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

Transcription

1 Even and Odd Funcions 23.3 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 On compleion ou should be able o... know how o obain a Fourier series be able o inegrae funcions involving sinusoids 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 3 HELM (28): Workbook 23: Fourier Series

2 1. Even and odd funcions We have shown in he previous Secion how o calculae, b inegraion, he coefficiens a n (n =, 1, 2, 3,...) and b n (n = 1, 2, 3,...) in a Fourier 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. In he previous Secion we showed ha he square wave (one period of which shown in Figure 12) has a Fourier series conaining a consan erm and cosine erms onl (i.e. all he Fourier coefficiens b n are zero) while he funcion shown in Figure 13 has a more complicaed Fourier series conaining boh cosine and sine erms as well as a consan Figure 12: Square wave 2 f() 2 2 Figure 13: Saw-ooh wave Conras he smmer or oherwise of he funcions in Figures 12 and 13. Your soluion The square wave in Figure 12 has a graph which is smmerical abou he -axis and is called an even funcion. The saw-ooh wave shown in Figure 13 has no paricular smmer. In general a funcion is called even if is graph is unchanged under reflecion in he -axis. This is equivalen o f( ) = f() for all Obvious examples of even funcions are 2, 4,, cos, cos 2, sin 2, cos n. A funcion is said o be odd if is graph is smmerical abou he origin (i.e. smmer abou he origin). This is equivalen o he condiion f( ) = f() i has roaional HELM (28): Secion 23.3: Even and Odd Funcions 31

3 Figure 14 shows an example of an odd funcion. f() Figure 14 Examples of odd funcions are, 3, sin, sin n. A periodic funcion which is odd is he saw-ooh wave in Figure f() Figure 15 Some funcions are neiher even nor odd. The periodic saw-ooh wave of Figure 13 is an example; anoher is he exponenial funcion e. Sae he period of each of he following periodic funcions and sa wheher i is even or odd or neiher. (a) f() (b) f() Your soluion (a) is neiher even nor odd (wih period 2) (b) is odd (wih period ). 32 HELM (28): Workbook 23: Fourier Series

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. 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 1 () So F 1 () is odd. Now suppose F 2 () = g() + h() (sum of odd and even funcions) F 2 ( ) = g( ) + h() = g() + h() We see ha F 2 ( ) F 2 () and F 2 ( ) F 2 () So F 2 () is neiher even nor odd. Invesigae he odd/even naure of sums and producs of (a) wo odd funcions g 1 (), g 2 () (b) wo even funcions h 1 (), h 2 () Your soluion HELM (28): Secion 23.3: Even and Odd Funcions 33

5 G 1 () = g 1 ()g 2 () G 1 ( ) = ( g 1 ())( g 2 ()) = g 1 ()g 2 () = G 1 () so he produc of wo odd funcions is even. G 2 () = g 1 () + g 2 () G 2 ( ) = g 1 ( ) + g 2 ( ) = g 1 () g 2 () = G 2 () so he sum of wo odd funcions is odd. H 1 () = h 1 ()h 2 () H 2 () = h 1 () + h 2 () A similar approach shows ha H 1 ( ) = H 1 () H 2 ( ) = H 2 () i.e. boh he sum and produc of wo even funcions are even. These resuls are summarized in he following Ke Poin. Ke Poin 5 Producs of funcions (even) (even) = (even) (even) (odd) = (odd) (odd) (odd) = (even) Sums of funcions (even) + (even) = (even) (even) + (odd) = (neiher) (odd) + (odd) = (odd) 34 HELM (28): Workbook 23: Fourier Series

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 16 f() d gives us he ne value of he shaded area, ha above he -axis 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 Your soluion We have a g() d = for an odd funcion a h() d = 2 a h() d for an even funcion (Noe ha neiher resul holds for a funcion which is neiher even nor odd.) HELM (28): Secion 23.3: Even and Odd Funcions 35

7 2. 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 2-periodic funcion b n = 1 f() sin n d If f() is even, 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() is even, 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 = 2 I should be obvious ha, for an odd funcion f(), a n = 1 b n = 2 f() cos n d = 1 f() sin n d f() cos n d. (odd funcion) d = Analogous resuls hold for funcions of an period, no necessaril HELM (28): Workbook 23: Fourier Series

8 For a periodic 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() 2 Figure 17: Square wave This funcion is neiher even nor odd and we have alread seen in Secion 23.2 ha is Fourier series conains a consan ( 1 2) and sine erms. This resul could be expeced because we can wrie f() = g() where g() is as shown: 1 2 g() Figure 18 Clearl g() is odd and will conain onl sine erms. The Fourier series are in fac f() = (sin + 13 sin ) sin and (sin + 13 sin sin ) g() = 2 HELM (28): Secion 23.3: Even and Odd Funcions 37

9 For each of he following funcions deduce wheher he corresponding Fourier series conains (a) sine erms onl or cosine erms onl or boh (b) a consan erm a Your soluion 1. cosine erms onl (plus consan). 5. sine erms onl (no consan). 2. cosine erms onl (no consan). 6. sine and cosine erms (plus consan). 3. sine erms onl (no consan). 7. cosine erms onl (plus consan). 4. cosine erms onl (plus consan). 38 HELM (28): Workbook 23: Fourier Series

10 Confirm he resul obained for he riangular wave, funcion 7 in he las, b finding he Fourier series full. The funcion involved is f() = < < f( + 2) = f() Your soluion Since f() is even we can sa immediael b n = n = 1, 2, 3,... Also a n = 2 Also a = 2 cos n d = n even 4 n 2 n odd d = so he Fourier series is (cos + 19 cos cos ) (afer inegraion b pars) f() = 2 4 HELM (28): Secion 23.3: Even and Odd Funcions 39