the method of discontinuities in fourier analysis bound... · the method of discontinuities in...

R 154 _!,hil~ps Res. Rep._..5, 4~5-460, 1950

THE METHOD OF DISCONTINUITIESIN FOURIER ANALYSIS

hy J. M. L. JANSSEN *) 511.512.2

SummaryA survey is given of the results arrived at in various publicationsdealing with the method of discontinuities in Fourier analysis. It isshown that a more general conception is practicable when starting fromthe Fourier integrals instead of the Fourier series. Then in a simple andnatural way also the frequency spectrum of continuous functionsmay be found by this method. It is investigated in how far rapidchanges may be construed as being real discontinuities. Correctionfactors are derived which take into account the shape of the rapidchanges.Several examples are given. Inter alia the method is applied to theproblem of the summation of a series, an Euler summation formulathen being found.

RésuméDiscussion des résultats de certaines puhlications concernant la-mëthode des discontinuités dans I'analyse de Fourier. L'auteurdémontre la practicabilitë d'une conception plus .gënërale partantdes intégrales de Fourier au lieu de la sërie de Fourier. Cette mëthodepermet aussi de déterminer, d'une façon simple et naturelle, le spectrede frëquence des fonetions continues. On examine dans quelle mesuredes variations rapides peuvent être considérées comme des disconti-nuités reëlles et on détermine les facteurs de correction tenant comptede la forme des variations rapides.Plusieurs exemples sont donnés. La méthode est; appliquée entreautres au problème de la' sommation d'une sërie. On arrive alorsà une formule de sommation de Euler.

ZusammenfassungDer Artikel gibt eine 'Übersicht der Ergebnisse von mehreren Ver-öffentlichungen über die Diskontinuitätenmethode in der FourierAnalyse. Gezeigt wird daB eine allgemeinere Auffassung möglich Ist,ausgehend von den Fourier Integralen statt der Fourier Reihe. Dannkann mit dieser Methode in einer einfachen und natürlichen Weiseauch das Frequenzspektrum stetiger Funktionen gefunden werden.Untersucht wird wiefern schnelle Änderungen als wirkliche Diskonti-nuitäten aufgefaBt werden können. Korrektionsfaktore werden gegebenahhängig von der Gestalt der schnellen Anderungen.Mehrere Beispiele werden gegeben; die Methode wird unter anderenangewendet auf das Problem der Summation einer Reihe. Eine Sum-mationsformel von Euler wird gefunden.

1. IntroductionWhen the picture of a periodic voltage on the screen of a cathode-ray

oscilloscopeshowsjumps and sharp hends this is an indication that thevoltage contains a large number of high harmonics, and the question then

• At present with KoninklijkefShell Laboratorium, Delft.

436 J. M. L. JANSSEN

arises whether a part of the Fourier spectrum can be particularly ascribedto these discontinuities. This problem has been dealt with in variouspublications 1), ... ,6) and the answer given to this question is in the affirm-ative. In many cases even the whole of the Fourier spectrum may hederived from the discontinuities. Of course the frequency spectrum of acontinuous periodic function cann?t be expressed in its discontinuities,and in the case of a function with discontinuities any continuous functionmay be added without contributing towards the discontinuitics.

Separate contributions of the discontinuities can be obtained by partialintegration of the equations used for calculating the Fourier coefficients.

For example let us take the coefficient ak of the cosine term with ordinalnumber k, If J(x) is periodic with period 2n then we have for this coeffi-cient

2"7tak= f J(x) cos kx dx .

o(1)

Suppose now that, in the interval 0 ~ x < 2n,J(x) has one discontinuity,at x = a (see fig. 1); ak can then he calculated by splitting the integralin the right-hand member of (1) into two parts:

a 2"nak = f J(x) cos kx dx + f J(x) cos kx dx.

o a(2)

Partial integration of (2) gives

~bal- lf- ~b21" 1fb::n:ak=J(x)-- -- 1'(x)sinkxdx+J(x)-- -- 1'(x)sinkxdx.k k '. . k ko 0 a+ a+

(3)

In the limits of integration a distinction has been made between a-and a+, since J (a) has not been unambiguously defined. As shown infig. 1, valuesJ(a-) and j'{c-l-] can indeed he defined.If

l' (a-) =l' (a+) =1'(a), (4)

"then in eq. (3) the two integrals can he added to give one single integral,where J'(x) is regarded as being continuous a~ x = a. Hence

2"sin ka 1 fnak = -k- ~J(a-) - J(a+)~ - k 1'(x) sinkx dx.

o(5)

It is seen that in (5) the first term of the right-hand member contributèstowards ak in proportion to the amplitude of the discontinuity. Thiscontribution appears, furthermore, to depend upon the position of the

METHOD OF DISCONTINUITIES IN FOURIER ANALYSIS

discontinuity (factor sin ka) and upon the ordinal number. In the second. term of the right-hand meinber the integral corresponds in shape to theright-hand member of eq. (1), the point from which we started. If, therefore,f'(x) also shows a discontinuity (sharp bend inf(x)), then the same processcan be applied again, leaving once more an integral, now with f" (x) in theintegrand but again of the same shape as that from which we started. Ifalso f"(X) has a discontinuity the process can be continued further, andso on for all derivatives.

f(x)

,

IIIrIIIfI

-------------~!- ------- ---.- If~+) I I

I II II I

21T65194

Fig. 1. One period of a fwlction that is periodic in x with period 2n.

a

For the sake of simplicity it has been assumed here that the function andits derivatives show not more than one discontinuity per period. Thisassumption does not imply any actual limitation. Where there are morediscontinuities the integrals have to be split up into a correspondinglylarger number of parts.The ultimate result of the process outlined here is easy to see. Each

new partial integration yields an additional factor Ilk, so that the contri-bution of a discontinuity in the nth derivative has a factor kn+l in thedenominator. Further, this contribution is proportional to the amplitudeof the discontinuity and dependent upon the position.Now there are several simple cases where the process of partial integration

comes to a natural end. As an example we may take a sawtooth function(fig. 2). In this case the function has one discontinuity per cycle andaccording to our line of thought the first-order derivative is continuousand constant. Thus the second term of the right-hand member of (5) isequal to zero.In this case all Fourier coefficients can be fully derived from the dis-

continuity (with the exception of the constant term of the Fourier series);

437

'438 J. M. L. JANSSEN

. .From this simple example the general conclusion can be drawn that theFourier coefficients can always be expressed in the discontinuities if oneof the derivatives of the function is a constant. This condition suffices.Also sufficient is that one of the derivatives is sectionally constant. THe.next .higher-order derivative is then everywhere equal to zero, so' thatif the partial integration is repeated far enough the remaining integralwill also he equal to zero.

The method of discontinuities in the form in which it 'has been outlinedhere will therefore rapidly lead to results, provided the condition justmentioned is satisfied and no discontinuities of high orders occur. Asimple example ofthis is found-in functions represented by curves composedof straight lines. These can only show discontinuities in the functionitself and in the first derivative, the second derivative always being equalto zero.

f(x)

~-----------+'=-----------Ä2~~~_'X65195

Fig. 2. Sawtooth function.

One drawback to the method is that the dependence on position of thediscontinuities is expressed in a somewhat complicated manner. Sine andcosine functions alternate. In (5) the contribution to ak of the disconti-nuity in the function itself contains a sine factor. The contribution tobk, the coefficient of the sine term in the Fourier series with ordinal numberk, will contain a cosine factor. For the contributions of the first derivativeit will be just the other way round. As will be explained below, this draw-back can he overcome by starting with the Fourier series in complex form.

There is a second drawback attaching to the method. As a rule one cannotderive the complete spectrum from the discontinuities. A solution to thisproblem has been indicated by Zech 6), based upon the following artifice:the method of discontinuities is also applied for values of k that are notintegers. Thus in (1) k is regarded as continuously variable. From (3)we do not then get eq. (5), since as a rule sin (k2n) '=i= O. The result is thatalso the value of the function and its derivatives for x = 2:n:(or x = 0)

METHOD OF DISCONTINUITIES IN FOURIER ANALYSIS 439

appear in the formula as if they were discontinuities. In this way it ispossible to treat also continuous functions according to the discontinuitymethod. As a rule with repeated partial integration for ak and bk series willbe found which in many practical cases converge and can be summed. Ifin the expression for the sum of the series the value of k is made integralthen the values of the Fourier coefficients are found also for a continuousfunction. If in such a case k had been made integral prior to the summationthen all terms of the series would, of course, have become equal to zero,since there are no other discontinuities than the "false" one for x = 2n.

By this artifice the method of discontinuities is made much more usefulbut its charm of simplicity is lost. This, however, can be saved by takingas starting-point the Fourier integral instead of the series. The frequencyis then continuously variable and thus a simple relation exists betweenthe frequency and the continuously variable k.

2. Theory of the method of discontinuities

As already stated in the introduction, we shall take the Fourier integralsas the starting-point, using these in the following forms:

+co2ng(w) = J h(x) e-j"'x dx,

-co+co ,

h(x) = J g(w) ei"'x dw.-co

(6)

(7)

The equations applying for the coefficients of the Fourier series can berelated to eq. (6). Whèn the Fourier series pertaining to the periodicfunction f (x) is written in the complex form

then we get for Ak

'+cof(x) = ~ Ak ei"",

k=-co

+n2nAk = J f(x) e-jkx dx.

-n

(8)

(9)

For the sake of symmetry, in (9) the values -n and +n have beentaken for the limits of integration.

Let ua now consider the case where h(x) (eqs (6) and (7)) is equal to 0for all values of x, except for the interval-n ~ x ~ n, where

h(x) =f(x), (10)

so that in point of fact for h(x) one separate period of the periodic functionf (x) is taken. The limits of integration in (6) can then be replaced by-n and +n. Comparing (9) with (6) we apparently have

, Ak = g(k) . \ . (11)


This mayalso he formulated as follows: the frequency spectrum of a periodicfunction can immediately he derived from that of a single period regardedas an aperiodic phenomenon. In the first case we have a line spectrum(discrete values of k) and in the second case a continuous spectrum (con-tinuously variahle (0). The two spectra have the same shape in the sensethat the continuous spectrum is the envelope of the line spectrum.In view of Zech's artifice it is natural to apply the method of discontinui-

ties to the Fourier integral. As a matter of fact this is what Zech himselfdoes, though he is apparently not aware of it. The values of the periodicfunction f (x) at the heginning and the end of a period are no longer"false" discontinuities hut real discontinuities of h(x). And the same holdsfor the derivatives.Thus we start from

+n2ng(w) = f h(x) e-joox dx.

-n(12)

Let us once more suppose that h(x) shows a discontinuity at x = a. Then,just as in the introduction, after partial integration we have

h(-n+) h(n-)2 () joon -joon +ngw =e . -e -.-JW JW

+nh(a+) -h(a-) 1 f+ e-jCl)(l . + -;- h'(x)e-jwx dx.

JW JW-n

Writing

then

LICO)(a) = h(a+) - h(a-),LlCO)(_n) = h(-n+) - 0,LlCO)(n) = 0 - h(n-) ,

LlCO)(a)e-jwa f+n h'(x) .2ng(w) = ~ . + -.- e-.Jwx dx,

Q JW JW-::r;

(13)

which contains a summation for all discontinuities, in this case for -n,a and +n. Naturally (13) also holds for a larger number of discontinuities.

Let us write

(14)

as representing a discontinuity in the rth derivative at the point a. Byrepeated partial integration we get

_ n-l LlCt)(a)e-jooQ r+::r; Mn)(x) -joox

2ng(w) - ~ ~ (' )'+1 + -(')n e dx.a .=0 JW . JW

-::r;

(15)


In order to make the application of (15) more general we shall includealso ideal impulses (delta functions) in our considerations. An impulse isideal when its width is zero, its height infinite and its surface area finite.Suppose that h(x) contains a delta function for x = a with surface areaLI{-l)(a). This yields a contribution to the right-hand member of eq. (12)equal to

+nJ ~LI{-l)(a) t5(x-aH e-jwx dx = LI{-l)(a) e-jwa•-n

(16)

With this extension, equation (15) becomes

_ "n-1 LI{r)(a) e-jwa j+.n h{n)(x) -jwx

2ng(co) - ~ ~ (' )r+l + (')n e dx.a r=-l JCO JCO

--n

(17)

Thus we arrive at a surveyable formulation of the contribution of thediscontinuities. From the discontinuities of h(x) it is possible to derive thecomplete spectrum if the remaining integral disappears with increasing nand the series converge. Some examples will show that this mayalso bethe case with a functionf(x) without discontinuities. We shall not enterinto a general discussion of the convergence of the right-hand member ofeq. (17), because in practice the method of discontinuities will be applied inthose cases where there are only few terms in the right-hand memberof (17) or where infinite series can easily be summed; in all other cases itis better to calculate the Fourier coefficients in the usual way. In casesof guaranteed convergence, however, the application of eq. (17) gives theassurance that the complete frequency spectrum is found, thanks to h(x)being made discontinuous for x= -n and x= +n .In the case where there is an infinite series in the right-hand member of

eq. (17) there is still the difficulty that whereas in a certain range of co-values they converge in another range they do not. If the series can besummed then for the frequency spectrum a closed expression in cois foundwhich is in any case valid within the range of convergence, thus within acontinuous range of co-values. Walther 4) has pointed out that his expres-sion must then represent the spectrum also outside that range. This isdemanded by the analytical continuation of the function, since thereare two expressions for g(co), viz. the sum of the series mentioned and cq.(12). Within the convergence range of the series the two expressions are inagreement, and they should also agree outside that range considering thataccording to (12) g(co) is an analytic function of ia for all finite co-values.Formula (17) lends itself not only for calculations but also for a graphical

treatment in a normal vector diagram, such as is used in the theory ofalternating currents. Since g(co) is complex it can be represented by a


vector, but this vector does not directly give the component of the fre-quency spectrum with frequency w. To find this, in the integral (7) thecontrihution of g(w) has to be combined with that of g(-w), which yields:

o +~ ~h(x) = f g(w)eiW"'dw + f g(w)eiw"'dw = f ~g(-w)e-iw", + g(w)eiw"'t dw.

=eo 0 0 (18)

The terms of the integrand being conjugate complex, the binomialcan be replaced by twice the real part of one of the two terms. Thus

~ ,h(x) = f Re ~2g(w) eiw",~dw.

o .(19)

The component with frequency w is therefore given by the projectionon the real axis of the rotating vector 2g(w)eiw",. Thus in the usual vectorrepresentation it is not g(w) but 2g(w) that gives the component withfrequency w.

In the vector. diagram the summation according to the formula (17)can be carried out for any frequency w.A discontinuity in the rth derivativeat the point x = a yields a component with amplitude LJ{r>(a)/wr+_land phase angle - oia - (r + 1)n/2.A drawback of this vector representation is the fact that the component

of zero frequency takes an exceptional position. This is given by g(O).In the Fourier series of the periodic function h(x) the constant term i?

equal to Ao = g(O). The kth harmonic can be represented by the vector2Ak = 2g(k).

3. Practical applications

Here two cases are to be distinguished: the function to be analysedmay be given in mathematical form but it mayalso he in graphical form.In the first case the method of discontinuities has to compete with the

conventional method of calculating the Fourier coefficients. The Fourieranalysis of many commonly occurring functions, such as a sawtoothfunction, a block function, is known and can be found in text-books,so that the method of discontinuities will not be used for these functions,which at the most can only serve as illustration.In the case of more complicated functions where the Fourier spectrum

is not so readily found the discontinuity method may render good service,especially in those cases.where the functions are built up from impulsesand straight lines.

In the second case, where the function is given in graphical form, -withthe known numerical methods of analysis one starts usually with the valuesof the function at a limited number of points. When we come to consider


especially the analysis of a periodic function it is clear that only a limitednumber of harmonics can be calculated from it, since each harmonic is~haracte.dzed by two constants (amplitude and phase angle). If, therefore,'it is desired to calculate n harmonics plus the constant term, we musthave at least (2n + 1) data. Inversely we might say that if we start fromthe function at (2n + 1) points and assume that the function has no morethan n harmonics then we have just sufficient data for the calculation.In general it is not known in advance whether the function has no morethan n harmonics, or, to put it in more practical words, whether the har-monics of a higher order than n are of negligible influence. Such a numericalmethod is only suitable for "smooth" functions, where it is known, orcan reasonably be assumed, that the terms of the Fourier series of higherordinal number quickly approach zero.Such, however, is certainly not the case with functions where peaks,

jumps or sharp bend~ occur. We mention here the discontinuities in theorder of their importance in this respect. It is often a matter of importanceto know to what extent such a function contains harmonics of a very highorder. Take, for instance, the case of a multivibrator voltage. A multi-vibrator is frequently used for measuring frequencies some hundreds oftimes higher than the fundamental frequency of the multivibrator. Theunknown frequency is then brought to interference with a very highmultivibrator harmonic.

In such cases the contents of very high harmonics in the function ismainly determined by the discontinuities, namely, by the discontinuitiesof the lowest order (exponent (r + 1) in the numerator in the case of eq.(17)). Here we have an excellent :field of application for the method ofdiscontinuities.Curves indicating the functional relation between two physical quantities

often do not show any real discontinuities, only rapid changes extendingover a range of :finite width. It is then worth while investigating in howfar such "discontinuities" may be regarded as being ideal ones. It is ob-vious to aS,sumethat the width of the "discontinuity" will be of no conse-quence in the case of harmonics where a period is long compared withthat width. This' will be further investigated in what follows. For this in-vestigation itself we can again apply the method of discontinuities.. Another typical application lies in the field of pure mathematics, in thesummation of series where the nth term tn is given as a function of n.Such a series can 'he represented by a modulated periodic impulse, withtn determining the form of the modulation. The sum of the series isthen equal to the component with frequency zero in the Fourier spectrum.When this is calculated by the method of discontinuities we arrive atEuler's summation formula.


4. Rapid changes of finite width

We shall now investigate in how far a rapid change in the function maybe construed as a discontinuity. For this purpose we shall take fig. 3a,where h(x) represents a rapid rise for Xl < X < X2•

It is to be pointed out that the following considerations are by no meansconfined to such a "discontinuity" in the functionitself; they are equallyapplicable to rapid changes in any derivative.

--x65196

Fig. 3a. Function h(x) with a non-ideal discontinuity in the interval Xl < X < ~.Fig. 3b. Functions h'(x) and H'(x). Outside the interval Xj < X < ~,h'(x) = H'(x);within the interval H'(x) is represented by the dashed line AB.Fig. 3c. Function HH(X).

The curve between the abscissa values Xl and X2 can be approximated. by a step function and to this we can apply the method of discontinuities.Corresponding to each step is a vector in the vector diagram and the result-ant of these vectors gives the total contribution to the Fourier spectrumof all steps between X = Xl and X = X2• This is illustrated in fig. 4a. IfLlx is the distance between two successive steps measured along the x-axisthen the angle between two successive vectors is equal to wLlx.


When the distance between two successive steps is made smaller andsmaller, thus the number of steps larger and larger, then in the limit thebroken line ofvectors merges into a continuous curve (fig.4b). The resultantis the line connecting the starting-point P to the final point Q. The anglebetween the tangents at the points Pand Q will be equal to w(x2 - xl)'

Within a factor l/w the length of the are will be equal to the total risebetween X = Xl and X = X2' thus equal to the amplitude of the "disconti-nuity".

+f

+1'

65197

Fig. 4. Vector diagram of 2ng(w) indicating the contribution of a non-ideal discontinuitytowards the frequency spectrum.' (a) Non-ideal discontinuity approximated by a step-like curve. (b) Not approximated.

The difference in length between the chord and the arc will be smallso long as W(X2 - xl) ~2:n;. This means that a rapid change may be regardedas a discontinuity so long as the width of the discontinuity is small comparedwith the wavelength of the vibration in question. In this case the differencein length is an effect of the second order. 'I'he decrease in length of thechord with increasing frequency will be as the square of w(x2 - Xl)'

Between the points Pand Q there is a point R on the are where thetangent is parallel to the chord. Without further data regarding the changeof the function between x = Xl and X = x2 it cannot be said exactly wherethat point R comes to lie. This means that in the vector corresponding to


the discontinuity there is an uncertainty in the phase angle of the ordera>(x2-xl)'

For a more formal treatment of the problem we start with eq. (13):

LI(O) (a) e-jwa (+31h'(x)2ng(a» = ~ . + -.- e-jw",dx.

a Ja> • Ja>-31

(13)

Since there are no real discontinuities in the function the first tèrm ofthe right-hand member is equal to zero and only the second term remains,for which we may writer

+31 '1'

j. h'(x) . nl"x h'(lLlx)e-Jw""Llx-.- e-Jwx dx = lim ~Ja> A",~ 1= -nfAx ja>-31

(20)

The sum in the right-hand member of (20) is of the same shape as thefirst term in the right-hand memb~r of (13), so that the quantities h'(lLlx) Llxcould be regarded as discontinuities. These quantities do indeed agreewith the steps in our previous consideration. An infinitesimal step yieldsa contribution to 2ng(a» equal to h'(x)dxjja>, and the right-hand memberof (13) is the integral to x of those quantities integrated over the wholeperiod. In our previous considerations we have done nothing else thanseparate from this integral the part between the limits Xl and x2• Thusthe contribution of the "discontinuity" to 2ng(a» is equal to

x, .

f h'(x) e-JWX

. dx.Ja>

""(21)

Let us now consider the course of h'(x) in fig. 3b. In the area where h(x)"rises sharply h'(x) has a high value, The v~riation of h'(x) is pulse-like for:Xl <X <x2.

It appears that simple results can be reached by imagining the contri-bution ofthe discontinuity as being derived, not from h'(x) between thelimits Xl <X <x2 in accordance with (21), but from another function whichdiffers little from h'(x) and can be derived therefrom in thé following way.We connect the points A and Bwith each other, for instance by a straight.

line, though this is not essential, and imagine the hatched impulse as being:exclusively responsible for the contribution of the discontinuity. Thuswe define a function H'(x) which is identical to h'(x) outside the intervalXl < X < x2 but which is represented inside that interval by the line con-necting the points A and B. Further we define

F'(x) = h'(x) - H'(x) . (22)


Equation (13) is now transformed into

+:n: • "" .<. r H'(x) e-]OJx f F'(x)e-'OJx2ng(a» = . dx + . dx.. Ja> Ja>

-:n: "'-

(23)

The first integral in the right-hand member is to be interpreted as thecontribution to the spectrum not derived from the "discontinuity".This integral does, it is true, extend over the range Xl < X < X2' but this

only counts for the fact that the function would also have risen in thisrange if there had been no "discontinuity", because as a rule the derivativeof h(x) differs from zero from x = Xl and x = x2• With the straight lineconnecting A and B we have defined within the interval Xl < x < X2 afictitious derivative connecting up with the variation outside that interval.

We shall now first deal with the second integral, introducing a newvariable y = x -. a, where Xl < a <Xz. Then we get

f""F' (x) e-jOJx ,1(O)(a) e-jwa rY.· G' (y) -jwy. dx=. A(O)() e dy,Ja> Ja>. LI a .

"'- n

(24)

in which G(y) = F(y + a),Y.

LI(O)(a) = I G'(y) dy. (25)Y.

LI(O)(a) is thus the area of the hatched impulse, or in other words: the am-plitude of the discontinuity. From (24) it follows that a rapid rise can becounted as a real discontinuity provided it is given a correction factoraccording to the integral in the right-hand member of (24). This oorreetionfactor admits of a simple interpretation. In the first place it is to he notedthat the value of this integral depends only on the shape of the impulseand not upon the amplitude. The integral is normalized by the term LIlO) (a)in the denominator. In the second place we see that this integral givesthe frequency spectrum of the impulse 2n G(y) / LI(O) (a). We may write:. .

1 Y. .

C(a» = ,1(O)(a) f G'(y) e-]OJY dy.

"1(26)

Table I gives the frequency spectrum for different forms of impulse.These functions are shown in fig. 5. For low frequencies we have

C(a» = 1 (27)

and for symmetrical impulses the phase angle is equal to zero. Withincreasing frequency the variation of C(a» is at first parabolic, as already


indicated in our qualitative considerations. For various impulse formsC(co) = 0 for certain frequencies, which means that for those frequenciesthe curve of fig. 4b closes itself. The points Pand Q coincide.We shall now consider more closely the first integral in the right-hand

member of eq. (23). H'(x) can be treated in exactly the same way as h(x),thanks to the connecting line between the points A and B, which makesH'(x) a continuous function just like h(x). If H'(x) rises sharply betweenthe points A and B this rise may be construed as a discontinuitity with acorrection factor determined by the pulse-like change of the derivativeof H'(x).

65198

Fig. 5. Frequency spectrum of some forms of impulse. Illustration of table I.On the leftthe correction factor C(w) plotted vertically for the cases A, B, E, F and G; lc(w)1 plottedfor the cases C and D. On the right the phase angle for the cases C and D; for' the othercases the phase angle is 0 for all frequencies. (For table I. see p. 460.)If the impulse D is given a width P then the correction factor lies within the hatchedarea for all the forms of impulse with width p.

As regards the practical application of these considerations it is to benoted that within the interval xl < X <x2 in most cases there will helittle difference between h'(x) and F'(x), so that it is of little consequencewhether for the "discontinuity" LI(Ol(a) (equation (25)) one takes the areaof the hatched impulse (fig. 3b) or the actual rise between 'Xl and x

2•

Working with the hatched impulse only has the great advantage, inthese theoretical discussions, that the same process can he continuedunchanged for the "discontinuities" in the various derivatives.

Finally it is to be noted that these considerations are not limited intheir application to sudden changes of a function. If, for instance, partof a function has a constant slope then the total rise can be considered as

METHOD OF DISCONTINUITIES IN .FOURIER ANALYSIS 449

being concentrated in the middle of that part and treated as a discontinuitywith a correction factor for the corresponding impulse in the derivative,which in this case may be very wide. Such a treatment has its advantagesif the impulse shape is of a known type (table I, p: 460):

5. Illustrative examples

Sawtooth function (fig. 6a)

The discontinuities of the function with their contributions to 2ng( co)are:

L1{O}(O) = - 1,

L1l1}(_n) = _!:_ ,2n

LI{l}(n) = -!_ ,2n

giving1

giving,

--;jco

1 .---el'''''·2nj2co2 '

1 .____ e-J"w•2nj2co2

giving

..I

65199

Fig. 6. (a) Sawtooth function. (b) Corresponding vector diagram of 2ng(ro). (c) Amplitudeof 2ng(ro). ..' .'

Fig. 6b is the corresponding vector diagram. The contrihutión :of thediscontuinity in the function itself lies along the imaginary axis, as isalso the case with the resultant of the cöntrihutions of the two disconti-nuities in the first derivative. It is seen at once that this resultant is equalto zero for nco= lat ; or co= 1 (l = 1,2,3, ... ).The contribution of this resultant to the Fourier series for the periodicsawtooth is therefore equal to zero. This is in agreement with the' factthat the periodic function is continuous at x = ± n. . ..

The frequency co= 0 calls for .special attention. Summation of the con-tributions of the discontinuities yields

j ( sin nco)g(co)= - 1--- ,2nco nco

(28).'

from whichfollows:g(O) = limg(co) = O.~o


Fig. 6c gives the ahsolute value of 2ng(w) as a function of w. Thecurve oscillates ahout the line I/w. These oscillations are due to the discon-tinuities in the fust derivative.

Fianlly we find for the Fourier series of the periodic function

+c:o 'k +c:o i :» 1 co sin kxf(x) = ~ gek) el" = ~ - el" = - - ~ k (29)

k=-c:o k=-co 2:rek :rek=l,*0

Block function (fig. 7a)

The discontinuities give the following contributions to 2:reg(w):

LI(Ol(_a) = 1, giving

LI(Ol(a)= -1, givingr:

aru

65200

Fig. 7. (a) Block function. (b) Corresponding vector diagram of 2ng(w). .~i J

Fig. 7h is the corresponding vector diagram. The resultant of the vectorslies along the real axis and is equal to zero for aw = lx (I = 1, 2, 3, ... ) .This means that the spectrum has zero points for those frequencies wherea whole numher of periods just fit in the width of the hlock. Again particularattention is directed towards the frequency w = O. For g(w) we find

a sin uwg(w)=---

:re aw (30)

and hence g(O) = a/:re.For the Fourier series of the periodic block function we find

+c:o a sin ka.kx a 2a co sin kaf(x) = ~ --- el = - + - ~ --cos kx, (31)

k=-co:re ka :re:re k-l ka

Parabolic impulse (fig.8a)

Between the limits - a ~ x ~ a this function is given hy

h(x) = 1- (xja)2.

METHOD OF DISCONTINUITIES IN FOURIER ANALYSIS

The parabolic impulse is a very good approximation of a "truncatedsine top" and lends itself excellently for Fourier analysis, as will be presentlyshown. The expression for the frequency spectrum of the parabolic impulseis much more surveyable .than that of the truncated sine top itself.

The discontinuities yield the following contributions to 2:ng(w):

2..,1 (ll (-a)= -,

a2-,a

..,1 (2l (-a)=-~,a2

2giving

65201

Fig. 8. (a) Parabolic impulse. (b) Corresponding vector diagram of 21lg(w).

Fig. 8b represents the corresponding vector diagram. Again there will be fre-quencies for which g(w) = 0, but these cannot be read directly from thevector diagram.

Summation of the various contributions gives

2 (Sin aw )g(w) = -- ----cos cao •a:nw2 aw

g(w) = 0 for tan ai» = aw, whilst for w = 0 we find

4ag(O)= limg(w) =-.

~O 3For the Fourier series of the periodic parabolic impulse we find

co 2 (Sin ak . )f(x) = ~ ----z -- - cos ak eJkx =k=-co a:nk ak4a 4 co 1 (Sin ak ) <=-+- ~- ---cosak coskx.3 a:n2 k_l k2 ak

451

(32)

(33)

452· J. M. L. JANSSEN

Step curve (fig. 9a)

Periodic functions of this form describe the variations of a voltageWith. time in certain frequency-dividing circuits. By way of examplelet us take the case where there are six steps per period. It is not difficultto generalize the treatment.The discontinuities yield the following contributions to 2ng(co):

1LI(O)(O) = -1 , giving --;

jco1 .2",

]-we 7 •6jco '

giving

and so on.Fig. 9b gives the corresponding vector diagram, from which it is seen

that g(co) has zero points for those values of cowhere 2nw/7 = l 2n (l =1, 2, 3, ... ) or co= 7 l.~~mt .... .

-!12-1( 0 +'j(

-21Th- -x

+T

Q 65202.

Fig. 9. (a) Step function. (b) Corresponding vector diagram of.2;rg(w).'. .Ó, ',': ' .... ' ••• " ". ' ,', :,' ••

This' means that in the periodic. step curve the 7th, 14th, ... harmonicsdo not occur.

Summation of the vario~s contributions gives-:

g(co) = :!__ _j_ (1- sin nco ).62nco 7sin(nco/7)

Attention is to be drawn to the similarity between eq. (34) and eq. (28)for the sawtooth function. Here again g(O) = O. Taking the number ofsteps equal to n we find for the Fourier series of the periodic step curve:

, (34)

.. n + 11 cof(x) = --- -:-~

n n k';'l*(n+1)1

sin kx(35)

k

Thus the frequency spectrum of this periodic step curve is obtained fromthe spectrum of the periodic sawtooth ftinction by taking the harmonicswith an ordinal number equal to a multiple of the number of steps plusone as being zero and multiplying the amplitude of the remaining har-monics by a factor (1 + l/n).

METHOD OF DISCONTINUITIES IN,FOURIER ANALYSIS

Sawtooth function with finite flyback (fig. 10)

. Fig. lOb gives the derivative of the sawtooth function. Accordingto .equation (25) we have to take for the "discontinuity" the valueLI(Ol (0) - -1, just the same as for the sawtooth function. The correctionfactor for multiplying the contribution of this non-ideal discontinuity isthe normalized frequency spectrum of a block-shaped impulse. Thus we find

( ) _ j (Sinaw Sinnw)gw --- -------,2nw aw stco

and for the Fourier 'series of the periodic sawtooth function

( )1 ee sin kx sin ka

fx =--~ ----.nk=l k ka

+*~------:?1h(x)Ot-~(2--+--t-t1

h'(i() aI

-7f -a o-ta +7f

-x 65203

Fig. 10. (a) Sawtooth function with finite flyback. (1) Derivative of this function,corresponding to fig. 3b.

Trapezoidal impulse (fig. 11)

Also in this case the correction factor is the normalized frequency spectrum of a block-shaped impulse. We now find

a sinaw sinfJwg(w)=-----.

n aw fJwIt is to be noted that this equation also holds when fJ = a, thus when

the impulse is triangular.

-x65204

Fig: 11. Trapezoidal impulse.

453

(36)

(37)

(38)

454 J.;fM. L. JANSSEN

For the Fourier series of the periodic trapezoidal impulse we have

a 2a ~ sin ka sin k{Jf(x) = - + - ~ ----.

:n; :n;k=l ka k{J(39)

Cosine functionWe now take as an example a function which when periodically repeated

is continuous, viz.,f(x) = 1 + cos x.

For the discontinuities and their contributions to 2:n;g(w) we find

LI{O)(-:n;) = LI{O)(:n;)= 0;

LI(l) (-:n;) = LI (1) (:n;)= 0;

LI(2)(_:n;) = 1, giving1 .__ el'''') •

(jW)3 '1 .___ e-J1fw•

(jW)3 'LI(2)(:n;) = -1, glVlng

LI(3)(_:n;) = LI(3)(:n;) = 0;

LI(4)(_:n;) = -1, giving1 .___ e)1fw.

(jW)5 '1 .

__ e-J1fw•

(jW)51, giving

Hence

1 ( .. )( 1 1 )2:n;g(w) = -- eJ1fW - e-J1fW 1- -- +-- _ ....(jW)3 (jW)2 (jW)4

The series in the right-hand member is absolutely convergent for w > l.Summation gives

2 12:n;g(w) = - - sin :n;w--

w3 11--

w2

orsin:n;w 1

g(w) = --:n;w- -w-2---1 (40)

For g(w) we have now found a closed expression in w which has itssignificanee also for w ~ 1 and therefore actually represents the completefrequency spectrum.

Both the numerator and the denominator of the right-hand memberof eq. (40) are equal to zero when w = ±l.We find

METHOD OF DISCONTINUlTIES IN FOURIER ANALYSIS

g(±l) = limg(w) = +!;<00+1

g(w) = 0 when w is an integer not equal to ± 1 or o.Thus for the Fourier series of the periodic cosine function we again

find the original function

f(x) = +i g(k)eikx = !e-ikx + 1+ !eJkx = 1 + cos x.k=-co

Asymmetrie multivibrator voltage

We now take as an example a function such as may be obtained from ameasurement, namely the shape of the periodic voltage of an asymmetricmultivibrator (fig. 12).

+10-----..,1+5-----1O...----H-rl/-=--v

t

-3£ I+1T'

65205-TT

Fig. 12. Voltage of an asymmetrie multivibrator.

For high frequencies the Fourier spectrum will be mainly determinedby the discontinuities in the function itself, in which case

1( .nk )2ng(k) = jk 5e} 10 - 40 .

The vector 5ejnk/10 rotates about the extremity of the vector -40and sets up waves in the spectrum. The first-mentioned vector makes onerevolution when k is increased by

2nL1k= -= 20.

n/l0

This is therefore the "wavelength" of the oscillations in g(k).The question is now in what frequency range equation (41) is valid. This

range is limited on both sides. On the high-frequency side a limit willhe set by the non-ideal character of the discontinuities in the function. Letus suppose that the multivibrator has a frequency of 100 kc/s and thatthe duration of the jumps can be estimated as 10-8 sec. The duration ofthe jumps in angular measure is then 2n/l000. From the frequency spectrum

455

(41)

45.6 J. M. L. JANSSEN

of fig. 5 it appears that the correction factor is 0·7-0·8 for k = 500(wP = n). For higher frequencies the estimation of the correction factorbecomes rapidly less accurate. For an accurate estimation it would benecessary to know the variation with time of the jump. Therefore we maysay that provided it is given a mean correction factor according to figure 5(34) can be applied up to the 500th harmonic.On the .low-frequency side a limit is set by the contributions of the

discontinuities in the derivatives.The discontinuities in the first derivative and their contributions to

2ng(w) are

..1(1)(_!!_) = 50, giving10 in:

(1) 250..1 (0) = -, giving

n

50 '1Ik/10'2k2 el ;

nJ

250nj2k2•

To arrive at an estimation of the effect of these contributions we compare..1(1)(0)with ..1(0)(0).The contributions of these discontinuities are mutuallyequal in amplitude when 250/nk2 = 40/k, from which we get k ~ 2.

For higher values of k the ratio ofthe contributions of ..1(1)(0)and ..1(0)(0)diminishes in inverse proportion to k. For the tenth harmonic the relativecontribution of the discontinuities in the first derivative is therefore ofthe order of 20%.In order to calculate, according to this method, the lower harmonics of

the spectrum it is necessary that the curve be given in mathematicalform. If we approximate the curved part by exp(-IOx/n) then the spec-trum can easily be found from the following discontinuities:

for x = - n/IO an ideal discontinuity ..1(O)(-n/IO) = 5;

for x = - n/20 a non-ideal discontinuity ..1(01(-n/20) = 5:

nsin- k

20correction factor: ----;

n-k20,

for x = 0, an ideal discontinuity ..1(0)(0)= - 10;:TO

for x = 0,a non-ideal discontinuity ..1(-1)(0,)= - 30 f e-lOx/ndx = -3n:o

1correction factor: ---- (see table I).

.nI+J-k10,


Thus nsin-k

5 i -"k 5 i!! k 20 10 3n2ng(k) --e 10 +-e 20 -~'--------ik ik . n ik

-k20

It can easily he verified that lim g(k) yields the average area between thek....O

curve and. the horizontal a:xis.

Application to the summation of series

Let t(n) be the nth term of a series. It is required to determine the sum Sof the first sterms.If s is a very large number the sum will be given to a first approximation

by ._+'/.

S(s) ~ J t(x) dx.'I.

(43)

The integration limits are relatively arbitrary. Instead of tand s + tonecould also take for instance 1 and s,The problem is now to find the correction terms completing eq. (43).

To solve this we regard the summation of the series as a "modulation"problem. The terms of the series may bé regarded as a periodic unit impulsemodulated with the function t(x) in the manner indicated in figure 13.Fig. 13a represents a periodic delta function. The period is equal to unity.

JilllllliLUU~_-/;-',-3 -2-1 0' 2:1 .5-2i-,S

s.

Fig. 13. (a) Periodic delta function with period equal to unity. (b) Function t(x) representingfor whole values of x = n the nth term of a series. (c) Periodic delta function modulatedwith t(:I:) between the limits E and s + d.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~----- --

458 J. M. L•.JANSSEN

Fig. 13b represents the function t(x). The modulating function is equal tot(x) within the interval e <x < s +~and equal to 0 outside it. Fig. 13cgives the amplitude-modulated impulses. The sum of the series will beequal to the sum of the areas of these impulses, which, within a factor 2.71,is equal to the component with frequency 0 in the Fourier spectrum. TheFourier spectrum of the modulated impulse can be found from the spectraof the non-modulated impulse and the modulation. Components of equalfrequency in the two spectra will together yield towards the spectrum ofthe modulated impulse a contribution with frequency o.

Now the spectrum of a periodic delta function is very simple, all com-ponents being of equal amplitude. '

For the sum of the series we shall therefore find an expression containingthe intensities of those frequencies in the Fourier spectrum of the modulationwhich are multiples of the impulse frequency. To calculate the Fourierspectrum of the modulation we shall apply the method of discontinuities.

These arguments will now be put in a mathematical form:

• • +<08(s) = ~ t(n) = ~ f t(x) ~(x-n) dx

n=1 ,.=l_co

or.+c5 +<0

8(s) = I ~ t(x) ~(x-n) dx8 n=-co

(44)

subject to the condition that

Now the Fourier series of the periodic delta function is given by

+<0 +<0~ ~(x - n) = ~ ej211kx•

n~-co h=-co(45)

Hence (44) becomes.+c5 +<0

8(s) = I ~ t(x) ej2nkx dx.B k=--co

(46)

Interchanging the order of summation and integration we get

(47)

or+<0

8(s) = 2.71. ~ g(- 2.71k),1<=-<0

in whicha+"

g(w) = 2~ f t(x) e-JO'% dx. (48)B

METHOD OF DISCONTIl'nJITIES IN FOURIER ANALYSIS 459

Thus g(-2nk) is a component of the frequency spectrum of the modulationthe frequency of which is a harmonic of the impulse frequency • We write(47) in another form:

. co8(s) = 2ng(0) + 2n ~ ~g(2nk) + g(- 2nk)t

k=1or

.+~ co8(s) = J t(x) dx + 4n ~ Re g(2nk).

s . 10=1(49)

Thus we arrive at the first solution of the problem set. The sum is ex-pressed in an integral with correction terms.

We express the spectrum of the modulation in the discontinuities:

co LI(r)(8) e-js2nk + LI(r) (s+<5) e-j(.+Ó)2nk

2ng(2nk) = ~- ('2 k)r+l ' (50)r=1 ] n

thereby assuming that the function t(x) and all derivatives are continuousb8<X<S+<5. .

We now have to make a choice for 8 and <5. It is obvious to choose

8=<5=t·Then

co LI(r) (1.) + LI(r) (s+1.)2n (2 k) = ~ (_1)10 2" 2" •

g ti r=1 (j2nk)'+1

We substitute (51) in (49). A real contribution is only yielded for oddvalues of r, Then (49) becomes

_ ,+'/. co ~ 1+1LI(21+1)(!)+ .1(21+1)(s+t) co (-1)"~8(5) - J t(x) dx + ~ (-1) 21+1 21+2 ~ k21+2 • (52)

'I. =0 2 n 10=1

(51)

coThe series .:£ (_I)kJk'+1 can be related to the Bernouilli numbers

10=1

When the discontinuities are expressed in the values of the functiont(x) and its derivatives eq. (52) becomes

11--.+'/. 221+1

8(s) = !. t(x) dx+ ~ ~t(21+1)(S+t)-t(21+1)(tH-( -l --)' B21+1' (53)I. 1=0 2+2.

Thus our object is achieved. We have found an Euler summation formula.By choosing different values for 8 and <5we can find other sum formulae.A difficulty arises when 8 and/or <5 are given. the value 0 or 1, because

460 J. 'M. L. JANSSEN

then the limit of the integration range coincides with the position' of animpulse, One must not, therefore, give e or ~ the value 0 or 1 but hy tran-sition allow e or ~ to approach such a value. Since our intention was onlyto give a practical example of the application of the method of discon-tinuities, and nof to deal with the summation of series, we have circum-vented this difficulty hy choosing e = ~= t.

Delft, September 1950

REFERENCES

1) A. Eagle, A practica1 treatise on Fourier's theorem and harmonic ana1ysis,London,1925.

B) G. Koehier and A. Walther, Arch. Elektrotech. 25, 747, 1931.8) R. Feinberg, Arch. Elektrotech. 27, 15-18, 1933.4) A. Walther, Arch. Elektrotech. 27, 19, 1933.6) A. Walther and K. Brinkman, Ingenieur Archiv 13, 1, 1942.6) Th. Zech, Arch. Elektrotech. 36, 322-328, 1942.

tFry}e·j<DydyCM=~---

(Fry)dy

FM YI Y2 C(w)

A 1 _n_ -l +4~-~ +~-y

BY+4 (y<O) _/\_ -4 +? (-~:/-y+ 4 (y>lJ) _I +~-y

-

C -y+(J .r-; 0 fJ 2J'-=<!!f!.)-i{.(l)(!_-sin'!!P)0 (;l -Y iJJ'(J'

0 e-1t~¥

.0 ,co

,+1!!!J-

E cos7f .r.. -f +~ =~_f! +1-Y ,-4 ..2 st

F -er .r-: -{- +.q. 3·sin !!'j -!!j!. cos 'E#

-~ +~-y (!!!f.t "

I+COS'_ _f\_ -4 +~ sin .!!!j ,G ,!!f- . ,-my-? +?-y

65207

TABLE I

the method of discontinuities in fourier analysis bound... · the method of discontinuities in...

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