Motivation. Section 5. Fourier Series. System. original signal. output. recombine. split into sine waves

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1 Moivaion Secion 5 We menioned a he sar of he las secion ha sine waves have a special propery in relaion o linear sysems. Fourier Series A sine wave a he inpu leads o a (possibly differen) sine wave a he oupu. Sine wave Linear Sysem Sine wave wih ampliude or phase changed The Fourier series is inroduced using an analogy wih spliing vecors up ino componens. I would herefore be useful o be able o express an arbirary signal in erms of a sum of sine waves. The symmery properies ha enable us o predic ha cerain coefficiens are zero are presened. original signal Sysem oupu spli ino sine waves recombine 61 62

2 Moivaion: Car Suspension Supposing we know ha our car suspension will sar o oscillae (bounce up and down uncomforably) a frequency f. Spliing up Vecors We wan o express a signal f() in he range in erms of some basic signals, i.e. sine waves. Le s look firs a how we do a similar hing wih vecors. Consider how we express he arbirary vecor r in erms of he basis vecors i and j. r = a i + b j We wan o measure a variey of ypical road profiles and calculae how much of frequency f hey each conain (wih he car ravelling a a paricular speed). This will ell us which combinaions of road profile and speed are likely o be a problem. j where a = r. i i. i b = r. j j. j The Fourier series enables us o represen he road profile as he sum of a se of sinusoidal componens a differen frequencies. i The basis vecors are orhogonal: i.j =

3 Basis Funcions Jus as we represen r using orhogonal basis vecors, we wan o represen f() in he range o using orhogonal basis funcions. We only need wo vecors, bu we need an infinie number of funcions. 1 (i.e. a consan erm) cos() cos(2) cos(3) cos(4)... sin() sin(2) sin(3) sin(4)... If n and m are posiive inegers greaer han zero. cos(n)sin(m) d = 0 cos(n) 1 d = 0 sin(n) 1 d = 0 { 0, n m cos(n)cos(m) d =, n = m { 0, n m sin(n)sin(m) d =, n = m 1 1 d = 2 So, using p()q() d as our do produc for funcions, he basis funcions are orhogonal. 65 Fourier Series The equivalens of our vecor do produc expressions o calculae he componen of r in each direcion (eg. a = (r.i)/(i.i)) are: a n = b n = d = cos(n)f() d = cos(n)cos(n) d sin(n)f() d = sin(n)sin(n) d 1 f() d d = 1 2 cos(n)f() d sin(n)f() d f() d The equivalen of our vecor expression for r in erms of i and j, (i.e. r = ai + bj) is an expression for f in erms of all he basis funcions. f() = a n cos(n) + b n sin(n) + d 1 66

4 Fourier Series Example 1 Fourier Series Properies Represen he square wave f() as a Fourier series. 2 f() We can use any range of lengh 2 insead of in he Fourier formulae. For example, 0 2 is equally OK. a n = 1 b n = 1 d = cos(n)f() d = 0 sin(n)f() d = 2(1 ( 1)n ) n 1 f() d = 0 2 Thus, we can model he square wave funcion f() using: f() = d + = = 4 [ (a n cos(n) + b n sin(n)) 2(1 ( 1) n ) n sin() + sin(3) 3 sin(n) + sin(5) ] 2. We are only modelling he funcion f() in he specified range (eg. o, or 0 o 2). Ouside his range he model will jus repea wih period 2. This is fine if he funcion we wish o model is periodic iself, bu if he funcion is no periodic he Fourier model will probably only be useful over he range on which i was buil. Original funcion 0 Fourier series represenaion based on range 0 o

5 Fourier Series Example 2 Represen f() = e as a Fourier series beween and. ( a n = 1 cos(n)e d = ( 1)n e e ) ( 1 + n 2) ( b n = 1 sin(n)e d = ( 1)n e e ) n ( 1 + n 2) 1 d = 2 e d = e e 2 Fourier Model of Exponenial Fourier model of e buil on range o repeas every 2 e Thus, in he range < < we can model he funcion f() = e using: f() = d + = e e (a n cos(n) + b n sin(n)) ( 1) n [cos(n) nsin(n)] 1 + n cos() sin() +1.47cos(2) 2.94sin(2)

6 Symmeric Signals ODD funcion f( ) = f() eg: sin() EVEN funcion f( ) = f() eg: cos() Avoiding Inegraion If we can spo a symmery in he funcion o be represened hen we can avoid evaluaing one or more of he Fourier inegrals. cos() sin() The a nerms model he EVEN componen in he funcion No even componen all a n = 0 No odd componen all b n = 0 Zero mean d = 0 EVEN funcion wih non zero mean: b = 0 n 1 The b n erms model he ODD componen in he funcion Purely ODD funcion wih zero mean: a n = 0 and d = 0 The d erm models he mean value of he funcion Funcion wih zero mean: d =

7 Fourier Series Example 3 Find he Fourier series represenaion for he funcion f() below. Fourier Series Example 4 Find he Fourier series represenaion for he funcion f() = cos( + /4). 2 f() EVEN funcion wih zero mean: b n = 0 and d = 0 This funcion has a mean value of zero so d = 0. a n = 1 cos(n)cos( + /4) d We only have o calculae a n = 1 2 cos(n ) + cos(n 4 ) d a n = 1 cos(n)f() d = cos(n)( /2) d 0 cos(n)( /2) d = 2 0 cos(n)( /2) d { = 2 0, n even n 2 (( 1)n 1) =, n odd so he Fourier series is: f() = 4 4 n 2 [cos() + 19 cos(3) cos(5) +... ] 73 = 1 2, when n = 1 and 0 oherwise. b n = 1 sin(n)cos( + /4) d = 1 2 sin(n ) + sin(n 4 ) d = 1 2, when n = 1 and 0 oherwise. so he Fourier series is: f() = cos() sin() 2 74

8 Secion 5: Summary Periodic funcions, (so far only wih period 2), can be represened using he he Fourier series. We can use symmery properies of he funcion o spo ha cerain Fourier coefficiens will be zero, and hence avoid performing he inegral o evaluae hem. Funcions wih zero mean have d = 0. Purely odd funcions have a n = 0. Purely even funcions have b n = 0. Segmens of non-periodic funcions can be represened using he Fourier series in he same way. The Fourier series represenaion jus repeas ouside he range on which i was buil. Secion 6 General Fourier Series The Fourier series for arbirary period is presened. We compare hree echniques for calculaing a general range Fourier series: direc inegraion, using a relaed series of dela funcions, and using he elecrical daa book. During he direc inegraion example, some symmery argumens for simplifying inegrals are illusraed