9. Fourier Series and Fourier Transforms

Transcription

1 9. Fourier Series nd Fourier Trnsforms The Fourier trnsform is one of the most importnt tools for nlyzing functions. The bsic underlying ide is tht function f(x) cn be expressed s liner combintion of elementry functions (specificlly, sinusoidl wves). The coefficients in this liner combintion cn be regrded s counterprt function, F (k), tht is defined in wve-number domin k R. This is helpful becuse certin mthemticl problems, such s differentil equtions, re esier to solve in terms of F (k) rther thn directly in terms of f(x). 9.1 Fourier series We begin by discussing the Fourier series, which is used to nlyze functions which re periodic in their inputs. A periodic function f(x) is function of rel vrible x tht repets itself every time x chnges by, s in the figure below: The constnt is clled the period. Mthemticlly, we write this condition s f(x + ) f(x), x R. (1) In physicl context, the vlue of f(x) cn be either rel or complex. We will ssume tht the input vrible x is rel, nd tht it refers to sptil coordinte. (Most of the following discussion cn lso pply to functions of time, with minor differences in convention tht we ll discuss lter.) We cn lso think of periodic function s being defined over domin of length, sy /2 x < /2. The periodicity condition is equivlent to joining the edges of the domin to form ring of circumference, s shown in the figure below. Then the position long the circumference of the ring serves s the x coordinte. Consider wht it mens to specify n rbitrry periodic function f(x). One wy to specify the function is to stte its vlue for every /2 x < /2. But tht s n uncountbly infinite set of numbers, which is cumbersome to del with. A better lterntive is to express the function s liner combintion of simpler periodic functions, such s sines nd cosines: ( ) 2πnx α n sin + n1 ( 2πmx β m cos m0 ). (2) This is clled Fourier series. If f(x) cn be expressed s such series, then it cn be fully specified by the set of numbers {α n, β m }, which re clled the Fourier coefficients. These coefficients re rel if f(x) is rel function, or complex if f(x) is complex-vlued. (Note tht the sum over n strts from 1, but the sum over m strts from 0; tht s becuse the sine term with n 0 is zero for ll x, so it s redundnt.) The justifiction for the Fourier 68

2 series formul is tht these sine nd cosine functions re themselves periodic, with period : ( ) ( ) ( ) 2πn(x + ) 2πnx 2πnx sin sin + 2πn sin (3) ( ) ( ) ( ) 2πn(x + ) 2πnx 2πnx cos cos + 2πn cos (4) Hence, ny such liner combintion stisfies the periodicity condition f(x + ) f(x) utomticlly. The Fourier series is nice wy to specify periodic functions, becuse we only need to supply the Fourier coefficients {α n, β m }, which re discrete set of numbers; then the vlue of f(x) is completely determined for ll x. Although the set of Fourier coefficients is formlly infinite, in mny cses the Fourier coefficients re negligible for lrge m nd n (corresponding to very rpidly-oscillting sine nd cosine wves), so we only need to keep trck of smll number of low-order coefficients Squre-integrble functions For function f(x) to be expressible s Fourier series, the series needs converge to f(x) s we sum to infinity. Under wht circumstnces does convergence occur? The full nswer to this question turns out to be long nd difficult, nd we will not go into the detils. Luckily, most periodic functions encountered in physicl contexts do hve convergent Fourier series. In fct, they re usully prt of clss of functions clled squre-integrble functions, which re gurnteed to hve convergent Fourier series. Squre-integrble functions re those for which the integrl /2 /2 dx f(x) 2 (5) exists nd is finite. From now on, we will simply ssume tht we re deling with functions of this sort, nd not worry bout the issue of convergence Complex Fourier series nd inverse reltions Using Euler s formul, we cn re-write the Fourier series s follows: n e 2πinx/ f n. (6) Insted of seprte sums over sine nd cosine functions, we sum over complex exponentil functions. We hve new set of Fourier coefficients, f n, nd the sum includes negtive integers n. This form of the Fourier series is lot more convenient to work with, since we now only hve to keep trck of single sum rther thn seprte sums for the sine nd cosine terms. (As n exercise, try working out the explicit reltionship between the old nd new coefficients.) The bove Fourier series formul tells us tht if the Fourier coefficients {f n } re known, then f(x) cn be determined. The converse is lso true: if we re given f(x), it is possible for us to determine the Fourier coefficients. To see how this is done, first observe tht /2 /2 where δ mn is the Kronecker delt, defined s: dx e 2πimx/ e 2πinx/ δ mn for m, n Z, (7) δ mn { 1, if m n 0, if m n. (8) 69

3 Due to this property, the set of functions exp(2πinx/), with integer vlues of n, re sid to be orthogonl functions. (We won t go into the detils now, but the term orthogonlity is used here with the sme mening s in vector lgebr, where set of vectors v 1, v 2,... is sid to be orthogonl if v m v n 0 for m n.) Using this result, we cn show tht /2 [ /2 ] dx e 2πimx/ dx e 2πimx/ e 2πinx/ f n (9) /2 /2 /2 n n /2 n dx e 2πimx/ e 2πinx/ f n (10) δ mn f n (11) f m. (12) The procedure of multiplying by exp( 2πimx/) nd integrting over x cts s kind of sieve, filtering out ll other Fourier components of f(x) nd keeping only the one with the mtching index m. Hence, we rrive t pir of reltions expressing f(x) in terms of its Fourier components, nd vice vers: f n 1 n /2 /2 e iknx f n dx e iknx f(x) where k n 2πn Here, the rel numbers k n re clled wve-numbers. They form discrete set, with one for ech Fourier component. In physics jrgon, we sy tht the wve-numbers re quntized to integer multiples of Exmple: Fourier series of squre wve (13) k 2π. (14) To get feel for how the Fourier series expnsion works, let s look t the squre wve, which is wveform tht tkes only two vlues +1 or 1, jumping discontinuously between those two vlues t periodic intervls. Within one period, the function is { 1, /2 x < 0 (15) +1, 0 x < /2. Plugging this into the Fourier reltion, nd doing the strightforwrd integrls, gives the Fourier coefficients [sin (nπ/2)]2 f n i nπ/2 { 2i/nπ, n odd 0, n even. As cn be seen, the high-frequency Fourier components re less importnt, since the Fourier coefficients go to zero for lrge n. We cn write the Fourier series s f(x) 4 sin(2πnx/). (17) nπ n1,3,5,... If this infinite series is truncted to finite number of terms, we get n pproximtion to f(x). The pproximtion becomes better nd better s more terms re included. This is illustrted in the following figure, where we show the Fourier series truncted up to N 5, nd the Fourier series truncted up to N 29: (16) 70

4 One musing consequence of this result is tht it cn be used s series expnsion for the mthemticl constnt π. If we set x /4, then f(/4) 1 4 [sin(π/2) + 13 π sin(3π/2) + 15 ] sin(5π/2) +, (18) nd hence π 4 ( ) +. (19) 9.2 Fourier trnsforms The Fourier series discussed in the previous section pplies to periodic functions, f(x), defined over the intervl /2 x < /2. This concept cn be generlized to functions defined over the entire rel line, x R, by crefully tking the limit. Consider wht hppens when we pply the limit to the right-hnd side of the complex Fourier series formul from Section 9.1.2: ( lim n e iknx f n ), where k n n k, k 2πn. (20) As, the wve-number quntum k goes to zero. Hence, the set of vlues of k n turns into continuum, nd we cn replce the discrete sum with n integrl over the vlues of k n. To do this, we multiply the summnd by fctor of ( k/2π)/( k/2π) 1: lim [ n k 2π eiknx ( ) ] 2πfn. (21) k If we now define then the sum becomes lim F (k n ) 2π k f n, (22) ( n ) k 2π eiknx F (k n ). (23) This limiting expression mtches the bsic definition of n integrl: 2π eikx F (k). (24) In cse you re wondering, the fctor of 2π is essentilly rbitrry, nd is just mtter of how we chose to define F (k n ). Our choice corresponds to the stndrd definition of the Fourier trnsform. 71

5 9.2.1 The Fourier reltions The function F (k) defined in the previous section is clled the Fourier trnsform of f(x). Just s we hve expressed f(x) in terms of F (k), we cn lso express F (k) in terms of f(x). To do this, we pply the limit to the inverse reltion for the Fourier series (see Section 9.1.2): 2π F (k n ) lim lim k f n (25) ( ) 2π 1 /2 dx e iknx (26) 2π/ /2 dx e ikx f(x). (27) Hence, we rrive t pir of equtions clled the Fourier reltions: F (k) dx e ikx f(x) 2π eikx F (k) (28) The first eqution is the Fourier trnsform, nd the second eqution is clled the inverse Fourier trnsform. These reltions stte tht if we hve function f(x) defined over x R, then there is unique counterprt function F (k) defined over k R, nd vice vers. The Fourier trnsform converts f(x) to F (k), nd the inverse Fourier trnsform does the reverse. It is importnt to note the smll differences between the two formuls. Firstly, there is fctor of 1/2π tht tgs long with, but not with dx; this is mtter of convention, tied to our definition of F (k), s mentioned before. Secondly, the integrl over x contins fctor of e ikx but the integrl over k contins fctor of e ikx. One wy to remember this is to think of the integrl over k, in the inverse Fourier trnsform eqution, s the continuum limit of sum over complex wves, with F (k) plying the role of the series coefficients; by convention, these complex wves hve the form exp(ikx). In our definition of the Fourier trnsform, it is cler tht the Fourier series needs to remin convergent s we tke the limit. Bsed on our erlier discussion of squreintegrble functions (Section 9.1.1), this mens we lwys del with functions such tht exists nd is finite A simple exmple Consider the function { e ηx, x 0 0, x < 0, dx f(x) 2 (29) η R +. (30) For x < 0, this is n exponentilly-decying function, nd for x < 0 it is identiclly zero. The rel prmeter η is clled the decy constnt; for η > 0, the function f(x) vnishes s x + nd cn thus be shown to be squre-integrble, nd lrger vlues of η correspond to fster exponentil decy. The Fourier trnsform cn be found by directly clculting the Fourier integrl: F (k) 0 dx e ikx e κx i k iη. (31) 72

6 It is useful to plot the squred mgnitude of the Fourier trnsform, F (k) 2, ginst k. This is clled the Fourier spectrum of f(x). In this cse, This is plotted in the right-hnd figure below: F (k) 2 1 k 2 + η 2. (32) We cll such grph Lorentzin curve. It consists of pek centered t k 0, whose height nd width re dependent on the decy constnt η. For smll η, i.e. wekly-decying f(x), the pek is high nd nrrow. For lrge η, i.e. rpidly-decying f(x), the pek is low nd brod. This kind of reltionship between the decy rte nd the Fourier spectrum pek width is very common. We cn quntify the width of the Lorentzin curve by defining the full-width t hlfmximum (FWHM), which is the width of the curve t hlf the vlue of its mximum. In this cse, the mximum of the Lorentzin curve occurs t k 0 nd hs the vlue of 1/η 2. The hlf-mximum, 1/2η 2, occurs when δk ±η. Hence, the originl function s decy constnt, η, is directly proportionl to the FWHM of the Fourier spectrum, which is 2η. Note lso tht this reltionship is dimensionlly consistent. In f(x), the exponent ηx needs to be dimensionless, so the decy constnt hs unit of [1/x]. This hs the sme units s the wve-number vrible k, which is the horizontl xis for the Fourier spectrum. To wrp up this exmple, let s evlute the inverse Fourier trnsform: i 2π e ikx k iη. (33) This cn be done by contour integrtion (Chpter 8). The nlytic continution of the integrnd hs one simple pole, t k iη. For x < 0, the numertor exp(ikx) blows up fr from the origin in the upper hlf of the complex plne, nd vnishes fr from the origin in the lower hlf-plne; hence we close the contour in the lower hlf-plne. This encloses no pole, so the integrl is zero. For x > 0, the numertor vnishes fr from the origin in the upper hlf-plne, so we close the contour in the upper hlf-plne (i.e., the contour is counter-clockwise). Hence, ( ) [ ] i e ikx (2πi) Res e ηx (x > 0) (34) 2π k iη kiη which is indeed the function tht we strted out with Fourier trnsforms for time-domin functions Thus fr, we hve been deling with functions of sptil coordinte x. Of course, these mthemticl reltions don t cre bout the physicl mening of the vribles, so the Fourier trnsform concept is lso pplicble to functions of time t. However, there is vextious difference in convention tht needs to be observed: when deling with functions of the time coordinte t, it is customry to use different sign convention in the Fourier reltions! 73

7 The Fourier reltions for function of time, f(t), re: F (ω) f(t) dt e iωt f(t) dω 2π e iωt F (ω). (35) These reltions differ in one notble wy from the Fourier reltions between f(x) nd F (k) discussed in Section 9.2.1: the signs of the ±iωt exponents re flipped. There s good reson for this difference in sign convention: it rises from the need to describe propgting wves, which vry with both spce nd time. As we discussed in Chpter 5, propgting plne wve cn be described by wvefunction f(x, t) Ae i(kx ωt), (36) where k is the wve-number nd ω is the frequency. We write the plne wve function this wy so tht positive k indictes forwrd propgtion in spce (i.e., in the +x direction), nd positive ω indictes forwrd propgtion in time (i.e., in the +t direction). This requires the kx nd ωt terms in the exponent to hve opposite signs, so tht when t increses by certin mount, corresponding increse in x leves the totl exponent unchnged. Now, s we hve seen, the inverse Fourier trnsform reltion describes how wve-form is broken up into superposition of elementry wves. In the cse of wvefunction f(x, t), the superposition is given in terms of plne wves: f(x, t) 2π dω 2π ei(kx ωt) F (k, ω). (37) To be consistent with this, we need to tret spce nd time vribles with oppositely-signed exponents: f(t) The other set of Fourier reltions follow from this choice Bsic properties of the Fourier trnsform 2π eikx F (k) (38) dω 2π e iωt F (ω). (39) The Fourier trnsform hs severl properties tht re useful to remember. These cn ll be directly proven using the definition of the Fourier trnsform. The proofs re left s exercises. (i) The Fourier trnsform is liner: if we hve two functions f(x) nd g(x), whose Fourier trnsforms re F (k) nd G(k) respectively, then for ny constnts, b C, f(x) + bg(x) FT F (k) + bg(k). (40) (ii) Performing coordinte trnsltion on function cuses its Fourier trnsform to be multiplied by phse fctor : f(x + b) FT e ikb F (k). (41) As consequence, trnsltions leve the Fourier spectrum F (k) 2 unchnged. 74

8 (iii) If the Fourier trnsform of f(x) is F (k), then f (x) FT F ( k). (42) As consequence, the Fourier trnsform of rel function must stisfy the symmetry reltion F (k) F ( k), mening tht the Fourier spectrum is symmetric bout the origin in k-spce: F (k) 2 F ( k) 2. (iv) When you tke the derivtive of function, tht is equivlent to multiplying its Fourier trnsform by fctor of ik: d dx f(x) FT ikf (k). (43) For functions of time, becuse of the difference in sign convention (Section 9.2.3), there is n extr minus sign: d dt f(t) Fourier trnsforms of differentil equtions FT iωf (ω). (44) The Fourier trnsform cn be very useful tool for solving differentil equtions. As n exmple, consider dmped hrmonic oscilltor tht is subjected to n dditionl driving force f(t). This force hs n rbitrry time dependence, nd is not necessrily hrmonic. The eqution of motion is d 2 x dx + 2γ dt2 dt + ω2 0x(t) f(t) m. (45) To solve for x(t), we first tke the Fourier trnsform of both sides of the bove eqution. The result is: ω 2 X(ω) 2iγωX(ω) + ω0x(ω) 2 F (ω) m, (46) where X(ω) nd F (ω) re the Fourier trnsforms of x(t) nd f(t) respectively. To obtin the left-hnd side of this eqution, we mde use of the properties of the Fourier trnsform described in Section 9.2.4, specificlly linerity (i) nd Fourier trnsformtions of derivtives (iv). Note lso tht we hve used the sign convention for time-domin functions discussed in Section The Fourier trnsform hs turned our ordinry differentil eqution into n lgebric eqution. This eqution cn be esily solved: F (ω)/m X(ω) ω 2 2iγω + ω0 2 (47) Knowing X(ω), we cn use the inverse Fourier trnsform to obtin x(t). To summrize, the solution procedure for the driven hrmonic oscilltor eqution consists of (i) using the Fourier trnsform on f(t) to obtin F (ω), (ii) using the bove eqution to find X(ω) lgebriclly, nd (iii) performing n inverse Fourier trnsform to obtin x(t). This will be the bsis for the Green s function method, method for systemticlly solving differentil equtions tht will be discussed lter. 9.3 Common Fourier trnsforms To ccumulte more intuition bout Fourier trnsforms, we will now study the Fourier trnsforms of few interesting functions. We will simply stte the results, leving the ctul clcultions of the Fourier trnsforms s exercises. 75

9 9.3.1 Dmped wves In Section 9.2.2, we sw tht n exponentilly decy function with decy constnt η R + hs the following Fourier trnsform: { e ηx, x 0 FT F (k) i 0, x < 0, k iη. (48) Observe tht F (k) is given by simple lgebric formul. If we extend the domin of k to complex vlues, F (k) corresponds to n nlytic function with simple pole in the upper hlf of the complex plne, t k iη. Next, consider decying wve with wve-number q R nd decy constnt η R +. The Fourier trnsform is function with simple pole t q + iη: Hence, the Fourier spectrum is { e i(q+iη)x, x 0 0, x < 0. F (k) 2 FT i F (k) k (q + iη). (49) 1 (k q) 2 + η 2. (50) This is Lorentzin peked t k q nd hving width 2η, s shown below: On the other hnd, consider wve tht grows exponentilly with x for x < 0, nd is zero for x > 0. The Fourier trnsform is function with simple pole in the lower hlf-plne: { 0, x 0 FT i e i(q iη)x F (k), x < 0. k (q iη). (51) The Fourier spectrum is, likewise, Lorentzin centered t k q. From these exmples, we see tht oscilltions nd mplifiction/decy in f(x) re relted to the existence of poles in the lgebric expression for F (k). The rel prt of the pole position gives the wve-number of the oscilltion, nd the distnce from the pole to the rel xis gives the mplifiction or decy constnt. A decying signl produces pole in the upper hlf-plne, while signl tht is incresing exponentilly with x produces pole in the lower hlf-plne. In both cses, if we plot the Fourier spectrum of F (k) 2 versus rel k, the result is Lorentzin curve centered t k q, with width 2η. 76

10 9.3.2 Gussin wve-pckets It is lso interesting to look t the Fourier trnsform of function tht decys fster thn n exponentil. In prticulr, let s consider function with decy envelope given by Gussin function: e iqx e γx2, where q C, γ R. (52) Such function is clled Gussin wve-pcket. The width of the Gussin envelope cn be chrcterized by the Gussin function s stndrd devition, which is where the curve reches e 1/2 times its pek vlue. In this cse, the stndrd devition is x 1/ 2γ. It cn be shown tht f(x) hs the following Fourier trnsform: π (k q) F (k) 2 γ e 4γ. (53) To derive this result, we perform the Fourier integrl s follows: F (k) dx e ikx f(x) (54) dx exp { i(k q)x γx 2}. (55) In the integrnd, the expression inside the exponentil is qudrtic in x. We complete the squre: { ( ) 2 ( ) } 2 i(k q) i(k q) F (k) dx exp γ x + + γ (56) 2γ 2γ { ( ) } 2 exp { γ x +. (57) } (k q)2 dx exp 4γ i(k q) 2γ The remining integrl is simply the Gussin integrl (see Chpter 2), with constnt shift in x which cn be eliminted by chnge of vribles. This yields the result stted bove. The Fourier spectrum, F (k) 2, is Gussin function with stndrd devition k 1 2(1/2γ) γ. (58) Thus, we gin see tht the Fourier spectrum is peked t vlue of k corresponding to the wve-number of the underlying sinusoidl wve in f(x). Moreover, stronger (weker) decy in f(x) leds to broder (nrrower) Fourier spectrum. 77

11 9.4 The delt function Wht hppens when we feed the Fourier reltions into one nother? Plugging the Fourier trnsform into the inverse Fourier trnsform, we get In the lst step, we hve introduced 2π eikx F (k) (59) 2π eikx δ(x x ) dx dx e ikx f(x ) (60) 2π eikx e ikx f(x ) (61) dx δ(x x ) f(x ). (62) ) 2π eik(x x, (63) which is clled the delt function. According to the bove equtions, the delt function cts s kind of filter: when we multiply it by ny function f(x ) nd integrte over x, the result is the vlue of tht function t prticulr point x. But here s problem: the bove integrl definition of the delt function is non-convergent; in prticulr, the integrnd does not vnish t ±. We cn get round this by thinking of the delt function s limiting cse of convergent integrl. Specificlly, let s tke δ(x x ) lim ) γ 0 2π eik(x x e γk2. (64) For γ 0, the regultor exp( γk 2 ) which we hve inserted into the integrnd goes to one, so tht the integrnd goes bck to wht we hd before; on the other hnd, for γ > 0 the regultor ensures tht the integrnd vnishes t the end-points so tht the integrl is welldefined. But the expression on the right is the Fourier trnsform for Gussin wve-pcket (see Section 9.3.2). Using tht result, we get δ(x x 1 ) lim e (x x ) 2 4γ. (65) γ 0 4πγ This is Gussin function, with stndrd devition 2γ nd re 1. Hence, the delt function cn be regrded s the limit of Gussin function s its width goes to zero while keeping the re under the curve fixed t unity (which mens the height of the pek goes to infinity). The most importnt feture of the delt function is it cts s filter. Whenever it shows up in n integrl, it picks out the vlue of the rest of the integrnd evluted where the delt function is centered: dx δ(x x 0 ) f(x 0 ). (66) Intuitively, we cn understnd this behvior from the bove definition of the delt function s the zero-width limit of Gussin. When we multiply function f(x) with nrrow Gussin centered t x 0, the product will pproch zero lmost everywhere, becuse the Gussin goes to zero. The product is non-zero only in the vicinity of x x 0, where the Gussin peks. And becuse the re under the delt function is unity, integrting tht product over ll x simply gives the vlue of the other function t the point x 0. 78

12 9.5 Multi-dimensionl Fourier trnsforms When studying problems such s wve propgtion, we will often hve to del with Fourier trnsforms cting on severl vribles simultneously. This is conceptully strightforwrd. For function f(x 1, x 2,..., x d ) which depends on d independent sptil coordintes x 1, x 2,... x d, we cn simply perform Fourier trnsform on ech coordinte individully: F (k 1, k 2,..., k d ) dx 1 e ik1x1 dx 2 e ik2x2 dx d e ik dx d f(x 1, x 2,..., x N ) (67) Note tht ech coordinte gets Fourier-trnsformed into its own independent k vrible, so tht the result is still function of d independent vribles. We cn compctly express such multi-dimensionl Fourier trnsform vi vector nottion. Let x [x 1, x 2,..., x d ] be d-dimensionl coordinte vector. The Fourier-trnsformed coordintes cn be written s k [k 1, k 2,..., k d ], nd the Fourier trnsform is F ( k) dx 1 dx 2 dx d e i k x f( x), (68) where k x k 1 x 1 + k 2 x k d x d is the usul dot product of two vectors. The inverse Fourier trnsform is f( x) 1 2π 2 2π d 2π ei k x F ( k). (69) A multi-dimensionl delt function cn be defined s the Fourier trnsform of plne wve: δ d ( x x ) 1 2π 2 2π d 2π ei k ( x x ). (70) Note tht δ d hs the dimensions of [x] d. The multi-dimensionl delt function hs filtering property similr to the one-dimensionl delt function. For ny f(x 1,..., x d ), dx 1 dx d δ d ( x x ) f( x) f( x ). (71) Finlly, if we hve mix of sptil nd temporl coordintes, then the usul sign conventions (see Section 9.2.3) pply to ech individul coordinte. For exmple, if f(x, t) is function of one sptil coordinte nd one temporl coordinte, the Fourier reltions re 9.6 Exercises F (k, ω) f(x, t) dx 2π dt e i(kx ωt) f(x, t) (72) dω 2π ei(kx ωt) F (k, ω). (73) 1. Find the reltionship between the coefficients {α n, β m } in the sine/cosine Fourier series nd the coefficients f n in the complex exponentil Fourier series: f(t) ( ) 2πnx ( ) 2πmx α n sin + β m cos (74) n1 m0 ( ) 2πinx f n exp. (75) n 79

13 2. Find the Fourier series expnsion of the tringulr wve { cx, /2 x < 0, cx, 0 x < /2 (76) where c is some rel constnt. Using the result, derive series expnsion for π Prove the properties of the Fourier trnsform listed in Section Find the Fourier trnsform of sin(κx)/x. 5. Prove tht if f(x) is rel function, then its Fourier trnsform stisfies F (k) F ( k). 80