Transforms, Convolutions, and Windows on the Discrete Domain

Transcription

1 Chapter 3 Transfors, Convoutions, and Windows on the Discrete Doain 3. Introduction The previous two chapters introduced Fourier transfors of functions of the periodic and nonperiodic types on the continuous space doain, as we as the principa and copeentary operations of convoution and truncation that have particuar interpretation in the frequency doain. In practice, however, we dea with data on a discrete, rather than continuous, doain. That is, a geophysica signa ay be defined on the basis of cassica physica aws for continuous tie or space, but we sape it, easure it, or reaize it at discrete points in space, or at discrete instants in tie. Therefore, data in practice aways constitute a sequence or an array of discrete vaues and a corresponding spectra anaysis ust be deveoped on the basis of this reaity. The degree to which the data represent the copete signa depends criticay on the saping interva and the significant part of spectra doain of the signa. The subject of this chapter, then, is not ony to deveop the transition of the Fourier and Legendre transfors and the attendant operations fro the continuous to the discrete doains, but aso to understand the consequent effects of saping on the spectra anaysis. The concepts of Fourier transfor, convoution (fiter, and window, discussed for functions on the continuous doain are readiy adapted to the discrete case with one iportant stipuation that the spatia doain is discretized at a constant interva. In practice, one aso has ony a finitey extended sequence of data, athough in soe instances where the extent of the data is sufficienty great, the approxiation by an infinite sequence is fuy justified. For periodic functions on the ine or pane and for spherica functions, the question turns on whether a fu period, or the copete spherica doain, of a function is saped. The first consideration in this chapter is of sequences defined on the infinite ine or pane. Ony after obtaining an appreciation of this practica ipeentation of continuous-doain theory, that is, of saping, do we enter the fina, abeit the ost usefu ipeentation of the theory: finitey extended sequences. Truncation of an infinite sequence is treated, as for the continuous case, using window sequences. However, it is aso convenient to interpret such finite Fourier Geodesy 3. Jeei, January 7

2 sequences as periodic with period equa to the ength of the sequence. Thus, the ast type of Fourier transfor to be exained is for functions that are both periodic and defined on a discrete doain, incuding arrays of function vaues covering the sphere that ay be viewed aso aternativey as the principa doain of a two-diensiona periodic function. 3. Infinite Sequences Spectra anaysis for infinite sequences can be deveoped without reference to a parent function fro which the sequence is derived. However, since geophysica data are aost aways sapes of a continuous geophysica signa, for the oent, a parent function is assued that has a Fourier integra. In one diension, suppose that vaues of g ( x are avaiabe at reguar intervas, x = x, where is an integer with < <. The resuting sequence of vaues is a set of discrete sapes of g, denoted sipy by the integer subscripts, g = g ( x, < <. (3. The saping interva, x, is constant and positive. Anaogous to absoute integrabiity, it is assued that the sapes are absoutey suabe, g = <, (3. which aso eans that they attenuate to zero in the iits, ±. To understand saping and the reationship between the discrete sequence and the originating continuous function one ay use the concept of ipuses defined by the Dirac deta δ x. Because the parent function is not periodic, aso the sequence, g, is not function, ( periodic, eading to the expectation that the spectra doain is continuous, as it is for nonperiodic functions. On further thought, however, the reader ay guess that there is a iit to the frequencies that coud be represented by the infinite sequence and that this depends in soe way on the saping interva. In fact, the sequence is band-iited, that is, the spectra doain is finite; and, in perfect duaity with the Fourier-series transfor pair, its spectru is periodic, as seen beow. The frequency iit in the spectra doain is tied intiatey to the saping interva and defines the resoution of the data (Section.5.. The concept of resoution is a-iportant when designing a easureent syste since it defines the tradeoffs between technica feasibiity and degree of reaization, or detai, of a function. Usuay, high-frequency noise in an instruent constrains the upper iit of usefu spectra content in the data. The constraints in data resoution then propagate to derived signas through processing operations, such as convoutions according to physica aws, as exepified by the force fieds. Fourier Geodesy 3. Jeei, January 7

3 3.. Fourier Transfors of Infinite Sequences One approach to the Fourier transfor of an infinite sequence is to use the specia case of the Fourier transfor for continuous functions appied to the Dirac deta function. Suppose one foray represents the infinite sequence of sapes as a function in the continuous variabe x by utipying the parent function with the saping function, (.96: gs ( x = g ( x δ ( x x x. (3.3 = The factor x is incuded so that the units of gs ( x atch those of g ( x (since the deta function has units inverse to those of its arguent. For any particuar x that is not an integer utipe of x (i.e., x x, for any, ( g x is a singe ipuse with apitude g x. Thus, g ( s g x = ; whie ( s s x is a train of ipuses with apitudes proportiona to the sape vaues. As before, when deaing with the deta function, one operates on it foray, recognizing that it is not a function in the conventiona eaning, and that one can aways approach the resut in the iit using the rectange function with vanishing base and unit rectanguar area. If G ( f is the Fourier transfor of g ( x, then since gs ( x = g ( x δ ( x x x, (3.4 = the dua convoution theore (. yieds F ( gs ( x = F ( g ( x * F δ ( x x x. (3.5 = It was shown with equation (. that the Fourier transfor of the saping function, again, is a train of ipuses, F δ ( x x = δ f x x = =. (3.6 Therefore, equation (3.5 becoes with the generaized convoution definition, equation (.4, Fourier Geodesy 3.3 Jeei, January 7

4 = = = F ( gs ( x = G ( f * δ f = G ( f * δ f x x = G ( f δ f f df x (3.7 The reproducing property of the Dirac deta function, equation (.89 then eads to a fundaenta resut, = = G ɶ s ( f F ( gs ( x = G f = G f + x x, (3.8 that reates the spectru of the sequence to the spectru of the parent function. The ast equation foows with a change in suation index,. Gɶ f is The ~ notation is appied to the Fourier transfor of the saped signa, because ( periodic with period in frequency equa to x, = Gɶ + s f + = G f + = Gɶ s ( f. (3.9 x x Thus, the sape spectru, G ( f principa part, ɶ, ay be defined soey on the interva of frequencies (the s f x < x, (3. ( ( ( or, any other interva of ength x. Periodicity ipies that G s ( x = G s x. The argest absoute frequency in this interva is f =. (3. x It is caed the yquist frequency (or, foding frequency and is the inverse of twice the saping interva. Gɶ f is a periodic function, with period, x, it can be represented as a Fourier Since ( series, equation (.8, s i x f Gɶ π s ( f x ce. (3. = = s Fourier Geodesy 3.4 Jeei, January 7

5 There is no oss in generaity in this case by choosing negative instead of positive exponents. It G f G f is copex. The Fourier is noted that ɶ ( generay is a copex function because ( s coefficients in equation (3. are given by equation (.3, with appropriate sign change in the exponentia, x s π ( ɶ ( c = Gɶ f e df = G f e df f i x f iπ xf s f. (3.3 With the representation of G ( f ɶ according to equation (3.8, one has s ( + x f iπ xf iπ xf c = G f + e df = G ( f e df x f = = x, (3.4 where the variabe of integration is changed to f = f + x. The su of integras is thus the infinite integra, and ( i π xf ( c = G f e df = g x = g. (3.5 That is, the Fourier series coefficients of the sape spectru, Gɶ ( f, are the sapes, theseves. Therefore, the Fourier transfor pair for infinitey extended sequences is ɶ ( iπ xf, (3.6 G f = x g e = f = i ( π xf g G f e df f ɶ, (3.7 where the subscript s is dropped because this transfor pair hods in genera, even if no specific reference is ade to an underying continuous function. The deta function is used ony a stepping stone to the fina resut that excudes this rather artificia function. On the other hand, it is instruenta in giving the fundaenta reationship, equation (3.8, between the spectru of the sequence and the Fourier transfor of the parent function. ote that the definition of the Fourier transfor, G ( f ɶ, agrees in ters of units with previous definitions. In the case that the sequence, g, has no underying parent function, a Fourier Geodesy 3.5 Jeei, January 7

6 constant saping interva is nevertheess assued and even if it is ipicit with x =, its units deterine the units of frequency. Because of the duaity between the Fourier series transfor pair, equations (.8, and the transfor pair for sequences, equations (3.6, (3.7, where the ony difference is in the (arbitrary convention of the sign in the exponentia, a the properties of one hod for the other with corresponding doains reversed. Therefore, the properties (. - (.4 of the periodic function and its Fourier transfor transate directy into the periodic spectru and the sequence of sapes. For exape, the syetry property is ( Gɶ f g. (3.8 Accounting for the change in sign of the exponentia, the transation properties are iπ xf g e G f + f ɶ (, g G ( f e π Aso, the syetry property anaogous to the equivaence (.9 is i x f + ɶ. (3.9 ( ( ( ( and if and ony if ɶ ɶ and ɶ ɶ ; (3. * * g = g g = g G f = G f G f = G f and, Parseva s theore in this case is f f Gɶ f df = x g ( =. (3. The discrete version of the Dirac deta function is the Kronecer deta. In order to be consistent with previous definitions in ters of units, we define a reated sequence by ( x x, = δ =, (3. Fro equation (3.6, its Fourier transfor is then sipy ( f ɶ =. (3.3 By the duaity with the Fourier series transfor, one then aso has a definition of the periodic deta function, ɶ δ ( x, that is zero everywhere except at od P ( x, such that P ( ɶ δ ( = ( gɶ x x x dx gɶ x, (3.4 Fourier Geodesy 3.6 Jeei, January 7

7 with spectru, =, for a. This aso eans that, using the first of equations (.8, π i x P e = Pɶ δ ( x. (3.5 = The rectange sequence, or discrete rectange function is defined here (with n even as n n ( n, b =, otherwise (3.6 where n ( T x = is a positive integer that divides the base, T, of the rectange into an odd nuber of intervas. The Fourier transfor of n n iπ x f ( ( = n ( n b, according to equation (3.6, is Bɶ f = x e ; (3.7 and, it can be shown that this sipifies to ( nπ xf ( π xf ( n sin i xf Bɶ π ( f = x e, sin Bɶ = n x. (3.8 ( n ( The vaue at f = is obtained by expanding nuerator and denoinator in series, dividing each n by f, and taing the iit as f. Equation (3.7 shows that Bɶ f approaches the periodic deta function, ɶ δ ( f, equation (3.5 (with period, ( ( x, as n, deonstrating aso with equation (3.8 that the periodic deta function is infinite at the origin. Unie the Fourier transfor for the continuous rectange function, the second of equations n (.77, Bɶ f is copex, which resuts fro the asyetry in the definition of the rectange ( ( sequence. The Gaussian sequence, corresponding to the Gaussian function, equation (.8, is ( β x π ( x β, γ = e, β >. (3.9 β Its Fourier transfor foows directy fro equation (3.6, Fourier Geodesy 3.7 Jeei, January 7

8 Γɶ ( ( x x f = e e e cos x f β = + β = = β, x π ( x β iπ xf π ( x β β π ( x β iπ xf π ( β f e e dx = e ( π (3.3 where the fina equaity coes fro the Fourier transfor of the scaed Gaussian function, equation (.83. The integra approxiation is vaid for sa x, or since β scaes the spatia doain, it is vaid for β x. A brief note on extensions to higher Cartesian diensions coses this section. Let g, be a two-diensiona, bounded, absoutey suabe array of vaues, with constant, but not necessariy equa sape intervas, x and x. The two-diensiona Fourier transfor and its inverse are =, = = (, ɶ iπ ( x f + x f, (3.3 G f f x x g e f f iπ ( x f + x f g G ( f, f e df df = ɶ, f f. (3.3 The spectru, Gɶ ( f, f, is a periodic function with periods, x and x, in respectivey. Corresponding yquist frequencies are f = ( x and f = ( x f and f, ; and the reationship between the spectra of the sapes and the parent function is given, anaogous to equation (3.8, by, =, x x = = Gɶ s ( f f G f f. (3.33 Corresponding properties foow iediatey fro the duaity to the two-diensiona Fourier series transfor pair; see Section Discrete Convoutions The concept of convoution carries over directy to the discrete doain fro the continuous doain with an appropriate definition. The discrete convoution of two infinite sequences, each absoutey suabe and each having the sae saping interva, x, is given by Fourier Geodesy 3.8 Jeei, January 7

9 g # h = x gnh n. (3.34 n= The incuded antecedent factor, x, aes this and subsequent foruations consistent with the continuous case in ters of units; see equation (.. Moreover, x is the reciproca of the period of the spectra of g and h, and thus equations (3.34 and (.35 are consistent. Again, there is a convoution theore, ( ( g # h Gɶ f Hɶ f, (3.35 which is proved easiy and anaogousy as in equations (.36 and (.37, by substituting equation (3.34 into equation (3. for the spectru of the convoution. The resut is ceary siiar to the Fourier transfor (.34 in the dua cycic convoution theore for periodic functions. On the other hand, the dua convoution theore for sequences is anaogous to the convoution theore for periodic functions, equations (.33, ( * ɶ ( g h Gɶ f Hɶ f, (3.36 where the cycic convoution, * ɶ, is defined by equation (.9. The spectra of the discrete convoution and of the discrete product both are periodic with period, x. The practica evauation of a geophysica ode that is a convoution of continuous functions usuay invoves a discretization using soe for of nuerica integration agorith. However, not a such agoriths preserve the ode as a convoution, which ay affect the efficiency of the cacuations if they rey on Fourier transfor techniques (Sections and 3.4. For exape, appying the rectange rue for nuerica integration to the convoution, T gt * h x gt x ' h x x ' dx ', (3.37 ( ( = ( ( where g ( T equation (3.34, T x is space-iited, as in equation (.3, sipy yieds a discrete convoution, as in T T n n n= g * h x g h,, (3.38 ( ( where ( g g ( n x additiona ters, T n =. Using the trapezoid rue aso yieds a discrete convoution, but with T Fourier Geodesy 3.9 Jeei, January 7

10 x g h g h g h ( ( * ( + ( T T n n T n+ n n= n= ( x = x ( g h ( g h + ( g h n T n T + T + (3.39 Both discrete convoutions in equations (3.38 and (3.39 can be evauated efficienty as discussed in Section However, the discretization of the integra convoution according to the we nown Sipson s rue presents a greater chaenge in preserving a convoution (in this case two convoutions and is eft to the reader. In two Cartesian diensions, the discrete convoution is, #, n, n n, n g h x x g h, (3.4 = n = n = with corresponding convoution theores, ( ( g # h Gɶ f, f Hɶ f, f, (3.4,, ( * ɶ ( g h Gɶ f f Hɶ f f. (3.4,, Both infinite arrays, g, and h,, ust have the sae respective saping intervas, x and x ; and, the spectru of the convoution is periodic with periods equa to x and x in the two coordinate diensions Aiasing of the Fourier Spectru Suppose that a function, g ( x, is band-iited, (, for or G f = f f f < f ; (3.43 and, further suppose that it is saped at an interva, x = ( f, so that the yquist frequency for the sequence, g, is f = f. On the principa doain, f f < f spectru is given by equation (3.8, =, its Fourier Gɶ s ( f = G f + = + G f + G ( f + G f + +. (3.44 x x x Fourier Geodesy 3. Jeei, January 7

11 Since the arguents, f = f ± x,,, =, for ( x f ( x < ipy , f, < f <, f <, f <,, (3.45 t t t t t t t t it is seen that a but the = ter of the series in equation (3.44 vanish in view of equation (3.43; and, therefore, in this case for f f < f s ( = (, Gɶ f G f. (3.46 Here, the spectru of the sape is denoted with the subscript s to distinguish it fro the ore genera periodic spectru, equation (3.6, where no specific reference to a continuous parent function is needed. Thus, according to equation (3.46, if the continuous function is bandiited to frequencies not greater than the yquist frequency as deterined by the saping interva, then the spectru of the saped sequence is equa to the spectru of the continuous function, over the doain of the iited band of frequencies. If G ( f for f f or f f g x, has spectra content beyond < then the function, ( the yquist iit; and, the spectru, Gs ( f, does not equa ( the overap (or, foding of the parts G ( f x, G ( f + x, etc., onto G ( f G f for f f < f due to, as indicated in equation (3.44. This effect is nown as aiasing, and the yquist frequency is aso nown as the foding frequency. That is, the spectru of the sape fro a parent function with spectra content beyond the yquist frequency is a corrupted version of the parent spectru. Attepting to deterine the spectru of the parent function fro the sequence of saped data is subject to the aiasing error, given by = Gɶ s ( f G ( f = G f +, f f < f x. (3.47 The generaization of aiasing to higher diensions foows the usua procedure. The aiasing error in the spectru derived fro the sape of a function on the two-diensiona Cartesian doain is s, = +, +, <, < x x = = ( ( Gɶ f f G f G f f f f f f f f (3.48 Representing a continuous geophysica signa by a discrete sape introduces a discretization error in the space doain. That is, the representation is not copete and this isrepresentation is best understood and foruated in the frequency doain in ters of the aiasing error. This error stands in contrast to the truncation error, or windowing effect (Section.6, in a function that is isrepresented due to the iited spatia doain. Aiasing can affect any part of the Fourier Geodesy 3. Jeei, January 7

12 spectru depending on its apitudes. However, for signas that have a continuous and attenuating spectru, such as the Earth s gravitationa fied, aiasing affects priariy the higher frequencies near the yquist frequency as shown in the foowing contrived exape. Figure 3. shows the spectru of a band-iited function, with f = 3, as we as the spectru of its sape, where the saping interva is x = 6. Thus, the yquist frequency is f = 3 and is not exceeded by the cutoff frequency of the parent function. Hence, there is no aiasing the spectru of the sape equas the spectru of the parent function in the band of Gɶ f in this case is 6. frequencies bounded by the yquist frequency. ote that the period of ( In Figure 3., by coparison, the saping interva for that sae continuous band-iited function is x = 3; and, consequenty, the yquist frequency is f = 3 < f. The period of Gɶ f is 3, and the enveoping ine in Figure 3. is the spectru of the sequence of sapes. It s ( deviates at the high frequencies near f fro the parent spectru due to aiasing. s Figure 3.: Spectra of a band-iited function (thic ine and of a sequence of its sapes (thin ine. The yquist frequency is f = 3. Fourier Geodesy 3. Jeei, January 7

13 Figure 3.: Spectru of a band-iited function (thic ine and the aiased spectru of a sequence of its sapes (thin soid ine. The yquist frequency is f = 3. The dashed ines are spectra of the band-iited function dispaced by twice the yquist frequency, and which are ters in the su of equation (3.44. Another way to view the aiasing error is to note that the sapes cannot distinguish between sinusoida coponents of the parent function at different frequencies, f, f, reated by f = f ± f, where is any integer, because the sinusoids at these frequencies pass through the sae sape points of g ( x ; that is, π ( ± ( π ± π iπ f x i f f x i f x iπ f x e = e = e = e. One can thin of the spectru of the saped sequence as having to accoodate the fu spectru (or, inforation of the parent function vaues in soe fashion, and it is accopished by foding Gɶ f, is caed the higher frequency coponents bac onto the principa part The spectru, ( aiased spectru of g. Two options exist to reduce the aiasing error. Either, the saped function shoud be fitered (ow-pass fiter, anti-aiasing fiter to reove the coponents with frequencies near and above the yquist frequency (if that is where ost of the aiasing error occurs; or, the saping interva shoud be decreased, thus increasing the yquist frequency to encopass a wider spectra band of the parent function. If the function is band-iited, it can be recovered fuy fro the sapes, provided the saping is done at intervas corresponding to twice the highest frequency contained in the function. Indeed, suppose that G ( f = for f > f ; then one can write on the basis of equations (3.43 and (3.44, ( ( ( G f = G ɶ f b x f, (3.49 s s Fourier Geodesy 3.3 Jeei, January 7

14 where x = ( f and b( x f is the rectange function, equation (.69, now a function in the frequency doain. That is, the spectru of the sape is truncated to just one period, and this truncated version then agrees with the tota spectru of the continuous band-iited function (Figure 3.. Taing the inverse Fourier transfor on both sides of equation (3.49 yieds, with the convoution theore (.7, ( s ( = F ɶ ( ( g x G f b x f = F s ( Gɶ s ( f * F ( b( x f ( * F ( ( = g x b x f (3.5 where gs ( x is given by equation (3.3. With the siiarity and duaity properties of the Fourier transfor, equations (.57 and (.56, appied to the rectange function, one obtains f x b x x b x f b x f x x x x ( sinc sinc ( = (, (3.5 where the ast equaity foows fro the syetry of the rectange function with respect to the origin. Therefore, equation (3.5 becoes x g ( x = g ( x δ ( x x x * sinc x x = = x x ' = g ( x ' δ ( x ' x sinc dx ' x (3.5 in view of the definition of convoution, equation (.4. Finay, with the reproducing property of the Dirac deta function, equation (.89, = x x g ( x = g ( xsinc x = x = g sinc x (3.53 This shows that if ( function can be reconstructed fro its saped vaues, g, if x = ( f g x is band-iited with no frequency content for f > f, then the entire. This resut is nown as the Whittaer-Shannon saping theore. ote, however, that the entire, infinite sequence of Fourier Geodesy 3.4 Jeei, January 7

15 vaues, g, is needed to reconstruct the parent function. Spectra eaage occurs if one uses ony a finite (truncated subset, as discussed in Section.6. However, it is shown in section 3.6 that an orthogona basis exists for the index-iited sequences that iniizes spectra eaage and thus offers a very good approxiation to a band-iited function. 3.3 Periodic Sequences In addition to being discrete, data coected using a easureent or observation syste usuay aso have finite extent in tie or space. Truncation or windowing of an infinite sequence ipies, according to the dua convoution theore, that the spectru of the finitey extended sequence is the convoution in the frequency doain of the spectru of the infinite sequence, G ( f ɶ, with the spectru of the discrete rectange function, equation (3.8, or other discretized window function. This is not considered further here since it is entirey equivaent to the windowing aready discussed in Section.6. If, on the other hand, the parent function is periodic and the finite extent of sapes covers a singe period, then the sapes, in fact, represent the fu ength of the parent function, if not its infinite resoution. Such is the case, in particuar, for functions on the sphere, which ay be regarded as periodic. In genera and aost without exception data generate a finite sequence (or array of sapes and, in order to ae use of the nuerica toos associated with the Fourier transfor, one ust often assue expicity that they actuay constitute one period fro a periodic sequence. Indeed, whie this sees to contradict the types of spatia geophysica or geodetic signas encountered in practice (i.e., they are patenty not periodic, it is a necessary assuption when using discrete Fourier techniques, such as the fast Fourier transfor (FFT, Section It causes an error in the anaysis, but if done propery iey not worse than using a truncated sequence without the nuerica benefit of the fast techniques. Before investigating these assuptions and their effect on operations such as convoution, the Fourier transfor is deveoped for the fourth fundaenta type of function that is periodic and defined on a discrete doain, that is, a periodic sequence (the other three fundaenta cases are the periodic and non-periodic functions on the continuous doain, and the non-periodic sequence. The periodic sequence, denoted, gɶ, ay be represented by a singe period, or a finite sequence. Because of this periodicity, the spectru of the finite sequence, in addition to being periodic (due to a discrete doain is, itsef, discrete, just ie the Fourier series transfor of a periodic function Discrete Fourier Transfor ɶ ɶ, =,,, be a finite sape fro a periodic function that has bounded Let g = g ( x variation, where the saping interva, x >, divides the period, P = x ; that is, gɶ + = gɶ for any integer,. Proceeding as for the sape of a non-periodic function, the finite periodic sequence ay be represented as a periodic function of ipuses (cf. equation (3.3, Fourier Geodesy 3.5 Jeei, January 7

16 s ( = ( ( gɶ x gɶ x δ x x x, (3.54 = where the infinite su is needed to ensure periodicity on the eft side. The Fourier series transfor, equation (.3, is an integra over one period, chosen for convenience to be x, P x, [ ] P x x s ( π i x P G = g x e dx ɶ. (3.55 Substituting equation (3.54, we derive ( π P x π = x π i x P = π i gɶ e = i x P ( ( G = x gɶ x e δ x x dx i x P ( ( = x gɶ x e δ x x dx = x gɶ x e = x P x = x (3.56 The second equaity foows fro δ ( x x = for x x P x, if or ; the third equaity is a consequence of the reproducing property of δ ( x, equation (.89; and P = x yieds the fourth equaity, which shows that this transfor is aso periodic in, and one can write G ɶ for the eft side. The fundaenta frequency of gɶ ( x is P = ( x ( corresponding to the sape interva, x, is f = ( x generaity that is even, the index, or wave nuber,, counts the frequencies, f ( x = ; and, the yquist frequency. Hence, assuing without oss in =, ; i.e., the principa part of the spectra doain for the periodic sequence coprises the finite set of discrete frequencies, f + =,,,,,,,,. (3.57 x x x x x x Fourier Geodesy 3.6 Jeei, January 7

17 If is odd, then ( (. In this case, the yquist frequency, as defined, is not actuay a constituent frequency of the principa part of the spectra doain. The foowing deonstrates the orthogonaity of the discrete functions, exp ( i =,,, π, + π i j e = ( ( j = ( j, od, od = (3.58 where is any integer. Mutipying both sides of the ast of equations (3.56 by exp iπ, suing over a wave nubers,, of the principa spectra doain, and ( aing use of equation (3.58 yieds π i ' i Gɶ e = x gɶ e = = = = xgɶ ' π ( ' (3.59 Therefore, the cobination of equations (3.56 and (3.59 is the discrete Fourier transfor (DFT pair, ( π i DFT gɶ Gɶ = x gɶ e,, (3.6 = π i DFT ( Gɶ g = Gɶ e, x = ɶ. (3.6 Both the sequence and its transfor are periodic with respective periods, x in spatia distance and x in frequency. The DFT is often given with no particuar reference to x by sipy setting it unity. Then, π i Gɶ = gɶ e,, (3.6 = π i ɶ gɶ = G e,, (3.63 = and the user provides the doain scae if needed as in equations (3.6 and (3.6. The given steps that ead to the DFT provide an instructive connection to previous Fourier transfors, but one coud as we sipy recognize that the DFT pair is a utuay consistent foruation for the Fourier Geodesy 3.7 Jeei, January 7

18 sets, { } g ɶ and { } G ɶ, based on the orthogonaity given by equation (3.58. o parent periodic function needs to enter the derivation, and thus there is no restriction on the vaues of gɶ ; they can be any copex nubers. However, in using the DFT it is aways assued that the finite sequence, gɶ, continues for a integer indices with period,. The DFT autoaticay ipies the inverse DFT, and vice versa, unie Fourier series or integra transfors, for exape, where the inverse Fourier transfor does not aways converge to the function. Since both the sequence of sapes and its DFT are periodic, one ay choose any consecutive vaues. For exape, an aternative, though ess coon, foruation is π i Gɶ = x gɶ e,, (3.64 = π i ɶ gɶ = G e,, (3.65 x = which assues that is even. This has the advantage of ore expicity expressing the yquist iits and conforing ore naturay to the other transfor types for continuous and nonperiodic functions and sequences. ote that G ɶ = G ɶ, provided the sae indexing is used for the sequence of sapes, gɶ, in both cases. Other variations in the definition of the DFT exist. For exape, the antecedent coefficient in the inverse transfor ight be repaced with for both the direct and inverse transfors, equations (3.6 and (3.63, to create even greater syetry. However, equations (3.6 and (3.6 are adopted in this text as the fora definition of the DFT pair. It has the advantage that ay be either even or odd., and incusion of the scaing by x identifies the units of frequency for any practica appication. Tabe 3. suarizes and copares the Fourier transfor pairs for the four types of functions in the Cartesian doain, where the doain is either discrete or continuous, and the function is either periodic or non-periodic. There is a perfect duaity in these characteristics aong the direct and inverse transfors: discreteness in the doain of either function or spectru ipies periodicity in its respective transfor, and continuity in the doain of either function or spectru ipies non-periodicity in its respective transfor. Each pair is foruated consistenty in ters of units, where the spectru has units of the function unit per frequency unit. Shown ony for the singy diensioned doain, these transfor pairs extend naturay to higher Cartesian diensions (Section Aso, they hod, in genera, for copex functions. Fourier Geodesy 3.8 Jeei, January 7

19 Tabe 3.: Fourier Transfor Pairs, g G. G : discrete doain g : G : periodic g : periodic discrete doain π i Gɶ = x gɶ e, = π i ɶ gɶ = G e, x = G : continuous doain G : periodic g : non-periodic = iπ xf ( =, Gɶ f x ge f x x ( x ( x ( = ɶ, < < iπ xf g G f e df g : continuous doain G : non-periodic g : periodic P π i x P G = g ( x e dx, < < ɶ = π i x P gɶ ( x = Ge, x P P G : non-periodic g : non-periodic i ( ( π f x G f = g x e dx, < f < i ( ( π f x g x = G f e df, < x < 3.3. FFT The fast Fourier transfor (FFT is an agorith to copute the DFT (usuay as defined by equations (3.6 and (3.63. As the nae ipies, it is a fast agorith. There is no additiona theory associated with the FFT, other than the deveopent of the agorith. It is a fact that FFT DFT, (3.66 as far as the nuerica resuts are concerned (up to a possibe scaing factor. In ters of properties the two are synonyous. It is no exaggeration to cai that the FFT nowadays is as ubiquitous and coon in spectra anaysis appications as the standard atheatica functions, ie sine or cosine. There are any good boos avaiabe (e.g., Brigha 988 that describe the detais of the agorith. This is not done here, just as the detais are not given for coputing the sine or cosine of an ange. We ay treat the FFT as a bac box, or a ibrary function we now what the resut, or output, shoud be for a given input. It is enough to now for our purposes that the FFT is the DFT; and that, when using the FFT function or subroutine fro a coputer software ibrary, it is iportant to deterine how the DFT is defined for that particuar FFT agorith in ters of the scaing Fourier Geodesy 3.9 Jeei, January 7

20 factor, data indexing, and units. That is, the antecedent factor ay be, or, or ; the data indices ay start at,, or ; and, usuay x ust be incuded anuay to ensure the appropriate units. In soe cases one ust aso verify the sign of the exponentia function in order to distinguish between the Fourier transfor and its inverse. The speed of the agorith depends on the prie factorization of the nuber,, of sape vaues. The fastest coputation occurs if is a power of, and this was assued for the origina foruation of the FFT. The speed in this case, as easured by the nuber of coputer utipications, is proportiona to og, copared to for the brute-force ethod according to the definition, equation (3.6. Thus, for exape, if = 4 =, there is 4 aready a treendous savings in coputation tie since the FFT requires ony ~ 6 utipications versus ~ for the nuber of utipications needed for the sus in equation (3.6. Agoriths exist for with arbitrary prie factorization (Singeton 969; but, the fewer the nuber of different factors, the faster is the coputation. Hence, when appying the FFT to a sequence of data, the greatest coputationa benefit is obtained if the nuber of data can be ade equa to a power of, or a product of powers of pries:, 3, 5, 7,, ; the fewer and saer, the better. The principa appication of the FFT is, of course, in coputing the spectru of a signa, or synthesizing the signa fro its spectru (inverse transfor. This hods for singy, as we as higher-diensioned discrete signas, since corresponding agoriths sipy ae use of the separabiity of the transfor in ters of the independent (Cartesian variabes. Siiary, the coputation of the Legendre spectru fro data given on a reguar spherica grid benefits fro the FFT, but ony in one diension (Section Another appication of the FFT concerns the coputation of convoutions, as enabed by the convoution theore. In particuar, equation (3.94 shows that the convoution can be perfored in about + 3 og utipications (if is a power of versus utipications using the definition ( Properties of the DFT and Higher Diensions The properties that were given for previous versions of the Fourier transfor, such as equations (. through (.4 for the Fourier series transfor, hod with corresponding odifications for the DFT, as we. They are isted here and easy to prove.. Fourier transfor pair: gɶ Gɶ ; (3.67. Proportionaity: agɶ agɶ, a is any constant ; ( Superposition: ( g + ( g ( Gɶ + ( Gɶ ɶ ɶ, (3.69 provided x and are the sae for ( ɶ and ( g 4. Syetry: gɶ Gɶ ; (3.7 g ɶ ; Fourier Geodesy 3. Jeei, January 7

21 5. Transation in : 6. Transation in : ɶ π i gɶ + G e ; (3.7 π i ɶ ɶ ; (3.7 ge G + 7. Duaity: G ɶ g ɶ. (3.73 In addition, the conjugate-syetry (Heritian property in the transfor hods for rea sequences (anaogous to equation (.7, * * gɶ = gɶ if and ony if Gɶ = Gɶ, (3.74 which ipies that G ɶ and G ɶ ( even are both rea, as can be seen fro equation (3.6, = = ( Gɶ = x gɶ, Gɶ = x gɶ, (3.75 since both sus are rea. By the duaity property, an anaogous resut hods for a rea-vaued DFT, * * gɶ = gɶ if and ony if Gɶ = Gɶ ; (3.76 so that, finay, gɶ = gɶ and gɶ = gɶ if and ony if Gɶ = Gɶ and Gɶ = Gɶ. (3.77 * * The for of the Parseva theore for the DFT is = = * ( ( = ( ɶ ( ɶ * x gɶ gɶ G G, (3.78 provided x and are the sae for ( gɶ and ( gɶ. The theore is easiy proved, as for the Fourier series transfor, equation (.3, by substituting equation (3.6 on the eft side, ( * * ( ( ( = π π π ( i i i G e G e G G e = = = = = = ɶ ɶ ɶ ɶ. (3.79 Using the orthogonaity, equation (3.58, then iediatey gives the resut. Fourier Geodesy 3. Jeei, January 7

22 Equation (3.74 shows that the nuber of independent spectra coponents (rea and iaginary is equa to the nuber of rea sape vaues, which is another way of saying that the DFT contains a the inforation contained in the periodic sequence, and vice versa. If a rea sequence is to be nuericay generated (synthesized fro a discrete spectru over a finite doain, then that spectru ust satisfy conjugate syetry; otherwise, the sequence wi not be rea. The DFT and its properties are extended naturay to higher diensions, as for a other Fourier transfors in the Cartesian doain. In two diensions, iπ + (, DFT, =,,, Gɶ gɶ x x gɶ e = = (3.8, iπ + (, DFT ɶ, = ɶ,,, gɶ G G e x x = = (3.8, The saping interva is constant in each diension, but x need not equa x. The discrete signa and its transfor are periodic in both diensions, with periods, and, respectivey. Equivaent expressions that ore ceary refect the yquist iits are, = ɶ, Gɶ x x g e = = iπ +,, (3.8, = ɶ, gɶ G e x x = = iπ +,, (3.83 The DFT as defined by equation (3.8 is identica to equation (3.8 provided the data arrays are consistent as indicated by their indices. Properties for the two-diensiona DFT are anaogous to equations (3.68 through (3.73 with obvious extensions. Of particuar note is the conjugate syetry (Heritian property of the spectru for rea signas. If and ony if ɶ is rea, then the spectru, equation (3.8 satisfies g, Fourier Geodesy 3. Jeei, January 7

23 Gɶ = Gɶ, or aso Gɶ = Gɶ, for a,, (3.84 * *,,,, the atter because of its periodicity (add, to the indices on the eft and reverse the sign of. Thus, anaogous to equations (3.75, four spectra vaues ust be rea, Gɶ = Gɶ, Gɶ = Gɶ, Gɶ = Gɶ, Gɶ = Gɶ. (3.85 * * * *,,,,,,,, And, there is conjugate syetry in for = and =, Gɶ = Gɶ, Gɶ = Gɶ ; (3.86 * *,,,, and in for = and =, Gɶ = Gɶ, Gɶ = Gɶ. (3.87 * *,,,, As in the one-diensiona case, there are as any independent spectra coponents (rea and iaginary in the discrete spectru, equation (3.8, as there are in the periodic sequence. If a rea, two-diensiona sape is to be synthesized fro a two-diensiona discrete spectru over a finite doain, then a these syetries ust be enforced; otherwise, the sape wi not be rea Discrete Cycic Convoution The definition of convoution of periodic sequences is anaogous to the definition for periodic continuous functions, equation (.8. Let gɶ and hɶ be discrete periodic sequences each having the sae period,. The discrete convoution of periodic sequences is denoted # ɶ and is given by cɶ = gɶ # ɶɶ h = x gɶ hɶ, =,,, (3.88 n= n n where the incusion of x is consistent with previous definitions in ters of units. The convoution is periodic with period, ( c+ = c for any integer,, and aes use of the fact that the sequence, hɶ, is extended periodicay beyond a singe period. For exape, Fourier Geodesy 3.3 Jeei, January 7

24 cɶ = gɶ hɶ + gɶ hɶ + gɶ hɶ + gɶ hɶ + + gɶ hɶ 3 + = gɶ hɶ + gɶ hɶ + gɶ hɶ + gɶ hɶ + + gɶ hɶ 3 (3.89 Hence, one coud aso write, gɶ # ɶɶ h = x gɶ hɶ, =,,, (3.9 n= ( + n od n which uses ony the vaues, h ɶ, for,, =. The convoution, equation (3.88 or (3.9, is nown as a discrete cycic, or circuar, convoution, since it is defined specificay for periodic sequences. As for continuous periodic functions, there is a cycic convoution theore and a dua cycic convoution theore for the discrete case. Discrete Cycic Convoution Theore The discrete Fourier transfor of the discrete cycic convoution (3.88 equas the product of the discrete Fourier transfors of the convoved periodic sequences. In sybos, gɶ ɶɶ h Gɶ Hɶ. (3.9 # The proof starts with the DFT of the convoution, equation (3.88, π i Cɶ = DFT ( gɶ # ɶɶ h = x x gnh n e ɶ ɶ. (3.9 = n= Interchanging suations and changing the index, = n, this is π n π i n i ɶ n ɶ Cɶ = x g e x h e n= = n = x g e x h e + h e n= = n = π π n π i n i i ɶ n ɶ ɶ (3.93 where the index of the second su is spit into parts with negative and non-negative indices,, which for the negative ones are repaced according to, noting that both ɶ ( i π and exp ( are periodic with period,. Cobining the ast two sus proves the DFT of the convoution. The inverse DFT foows autoaticay. Fro a coputationa viewpoint, therefore, the discrete cycic convoution is equivaent to h ' Fourier Geodesy 3.4 Jeei, January 7

25 ɶ ɶɶ ( ɶ ɶ g # h = DFT G H = ( ( gɶ ( hɶ DFT DFT DFT (3.94 It can be cacuated by utipying two discrete spectra and appying three DFTs. Dua Discrete Cycic Convoution Theore The discrete Fourier transfor of the product of two periodic sequences having the sae period equas the discrete convoution of their discrete Fourier transfors. In sybos, gɶ hɶ Gɶ # ɶ Hɶ, (3.95 where the convoution for discrete periodic spectra is defined by Gɶ # ɶ Hɶ = Gɶ nh ɶ n, =,,, (3.96 x n= The proof foows easiy, starting with equations (3.6 and (3.6, DFT g h = x G e h e x = n= ( ɶ ɶ = π π i n i ɶ ɶ n i Gɶ n hɶ e n= = π ( n and the resut is obtained directy by using equation (3.6 for foows autoaticay. H n (3.97 ɶ. The inverse DFT The convoution in the frequency doain, equation (3.96, is anaogous to equation (3.88 with f = x. the recognition that the saping interva in the frequency doain is ( For two-diensiona Cartesian doains, the discrete cycic convoution is ɶɶ ɶ n n n n gɶ # h = x x gɶ h, =,,, =,, ; (3.98,,,, n = n = and, corresponding convoution theores are gɶ ɶɶ h Gɶ Hɶ, (3.99, #,,, gɶ hɶ Gɶ # ɶ Hɶ, (3.,,,, Fourier Geodesy 3.5 Jeei, January 7

26 where ɶ ɶ Gɶ # Hɶ = G Hɶ, =,,, =,,.,, n, n n, n x x n = n = (3. Expicity, the Fourier transfor pair (3.99 shows how to copute the convoution efficienty using three FFTs, ( ( ɶ (,, gɶ # ɶɶ h = DFT DFT gɶ DFT h. (3.,,,, These resuts are sef-evident based on the previous generaizations to higher diensions, as discussed in Sections.3 and 3..., Aiasing of the Discrete Fourier Spectru Let gɶ ( x be a periodic function with period, P, that is band-iited, G =,, ; (3.3 and, suppose it is saped at an interva, x = P ɶ ɶ, is sapes, g = g ( x. Using equation (3.64, the DFT of the π i ɶ Gɶ = x g e = = x G e e P = = = π π i x i P ' π i ( G e = = (3.4 = G for =,,, where the second equaity coes fro equations (.8 and (3.3, and the ast equaity is due to the orthogonaity, equation (3.58. Equation (.8 then aso gives Fourier Geodesy 3.6 Jeei, January 7

27 ( π i x P Ge gɶ x = P ɶ, (3.5 = for a x ; in other words, the periodic function, gɶ ( x, is deterined copetey by its sapes, gɶ ( x, provided x = P. This is another case of the Whittaer-Shannon theore. If a periodic, non-band-iited signa is saped at constant interva, x, then the DFT is gɶ x, equation (.8, the subject to an aiasing error. Starting with the Fourier series for ( sapes are given by π π i x i ( j + x P P j + gɶ = gɶ ( x = G e = G e, (3.6 P P = j= = which is a ore eaborate, but usefu way of writing the infinite su. suations, Interchanging the π i j + gɶ G e, (3.7 x = where P x = j= =, and because ( i exp π =. A coparison of equations (3.7 and (3.65 yieds the reationship between the spectru of the sape and the spectru of the periodic (continuous parent function, G ɶ = G, (3.8 j + j= anaogous to equation (3.8 for the continuous spectru. The aiasing error is given, as in equation (3.47, by G ɶ G = G,. (3.9 j + j= j It is zero if the periodic parent signa is band-iited by the yquist wave nuber, = = P x, as shown by equation (3.5 Again, for two-diensiona doains, the ( anaogous aiasing error in the DFT is,, j +, j + j = j= j and j Gɶ G = G,,.(3. Fourier Geodesy 3.7 Jeei, January 7

28 Throughout this section it is assued that the period of the parent function is an integer utipe of the saping interva and that the function is saped over the entire period ( x = P. Generay, this is not a serious constraint since the tota period, or interva, of an avaiabe geophysica signa is defined by a finite nuber of saping intervas and x = P is autoaticay satisfied; this signa is then assued to be periodic with period, P. However, it ay be noted that the case, x < P, represents a truncation of the doain and spectra eaage (in addition to possibe aiasing corrupts the estiation of the parent signa spectru. If x > P and od x ( P, then the assuptions ipicit in the appication of the discrete techniques such as the DFT and the discrete cycic convoution no onger hod. This eans, again, that the parent signa has been truncated to ess that a fu nuber of periods and spectra eaage corrupts the spectru estiated fro the sapes. 3.4 Cycic Versus Linear Discrete Convoution The concept of convoution, first introduced in Chapter, is ade practicabe and coputationay efficient using the FFT by saping over a finite interva that is assued to be the period of the parent function. These assuptions and restrictions cause errors when appied to actua discrete data because, in fact, geophysica odes usuay are based on continuous functions over doains typicay arger than the data doain. Truncation of the doain causes spectra eaage and finite saping causes aiasing. Using the FFT to copute the convoution assues periodicity in the data (hence, in the parent function and introduces yet another ind of error, caed the cycic convoution error. Spectra eaage and aiasing coud be reduced, respectivey, with additiona data on an extended doain and finer saping. However, the cycic convoution error can be eiinated entirey without obtaining additiona data. To put these various errors in proper perspective, consider the typica situation where the function, g, in the convoution is the data function (or signa function and h is a nown erne function, representing the convoution (or, a fiter, that transfors the data in soe physicay eaningfu way. The convoution, equation (.3, representing the true ode for the data and erne functions is repeated here for convenience, g ( x * h( x = g ( x h( x x dx. (3. Let g, =,, be discrete vaues of the data function saped at unifor spacing, where, without oss in generaity, is even.. Regarding first the finite extent of the function, define a new data function, g T, given by g T ( x g ( x, x T =, x > T (3. Fourier Geodesy 3.8 Jeei, January 7

29 that refects its truncation to a doain, [ T, T ]. The approxiation of the true convoution by the convoution with T T g, thus generates a truncation error, ( x ( * ( = ( ( + ( ( g x h x g x h x x dx g x h x x dx T x > T T ( * ( ε ( = g x h x x trunc ε, (3.3 This error is aso caed an edge-effect error if the erne function attenuates away fro the origin, since in that case the error is significant ony when the coputation point approaches the edge of the data doain and the significant part of the erne acs data beyond the edge (see Figure 3.3. Assuing that the saping interva divides the truncated doain, T = x, define a data g, sequence, ( trunc ( g g, =, otherwise (3.4 ote that ( g is not periodic; it sipy ebodies the coection of avaiabe sapes of the continuous function. Further approxiating the continuous convoution by a discrete convoution, equation (3.3 becoes ( * ( = ( # ε ( ε ( g x h x g h x x discrete trunc, (3.5 where the discrete convoution on the right side is defined by equation (3.34, now given by g # h = x gnh n. (3.6 ( n= This can be cacuated for any fro the data and a discretization or saping of the nown erne function, h. With the ai of ipeenting the DFT (FFT, ony the sipe rectange rue of nuerica integration is assued; other ethods of nuerica integration coud be considered if they yied a convoution, such as the trapezoid rue, given by equation (3.39. Finay, in order to evauate the discrete convoution using the DFT, according to equation (3.94, one ust assue that both the data and the erne sequences are periodic with the sae period,, ɶ +, (3.7 ( g = g, Fourier Geodesy 3.9 Jeei, January 7

30 ( h = h, ɶ, (3.8 + for any integer,. Owing to their periodic nature, the product, ( g ( hɶ ɶ is aso periodic in n (period,, and the discrete cycic convoution, equation (3.88, ay be written as ɶ # = ɶ,. (3.9 ( g ɶ ( hɶ x ( g ( hɶ n n n= Using this cycic convoution instead of the inear convoution, equation (3.6, introduces an additiona cycic convoution error if one sipy evauates the convoution without odification, ( * ( = ( # ɶ ( ɶ ( ε ε ( ε ( g x h x gɶ h x x, (3. where the cycic convoution error is given by ( ( ɶ ( ɶ ( cyc discrete trunc ε cyc = gɶ # h g # h,. (3. A ore expicit expression for the cycic convoution error gives further insight into its source, and offers aso the opportunity to anayze its agnitude for particuar appications. Fro the convoution definitions given by equations (3.34 and (3.9, and the definitions of the sequences, equations (3.4 and (3.7, equation (3. becoes n n n n n n n= n= ( ε cyc = ( ɶ ( ɶ ( x g h x g h n= ( ɶ ( = x g h h n n n (3. The error is depicted graphicay in Figure 3.3. It is readiy verified that it sipifies to ( ε cyc x gn ( h n+ h n, n= + + =, = + x gn ( h n h n, n= (3.3 Fourier Geodesy 3.3 Jeei, January 7

31 That the error is zero at = instead of = is due to the sighty asyetric anner in which the truncation is defined. Figure 3.3: Truncation and cycic convoution errors (shaded regions of ( h ɶ utipied by the saped signa, ( gɶ for the convoution ( g # n Soid ines connect the sequence points for carity. n h n, respectivey, h evauated at. The suand in ( ε cyc For exape, when = invoves vaues of the erne function potentiay cose to the origin., the error incudes the product, g ( h h. Hence, if the erne is argest near the origin (as shown in Figure 3.3 and attenuates with distance fro the origin then the cycic convoution error can be significant. Many geodetic and geophysica ernes, in fact, behave ie a power of the reciproca of the distance between the evauation point and the integration point. On the other hand, fro Figure 3.3 it is evident that the truncation error and the cycic convoution error have siiar characteristics; both are argest when the coputation point,, of the convoution is cose to the edge of the data doain. Therefore, in avoiding the truncation error by restricting the coputation point to a neighborhood of the origin of the data doain, one aso tends to avoid the cycic convoution error. It is possibe, on the other hand, to construct a discrete cycic convoution fro the given data sequence that exacty equas the discrete inear convoution. Consider the discrete cycic g # ɶ hɶ ɶ is a periodic sequence whose principa part is convoution, ( ɶ (, where ( g defined over the doain,, by extending (padding ( g side of its own principa interva,, ɶ with zeros on either ( gɶ g, =, and (3.4 ( ( gɶ = gɶ,, is any integer ; (3.5 + Fourier Geodesy 3.3 Jeei, January 7

32 where the second equation defines the periodic extension of ( g extended periodic sequence, h ɶ, is defined by ( ( ( + ɶ over a integers. The hɶ = h, hɶ = hɶ,, is any integer. (3.6 Thus, whie the data are extended with zeros, the erne sequence, though assued periodic (period, is extended naturay using the actua nown vaues of h. The foowing shows that the discrete cycic convoution of these extended sequences equas the discrete inear convoution of the origina finite sequences for. First, note that ( ( gɶ = g, n ; (3.7 n n ( ( hɶ = h = h n, n and n n. (3.8 Hence, for, ɶ = ɶ, fro equation (3.9; ( g # ɶ ( hɶ x ( g ( hɶ That is, n n n= = x ( g n ( h n n= = x ( g n ( h n n= n= ( n ɶ, fro equation (3.7; ɶ, fro equation (3.4; = x g h, fro equation (3.8; ( g # = h ( g h ( g ɶ ( hɶ n, fro equations (3.4 and (3.34. # = ɶ #,, (3.9 Fourier Geodesy 3.3 Jeei, January 7

33 and the cycic convoution error, equation (3., is zero in this case. Figure 3.4 shows gɶ # ɶ hɶ, provided that scheaticay how the cycic convoution error vanishes for ( (. The vaues of the cycic convoution of the extended sequences for other are discarded. Figure 3.4: Truncation and cycic convoution error (shaded regions of ( h ɶ utipied by the saped signa, ( gɶ for the convoution ( g # n n h n, respectivey, h evauated at ; the cycic convoution error is zero. Soid ines connect the sequence points for carity. Equations (3.4 and (3.6 indicate precisey how the periodic sequences, ( g ( h ɶ and ɶ, ust be constructed so that their cycic convoution equas the inear convoution of the origina sequences. Once constructed, any period coud be used in the FFT agorith, e.g., aso. Using this ethod to estiate the true convoution thus eaves ony the discretization and truncation errors as indicated on the right side of equation (3.5. However, it invoves twice as uch coputer storage space for the extended sequences (but a reativey insignificant increase in coputation tie. It ay not be justified fro a nuerica standpoint to nuify the cycic convoution error, if the truncation error, usuay of siiar agnitude, is aready avoided by restricting the doain of the coputation points of the convoution, as this aso reduces at the sae tie the cycic convoution error (Jeei 998. If the extra coputer storage space is not an issue, it is prudent to eiinate the cycic convoution error as a atter of routine practice. The erne function is not padded with zeros (as soeties erroneousy advocated, as this woud cause the cycic convoution of the extended sequences to differ fro the inear convoution of the origina sequences. However, if the function, h, is not nown beyond soe finite interva, then zero padding of this function aso heps to reduce (not eiinate the cycic convoution error. The two-diensiona, inear, discrete convoution of a truncated data array and a saped erne function is Fourier Geodesy 3.33 Jeei, January 7

34 (, #, =,,, g h x x g h n n n n n = n =. (3.3 The corresponding discrete cycic convoution is ( gɶ ɶ ( ( (, hɶ, = x x gɶ, hɶ, #,,, n, n n, n n = n =, (3.3 They are equa over a iited doain for an appropriatey zero-padded data array and the extended erne array, anaogous to equation (3.9, # = ɶ #,,, (3.3 ( g ( ( h g ɶ hɶ,,,,,,,, where the zero padding is defined by ( gɶ,, and,, gɶ, =, or or, or ( (3.33,, ( ( g ɶ = g ɶ,, (3.34,, +, +, where,, are any integers. That is, the zero-padded data array, ( gɶ,,, is the origina array pus a border of zeros, whose width is either or, depending on the coordinate direction. This extended array is defined periodicay over the entire pane. The periodic erne array is defined, anaogous to equation (3.6, by (,,, h ɶ = h,,, (3.35 ( ( = h ɶ h ɶ,, are any integers, (3.36,, +, +, Fourier Geodesy 3.34 Jeei, January 7

35 where the extension to the arger grid is accopished using the actua nown vaues of the erne function. The proof of equation (3.3 proceeds exacty as for the onediensiona case. Most FFT agoriths assue the DFT is defined with indices starting at zero, as in equation (3.8. Since it is periodic, the cycic convoution that is identica to the inear convoution is aso in this case given by equation (3.3, but for indices, =,, and =,,. The essentia difference is in the padding of the extended erne array prior to convoution. By shifting the index to start at zero, the extended part of the array ust be such that when viewed as periodic over the pane it is sti propery defined in the doain that is hɶ in its principa syetric with respect to the origin. That is, one ust aways use (, doain, equation (3.35, when extending it periodicay over the pane. The foruas for padding the data and the erne arrays, in this case, are, ( gɶ,,, g,,,,, =,,,, (3.37 and ( hɶ,, h,,, h,,, = h,,, h,,, ( Discrete Functions on the Sphere Sapes on the sphere fro a parent function, g ( θ, λ, such as the gravitationa fied or the agnetic fied, are usuay gridded at equi-anguar intervas corresponding to spherica coordinate differences (Leoine et a. 998, Maus et a. 9, for exape, M M g j, g ( θ j, λ = g j + θ, + λ, j =,, K, =,,, (3.39 where K = π θ, M = π λ, and θ, λ are intervas, respectivey, in co-atitude and ongitude. Incuding the in the definitions of θ j and λ paces the sapes of gɶ at the Fourier Geodesy 3.35 Jeei, January 7

36 center of each ce of a grid defined by the coordinate ines, thus avoiding a utipicity at the poes. The copex Legendre transfor pair of a continuous function on the sphere, given by equations (.8 and equations (.85 ay be re-written using equation (.83 as π π c ε iλ Gn, = g ( θ, λ P, ( cosθ e sinθdθdλ, (3.4 n 4π c i g ( θ, λ ε Gn, P, ( cosθ n e λ =, (3.4 = n= where, as before, n and represent wave nubers, and ε, < n =, = (3.4 The rarey utiized, but nonetheess equay vaid, Fourier representation, equation (.9, is obtained by interpreting, g θ, λ, as θ λ as Cartesian coordinates and viewing the function, ( periodic in the pane, with respective periods, π in θ and π in λ. It is repeated here for convenience and with an interchange of suations, i F + π π, = = gɶ ( θ, λ G e π θ λ, (3.43 π = with Fourier transfor, equation (.9, F, π π λ= θ = ( θ, λ iπ θ + λ π π G g e dθdλ = ɶ. (3.44 ote that gɶ ( θ, λ g ( θ, λ on Ω, and gɶ ( θ, λ is the periodic extension of (, infinite pane defined by g θ λ on the < λ < and < θ <. For the saped array, gɶ j, = g j,, the gɶ θ, λ corresponding truncated series represents the axiu recoverabe resoution of ( according to the Fourier yquist iits, where ɶ M K i F + π π, = M = K gɶ ( θ, λ G e π θ λ, (3.45 π Fourier Geodesy 3.36 Jeei, January 7

37 M K j iπ F + K M, θ λ j, Gɶ gɶ e, (3.46 = = M j= and where the yquist wave nubers are K and M, respectivey, for the coordinates, θ and λ, with corresponding yquist frequencies, ( θ and ( λ. Questions then naturay arise: what are the yquist wave nubers for the Fourier--Legendre spectru and what is the optia truncated Fourier-Legendre series in ters of axiu recoverabe resoution? c F The answers ay be obtained by reating G n, and G,, since the yquist frequencies of the c atter are aready nown. Utiatey, one aso needs the reationship between G n, and the F discrete Fourier transfor, G ɶ,, to deterine the axiu resoution recoverabe fro the sapes. However, transating the Fourier foruation directy to the spherica case is copicated by the topoogy of the sphere on which the orthogona basis functions, in contrast to the sinusoida functions, do not create a periodic spectru of the sape. That is, whie the i spherica haronics, P, ( cosθ n e λ, are periodic in ongitude, with period, π, equa to the principa doain of λ, they are not periodic in co-atitude with period equa to its principa,π. Moreover, the condition, n, iposes a dependency between the axiu doain, [ ] degree and order, even if the sape nubers, K and M, are independent. F F It is ore convenient to find a reationship between G, (or, G ɶ c, and G n, instead of G n,. Ipications for G n, then foow iediatey fro equation (.88. Toward this objective, define the periodic function, with period, π, ( P ( sɶ n, θ = n, cosθ sin θ, θ π, (3.47 such that ( θ + π = ( θ sɶ sɶ, for any integer,. (3.48 n, n, It can be expressed as a Fourier series, equation (.9, π ( n, i θ sɶ π n, ( θ = S e π, (3.49 = with Fourier coefficients, equation (.3, ( n, π π θ i π n, ( cosθ sin θ θ,. (3.5 S = P e d < < Fourier Geodesy 3.37 Jeei, January 7

38 Substituting equations (3.43 and (3.49 into equation (3.4 and re-arranging suations and integrations yieds ε π π π θ ( n, i ( ( c F iλ π Gn, G, S e dλ e d 4 π π π =, (3.5 = = = where the reversa in the sign of is in accord with the syetry of this suation index. By the orthogonaity, equation (., the integras are πδ and πδ ; hence, θ ε c F ( n, Gn, = G, S 4π. (3.5 If g ( θ, λ = ɶ is Fourier band-iited in the sense, F M M K K G, =, <, >, <, >, (3.53 c then ceary aso G, =, < M, > M ( n n, and (, g θ λ is band-iited in the sae sense with respect to the order,, in its Fourier-Legendre spectru. However, the Fourier band-iit in wave nubers,, does not iit the spectru in degrees, n. Indeed, cobining equations (3.5 and (3.53, we have ε K c F ( n, Gn, = G, S 4, (3.54 π = K (, which is not zero for any degree, n, since the Fourier spectru, { S } n, has at east one nonzero coponent for every n. Thus, the Legendre yquist iit, stricty speaing, does not exist with respect to the co-atitude even if the parent function is Fourier band-iited in co-atitude (according to equation (3.53. Yet, it is desired to truncate the Fourier-Legendre series in an optia way, recognizing that G c. Since usuay the sapes cannot deterine the entire function, or the entire spectru, { n, } λ = θ, et us assue for the oent that M = K. Considering the yquist iit in ongitude, it is reasonabe to truncate the Fourier-Legendre series at the yquist order, = M (see Section 3.3. with corresponding axiu degree, nax = K = M. That this is the yquist iit in haronic degree ight aso be argued fro the viewpoint that the, π. In fact, Legendre functions, with arguent cosθ, are periodic on the principa doain, [ ] this is how the standard Fourier-Legendre series odes of the potentia fieds of the Earth and other panets are truncated. But, this does not give the copete characterization of yquist iit in co-atitude. If M, and n M, then the roughy ( M resuting Fourier Geodesy 3.38 Jeei, January 7

39 independent spectra coponents do not fuy represent the greater nuber, rea-vaued sapes, ɶ j,. g On the other hand, the discrete Fourier spectru, { F, } MK = M, of G ɶ, as in equation (3.46, is the fuest representation of the function based on the sapes. In order to obtain an equivaent representation in ters of the Legendre spectru and to attept answering the questions posed rɶ θ be a periodic function with period, π, defined by above, et ( n, ( P ( rɶ n, θ = n, cos θ, θ π, (3.55 such that ( θ + π = ( θ rɶ rɶ, (3.56 n, n, for any integer,. The discrete Fourier transfor pair for the sapes of this function, rɶ n, = Pn, cosθ j, j K, is (equations (3.6 and (3.6 ( ( j K π ( n, i j K θ ( n, j j= Rɶ = rɶ e, (3.57 K π i j K ( n, ( n, = ɶ rɶ R e. (3.58 j π = Fro equations (3.4, (3.55, and (3.58, the sapes, g j,, are foruated as K π i j j, ε n, c ( n, K iλ g = G Rɶ e e. (3.59 π = n= = Substituting these into equation (3.46 yieds π K K π M π i ( n, i j( i ( F c M K M Gɶ = G e Rɶ e e, (3.6 θ λ ε, n, where λ = π ( + π = n= = j= = M M. By orthogonaity, equation (3.58, the ast two sus are Kδ and Mδ, respectivey; hence, od M ( = = n= π i F c M ( n, G ɶ, π ε Gn, e R ɶ δod (. (3.6 M Fourier Geodesy 3.39 Jeei, January 7

40 od M = hods for any integer, µ, such that = µ M +. Therefore, ow, ( π, π ( ε µ M + n, µ M + µ = n= µ M + i F c M ( n, M G ɶ µ µ + = G e R ɶ, (3.6 which is the desired reationship between the discrete Fourier transfor and the copex Fourier-Legendre transfor. It is not quite the inverse to equation (3.5 since the atter reates c the Fourier-Legendre transfor, G,, to the Fourier-series transfor; and, the difference F between the discrete Fourier transfor, G ɶ F,, and the Fourier-series transfor, G,, is the aiasing error, equation (3.. To deterine the resoving power of a discrete array of spherica data in ters of spherica wave nubers, n and, consider isoating the µ = ter in equation (3.6, and further spitting this into two ters, + K π i (, i F c M n c M ( n, Gɶ = G e Rɶ + G e Rɶ π ε π ε, n, n, n= n= + K µ = n= µ M + µ ( + π If the function is Fourier-Legendre band-iited in the sense, π (3.63 π i c M ( n, M + G e Rɶ µ µ ε µ M + n, µ M + G c n, M M, n + K =, M M or, n (3.64 then the ast two sus in equation (3.63 vanish; and, + K π i F c M ( n, Gɶ, = π ε Gn, e Rɶ, (3.65 n= F which gives a reationship between the KM coefficients, G ɶ c,, and the KM coefficients, G n, ( K degrees, n, for each of the M orders,. Figure 3.5 shows the band-iited doain of the Fourier-Legendre spectru according to equation (3.64. The reationship, equation (3.65, is invertibe, as indicated in Section This eans that the inforation contained in the sapes, { g j, }, is fuy captured either by the DFT, Fourier Geodesy 3.4 Jeei, January 7

41 { G ɶ F, }, or by the set, { G c n, n K, M M } +. On this basis, one ay define the yquist wave nubers that specify the iits of recoverabe Fourier-Legendre spectra coefficients fro the sapes and that deineate the high-frequency spectru that causes aiasing errors. Referring to Figure 3.5, the yquist wave nubers in ongitude and co-atitude are M =, (3.66 (, n = + K <, (3.67 where is soewhat conservative since aso coefficients with orders, = M, technicay are recoverabe. Figure 3.5 shows the case for M = K, but it is cear fro the derivation of the yquist iits that M and K ay be independent. In suary, whie the yquist iit associated with the saping interva, λ, corresponds essentiay to the Fourier (Cartesian case, it is ess definite with respect to the saping interva, θ. This aso affects the defined spatia resoution (iniu haf-waveength of a spherica haronic series truncated at axiu degree, n. Often it is sipy quoted as the reciproca of the yquist iit in λ, which, oreover, is aso casuay identified with the yquist iit in θ, assuing θ = λ. Under this adopted convention and the definition of the extent-bandwidth product, equation (., the spatia resoution is stated as the reciproca of the bandwidth, M ( π = K π, where K n, hence as θ = λ = π n. However, this underestiates the resoution as given perhaps ore accuratey by equation (. that better refects the spatia decrease in saping interva, λ, as one approaches either poe. Fourier Geodesy 3.4 Jeei, January 7

42 Figure 3.5: Large dots denote the doain of the Fourier-Legendre spectru of a band-iited spherica function as defined by the iits of equation (3.64. The arge and sa dots together denote the entire doain, extending to n with n Aiasing of the Fourier-Legendre Spectru For a continuous function on the sphere that is Fourier band-iited in the sense of equation (3.53, the Whittaer-Shannon saping theore for periodic functions, equation (3.4, states F that the DFT, G ɶ,, of its sapes copetey defines the function. ow suppose that the function is Fourier-Legendre band-iited as given by equation (3.64, and define the K vectors, G ɶ = ɶ ( G T, G ɶ, F F F K K, (3.68 ( G G T, +, G =, (3.69 and the K c c c K K atrix, Fourier Geodesy 3.4 Jeei, January 7

43 R = π ε e π i M (, ( K, Rɶ + K Rɶ K. (3.7 (, ( + K, Rɶ K R ɶ K Then equation (3.65, under the assuption of equation (3.64, in vector-atrix for is F c M M Gɶ = RG,. (3.7 The atrices, R, defined by equations (3.7 and (3.57, are assued to be invertibe since the couns are the DFTs of unifor sapes of associated Legendre functions of different degrees (and the sae order that by equation (.7 are orthogona. Thus, we have the Whittaer-Shannon theore for functions on the sphere. If the iited Fourier-Legendre spectru is coputed fro the sapes, gɶ j,, of a Legendre band-iited function (equation (3.64 according to c F M M G = R G ɶ,, (3.7 F where the eeents, G ɶ,, are given by equation (3.46, then this copetey deterines the function for a points on the sphere, M + K ( n c c ( θ, λ = ( θ, λ g G Y = = M n= M + K = M n= G n,, Y n, n, ( θ, λ * (3.73 where the second equation hods in view of the one-to-one reationship between the rea and copex Legendre spectra, equations (.87 and (.88. It is noted that the Fourier-Legendre spectru coputed aternativey by sipy discretizing the integras of equation (.8 according to the rectange rue, K M c θ λ c n, j, n, θ j, λ sinθ j G g Y 4π j= = M (, (3.74 does not reproduce the entire band-iited function exacty. If the function is not band-iited, then the coputation of the spectru fro the sapes according to equation (3.7 is corrupted by aiasing, that is, by the spectra content that cannot be resoved fro the finite saping of the signa. The aiasing error ay be inferred fro equation (3.63. Define the sei-infinite vectors, Fourier Geodesy 3.43 Jeei, January 7

44 ( G G T,, G, (3.75 c c c δ = + K + K + ( G G T G, (3.76 c c c µ, = µ M +, µ M + µ M + +, µ M + and the sei-infinite, K, atrices, δ R = π ε e ( π i M µ µ, = µ M + R π ε ( K, ( K, Rɶ + K Rɶ + + K, (3.77 ( + K, ( + K +, Rɶ K Rɶ K e π i M Then the aiasing error, or the difference between equation (3.63, is ( M, ( M, Rɶ µ + K Rɶ µ + + K. (3.78 ( µ M +, ( µ M + +, Rɶ K Rɶ K c G derived fro equation (3.7 and fro F c c c M M R Gɶ G = R δ RδG + Rµ, Gµ,,. (3.79 µ = µ A ess optia aiasing error resuts if the spectru is derived in other ways fro the sapes, for exape, using equation ( Spectra Anaysis and Synthesis on the Sphere Soving equation (3.7 for the M vectors, c equation (3.39, is equivaent to soving for the KM coefficients, G, syste of KM equations, + K M π i c M = ε ( j, n, n, j = M n= c G, fro the goba array of sapes, { j, } n g,, by inverting the inear g G rɶ e, (3.8 Fourier Geodesy 3.44 Jeei, January 7

45 and aso invoves the inversion of M K K atrices (Jeei 996. The spectru is thus deterined to the fu extent aowed by the yquist iits, equations (3.66 and (3.67. However, it is coon practice to copute the Fourier-Legendre spectru fro the goba grid ony up to the fixed degree, K. The ode for the rea-vaued spectru in this case is M K M M g j, = Gn, Yn, ( θ j, λ, j =,, K ; =,, ; (3.8 = M + n= and, equivaenty, anaogous to equation (3.73, the ode with copex spherica haronics, equation (.85, is M K iπ iπ c M M j, = ε n,, ( cosθ n j = M + n= g G P e e, (3.8 where M = K ( λ = θ, and where in each case the upper iit, K, in the degree, n, aso constrains the ower iit in the order,, to be M +. Since the nuber of unnown coefficients, K, now is ony haf the nuber of equations, K, they are estiated as a soution to an over-deterined probe. For convenience, the foowing is foruated for the copex spectru, with a transforation to the rea spectru at the end. Restricting the vectors, G, equation (3.69, to the first K eeents, c ( T, c c c M M G = G G,, K +, (3.83 and defining the ( K vectors, ( T ( j ( cos = P θ, j P, ( cosθ K j r, (3.84 equation (3.8 becoes M M j, =, (3.85 ( j i g C e π = M + where, for j =,, K, j j c iπ ( ( ( T M M, M C = ε e r G + < (3.86 Fro equations (3.64 and (3.65, it is cear that Fourier Geodesy 3.45 Jeei, January 7

46 C =, (3.87 M λ ( j DFT ( g j, except that the coponent, ( j ( j C M, thus coputed, is not used in the ode given by equation (3.85. M λ C is the DFT of the data for each zone of ongitudes at co-atitude, θ j. For each, M + M, the set of K equations (3.86 ay be written in vectoratrix for as where iπ c M M M C = ε e rg, + <, (3.88 ( ( ( C C K T C = is a K vector with eeents given by equation (3.87, and where, fro equation (3.84, r ( ( r T P ( cosθ P,, ( cosθ K = = T ( K P, ( ( cosθk P, ( cosθ K K r (3.89 is a K ( K atrix. c The set of inear equations (3.88 can be used to sove for the eeents of the vectors, G, fro the DFT of the sapes. There are K equations for any, M + M, but ony K unnown coefficients, equation (3.83. Thus, except for =, the systes of equations are over-deterined and soe for of constraint is needed to obtain a unique soution. Specificay, it is desired to find the soution, G ˆ c, that iniizes the difference between eft and right sides in the sense of the iniu squared-nor, T iπ iπ M c M c c C ε e rg C ε e r G in with respect to G. (3.9 The soution, G ˆ c, that satisfies this iniization is obtained by appying the Moore-Penrose pseudoinverse of r (Goub and Van Loan 996 to equation (3.88, iπ M ˆ c e T ( T M M = r r r, + ε G C, (3.9 Fourier Geodesy 3.46 Jeei, January 7

47 where atrix that is invertibe because the couns of r are a assued to be independent of each other, coprising unifor sapes of orthogona associated Legendre functions ( r is assued to have fu coun ran. Fro equation (.88, and the T r r is a ( K ( K fact that ( * C = C (for rea-vaued j, Legendre spectru is g, the iniu-nor estiate of the rea Fourier- iπ T M r C T ( r r Re e, M T ( r r ˆ T = r C, = G T ( r r I, iπ T M r e C M + (3.9 where ( T,, ˆ ˆ ˆ, M M G = G GK +. (3.93 The entire nuerica process invoves the inversion of K atrices that range in size fro ( = K to K K ( =. If the data errors are considered as offering weights to the data, then a corresponding eastsquares soution ay be foruated if the covariance atrix of the errors, D = P, is nonsinguar. The soution is then based on the iniization, T ( P ( c g gˆ g gˆ in with respect to G, (3.94 where the KM data vector is ( g T, M g, M gk, M gk, M g =, (3.95 and the corresponding ode vector, ĝ, has eeents (upon the iniization, ( j ( M iπ T iπ M ˆ c M j, = ε r G gˆ e e. (3.96 = M + The east-squares soution is given by H ( ˆ c H = A PA A P G g, (3.97 ε g Fourier Geodesy 3.47 Jeei, January 7

48 where H A is the copex transpose of the iπ M (,,(, = ε, ( cosθ j n n j iπ M KM K atrix, A, that contains the eeents, a e P e. (3.98 In order to preserve the coputationa efficiency of the DFT, the error covariance atrix ust be ( diagona and the variances for any particuar ongitude ust be identica; thus P = diag p( j,, where the diagona eeent is p( j, = p j. Then an eeent of the K vector, b H = A P g, is K M j j j, j= = M iπ M (, = ε, ( cosθ n n b e P p g e = K iπ M e Pn, j p j g j j= iπ M ( cos DFT (, ε θ λ (3.99 H and the nora atrix, A PA, is boc diagona with K bocs ranging in size, for = M +,, M, fro to K K and bac to, a ( ε K K p jp ( cosθ, j P, ( cosθ j, ( cos, ( cos p jp θ j P θ K j j= j=. p jp, ( cosθ j P, ( cosθ j p jp, ( cosθ j P, ( cosθ K K K j j= j=. (3. = M K K With carefu attention to the indices it is readiy shown that the east-squares soution reduces to iπ M ˆ c e T ( T M M = r pr r p, + ε G C, (3. p and the vector, C, has eeents given by equation (3.87. If p is proportiona to the identity atrix (equa weights on the data, then the east-squares estiation for the Fourier-Legendre spectru is identica to the iniu-nor soution, equation (3.9. For variations on these ethods the reader ay consut (Leoine et a. 998, Ch.8; Coobo 98. The aternative, non-optia ethod of spectra anaysis on the sphere is based on nuerica quadratures, equation (3.74 (see aso Schitz and Cain 983, Rapp 969, which can aso be where = diag ( p j Fourier Geodesy 3.48 Jeei, January 7

49 foruated in ters of the DFT of the sapes, g j,. Substituting equation (.83 into equation (3.74, an estiate of the copex spectru is K c θ n, ε ( j,, ( θ n j θ j 4π j= Gˆ = DFT g P cos sin. (3. Then, with equation (3.87 and by defining s ( P ( P cosθ sinθ, cosθ, sinθ K =, (3.3 P, ( cosθk sinθk P, ( cosθk sinθ K K this becoes in vector-atrix for, ˆ c θ T G = ε sc, K + K. (3.4 For the rea spectru, θ T s Re ( C, K ˆ θ T G = sc, = θ T s I ( C, K + (3.5 where, as before, G ˆ c and G ˆ are vectors whose eeents constitute the Fourier-Legendre spectru, as in equations (3.83 and (3.93. Ceary, the nuerica-quadratures estiates, equation (3.4 (or, (3.5, though not optia, are coputationay ore efficient and stabe for high haronic degrees since no atrix inversion is required. Sipe integra discretizations for the anaysis, such as equations (3.8 or (3.8, using data vaues at points of the spherica grid, ay be ess appropriate if the sapes are averages over the grid ces. In this case, the right sides of these equations coprise sus of integras of the basis functions. This is a straightforward odification that can be derived fro the basic anaysis equations above. Reference is ade to (Coobo 98 for the detais. The synthesis of the function fro a given Fourier-Legendre spectru is straightforward and nuericay efficient when coputed on a reguar grid over the entire sphere using the DFT in ongitude. If the spectru is nown and band-iited and the function vaues are desired on a grid as in equations (3.8 or (3.8, then for the atter it is easiy shown that ( j ( g C j K, (3.6, DFT j =, =,, Fourier Geodesy 3.49 Jeei, January 7

50 where M, = ( j C = iπ M ( j ( T c M M ε e r G, + (3.7 ( j c and the vectors, G, r, are given by equations (3.83 and (3.84. For a given rea-vaued Fourier-Legendre spectru, equation (.87 yieds to corresponding copex spectru to be used in these equations. Reated efficient agoriths have been deveoped for anaysis and synthesis, for exape using the Censhaw suation ethod (Geason 985, Hoes and Featherstone DFT of Convoutions on the Sphere Geodetic and geophysica convoutions on the sphere often invove a data function and a erne function that depends ony on a singe variabe, as in equation (.5, g ( θ, λ * h( θ = g ( θ, λ h( ψ sinθ dθ dλ, (3.8 4π Ω where h( ψ typicay is a function of cosψ = cosθ cosθ + sinθ sinθ cos( λ λ (3.9 and attenuates to zero as the spherica distance, ψ, between evauation and integration points increases. Then it is nuericay justified that the integration region ay be truncated to a θ, λ. Moreover, it is coon that discrete data of a neighborhood of the coputation points, ( function are distributed on a unifor grid such as defined by equation (3.39. If one coud write the spherica distance, ψ, just in ters of atitude differences and ongitude differences, then any convoution with such a erne woud have the for of equation (3.98 that woud be aenabe to fast coputation using the FFT, as described by equation (3.. Equation (3.9 aready shows that the erne depends on the difference in ongitude. For iited spherica regions, it is aso possibe to approxiate the erne as depending on atitude differences. Adding and subtracting sinθ sinθ, one obtains ( ( θ θ θ θ ( λ λ ( cosψ = cosθ cosθ + sinθ sinθ sinθ sinθ + sinθ sinθ cos λ λ ' = cos sin sin cos ' (3. Fourier Geodesy 3.5 Jeei, January 7

51 and ( ( sinθ sinθ = sinθ sin θ θ θ ( ( = sin θ cos θ θ sinθ cosθ sin θ θ (3. ow, if the erne is approxiated by setting the co-atitudes of the evauation points to the average co-atitude, θ θ, of the integration region under consideration, then it depends ony on θ θ and λ λ. Denoting the approxiate erne by ĥ, the convoution, equation (3.8, then becoes g ( θ, λ * h( θ g ( θ, λ sin θ ' hˆ ( θ θ, λ λ dθ dλ, (3. 4π Ω where Ω is the truncated spherica doain on which the approxiation is nuericay justified. It is noted that aside fro the truncation of the integra this convoution is exact for the evauation points with co-atitude, θ = θ. This suggests a coproise in coputationa efficiency by appying the DFT ony with respect to the ongitude dependence, but perforing the convoution with standard nuerica integration in the space doain for the co-atitude so as to avoid the error associated with the approxiation, θ θ. Thus, one ay re-write equation (3.8 as c( θ, λ = g ( θ, λ * h( θ = g ( θ, λ h( θ, θ, λ λ sinθ dθ dλ. (3.3 4π Ω A discretization and truncation of this integra proceeds under the assuption that the data function, g, is saped at a unifor interva, λ, in ongitude as in equation (3.39, but for any arbitrary discrete co-atitudes, L L g j, = g ( θ j, λ, j =,, J, =,,, (3.4 where, athough the co-atitudes, θ j, need not be equay spaced, we ay assue a constant interva, θ, and J, L deiit the area of integration. Then, J L θ λ j, = sin θ j j, n j, j 4π j = n= L c g h ( θ θ n, (3.5 where, in order to utiize the DFT, the erne function vaues ust for a periodic sequence in ongitude. Thus, one defines Fourier Geodesy 3.5 Jeei, January 7

52 with h L L, =,,, (3.6 ( θ j θ j h( θ j θ j λ ( θ j θ j = ( θ j θ j h, h, for any integer. (3.7 + L ow, fro equations (3.88 and (3.94, the discrete convoution (3.5 becoes ( ( ( ( θ θ J θ j, = sinθ j DFT DFT j, n DFT j, j 4π n j = c g h, (3.8 where the DFTs are one-diensiona with respect to the ongitude. The integration with respect to atitude is foruated here sipy as the rectange rue for nuerica integration. Thus, one achieves at east soe coputationa efficiency without introducing additiona errors, whie aso eeping true to the spherica curvature of the integration doain. This procedure of appying the DFT (or, FFT to the spherica convoution has acquired the nae of the -D spherica FFT ethod (Haagans et a It shoud be noted that the DFT foruation above sti contains truncation and cycic convoution errors, where the atter can be eiinated as discussed in Section 3.4, and the forer occurs with any other nuerica integration of the convoution over a iited area. 3.6 Discrete Fiters and Windows The discrete versions of the fiters and windows introduced in Chapter, with soe exceptions, foow directy fro a saping of the corresponding continuous functions. In eectrica and counications engineering fiters have a uch greater iportance as signas are assaged and anipuated to have various desirabe characteristics. They are expounded invariaby in ters of the tie and tepora frequency doains. The discrete fiters are caed digita fiters and the ethods of their design in the engineering appications occupy any textboos. Therefore, it woud be reiss not to ention the here, at east to introduce coon terinoogy, but for ost geodetic and geophysica appications it suffices to adopt discretized versions of the fiters and windows aready discussed. In genera, a digita syste is any operation that taes a digita input (i.e., a sequence of sapes or data and produces a digita output (cf. equation (.6: ( y = h g, (3.9 Fourier Geodesy 3.5 Jeei, January 7

53 where h is the syste function and the sequences, g and y, are infinite ( < <, bounded, and absoutey suabe. As usua, it is assued that the saping interva is constant and the sae for the input and the output. A inear, tie-invariant digita syste is caed a digita fiter; and, anaogous to the case of continuous functions a digita fiter is a discrete convoution (cf. equation (.6, y = h # g. (3. The fiter is characterized by the (possiby infinite sequence, h. Shift-invariance (caed tieinvariance in tepora doain appications ipies that shifting the index of the input sequence by a certain nuber aso shifts the index of the output sequence by that sae nuber; in other words, the fiter is independent of the index origin. The output sequence, y, is equa to the fiter sequence, h, if the input is a digita ipuse, that is, the Kronecer deta, equation (3.. Fro equation (3.34, there is n= ( x y = h # δ = x hnδ n = h ; (3. and, therefore, the fiter sequence, h, is aso caed the ipuse response of the fiter. If the sequence, h, consists of a finite nuber of non-zero eeents, then the fiter is nown as a finite-ength ipuse response (FIR fiter; and, if h is an infinite sequence of non-zero eeents then the fiter is an infinite-ength ipuse response (IIR fiter. For the IIR fiter, it is required, of course, that the convoution (3. exists. The fiter is said to be stabe if the output exists (is bounded for every bounded input. It can be shown that a fiter is stabe if and ony if the ipuse response sequence, h, is absoutey suabe, equation (3.. The discussion here is restricted to a few exapes of the FIR fiter, aso caed a oving-average fiter, that are owpass fiters adapted fro the fiters discussed in Chapter 3. Typica fiters appied to rea-vaued data are aso rea and syetric (see Chapter. In order to preserve the average of the origina data sequence the zero-frequency spectra coponent of the fiter shoud be unity, and this then dictates the scae of the fiter sequence. By equation (3., the spectru of the fiter is aso rea and syetric. The rectange fiter, aready encountered in Section 3.., as the rectange sequence, is repeated here with sighty different definition (for even n, cf. equation (3.6, h ( n n n, = n x, otherwise (3. recaing that a fiter is a convoution, here given by equation (3.34, hence, the factor, x, in the denoinator. Its frequency response, using equation (3.8, is given by Fourier Geodesy 3.53 Jeei, January 7

54 ( nπ xf ( π xf ( n sin i xf Hɶ π ( f = e, (3.3 n sin which equas unity at zero frequency, which is obtained, for exape, by appying Hôpita s rue to the iit, f. Siiary, the discrete triange (Bartett or Fejér fiter ay be defined by n n n, n n h = h h = n x n, otherwise ( ( ( ( Bartett # (3.4 With the transation property (3.9 and the convoution theore (3.35, the frequency response is given by H Bartett f Fn xf n ( ( ɶ ( n = π, (3.5 where the Fejér erne, F ( F n ( x n ( nx ( x sin x, aso periodic, is defined by = n sin. (3.6 ( ɶ ( n. The response at the origin is unity, H Bartett = The discrete rectange and Bartett fiters are FIR fiters. Other exapes incude the cosine window functions, equation (., such as the Hann window, treated as fiters. The discretization of these functions is straightforward and foows the discretization of the window sequences (beow with appropriate odification to accoodate the desideratu for the spectra coponent at the origin to equa to. On the other hand, the discretized Gaussian function, stricty speaing is an IIR fiter, athough practicay its non-zero vaues are sufficienty sa beyond soe finite so as to be negigibe. Taen as in equation (3.9, ( β π ( x β ( h Gauss e =, β >, (3.7 β where the paraeter for the fiter function β x, it does preserve the average of the fitered data, since the frequency response, given by equation (3.3, or its approxiation, ( ɶ ( β π ( β xf, (3.8 H f e Gauss equas unity at f =. Fourier Geodesy 3.54 Jeei, January 7

55 Higher-diensioned fiters in Cartesian coordinates foow directy as for the continuous case, fored by the product of one-diensiona fiters according to the separation of variabes; as in equation (.34. The window (or taper functions ay be discretized just ie the fiter functions, athough one ay prefer that in order to appy the DFT, it ie the data sequence shoud be periodic. Aso, unie the fiter sequences whose frequency responses are unity at the frequency origin, the window sequences defined here are equa to at the origin in the space doain. In this case, for ( exape, the discrete rectange window, b, is given by equation (3.6, the discrete Bartett window is given by (copare with equation (3.4 ( ( ubartett, =, otherwise (3.9 and, the discrete Hann window is ( ( uhann π + cos, =, otherwise (3.3 Their Fourier transfors for f f = ( x ( ( π are, respectivey, ( π xf ( π xf sin ɶ ( = =, (3.3 U f xf xf Bartett ( ( UHann f B ( f B f B f 4 x x sin ɶ ( ( ( = ɶ + ɶ + ɶ +, (3.3 where F is the Fejér erne, equation (3.6, and ( B ɶ is the spectru of ( b. Due to the discretization they are very sighty aiased versions of the transfors of their continuous cousins (Section.6. If these window sequences are used as data tapers and assued to be periodic, together with a finite sequence of data vaues, then the DFTs of the window sequences are sipy ( ( ( Bartett = Bartett ( Uɶ Uɶ f and ( ( ( Hann = ( ɶ ɶ Hann, where f U U f = x, =,.,. The discrete version of the proate spheroida wave functions, considered aso as tapers, is derived by setting up a concentration probe for sequences, anaogous to the case for continuous functions (Section.6.. For space- (index- iited sequences, ( g, where ( g =, < or, it is desired to find the sequence with axiu energy Fourier Geodesy 3.55 Jeei, January 7

56 concentrated in a particuar bandwidth, f f < f, with f equa to the yquist frequency. In other words (cf. equation (.45, the ratio, λ f f ɶ ( f = = f = = f Gɶ ( x ( g f df = f ( f π ( ( ( i x f G f df x g g e df, (3.33 shoud be axiized. The nuerator on the far right foows fro equation (3.6 and the denoinator refects Parseva s theore, equation (3.. Fro equation (.44, f f ( ( π xf ( df = π x ( iπ x f sin e hence, equation (3.33 becoes λ ( f ( g ( g sin ; (3.34 ( π xf ( π ( T = = g = = T ( g = where A is an atrix with eeents, Ag, (3.35 g g ( π ( π ( sin xf, and g is the vector of sequence eeents. Assuing for the oent that g is a continuous variabe, taing the differentias of λ ( f and g, i.e., δ λ ( f derivatives, one obtains ( T T ( f = A ( f T T ( = δ ( A g g g g, and appying the product rue for δλ g g δ g g λ g. (3.36 Thus, λ ( f is axiu ony if ( f ( f δλ vanishes, or Ag λ g =. (3.37 This is now a standard eigenvector/eigenvaue probe. Because A is syetric and invertibe, the eigenvaues (concentration ratios as in the continuous case (Section.6. are finite in nuber, rea, distinct, and positive by equation (3.33. Thus, they can be ordered, Fourier Geodesy 3.56 Jeei, January 7

57 ( f ( f > λ > > >. (3.38 λ The eigenvector, (, f g = ψ, associated with the axiu eigenvaue, λ ( f, soves the concentration probe. However, the atrix, A, is poory conditioned, where the ratio of argest to saest eigenvaues typicay is any orders of agnitude. Indeed, the soution to equation (3.37 is aso a soution to a finite difference equation anaogous to the Stur- Liouvie differentia equation, which is then used to deterine the eigenvectors (Perciva and Waden 993, p.386. The eeents of the eigenvector for a discrete proate spheroida sequence (dpss. The agoriths to copute these are outside the present scope; see, e.g., (Gruenbacher and Hues 994. Figure 3.6 shows exapes of the dpss, spectru, equation (3., ( Ψ ( = ψ ( ( (, f ( ψ, and its ɶ, f, f iπ xf f x e, (3.39 = for the cases, = 5, x =, f = =.4 f and f = =.8 f. Each such dpss is essentiay iited in energy to the band of frequencies, [ f, f ]. Figure 3.6: Discrete proate spheroida sequences, (, f ( f (, f ( ψ Ψ ɶ, (right for f = = 5. and f = = 5.4., (eft and their apitude spectra, The dpss ay be viewed as the discretization of the proate spheroida wave functions encountered in Section.6.. Indeed, the eeents of the atrix, A, equation (3.35, are x d x, x ; f, equation (.44. It is noted in Section discrete sapes of the erne function, (.6. that the one-diensiona case and generaizations to the Cartesian pane and to the sphere Fourier Geodesy 3.57 Jeei, January 7

58 share siiar properties, since the corresponding erne functions are essentiay identica in the sense of being the transfor of either a rectange or cap function (assuing the spectra concentration region is a rectange or cap. The sae hods for the discretization in higherdiensioned doains, and the detais are ony in the nuerica coputation of the sequences, which are eft to other sources, notaby Sions and Wang (. As in the case of continuous space-iited functions, the finite eigenvectors, ( ( ( ( ψ ( ψ =, f, f, f j j j T ψ, (3.4 corresponding to the eigenvaues, λ j ( f, for an orthogona basis for the index-iited sequences. These basis sequences are aso nown as Sepian sequences. Since the eigenvaues are osty either cose to unity or cose to zero, where the nuber of significant ( near-unity vaues is the Shannon nuber, xf, the space of index-iited sequences is approxiatey spanned by these corresponding eigenvectors if f is reativey arge. And, because the eigenvaues are near unity, these eigenvectors aso have axia concentration in the frequency (, f doain, with ψ having the highest concentration. Thus each one coud serve as a taper sequence that has inia spectra eaage, which has iportant appication in power spectra density estiation (Chapter 4. Figure 3.7 shows the apitude spectra of severa ow-order (, f dpss s, ψ, for x =, = 6,.5 j xf = 6 ( j =,,5 are cose to in vaue. f =, and corresponding eigenvaues, λ ( j f ; the first (, f ( Figure 3.7: DPSS apitude spectra, Ψ ɶ f, j =,, 4,6, for = 6 and f =.5 (eft; and eigenvaues, λ j ( f,,,, j j = (right. The Shannon nuber is xf = 6. ote that on the eft ony part of the frequency doain, f f =.5, is shown. Fourier Geodesy 3.58 Jeei, January 7