Fourier Transforms. L is given by

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1 Royal Holloway Universiy of ondon Fourier Transforms Inroducion The imporan discovery of Fourier was ha an arbirary waveform could be consruced as a superposiion of sine (and cosine) funcions. We have already considered a firs generalizaion of ha principle ha he waveform can be consruced from a superposiion of any appropriae complee orhonormal se of funcions. These superposiions consruc waveforms over a finie and predefined inerval. Alernaively we may consider funcions of infinie exen ha are periodic on he inerval. In his secion we consider a differen generalizaion of Fourier s principle where he waveform is non-repeiive and specified over an infinie range. In his case, as we shall see, he Fourier sum of sinusoids will become a Fourier inegral. There is, of course, he final generalizaion where he sinusoids are replaced by oher orhonormal funcions; ha is no reaed in his course. Fourier series summary of relevan knowledge The Fourier series on he inerval x is given by nπ nπ F( x) = ancos x + bnsin x n= 0 where he n = 0 erm is subsumed in he sum. The Fourier componens a n and b n are given by he Euler formulae 1 nπ an = F( x) cos x dx 1 nπ bn = F( x) sin x d x. The Fourier componens are deermined by he funcion F( x ) only over he inerval x. Ouside his inerval we can say eiher ha he funcion is undefined or we F x+ = F x ; his is a maer of choice. can say ha here is periodic behavior ( ) ( ) A sligh change of noaion will make closer connecion wih physical applicaions. In he argumen of he sin and cos, we can inerpre nπ as a wave number. Accordingly we wrie he n h wave number as π = n. kn And hen we can wrie he Fourier series and he inversion formulae as k F( x) = { akcos( kx) + bksin( kx) } (1.1) and k= k0 PH130 Mahemaical Mehods 1

2 Royal Holloway Universiy of ondon 1 ak = F( x) cos( kx) dx 1 bk = F( x) sin ( kx) d x. The Fourier Inegral We now consider he case where inerval over which he funcion is specified becomes indefiniely large here is no periodiciy and he funcion exends over all x. In oher words, we wan o ake he limi. This is no quie so simply done since he various facors 1 will become problemaic. The key o he soluion is o approximae he Fourier sum, Eq. (1.1), by an inegral. a k coskx a k coskx n Fourier sum: sum of heighs k Inegral approximaion k n The Fourier sum is given by he sum of he heighs of he lines. Bu he sum of heighs muliplied by he spacing is he area under he curve: or area heighs = widh of slices dk =. k k And he widh of he slices is π k = kn kn 1 =. The Fourier sum is hen approximaed by he inegral: as. F( x) { ak cos( kx) + bksin ( kx) } dk π 0 As we see ha he widh of he slices ends o zero so in ha limi he inegral approximaion becomes more accurae. Bu of course in his limi he Fourier componens PH130 Mahemaical Mehods

3 Royal Holloway Universiy of ondon 1 ak = F( x) cos( kx) dx 1 bk = F( x) sin( kx) dx end o zero because of he 1 pre-facor. In his case we define he Fourier sine and cosine ampliudes ( ) s f k and fc ( k) ( ), ( ) Thus F( x) is expressed as a Fourier inegral f k = a f k = b. c k 1 F ( x) = { fc( k) cos( kx) fs( k) sin( kx) } dk π + (1.) 0 and he Fourier ampliudes are hen given by f k = F x cos kx dx c s ( ) ( ) ( ) ( ) = ( ) ( ) f k F x sin kx d x. s k Complex form of he Fourier Inegral I is ofen convenien o express he Fourier inegral relaions in complex form. We define f k as he complex Fourier ampliude ( ) f ( k) = f ( k) + if ( k) so ha and hen c ( ) ( ) i kx d f k = F x e x 1 ikx F( x) f ( k) e dk π =. This resul is obained from Eq. (1.) by subsiuing kx ikx ( e ikx e ) ikx ikx sin kx = ( e e ) i. s cos = + and We observe here he emergence of a symmery beween he Fourier ransform relaions F x. The posiion of he π facor is for F( x) in erms of f ( k) and for f ( k) in erms of ( ) arbirary; by a redefiniion of f ( k) one can obain he more symmeric form of he Fourier inegrals wih a pre-facor 1 π accompanying boh inegrals. We have adoped he convenion mos used in Physics applicaions. The posiion of he minus sign muliplying i is also arbirary; here is a minus in one inegral and no he oher. The Fourier pair PH130 Mahemaical Mehods 3

4 Royal Holloway Universiy of ondon 1 ikx F x f k e k ( ) = ( ) π ikx f k F x e x ( ) = ( ) d d may be inerpreed as elling us ha a given funcion has a descripion in x-space as F( x) and a descripion in k-space as f ( k ). These are equally valid represenaions of he funcion. Thus far we have considered spaial funcions. The same consideraions apply o emporal funcions; an arbirary funcion of ime may be expressed as a superposiion (inegral) of differen frequencies. Thus replacing x by and k by ω we obain he Fourier pair 1 F() = f ( ω) e π iω f F e ( ω ) = ( ) An example of his Fourier ransform pair is in he area of acousics; any sound can be represened as a superposiion (inegral) of pure ones of differen frequencies. d. iω dω A gun sho A gun sho is an example of a sharp impulse. This may be approximaed by a high recangular pulse of very shor duraion. We specify he magniude of he pulse by a and is duraion by τ. We assume he pulse occurs a ime = 0. F() W a W/ +W/ = 0 recangular pulse approximaing a gun sho The mahemaical specificaion of he pulse, he funcion F() is F() = a τ τ = 0 oherwise. PH130 Mahemaical Mehods 4

5 Royal Holloway Universiy of ondon Thus he Fourier inegral expression for he specrum f ( ω ) is given by This is shown in he figure. ( ω ) = ( ) i ω f F e d τ iω τ iω ae a ωτ = a e d = = sin. iω ω τ τ f( Z) Z envelope Z specrum of a recangular pulse The funcion sin( x) xis known as he sinc funcion. When he duraion of he impulse ges shorer and shorer he specrum ges broader and broader a manifesaion of he Uncerainy Principle. You can invesigae his by making plos using Mahemaica. be e limi τ 0 will be reaed in a laer secion. Idealizaion of he gun sho he dela funcion The duraion of he gun sho, τ, is very shor and is insananeous magniude, a, is very large. As an idealizaion we may consider he limi where he duraion ends o zero, while he insananeous ampliude ends o infiniy. A he same ime we shall specify a finie value for he inensiy of he pulse, he area aτ. For simpliciy we will consider a sho of uni inensiy: aτ = 1. This is a minimalis descripion; we srip he sysem down o is bare essenials. This idealized gun sho is hus defined as he limi of he box funcion F( x) above, when τ 0 a bu aτ = 1. In his limi he box becomes a spike, known as Dirac s dela funcion wih symbol δ (). PH130 Mahemaical Mehods 5

6 Royal Holloway Universiy of ondon F() = 0 dela funcion idealizaion of a gun sho The dela funcion is very useful, wih many ineresing properies. I should be menioned, however ha i is no a mahemaically respecable funcion! δ ( ) should be undersood, raher, as he limi of a sequence of funcions. The specrum of he spike may be found from he specrum of he box funcion upon aking he appropriae limi. We found a ωτ f ( ω ) = sin ω. Since τ is small he argumen of he sine is small and so a series expansion is appropriae: a ωτ ( ωτ ) 3 f ( ω ) ~ +! ω 3! 3 aωτ = aτ +!. 4 Bu we have he normalizaion condiion aτ = 1, which hen leads o ωτ f ( ω ) ~1 +! 4 and now aking he limi τ 0, his gives f ( ω ) = 1. f( Z) specrum of dela funcion This is a remarkable resul. I is elling us ha he specrum of a dela funcion spike is a uniform superposiion of sinusoids of all frequencies. This is known as a whie specrum by analogy wih ligh, where whie ligh is he resul of a superposiion of colours from all regions of he visible specrum. Z PH130 Mahemaical Mehods 6

7 Royal Holloway Universiy of ondon When we looked a he finie-widh box funcion we saw ha as F() became narrower, ha f ( ω ) became broader. This, i was saed, was an example of he Uncerainy Principle. We now have a limiing example of his: when he widh of F() goes o zero, hen he widh of f ( ω ) becomes infinie. Frequency response of an amplifier The human ear responds o frequencies from abou 30 Hz o abou 15 khz. An audio amplifier needs o cover his frequency range for faihful sound reproducion. In order o measure he frequency response of such an amplifier one would connec a signal generaor o he amplifier s inpu and an AC volmeer o he oupu. The ampliude of he generaor is kep consan. The generaor s frequency is varied and he volmeer reading is recorded as a funcion of he frequency. signal generaor am plifier in ou AC volm eer measuring he frequency response of an amplifier When he gain is ploed as a funcion of frequency here will be a fla region corresponding o he pass band and he gain will drop a higher and lower frequencies. a ( Z) Z S.30Hz S.15kHz amplifier frequency response I akes ime o change he generaor frequency, le i sele down and hen measure he amplifier oupu. All he daa poins mus be colleced and hen ploed. Could he process be done quicker? Perhaps we could apply and measure all frequencies a he same ime! If we pu all frequencies ogeher hen we will produce a dela funcion. This could be implemened as a gun shooing in he viciniy of a microphone. Then he oupu, ransien response, of he amplifier conains he superposiion of he responses o he consiuen sinusoids of he gun sho all frequencies in equal measure. PH130 Mahemaical Mehods 7

8 Royal Holloway Universiy of ondon am plifier oscilloscope in ou ransien response of amplifier The frequency response of he amplifier is hus given from he specrum of sinusoids in he ransien response. To find his we need o Fourier analyze he ransien response. A() a ( Z) FT Z Fourier analysis of he ransien response gives he frequency response This demonsraes an alernaive way of measuring he frequency response of an amplifier. Raher han painsakingly sweeping he inpu frequency and recording he oupu signal, we may apply a dela funcion inpu and Fourier analyze he oupu. In oher words we apply all sine waves ogeher and sor hem a he oupu. Specrum of a finie sine wave A sine wave of infinie duraion has a specrum comprising a single frequency. The lesson of Fourier heory is ha a sine wave of finie duraion may be synhesized by combining a range of (infinie duraion) sine waves over a range of frequencies. sw ich on sw ich off a W/ 0 W/ finie duraion cosine wave We shall represen he wave form as iω F = ae τ < < τ () 0 = 0 oherwise. PH130 Mahemaical Mehods 8

9 Royal Holloway Universiy of ondon Here a is he ampliude of he cosine burs and we have chosen he complex form for mahemaical simpliciy. Because of he lineariy of he Fourier ransform relaions, he specrum of he cosine is found from he real par and ha of he sine from he imaginary par of he complex specrum. The Fourier specrum is easily found: iω f ( ω ) = F( ) e d τ i( ω ω0 ) = a e τ And his is inegraed o give a f ( ω) = sin( ω ω0 ) τ. ω ω0 This is he sinc funcion we have encounered before. Now however, i is cenred on and peaked a ω = ω0. f( Z) d. Z specrum of a finie sinusoid Z The value of f(ω) a is peak a ω = ω 0 is found by expanding he sin before seing ω = ω 0 ; his gives f ω = aτ. ( ) 0 Very long sinusoid Now a he ampliude of he sine wave is a finie quaniy. I hen follows ha as he sinusoid is of longer and longer duraion, he peak f(ω 0 ) becomes infiniely large, bu a he same ime i ges narrower and narrower. In paricular, when he rain is of infinie duraion he heigh becomes infinie as he widh ends o zero. Bu his is reminiscen of he dela funcion. e us check on he area of f(ω) o see if i remains finie. We can find he area of f(ω) by using a rick. We sar wih he inverse Fourier ransform, giving F() in erms of f(ω): F 1 i () ( ) d f ω e = ω ω π. The area is given by he inegral wih = 0. Thus we have ( ω) dω = πf( 0) f. PH130 Mahemaical Mehods 9

10 Royal Holloway Universiy of ondon Bu F(0) = a, so ha he area is indeed finie: πa. Since he area under he dela funcion is uniy, by definiion, i follows ha he specrum for an infinie sinusoid of (angular) frequency ω 0 and ampliude a is given by f ( ω) = πaδ( ω ω0 ). Of course wriing he specrum as a dela funcion is equivalen, physically, o saying ha he waveform comprises a sinusoid a a single frequency only. Incidenally, his gives an alernaive mahemaical represenaion of he dela funcion as 1 δ( ω) = lim sinωτ. τ 0 πω A paradox e us reurn o a sine rain of finie duraion. We have seen how is specrum is described by he sinc funcion a f ( ω) = sin( ω ω0 ) τ. ω ω0 In paricular, he here indicaes he duraion of he rain. There is a laboraory insrumen known as a specrum analyzer. Is funcion is o display he specrum of is inpu signal. Now consider he following: You have a specrum analyzer se up in he laboraory. A a given insan you urn on a sine wave generaor conneced o he analyzer. If you look a he analyzer display you will see he signal s specrum. Then by sudying he shape of he specrum you can exrac he value of. So you will know how ling he rain will las, and when he signal generaor will be swiched off. In oher words a specrum analyzer can enable you o see ino he fuure! The paradox is resolved by appreciaing ha he specrum analyzer has a frequency resoluion deermined by he lengh of ime i receives is inpu. An infinie duraion is required for infinie resoluion. And when he signal is observed for a ime only here is a corresponding uncerainy in he frequency resoluion of ω ~1/, meaning ha a dela funcion spike is broadened by his amoun. Thus he analyzer is no able o ell you when o urn he generaor off. The Gaussian curve One funcion is he Fourier ransform of iself. This is he Gaussian funcion F() e =. The Fourier ransform is calculaed as PH130 Mahemaical Mehods 10

11 Royal Holloway Universiy of ondon iω f F e ( ω ) = ( ) = e iω This can be evaluaed by hand, by compleing he square in he exponen o give ( ) f ω = π e ω. This propery is unique o he Gaussian funcion. You shouldn be concerned wih he facors of π which are doed around; heir precise posiions are purely a maer of convenion. Things become more ineresing when we specify he widh of he Gaussian curve. If we wrie () F = e a hen a is he rms widh of he curve. The Fourier ransform of his can be obained from he previous resul by a change of variables; he resul is a f ( ω) = πa e ω. So when he widh of F() is a, he corresponding widh of f(ω) is 1/a. d d. = a Z = 1/a F() f( Z) Gaussian funcion and is Fourier ransform Ye again we have encounered he resul ha he widh of a funcion and of is Fourier ransform have an inverse relaion beween hem. For he ime-frequency Fourier pair we express his as ω = 1 while for he posiion-wave number Fourier pair we have correspondingly x k = 1. These resuls hold for a Gaussian curve. Now he Gaussian has he imporan propery ha he uncerainy produc is a minimum. For general funcions he uncerainy produc is greaer han his. Thus in he general case we wrie he Fourier ransform uncerainy relaions as PH130 Mahemaical Mehods 11

12 Royal Holloway Universiy of ondon ω 1 x k 1. Physically, his is seing a fundamenal limi of resoluion in any specroscopic-ype of measuremen. Connecions wih Quanum Mechanics Cenral o quanum mechanics are he Einsein relaion and he debroglie relaion: E = = ω p = = k where E is energy, p is momenum, ω is angular frequency and k is wave number. Very simply, we may subsiue for ω and k in he uncerainy relaions using he Einsein and debroglie relaions o give E = x p =. These are he convenional quanum uncerainy relaions. We noe ha he wave funcion for a paricle a a specified posiion x 0 is he dela funcion Ψ ( x) = δ ( x x 0 ). This follows from he probabilisic inerpreaion of he wave funcion he uncerainy in x is zero. If we hen ask abou a momenum measuremen, we mus express he wave funcion as a funcion of p. This is done using a Fourier ransform. ( ) ( ) i px = = Ψ d ψ p x e x. So when Ψ(x) is a dela funcion hen ψ(p) is a consan; he uncerainy in p is infinie. On he oher hand, if he paricle has a well-defined momenum p 0 hen ψ ( p) = δ ( p p 0 ), he posiion wave funcion is a consan and so he uncerainy in posiion is infinie. The same argumens apply o ime and energy. In paricular, he ime-energy uncerainy relaion ells us ha energy conservaion may be violaed for shor imes. You may borrow a large amoun of energy for a shor ime or a small amoun for a longer ime. One also concludes ha i akes a long ime o make an accurae measuremen of energy. PH130 Mahemaical Mehods 1