Chapter 3 Discrete Techniques 3. Introduction In the previous two chapters we introduced Fourier transforms of continuous functions of the periodic and non-periodic (finite energy) type, we as various operations, principay the convoution, that have specia meaning in the frequency domain. In practice, however, we dea with discrete, rather than continuous, signas. That is, the signa may be defined physicay for continuous time or space (we aso say it is an anaog signa), but we sampe it (or measure it, or reaize it) at discrete epochs in time, or at discrete points in space. The resut is a (discrete) sequence of data that captures some but not necessariy a the information contained in the signa. ote that ony the domain (e.g., time-axis, or spatia area) of the signa is discretized. The eements (data) of the sampe sequence may assume any vaue aowed by the continuous signa. In the modern parance, one usuay aso uses the term digita signa to refer to a discrete signa as defined here. These two terms, digita and discrete, are synonymous in the present discussion. However, one can consider, in addition, the discretization, or quantization, of the vaues or ampitudes of the signa. Then one may need to mae a distinction between continuous and discrete ampitude signas and use, e.g., the terms discrete and digita, respectivey, for that purpose. Ceary, the quantized sequence is a specia case of the continuous-ampitude sequence; and, the consideration of the former with respect to the atter invoves primariy the incorporation of the quantization effect. We do not treat this here and restrict our attention to continuousampitude, discrete signas, or sequences. The concepts of Fourier transform, convoution, windows, and fiters, discussed before for the continuous case, can be adapted to the discrete case, athough one important premise is that the signa shoud be discretized (or, samped) at a constant interva. In practice, we usuay aso have ony a finitey extended sequence of data. However, we consider both infinite as we as finite sequences, the former first. The truncation to a finite extent wi be treated much ie it was for continuous signas. Fourier Geodesy 3. Jeei, Apri 7
3. Transforms of Infinite Sequences First, we must understand samping and the reationship between the discrete sequence and the continuous signa from whence it originates. To do this we mae use of the concept of impuses δ t. defined by the Dirac function, ( ) Let g( t ) be a continuous function of t (as before, t may be interpreted as a time variabe or a space variabe; and ater, we wi generaize our deveopment to higher dimensions). Suppose that g( t ) is samped at reguar intervas, t =, where is an integer with < <. We have the foowing sequence of sampes: g = g( ), < <. (3.) t is constant and is caed the samping interva. If we assume that the origina continuous signa has finite energy, then its (continuous) Fourier transform exists. We woud ie to find a simiar transform for the discrete sequence of sampes. Because the parent signa is not periodic, we wi assume that the sequence, g, aso is not periodic. We may expect that the transform wi not be discrete, since it was discrete ony for periodic functions. In fact, the transform must be continuous because in a certain given band of frequencies, every frequency may be represented by the infinite sequence. On further thought, the reader may aready guess that the truth of the atter part of this statement depends in some way on a reationship between the band of frequencies and the samping interva. This is actuay the case, as seen beow. In order that we may use the Fourier transform previousy defined for continuous signas, we formay represent the infinite sequence of sampes as a function in the continuous variabe t by mutipying the continuous signa with the samping function, (.53): gs ( t) = g( t) δ ( t ). (3.) = The factor t s g t (since the deta function has units inverse to those of its argument). For any particuar integer such that t, ( ) g t is a singe impuse with ampitude g g t is a s is incuded so that the units of g ( t ) match those of ( ) g t = ; whie ( ) s. Thus, ( ) train of impuses with ampitude at each point, t, proportiona to the vaue of the signa at this point. Aso, by appying the imit to equation (3.) as the samping interva approaches zero, we return to the origina signa: im g ( ) ( ) ( ' ) ' s t = g t δ t t dt = g( t ), (3.3) s where t'. Again, equation (3.) is a more natura representation of the discrete sequence Fourier Geodesy 3. Jeei, Apri 7
if t is incuded. As before, when deaing with the deta function, we operate on it formay, recognizing that it is not a function in the traditiona meaning, and that one can aways approach the resut in the imit using the rectanguar function with vanishing base and unit rectanguar area. But one needs to be carefu! g t = G f g t, then since ( ) ( ) If F ( ) is the Fourier transform of ( ) gs ( t) = g( t) δ ( t ), (3.4) = by the Dua Convoution Theorem (.5), we have F ( gs ( t) ) = F ( g( t) )* F δ ( t ). (3.5) = It was shown with equation (.57) that the Fourier transform of a train of impuses, the samping function, is, again, a train of impuses: F δ ( t ) = δ f. (3.6) = = Therefore, equation (3.5) becomes F = = g t = G f * f G f * δ f s = ( ( )) ( ) δ ( ) = = G( f ') δ f f ' df ' (3.7) where the definition of convoution, equation (.) (see aso equation (.)), was used. With the definition of the deta function, equation (.46), we obtain a fundamenta resut: = = G % s ( f ) F ( gs ( t) ) = G f = G f +. (3.8) We appy the ~ notation to the spectrum (Fourier transform) of the samped signa, because G% f is periodic with period in frequency of t : s ( ) = G% + s f + = G f + = G% s( f ). (3.9) Fourier Geodesy 3.3 Jeei, Apri 7
Thus, the spectrum, G ( f ) part): %, may be defined soey on the interva of frequencies (the principa s f, (3.) or, any other interva of ength t. The argest frequency in this interva is f =. (3.) It is caed the yquist frequency (or, foding frequency - see beow) and is one haf of the inverse samping interva. Since G% s ( f ) is a periodic function, with period, t, it can be represented as a Fourier series, equation (.6): % s ( ) iπf. (3.) G f = t ce = There is no oss in generaity by choosing negative instead of positive exponents. The Fourier coefficients are given by equation (.), with appropriate sign change in the exponentia: % iπf s ( ). (3.3) c = G f e df Because of the periodicity of G ( f ) obtain iπf i tf c Gs ( f ) e df G f π = = + e df = %, and with its representation given by equation (3.8), we s %. (3.4) ow changing variabes of integration, see that f ' = f +, and then performing the summation we + = + i πf ' i πf' ( ') ' ( ') ' ( ) c = G f e df = G f e df = g t = g. (3.5) That is, the Fourier series coefficients of the sampe spectrum are the sampes, themseves. Fourier Geodesy 3.4 Jeei, Apri 7
Therefore, we have the foowing Fourier transform pair for discrete (infinitey extended) signas, i.e., for infinite sequences: iπf iπf G( f ) = ge g = G( f ) e df = % %, (3.6) where we have dropped the subscript s because this transform pair hods in genera, even if no specific reference is made to an underying continuous signa. Aso, it is seen that the use of the deta function was ony a stepping stone to our fina resut that excudes this rather artificia function. On the other hand, it was instrumenta in giving us the fundamenta reationship, equation (3.8), between the spectrum of the samped process and the Fourier transform of the continuous, parent signa. ote that the definition of the Fourier transform, G% ( f ), agrees in terms of units with previous definitions. The transform pair, equations (3.6), is the dua form of the transform pair (.), where the continuous, periodic signa has a discrete Fourier spectrum. Ceary, a anaogous properties (.4) - (.8), hod for the pair (3.6), and the reader is invited to correspondingy formuate and verify them; for exampe, the transation property is in this case: % i t f. (3.7) π ( ) g G f e Finay, to cose this section, we define the discrete convoution of two infinite sequences, each having the same samping interva, t, (see aso equation (.35)): g h = gh #, (3.8) n n n= where, of course, the sum is assumed to exist for a. Again, there is a convoution theorem: g # h G% ( f) H% ( f), (3.9) which can be proved easiy, anaogous as in equations (.4), by substituting equation (3.8) into equation (3.) to express the spectrum of the convoution. ote the simiarity to transform (.34), which is the dua cycic convoution theorem for continuous periodic signas. The dua convoution theorem for discrete signas is (again, referring to equations (.33)): gh G% ( f )*% H% ( f ), (3.) where the cycic convoution, * %, is defined by equation (.3) and the period in this case is t. Fourier Geodesy 3.5 Jeei, Apri 7
3.3 Aiasing We say that a signa, g( t ), is band-imited if its spectrum beyond a certain frequency is zero. Consider the specia case that g( t ) contains no spectra components at frequencies arger than the yquist frequency. To be specific, suppose G( f ) =, for f > f = and f f =. (3.) In this case, for ( ) f ( ) <, the spectrum of the corresponding samped signa is, from equation (3.8): F gs G% s f = G f + = L+ G f + G f + G f + + L ( ) ( ) ( ) ( L G( f ) L) ( ) = G f = = + + + + (3.) since the arguments, f ±, f ±, etc., for ( ) < f ( ) are a outside the (, ], and so G is zero there, by equation (3.). Here, again, we denote the spectrum of the sampe with the subscript s to distinguish it from the more genera periodic spectrum, equation (3.6), where no specific reference to a continuous signa is needed. Thus, according to equation (3.), if the continuous signa is band-imited to frequencies not greater than the yquist frequency as determined by the samping interva, then the spectrum of the samped signa is equa to the spectrum of the continuous signa (see Figure frequency interva ( ) ( ) 3.). If G( f ) for f > f (the signa has energy beyond the yquist imit), then the spectrum, Gs ( f ), is different from G( f ), for f f, due to the overap (or, foding) of the parts G( f ), G( f + ), etc., onto ( ) G f (see Figure 3.). This effect is nown as aiasing. That is, the spectrum of the sampe is a corrupted version of the spectrum of the parent signa. Attempting to determine the spectrum of the continuous signa from the sequence of samped data is subject to the aiasing error if the continuous signa has spectra content beyond the yquist frequency. The aiasing error is given by = G% s ( f ) G( f ) = G f +, f f. (3.3) In Figure 3., the samping interva is t = 6(the reciproca of the period of the spectrum Fourier Geodesy 3.6 Jeei, Apri 7
of the sampe spectrum). Thus, the yquist frequency is f = 3. Aso, it is seen that the parent signa has no frequency content beyond f max = 3, hence there is no aiasing the spectrum of the sampe equas the spectrum of the parent signa in the band of frequencies bounded by the yquist frequency. In Figure 3., by contrast, the samping interva for that same continuous band- imited signa is ony t 3 G% f is 3). Therefore, = (and, consequenty, the period of ( ) the yquist frequency is f = 3< fmax. The enveoping ine in Figure 3. is the spectrum of the sequence of sampes and deviates at the high frequencies from the parent signa spectrum due to aiasing.. G % s ( f ) s.8 G ( f ).6.4. 8 6 4 frequency 4 6 8 Figure 3.: Spectra of a band-imited signa and of a sequence of its sampes...8.6 G % s ( f ).4. ( ) G f 5 4 3 3 4 5 frequency Figure 3.: Spectrum of a band-imited signa and aiased spectrum of a sequence of its sampes. Fourier Geodesy 3.7 Jeei, Apri 7
Another way to view the aiasing error is to note that the samped signa cannot distinguish between sinusoida components of the parent signa at different frequencies f, f reated by f = f ± f, where is any integer, because the sinusoids at these frequencies pass through g t. One can thin of the spectrum of the samped signa as having to accommodate the fu spectrum in some fashion, and it is accompished by foding higher frequency components bac onto the principa part. The spectrum, G% s ( f ), is caed the aiased spectrum of g. Thus, the estimation, or computation, of the spectrum of g based on a samped sequence is in error by this aiasing effect, since the higher-frequency components, especiay those near and beow the yquist frequency cannot be separated from frequencies near and above the yquist frequency. In order to reduce the aiasing effects, a samped signa shoud be fitered (ow-pass fiter) to remove the components with frequencies near the yquist frequency; or, the samping interva shoud be decreased. If the signa is band-imited, it can be recovered fuy from the sampes provided the samping is done at intervas corresponding to twice the highest frequency contained in the signa. Specificay, suppose that G( f ) = for f > fc ; then we can write on the basis of equations (3.) and (3.) the same sampe points of ( ) ( ) ( ) ( ) G f = G % f b tf, (3.4) s where t = ( f ) and b ( tf) c is the rectanguar function (.38), now a function in the frequency domain. That is, the spectrum of the sampe, being periodic, is truncated to just one period, and then this truncated version agrees with the tota spectrum of the continuous signa that is supposed to be band-imited. Taing the inverse Fourier transform on both sides yieds, with the convoution theorem (.), ( % s ) ( ) = F ( ) ( ) g t G f b tf = F s ( G% s ( f ))* F ( b ( f )) ( )* F ( ( )) = g t b tf (3.5) With Properties 4 and 5 of Fourier transforms (equations (.3) and (.3)) appied to transform (.36), we recognize that: t sinc b ( f). (3.6) Since the rectanguar function is symmetric with respect to the origin, its inverse Fourier transform in equation (3.5) is the sinc function. We now have with equation (3.): Fourier Geodesy 3.8 Jeei, Apri 7
t g( t) = g( t) δ ( t ) * sinc = = t t' = g( t' ) δ ( t' ) sinc dt ' (3.7) in view of the definition of convoution, equation (.). Finay, with the deta function definition, equation (.46), we obtain t = ( ) g( )sinc g t = = t = g sinc (3.8) Thus, if g( t ) is band-imited (frequency content is zero for f reconstructed from its samped vaues, g, if t = ( f ) (Whittaer-Shannon) samping theorem. c > f ), the entire function can be c. This resut is nown as the 3.4 Discrete Fourier Transform In addition to being discrete, our data usuay have finite extent. In essence, we may thin of this situation as appying a rectanguar function (a window function) to the infinite sequence of sampes. This impies, according to the dua convoution theorem, that the spectrum of the finitey extended sequence is the convoution in the frequency domain of the spectrum of the infinite sequence, G% ( f ), with the sinc function. We wi not consider this particuar case further since it is ony an appication of the previous cass of functions infinitey extended sequences defined on a discrete domain, where most of the data vaues happen to be zero. Instead, we consider the fourth fundamenta case (the other three being continuous periodic and non-periodic functions, and non-periodic sequences), where the data, constituting a finitey extended sequence on a discrete domain, aso form a periodic sequence. In other words, we assume that the finite sequence of data, g, repeats itsef for < <. This case is of fundamenta importance in our consideration of the practica situation. That is, given a finite sequence of sampes, we assume that it actuay comprises one period from a periodic sequence. Whie this seems to contradict the types of signas we usuay see in practice, it is a necessary condition that wi aow us to utiize the discrete Fourier techniques. Thus, the ony difference between this case and the previous case of Section 3.3 is the periodicity of the sequence. This means that the spectrum of the sequence, in addition to being periodic (because the data are discrete), is, itsef, discrete (because the data are periodic, just ie for the Fourier series). Fourier Geodesy 3.9 Jeei, Apri 7
We write the samped signa as a finite sum (Fourier series) of (discrete) spectrum vaues as foows: π i g% ( ) g% = Ge %, (3.9) = where the tota number of sampes is (we may assume is even, without oss in generaity). The tota ength of the data record is t. Again, we assume that even though ony data have been samped, the sequence is extended periodicay with these sampes to ±. Equation (3.9) hods for a integers,, and it represents a periodic sequence. We assume, furthermore, that the continuous parent signa (from which the data were extracted) is aso periodic with period, T =. That is, t divides the period without remainder. The yquist frequency, in this case, is f = ( ) ; whie the fundamenta frequency is f = ( ). Hence, the index counts off frequencies: f + =,, K,,,,, K,. (3.3) If is odd, then ( ) ( ). In this case the yquist frequency, as defined, is not actuay a constituent frequency of the discrete spectra domain. The foowing sum can easiy be verified and demonstrates the orthogonaity of the functions exp iπ : ( ) + π i j, mod ; e, mod = ; = ( ) ( j) = ( j) (3.3) where is any integer. Since the data sequence is periodic, we may choose any consecutive sampes. Suppose that the index for the discrete data runs as =, K, ). If we mutipy both sides of equation (3.9) by exp( iπ j ) equation (3.3) and = : and sum over a, we get, with π π π i j i i j ge % = Ge % e = = = π i G% e = = = ( j ) (3.3) = G% j Fourier Geodesy 3. Jeei, Apri 7
Therefore, renaming the index on G %, we have the foowing pair of expressions: π i % g% = Ge, ; = π i G% = ge %, ; = (3.33) which simpy says that { } g other. % and { } G % are mutuay consistent and Fourier transforms of each It is easiy seen that G % is a periodic sequence, with the anaogous period on. (Actuay, we may prove that G % G % ( ( ) ), and the period in frequency is t, which corresponds to wavenumber, =.) Since both the data sequence and its Fourier transform are periodic, we may choose any consecutive vaues. Typicay, one may choose =, K, and =, K, ; and we then have: π i G% = ge %, (3.34) = π i % g% = Ge,. (3.35) = This is our definition of the discrete Fourier transform (DFT) pair. The samping interva, t, is often normaized: t =, athough this tends to hide the units invoved. evertheess, the DFT is usuay seen in the iterature as: ( ) π i G% = DFT g% = ge %, ; (3.36) ( ) = π i g% = DFT G% = Ge %,. (3.37) = We note that the pairs shown by equations (3.33) are aso DFT pairs, just differenty defined. In fact, additiona definitions exist where the antecedent coefficient in the inverse transform is appied instead to the direct transform, or the square root is appied to both for maximum symmetry. In a cases, the direct and inverse transforms are mutuay consistent (an exercise eft to the reader); we wi adopt the definitions (3.36) and (3.37) because they impy units for the direct transform consistent with our previous conventions (see aso equations (3.34) and (3.35)). Fourier Geodesy 3. Jeei, Apri 7
This competes the possibiities for defining Fourier transform pairs for continuous or discrete signas that are either periodic or non-periodic. The definitions of the various forms of the Fourier transform may be summarized in Tabe 3.. We see the perfect duaity in characteristics among the transforms: discreteness in either signa or spectrum impies periodicity in its respective transform, and continuity in either signa or spectrum impies nonperiodicity in its respective transform. Tabe 3.: Fourier Transform Pairs g G. g: periodic g: non-periodic (finite energy) g: discrete G: discrete and periodic G: continuous and periodic π i G% = ge %, = = i tf G% π ( f ) = ge, f π i % g% = Ge, = = %, < < ( ) iπf g G f e df g: continuous G: discrete and non-periodic T π i t T G = g( t) e dt, < < % = π i t T g% ( t) = Ge, t T T G: continuous and non-periodic ( ) ( ) iπ ft G f = g t e dt, < f < ( ) ( ) iπ ft g t = G f e df, < t < 3.5 Properties of the DFT The properties that were given for previous versions of the Fourier transform, such as equations (.3) - (.8) for the Fourier series transform and equations.6-.35 for the continuous Fourier transform, hod with corresponding modifications for the DFT, as we. In addition, we have, for exampe (an anaogous property coud be found for the other transforms): g% %. (3.38) * * G The foowing conjugate-symmetry property is aso noted: Fourier Geodesy 3. Jeei, Apri 7
g% rea G% G%, (3.39) * is if and ony if = which impies that G % and G % ( even) are both rea. The proof foows directy from equation (3.36). The property (3.39) shows that there are as many independent frequency components as data vaues, which is another way of saying, as for a types of transforms, that the DFT contains a the information contained in the periodic sequence. If g% is a sequence of rea vaues, then the compex sequence of its discrete spectrum has ony haf as many independent components, according to equation (3.39) (but, of course, each component has two independent vaues rea and imaginary). The convoution of discrete periodic sequences is defined simiary to periodic continuous functions, equation (.3). Let ( ) ( % ) G % = DFT g%, H % = DFT h (3.4) be discrete Fourier transforms of discrete sequences each having the same period,. Identica periods is a requirement for the convoution to be defined. The discrete convoution of periodic sequences is denoted # % and is given by f% = g% #%% h = gh % %, =, K,, (3.4) n n n= where, ceary, we must mae use of the fact that the sequence, h%, is extended periodicay beyond a singe period. For exampe, f% = gh % % + gh % % + gh % % + gh % % + L 3 = gh % % + gh % % + gh % % + gh % % + L 3 (3.4) Hence, we coud aso write the convoution (3.4) as f% = gh % %, =, K,, (3.43) n= ( + ) n mod n if we wish to use ony h % -vaues for,, = K. This convoution, equation (3.4) or (3.43), is nown as a cycic, or circuar, convoution, since we are deaing with periodic sequences. ote that the units of f% are the same as the product of units of g% and h%. Just ie for continuous functions, we aso have a convoution theorem and a dua convoution theorem for the discrete case. Fourier Geodesy 3.3 Jeei, Apri 7
Theorem 3.: Discrete Convoution Theorem The discrete Fourier transform of the discrete convoution (3.4) equas the product of the discrete Fourier transforms of the convoved periodic sequences. In symbos: F% = GH % %. (3.44) Proof: Consider the DFT of the convoution, equation (3.4): π i F% = DFT( f% ) = gh % % n n e. (3.45) = n= Interchanging summations, we get π π i n i ( n ) % n n F% t ge % h e. (3.46) = n= = For each n in the first sum, we have the second sum of a the ( m ) mutipied by the corresponding exp i( π ) h % s, m =, K,, each (which is aso periodic with period, ). Hence, each sum in equation (3.46) is the DFT of the corresponding discrete signa; and with equation (3.34) we have proved the convoution theorem. More expicity, the discrete convoution is therefore equivaent to m % %% = % % g# h DFT GH ( g% ) ( h% ) = DFT DFT DFT (3.47) Again, anaogous to the continuous case, the discrete convoution can be performed by mutipying two discrete spectra and appying three DFT s. We aso have the dua convoution theorem. Theorem 3.: Dua Discrete Convoution Theorem The discrete Fourier transform of the product of two periodic sequences having the same period equas the discrete convoution of their discrete Fourier transforms. In symbos: Fourier Geodesy 3.4 Jeei, Apri 7
( ) DFT gh % % = G% # % H%, (3.48) where the discrete convoution is defined by equation (3.4). Proof: Consider the DFT of the product of sequences: DFT ( ) = π i gh % % = ghe % % = π π i j i Ge % j he % = j= (3.49) where equation (3.35) was used. Interchanging sums and again using the definition of the DFT: DFT ( % ) π i ( j) % j j= = gh% = G he % = j= And, the proof is compete. GH % % j j (3.5) It is easy to see that the discrete convoution (3.4) is a periodic sequence (period, t); hence, aso its spectrum, F %, is periodic (period,, in terms of the wavenumber, ). Equation (3.48) gives us directy Parseva s Theorem for discrete, periodic signas. On the eft-hand side, consider the DFT of at zero frequency ( = ). By the periodicity of the DFT, property (3.38) gives us: % %. (3.5) * * h H Then, we have, using equation (3.5) - ( ) * * * gh % % =DFT gh % % = GH % % j + j = j= = j= GH % % j * j which is Parseva s Theorem in this case. = (3.5) Fourier Geodesy 3.5 Jeei, Apri 7
The aiasing error for discrete, periodic sequences can be formuated in a fashion simiar to non-periodic sequences. We start with equation (.6), the Fourier series representation of the periodic, continuous signa: π i t T g% ( t) = Ge T. (3.53) = Then, the sampes are given by g% = g% ( ) = Ge T = T = π i T π i ( + j) G e T + j = j= (3.54) with a more eaborate, but usefu way of writing the infinite sum (the reader is ased to verify the equivaence of these sums). The summations can now be interchanged, so that π i j + j g% G e, (3.55) = j= = where T exp iπ =. Comparing equations (3.55) and (3.35), we find that the spectrum of the sampe is reated to the spectrum of the periodic parent signa by =, and since ( ) G% j = G. (3.56) + j = This is anaogous to equation (3.8) for the continuous spectrum. The aiasing error is given, as in equation (3.3) by G % G = G, j. (3.57) j j + j = 3.6 FFT The fast Fourier transform (FFT) is an agorithm to compute the DFT. As the name impies it is a fast agorithm. There is no additiona theory associated with the FFT, other than the deveopment of the agorithm. We may say that Fourier Geodesy 3.6 Jeei, Apri 7
FFT DFT, (3.58) as far as the numerica resuts are concerned. To now what the DFT is a about is to now what the FFT is in terms of properties, theorems, etc.; the two are synonymous from that perspective. It is amost no exaggeration to caim that the FFT is as ubiquitous nowadays in spectra anaysis appications as the standard mathematica functions, ie sine, cosine, etc. There are many good boos avaiabe (e.g., Brigham, 988) that go into the detais of the agorithm. We do not do this here, any more than we woud consider the detais of computing the sine or cosine of an ange. We treat the FFT as a bac box, or a ibrary function we now what shoud come out for a given input. It is enough to now for our purposes that the FFT is the DFT; and that, when using computer agorithms, it is important to determine how the DFT is defined for that particuar FFT agorithm (what scaing factor is used). That is, is the antecedent factor, or, or? The speed of the agorithm depends on the prime factorization of the number,, of sampe vaues. The fastest computation occurs if is a power of, and this was assumed for the origina formuation of the FFT. The speed in this case, as measured by the number of mutipications, is proportiona to og, compared to for the brute-force method according to the definition (3.36). Thus, for exampe, if = 4, we see a tremendous savings 4 6 in computation as ony ~ mutipications are needed versus ~. owadays, agorithms exist for with arbitrary prime factorization; but, the fewer the number of different factors, the faster is the computation. The principa appication of the FFT is, of course, in computing the spectrum of a signa, or synthesizing the signa from its spectrum (inverse transform). We note that this hods for -D as we as higher-dimensioned discrete signas, since corresponding agorithms simpy mae use of the separabiity of the transform in terms of the independent (Cartesian) variabes. This separabiity was aready evident with the transforms of continuous functions, equations (.59) and (.6), and is a property in the discrete case as we (see Section 3.8). Simiary, the computation of the Legendre spectrum from data given on a reguar spherica grid benefits from the FFT, but ony in one dimension (see Section 3.9). Another appication of the FFT is concerned with the computation of convoutions, as enabed by the convoution theorem. In particuar, equation (3.47) shows that the convoution can be performed in about + 3og mutipications (if is a power of ) versus mutipications using the definition (3.4). Specific appications in geodesy abound, e.g., Stoes integra, the terrain correction, and other convoutions incuding east-squares coocation. These are the topics of ater Chapters. 3.7 Cycic Versus Linear Convoution In Chapter we introduced the concept of convoution and saw exampes in physica geodesy, ie (.6), the disturbing potentia (or, geoid unduation), and (.3), the defection of the vertica, that invoved gravity anomaies and the Stoes function. There are many other simiar types of convoutions in geodesy and geophysics. We ca the two functions being convoved the Fourier Geodesy 3.7 Jeei, Apri 7
signa function (or data) and the erne function, where the atter represents the mode, or system, that transforms the data in some physicay meaningfu way. If the erne function in a spherica convoution attenuates rapidy from the origin, one may usuay approximate the convoution in the panar domain. We wi concentrate on the convoution in Cartesian coordinates from the point of view of rapid cacuation made possibe by the FFT. It is we nown that, because the data that enter the convoution in practica appications are necessariy of finite extent and discrete; and, because certain fast computationa techniques (i.e., the FFT) require it, the convoution theoreticay defined on the entire pane must be repaced with a convoution for periodic functions. This use of the so-caed cycic convoution, instead of the inear convoution, therefore, naturay introduces corresponding errors in the mode for the desired quantity; we ca this the cycic convoution error. Beow it is shown how one can design a cycic, discrete convoution that exacty equas the inear, discrete convoution. The proof of this equaity is given for -D convoutions, and the resut is stated for -D convoutions, where the proof is eft to the reader. To give a proper perspective of the cycic convoution error among a other errors committed when evauating a convoution, we start with the continuous convoution, equation (.), that represents the truth (the mode) of combining the signa and erne functions, g and h. This convoution mode is repeated here for convenience: g( t) * ht ( ) = g( t' ) h( t t' ) dt '. (3.59) In practice, one has a finite number of discrete vaues of the signa, that is, a finite sequence of (equay spaced) data, g. Considering first the finite extent of the signa, we define a new signa function, g T, given by equation (.36), that refects its truncation to a domain given by [ T, T ]. Then, the true convoution is approximated by the convoution using, g T, thus generating a truncation error. This error is usuay aso caed an edge effect error if the erne function attenuates rapidy away from the origin, since in that case the error is significant ony when the computation point approaches the edge of the data domain. That is, as t ± T the convoution is corrupted by the ac of data beyond the edge (see Figure 3.3). Formay, we have from equations (3.59) and (.36) T (3.6) ( ) ( ) = ( ) ( ) + ( ) ( ) g t * ht g t' ht t ' dt' g t' ht t' dt ' T t' > T T ( )* ( ) = g t ht ε edge where ε edge is the edge effect error. ext, consider the fact that the data are discrete sampes of the signa with (presumaby) constant samping interva, t, such that T =. Corresponding to the truncation, we define a, that is imited to the interva, : g new data sequence, ( ) Fourier Geodesy 3.8 Jeei, Apri 7
( g ) g, ; =, otherwise; (3.6) where, without oss in generaity, we may assume that the integer,, is even. ote that ( g ) is not periodic; it simpy embodies the coection of avaiabe sampes of the continuous signa. Further approximating the continuous convoution, gt ( t) * ht ( ), by a discrete convoution, equation (3.6) becomes ( )* ( ) = ( ) # discrete edge g t ht g h ε ε, (3.6) where the discrete convoution on the right side is defined by equation (3.8). It can be cacuated for any from the data and a discretization (samping) of the erne function, h. The second and third parts of equation (3.6), ε discrete and ε edge, represent errors due to the discreteness and finite extent of the data, respectivey. Finay, in order to evauate the discrete convoution using discrete Fourier transforms (via the convoution theorem, equation (3.47)), we must assume that both the data and the erne sequences are periodic with the same period,. Using this cycic convoution introduces the additiona cycic convoution error if we simpy evauate the convoution without modification. In order to maintain the sense of successive approximations considered so far (and with the assumption that the most significant part of the erne function is near the origin) we re-write the discrete cycic convoution, equation (3.4), using indices symmetric with respect to zero. This is permitted because the sequence of products in the discrete convoution, gh % %, gh % %, K, g% h % +, is periodic, as is the convoution itsef. Thus, we have: g% #%% h = gh % % n n,. (3.63) n= Consider the foowing assignments that create periodic sequences from the given data and erne function vaues: g % = g, g % = g,, n is any integer + n ; (3.64) ( ) ( ) ( ) ( ) h% = h, h% = h,, n is any integer. (3.65) + n The compete reationship between the true convoution mode and the approximation used in practice is then g t ht g% % h% ε ε ε, (3.66) ( )* ( ) = ( ) #( ) discrete edge Fourier Geodesy 3.9 Jeei, Apri 7
where ε is the (-D) cycic convoution error, given by ε = g% #% h% g # h, (3.67) ( ) ( ) ( ) ( ) for. The edge effect can be reduced by imiting the domain of computation of the convoution to an interva smaer than the data domain (the edge effect is smaest near =, the center of the computation domain), or by increasing the data domain (eeping the computation domain fixed). The discretization error can be controed by the samping interva (it decreases as t becomes smaer). either the discretization error nor the edge effect is within the scope of further discussion; and neither one can be totay eiminated. Instead, the cycic convoution error is now investigated, because it, in fact, can be eiminated with an appropriate modification of the data. It is assumed that the erne function, h, is nown for a t; therefore, it can be samped for a. We first find an expression for the cycic convoution error, which is usefu when anayzing its effect in a particuar appication. Using the convoution definitions given by equations (3.8) and (3.63), and the sequence definitions, given by equations (3.6) and (3.64), we have ( ε ) = ( % ) ( ) ( ) g h% n g h n n n n= n= n= (( % ) ) = g h h n n n (3.68) ow, it can be verified (an exercise eft to the reader) that ( ε ) gn( h n+ h n), ; n= + + =, = ; + gn( h n h n). n= (3.69) The fact that the error is zero at = ε ote that the summand in ( ) instead of = is an artifact of being even. (see Figure 3.3) invoves the vaues of the erne function potentiay cose to the origin (e.g., when =, g is mutipied by ( h h ) ). Hence, if the erne is argest near the origin and attenuates with distance from the origin (many geodetic ernes, in fact, behave ie the reciproca distance), then the cycic convoution error, ( ε ), can be quite arge. On the other hand, from Figure 3.3 we see that the edge effect error and the cycic convoution error have simiar characteristics; both are argest when the Fourier Geodesy 3. Jeei, Apri 7
computation point,, of the convoution is cose to the edge. Therefore, in avoiding the edge effect by restricting the computation point to an interior sub-domain of the signa, one aso tends to avoid the cycic convoution error. Under fortuitous conditions, the cycic convoution error may actuay cance part of the edge effect however, this shoud never be assumed from the start and depends entirey on the signa beyond the window of avaiabiity. cycic convoution error g % ( ) n (g ) n edge effect h n / / 3/ edge effect = Figure 3.3: Edge effect for the convoution ( g ) # h evauated at. Aso shown is the cycic g% #% h%. Soid ines are convoution error committed by computing this convoution as ( ) ( ) used to depict the discrete sequences for carity. It is possibe, on the other hand, to construct a discrete, cycic convoution from the given data sequence that exacty equas the discrete, inear convoution. This does not eiminate the edge effect, just the cycic convoution error. Consider the discrete, cycic convoution, g #% h% % is a periodic sequence whose principa part is defined over the ( % ) ( ), where ( g ) domain, interva, : ( g% ), by extending ( g ) g, ; =, and ; ( ) ( ) + n % with zeros on either side of its own principa (3.7) g% = g%,, n is any integer ; (3.7) where the second equation defines the periodic extension of ( g ) that ( g% ) is the same as ( g ) % over a integers. We see % but padded with zeros on either side of its principa part. Aso, over the interva,, it is the same as g, given by equation (3.6). The extended periodic sequence, h %, is defined by Fourier Geodesy 3. Jeei, Apri 7
( ) ( ) ( ) h% = h, h% = h%,, n is any integer. (3.7) + n Thus, whie the data are extended with zeros, the erne sequence, though assumed periodic (period ), is extended naturay using its own vaues. It is now proved that the discrete, cycic convoution of these extended sequences equas the discrete, inear convoution of the origina finite sequences. First note that ( ) ( ) g% = g, n ; (3.73) n n ( ) ( ) h% = h = h n, n and n n. (3.74) Therefore we have, for : % = %, from equation (3.63); ( g ) #% ( h% ) ( g ) ( h% ) That is, n n n= = ( g) n ( h) n n = n= ( g) ( h n ) %, from equation (3.73); = %, from equation (3.6); n n= ( g ) = h, from equation (3.65); ( g ) # = h ( g) h ( g) %( h% ) n n, from equations (3.6) and (3.8). # = % #,. (3.75) Equation (3.75) indicates precisey how the periodic sequences must be constructed so that their % cycic convoution equas the inear convoution of the origina sequences, that is, ( g ) according to equation (3.7) and ( h ) schematicay how the cycic convoution error vanishes for ( g ) #% ( h% ) % according to equation (3.7). Figure 3.4 shows % ; provided Fourier Geodesy 3. Jeei, Apri 7
. We discard the vaues of the cycic convoution of the extended sequences for other. Using this method to estimate the true convoution thus eaves ony the discretization and edge effect errors as indicated on the right side of equation (3.6). Again, it may not be justified from a numerica standpoint to try to avoid the cycic convoution error, because the edge effect remains and shoud anyway be avoided by restricting the domain of the computation points of the convoution, and this reduces at the same time the cycic convoution error (Jeei, 998). However, because the extra storage space needed to convove the extended sequences is usuay not a computationa issue, nor is the extra time required (using FFTs), it is prudent to eiminate the cycic convoution error as a matter of routine practice. It is re-emphasized that the erne function is not padded with zeros (as sometimes erroneousy advocated), as this woud cause the cycic convoution of the extended sequences to differ from the inear convoution of the origina sequences. However, if the function, h, is not nown beyond some finite interva, then zero padding of this function woud aso hep to reduce (not eiminate) the cycic convoution error. cycic convoution error =, since ( % ) ( ) g% n g % = h n n edge effect / / 3/ edge effect = Figure 3.4: Truncation error and edge effect for the convoution ( g ) # h evauated at, and g% #% h% (it is the cycic convoution error committed by computing this convoution as ( ) ( ) zero). Soid ines are used to depict the discrete sequences for carity. 3.8 -D Discrete Fourier Transform The DFT and a its properties and associated theorems and resuts can be extended naturay to higher dimensions. This is usuay done by simpy incuding an extra index, summation, scaing factor, etc., in a the formuas of the previous sections. We begin with the basic definition of the DFT pair in -D: iπ + ( ), = DFT, =,, G% g% x x g% e = = (3.76), ; Fourier Geodesy 3.3 Jeei, Apri 7
iπ + ( ), = DFT %, = %,, g% G G e x x = = (3.77),. Ceary, the discrete signa and its transform are periodic in both dimensions, with periods and, respectivey. Aso, the samping interva shoud be constant in each dimension, but x need not equa x. If (and ony if) the signa is rea, then the spectrum, equation (3.76) has the conjugate symmetry property, which now is: G% = G%, or aso G% = G%. (3.78) * *,,,, These two equations are equivaent because of the periodicity of the spectrum. Equation (3.78) impies, again because of the periodicity, that four spectra vaues must be rea: G% = G%, G% = G%, * *,,,, G% = G%, G% = G%. * *,,,, (3.79) In addition, there is conjugate symmetry in for = and = : G% = G%, G% = G% ; (3.8) * *,,,, and in for = and = : G% = G%, G% = G%. (3.8) * *,,,, As in the one-dimensiona case there are as many independent spectra components in the discrete spectrum, equation (3.76), as there are in the discrete signa. The symmetries above ensure this fact for rea signas. The two-dimensiona, inear, discrete convoution of a truncated data array and a erne function is equa to the cycic convoution of an appropriatey zero-padded signa array and the extended erne array, just ie in the one-dimensiona case. Corresponding to equation (3.75), we have # = % #,,, (3.8) ( g ) ( ) ( ) h g % h%,,,,,,,, Fourier Geodesy 3.4 Jeei, Apri 7
where the zero padding is done as foows, ( g%, ), ( g% ), and ;,, = or or, or ; (3.83),, ( ) ( ) g % = g %,, (3.84) where,,, + n, + n,, n n are any integers. That is, the zero-padded signa array, ( g ), %, is the origina array pus a border of zeros, whose width is either or, depending on the coordinate direction. This extended array is continued periodicay over the entire pane. The periodic erne array is defined anaogous to equation (3.7):, ( ) h% = h,, ;,,, ( % ) = ( % ) h h, n, n are any integers;,, + n, + n, (3.85) where the extension to the arger grid is accompished using the actua vaues of the erne function. Most FFT agorithms assume the DFT is defined with indices starting at zero, as in equation (3.76). Since it is periodic, the cycic convoution that is identica to the inear convoution is aso in this case given by equation (3.8), but for different domains of the indices: =, K, and =, K,. 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 must be such that when viewed as periodic over the pane it is sti we defined (e.g., h % to symmetric) near the origin. That is, we must aways use the principa domain of ( ), extend the erne periodicay over the pane. We have the foowing agorithm for padding the data and the erne, respectivey:, ( g%, ),, g,,, ;,, ; =,, ;,, ; (3.86) and Fourier Geodesy 3.5 Jeei, Apri 7
( h% ),, h,,, ; h,,, ; = h,,, ; h,,,. (3.87) An exampe of this wi be given in a ater chapter. 3.9 Discrete Signas on the Sphere On the sphere data are usuay gridded at equi-anguar intervas: 8 36 g, g( θ, λ) = g + θ, + λ, =,,, =,, K K, (3.88) θ λ where θ, λ are intervas, respectivey, in co-atitude and ongitude, in degrees. Incuding the in the definitions of θ and λ paces the sampes of g at the center of each grid ce, thus avoiding a mutipicity at the north poe and yieding symmetry with respect to both poes. Reca that the inverse Legendre transform of a signa on the sphere is given by (.99) or (.6); and, we re-write the atter as foows: im g( θλ, ) γnm, P, ( cosθ n m ) e λ m= n= m =, (3.89) where, as before, n and m represent frequencies (or wavenumbers) of the spectrum, γ nm,. ow, the yquist wavenumber in ongitude is based on the samping interva aong any parae: 36 8 m = =. (3.9) λ λ Since, in principe, the samping of g aong a meridian (i.e., in atitude) occurs ony for < θ < 8, the yquist frequency in atitude woud be given by 9 θ. But the topoogy of the sphere invaidates this Cartesian approach. Specificay, a function, g, that is band-imited with γ nm, = for m m and n 9 θ, contains ess information than a grid of vaues as represented on a sphere by equation (3.88). It can be shown (Jeei, 996) that the yquist imit for the degree, n, is a function of the order, m, as foows (see Figure 3.5): Fourier Geodesy 3.6 Jeei, Apri 7
n ( m) 8 m +, m < m ; = θ m, m m. (3.9) The function, g ( θλ, ), that is band-imited in the sense that γ nm, = for m m and n n ( m) is competey and without aiasing error represented by its vaues on a spherica grid, as given by equation (3.88). m, order n = m 8º/ λ m γ n, m 8º/ θ n, degree Figure 3:5: Domain of Legendre spectrum uniquey consistent with a grid of vaues as given by equation (3.88). Generay, λ = θ, and it is common practice to determine the Legendre spectrum up to degree nmax = m. Then, substituting equation (3.88) into equation (3.89), truncated appropriatey, the mode for the observations (or the data) on the grid is given by nmax nmax m m iπ iπ M, = nm,, ( cos nm ) m θ = n n= m ˆ M g γ P e e, (3.9) max where M = 36 λ and γ ˆn, m are estimates of the spectrum components (to be determined or given). We can write this as M m i M g, = Gm, e π, (3.93) m= M where We note that in actuaity this is ony an approximate resut, since the number of data vaues is not exacty equa to the number of independent parameters in the spectrum indicated in Figure 3.5. The reader is ased to provide an exact count in both domains; see (.9) Fourier Geodesy 3.7 Jeei, Apri 7
M m iπ ˆ M γ, ( cosθ ) ( with n m ). (3.94) G = P e G m, nm, M, n= m Using g = DFT ( M λ G ) or, aso, m, (see equation (3.9)), we aso have M m i ˆ M g, = π DFT γ nm, Pnm, ( cosθ ) e π ; (3.95) n= m M m i DFT( ) ˆ M g = π γ nmp ( cosθ) e π. (3.96),, nm, n= m where DFT refers to the transform with respect to the ongitude index,. These equations are the basis for fast anaysis of sphericay gridded data to determine the Legendre spectrum, γ ˆn, m (using equations (3.96) and (3.94)) and for fast synthesis of the Legendre spectrum to determine a grid of mode vaues on the sphere (using equation (3.95)). The computationa speed comes from the appication of the FFT to perform the DFT. 3. Digita Fiters The concepts of fiters for continuous signas briefy described in Chapter carry over directy for discrete, or digita signas. In genera, we consider a digita system to be an operation that taes a digita input and produces a digita output (cf. (.49)): ( ) y = h g, (3.97) where h is the system function and the sequences g and y, in genera, are infinite. We assume that the samping interva is the same for the input and the output. In fact, the digita signas usuay are denoted simpy as indexed sequences, as above, with no expicit samping interva specified, where the impicit assumption is that the samping is done at equa and uniform intervas. Athough the terminoogy is not universa, we ca a inear, time-invariant digita system a digita fiter. A digita fiter is a discrete convoution (cf. (.5)): y = h # g. (3.98) Fourier Geodesy 3.8 Jeei, Apri 7
The fiter is characterized by the (possiby infinite) sequence, h. Time-invariance is aso caed shift-invariance, impying that if we shift the index for the input sequence by a certain number then the index of the output sequence is shifted by that same number; 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 impuse, defined as a deta function anaogous to the Kroenecer deta: δ, = ; =,. (3.99) We have from equation (3.8) # δ nδ n y = h = h = h ; (3.) n= and, therefore, the fiter sequence, h, is aso caed the impuse response of the fiter. If the sequence, h, consists of a finite number of non-zero eements, then the fiter is nown as a finite-ength impuse response (FIR) fiter; and, if h is an infinite sequence of non-zero eements then the fiter is an infinite-ength impuse response (IIR) fiter. For the IIR fiter we require, of course, that the convoution (3.98) exists; we say that the fiter is 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 impuse response sequence is absoutey summabe: h = <. (3.) (A sufficient condition for absoute summabiity is finite energy.) An FIR fiter is aso nown as a moving-average (MA) fiter; whie an IIR fiter is the genera type for an autoregressive (recursive) fiter. We wi not dea with these genera autoregressive - moving-average (ARMA) fiters; nor wi we deve into the methods of designing specific digita fiters, a topic that in itsef occupies many textboos. We restrict the discussion to a few exampes of FIR, or MA fiters that are ow-pass fiters adapted from the windows and fiters discussed in Chapter. Exampe: Rectanguar fiter, -D and -D (cf. (.38) and (.66)): h, ; =, otherwise; (3.) Fourier Geodesy 3.9 Jeei, Apri 7
h,, and ; =, otherwise. (3.3) The rectanguar fiter, therefore, is an FIR fiter. Other exampes of FIR fiters incude the Hann and Hamming functions treated as fiters. The discretization of these functions is straightforward. The Butterworth fiter (whose frequency response is given by equation (.64)) is an IIR fiter. Determining its digita fiter function, which in principe has an infinity of nonzero vaues, is beyond the present scope. Fourier Geodesy 3.3 Jeei, Apri 7