ES.83 Topic Noes Jeremy Orloff Fourier series inroducion: coninued. Goals. Be able o compue he Fourier coefficiens of even or odd periodic funcion using he simplified formulas.. Be able o wrie and graph he funcion a Fourier series converges o. 3. Be able o deermine he decay rae of he coefficiens of a Fourier series. 4. Be able o predic he decay rae of he Fourier coefficiens based on how smooh he original funcion is.. Inroducion In his opic we coninue our inroducion o Fourier series. We sar by looking a some ricks for compuing Fourier coefficiens. Then we will alk abou more concepual noions, including he convergence properies of Fourier series and he decay rae of Fourier coefficiens. A he end we will look a he orhogonaliy relaions which eplain he our formulas for Fourier coefficiens..3 Calculaion ricks: even and odd funcions.3. Even and odd funcions A funcion is an even funcion if f( = f( for all. The graph of an even funcion is symmeric abou he y-ais. Graphs of some even funcions Eamples of even funcions:,, 4,..., cos(ω. In general, even funcions are buil ou of even powers of. Noe ha, he power series for cos(ω has only even powers. By symmery we have he following key inegraion fac for even funcions: f( d = f( d for any even f(. A funcion is an odd funcion if f( = f( for all.
FOURIER SERIES INTRODUCTION: CONTINUED The graph of an odd funcion is symmeric abou he origin. Graphs of some odd funcions Eamples of odd funcions:, 3, 5,..., sin(ω. In general, odd funcions are buil ou of odd powers of. Noe ha, he power series for sin(ω has only odd powers. By symmery we have he following key inegraion fac for odd funcions: f( d = Producs of even and odd funcions for any odd f(. We give he rules in a kind of shor-hand. You can remember hese rules by hinking abou powers of, e.g. 4 7 =, so even odd is odd. even even = even odd odd = even odd even = odd.3. Fourier coefficiens of even and odd funcions If f( even hen b n = and a n = If f( is odd hen a n =, and b n = f( cos d. f( sin d. Reason. If f( is even hen by he muliplicaion rules for even funcions f( cos(ω, so a n = f( cos d = f( cos d. ikewise, f( sin(ω is odd, so b n = f( sin d =. Eample.. In he las opic we me he period π square wave, which over one period for π < has he formula sq( = for < π. 3π π π π π 3π Graph of sq( = square wave
FOURIER SERIES INTRODUCTION: CONTINUED 3 Since he period is π we have = π. Since sq( is odd, we know ha a n = b n = π sq( sin(n d = π We have found he Fourier series for sq(: sq( = b n sin(n = 4 π n= sin(n d = nπ cos(n π = 4 nπ ( sin( + sin(3 + sin(5 + 3 5 and for n odd for n even. Eample.. Triangle wave funcion (also called he coninuous sawooh funcion. e f( have period π and f( = for π π. π 3π π π π π 3π Graph of f( = riangle wave Since f( is an even funcion we know ha b n = and for n we have a n = cos(n d = π π π = [ sin(n + cos(n π n n ] π As always we compue a separaely: a = π Thus we have he Fourier series for f(: f( = a + a n cos(n = π 4 π n=.4 Summing Fourier series cos(n d = n π (( n = π d = π 4 n π for n odd for n even. d = π. ( cos( + cos(3 3 + cos(5 5 + We can use he sum of a finie number of erms from a Fourier series o approimae he original funcion. The apple hp://web.mi.edu/jorloff/www/jmoapples/hml5/ fourierapproimaion.hml illusraes his. In he following secions we will bring ou he following key poins: The firs few erms of he Fourier series approimae he shape of he funcion, no necessarily he value of he funcion a any one poin. A poins of coninuiy he Fourier series converges o he original funcion. The smooher he funcion he faser he Fourier series converges o he funcion.
FOURIER SERIES INTRODUCTION: CONTINUED 4 A jumps in he graph, no maer how many erms you use he Fourier series always overshoos he graph near ha poin. The smooher he funcion he faser he Fourier series converges o he funcion..5 Convergence of Fourier series Piecewise smooh: The period funcion f( is called piecewise smooh if here are only a finie number of poins < <... < n where f( is no differeniable and a each of hese poins he lef and righhand limis f( + i = lim + i eis (alhough hey migh no be equal. f( and f( i = lim i In shor, a funcion is piecewise smooh if i is smooh ecep a a discree se of poins where is has jump disconinuiies. Here is our main heorem abou convergence of Fourier series. ES.83. f( We will no prove i in Theorem: If f( is piecewise smooh and periodic hen he Fourier series for f. converges o f( a values of where f is coninuous. converges o he average of f( and f( + a values of where f( has a jump disconinuiy. Eample.3. Square wave. The square wave in he eample above has jump disconinuiies. No maer how we specify he endpoin behavior of sq( he Fourier series converge o a he disconinuiies. Original sq( Fourier series Eample.4. The riangle wave in he eample above is coninuous so is Fourier series converges o he original funcion f(. Eample.5. We give one more graphical eample. Here he original funcion has disconinuiies admiedly somewha arificial. Since he lef and righhand limis are he same a each disconinuiy he Fourier series is coninuous. Original f( Fourier series
FOURIER SERIES INTRODUCTION: CONTINUED 5.5. Decay rae of Fourier coefficiens Sequences like a n = /n and b n = /n go o as n goes o infiniy. We say hey decay o. Clearly b n goes o faser han a n. We will say b n decays like /n. In general we will ignore consan facors, so, for eample, we say 4/(nπ decays like /n. Eample.6. The Fourier coefficiens of sq( are 4/(nπ for n odd a n = and b n = for n even. We say hese coefficiens decay like /n. Eample.7. The riangle wave looked a above has Fourier coefficiens 4/(n π for n odd b n = and a n = for even n. So hese coefficiens decay like /n. Eample.8. The coefficiens a n = /(n + n decay like /n. Eample.9. If a Fourier series has a n = /n and b n = /n we say he Fourier coefficiens decay like /n. Tha is, he decay rae is he slower of he wo decay raes. Eample.. The funcion f( = 3 cos( + 5 cos( is a already given as a (finie Fourier series. The coefficiens are a =, a = 3, a = 5, a 3 =, a 4 =,... We say hese coefficiens have infinie decay rae. Tha is he decay faser han /n p for any p..5. Imporan heurisics If a funcion has a jump disconinuiy hen is Fourier coefficiens decay like n, e.g. he square wave. If a funcion has a corner hen is Fourier coefficiens decay like, n wave e.g. he riangle A smooh funcion has Fourier coefficiens ha decay like n 3 or faser. The smooher he funcion, he faser he coefficiens decay..6 Gibbs phenomenon Non-local naure of Fourier series They approimae over he whole inerval, no jus near. (Analogy: leas squares fi of daa poins. Gibbs phenomenon For he square wave he Fourier ransform fis well a poins of coninuiy by here is always abou a 9% error near poins of disconinuiy no maer how many erms of he series you use. I won give physical eamples for ES.83, bu i is eremely imporan for many hings, e.g. digiial filering of signals.
FOURIER SERIES INTRODUCTION: CONTINUED 6.8.8.8.8.8.8.8.8 Gibbs: ma n = Gibbs: ma n = 3 Gibbs: ma n = 9 Gibbs: ma n = 33.7 Orhogonaliy relaions This will probably no be alked abou in class..7. The orhogonaliy relaions The key o he inegral formulas for Fourier coefficiens are he orhogonaliy relaions. These are he following inegral formulas ha say cerain rigonomeric inegrals are eiher or. The erm orhogonaliy refers o he fac ha if you have a se of muually orhogonal uni vecors in a vecor space hen he do produc of any wo is eiher (if hey re differen or (if hey re he same. ( mπ cos cos d = ( mπ sin cos sin sin d = ( mπ d = n = m n m n = m = n = m n m Proof. We have wo mehods o do his. We will carry ou he firs, bu only menion he second. Mehod : Use he following rig. ideniies cos(α cos(β = sin(α cos(β = sin(α sin(β = (cos(α + β + cos(α β (sin(α + β + sin(α β (cos(α β cos(α β Mehod : Use cos(a = eia +e ia ec.
FOURIER SERIES INTRODUCTION: CONTINUED 7 Using mehod we ge he following if n m: ( mπ cos cos d = = =. ( ( cos (n+mπ + cos (n mπ d ( ( sin (n+mπ sin (n mπ (n + mπ/ + (n mπ/ The las equaliy is easy o see since every erm is when = ±. The case n = m is special because hen n m =. I is easy o use he rig. ideniy o see ha he inegral in his case is. All he oher orhogonaliy relaions are proved in a similar fashion..7. Using orhogonaliy relaions o show he formula for Fourier coefficiens The orhogonaliy relaions allow us o see ha if f( is wrien as a Fourier series hen he coefficiens mus be given by he inegral formulas we ve been using. So, suppose f( has Fourier series : f( = a ( π ( π ( π ( π + a cos + a cos + + b sin + b sin + Then for n > f( cos d = [ a cos ( π + a cos ( π +b sin cos cos ( π + a cos cos + + b sin ( π ] cos + d Now we can apply he orhogonaliy relaions o each erm. All of hem are, ecep he erm wih a n cos cos which, again by he orhogonaliy relaions, inegraes o a n. Thus f( cos d = a n. Which is eacly he formula for he Fourier coefficien. The formulas for a and b n are found in he same way..8 Hearing a musical riad: C-E-G Here is a simplified Fourier-cenric view of how humans hear sound. Sound reaches your ear as a pressure wave. For eample f( = a cos(ω + a cos(w +
FOURIER SERIES INTRODUCTION: CONTINUED 8 Do he ears do Fourier analysis? Answer: Yes! The ear conains hair-like srucures called sereocilia. These are differen sizes and, so, resonae a differen frequencies. As hey vibrae hey simulae nerves which hen send signals o he brain. Thus, for each frequency in he pressure wave he brain is geing a signal from he nerves aached o he sereocilia which vibrae a ha frequency. The greaer he ampliude in he inpu wave he greaer he ampliude of he signal sen o he brain. Does he brain do Fourier synhesis? Answer: Yes! I is up o he brain o combine all he nerve signals a differen frequencies ino a single signal which i hen inerpres.