On Choosing Fourier Transforms for Practical Geoscience Applications

Transcription

1 International Journal of Geosciences, 0, 3, Published Online October 0 ( On Choosing Fourier Transfors for Practical Geoscience Applications David Boteler Geoagnetic Laboratory, atural Resources Canada, Ottawa, Canada Eail: dboteler@nrcan.gc.ca Received July 3, 0; revised August 9, 0; accepted Septeber 7, 0 ABSTRACT The variety of definitions of Fourier transfors can create confusion for practical applications. This paper exaines the choice of forulas for Fourier transfors and deterines the appropriate choices for geoscience applications. One set of Discrete Fourier Transfors can be defined that approxiate Fourier integrals and provide transfors between sapled continuous functions in both doains. For applications involving transfors between a continuous function and a discrete function a second set of Discrete Fourier Transfors is needed with different scaling factors. Two classes of application are presented: those where either for of transfors can be used and those where it is necessary to use a particular transfor to obtain the correct results. Keywords: Fourier Transfors; Filtering; Spectra; Ipulse Response. Introduction The Fourier transfor is a widely used tool for any applications. Its value in physics is best described by Lord Kelvin (in a quote reproduced in the frontpiece of the book by Bracewell []): Fourier s theore is not only one of the ost beautiful results of odern analysis, but it ay be said to furnish an indispensable instruent in the treatent of nearly every recondite question in odern physics. This is all the ore fulsoe praise when it is reebered that Kelvin ade his coents long before the introduction of the Fast Fourier Transfor by Cooley and Tukey []. Since then the efficient coputation of the Fourier Transfor has led to its widespread use in signal processing and incorporation into any software packages. Surprisingly, considering its extensive use, there are no standard definitions for the Fourier transfor. Different software packages ipleent different definitions of the Fourier transfor, so that a forward transfor in one package can be the sae as an inverse transfor in another package. This can lead to considerable confusion aongst users and interferes with the efficient use of this valuable tool. This paper reviews the different fors of the Fourier transfor and identifies the particular choice for geoscience (and other) applications. First we consider Fourier s theore and the definition of the Fourier integral and then exaine the constraints iposed by using a discrete Fourier transfor and how the discrete Fourier transfor can approxiate the Fourier integral. This in- cludes consideration of the different Fourier transfor forulations for continuous and discrete functions. Several exaples are given to illustrate the application of the different Fourier transfor forulations. These applications are drawn fro studies of electroagnetic induction in the Earth, agnetotellurics and geoagnetically induced current effects on power systes.. Fourier Series and Fourier Integral The fundaental idea described by Fourier is that any function can be represented as a su of cosine and sine functions. For the applications considered in this paper we will deal with functions varying in tie so the cosine and sine functions represent different frequencies. In the general case, any tie varying function can be represented by an infinite su of cosine and sine waves where []: 0 f t a a cos ft b sin ft () 0 T a f t dt (a) T T T T T a f t )cosftdt T T T b f()sin t ftdt (c)

2 D. H. BOTELER 953 If we copute the partial su of the Fourier series then we obtain a function that approxiates the original tie variation 0 f t a acos ft bsin ft (3) How good of an approxiation that is achieved depends on the length of the series and the frequency content of the tie doain function. For practical applications with sapled data we are dealing with a tie doain function for which frequentcies above the yquist frequency have been reoved to avoid aliasing probles. This band-liited signal is only an approxiation of the coplete true natural variation that occurred, but can be considered a true representation of the natural variation within the frequency range of interest for which the recordings were ade. We will denote this band-liited signal by s(t). ow the signal s(t) can be represented by the partial Fourier series without any approxiation 0 s t a acos ftbsin ft (4) In preparation for introducing the standard fors of the Fourier integral and Discrete Fourier Transfor it is useful at this stage to introduce the exponential fors of the cosine and sine ters ix ix ix ix e e e e cos x and sin x (5) i Equation (4) can then be rewritten as ift ift ift ift e e e e s t a0 a b (6) i ift Grouping ter s in e ift and e gives n n a ib ift a ib e e a0 s t ift (7) All the ters in this equation can be cobined into one suation fro to where s t c (8) e ift a ib c for positive (9a) a ib c for negative (9b) c a for = 0 (9c) 0 0 The use of the exponential for leads to the copact ift notation of Equation (8). The ters involving e and e ift are often referred to as the ters for positive and negative frequencies. This ay be confusing for anyone who tries to ascribe a physical eaning to the ter negative frequency. Instead the ters positive frequency and negative frequency should just be treated as labels for the ters involving positive and negative signs in the exponents of the exponentials aking up the cosine and sine ters. In the liit as the interval between frequencies goes to zero, the Fourier series goes to the Fourier integral. The Fourier integral has a variety of fors, for exaple as shown by Bracewell []. The custoary forulas for the Fourier transfor and the inverse Fourier transfor given by Bracewell are i e ft F f f t dt (0a) i e ft f t F f df (0b) where Bracewell s forulas have been rewritten in ters of functions f(t) in the tie doain and F(f) in the frequency doain. If the integrals are written in ters of ω there is in the inverse transfor. Thus the transfor pair is: it )e d F f t t F f t e it d The forward and inverse transfors are essentially to do the sae thing so it could be expected that the forward and inverse transfors would be syetrical. To achieve this syetry people often write the transfor pair as it F f t e dt (a) F f t e it d Thus we already have three definitions of the Fourier integral. Brigha [3] exaines this difficulty and considers the appropriate factors if the transfors are to coply with Parseval s theore that the energy coputed in the tie doain is equal to the energy coputed in the frequency doain and for consistency with the Laplace transfor and found that these requireents were in conflict with the various definitions of the Fourier transfor pair written in ters of ω. Brigha concludes that a logical way to resolve this conflict is to define the Fourier transfor pair in ters of frequency f as done in Equation (0) above. With this definition Parseval s theore becoes

3 954 D. H. BOTELER f t dt Ff df (3) As long as integration is with respect to f, the scale factor never appears. Equation (0) is the syste of Fourier integrals also adopted by Bracewell []. 3. Discrete Fourier Transfor The Discrete Fourier Transfor is defined in a variety of ways. All are basically the sae but have a constant / either in the forward transfor or inverse transfor and also differ in the sign of the exponent of the exponential ter in each transfor. Proakis and Manolakis [4] use the definitions ikn xne (4a) n0 k 0 e i kn (4b) For the forward and inverse transfors. In contrast, Bracewell [] uses / and e i kn in the forward transfor, whil e Press et al. [5] use these sae ters in the inverse transfor. This raises the questions as to which ters are the ost appropriate ones to use. The criteria that we will use for answering these questions are: ) to provide the best approxiation to the Fourier integral and ) for consistency with standard practice in geoscience applications. The first choice to ake concerns the sign of the exponent in the exponential ter. The choices are e ikn cos kn i sin πkn (5a) kn i kn ikn e cos sin π (5b) where n is the index for saples in the tie doain and k is the index for saples in the frequency doain. The first choice (Equation (5a)) is the sipler for with a siple addition of the cosine and sine ters. Also e i kn corresponds to the tie dependence of the for e i ft which is ost coonly used in agnetotellurics, as shown by the standard papers and textbooks Price [6], Wait [ 7], Ward and Hohann [8], and Chave and Weidelt [9]. Therefore we will choose the inverse transfor with the exponential ter e i ft. This autoatically requires that the forward transfor be written with the ex- ponential ter e i ft. The other consideration in our choice of Fourier transfor is the placeent of any scaling factors (usually /) used with the suations. For the geoscience applications considered here the forward discrete Fourier transfor can be considered to produce one of two classes of outputs in the frequency doain: ) saples of a continuous function, or ) a set of discrete frequency coponents. Obtaining these two results ay be considered to be using the discrete Fourier transfor either as an approxiation to the Fourier integral or as an approxiation to a Fourier series. 3.. Continuous-Tie, Continuous-Frequency Functions Considering first the discrete Fourier transfors as an approxiation to the Fourier integral pair in Equation (0) we can write xne i kn t n0 (6a) ikn xn e f k 0 (6b) Press et al. [5] also use the additional ter t in approxiating the Fourier integral, but exclude this fro their foral definition of the Fourier transfor. In this case there is a continuous function in the tie doain that has been regularly sapled with a sapling interval t and there is a continuous function in the frequency doain sapled with a sapling interval f. Thus the tie doain and frequency doain functions have equivalent fors and correspondingly there is a syetry in the transfors to go between these doains. 3.. Continuous-Tie, Discrete-Frequency Functions Let us now consider a situation where the tie doain function is considered to be the suation of a set of discrete frequencies. The inverse discrete Fourier transfor is then ore appropriately written as a siple suation of these ters without the f ter: ikn e (7a) k 0 The f ter then needs to be included in the forward transfor which, with the relation t f allows the forward Fourier transfor to be written ikn xne (7b) n0 This transfor pair is not syetrical but the transfors now connect two dissiilar functions: a continuous function in the tie doain and a discrete function in the frequency doain, so it is reasonable that syetry not occur in this case. Equations (7a) and (7b) represent one of the standard discrete Fourier transfor pairs coonly used. Thus this is an appropriate choice when transforing between a continuous function in the tie doain and a discrete function in the frequency doain. However, if

4 D. H. BOTELER 955 the values in both doains are considered as saples of continuous functions then the syetrical pair of transfors in Equation (6) should be used Practical Considerations Bracewell starts with the standard definitions of the Fourier transfors, Equations (0a) and (0b), and then shows they ay be written as a suation over positive and negative ties and frequencies. This does not show where the discrete Fourier transfor coes fro. Here we choose to start with the suation of positive and negative tie and frequency as an approxiation of the Fourier integrals, then show that because the functions are periodic we can write the suations starting at zero. Discrete Fourier transfors convert between an array of values in the tie doain and an array of values in the frequency doain. Discrete Fourier transfors produce an output array in the frequency doain, starting with zero frequency, then positive frequencies and then negative frequencies, as shown in Figure. This placeent of negative frequencies at the end of the Fourier transfor array is explained in a nuber of books (e.g. [5]); however, the placeent of negative ties is not usually discussed. orally it is not a consideration because we are dealing with a tie series and it is easier to think 0 / - -f/ 0 f/ Figure. Scheatic showing the output array fro the Fast Fourier Transfor (solid line) and its extension as a periodic function (dashed line); and Allocation of values to positive and negative frequencies. of the tie series starting at t = 0 as in Figure rather than starting at t T as in Figure. However, knowledge of the negative t placeent is shown to be necessary in the exaple considered later in Application where we do an inverse Fourier transfor of a transfer function in the fr equency doain to obtain the ipulse response in the tie doain. 4. Applications To illustrate the application of the different forulations of the Discrete Fourier Transfors three applications are presented: the first involving a pair of transfors in which either forulation can be used, the second involving the continuous-discrete forward transfor, and the third involving the continuous-continuous inverse transfor. 4.. Application : Low Pass Filter In any geoscience applications it is necessary to filter a dataset or to cobine an input signal with a transfer function of a physical syste (e.g. induction in the Earth) which is equivalent to a filter response. This can be done by convolution of the input signal with the filter ipulse response in the tie doain or using ultiplication by the filter transfer function in the frequency doain, as shown in Figure 3. The frequency doain ethod is often the preferred process because of the coputational efficiencies obtained by using the Fast Fourier Transfor. This involves taking a Fourier Transfor of the input tie series to obtain the spectru of the input signal. Then ultiply this spectru by the filter transfer function to obtain the output spectru. An inverse Fourier Transfor is then used to obtain the output tie series. In this application we consider filtering of geoag- TIME DOMAI FREQUECY DOMAI -T/ 0 T/ 0 Figure. Scheatic showing variations in the tie doain taken fro T/ to T/ (solid line) and its extension as a periodic function (dashed line); The output array fro the Fast Fourier Transfor containing the variations fro 0 to T. T/ T Figure 3. Filtering of a signal via convolution in the tie doain or ultiplication in the frquency doain. The Fast Fourier Transfor (FFT) is used to go between the tie doain and frequency doain functions.

5 956 D. H. BOTELER netic data. Figure 4 shows the agnetic field variations recorded at the Victoria Magnetic Observatory during a agnetic stor in 980. This shows a ixture of rapid and slow variations and, as in any other geoscience applications, it is useful to filter out soe of the variations to ore clearly show the other coponents. In this case we wish to apply a low pass filter with transfer function, T(f), defined by f T f f f (8a) c T f 0 f f and f f (8b) c where f c = Hz (corresponding to a period of hour). Using this filter response with the Fourier Transfors as described above gives the soother signal shown in Figure 4. In this application using the transfor pairs in Equation (6) or using the transfor pair in Equation (7) gives the sae result. This is because using a pair of transfors involves applying the sae overall scaling factor, either t and f in Equation (6), where t f, or by applying alone in equation (7). Thus in this case either transfor pair is satisfactory. However, in the next two applications we will see that a particular choice of transfor is necessary in order to obtain the correct results. 4.. Application : Ipulse Response In soe cases it is ore appropriate to use the tie doain ethod shown in Figure 3 for filter calculations. The frequency response of the filter is defined in the frequency doain and an inverse Fourier transfor used to c c deterine the ipulse response in the tie doain. Consider the frequency response already used in Application, defined in Equation (8). Figure 5 shows this frequency response, both in ters of positive and negative frequencies and as it is ordered in the FFT array. The Fourier integral of such a rectangular (boxcar) function in the frequency doain is a sinc function in the tie doain. Because we want to approxiate the Fourier integral in this case we use the continuous-continuous version of the inverse transfor (Equation (6b)). This gives the results shown in Figure 6. Where, again, the function is shown as ordered by position in the FFT output array and as ordered in ters of positive and negative tie. The values for the sinc function look sall, but integrating the sinc function (su of values ultiplied by t (=60 sec) gives a value of.0 as required. This is confiration that the scaling factor used in the inverse transfor was correct. Use of the other version of the transfor would have applied a different scaling factor and given an incorrect result Application 3: Spectru Deterination There are applications that do not involve filtering but where it is useful to know the spectru of the signal. One such exaple concerns the partial saturation of transforers produced by a cobination of AC and DC currents flowing through transforer windings. This can occur because of geoagnetically induced currents (GIC) in power systes [0]. Consider a power transforer with noral agnetising current I AC subjected to a DC current I DC. The extra agnetic field created by the DC current creates an offset in the agnetic field inside the Figure 4. Magnetic stor recorded at victoria agnetic observatory. Original data; Filtered data.

6 D. H. BOTELER 957 Figure 5. The frequency response of the boxcar filter. As a function of positive and negative frequencies; As ordered in the array for input to the FFT. transforer that pushes the transforer into the saturation region of the hysteresis curve for part of each cycle (Figure 7). The partial satuation of the transforer during each cycle creates the distorted agnetising current wavefor shown in Figure 8. For analysing the ipact on power syste operation it is necessary to deterine the spectral content of the distorted wavefor. This can be done by a discrete Fourier transfor of the wavefor which shows that the signal is coprised of the fundaental plus haronics of the 60 Hz AC frequency (Figure 8). Thus the frequency doain function is not continuous and only contains discrete frequencies. In this case it is appropriate to use the discrete Fourier transfor in Equation (7a). The appropriateness of the continuous-discrete transfor in such a case can also be seen by taking the discrete Fourier transfor of a cosine wave of frequency, f and aplitude, A. This results in a frequency spectru with spikes of aplitude A/ at frequencies ±f. Cobining these two, as in Equation (5), gives the expected aplitude of the cosine wave. 5. Discussion Filtering of a signal can be done by taking the Fourier transfor of the input tie series, ultiplication by the transfer function in the frequency doain, and then taking the inverse Fourier transfor to obtain the output in the tie doain (Figure 3). An alternative, equivalent procedure is to convolve the input with the filter ipulse response to directly obtain the output in the tie doain. The frequency doain e thod is often used because of the coputational efficiencies provided by the Fast Fourier Transfor; however, there are occasions where the tie doain ethod is preferable. The ipulse response is obtained by taking the inverse Fourier transfor of the frequency doain transfer function (T.F.) and this has b) Figure 6. Ipulse Response (sinc function). As it appears in the output array fro the FFT; As a function of positive and negative tie. Figure 7. DC offset in agnetisation of transforer producing a distorted current wavefor.

7 958 D. H. BOTELER Figure 8. Distorted AC wavefor produced by transforer saturation; Spectru of the distorted wavefor. already been explained in Application. ow we need to consider the appropriate forulas to use for the convolution calculation. The ipulse response values in the tie doain are saples of a contiuous function. This is to be convolved with the tie doain signal which is itself a tie series of values that are saples of a continuous function. Thus we need to perfor a discrete convolution that is an approxiation of the convolution integral. The convolution of two functions f(t) and g(t) is gt d f t g t f (9) Discrete convolution as an approxiation of this integral is then given by i ib b (0) f g f g t Coparing the steps involved in the frequency doain calculation (Figure 9) and tie doain calculation (Figure 9) we can see that both calculations involve two suations and a ultiplication with the transfer function or ipulse response. Figure 9 also shows that both calculations involve the scaling factors t and f. Thus all the sae factors are involved in the two calculations showing the equi valence of the two procedures. Figure 9. Coputations for filtering of a signal using a filter transfer function (T.F.) by: Multiplication in the frequency doain; Convolution in the tie doain. 6. Conclusions There are a variety of definitions for Fourier integrals and discrete Fourier transfors. This situation is further confused by different software packages using different definitions so that a forward transfor in one package can be the sae as an inverse transfor in another package. This all hinders the production of rigorous reproducile results when using Fourier transfors for practi- cal geoscience applications. The choice of Discrete Fourier Transfor pair reduces to selection of the signs of the exponent in the exponenttial ters and distribution of the scaling factors between the forward and inverse transfors. Here the selections are ade based on coon practice and to provide the best approxiation to Fourier series and Fourier integrals. The tie dependence is chosen as e ift which is ost coonly used in geosciences. This also represents the siple suation of cosine and sine ters, ift e cosft isin ft. This choice eans that e i ft appears in the inverse transfor and consequently the forward transfor contains the ter e i ft. The chosen discrete Fourier transfor is defined as an approxiation to the Fourier integral to transfor between saples of continuous functions in both the tie

8 D. H. BOTELER 959 and frequency doain e i kn t n0 (a) e i kn f k 0 In soe applications the (continuous) tie doain function is known to be coprised of discrete frequency coponents instead of a continuous spectru. In these cases the Discrete Fourier Transfors to go between the continuous and discrete fun ctions are defined as k 0 n0 e i kn (a) e i kn Applications that involve use of a Fourier transfor pair, e.g. filtering by a DFT to the frequency doain, ultiplication by the frequency response, followed by inverse DFT, can use either transfor pair because the cobined scaling factors are the sae in each case, t f. Applications that involve use of a single transfor, either forward or inverse, ust use the appropriate transfor () or () to obtain the correct results. 7. Acknowledgeents I a grateful to Risto Pirjola and a referee for useful coents on the anuscript. This work was perfored as part of atural Resources Canada s Public Safety Geoscience progra with additional support fro the Canadian Space Agency and Hydro One. REFERECES [] R.. Bracewell, The Fourier Transfor and Its Applications, McGraw-Hill, ew York, 978. [] J. W. Cooley and J. W. Tukey, An Algorith for the Machine Coputation of Coplex Fourier Series, Matheatics of Coputation, Vol. 9, 965, pp doi:0.090/s [3] E. O. Brigha, The Fast Fourier Transfor, Prentice- Hall, Upper Saddle River, 974. [4] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Principles, Algoriths, and Applications, 3rd Edition, Prentice Hall, Upper Saddle River, 996. [5] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, uerical Recipes in Fortran 77: The Art of Scientific Coputing, nd Edition, Cabridge Univer- Methods sity Press, Cabridge, 999. [6] A. T. Price, The Theory of Magnetotelluric When the Source Field Is Considered, Journal of Geodoi:0.09/JZ067i005p0907 physical Research, Vol. 67, o. 5, 96, pp [7] J. R. Wait, Electroagnetic Waves in Stratified Media, nd Edition, Pergaon Press, Oxford, 970. [8] S. H. Ward and G. W. Hohann, Electroagnetic Theory for Geophysical Applications, in Electroagnetic Methods in Applied Geophysics Theory, Society of Exploration Geophysicists, Tulsa, Vol., 988, pp [9] A. D. Chave and P. Weidelt, The Theoretical Basis for Electroagnetic Induction, in The Magnetotelluric Method: Theory and Practice, Cabridge University Press, Cabridge, 0. [0] D. H. Boteler, R. M. Shier, T. Watanabe and R. E. Horita, Effects of Geoagnetically Induced Currents in the BC Hydro 500 kv Syste, IEEE Transactions on Power Delivery, Vol. 4, o., 989, pp doi:0.09/6.975