Fourier series. Learning outcomes


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1 Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series Learning oucomes needs doing Time allocaion You are expeced o spend approximaely hireen hours of independen sudy on he maerial presened in his workbook. However, depending upon your abiliy o concenrae and on your previous experience wih cerain mahemaical opics his ime may vary considerably.
2 Periodic Funcions 23. Inroducion You should already know how o ake a funcion of a single variable f(x) and represen i by apower series in x abou any poin x 0 of ineres. Such a series is known as a Taylor series or Taylor expansion or, if x 0 =0,asaMaclaurin series. This expansion is only possible if he funcion is sufficienly smooh (ha is, if i can be differeniaed as ofen as required). Geomerically his means ha here are no jumps or spikes in he curve y = f(x) near he poin of expansion. However, in many pracical siuaions he funcions we have o deal wih are no as well behaved as his and so no power series expansion in x is possible. Neverheless, if he funcion is ic, so ha i repeas over and over again, hen, irrespecive of he funcion s behaviour, (ha is, no maer how many jumps or spikes i has) he funcion may be expressed as a series of sines and cosines. Such a series is called a Fourier series. Fourier series have many applicaions in mahemaics, in physics and in engineering. For example hey are someimes essenial in solving problems (in hea conducion, wave propagaion ec) ha involve parial differenial equaions. Also, using Fourier series he analysis of many engineering sysems (such as elecric circuis or mechanical vibraing sysems) can be exended from he case where he inpu o he sysem is a sinusoidal funcion o he more general case where he inpu is ic bu nonsinsusoidal. Prerequisies Before saring his Secion you should... Learning Oucomes Afer compleing his Secion you should be able o... be familiar wih rigonomeric funcions recognise ic funcions be able o deermine he frequency, he ampliude and he of a sinusoid be able o represen common ic funcions by rigonomeric Fourier series.
3 . Inroducion Youhave me in earlier Mahemaics courses he concep of represening a funcion by an infinie series of simpler funcions such as polynomials. For example, he Maclaurin series represening e x has he form e x =+x + x2 2! + x3 3! +... or, in he more concise sigma noaion, e x = n=0 (remembering ha 0! is defined as ). The basic idea is ha for hose values of x for which he series converges we may approximae he funcion by using only he firs few erms of he series. Fourier series, which we discuss in his and he following Secions, are also usually infinie series bu involve sine and cosine funcions (or heir complex exponenial equivalens) raher han polynomials. They are widely used for approximaing ic funcions. Such approximaions are of considerable use in science and engineering. For example, elemenary a.c. heory provides echniques for analyzing elecrical circuis when he currens and volages presen are assumed o be sinusoidal. Fourier Series enable us o exend such echniques o he siuaion where he funcions (or signals) involved are ic bu no acually sinusoidal. You may also see in Workbook 25 ha Fourier series someimes have o be used when solving parial differenial equaions. x n n! 2. Periodic Funcions A funcion f() is ic if he funcion values repea a regular inervals of he independen variable. The regular inerval is referred o as he. See Figure. f() If P denoes he we have Figure f( + P )=f() for any value of. The mos obvious examples of ic funcions are he rigonomeric funcions sin and cos, boh of which have 2 (using radian measure as we shall do hroughou his uni.) This follows since sin( + 2) = sin and cos( + 2) = cos 3 HELM (VERSION : April 8, 2004): Workbook Level 2 23.: Periodic Funcions
4 y = sin y = cos 2 2 Figure 2 The ampliude of hese sinusoidal funcions is he maximum displacemen from y = 0and is clearly. (Noe ha we use he erm sinusoidal o include cosine as well as sine funcions.) More generally we can consider a sinusoid y = A sin n which has maximum value, or ampliude, A and where n is usually a posiive ineger. For example y = sin 2 is a sinusoid of ampliude and 2 2 =. The fac ha he is follows because sin 2( + ) =sin(2 +2) =sin 2 for any value of. y = sin 2 2 Figure 3 We see ha y = sin 2 has half he of sin ( as opposed o 2). This can alernaively be phrased by saing ha sin 2 oscillaes wice as rapidly (or has wice he frequency) ofsin. HELM (VERSION : April 8, 2004): Workbook Level 2 23.: Periodic Funcions 4
5 y = sin y = sin 2 2 Figure 4 In general y = A sin n has ampliude A, 2 and complees n oscillaions when changes by 2. n Formally, we define he frequency of a sinusoid as he reciprocal of he : frequency = and he angular frequency (ofen denoed he Greek Leer ω (omega)) as Thus has frequency n 2 angular frequency = 2 frequency = 2 and angular frequency n. y = A sin n Sae he ampliude,, frequency and angular frequency of (i) y =5cos 4 (ii) y =6sin 2. 3 Your soluion For (i) we have Your soluion, angular frequency 4 ampliude 5, 2 4 = 2, frequency 2 For (ii) we have ampliude 6, 3, frequency, angular frequency HELM (VERSION : April 8, 2004): Workbook Level 2 23.: Periodic Funcions
6 Harmonics In represening a nonsinusoidal funcion of 2 by a Fourier Series we shall see shorly ha only cerain sinusoids will be required: (a) A cos (and B sin ) These also have 2 and ogeher are referred o as he firs (or fundamenal) harmonic. (b) A 2 cos 2 (and B 2 sin 2) These have half he, or double he frequency of he firs harmonic and are referred o as he second harmonic. (c) A 3 cos 3 (and B 3 sin 3) These have 2 and consiue he hird harmonic. 3 In general he Fourier Series of a funcion of 2 will require harmonics of he ype A n cos n (and B n sin n) where n =, 2, 3,... Nonsinusoidal ic funcions. The following are examples of such funcions (hey are ofen called waves ): Square Wave f() 2 Figure 5 Analyically we can describe his funcion as follows: { <<0 f() = + 0 << (which gives he definiion over one.) f( +2) =f() (which ells us ha he funcion has 2). HELM (VERSION : April 8, 2004): Workbook Level 2 23.: Periodic Funcions 6
7 Sawooh wave f() Figure 6 In his case we can describe he funcion as follows: f() = 2 0 <<2 f( +2) = f() Here he is 2, he frequency is 2 and he angular frequency is 2 2 =. Triangular wave f() 2 Figure 7 Here we can convenienly define he funcion using <<as he basic : { <<0 f() = 0 << or, more concisely, f() = << ogeher wih he usual saemen on iciy f( +2) =f(). Wrie down an analyic definiion for he following ic funcion: 2 f() Figure 8 7 HELM (VERSION : April 8, 2004): Workbook Level 2 23.: Periodic Funcions
8 Your soluion We have f() = { 2 0 <<3 3 <<5 f( +5)=f() Skech he graph of he following ic funcions showing all relevan values: (i) f() = 2 0 << <<6 0 6 <<8 f( +8)=f() (ii) f() = <<2 f( +2)=f() Your soluion (i) 8 f() (ii) f() 2 Figure 9 HELM (VERSION : April 8, 2004): Workbook Level 2 23.: Periodic Funcions 8
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