Fourier Series And Transforms

Transcription

1 Chapter Fourier Serie And ranform. Fourier Serie A function that i defined and quare-integrable over an interval, [,], and i then periodically extended over the entire real line can be expreed a an infinite erie of ine and coine. hi fact will be aumed without proof for now. hu, uppoe g% ( t) i a function that i periodic with period. Periodicity implie that the function i completely defined by it value over jut g% t a the following erie: the ingle interval, [,]. We may expre ( ) = = g % π π ( t) = a + a co t+ bin t, (.) where a and b are contant coefficient. An example, where mot of the coefficient are equal to zero, i given in Figure.. he repreentation (.) of g% ( t) i called a Fourier erie. Note that g% ( t) written thi way i, in fact, periodic with period : g% ( t+ ) = g% ( t), alo g( t± n) = gt ( ), n=,,, % % K. (.) he proof of thi i left to the reader. Since pectral analyi originated in the theory of communication with electromagnetic ignal, the function g, whether periodic or not, often refer to a ignal varying with repect to time, and the domain of definition of g, i.e., < t <, i uually called the time domain. In geodey, and many other field, the domain doe not have to be time, but could be ditance or everal patial dimenion; in general, it may be any other independent variable; for patial variable, the domain i called the pace domain. At preent, we aume that the domain i one Fourier Geodey. Jeeli, March 7

2 dimenional and infinite: (, ) ; and, for the moment, we will retain the notation, t, allowing it to denote either time or ditance, or any other one-dimenional variable.. 4 ~ 6 g(t) time, t Figure.: ( ) 7 3 g % t.5 a co π π = + t+ b in t ; = = { a } = {,,.5,3.,, 3,.7}, { b } = {,.4,3} he expreion (.) for g% ( t) in term of a Fourier erie involve quantitie,. hee are the harmonic frequencie of the ignal. he fundamental frequency i ( = ). Again, the term frequency i generally aociated with time ignal and ha unit of cycle per unit of time, e.g., cycle per econd, or Hz (Hertz). Frequency i the invere of wavelength high frequency (large ) implie hort wavelength; low frequency (mall ) implie long wavelength. hi can be viualized (left to reader) by examining a graph of ( ) = inπ g% t t, (.3) by varying the parameter,, ( weeping through frequencie). In geodey, where t i in the pace domain, we till ue the term frequency, although wavenumber (referring to the integer ) i alo ued. In thi cae, the unit of frequency are cycle per unit of ditance, e.g., [cy/m]. In cae the frequency tae on continuou real value, we denote it by f (cycle-frequency) having the ame unit a, or alo, ω (radian-frequency) ω = π f, (.4) Fourier Geodey. Jeeli, March 7

3 which ha unit [rad/], or [rad/m], a the cae may be. We will retain the cyclical frequency, although it mean that we alway have to include π [rad/cy], ince the argument of inuoidal function mut be unitle (or in radian). Sometime, the Fourier erie i written a = = g % π π ( t) = a' co t+ b' in t, (.5) or, even more compactly a where π i t g% ( t) = Ge, (.6) = i =, i i the imaginary unit, and the coefficient G are (in general cae) complex a', b ' in number. he relationhip between the coefficient { a, b } in equation (.) and { } equation (.5) i obviou. Since π co t = e + e π π i t i t and π i in t = e e π π i t i t, (.7) one can eaily derive the correponding relationhip between the coefficient {, } equation (.) and { G } in equation (.6): a b in a ib, > ; G = a, = ; a + ib, <. (.8) he invere relationhip i given by a = ( G + G ), > ; a = G, = ; i b = ( G G ), >. (.9) We pre-multiply the um in equation (.6) by to be conitent with the unit ued later for tranform of non-periodic function. hu, the unit of G are the unit of g% ( t) per frequency unit, ince the unit of are invere to thoe of frequency. For convenience of notation, we will Fourier Geodey.3 Jeeli, March 7

4 ue only the complex-coefficient Fourier erie (.6), noting that with equation (.8) and (.9) one could alway revert to the erie (.) in term of the ine and coine function. he et of coefficient, { a, b } or { G }, i nown a the (Fourier) pectrum of g. Given g, one can compute it pectrum uing the orthogonality of the ine and coine on the interval [,]. For the complex exponential, there i the following orthogonality relationhip: π π π i t i t i t π ( ) e e l dt e l π = dt = co ( l) t dt+ i in ( l) t dt, l; =, = l. (.) herefore, multiplying equation (.6) on both ide by equation (.), e π i lt and integrating yield, in view of ( ) π i t G = g t e dt %. (.) hu, given g% ( t) in the interval, [,], one can find it pectrum according to equation (.); and, given it pectrum, one can compute g% ( t) for any t uing equation (.6). he pectrum and it function are dual repreentation of the ame information - one i equivalent to the other (a long a the function i continuou). he pectrum of a function diplay the ame information, but in a different domain the frequency domain, where it i often more ueful than in the time- (or pace-) domain. he two relationhip, (.6) and (.), contitute a Fourier ranform pair for periodic function. We will denote thi relationhip a ( ) G g% t. (.) Although it may be viewed legitimately a a tranform, we will ue thi terminology for other type for Fourier tranform, a well. herefore, to be more pecific we hould ay that it i the Fourier erie tranform pair.. Propertie of Fourier Serie he Fourier erie tranform pair (.) may be manipulated uing everal linear operation whoe reult are ummarized with the following (non-exhautive lit of) propertie. hey can be proved with relative eae from the baic tranform pair, (.), and the definition (.6) and (.). It i noted that while we generally ue only real-valued function, g, all definition and propertie hold equally for complex-valued function and are written a uch. Fourier Geodey.4 Jeeli, March 7

5 g% t G ; (.3). ( ) ag% t ag : proportionality; (.4). ( ) % % : uperpoition; (.5). g ( t) + g ( t) ( G ) + ( G ) provided g% and 3. ( ) g% have the ame period, ; g% t G : ymmetry; (.6) % : tranlation. (.7) 4. ( ) g t t Ge π i t here i alo an important theorem nown a Pareval heorem, which tate that = % % (.8) * * 5. g( t) g( t) dt ( G) ( G) = provided g% and g% have the ame period, (* denote complex conjugate). he proof of thi i eaily done by ubtituting equation (.6) on the left ide, a follow: π π π i t i t ' i ( ' * * ) t ( G) e ( G) e dt = ( G) ( G) e dt. (.9) ' ' = ' = = ' = With the orthogonality of complex exponential, equation (.), the reulting equation (.8) follow immediately..3 Fourier ranform We now find correponding Fourier tranform pair for certain non-periodic function. Specifically, we retrict ourelve to quare-integrable function defined on the real line, which are function, g, uch that g ( t ) dt <, (.) Fourier Geodey.5 Jeeli, March 7

6 * where g( t) g( t) g ( t) with equation (.)): =. hen it (continuou) Fourier tranform i defined by (compare thi iπ ft F ( g) G( f ) = g( t) e dt. (.) It can be hown that F ( g) i alo quare-integrable. Becaue of the quare-integrability, it i aid that g and G each ha finite energy. In addition, the invere (continuou) Fourier tranform i given by iπ ft F ( G) g( t) = G( f ) e df. (.) G( f ) i alo nown a the pectrum (or pectral denity) of g. In general, G i complex, even if g i real (again, we will deal only with real function, g). he pectrum can be decompoed into the amplitude pectrum: (( ) ( ( )) ) Im ( ) Re ( ) A f = G f + G f, (.3) and the phae pectrum: ϕ ( f ) ( ) ( ) Im G f = tan. (.4) Re G f Amplitude and phae together yield the pectrum in the form: i ( f ) ( ) A( f ) e φ G f = (.5) he quare of the amplitude pectrum i alo nown a the energy pectrum. mgal/(cy/m), If g ha unit [mgal], and t ha unit [m], then the pectrum, G, ha unit [ ] whence it alternative name, pectral denity. he unit of the coefficient G, equation (.), are conitent with thi. Note that equation (.) and (.) could be written with radian frequency, ω, where from equation (.4), df = dω π. Finally, we ee that periodic function are not included in the definition of the Fourier tranform (.), becaue they are not quare-integrable they have infinite energy! (However, we can get around thi, later, in a formal ene.) Fourier Geodey.6 Jeeli, March 7

7 .4 Propertie of Fourier ranform A with Fourier erie, there are a number of propertie aociated with the Fourier tranform of a function and of linear operation performed on the function. hee are lited a follow for the Fourier tranform pair:. g( t) G( f ) ; (.6). ag( t) ag( f ): proportionality; (.7). g ( t) g ( t) G ( f ) G ( f ) + + : uperpoition; (.8) 3. g( t) G( f ): ymmetry; (.9) 4. G( t) g( f ) : duality; (.3) f g at G, a a a : imilarity theorem; (.3) 5. ( ) iπ ft 6. g( t t ) G( f ) e : tranlation; (.3) d g t i fg f : differentiation in time; (.33) dt 7. ( ) π ( ) d : differentiation in frequency; (.34) π df 8. itg( t) G( f ) g t g t dt G f G f df : Pareval heorem. (.35) * * 9. ( ) ( ) = ( ) ( ) here are everal other propertie; ome of thee will be noted later on. Pareval heorem will be proved later (ee Section.) on the bai of the Convolution heorem. A proof of Property 5, equation (.3), i given below. Proof: Let a be a non-zero contant; and let t' = at. hen Fourier Geodey.7 Jeeli, March 7

8 F t' i π f i ft a π ( ( )) = ( ) = ( ') g at g at e dt g t e dt ' a f = G a a (.36) Example: We can ue Property 5 to determine the Fourier tranform pair of a function if we change unit through a change of variable, uch a a change from patial to temporal domain. Uually thi i needed if a velocity, v, i involved, and we wih to change from the time domain (t) to the ditance domain (), and vice vera. Auming the velocity i contant, the change of variable i given by = vt. (.37) hen if G ( µ ) i the pectrum of g( ), the pectrum, G' ( f ), of g' ( t) g( vt) G'( f ) = G( f v) v. Suppoe g ha unit [mgal] and ha unit [m], then ( ) [mgal/(cy/m)]. Now if v ha unit [m/], then t ha unit [], f ha unit [cy/], g' ( ) unit [mgal], and it pectrum, G' ( f ), ha unit [mgal/(cy/)]. = i given by G µ ha unit t till ha.5 Some Important Example he following example of Fourier tranform of particular function will be very ueful. Example : he rectangular ("box-car") function (Figure.). Let b t ( ), t < ; =.5, t = ;, t >. (.38) hen the Fourier tranform, according to the definition (.) i given by iπ ft iπ ft ( ) = F ( ( )) = ( ) = = co( π ) in = π f ( π f ) B f b t b t e dt e dt ftdt (.39) Fourier Geodey.8 Jeeli, March 7

9 We define the value of the rectangular function at t = even though it i of no conequence for the direct Fourier tranform (.39). However, the invere Fourier tranform yield F iπ f ± ( ( )) ( ) t=± ( π f ) in B f = B f e df = co( π f ) df =, (.4) π f which agree with the defined value b (t) t inc(f) f Figure.: Rectangular function and it Fourier tranform, the inc function. he Fourier tranform of the rectangular function i the o-called "inc" function of f (formally the ine cardinal function): inc ( f ) in( π f ) =. (.4) π f Becaue thi function appear o frequently, it i given thi pecial name. Note that ince the invere Fourier tranform of the inc function i the rectangular function, we have with equation (.) and t = : inc( f ) df =. (.4) hat i, the area encloed by the inc function in the frequency domain i alo unity, a it i for the box-car function in the time domain. By the imilarity theorem (.3) and proportionality (.7), we have Fourier Geodey.9 Jeeli, March 7

10 ( π ) t in f b B f = πf ( ). (.43) From thi we ee that a the rectangular function bae hrin ( decreae), the main lobe of the inc function expand (firt zero of the inc function i at f =± ); the oppoite clearly hold. Example : he Dirac function (alo nown a the delta function, or the impule function). hi i not really a function becaue it i not well defined for all t. Specifically, the Dirac function, δ t, i the "function" that atifie the following: ( ) a) ( t), for all t δ = ; (.44) ; (.45) b) δ ( t) dt =. (.46) c) δ ( t t ) g( t) dt = δ ( t t) g( t) dt = g( t ) In order for the area under the delta function to be non-zero, according to equation (.45), even though it i zero almot everywhere, in view of equation (.44), it "value" at t = mut be infinite. hi can be een by uing the following approximate form of the delta function. b t with < (ee Figure.3). Note that the area under each Conider the function ( ) of thee function i equal to. A approache zero, the magnitude of b ( t) around t = approache infinity, while it approache zero everywhere ele. herefore, we have δ t =. (.47) ( t) lim b Again, thi i jut a formal expreion becaue, of coure, the limit doe not exit for t =. But, we can now (formally) determine the Fourier tranform of the delta function by determining firt the Fourier tranform of the function b ( t) approache zero. With equation (.43) we have and then taing the limit a F ( δ ( t) ) = lim F b = lim inc ( f ) =, for all f t ( ) (.48) Fourier Geodey. Jeeli, March 7

11 hat i, the Fourier tranform of the delta function i a contant (equal to ). he unit of the δ t, according to equation (.45) are /(unit of t), whatever the unit of t. delta function, ( ) 9 8 = / = /4 4 3 = / = t Figure.3: he function b ( t) with. We ay that δ ( t ) and are a Fourier tranform pair in the limit. We will be able to manipulate the delta function and it tranform according to the above, alway remembering that rigorouly, it hould be done firt with the rectangular function, and then, by applying the limit. Here we alway aume the interchangeability of the limiting proce and whatever other operator occur, provided the reult either exit or can be interpreted in term of the delta function. Example 3: Fourier tranform of a periodic function. When defining the Fourier tranform for continuou function, we pecifically excluded periodic function becaue they do not have finite energy. However, with the concept of the delta function, we can formally determine the Fourier tranform of a periodic function. hi i done by noting that ince, from equation (.48), δ t = F = δ t ; or F ( ( )), we have ( ) ( ) δ ( ) iπ ft t = e df. (.49) Now change variable in equation (.49) a follow: firt replace t by /; ubequently replace f by t. hen π i t e dt = δ. (.5) Fourier Geodey. Jeeli, March 7

12 Now for periodic function expreed a Fourier erie (.6), the Fourier tranform, formally, can be expreed a F ( % ( )) ( ) π i t iπ f t iπ ft gt = G f = Ge e dt = G e dt = = = = Gδ f (.5) hat i, the Fourier tranform of a periodic function with Fourier coefficient equence of impule caled by G G i an infinite and paced along the frequency axi at the dicrete frequencie (Figure.4). Note that the unit of the delta function in (.48) are /(unit of frequency), canceling the unit of ; o, G( f ) and G have the ame unit. he Fourier tranform of a periodic function, a expreed in equation (.5), i formulated thi way only to mae it conitent with our previou definition of a Fourier tranform of nonperiodic and quare-integrable function (i.e., finite-energy function). In that ene it ha ome uefulne, but mainly we will conider the Fourier erie pair, equation (.6) and (.) to be a ind of Fourier tranform pair, a noted earlier, thu avoiding the ue of the delta function f Figure.4: Fourier tranform (amplitude pectrum) of g% ( t), hown in Figure. Example 4. he ampling function. Conider the periodic function that i an infinite equence of identical rectangular function, each having a it bae: Fourier Geodey. Jeeli, March 7

13 = t t g% ( t) = b, (.5) where t > i the pacing between them (Figure.5a), and hence the period of g( t) We define t ( ) to be the limit of g% ( t) a to. From equation (.47), t ( ) i given by (ee alo Figure.5b) ( ) = lim ( ) = δ ( ) = % i t. while the area of each rectangle remain equal t g% t t t. (.53).5 t = =.5 t = = t Figure.5: a) Infinite equence of rectangular function. b) Sampling function. t Being periodic, with period, t g% t, given by equation(.5), can be repreented a a Fourier erie (.6), where the Fourier coefficient are given by equation (.), which i an, t : integral over any period of g( t), ( ) %. We chooe [ ] t π t i t t G = b e dt. (.54) Now, outide the integration interval the rectangular function, b, i zero, o the integration can be extended to ±, and with a uitable change of integration variable ( t' = t), we get, uing the reult (.39) and the definition (.4), Fourier Geodey.3 Jeeli, March 7

14 π π t i t i t' t t ( ') G = b e dt = b t e dt ' = inc t (.55) Formally, the Fourier tranform of g% ( t), according to equation (.5) i an infinite equence of impule having magnitude, equation (.5), we have G. Subtituting equation (.55) into the right ide of F ( g% ( t) ) = inc δ f t. (.56) t t = hen we apply the limit a ; and, noting that (.53): lim inc =, we obtain with equation t F δ ( t t) = δ f t t. (.57) = = he infinite equence of impule, t, ( ) being the limit of the equence of rectangular function, i called the ampling function, becaue if multiplied with ome arbitrary function it ample the latter uing impule (ee Chapter 3). he Fourier tranform of the ampling function i, again, a ampling function, but now in the frequency domain..6 -D Fourier Serie and ranform In geodey and geophyic we deal with ignal on higher dimenioned domain uch a the plane, the phere, and three-pace. All the concept of Fourier erie and tranform eaily carry over into thee higher dimenion epecially if the underlying coordinate frame i Carteian. he Fourier erie of a function periodic in two (Carteian) variable i given by =, = = (, ) iπ x+ x g% x x G e, (.58) where the coefficient are given by Fourier Geodey.4 Jeeli, March 7

15 iπ x+ x, (, ) G = g x x e dxdx %. (.59) We ue the complex notation for convenience, but one could a well write equation (.58) and (.59) in term of ine and coine. Note that the periodicity of g may be different in the two dimenion, viz. and ; g% ( x ± n, x ± n ) = g% ( x, x), where n and n are integer. Clearly, higher-dimenioned function have analogou erie expanion where the generalization hould be obviou. g x, x Auming finite-energy function in two dimenion, i.e., ( ) have the (continuou) Fourier tranform pair: iπ ( fx + fx) (, ) (, ) G f f g x x e dxdx dxdx <, we =, (.6) iπ ( fx + fx) (, ) (, ) g x x G f f e df df =. (.6) If the unit of g are [mgal] and thoe of x, x are each [m], then G ha unit of [mgal/(cy/m) ]. he propertie for the continuou -D Fourier tranform, equation (.6) through (.35), can naturally be extended for the -D tranform. Only the following are mentioned in particular, tarting with the tranform pair:. g( x, x ) G( f, f ) ; (.6) f f g ax, ax G,, a, a : imilarity theorem; (.63) aa a a. ( ) p q g x x i f i f G f f p q x x p q. (, ) ( π ) ( π ) (, ) g x x g x x dxdx G f f G f f dfdf : 3. (, ) * (, ) = (, ) * (, ) : differentiation; (.64) Pareval heorem. (.65) Pareval heorem, again, will be proved later uing the convolution theorem. We can now alo define the rectangular (box-car) function in two dimenion: Fourier Geodey.5 Jeeli, March 7

16 , x < and x < ; b =.5, x = and x or x and x = ;, x > or x > ; (.66) and it Fourier tranform i imply the product of inc function (the proof i left to the reader): ( b ( x, x )) = B ( f, f ) = inc( f ) inc ( f ) F. (.67) Analogou to one dimenion, the Dirac delta function i defined by δ x, x =, if x or x ; (.68) a) ( ) x, x dxdx = ; (.69) b) δ ( ) ; (.7) c) δ ( x' x, x' x ) g( x', x' ) dx' dx' = g( x, x ) and it Fourier tranform i ( ( )) F δ x, x =, for all f, f (.7).7 -D ranform of the Potential Function Conider a function defined in three-dimenional pace, but retrict the Fourier tranform to the firt two dimenion with the third variable held fixed. For certain function thee -D tranform, for different value of the third variable, can be related to each other. hi i the cae, for example, for the gravitational potential, v, that atify Laplace equation in the half pace z > : v= + + v =. (.7) x x z Solution to thi equation are aid to be harmonic. We ue the notation z, rather than 3 x, for the third dimenion to emphaize that the Fourier tranformation are retricted to the firt two Fourier Geodey.6 Jeeli, March 7

17 dimenion and z may be viewed more a a parameter. Since v i a potential, we have v a z. Let V be the -D Fourier tranform of v with repect to x, x : iπ ( fx + fx) (, ; ) ( ) (,, ) V f f z v v x x z e dxdx he invere i given by = F =. (.73) iπ( fx + fx) (,, ) ( ) (, ; ) v x x z V V f f z e df df = F =. (.74) Applying equation (.7) to (.74) yield iπ( fx + fx) ( V( f, f; z) e ) dfdf =. (.75) Hence, performing the differentiation of the integrand according to equation (.7), we get for all z > : ( ) ( π ) ( ) ( ) z ( + ) iπ fx fx V f, f; z f + f V f, f; z e dfdf =. (.76) hi ha to hold for all x, x, which implie that we have the following differential equation for V in term of the variable z: z V ( f, f; z) ( π ) ( f + f ) V ( f, f; z) =. (.77) he olution i given by (a can eaily be verified by bac-ubtitution into equation (.77)) (, ; ) π πz f + f z f + f V f f z = Ce + Ce, (.78) where C and C are contant. We mut have C = ; otherwie, v a z, which contradict our aumption that v i a Newtonian potential. Alo, uppoe that on the plane z = we have (, ;) (, ) V f f = V f f (.79) hen, finally, the olution (.78) i given by Fourier Geodey.7 Jeeli, March 7

18 (, ; ) (, ) V f f z V f f e π zf =, (.8) where we define the radial frequency f = f + f. (.8) he -D pectra of v at two level, z and z, are thu related according to equation (.8) by the ( ) z z f e π. factor he invere Fourier tranform (.74) now become π zf iπ ( fx + f x ) (,, ) ( ) (, ) v x x z V V f f e e df df hen, for any n = F =. (.8) n n π zf iπ ( fx + fx) v( x, x, z) = ( π f ) V( f, f n ) e e dfdf, (.83) z which implie that the Fourier tranform of the z-derivative of v are given by n v ( x, x, z n F ) ( π f ) ( v ( x, x, z n = F )). (.84) z Equation (.84) hold only for thoe finite energy (in -D) function atifying our original aumption of harmonicity and vanihing at infinity. Clearly, the pectrum of the derivative i amplified at high frequencie (large n). hi feature may be advantageou, a in the cae of gravity gradiometry, where it i ought to detect the high-frequency component of the gravity field; or, it may be deleteriou, a when one numerically differentiate a noiy potential etimate (and there i ignificant noie at the high frequencie)..8 he Hanel ranform We now conider the pecial cae of a function defined on the -D plane, but depending on jut a ingle variable, the radial ditance from the origin. hu, let r = x + x, (.85) and then uppoe that Fourier Geodey.8 Jeeli, March 7

19 (, ) g( r) g x x =. (.86) he Fourier tranform in thi cae i, according to equation (.6) F π iπ( fx + fx) iπ frco( φ α) ( (, ) ) = (, ) = ( ) g x x g x x e dxdx g r e rdφdr = π iπ frco( φ α) rg( r) e dφ dr (.87) where x = rcoφ, x = rinφ and f = f coα, f = f inα. he integral in the lat line of equation (.87) with repect to φ i nown to be π iπ frco( φ α) e dφ = πj ( π fr ), (.88) for any α, where J i the zero-order Beel function of the firt ind. We have, therefore, ( ) π ( ) ( π ) G f = rg r J frdr, (.89) which i called the Hanel tranform of g. he Hanel tranform i jut a pecial cae of the Fourier tranform, where the function being tranformed ha circular ymmetry, given by equation (.86). We then have alo the invere Hanel tranform, derived from the invere Fourier tranform in exactly the ame way. he reult i: ( ) π ( ) ( π ) g r = fg f J fr df. (.9) Note that although G( f ) i a function of only one frequency (the radial frequency, f, equation (.8)), it unit till contain the invere quare of frequency unit. For example, if g ha unit [mgal] and r ha unit [m], then the unit of G are [mgal/(cy/m) ]. Fourier Geodey.9 Jeeli, March 7

20 .9 Propertie of the Hanel ranform Being a pecial cae of the Fourier tranform, there are imilar propertie for the Hanel ranform. We tart with the baic Hanel tranform pair:. g( r) G( f ); (.9). ag( r) ag( f ) : proportionality; (.9). g ( r) g ( r) G ( f ) G ( f ) + + : uperpoition; (.93) 3. G( r) g( f ) duality; (.94) f g ar G, a> : imilarity theorem; (.95) a a 4. ( ) g r g r rdr G f G f fdf : Pareval heorem. (.96) * * 5. ( ) ( ) = ( ) ( ) Pareval heorem in thi cae derive directly from the correponding theorem for function of two independent variable, equation. he rectangular function in thi cae i given by the cylinder function, having circular ymmetry: ( ), r< a ; bc r = π a, r> a; (.97) where a i the radiu of the cylinder. he Hanel tranform of b ( ) a r Bc ( f ) = π J ( π frdr ) π a ( π f ) a c r i given by π fr a J ( π fr ) d ( π fr ) π frj ( π fr ) πa πa f ( π f (.98) ) π = = = J π af r= ( π fa) Fourier Geodey. Jeeli, March 7

21 where d xj x = xj x. dx J i the firt-order Beel function of the firt ind, and ( ( )) ( ). Spherical Harmonic Serie and the Legendre ranform For a function defined on a phere, we have coordinate ( θλ, ) - geocentric co-latitude and longitude (we could alo ue geocentric latitude, φ, intead, where φ = 9 θ ). We note that any uch function i periodic in λ with period π and (technically) periodic in θ with period π (although it i defined only for θ π ). herefore, we expect that reaonably well behaved function might be expanded a erie of inuoidal function in thi cae, however, in addition to the uual inuoidal function of longitude, the pecial geometry of the phere call for Legendre function of coθ. We have the well nown expanion, where we omit the ~ notation ued earlier for periodic function becaue the argument, ( θλ, ), already ignify periodicity: n ( θλ, ) nm, nm, ( θλ, ) g = G Y (.99) n= m= n where the pherical harmonic function are given by Y ( θλ, ) P ( coθ) nm, nm, co mλ, m ; = in m λ, m< ; (.) and where P ( ) coθ i the fully-normalized, aociated Legendre function of the firt ind of n, m degree n and order m, with n m n and n. hi function i defined a follow: P n, m ( coθ ) = n, ( θ ) n+ P co, m= ; ( + )( ) ( n+ m)! n n m! P co, m > ; nm, ( θ ) (.) where the un-normalized Legendre function, P ( ) ( coθ ), coθ, are given by nm m m d Pnm, ( coθ) = in θ Pn( coθ m ), (.) d and the Legendre polynomial, Pn = Pn,, are given by Fourier Geodey. Jeeli, March 7

22 P d n co = co. (.3) n! d co ( θ) n n n n ( ) ( θ ) θ he function, P nm,, i normalized uch that the pherical harmonic function, Y, orthogonal, are, in fact, orthonormal: nm, which are 4π π π Ynm, ( θλ, ) Ypq, ( θλ, ) inθdθdλ = (.4), n= p and m= q., n p or m q; It i readily hown uing thi orthogonality relationhip that Gnm, = g( θλ, ) Ynm, ( θλ, ) dσ, (.5) 4π σ where dσ inθdθdλ σ = θλ, θ π, λ π repreent the urface of the unit phere. Equation (.5) i called the Legendre tranform of g, or the Legendre pectrum of g. g θλ, are the Legendre tranform pair. We will deal only with real function, g; and G and ( ) nm, =, and ( ) { } the coefficient, G nm, are then all real, a well. An alternative form of the Legendre tranform pair may be found in the literature, although it i not a often ued in geodey: where n im g( θλ, ) γnm, Pnm, ( coθ) e λ =, (.6) n= m= n ππ imλ γ = g θλ, P coθ e inθdθdλ, (.7) ( ) ( ) nm, nm, 4πε m ε m, m = ; =, m. (.8) he relationhip between the coefficient γ nm, and G nm, i given by Fourier Geodey. Jeeli, March 7

23 γ ( Gnm, ign, m), m> ; = G, m= ; ( Gn, m + ignm, ), m<. nm, n, (.9) In thi cae, the Legendre pectrum, defined by γ nm,, i complex, but it atifie γnm, = γnm, if g i a real function. Many function in geodey are alo defined in term of jut one variable, ψ, the pherical ditance, that technically can be conidered a an angle in the domain of a emi-circle: { ψ ψ π}. In thi cae, one could expand the function a a erie of inuoid a before; or, to be conitent with pherical harmonic, one can alo ue Legendre polynomial of coψ, P, form a complete bai for function defined on [,] ince the Legendre polynomial, n ψ π. We have, therefore, for a function, g, defined on ψ π : and n= *, or on g( ψ ) = ( n+ ) GP n n( coψ ), (.) π Gn = g( ψ ) Pn( coψ ) inψdψ, (.) which, together, give the one-dimenional Legendre tranform pair. Here, G n i the Legendre pectrum; it i not exactly conitent in term of cale with the pectrum of function defined on the phere, due to the normalization of P n, : Gn, = n+ Gn. (.) Although the above definition (.) i omewhat le cumberome, it i not univerally ued in the literature.. Propertie of Legendre ranform Some eential propertie aociated with the Legendre tranform are analogou to propertie of the Fourier tranform. hoe given for the -D Legendre tranform hold equally for the -D tranform on the bai of equation (.). Again, we aume the baic Legendre tranform pair: Fourier Geodey.3 Jeeli, March 7

24 g θλ ; (.3), Gnm. ( ), ag θλ : proportionality; (.4), agnm. ( ),. g( θλ, ) g( θλ, ) ( G) ( G) 4π 3. ( θλ, ) r p + + : uperpoition; (.5) n = n= m= n nm, nm, ( g ) dσ ( G ) nm, : Pareval heorem; (.6) σ 4. g( r, θλ, ) ( ) p r= R p ( n+ ) L( n+ p) R p G nm, radial differentiation of a harmonic function. (.7) he proof of Pareval heorem (.6) come directly from the orthogonality of the pherical harmonic function, equation (.4)). Property 4 applie to the radial extenion of a function on g θλ,, the phere according to potential theory. hat i, we aume that the function, ( ) repreent the boundary value on the phere (radiu, R) of a function, g( r, θλ, ), that i harmonic in the pace external to the phere (recall equation (.7)): ( θλ) g r,, =, r>, (.8) where, in thi cae, the Laplacian operator i given by cotθ r r r r θ r θ r in θ λ = (.9) It can be hown that the olution to the partial differential equation (.8) i given by n n+ R. (.) r n= m= n (, θλ, ) nm, nm, ( θλ, ) g r = G Y We ee that { G nm, } i the Legendre pectrum of the function, gr ( θλ, ) g( R, θλ, ) n+ phere of radiu, R; and that ( R ( R h) ) G nm, ( θλ, ) (, θλ, ) { } =, on the + i the Legendre pectrum of the function, gr+ h = g R+ h, on the phere of radiu, R+ h, h. Property 4, above, i eaily verified by applying the derivative to equation (.). Fourier Geodey.4 Jeeli, March 7

25 . Some Important Example he rectangular function in the cae of function on the phere i defined for convenience in term of jut one variable, the angular ditance from the center, a b ( ψ ) 4 π, ψ < ψ ; = σ, ψ > ψ ; (.) where σ i the area of the pherical cap { ψ ψ ψ } with radiu ψ (ee Figure.6): π ψ ψ d in d ( co ). (.) σ = σ = π ψ ψ = π ψ Uing the recurion formula d + n = ( n + n ), (.3) dy ( n ) P ( y) P ( y) P ( y) the pectrum of b i found to be 4π Bn = Pn( coψ ) inψdψ σ ψ = ( Pn ( coψ) Pn+ ( co ψ) ), n coψ n+ (.4) Fourier Geodey.5 Jeeli, March 7

26 ψ.. (θ,λ) ψ (θ,λ ) Figure.6: Geometry of pherical cap. In the geodetic literature, the pectral component, B n, are denoted β n ( Bn βn) and are called Pellinen moothing factor; we ll ee why moothing later on. Recurion formula exit for β n ; they are derived from correponding recurion formula for the Legendre polynomial, P n. For example, ubtituting the recurion formula ( ) ( ) ( ) ( ) ( ) n+ P y = n+ yp y np y, n, (.5) n+ n n into the integral of equation (.4), and uing equation (.3), we obtain β n n n n co n ; n ; n β ψ = n β + + β = ; β = ( + co ψ ). (.6) A plot of thee moothing factor for ψ = i hown in Figure.7. Note that the firt zerocroing occur at n = 9 8 ψ (roughly). he imilarity to the inc function (Figure.), which ha it firt zero-croing at, i evident. Fourier Geodey.6 Jeeli, March 7

27 ..8.6 β n n Figure.7: Pellinen moothing factor, β, ψ =. n Finally, we can define alo the Dirac function, again, conveniently only a a function of ψ : a) δ ( ψ), if ψ 4π b) ( ) 4π = ; (.7) δ ψ dσ = ; (.8) σ g d g ; (.9) c) δ ( ψ) ( θ', λ' ) σ = ( θλ, ) σ where from pherical trigonometry (ee alo Figure.6): ( ) coψ = coθco θ' + inθin θ 'co λ λ', (.3) and ψ i the angle at the center of the unit phere between ( θλ, ) and ( θ', λ ') pectrum of δ ( ψ ) i given by. he Legendre Fourier Geodey.7 Jeeli, March 7

28 Dn = δ( ψ ) Pn( coψ ) inψdψ = 4π δ ( ψ ) Pn( coψ ) dσ (.3) ( ) = P co =, for all n; n π σ where, in thi cae, ψ, trivially i the angle between and ψ..3 From Sphere to Plane here i an approximate relationhip between the Legendre tranform and the Hanel tranform for local application (i.e., the phere i approximated by a plane, locally). hi allow the tranition from the pherical pectral domain to the Carteian pectral domain and the from harmonic degree to patial frequency on the plane. Uing the aymptotic relationhip between Legendre polynomial and Beel function, Forberg (OSU Report No.356, 984) derive ω = π f n+ R, (.3) where R i the mean radiu of the Earth. An alternative formula wa given by Echardt (983). In Carteian coordinate, conider a v x, y, z = u x, y w z, harmonic in the half pace z, i.e., v =. It i readily function, ( ) ( ) ( ) hown that the part u( x, y ) atifie the differential equation + u( x, y) + ω uxy (, ) =, (.33) x y where ω i ome contant with repect to u( x, y ). Solution of equation (.33) are harmonic ocillator with frequency, ω. Analogouly, in pherical coordinate, a harmonic function can be eparated into v( r, θλ, ) = p( r) q( θλ, ), where q( θλ, ) atifie the differential equation ( + ) nn + cot θ + q ( θλ, ) + q ( θλ, ) =. (.34) R θ θ in θ λ R he olution of thi equation are the pherical harmonic function of degree, n. he derivative in equation (.34) are equivalent to the derivative in equation (.33) in a local coordinate ytem with horizontal and vertical direction (ay, north, eat, down). herefore, we can approximate by comparion: Fourier Geodey.8 Jeeli, March 7

29 ω ( + ) nn (.35) R which i Echardt alternative relationhip. Note that Forberg approximation doe not wor for n =, wherea, Echardt doe. However, both approximation are found to be equally ueful for the higher degree (they are baed, after all, on a local approximation); and both are generally better than the approximation ω nr..4 Example of ranform In thi ection, everal of the more common Fourier (-D and -D), Hanel, and -D Legendre tranform are given. Mot can be obtained by conulting a good able of Integral, e.g., Gradhteyn and Ryzhi (98). Some of thee were encountered already in the dicuion of previou ection; and other are eaily derived from thee uing the propertie of tranform. Still other, epecially the Legendre tranform pair, are familiar to geodeit. In each cae, we ue the generic ymbol to denote the tranform pair, and the type of tranform hould be obviou from the aociated independent variable.. b ( t) ( f ) inc (ee equation (.39)) (.36). δ ( t) (ee equation (.48)) (.37) 3. e πt π f e ( - D) (.38) 4. e t ( - D) (.39) + ( π f ) 5. b ( t, t ) inc( f ) inc( f ) (ee equation (.67)) (.4) 6. ( t t ) δ, (ee equation (.69)) (.4) ( x + x + a ) π π a( f ) / + f e ; a> 3/ a x iπ f x iπ f ; 3/ / 3/ / ( x + x) ( f + f ) ( x + x ) ( f + f ) (.4) (.43) Fourier Geodey.9 Jeeli, March 7

30 (ee equation (.98)) (.44) π fa 9. b ( r) J ( π fa) c. (Hanel tranform) (.45) r f π af / + a f. e ( ) r (Hanel tranform) (.46). e πr π f e (Hanel tranform) (.47) 3. e ar π a ( 4π f + a ) 3/ (Hanel tranform) (.48) ar π a r e f a ( 4π + ) / π π ar f re f e π (Hanel tranform) (.49) (Hanel tranform) (.5) 6. l R n+ =, R > R R + R RR coψ n+ R R, (-D) (.5) n+ R Y ', R > R l n R R 7. ( θ', λ ) + nm, (-D) (.5) 8. ( ) R R R R 3 l R n+, R > R (-D) (.53) 9. n+ R R R 3Rl R coψ l+ R Rcoψ, n ; + 5 3ln + n R R > R l R R R R, n =,; (.54) hi i the -D Legendre tranform of the generalized Stoe function S( R, ψ ). Fourier Geodey.3 Jeeli, March 7