Examples of odd funcions are, 3, sin, sin n. Aperiodic funcion which is odd is he saw-ooh wave in Figure 3. 1 f() Figure 3 Some funcions are neiher even nor odd. The periodic sawooh of Figure 1(b) is an example, as is he exponenial funcion e. Sae wheher each of he following periodic funcions is even or odd or neiher. (i) f() (ii) f() 4 4 Figure 4 Your soluion (i) is neiher even nor odd (wih period ) (ii) is odd (wih period ). 3 HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions

4 A Fourier Series conains a sum of erms while he inegral formulae for he Fourier coefficiens a n and b n conain producs of he pe f() cos n and f() sin n (for he case where f() has period.) We need herefore resuls for sums and producs of funcions. Suppose, for example, g() is an odd funcion and h() is an even funcion. Le F 1 () = g() h() (produc of odd and even funcions) so F 1 ( ) = g( )h( ) (replacing b ) = ( g())h() (since g is odd and h is even) = g()h() = F () so F 1 () isodd. Now suppose F () = g()+h() (sum of odd and even funcions) F ( ) = g( )+h() = g()+h() We see ha F ( ) F () and F ( ) F () So F () isneiher even nor odd. Use similar approaches o he above o invesigae sums and producs of (i) wo odd funcions g 1 (),g () (ii) wo even funcions h 1 (),h () Your soluion HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions 4

5 so he produc of wo odd funcions is even. If G 1 () = g 1 ()g () G 1 ( ) = ( g 1 ())( g ()) = g 1 ()g () = G 1 () G () = g 1 ()+g () G ( ) = g 1 ( )+g ( ) = g 1 () g () = G () so he sum of wo odd funcions is odd. If H 1 () = h 1 ()h () H () = h 1 ()+h () a similar approach shows ha H 1 ( ) = H 1 () H ( ) = H () i.e. boh he sum and produc of wo even funcions are even. These resuls are summarized in he following Ke Poin. If Producs of funcions Ke Poin (even) (even) = (even) (even) (odd) = (odd) (odd) (odd) = (even) Sums of funcions (even) + (even) = (even) (even) + (odd) = (neiher) (odd) + (odd) = (odd) 5 HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions

6 Useful properies of even and of odd funcions in connecion wih inegrals can be readil deduced if we recall ha a definie inegral has he significance of giving us he value of an area: = f() a b b a Figure 5 f()d gives us he ne value of he shaded area, ha above he xis being posiive, ha below being negaive. For he case of a smmerical inerval (, a) deduce wha ou can abou a g()d and a h()d where g() is an odd funcion and h() is an even funcion. g() h() a a Figure 6 Your soluion We have a a g()d = for an odd funcion h()d = a h()d for an even funcion (Noe ha neiher resul holds for a funcion which is neiher even nor odd.) HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions 6

7 . Fourier Series implicaions Since a sum of even funcions is iself an even funcion i is no unreasonable o sugges ha a Fourier Series conaining onl cosine erms (and perhaps a consan erm which can also be considered as an even funcion) can onl represen an even periodic funcion. Similarl a series of sine erms (and no consan) can onl represen an odd funcion. These resuls can readil be shown more formall using he expressions for he Fourier coefficiens a n and b n. Recall ha for a periodic funcion b n = 1 f() sin n d If f() iseven, deduce wheher he inegrand is even or odd (or neiher) and hence evaluae b n. Repea for he Fourier coefficiens a n. Your soluion We have, if f() iseven, f() sin n = (even) (odd) = odd hence b n = 1 (odd funcion) d = Thus an even funcion has no sine erms in is Fourier Series. Also f() cos n = (even) (even) = even a n = 1 (even funcion) d = f() cos n d. I should be obvious ha, for an odd funcion f(), a n = 1 b n = f() cos n d = 1 f() sin n d (odd funcion) d = Analagous resuls hold for funcions of an period, no necessaril. 7 HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions

8 Foraperiodic funcion which is neiher even nor odd we can expec a leas some of boh he a n and b n o be non-zero. For example consider he square wave funcion: 1 f() Figure 7 This funcion is neiher even nor odd and we have alread seen in he previous Secion ha is Fourier Series conains a consan ( 1 ) and sine erms. This resul could be expeced because we can wrie f() = 1 + g() where g() is as shown: 1 g() 1 Figure 8 Clearl g() isodd and will conain onl sine erms. The Fourier series are in fac f() = 1 + (sin + 13 sin ) sin and (sin + 13 sin sin ) g() = HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions 8

9 For each of he following funcions deduce wheher he corresponding Fourier Series conains (i) sine or cosine erms or boh (ii) a consan erm a Figure 9 Your soluion 9 HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions

10 1. cosine erms onl (plus consan).. sine erms onl (no consan). 3. sine erms onl (no consan). 4. cosine erms onl (no consan). 5. cosine erms onl (plus consan). 6. sine and cosine erms (plus consan). 7. cosine erms onl (plus consan). Confirm he resul obained for par 7 in he las exercise b finding he Fourier Series full. The funcion involved is f() = << f( +) = f() Your soluion Since f() is even we can sa immediael b n = n =1,, 3,... Also a n = cos n d = n even 4 n odd n (afer inegraion b pars). Also a = d= so he Fourier Series is f() = 4 (cos + 19 cos ) cos HELM (VERSION 1: March 18, 4): Workbook Level 3.3: Even and Odd Funcions 1